Handbook of Asynchronous Machine with Variable …...Razik, Hubert. Handbook of asynchronous...
Transcript of Handbook of Asynchronous Machine with Variable …...Razik, Hubert. Handbook of asynchronous...
Handbook of Asynchronous Machine with Variable Speed
Handbook of Asynchronous Machine with Variable Speed
Hubert Razik
First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc Adapted and updated from La machine asynchrone agrave vitesse variable 1amp2 capteurs modegraveles controcircle et diagnostic published 2006 in France by Hermes ScienceLavoisier copy LAVOISIER 2006
Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address
ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom
copy ISTE Ltd 2011 The rights of Hubert Razik to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988 ____________________________________________________________________________________
Library of Congress Cataloging-in-Publication Data Razik Hubert Handbook of asynchronous machines with variable speed Hubert Razik p cm Includes bibliographical references and index ISBN 978-1-84821-225-1 1 Electric machinery Induction--Automatic control 2 Electric motors Induction--Automatic control 3 Electric driving Variable speed I Title TK2731R37 2011 6213136--dc22
2010048625
British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-225-1 Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne
Table of Contents
Foreword xiii
Introduction xvii
Chapter 1 Sensors and Electrical Measurements 1
11 Optical encoder 2111 Technical aspect 2112 Absolute encoder 3113 Incremental encoder 6
12 The velocity measurement 7121 Method of the frequency counter 7122 Method of the period measurement 8
13 The resolver 914 The isolated measurement 14
141 The isolated ammeter 14142 The isolated voltmeter 15
15 The numerical aspect 1516 The analog to digital converter 16
161 Principle of the flash converter 17162 Principle of the successive approximation converter 18163 The zero-order hold 18164 The multiplexer 19165 Principle of converter using slope(s) 20
17 The digital-to-analog converter 2118 The digital output 2219 The arithmetic logic unit 22110 Real time or abuse language 23111 Programming 24
v
vi Asynchronous Machine with Variable Speed
Chapter 2 Analog Numerical Control 25
21 Structure of a regulator 2522 Stability of a system 26
221 Introduction 26222 A formal criterion 27223 A graphical criterion 28224 The stability criterion 29
23 Precision of systems 30231 The initial and final value 30232 The precision of systems 31
24 Correction of systems 31241 The lag and lead corrector 32242 Other correctors 33
25 Nonlinear control 34251 First harmonic method 34252 The oscillation stability 34
26 Practical method of identification and control 35261 Broiumldarsquos method 35262 Zieglerrsquos and Nicholsrsquos method 36
27 The digital correctors 36271 Digital controller 36272 The Z-transform 37273 The Z-transform of a function 38274 Advanced Z-transform 39275 The Z-transform of a loop 40276 Some theorems 41
2761 The initial and final value 412762 The recurrence relation 412763 The fraction expansion 42
277 The Jury stability criterion 42278 Stability graphical criterion 43
2781 The bilinear transform 442782 The formal criterion 442783 The graphical criterion 45
28 Classical controllers 45281 The PID structure 46282 The PI anti-windup structure 46283 Conversion of an analog controller to a digital controller 48
2831 Approximation of the integrator 482832 Use of the bilinear transform 49
29 Disadvantages of digital controller 52291 Choice of the sampling period 52292 Noise 53
Table of Contents vii
2921 Reminder of some concepts 532922 Quantization by truncation 532923 Quantization by rounding 542924 Quantization of a product using tworsquos complement 552925 Quantization of a product by truncation 562926 The signal-to-quantization noise ratio 57
293 Cycles limits and limitations 58
Chapter 3 Models of Asynchronous Machines 59
31 The induction motor 59311 The electromagnetic torque 62312 The equivalent scheme 63
32 The squirrel cage induction motor 66321 The stator inductances 67322 The stator mutual inductances 69323 The rotor inductances 70324 The rotor mutual inductances 72325 The stator-rotor mutual inductances 73326 The rotor voltage equations 75327 The voltage and mechanical equations 75328 Reduction of the model 77
33 The static and dynamic behavior 82331 The steady state of the induction machine 82
3311 Assessment of the power 823312 Characteristics of the electromagnetic torque 84
332 Some practical characteristics 88333 The dynamics of the induction motor 92
3331 No choice of reference frame 943332 Choice of rotor reference frame 943333 Choice of stator reference frame 953334 Choice of synchronous reference frame 953335 Arrangement of variables 96
334 Some electromagnetic torque expressions 9834 Winding and induced harmonics 99
341 Principle of the rotating field 100342 The effect of currents 103
3421 Effect of unbalanced currents 1043422 Effect of non-sinusoidal currents 1063423 Effect of non-sinusoidal winding 1073424 Effect of harmonic components and winding 108
343 Choices of winding 1083431 Single-layer winding 1093432 Concentric and distributed winding 112
viii Asynchronous Machine with Variable Speed
3433 Double-layer winding 11335 Squirrel cage 115
351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117
36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120
37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124
38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126
3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128
382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129
39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134
Chapter 4 Speed Variation 137
41 Cases of multiphase machines 137411 Motors with a high number of phases 138
4111 Type-I motors 1384112 Type-II motors 140
412 Interactions between harmonics 141413 Three-phase induction machine 144
4131 Three-phase model 1444132 Application in another frame 146
414 Five-phase induction machine 151415 Double-star induction motor 155
4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161
42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Handbook of Asynchronous Machine with Variable Speed
Handbook of Asynchronous Machine with Variable Speed
Hubert Razik
First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc Adapted and updated from La machine asynchrone agrave vitesse variable 1amp2 capteurs modegraveles controcircle et diagnostic published 2006 in France by Hermes ScienceLavoisier copy LAVOISIER 2006
Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address
ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom
copy ISTE Ltd 2011 The rights of Hubert Razik to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988 ____________________________________________________________________________________
Library of Congress Cataloging-in-Publication Data Razik Hubert Handbook of asynchronous machines with variable speed Hubert Razik p cm Includes bibliographical references and index ISBN 978-1-84821-225-1 1 Electric machinery Induction--Automatic control 2 Electric motors Induction--Automatic control 3 Electric driving Variable speed I Title TK2731R37 2011 6213136--dc22
2010048625
British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-225-1 Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne
Table of Contents
Foreword xiii
Introduction xvii
Chapter 1 Sensors and Electrical Measurements 1
11 Optical encoder 2111 Technical aspect 2112 Absolute encoder 3113 Incremental encoder 6
12 The velocity measurement 7121 Method of the frequency counter 7122 Method of the period measurement 8
13 The resolver 914 The isolated measurement 14
141 The isolated ammeter 14142 The isolated voltmeter 15
15 The numerical aspect 1516 The analog to digital converter 16
161 Principle of the flash converter 17162 Principle of the successive approximation converter 18163 The zero-order hold 18164 The multiplexer 19165 Principle of converter using slope(s) 20
17 The digital-to-analog converter 2118 The digital output 2219 The arithmetic logic unit 22110 Real time or abuse language 23111 Programming 24
v
vi Asynchronous Machine with Variable Speed
