In The Name of God The Most Compassionate, The Most...

General Theory of Electric Machines

2017 Shiraz University of Technology Dr. A. Rahideh

In The Name of God The Most

Compassionate, The Most Merciful

2

Table of Contents


1. Introduction

2. Transformers

3. Reference-Frame Theory

4. Induction Machines

5. Synchronous Machines

3

References

1. Chee-Mun Ong, "Dynamic Simulation of Electric Machinery Using MATLAB/SIMULINK", 1998, Prentice Hall PTR.

2. Paul C. Krause, Oleg Wasynczuk, Scott D. Sudhoff, "Analysis of Electric Machinery and Drive Systems", 2nd Edition, 2002, John Wiley & Sons Inc. Publication.


4

Evaluation

Homework and Quizzes

15 %

Midterm 25 %

Final 35 %

Project 25 %


5

Table of Contents


1. Introduction

2. Transformers

3. Reference-Frame Theory

4. Induction Machines

5. Synchronous Machines

6

Electric Machines

Classification in terms of motion and flux direction

Radial flux Axial flux

Linear motion Rotary motion

Inner rotor Outer rotor Single sided Double sided Multi-stage

Slotless stator

Slotted stator Slotted stator

Electric machines

Slotless stator

Slotted stator

Salient-pole stator

Internal stator

Internal rotor Slotless stator

Transverse flux Longitudinal flux Transverse flux

Flat (planar) Tubular

Single sided

Double sided

Surface

mounted

Flux

focusing

Axis

Rotor

iron

Magnets Winding Stator

iron

Axis Axis

Rotor

iron Magnets Winding Stator

iron


7

Electric Machines Classification in terms of brush and input sources

Electric Motors

Brushed Brushless

Self excited DC Separately excited DC

Series Shunt Wound PM

Sinusoidal fed Square fed

SRM BLDC Induction Synchronous

Squirrel cage Wound rotor Wound rotor PM rotor Reluctance

Compound

PM on stator

DSPM Flux-reversal DSPM Flux-switching DSPM


8

Electric Machines Applications • Automobiles with combustion engines:

• Transportation:

– elevators and escalators

– people movers

– light railways and streetcars (trams)

– electric road vehicles

– aircraft flight control surface actuation

– electric ships

– boats

• Defence forces:

– tanks

– missiles

– radar systems

– submarines

– torpedoes

– rockets

– space shuttles

– satellites


• Medical and healthcare equipment:

– dentist’s drills

– electric wheelchairs

– air compressors

– rehabilitation equipment

– artificial heart motors

• Power tools:

– drills

– hammers

– screwdrivers

– grinders

– polishers

– saws

– sanders

– sheep shearing hand-pieces

• Renewable energy systems

• Research and exploration equipment 9

Electric Machines Applications


10

Electric Machines Applications • Industry:

– industrial drives, e.g., pumps, fans, blowers, compressors, centrifuges, mills, hoists, handling systems, etc.

– machine tools

– servo drives

– automation processes

– internal transportation systems

– robots

• Public life:

– air conditioning systems

– catering equipment

– coin laundry machines

– autobank machines

– automatic vending machines

– money changing machines

– ticketing machines

– bar-code readers at supermarkets

– environmental control systems

– amusement park equipment


11

Electric Machines Applications • Domestic life:

– clocks

– kitchen equipment (refrigerators, microwave ovens, mixers, dishwashers, etc.)

– bathroom equipment (shavers, hair dryers, tooth brushes)

– washing machines and clothes dryers

– heating and air conditioning systems

– vacuum cleaners

– lawn mowers

– swimming pool pumps

– toys

– vision and sound equipment

– security systems (automatic garage doors, automatic gates)

• Information and office equipment:

– computers

– printers

– plotters

– scanners

– facsimile machines

– photocopiers


12

Modelling & Simulation

• Model: a set of equations which represent the behaviour of a system.

• Modelling: the process to derive a set of governing equations which represent the behaviour of a system

• Simulation: the implementation of the derived equations to find the response of the system to a given input


13

Electric Machine Modelling

Models can be classified in terms of being

• Linear or Nonlinear

• Lumped or Distributed Parameter

• Static or Dynamic

• Continuous or Discrete

• Deterministic or Stochastic


14


Linear vs. Nonlinear Models

• A model is linear if the superposition and homogeneity rules

are held;

• Otherwise, the model is nonlinear.

• A nonlinear model can be linearized around an operating point.

Both linear & nonlinear models are discussed.


