ENHANCEMENT MATHEMATICAL MODEL OF IM DRIVE USING … · Cite this Article: Dr Mohammed Saadi Hasan,...

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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 9, Issue 9, September 2018, pp. 851–865, Article ID: IJMET_09_09_094

Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=9

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

ENHANCEMENT MATHEMATICAL MODEL OF

IM DRIVE USING ROTATION COORDINATE

FOR BETTER OUTPUT PERFORMANCE

Dr Mohammed Saadi Hasan, Dr Raghad Ali Mejeed and Rasha Yasen Abd

Collage of Engineering,Elect. Power & Machines department,Diyala, Iraq

University of Diyala, Iraq –Diyala

ABSTRACT

This paper has been developed the mathematical model of three phase induction

model and Simulink Leads to describe the demeanor of AC machine adopting the

rotating frame in the primary and secondary winding, in adding to describe the

mathematical approach to and analyze the interacting the magnetic field and current

variation for the three-phase induction motor control featured by the rotary frame.

The advantages of adopting the induction motor taking in account flexibility in

manufacturing application, economical maintenance and quite simple response

minimum costs are favorable. In this paper a mathematical development to general

model of three phase AC induction to MATLAB Simulink. The development applying

on model of a real drive of 1.1 kW motor has been achieved. To analysis the axis,

three-phase coordinate (A, B, C) to a two axis (β - α) as a (rotationary coordinate

system) representation. Through this drive conversion torque and speed can be

controlled easily and control optimization induction motors. MATLAB/ SIMULINK is

used to simulate the model of (IM) as tool and for study of specific characteristics of

the motor and estimated leakage flux in rotor and stator equations can be used FOC,

DTC and SLS field-oriented control, direct torque control and, sensor less speed

control of induction.

Keywords: Induction motor; coordinate β, α; MATLAB/ SIMULINK; mathematical

driving analysis; rotationay system; matrix form.

Cite this Article: Dr Mohammed Saadi Hasan, Dr Raghad Ali Mejeed and Rasha

Yasen Abd, Enhancement Mathematical model of IM Drive Using Rotation

Coordinate for Better Output Performance, International Journal of Mechanical

Engineering and Technology, 9(9), 2018, pp. 851–865.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=9

Enhancement Mathematical model of IM Drive Using Rotation Coordinate for Better Output

Performance


1. INTRODUCTION

Normally IM starts with high amount of current, as well during transient operation. Thus,

there are some draw back would be happening, like voltage sag, oscillation of torques and can

even to produce harmonics in the power systems. To simplify these problems found d-q

cardiant axis to prove and find precision and reliable tests [1-3]. This model illustrates the

basic concepts of the transient state in electrical machines and through which the dynamic

behavior of the machines is analyzed using any one of following the reference frames

a. Stationary reference frame

b. Rotor reference frame

c. Synchronous reference frame

[4-5] The tests have been recommended individually to approximate the machine

parameters to proceed with the transient model. [6-11] The rotationally synchronous reference

frame proved its efficiency in manipulating with induction machines. [12] as well as the

effectiveness of different reference frame for different purposes. However, the time period for

the simulation was found to be very short to determine the more accurate behaviors of the

induction motor.

Researchers [13-15] have developed their own software packages for the modeling and

simulation of induction machines. In [13], a software package was developed, using the

FORTRAN programming language, this for the steady-state and dynamic simulation based on

induction motor motivations. Arithmetical simulation of field-oriented control (FOC) for an

induction motor, alternatively applied in FORTRAN, is described in [14]. Both of these

approaches required the development of numerical integration routines for the solution of the

resultant state-space equations that describe the systems. A general-purpose program, the

electromagnetic transients program (EMTP), is described in [15] for the digital simulation of

field-oriented control of induction motor drives. This program has the facility to extend

simulation to other machine types.

Recently MATLAB/Simulink is a very useful tool for modulation of electrical machine

drives and it can be used to predict the dynamic performance of the machine [16-18]. In this

paper MATLAB/Simulink based modeling of induction motor drives by rotation coordinate

system.

In this paper, focus attention on the general principles of 3-phase induction motors

Modeling and simulation of a three-phase induction motor and its electronic drive are

considered using MATLAB/Simulink. The induction motor is modeled in syncrotating

rotating frame coordinates.