Chapter 2 Analog Numerical Control 25
21 Structure of a regulator 2522 Stability of a system 26
221 Introduction 26222 A formal criterion 27223 A graphical criterion 28224 The stability criterion 29
23 Precision of systems 30231 The initial and final value 30232 The precision of systems 31
24 Correction of systems 31241 The lag and lead corrector 32242 Other correctors 33
25 Nonlinear control 34251 First harmonic method 34252 The oscillation stability 34
26 Practical method of identification and control 35261 Broiumldarsquos method 35262 Zieglerrsquos and Nicholsrsquos method 36
27 The digital correctors 36271 Digital controller 36272 The Z-transform 37273 The Z-transform of a function 38274 Advanced Z-transform 39275 The Z-transform of a loop 40276 Some theorems 41
2761 The initial and final value 412762 The recurrence relation 412763 The fraction expansion 42
277 The Jury stability criterion 42278 Stability graphical criterion 43
2781 The bilinear transform 442782 The formal criterion 442783 The graphical criterion 45
28 Classical controllers 45281 The PID structure 46282 The PI anti-windup structure 46283 Conversion of an analog controller to a digital controller 48
2831 Approximation of the integrator 482832 Use of the bilinear transform 49
29 Disadvantages of digital controller 52291 Choice of the sampling period 52292 Noise 53
Table of Contents vii
2921 Reminder of some concepts 532922 Quantization by truncation 532923 Quantization by rounding 542924 Quantization of a product using tworsquos complement 552925 Quantization of a product by truncation 562926 The signal-to-quantization noise ratio 57
293 Cycles limits and limitations 58
Chapter 3 Models of Asynchronous Machines 59
31 The induction motor 59311 The electromagnetic torque 62312 The equivalent scheme 63
32 The squirrel cage induction motor 66321 The stator inductances 67322 The stator mutual inductances 69323 The rotor inductances 70324 The rotor mutual inductances 72325 The stator-rotor mutual inductances 73326 The rotor voltage equations 75327 The voltage and mechanical equations 75328 Reduction of the model 77
33 The static and dynamic behavior 82331 The steady state of the induction machine 82
3311 Assessment of the power 823312 Characteristics of the electromagnetic torque 84
332 Some practical characteristics 88333 The dynamics of the induction motor 92
3331 No choice of reference frame 943332 Choice of rotor reference frame 943333 Choice of stator reference frame 953334 Choice of synchronous reference frame 953335 Arrangement of variables 96
334 Some electromagnetic torque expressions 9834 Winding and induced harmonics 99
341 Principle of the rotating field 100342 The effect of currents 103
3421 Effect of unbalanced currents 1043422 Effect of non-sinusoidal currents 1063423 Effect of non-sinusoidal winding 1073424 Effect of harmonic components and winding 108
343 Choices of winding 1083431 Single-layer winding 1093432 Concentric and distributed winding 112
viii Asynchronous Machine with Variable Speed
3433 Double-layer winding 11335 Squirrel cage 115
351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117
36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120
37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124
38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126
3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128
382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129
39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134
Chapter 4 Speed Variation 137
41 Cases of multiphase machines 137411 Motors with a high number of phases 138
4111 Type-I motors 1384112 Type-II motors 140
412 Interactions between harmonics 141413 Three-phase induction machine 144
4131 Three-phase model 1444132 Application in another frame 146
414 Five-phase induction machine 151415 Double-star induction motor 155
4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161
42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Handbook of Asynchronous Machine with Variable Speed
Hubert Razik
First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc Adapted and updated from La machine asynchrone agrave vitesse variable 1amp2 capteurs modegraveles controcircle et diagnostic published 2006 in France by Hermes ScienceLavoisier copy LAVOISIER 2006
Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address
ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom
copy ISTE Ltd 2011 The rights of Hubert Razik to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988 ____________________________________________________________________________________
Library of Congress Cataloging-in-Publication Data Razik Hubert Handbook of asynchronous machines with variable speed Hubert Razik p cm Includes bibliographical references and index ISBN 978-1-84821-225-1 1 Electric machinery Induction--Automatic control 2 Electric motors Induction--Automatic control 3 Electric driving Variable speed I Title TK2731R37 2011 6213136--dc22
2010048625
British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-225-1 Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne
Table of Contents
Foreword xiii
Introduction xvii
Chapter 1 Sensors and Electrical Measurements 1
11 Optical encoder 2111 Technical aspect 2112 Absolute encoder 3113 Incremental encoder 6
12 The velocity measurement 7121 Method of the frequency counter 7122 Method of the period measurement 8
13 The resolver 914 The isolated measurement 14
141 The isolated ammeter 14142 The isolated voltmeter 15
15 The numerical aspect 1516 The analog to digital converter 16
161 Principle of the flash converter 17162 Principle of the successive approximation converter 18163 The zero-order hold 18164 The multiplexer 19165 Principle of converter using slope(s) 20
17 The digital-to-analog converter 2118 The digital output 2219 The arithmetic logic unit 22110 Real time or abuse language 23111 Programming 24
v
vi Asynchronous Machine with Variable Speed
Chapter 2 Analog Numerical Control 25
21 Structure of a regulator 2522 Stability of a system 26
221 Introduction 26222 A formal criterion 27223 A graphical criterion 28224 The stability criterion 29
23 Precision of systems 30231 The initial and final value 30232 The precision of systems 31
24 Correction of systems 31241 The lag and lead corrector 32242 Other correctors 33
25 Nonlinear control 34251 First harmonic method 34252 The oscillation stability 34
26 Practical method of identification and control 35261 Broiumldarsquos method 35262 Zieglerrsquos and Nicholsrsquos method 36
27 The digital correctors 36271 Digital controller 36272 The Z-transform 37273 The Z-transform of a function 38274 Advanced Z-transform 39275 The Z-transform of a loop 40276 Some theorems 41
2761 The initial and final value 412762 The recurrence relation 412763 The fraction expansion 42
277 The Jury stability criterion 42278 Stability graphical criterion 43
2781 The bilinear transform 442782 The formal criterion 442783 The graphical criterion 45
28 Classical controllers 45281 The PID structure 46282 The PI anti-windup structure 46283 Conversion of an analog controller to a digital controller 48
2831 Approximation of the integrator 482832 Use of the bilinear transform 49
29 Disadvantages of digital controller 52291 Choice of the sampling period 52292 Noise 53
Table of Contents vii
2921 Reminder of some concepts 532922 Quantization by truncation 532923 Quantization by rounding 542924 Quantization of a product using tworsquos complement 552925 Quantization of a product by truncation 562926 The signal-to-quantization noise ratio 57
293 Cycles limits and limitations 58
Chapter 3 Models of Asynchronous Machines 59
31 The induction motor 59311 The electromagnetic torque 62312 The equivalent scheme 63
32 The squirrel cage induction motor 66321 The stator inductances 67322 The stator mutual inductances 69323 The rotor inductances 70324 The rotor mutual inductances 72325 The stator-rotor mutual inductances 73326 The rotor voltage equations 75327 The voltage and mechanical equations 75328 Reduction of the model 77
33 The static and dynamic behavior 82331 The steady state of the induction machine 82
3311 Assessment of the power 823312 Characteristics of the electromagnetic torque 84