• Superposition

• Homogeneity or Scaling

• Derivative

• Integration

15

Properties of Linear models

Linear

System

Input Output

)(1 tu )(1 ty

Linear

System

Input Output

)(2 tu )(2 ty

Linear

System

Input Output

)()( 21 tutu )()( 21 tyty

Linear

System

Input Output

)(1 tu )(1 ty

Linear

System

Input Output

)(1 tu )(1 ty

Linear

System

Input Output

)(1 tu )(1 ty

Linear

System

Input Output

)(1 tudt

d )(1 tydt

d

Linear

System

Input Output

)(1 tu )(1 ty

Linear

System

Input Output

dtu 1 dty1


16


Lumped vs. Distributed Parameter Models

• Lumped parameter models are described by ordinary

differential equations with only one independent variable which is time.

• Distributed parameter models are described by partial differential equations with time and one or more spatial coordinates as independent variables.

We focus on lumped parameter models.


17


Static vs. Dynamic Models

• Static models do not take time variations into account.

• Static models are normally used for machine design purposes.

• Dynamic models take time-varying characteristics into account.

• Dynamic models are employed to dynamically analyse electric machines.

• Dynamic models are used for control design purposes.

We focus on dynamic models.


18


Continuous vs. Discrete Models

• Continuous time models are described by equations in which the dependent variables are continuous in time.

• Discrete time models are described by difference equations whose dependent variables are defined at distinct instances.

We focus on continuous models.


)()( tt uBxAx

)()()1( kkk uBxAx

19


Deterministic vs. Stochastic Models

• A model is deterministic if there are no chance factors.

• A model is stochastic if chance factors are taken into account.

We focus on deterministic models.


20

Simplicity vs. Accuracy

• Increasing complexity of a model can improve the accuracy.

• There should be a compromise between the simplicity and accuracy of the model.

• Therefore it may be necessary to ignore some inherent physical property.

• The ignored properties should not have significant effect on the model response.


21

Different ways of model representation

1. Time domain

• Differential equations (for both linear & nonlinear systems)

• State-space equations (for both linear & nonlinear systems)

• Impulse response (only for linear systems)

2. Frequency domain

• Transfer function (only for linear systems)

• Frequency response (only for linear systems)


22

Time Domain Modelling 1. Differential equations

• For a SISO linear time-invariant system, the model can be represented by the following differential equation: where y is the output and u is the input.

• If one or more coefficients of the above differential equation

are time-dependent the system is a SISO linear time-varying system:

ubdt

dub

dt

udb

dt

udbya

dt

dya

dt

yda

dt

yda

dt

ydm

m

mm

m

mnnn

n

n

n

n

n

011

1

112

2

21

1

1

ubdt

dub

dt

udb

dt

udbya

dt

dya

dt

yda

dt

ydta

dt

ydm

m

mm

m

mnnn

n

n

n

n

n

011

1

112

2

21

1

1


23

Time Domain Modelling 1. Differential equations

• The following differential equation is, for example, for a SISO nonlinear time-invariant system:

• The following differential equation is, for example, for a SISO nonlinear time-varying system:

• It is impossible to represent a general differential equation for nonlinear systems.

uydt

dy

dt

yd

dt

yd5364

2

2

2

3

3

uydt

dyt

dt

yd

dt

yd5364 3

2

2

2

3

3


• State: The state of a dynamic system is the smallest set of variables (called state variables) such that the knowledge of these variables at t=t0, together with the knowledge of the input for , completely determines the behaviour of the system for any time .

• State vector: if n state variables are needed to completely describe the behaviour of a given system therefore:

24

Time Domain Modelling 2. State space equations

0tt

T

21 nxxx x

0tt


• State space equations can be defined for both linear and nonlinear with time-invariant or time-varying systems:

where x is the state vector, u is the input vector and y is the output vector

Time-invariant Time-varying

Linear

Nonlinear

25

Time Domain Modelling 2. State space equations

)()(

)()(

tt

tt

DuCxy

BuAxx

)()()()(

)()()()(

tttt

tttt

uDxCy

uBxAx

)(),(

)(),(

tt

tt

uxgy

uxfx

ttt

ttt

),(),(

),(),(

uxgy

uxfx


26

Time Domain Modelling 3. Impulse response

• Having the impulse response, g(t), the response of the system, y(t), with any other input, u(t), can be obtained:

t

dutgty0

)()()(

t

)(t

Process

Impulse

input

Impulse

response

)(t )(tg


27

Frequency Domain Modelling 1. Transfer function

• The transfer function of a linear, time-invariant differential equation system is defined as the ratio of the Laplace transform of the output to the Laplace transform of the input under the assumption that all initial conditions are zero:

