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DYNAMIC SIMULATION OF THREE PHASE INDUCTION MOTOR DRIVING A MECHANICAL LOAD

BY

OFFORDILE EUCHARIA ONYEKACHI

2006/142185

A PROJECT SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENT FOR BACHELOR OF

ENGINEERING (B.ENG) IN ELECTRICAL ENGINEERING UNISERTY OF NIGERIA, NSUKKA

SEPTEMBER,2011.

Author s Signature : ........................................................ Date......................................... ..............

Offordile, Eucharia, O

Student

Certified by:.........................................................................Date................................... .......................

Engr. Ogbuka, Cosmas Uchenna,

Project Supervisor

Accepted by:.......................................................................Date...................................... ..................

Engr. Dr. L. U. Anih

Head of Department

External Examiner: ...........................................................Date..............................................................


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CERTIFICATION

This is to certify that the project title DYNAMIC SIMULATION OF A THREE INDUCTION MOTOR

DRIVING A MECHANICAL LOAD is original and was carried out by OFFORDILE, EUCHARIA, O. with

Registration number 2006 /142185 and was submitted to the Department of Electrical Engineering,

University of Nigeria ,Nsukka.

Author s Signature:.................................. ........................................Date.......................................

Offordile , Eucharia, O

Student

Certified by:........................................................ ..................................Date...........................................

Engr. Ogbuka Cosmas Uchenna

Project Supervisor

Accepted by: ..................................................... ....................................Date...........................................

Engr. Dr.L.U .Anih

Head of Department


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ABSTRACT

The qdo transformation theory is applied in the dynamic modelling and

simulation, on the stationary reference frame, of a 3- phase motor driving a

20N -m mechanical load. The system of differential equations representing the

dynamic state behaviours of a machine, as developed are implemented in

SIMULINK. The effect of the programmed sequence of mechanical loading on

the motor output variable namely: phase currents, motor speed and

electromechanical torque are examined. The result obtained earlier shows the

elegance of the qdo transformation theory in machine modelling and the

inherent limitation of the direct -on - line starting of synchronous motor as

evident in the excess starting currents.


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CHAPTER ONE

1.0 INTRODUCTION

The three phase induction machines are used in wide variety of application as

a means of converting electric power to mechanical work. They are

asynchronous speed machines, operating below the synchronous speed when

motoring and above synchronous speed when generating. They are

comparatively less expensively to equivalent size synchronous or d.c machines

and range in size from a few watts to 10,00hp. They, indeed, are the

workhorse of today s electric power industry. As a motor they are rugged and

required very little maintenance. However their speeds are not as easily

controlled as with d.c motors. They draw large starting currents; typical about

six to eight time their full load values and operates with a poor lagging power

factor when lightly loaded. The idealized three phase induction machine is

research is assumed to have symmetrical air gap.

1.1 Background

In the late 1920s, R.H. Park [7] introduced a new approach to electric

machine analysis. He formulated a change of variable which, in effect, replaced

the variables (voltages, current and flux linkages) associated with the stator

winding of synchronous machine with variable associated with fiction winding


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rotating with the rotor. In order words, he transfer the stator variables to a

frame work of reference fixed in the rotor. Park s transformation, which

revolutionized electric machine analysis, has the unique property of

eliminating all time varying inductance from the voltage equation of the

synchronous machine which occur due electric circuit in relative motion [7]

and electric circuit with varying electric reluctance [8]

In late 19 3 0s, H.C.Stanley [8] also employed a change of variable in the

analysis of the Induction machines. He showed the time varying inductance in

a voltage equation of an induction machine due to electric circuit in relative

motion could be eliminated by transforming the variable associated with the

fiction stationary windings. In this case the variables are transformed to frame

of reference fixed in the stator.

G.Kron [9] introduced a change of variables which eliminated the time

varying inductance of a symmetrical induction machine by transforming both

the stator variables and the rotor variables to a reference frame rotating in

synchronism with the rotating magnetic field. The frame of reference is called

synchronously rotating reference frame.

Park, Stanley, Kron and Brereton et al. developed changes of variable

each of which appeared to be uniquely suited for a particular application.

Consequently, each transformation was derived and treated separately in


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literature until it was noted in 1965 that all known real transformation used in

induction machine analysis are contained in one general transformation which

eliminates all time - varying inductances by referring the stator rotor variables

to a frame of reference which may rotate at any angular velocity or remain

stationary. All known real transformation may then be obtained by simple

assigning the appropriate speed of rotation of the so -called arbitrary frame

work.

1.2 STATEMENT OF PROBLEM

The asynchronous (induction) machines like other electrical machines with

rotating part are not 100% efficient and thus have limitations which have

limited its efficiency. Thus this research work sets out address some of the

problem which includes amongst others, as stipulated below:

(1) The effect of direct - on - line starting of asynchronous motors as evident

in the excess starting current which is often about 10 times its rated

value.

(2) Voltage dips produced, oscillatory torques and harmonics generated in

the power systems during start - up and other severe motoring

operations.

(3 ) Poor power factor operation when lightly loaded.