2. MATHEMATICAL DESCRIPTION OF AC INDUCTION MOTORS

BY ROTATION COORDINATE

The three-phase motor all component (voltage, current, magnetic flux, and etc.). They are

demonstrated in the form of transverse space vectors. This model is valid for any

instantaneous voltage and current variation and sufficiently illustrates the device behaviors

under both stable and transient condition. Diagonal space vectors can be described using only

two orthogonal axes d-q and a second-phase machine. Using a two-step equation model

reduces the number of equations and simplifies control design [2]. These mathematical

equations can be represented in the following expressions:



11 1 1 1 0

2 2 2 2 2

1 1 1 2

2 1 2 2

e

j ;

;

;

.

T,

J

r

m

m

L

dU R I

dt

dU R I j

dt

L I L I

L I L I

T

(1)

When Te, represents electromagnetic torque, TL, Load torque, ω0 rated speed of the motor

at nominal frequency, and ωr rotor angular speed. Let

, when U2 =0 because the

closed rotor the equation of space vector of leakage flux of stator part can be given by

equation 2

101111

~~P jIRU

(2)

Also, the equations 3 represents space vector of leakage flux in rotor part

2r222

~~P jIR

(3)

The space vector of stator and rotor currents can be written given in equation 4 and 5

)~~

(L

1221

1

1 KI

(4)

)~~

(L

1112

2

2 KI

(5)

The equation 6, represents electromagnetic torque of the motor

];)[(2

3211e

IIILPT mm

(6)

The projection of stator voltage in rotating coordinate frame to equal equations below

111U~

jUU (7)

Also, the projections of leakages flux in stator and rotor by rotating frame shown in below

8,9

111~ j

(8)

222~ j

(9)

Space vector of stator and rotor current by rotating frame illustrate in equation 10,11

111

~jiiI

(10)

222

~jiiI

(11)

They can rebuild the mathematical expression when submitted equations8,9,10 and 11 in

2 and 3


Performance


)()()~~

P( 1101111111 jjjiiRjUUj

(12)

101111

~P jiRU

(13)

The space vector of leakage flux can by written like equation below 14

101111 I~

P BjRU (14)

Where the column matrices of voltages, currents and flux linkages in stator by rotating

coordinate frame as in equation 15

1

1

1UU

U

;

1

1

1Ii

i

;

1

1

1

(15)

The mathematical in rotating coordinate shown in equation 1-15 in which variable output

are the projection of stator and rotor current , leakage flux After the transition to the

projections of the vectors on the coordinate axes and the starter of the scheme, 111 / RLT and

222 / RLT , depending on the equations and the derivative of those equation we obtain a

system of equations 16,17,18,19 and 20 To develop rotating model of induction motor by

rotating coordinate frame (α, β) shown in figure(1,2,3,4 and 5) respectively.

1 1 1 1 1 1 0 1 1 2 0 1 2/ ( ) / ;pi u RT i T i k pi k i (16)

1 1 1 1 0 1 1 1 0 1 2 1 2/ ( ) / ;pi u RT i i T k i k pi (17)

2 2 1 р 2 1 2 2 2/ ;rpi k pi k i i T i (18)

2 р 2 1 2 1 2 2 2/ ,rpi k i k pi i i T (19)

).()2/3(Te 2112p iiiiLp m (20)

Figure 1 The structure of development model to find 1i


Performance


Figure 5 The structure of development model to find electromagnetic torque for IM drive by

rotationary coordinate frame

To develop the transformation current coordinate rotating model of induction motor, from

the two axis (β, α) to three phase stator currents ABC is needed which is shown in figure 6 by

the equations (21-24) shows that the transformation two axes into three phases.

(

) √

[

] (

) (21)

√

(22)

√

(23)

√

(24)

Figure 6 The structure of development model to find Transformation current two axis to three

phase stator current ABC

Figure 7 represented the general model of three phase induction motor by rotating frame

used MATLAB/Simulink.



Figure 7 The development general computer model of three phase induction motor drives by rotating

frame

Where R1-phase resistance of the stator winding; R2 – phase resistance of rotor winding

referred to the stator; Tl load torque; Ψ1α-β - vector of the stator flux linkage; Ψ2α-β - vector of

the rotor flux linkage; J - moment of inertia reduced to the motor; U1α-β. stator voltage vector;

U2α-β - vector rotor voltage; ωo - nominal angular velocity; ωr - speed of the rotor; L1 - total

inductance phase stator winding; L2 - total inductance phase rotor winding; Lm-magnetizing

inductance circuit.