)(

)(

tuL

tyLsG

sU

sYsG

Process

Input Output

)(tu )(ty

)(sU )(sY


28

Frequency Domain Modelling 2. Frequency response

• The input to the system is a sinusoidal signal in which the frequency is variable within a range:

• The corresponding output

)sin( tAu

tBy sin

maxmin ,

Process

Input Output

u y


29

Modelling & Simulation Procedure

1. Identify the purpose of the model and its constraints.

2. Identify the simplifying assumptions.

3. Determine the inputs, outputs and (state variables) of the system.

4. Derive the governing equations of the system based on the above two items.

5. Determine the model parameters based on physical means.

6. Implement the derived equations using the obtained parameters in a simulation environment.

7. Validate the model using the data extracted from the real system.

8. If the model response is within the required accuracy range, terminate the procedure; otherwise, go to step 2 for modification.


30


Example: Assume a separately excited DC motor is available and we need to derive a dynamic and continuous-time model of the system.

1. Identify the purpose of the model and its constraints.

• The purpose of the model is to design a control system.

• The model should be dynamic, continuous, deterministic and lumped parameter (constraint).

2. Identify the simplifying assumptions.

• The saturation effects are neglected.

• The core losses are neglected.

• Armature reaction is neglected. 2017 Shiraz University of Technology Dr. A. Rahideh

31


Example:

3. Determine the inputs, outputs and (state variables) of the system.

• The inputs are the armature voltage and the field circuit voltage.

• The output is the rotor velocity.

• The state variables are the armature current, field circuit current, rotor velocity and rotor position.


Inputs va & vf

Outputs

State variables ia , if , & q

32


Example:

4. Derive the governing equations of the system.


av ai

+ _

aR

+

ae

_

aL

J

q

B

fi fL

fR

fv +

_

Why there is no induced voltage in the field circuit?

33


Example:


4.1. KVL in armature loop:

4.2. KVL in field circuit loop:


av ai

+ _

aR

+

ae

_

aL

J

q

B

fi fL

fR

fv +

_

aa

aaaa edt

diLiRv

dt

diLiRv

f

ffff

34


Example:


4.3. Torque relation:

4.4. Back-EMF relation:

4.5. Newton’s 2nd law:


av ai

+ _

aR

+

ae

_

aL

J

q

B

fi fL

fR

fv +

_ af iikT 1

fa ike 2

dt

dq

B

dt

dJT

4.6. Velocity-position relation:

35

Modelling & Simulation Procedure Example:


• Representing in the form of state-space equations:


q

dt

d

J

Bii

J

k

dt

daf 1

a

a

f

a

a

a

aa vL

iL

ki

L

R

dt

di 12

f

f

f

f

ffv

Li

L

R

dt

di 1

Tfa ii qx

Tfa vvu

y

uxx ,f

uxy ,g

36



• As evident, in general, the system is nonlinear


q

dt

d

J

Bii

J

k

dt

daf 1

a

a

f

a

a

a

aa vL

iL

ki

L

R

dt

di 12

f

f

f

f

ffv

Li

L

R

dt

di 1

av ai

+ _

aR

+

ae

_

aL

J

q

B

fi fL

fR

fv +

_

37



If the field current is assumed to be constant, the model has one input (va), the order of the system will be 3 and the system will be linear:


q

dt

d

J

Bi

J

k

dt

da 1

ˆ

a

aa

a

a

aa vLL

ki

L

R

dt

di 1ˆ2

a

aa

aa

a

a

v

Li

J

B

J

k

L

k

L

R

i

dt

d

0

0

1

010

0ˆ

0ˆ

1

2

q

q

38



The constant matrices are:


uBxAx

uDxCy

a

aa

a

J

B

J

k

L

k

L

R

010

0ˆ

0ˆ

1

2

A

0

0

1

aL

B

010C 0D

Tai qx

avu

y

39


5. Determine the model parameters based on physical means.

• The model parameters are , , , , and .

• The armature resistance and inductance can be easily measured from the real machine.

• The torque and back-EMF constants can be obtained by experimental means or by calculation from the machine structure.

• The moment of inertia and friction coefficient can be calculated.


2k̂aRaL 1k̂ J B

8 0.86 0.02 0.02 2.2x10-4 4.5x10-5

)(aR )mH(aL )N.m/A(ˆ1k )V.s/rad(ˆ

2k )kg.m( 2J )N.m.s/rad(B

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6. Implement the derived equations using the obtained parameters in a simulation environment.


Avoid the derivative in the simulation; integration to be used.

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