332 Some practical characteristics 88333 The dynamics of the induction motor 92
3331 No choice of reference frame 943332 Choice of rotor reference frame 943333 Choice of stator reference frame 953334 Choice of synchronous reference frame 953335 Arrangement of variables 96
334 Some electromagnetic torque expressions 9834 Winding and induced harmonics 99
341 Principle of the rotating field 100342 The effect of currents 103
3421 Effect of unbalanced currents 1043422 Effect of non-sinusoidal currents 1063423 Effect of non-sinusoidal winding 1073424 Effect of harmonic components and winding 108
343 Choices of winding 1083431 Single-layer winding 1093432 Concentric and distributed winding 112
viii Asynchronous Machine with Variable Speed
3433 Double-layer winding 11335 Squirrel cage 115
351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117
36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120
37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124
38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126
3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128
382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129
39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134
Chapter 4 Speed Variation 137
41 Cases of multiphase machines 137411 Motors with a high number of phases 138
4111 Type-I motors 1384112 Type-II motors 140
412 Interactions between harmonics 141413 Three-phase induction machine 144
4131 Three-phase model 1444132 Application in another frame 146
414 Five-phase induction machine 151415 Double-star induction motor 155
4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161
42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
First published 2011 in Great Britain and the United States by ISTE Ltd and John Wiley amp Sons Inc Adapted and updated from La machine asynchrone agrave vitesse variable 1amp2 capteurs modegraveles controcircle et diagnostic published 2006 in France by Hermes ScienceLavoisier copy LAVOISIER 2006
Apart from any fair dealing for the purposes of research or private study or criticism or review as permitted under the Copyright Designs and Patents Act 1988 this publication may only be reproduced stored or transmitted in any form or by any means with the prior permission in writing of the publishers or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address
ISTE Ltd John Wiley amp Sons Inc 27-37 St Georgersquos Road 111 River Street London SW19 4EU Hoboken NJ 07030 UK USA
wwwistecouk wwwwileycom
copy ISTE Ltd 2011 The rights of Hubert Razik to be identified as the author of this work have been asserted by him in accordance with the Copyright Designs and Patents Act 1988 ____________________________________________________________________________________
Library of Congress Cataloging-in-Publication Data Razik Hubert Handbook of asynchronous machines with variable speed Hubert Razik p cm Includes bibliographical references and index ISBN 978-1-84821-225-1 1 Electric machinery Induction--Automatic control 2 Electric motors Induction--Automatic control 3 Electric driving Variable speed I Title TK2731R37 2011 6213136--dc22
2010048625
British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-225-1 Printed and bound in Great Britain by CPI Antony Rowe Chippenham and Eastbourne
Table of Contents
Foreword xiii
Introduction xvii
Chapter 1 Sensors and Electrical Measurements 1
11 Optical encoder 2111 Technical aspect 2112 Absolute encoder 3113 Incremental encoder 6
12 The velocity measurement 7121 Method of the frequency counter 7122 Method of the period measurement 8
13 The resolver 914 The isolated measurement 14
141 The isolated ammeter 14142 The isolated voltmeter 15
15 The numerical aspect 1516 The analog to digital converter 16
161 Principle of the flash converter 17162 Principle of the successive approximation converter 18163 The zero-order hold 18164 The multiplexer 19165 Principle of converter using slope(s) 20
17 The digital-to-analog converter 2118 The digital output 2219 The arithmetic logic unit 22110 Real time or abuse language 23111 Programming 24
v
vi Asynchronous Machine with Variable Speed
Chapter 2 Analog Numerical Control 25
21 Structure of a regulator 2522 Stability of a system 26
221 Introduction 26222 A formal criterion 27223 A graphical criterion 28224 The stability criterion 29
23 Precision of systems 30231 The initial and final value 30232 The precision of systems 31
24 Correction of systems 31241 The lag and lead corrector 32242 Other correctors 33
25 Nonlinear control 34251 First harmonic method 34252 The oscillation stability 34
26 Practical method of identification and control 35261 Broiumldarsquos method 35262 Zieglerrsquos and Nicholsrsquos method 36
27 The digital correctors 36271 Digital controller 36272 The Z-transform 37273 The Z-transform of a function 38274 Advanced Z-transform 39275 The Z-transform of a loop 40276 Some theorems 41
2761 The initial and final value 412762 The recurrence relation 412763 The fraction expansion 42
277 The Jury stability criterion 42278 Stability graphical criterion 43
2781 The bilinear transform 442782 The formal criterion 442783 The graphical criterion 45
28 Classical controllers 45281 The PID structure 46282 The PI anti-windup structure 46283 Conversion of an analog controller to a digital controller 48
2831 Approximation of the integrator 482832 Use of the bilinear transform 49
29 Disadvantages of digital controller 52291 Choice of the sampling period 52292 Noise 53
Table of Contents vii
2921 Reminder of some concepts 532922 Quantization by truncation 532923 Quantization by rounding 542924 Quantization of a product using tworsquos complement 552925 Quantization of a product by truncation 562926 The signal-to-quantization noise ratio 57
293 Cycles limits and limitations 58
Chapter 3 Models of Asynchronous Machines 59
31 The induction motor 59311 The electromagnetic torque 62312 The equivalent scheme 63
32 The squirrel cage induction motor 66321 The stator inductances 67322 The stator mutual inductances 69323 The rotor inductances 70324 The rotor mutual inductances 72325 The stator-rotor mutual inductances 73326 The rotor voltage equations 75327 The voltage and mechanical equations 75328 Reduction of the model 77
33 The static and dynamic behavior 82331 The steady state of the induction machine 82
3311 Assessment of the power 823312 Characteristics of the electromagnetic torque 84
332 Some practical characteristics 88333 The dynamics of the induction motor 92
3331 No choice of reference frame 943332 Choice of rotor reference frame 943333 Choice of stator reference frame 953334 Choice of synchronous reference frame 953335 Arrangement of variables 96
334 Some electromagnetic torque expressions 9834 Winding and induced harmonics 99
341 Principle of the rotating field 100342 The effect of currents 103
3421 Effect of unbalanced currents 1043422 Effect of non-sinusoidal currents 1063423 Effect of non-sinusoidal winding 1073424 Effect of harmonic components and winding 108
343 Choices of winding 1083431 Single-layer winding 1093432 Concentric and distributed winding 112
viii Asynchronous Machine with Variable Speed
3433 Double-layer winding 11335 Squirrel cage 115
351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117
36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120
37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124
38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126
3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128
382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129
39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134
Chapter 4 Speed Variation 137
41 Cases of multiphase machines 137411 Motors with a high number of phases 138
4111 Type-I motors 1384112 Type-II motors 140
412 Interactions between harmonics 141413 Three-phase induction machine 144
4131 Three-phase model 1444132 Application in another frame 146
414 Five-phase induction machine 151415 Double-star induction motor 155
4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161
42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Table of Contents
Foreword xiii
Introduction xvii
Chapter 1 Sensors and Electrical Measurements 1
11 Optical encoder 2111 Technical aspect 2112 Absolute encoder 3113 Incremental encoder 6