3. CALCULATION OF THE VALUES OF FLUX LINKAGES

To calculate the flux linkages in the nominal mode, the equations can be written in matrix

form:

BAΨ1 , where the input real parameters given as the table 1 below

Table 1 motor parameters

Types of

motor

Effi,

%

соsφ

Pnom,

KW

Snom,

pu

Xm

pu

R1,

pu

X1

pu

R2

pu

X2

pu

J

Kg·m

4А80А4 75 0,81 1,1 0,054 1,7 0,12 0,078 0,068 0,12 0,033


Performance


1 1

2 1

0

11

0

11002.74 ;

3 cos 3 220 0.75 0.81

22080.29 ;

2.74

0.078*80.29 6.26 ;

0.12 80.29 9.634 ;

1.7 80.29 136.493 ;

136.4930.434 ;

314

6.

ratednom

no

no

no

m m

mm

PI A

U

UR Ом

I

X x R Ом

X x R Ом

X x R Ом

XL Гн

XL

22

0

1 1

2

1 1

2 2

11

1

22

2

260.02 ;

314

9.6340.03 ;

314

0.434 0.02 0.454 ;

2 0.434 0.03 0.464 ;

* 80.29*0.12 9.6348;

* 80.29*0.068 5.46;

0.4540.047 ;

9.6348

0.4640.0

5.46

m

m

pu

pu

Гн

XL Гн

L L L Гн

L L L Гн

R R R

R R R

LT s

R

LT

R

1

1

2

1

1 2

8443 ;

0.4340.9559;

0.454

0.4340.9353;

0.464

1 1 0.9559*0.9353 0.106;

m

m

s

Lk

L

Lk

L

k k

A=

1

2

r

2

1

r

22

1

1

2

1

0

1

20

1

10

10

01

01

TT

k

TT

k

T

k

T

T

k

T

,

B =

0

0

0

m1U

.,



Then

-1

2

1 1111.7768 s ;

0.106 0.0844T

-11

2

0.9559106.847447 s ;

0.106 0.0844

k

T

Where angular rotor speed:

r 0 = 314 0,0746 23.42rad/snoms .

A=

200.722 314 104.5144 0

314 200.722 0 139.0347

106.87 0 111.776 23.42

0 106.87 23.42 111.776

;

B =

0

0

0

220

.;

And dissipation Coefficient, k1, k2 design factor and. From this calculation, the values of

the estimates of flux linkages on the axis of the rotating coordinate system -β:

1

1

2

2

0.0511; .

0.6326; .

0.0992; .

0.5581 .

Wb

Wb

Wb

Wb

Effective value of the stator and rotor linkage modules

2 2

1 0.0511 0.6326 0.6347 ;Wb

2 2

2 0.0992 0.5581 0.5668wb

The corresponding magnetizing amplitude values in stator and rotor by rotationary frame:

1 20.8976; 8016m m Wb

4. MODELING OF THE MOTOR

Based on the mathematical description of computer model developed in package MATLAB /

Simulink by rotationary coordinate system. Where the equations 1-15 derived and analyses.

To developed this mathematical equation of three phase induction motor by rotating farm

shown in equation 16-20 used the modulation IM model system given by figure 7.

5. SIMULATION RESULTS

The figure below (8) describe the three-phase stator current of (ABC). The figures below

(9,10 and 11) represents the current phase A, phase B, and phase C respectively, in all the

curve can notice the change phases current of the stator part when the load enters at varying

times.

The figure below (12) represented (Torque and Speed) with change in the value of torque

with time. The figure in the first (which have transient instantaneous represented torque


Performance


curve). The second figure curve represented (speed curve). Notices that when change the load

torque in certain times a drop rotor speed in that specific time. Also, to enhance the

electromagnetic torque multiplying by gain ten to clear out line results, in this case the speed

machine varying rapidly as a result of the different between the electrical torque and load

torque taking in account the low fraction machine losses.

The figure (13,14) represented (I1α, I1β) in the stator by rotating frame system. The

variable load proportional with I1α so the I1β well response the change in I1α due to the

coupling side IM. And the complaining behavior Iα and Iβ stator and rotor.