12 The velocity measurement 7121 Method of the frequency counter 7122 Method of the period measurement 8
13 The resolver 914 The isolated measurement 14
141 The isolated ammeter 14142 The isolated voltmeter 15
15 The numerical aspect 1516 The analog to digital converter 16
161 Principle of the flash converter 17162 Principle of the successive approximation converter 18163 The zero-order hold 18164 The multiplexer 19165 Principle of converter using slope(s) 20
17 The digital-to-analog converter 2118 The digital output 2219 The arithmetic logic unit 22110 Real time or abuse language 23111 Programming 24
v
vi Asynchronous Machine with Variable Speed
Chapter 2 Analog Numerical Control 25
21 Structure of a regulator 2522 Stability of a system 26
221 Introduction 26222 A formal criterion 27223 A graphical criterion 28224 The stability criterion 29
23 Precision of systems 30231 The initial and final value 30232 The precision of systems 31
24 Correction of systems 31241 The lag and lead corrector 32242 Other correctors 33
25 Nonlinear control 34251 First harmonic method 34252 The oscillation stability 34
26 Practical method of identification and control 35261 Broiumldarsquos method 35262 Zieglerrsquos and Nicholsrsquos method 36
27 The digital correctors 36271 Digital controller 36272 The Z-transform 37273 The Z-transform of a function 38274 Advanced Z-transform 39275 The Z-transform of a loop 40276 Some theorems 41
2761 The initial and final value 412762 The recurrence relation 412763 The fraction expansion 42
277 The Jury stability criterion 42278 Stability graphical criterion 43
2781 The bilinear transform 442782 The formal criterion 442783 The graphical criterion 45
28 Classical controllers 45281 The PID structure 46282 The PI anti-windup structure 46283 Conversion of an analog controller to a digital controller 48
2831 Approximation of the integrator 482832 Use of the bilinear transform 49
29 Disadvantages of digital controller 52291 Choice of the sampling period 52292 Noise 53
Table of Contents vii
2921 Reminder of some concepts 532922 Quantization by truncation 532923 Quantization by rounding 542924 Quantization of a product using tworsquos complement 552925 Quantization of a product by truncation 562926 The signal-to-quantization noise ratio 57
293 Cycles limits and limitations 58
Chapter 3 Models of Asynchronous Machines 59
31 The induction motor 59311 The electromagnetic torque 62312 The equivalent scheme 63
32 The squirrel cage induction motor 66321 The stator inductances 67322 The stator mutual inductances 69323 The rotor inductances 70324 The rotor mutual inductances 72325 The stator-rotor mutual inductances 73326 The rotor voltage equations 75327 The voltage and mechanical equations 75328 Reduction of the model 77
33 The static and dynamic behavior 82331 The steady state of the induction machine 82
3311 Assessment of the power 823312 Characteristics of the electromagnetic torque 84
332 Some practical characteristics 88333 The dynamics of the induction motor 92
3331 No choice of reference frame 943332 Choice of rotor reference frame 943333 Choice of stator reference frame 953334 Choice of synchronous reference frame 953335 Arrangement of variables 96
334 Some electromagnetic torque expressions 9834 Winding and induced harmonics 99
341 Principle of the rotating field 100342 The effect of currents 103
3421 Effect of unbalanced currents 1043422 Effect of non-sinusoidal currents 1063423 Effect of non-sinusoidal winding 1073424 Effect of harmonic components and winding 108
343 Choices of winding 1083431 Single-layer winding 1093432 Concentric and distributed winding 112
viii Asynchronous Machine with Variable Speed
3433 Double-layer winding 11335 Squirrel cage 115
351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117
36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120
37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124
38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126
3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128
382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129
39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134
Chapter 4 Speed Variation 137
41 Cases of multiphase machines 137411 Motors with a high number of phases 138
4111 Type-I motors 1384112 Type-II motors 140
412 Interactions between harmonics 141413 Three-phase induction machine 144
4131 Three-phase model 1444132 Application in another frame 146
414 Five-phase induction machine 151415 Double-star induction motor 155
4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161
42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
vi Asynchronous Machine with Variable Speed
Chapter 2 Analog Numerical Control 25
21 Structure of a regulator 2522 Stability of a system 26
221 Introduction 26222 A formal criterion 27223 A graphical criterion 28224 The stability criterion 29
23 Precision of systems 30231 The initial and final value 30232 The precision of systems 31
24 Correction of systems 31241 The lag and lead corrector 32242 Other correctors 33
25 Nonlinear control 34251 First harmonic method 34252 The oscillation stability 34
26 Practical method of identification and control 35261 Broiumldarsquos method 35262 Zieglerrsquos and Nicholsrsquos method 36
27 The digital correctors 36271 Digital controller 36272 The Z-transform 37273 The Z-transform of a function 38274 Advanced Z-transform 39275 The Z-transform of a loop 40276 Some theorems 41
2761 The initial and final value 412762 The recurrence relation 412763 The fraction expansion 42
277 The Jury stability criterion 42278 Stability graphical criterion 43
2781 The bilinear transform 442782 The formal criterion 442783 The graphical criterion 45
28 Classical controllers 45281 The PID structure 46282 The PI anti-windup structure 46283 Conversion of an analog controller to a digital controller 48
2831 Approximation of the integrator 482832 Use of the bilinear transform 49
29 Disadvantages of digital controller 52291 Choice of the sampling period 52292 Noise 53
Table of Contents vii
2921 Reminder of some concepts 532922 Quantization by truncation 532923 Quantization by rounding 542924 Quantization of a product using tworsquos complement 552925 Quantization of a product by truncation 562926 The signal-to-quantization noise ratio 57
293 Cycles limits and limitations 58
Chapter 3 Models of Asynchronous Machines 59
31 The induction motor 59311 The electromagnetic torque 62312 The equivalent scheme 63
32 The squirrel cage induction motor 66321 The stator inductances 67322 The stator mutual inductances 69323 The rotor inductances 70324 The rotor mutual inductances 72325 The stator-rotor mutual inductances 73326 The rotor voltage equations 75327 The voltage and mechanical equations 75328 Reduction of the model 77
33 The static and dynamic behavior 82331 The steady state of the induction machine 82
3311 Assessment of the power 823312 Characteristics of the electromagnetic torque 84
332 Some practical characteristics 88333 The dynamics of the induction motor 92
3331 No choice of reference frame 943332 Choice of rotor reference frame 943333 Choice of stator reference frame 953334 Choice of synchronous reference frame 953335 Arrangement of variables 96
334 Some electromagnetic torque expressions 9834 Winding and induced harmonics 99
341 Principle of the rotating field 100342 The effect of currents 103
3421 Effect of unbalanced currents 1043422 Effect of non-sinusoidal currents 1063423 Effect of non-sinusoidal winding 1073424 Effect of harmonic components and winding 108
343 Choices of winding 1083431 Single-layer winding 1093432 Concentric and distributed winding 112
viii Asynchronous Machine with Variable Speed
3433 Double-layer winding 11335 Squirrel cage 115
351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117
36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120
37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124
38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126
3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128
382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129
39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134
Chapter 4 Speed Variation 137
41 Cases of multiphase machines 137411 Motors with a high number of phases 138
4111 Type-I motors 1384112 Type-II motors 140
412 Interactions between harmonics 141413 Three-phase induction machine 144
4131 Three-phase model 1444132 Application in another frame 146
414 Five-phase induction machine 151415 Double-star induction motor 155
4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161
42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Table of Contents vii
2921 Reminder of some concepts 532922 Quantization by truncation 532923 Quantization by rounding 542924 Quantization of a product using tworsquos complement 552925 Quantization of a product by truncation 562926 The signal-to-quantization noise ratio 57
293 Cycles limits and limitations 58
Chapter 3 Models of Asynchronous Machines 59
31 The induction motor 59311 The electromagnetic torque 62312 The equivalent scheme 63
32 The squirrel cage induction motor 66321 The stator inductances 67322 The stator mutual inductances 69323 The rotor inductances 70324 The rotor mutual inductances 72325 The stator-rotor mutual inductances 73326 The rotor voltage equations 75327 The voltage and mechanical equations 75328 Reduction of the model 77
33 The static and dynamic behavior 82331 The steady state of the induction machine 82
3311 Assessment of the power 823312 Characteristics of the electromagnetic torque 84
332 Some practical characteristics 88333 The dynamics of the induction motor 92
3331 No choice of reference frame 943332 Choice of rotor reference frame 943333 Choice of stator reference frame 953334 Choice of synchronous reference frame 953335 Arrangement of variables 96
334 Some electromagnetic torque expressions 9834 Winding and induced harmonics 99
341 Principle of the rotating field 100342 The effect of currents 103
3421 Effect of unbalanced currents 1043422 Effect of non-sinusoidal currents 1063423 Effect of non-sinusoidal winding 1073424 Effect of harmonic components and winding 108
343 Choices of winding 1083431 Single-layer winding 1093432 Concentric and distributed winding 112
viii Asynchronous Machine with Variable Speed
3433 Double-layer winding 11335 Squirrel cage 115
351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117
36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120
37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124
38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126
3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128
382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129
39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134
Chapter 4 Speed Variation 137
41 Cases of multiphase machines 137411 Motors with a high number of phases 138
4111 Type-I motors 1384112 Type-II motors 140
412 Interactions between harmonics 141413 Three-phase induction machine 144
4131 Three-phase model 1444132 Application in another frame 146
414 Five-phase induction machine 151415 Double-star induction motor 155
4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161
42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
viii Asynchronous Machine with Variable Speed
3433 Double-layer winding 11335 Squirrel cage 115
351 The fundamental component of MMF 115352 Effect of harmonics due to slots 116353 Effect of harmonic components on the torque 117
36 Variation in air-gap permeance 118361 Effect of the rotor and stator slots 119362 Effect of magnetic saturation 120363 Effect of eccentricity 120
37 Noise and vibrations 121371 The first harmonics approach 122372 Choice of the number of rotor bars in squirrel-cage induction 124
38 Influence of rotor frequency 125381 One ideal rotor bar at null frequency 126
3811 Aspects of the rotor bar 1263812 The aspect of the isthmus 1273813 Synthesis 128
382 One ideal rotor bar at non-null frequency 1283821 The aspect of inductance 1293822 The aspect of resistance 1293823 Synthesis 129
39 Thermal behavior 130391 Insulation classes 131392 Static thermal model 132393 A dynamic hybrid thermal model 134
Chapter 4 Speed Variation 137
41 Cases of multiphase machines 137411 Motors with a high number of phases 138
4111 Type-I motors 1384112 Type-II motors 140
412 Interactions between harmonics 141413 Three-phase induction machine 144
4131 Three-phase model 1444132 Application in another frame 146
414 Five-phase induction machine 151415 Double-star induction motor 155
4151 Six-phase induction motor version 1 1564152 Six-phase induction motor version 2 161
42 Control of asynchronous motors 164421 The basic environment 166422 Scalar control Vf 167423 Vector control Vf 169
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Table of Contents ix
4231 A classical approach 1714232 Variant without a speed sensor 172
424 Direct torque control (DTC) 1754241 The concept 1784242 Strategy of vector choice 1814243 Torque ripple 1824244 Three-level inverter 1844245 Influence of voltage limitation 1894246 The DTC-SVM approach 1894247 Prediction of the torque ripple 1924248 Application to a five-phase induction motor 193
425 Direct self-control approach (DSC) 194426 Vector control FOC 197
4261 Application to three-phase induction motors 2004262 Application to five-phase induction motors 2034263 Application to six-phase induction motors 207
427 Control without a position sensor 208428 Exploitation of natural asymmetries 209
4281 The static and dynamic eccentricity 2094282 The rotor slots effect 2104283 The magnetic saturation effect 2114284 The estimation of the velocity 2114285 Spectrum estimation 213
429 Estimation by high-frequency injection 21343 Identification of parameter aspects 216
431 Classical methods 2164311 The step method 2174312 Empirical method 219
432 Generic methods 2214321 Principle of the method based on the model 2214322 The gradient method 2224323 The Newton-Raphson method 2224324 The Marquardt-Levenberg method 2224325 The genetic algorithm 2234326 Identification of electrical and mechanical parameters 225
433 Conclusion 22644 Voltage inverter converters 227
441 Inverters using the pulse width modulation technique 2274411 Two-level inverter 2284412 Over-modulation 2324413 Three levels inverter 2334414 Three-level inverter using clamped capacitor 2354415 Four-level inverter 236
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
x Asynchronous Machine with Variable Speed
4416 Multi-levels inverter 239442 The inverters using the space vector modulation 243
4421 Application to the three-phase induction motor 2454422 Application to the five-phase induction motor 2494423 Application to the six-phase induction motor 2534424 Multilevel aspect 257
443 The matrix converter 2614431 Direct matrix converter 2634432 Indirect matrix converter 266
45 Rectifiers based on the PWM 268451 Two-level rectifier 268452 Three-level rectifier 270
Chapter 5 Tools of Fuzzy Logic 273
51 Preamble 27352 Introduction 27453 Fuzzy logic 275
531 Definitions and norms 275532 Some variants 276533 T -norm and T -conorm 276534 Membership functions 277535 Inference engine 278536 Defuzzification 280
54 Fuzzy logic controller 28055 Fuzzy and adaptive PI 284
551 Examples of programs to calculate a fuzzy surface 2865511 The layout of a fuzzy surface 2865512 Routine of a PI-fuzzy controller 287
552 Examples of application 288553 Examples of simulation results 289
5531 Controller based on a fuzzy PI 2895532 A controller based on a fuzzy PID 291
554 Examples of tables of rules 29156 Conclusion 295
Chapter 6 Diagnostics and Signals Pointing to a Change 297
61 Signals and measurements 29862 Defects 299
621 Problems with broken bars 300622 Problems in the stator 302623 Problems due to eccentricities 304624 Problems due to speed ripples 307625 Problems with ball bearings 307
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Table of Contents xi
63 Analysis of signals 309631 Fast Fourier transform analysis of the stator current 309632 Fast Fourier transform 309633 Discrete fast Fourier transform 311634 Windows functions 312
6341 The Hamming function 3136342 The Hanning function 3136343 The Blackmann function 3136344 The Bartlett function 3136345 The Kaiser function 313
635 Sliding discrete fast Fourier transform 3146351 Zoom effect 316
64 Some considerations regarding broken bar defects 317641 Model of the induction motor 317642 Inherent frequencies in the broken bar defect 318643 Evaluation of the magnitude of the left line 320644 Equivalent model in the steady state 320