The figure (15,16) represented rotor current in rotationary coordinate system the current of

(I2α, I2β) the same analysis could be in valve it in the rotor side to justify the variation the level

of (I2α, I2β).

Figure 8 three phase stator current ABC for induction motor

Figure 9 phase stator current A for induction motor

0 1 2 3 4 5-15

-10

-5

0

5

10

15

TIME/S

TH

RE

E P

HA

SE

ST

AT

OR

CU

RR

EN

T A

BC

/A

relation between three phase stator current with time

A

B

C

0 1 2 3 4 5-15

-10

-5

0

5

10

15

TIME/S

ST

AT

OR

PH

AS

E C

UR

RE

NT

A /

A

relation between phase stator current Awith time

A



Figure 10 phase stator current B for induction motor

Figure 11 phase stator current C for induction motor

In adding the change in the amount of amplitude of the current due the increasing of load

the other parameter will associated this change that is frequency shown in figure 7, 8,9 and

10.

Three phase stator currents ABC shown in figures 7, 8, 9 and 10 behaviors, at starting

equal to 11 A. Notice that current increases when varying TL (2 4 6) Nm increases, at time 1

to2.5 s the current equal 3 amper, also at time 2.5 to 3.8 s the load torque increases and at time

equal 3.8 to 5 s the stator current increases to 6 amper.

0 1 2 3 4 5-15

-10

-5

0

5

10

15Relation between phase current with time

TIME/S

PH

AS

E B

ST

AT

OR

CU

RR

EN

T/A

B

0 1 2 3 4 5-15

-10

-5

0

5

10

15Relation between phase current with time

TIME/S

ST

AT

OR

PH

AS

E C

UR

RE

NT

C/A

C


Performance


Figure 12 Waveforms for motor torque the /Nm. and Speed rotor speed rps

The figure 12 describing speed and torque behavior reflect the fundamental relation

between them so the increasing take torque phased high speed and respectively growth the

torque will decreases the speed level. Reminder that when load torque increases, the speed a

touch reduces from 160, 155 and 149 rps.

Figure 13 Waveforms for stator current by rotating frame current I1a

0 1 2 3 4 5-50

0

50

100

150

200Relation between motor torque and speed with time

TIME/S

ELE

CT

RO

MA

GN

AT

IC T

OR

QU

E,N

.M A

ND

RO

TO

R S

PE

ED

rps

Torque

Speed

0 1 2 3 4 50

2

4

6

8

10

12Relation between stator current axis with time

TIME/S

I1a/A

I1a



Figure 14 Waveforms for stator current by rotating frame current I1b

Figure 15 Waveforms for rotor current by rotating frame current I2a

Figure 16 Waveforms for rotor current by rotating frame current I2b

0 1 2 3 4 5-12

-10

-8

-6

-4

-2

0

2Relation between stator current axis beta with time

TIME/S

I1b/A

I1b

0 1 2 3 4 50

2

4

6

8

10

12Relation between rotor I2a current axis with time

TIME/S

I2A

/A

I2a

0 1 2 3 4 5-12

-10

-8

-6

-4

-2

0

2Relation between rotor axis current I2b with time

TIME/S

I2b/A

I2b


Performance


6. CONCLUSION

In this paper, development of a mathematical model for 3 phases induction motor drive

system and grained new characteristics algorithm by rotating coordinate systems by using

MATLAB/SIMULINK. This development has been achieved by a real drive of 0.75 kW

motor. Simulation result shows that the model was developed to reduce the cost and increases

the overall efficiency. To analysis the axis, the conversion from three-phase coordinate (A, B,

C) to a two axis (β - α) as a (rotationary part) represented and achieved. This converted was a

control method on the torque and speed for the motor, and it was a good selected and easily

way for control. The results show a good agreement to the theoretical background of the drive

system. Further to distinct calculation in stator and rotor leakage flux equations to implement

in FOC, this model not only can be used alone but also can be used in high performance

motor drive systems such as FOC, DTC and SLS field-oriented control, direct torque control

and, sensor less speed control of induction motor.

ACKNOWLEDGMENT

I sincerely acknowledge the support of received from Dr. M.S. Hasan and all those whose

helped me at diyala university during may writing possesses.

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http://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=9135



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