65 Evaluation of the severity of broken bars 322651 Some spectra results 322652 Evaluation of the severity of broken bars 326
6521 Analytical approach 3266522 Artificial intelligence approach 3286523 Self-extraction of signatures an application of PSO 330
653 Wireless communication 335
Exercise No 1 Fuzzy Logic 337
11 Adaptive k and ki coefficients in function of the error 33712 Adaptive k and ki coefficients in function of the error and
its derivative 33813 Answers 339
Exercise No 2 The Stator Defect 345
21 Equations of the induction motor under stator defect 34722 Torque ripple due to a stator defect 34823 Fault current estimation 34924 Schematic model of three-phase induction motor under a stator
defect 35025 Answers 351
Exercise No 3 The Control of Five-Phase Induction Motors 357
31 The five-phase system 35832 Distribution of active currents 359
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
xii Asynchronous Machine with Variable Speed
33 A model for control 36234 Answers 364
Exercise No 4 The Control of Serial Connected Induction Motors 373
41 Study about the serial connection of two five-phase inductionmotors 374
42 Study on the serial connection of several seven-phaseinduction motors 375
43 Study on the serial connection of multi-phase induction motors 37744 Answers 378
Exercise No 5 Fault Detection of a Three-Phase Voltage InverterConverter 385
51 A conducting fault 38652 Fault detector 38753 Monitoring of the DC component 38954 Answers 390
Appendix Some Mathematical Expressions 393A1 Laplace transforms 393A2 Z transforms 394A3 W transforms 395A4 Common expressions 395A5 Trigonometric identities 395
A51 Addition 396A52 Sum identities 396A53 Product identities 397A54 The product 397A55 Sum of sinus and cosinus 397
A6 Mathematical series 398A7 Greek numbers 398
Bibliography 399
Index 407
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Foreword
The asynchronous machine also known as the induction machine has been anindustry workhorse for more than 120 years The first rotating machine to work usingalternating current its advent started a new chapter in the evolution of our societywith the propagation of long-distance electrical lines ensuring a large distributionof electrical energy and a systematic replacement of steam-powered machines withelectrical ones This expansion still continues today as much in industrial applicationsand transport as home automation The induction machine is a part of our everydaylives in the broadest sense it is present in all Western households and emerging anddeveloping countries strongly depend on its use
Naturally since its first appearance the induction motor has undergonerevolutions and changes through the development of techniques and progress inthe field of magnetic or dielectric materials This can easily be seen from the sizeof the machines with a power-to-mass ratio that has considerably evolved and canoperate at variable speeds which makes it a broad-spectrum actuator The intrinsicperformances of this machine have considerably progressed first through a moreoptimized construction of its structure and by the use of better quality materials butmostly from the set-up of an adapted electronic environment allowing the optimizedmanagement of energy processing
To this end this book by Professor Hubert Razik provides an eloquent and originaltestimonial by discussing in an educational and rational way all the constituentelements of a variable speed drive at the base of the asynchronous machine In thisway we can easily navigate between methods and tools principles and rules in orderto cover all the angles of modern control of this machine by making the link betweensignal processing control and diagnostics
This book is split into six chapters and accompanied by a bibliography consistingof the greatest contributions to the field
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
xiv Asynchronous Machine with Variable Speed
Chapter 1 covers sensors and electrical measurements and brings to light in aconcise way the notions of coding and restitution of information that can be read by acontroller
Chapter 2 clarifies the fundamental elements for the control of the systems througha review of the representation and synthesis of corrector tools in the context of electricmotor control (identification nonlinear control stability data quantization etc)
Chapter 3 covers the modeling of induction motors with a gradual evolutionfrom simple models valid in nominal cases up to taking into account very unusualphenomena such as magnetic saturation rotor eccentricity or thermal incidence Itcontains a very explicit and relevant contribution
Chapter 4 the longest of all splits speed variation into its different aspects Itdiscusses scalar then vector control then passes through direct torque control andcontrols without mechanical sensors This part is subject to several declinations beit over the phases of the motor (five- or six-phase motor) or even over the numberof inverter output levels (three level four or multi-level) This chapter is even morevaluable since it is based on the real know-how of the author It is the result of longand meticulous practice searching for better solutions
Chapter 5 covers fuzzy logic by simply setting down the meets and bounds ofthis approach and its applications for control of the asynchronous motor The basis iscarefully shown allowing quick and relevant applications for the given potentials
The final chapter ndash Chapter 6 ndash covers the delicate problem of diagnostics ofelectric motors which is currently the subject of ever more numerous and openresearch Professor Razikrsquos contribution to the field is very well developed He hascompiled an inventory of different problems and proposed methods of detectingand identifying defects Ultimately machine monitoring is proposed via the use ofwireless communication
The authorrsquos educational background is present throughout this book Forcompletion the book proposes a series of corrected exercises that allow theassimilation of the concepts needed for this discipline
The operation and diagnostics of electric motors such as induction motorsremains a very complex discipline since it brings into play conversion and processingphenomena of electrical energy with a distinct nonlinear character This is an exampleof an application that joins significant know-how and we thank Professor Razikfor this accessible work demystifying the laws of control and diagnostics of thesedevices
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Foreword xv
This book will be useful for postgraduate students engineering students theirtutors and other young researchers starting a career in this field where much progressis awaited The interconnection of the notions covered make this a complete book forthe field of electric motors
Professor Maurice FADELENSEEIHT
Deputy Director of LAPLACE
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
xvi
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Introduction
This book is aimed as much at students and postgraduates whether working inresearch or not as it is for tutors and engineers
It offers students knowledge that goes from the basics to a much more advancedlevel the broad spectrum offered to readers allows easy comprehension of morespecialized books Exercises are included at the end of the book
Engineers will be able to find elements corresponding to their needs whetheracademic or providing the in-depth knowledge that will give them the tools for broaderthinking
This book covers the asynchronous machine in its immediate environment Itstarted out as a reflection on the electromagnetic converter whose integration inindustrial environments is taking on an increasingly important role Historically thismotor was used in a chain of a fixed speed process Now it is increasingly beingintegrated in variable speed processes It is for this reason it seemed useful if notnecessary to write a book covering the different aspects from the motor itself passingthrough control and finishing with the diagnostic aspect In fact asynchronousmotors are today in an industry where variation of speed and reliability are requiredFor sensitive systems we should always know the state of the conversion chain andinform the operator of the appearance of any anomaly and its severity
All these approaches are described in a rigorous manner and numerous referenceswill allow readers to further develop their comprehension and knowledge
I would like to thank my wife and my daughters for their loving support throughoutthis endeavour
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
xviii
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Chapter 1
Sensors and Electrical Measurements
Measurement is an obligatory phase that we cannot avoid In fact in allelectromagnetic processes ndash whether at a variable or constant speed ndash instructedmeasurements are carried out These measurements are compared to those taken tocorrect the process so that they conform to the desired values This phase is used as acontrol and in some cases is incorporated into regulations Such a system can generallybe split into four parts as we see in Figure 11 which are
(1) instruction(s) and protection these different parts allow us to take into accountthe needs and constraints
(2) control and regulation(s) this is dedicated to control loop that can consist ofseveral interleaved loops
(3) electric supply this is an interface allowing application of the desired voltagesto the motor in alternating current
(4) electric motor this converts electrical energy into mechanical energy
Between these parts connections are necessary either to transmit information(instructions between parts) or to transmit energy from a source of power towards theelectric motor For this analog lines such as coaxial or fiber optics are used for powerFor instruction(s) and regulation(s) digital connections are preferred It will benecessary to represent the numbers in digital form which will lead to a quantizationthat will be represented in binary form The choice is not without consequencesince this noise of quantization will alter the performance of the system (closed loopprocess) Nevertheless we can reasonably say that a representation between 12 and16 bits is acceptable for the majority of microcontrollers for regulation
1
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
2 Asynchronous Machine with Variable Speed
Figure 11 Flow diagram representing the process with regulation
Let us approach the measurement aspect of the angular position of a mechanicalshaft then its rotation speed For this two categories of coders exist one is opticaland the other is electromagnetic
11 Optical encoder
Different types of optical coders exist Certain ones have many advantages andothers fewer along with the inherent constraints of the choice of technology used Forthis reason we will first cover the measurement of the absolute position and secondthe measurement of the relative position of the mechanical shaft
111 Technical aspect
What is an optical encoder
An optical encoder is a system that allows us to provide logic level transitions 1and 0 depending on the mechanical rotation It consists of a disc with cuts similar to amultitude of lines that allow light to pass though In this way a beam of light will excitea receiver which will provide either level 1 or level 0 transitions after transformationof the collected signals Figure 12 is a synoptic diagram of an optical encoder wherewe can discern the disc itself with it axis of revolution light emitter receiver and spaceletting the beam of light pass through
The optical encoder is free of restraints since mechanical disturbances interferewith its lifespan and its performance For this reason overspeeds are not allowed sincethis will destroy the cuttings as well as causing mechanical vibrations that will vibratethe disc bringing it into involuntary contact with the receiver or the emitter In bothcases the defect encountered is irreversible since the cutting is affected by an anomalyBefore mentioning some of the defects of encoders let us now look at the differenttypes of optical encoders starting with the absolute encoder
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Sensors and Electrical Measurements 3
Figure 12 Representative diagram of an optical encoder
112 Absolute encoder
The advantage of this encoder is to provide precise information of the angularposition since during the operation of the sensor it is the real position of the shaftthat is provided Nevertheless for problems of cutting quality only the Gray codeis recommended In fact there is one and only one transition of logic state 0 to 1or 1 to 0 This is not the case for classic binary code For the purposes of structuralcomparison Figure 13(a) shows the said relative absolute encoder and Figure 13(b)the Gray encoder The angular representation is given in Table 11
(a) The natural code (b) The Gray code
Figure 13 Different types of absolute encoder (a) the natural or binary code and (b) theGray encoder whose angular resolution is Δθ = 360˚23
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
4 Asynchronous Machine with Variable Speed
These two coders have a resolution of eight positions This means that we will onlyhave the position of the measurable shaft in the sectors that have an angular definitionof 2π8 radians or 45˚ For this reason Table 12 lists all the values for this type ofcoder as well as the codes and angular sectors For illustrative purposes code No 2corresponds to an angular position of the shaft of 112˚ plusmn 225˚ The correspondingGray code is 010 and the natural binary code is 011 It is obvious that the Gray codesuch as it is does not give the direct correspondence of the angular position Also weare brought to convert the Gray code to a natural binary code This can easily be doneby using logic gates whose function is ldquoor exclusiverdquo (see Figure 14) represented bythe symbol oplus
Number of bits Number of sectors Angle in radian Angle in degrees
1 2 3141592 180
4 16 0392699 225
8 256 0024543 140625
10 1024 0006135 0351562
12 4096 0001534 0008789
Table 11 Table illustrating the angular resolution of an optical encoder
Decimal code Binary code Gray code Angular sector
0 000 000 0ndash45˚
1 001 001 45ndash90˚
2 010 011 90ndash135˚
3 011 010 135ndash180˚
4 100 110 180ndash225˚
5 101 111 225ndash270˚
6 110 101 270ndash315˚
7 111 100 315ndash360˚
d2 d1 d0 g2 g1 g0
Table 12 Table illustrating the conversion of the Gray code to a natural one
For a number n of bits
ndash dn = gn
ndash di = gn oplus gnminus1 oplus gnminus2 middot middot middot gi so that i lt n
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Sensors and Electrical Measurements 5
Figure 14 shows the use of these logic gates
Figure 14 Conversion of the Gray code to a natural code
In the case of a cutting fault the absolute encoder using the natural code will beinterrupted Figure 15 shows two examples of interruptions
Case 1 (the code changes from 111 to010 then 000) The angular sectors are
therefore 315ndash360˚90ndash135˚ then 0ndash45˚
Case 2 (the code changes from 001 to000 then 010) The angular sectors are
therefore 45ndash90˚0ndash45˚ then 90ndash135˚
Figure 15 Absolute natural encoder with two cutting faults
In any event we note a failure in the transitions For this the Gray code will befavored and only one transition at a time can occur Therefore we can overcome thesector imperfections due to imprecise cutting through an adapted digital system It is
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
6 Asynchronous Machine with Variable Speed
to be noted that with an increasing number of tracks the angular resolution increasesin the same way leading to a significant number of light emitters and receivers nearthe disc In order for an encoder to be of a smaller size the collection of componentsmust coexist in a confined space
113 Incremental encoder
A relative encoder is an encoder that cannot provide the exact position of the shaftduring the operation of the device In fact it provides a position relative to that of itsstarting point As a result its simplicity comes down to the requirement for one cuttrack instead of a huge number as in the case of an absolute encoder However we adda second track shifted by 90˚ electric in such a way as to discern the rotation of theshaft As we have shown in Figure 16 these two tracks are shifted in relation to eachother but with a surprising allowance (manufacturerrsquos data) Track B does not requireparticular attention during cutting since it is only dedicated to obtaining the directionof rotation A third track commonly called zero generates one pulse per revolutionThis allows us to lock a position
Figure 16 The relative or incremental encoder
The angular resolution is therefore inversely proportional to the number of pointsper revolution In order to obtain the angular movement it is necessary to use acountercount-down counter that track A will excite Obtaining the direction ofrotation will be achieved using an RS edge latch Track B connected to the input ofthe clock will change output Q of the latch to the value of track A connected to thedata input at each rising edge of B
In order to measure the position we can opt for several methods that all haveadvantages and disadvantages We can proceed as follows
(1) s = A simple coding on level A
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Sensors and Electrical Measurements 7
(2) s =uarr A simple coding on rising edges of A
(3) s =uarr A and uarr B double coding on rising edges
(4) s =uarr A and darr A uarr B and darr B quadruple encoding on rising and falling edges
The HCTL 2000 is one of the circuits allowing the measurement of positionIt is very versatile in its use while requiring the minimum number of componentsThis integrated circuit (Hewlett Packard) is a decodercounter interface for a relativeencoder We can also refer to Texas Instrumentsrsquo THCT 12316 circuit which presentsa high quality interface allowing us to decode three simultaneous tracks Each trackis compatible with the basic TCHT 2000 circuit It detects the direction of rotationmeasures the position in simple double or quadruple coding either length of thepulses or frequency and provides the result on a number of 16 bits This circuit iscommercialized with a PLCC casing with 68 pins The monotrack version is calledTHCT 12016 (dual-in-line 28 PIN casing)
12 The velocity measurement
The measurement of rotation speed can be done in three ways The first consistsof calculating the difference in position between two precise instants and dividingthe result by the measurement time interval The second calls upon the method of afrequency indicator the third the method of a period meter These two last methodshave the advantage of directly providing information relative to the speed of rotationon a while number of 16 bits
121 Method of the frequency counter
The method of the frequency counter is based on counting the number pulses ina given time frame From a precise time base we count the number of pulses Nappearing from the encoder over an interval Tm In this way the frequency of thesignal received has for a relationship
F = NTm (11)
If the number of pulses per revolution of the encoder is Ni then the rotationfrequency Fr in Hz is
Fr = NNiTm (12)
We would not be able to finish this approach to measurement without concerningourselves with the precision of the measurement In fact if we commit a counting
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
8 Asynchronous Machine with Variable Speed
error of a pulse then the measurement error of the frequency of rotation is written
Error(N) = Fr(N + 1)minus Fr(N) =1
NiTm(13)
or in percent
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=
100N
(14)
As a result the higher the number of pulses counted the lower the error on thefrequency of rotation It is convenient to change the measurement method when thisreaches excessive values Now we will look at the method of the measuring the period
122 Method of the period measurement
As with the method of frequency counting period measurement is based oncounting the number of pulses in a given time frame The difference resides in thesignal to be analyzed It concerns taking the signal of track A as a counting frameThe precise time base is a very high frequency signal that stable in the time Fhf We therefore count the number of these high frequency pulses N appearing duringa measurement time interval Tm linked to the speed of rotation of the shaft In thisway the frequency observed is
F = 1Tm (15)
If the number of pulses per rotation of the encoder is Ni the rotation frequency Fr
in Hz is reduced to
Fr = FhfNiN (16)
As a result the lower the rotation speed the higher the number of pulses Thismethod is therefore adapted for the measurement of low rotation speeds Once againit is necessary to concern ourselves with measurement error In the same way as beforewe will proceed to a variation of a unit over the number of pulses We therefore get
Error(N) = Fr(N + 1)minus Fr(N) =Fhf
Ni
minus1N(N + 1)
(17)
or in percentage
Error(N) =100Fr(N + 1)minus Fr(N)
Fr(N)=minus100N + 1
(18)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
Sensors and Electrical Measurements 9
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
N
Rotation speed
Frequency counter
Speed limitation
Transitionzone
Period measurement
Limitation of N
Figure 17 The number of pulses counted as a function of rotation frequency
The result is the same as with the frequency counter the higher the number ofpulses counted the lower the relative error
Figure 17 contains numerous data relative to precision and limits The countingof N is limited by the number of bits of the counter (Nmax) The frequency ofthe reproduced speed is also limited (Fmax) This figure illustrates a remarkablephenomenon created by the transition zone This zone defines the operating modegiving the best precision We therefore find that at low speed the method of periodcounting is advised At high speeds it is the method of frequency counting that isrecommended There is therefore a critical frequency that if the electronics permitwill change the measurement mode of the rotation speed
13 The resolver
A relevant question here is what is a resolver
A resolver is an analog sensor of angular position Its make-up is similar to thatof a rotary transformer Traditionally the measurement of speed and position is doneby the use of an optical encoder Nevertheless this technique is not without its limitsThe precision of the measurement is inversely proportional to the number of lines perrotation The most frequently used have between 500 and 4096 lines per revolutionOne of the disadvantages is related to the cost which is exponential depending on thenumber of lines and therefore on the angular resolution Its use requires mechanicalprecautions It is necessary to avoid mechanical disturbances (vibrations) that lead to
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)
10 Asynchronous Machine with Variable Speed
premature wearing of the encoder and implicates the use of specific coupling (and istherefore expensive) that allows mechanical defaults It is also necessary to ensure thatthe maximum rotation speed is not exceeded due to the risk of irreversible alterationsto the lines leading to irreparable distortion of the measurements We must not forgetthat that an encoder whichever it is requires an electronic conversion interface
Before going into more detail on the operation and the interface necessary for thereconstruction of the angular position we can set out some properties of this device
A resolver is a rotary transformer whose rotor and stator are both coiled (seeFigure 18)
Figure 18 Principle diagram of a resolver
There is an advantage in the external aspect of the resolver In effect this isjoined to the electric motor If this is explosion-proof then the resolver will alsobe so The axis of the said rotor will be coupled with the machine whose speed wewant to measure This will be excited by sinusoidal voltage of constant amplitude andfrequency (for example Vexc = V cos(ωt) with V = 7 v effective and f = 5 kHz)
As shown in Figure 19 the analog-digital converter has three basic components
ndash a phase comparator
ndash a ldquolooprdquo filter
ndash a voltage controlled oscillator
The phase comparator consists of a ldquocosinusoidalrdquo multiplier a sinusoidalmultiplier a subtractor and a demodulator The basis function of the voltagecontrolled oscillator (VCO) is a digital integrator
The voltage v(t) from
v(t) = V sin θ cos(ωt) cosφminus V cos θ cos(ωt) sinφ (19)