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(4) A difficult combination of high starting staring torque and good

efficiency.

1.3 OBJECTIVE OF STUDY

The quest to achieve a maximum efficiency out of the asynchronous

machines during operation is among the thrust that propelled this research

work. Others include:

(1) To study the effect of the mechanical loading on the motor output

variables namely: phase currents, motor speed and Electromechanical

Torque.

(2) To study how high starting currents can be reduced through the use of

reduced voltage starting methods such as soft start method, start/delta

method and autotransformer method.

(3 ) To obtain a high Torque to current ratio, a good power factor and a high

efficiency.


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1.4 SCOPE OF THE STUDY

The scope of this work cover the simulation of three phase star connected

induction machine driving a mechanical load of say 20N using MATLAB

Under the following conditions:

Rated voltage =220V. Rated speed =960 revolution per minute,

Rated power = 3 Horse power. Number of poles=4

Rated frequency=60Hz. Rotor referred resistance =0.4 3 5

Magnetising reactance =26.1 3 . Stator reactance= 0.18

Stator resistance=0.4 3 5

Rotor referred reactance=0.754

Moment of inertia=0.089kg.m

1.5 SIGNIFICANCES

The significances of the study area as follows

(1) The discovery of ac machines has gradually offered a permanent

solution to huge amount constantly spent in maintaining dc machines .

(2) The knowledge of the operational principle of an induction machines

help the work men minimize cost and maximize gain this helps to increase

productivity.


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n is the number of the stator slots per pole per phase. An induction machine

can be considered as asynchronous machine because the machine can never

run at the synchronous speed of the rotating parts. Induction machine can

alternatively be used as a motor or a generator depending on the mode. The

polyphase induction machine find wide application in industries say about 90%

of the whole machine use in the industries are induction machines due to its

i. Low cost

ii. Simple

iii. Rugged construction

iv. Absence of commutator

v. Good operating characteristics such reasonably good power factor,

sufficient high efficiency and good speed regulation

Generally an induction machine consist primarily of stator and rotor

{ as shown below} enclosed in a frame, there are also the stator

cores, the rotor cores, the stator and rotor winding, the air gap, Shaft

and bearings, fans and slip rings and the slip _ring enclosure.

The rotors employed in polyphase induction machine are

basically of two types

1. The squirrel case rotor

2. The wound rotor {slip ring}


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The formal is most commonly use because it is cheaper and more robust. It

also designed in a way that a very high current is obtained with a low

resistance due to the fact that the slots are placed parallel to the shaft. The

later is use when variable speed starting torque is desired. In both cases,

power is transferred from the primary to the secondary by mutual induction.

Figure 1 : Cross Section of a cut Away Polyphase Induction Motor

2.2 Principle of Operation of a Polyphase Induction Machine

The electrical behaviour of an induction machine is similar to that of a

transformer but with additional feature of frequency transformation produced


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by the relative motion of the stator and the rotor winding. And as such the

wound rotor type induction motor can be used as frequency changer. The

relative motion of the stator flux and the rotor conductors induced voltage of

the frequency

=

Called the slip frequency in the rotor. One can say that the electrical

behaviour of an induction motor is similar to that of the transformer except

with additional feature of frequency transformation produced by the

relative when the stator of three phase motor is fed from the three

phase supply, the flux of a constant magnitude but rotating with

synchronous speed is developed. The flux cut the rotor into vibration

with a speed close to that of the synchronous speed. It tries to catch up

with the synchronous speed. The application of Right Hand Rule shows

that the induced EMF in the rotor that the magnitude of the rotor EMF,

rotor current and torque depend on the relative motion between the

rotor field and the stator. If the relative motion is zero, the rotor runs at

zero speed, no emf will be induced and the torque is zero. The motor

cannot run at synchronous speed. In fact, a wound rotor machine can be

used as a frequency changer. The primary winding { stator } is stationary


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motion. As the mechanical load is applied on the motor shaft, it must slow

down because the torque developed at no load will not be suff icient keep

rotor revolving at the no load speed against the additional opposing torque. As

the motor slows down, the relative motion between the magnetic field and

rotor is increased [2]. This gives rise to greater rotor current and greater

developed torque. Thus as the load is increased, the motor slow down until the

relative motion between the rotating magnetic field is just sufficient to result

in the development of torque necessary for the particular load. The decrease in

speed from no load to full load is usually 4 to 5 per cent in small and medium

side induction machine and in case of large size machine, it varies from 2 to 2.5

in large side induction motion. In respect to speed load characteristics a three

phase induction machine is quite similar to a dc shunt motor.

The direction of revolution of the field will depend upon the phase

sequence of the primary current and therefore will depend upon the other of

connection of the primary terminal to the supply. The direction can also be

changed by interchanging any of the loads. The speed at which the field

produced by primary current will revolve is called synchronous speed given as

Ns = . The whole idea of an induction machine is to keep the rotor speed

close and possible very close to the synchronous speed but never to equal it. If

the synchronous speed Ns equal to the rotor speed Nr, there will be


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increase until the driving torque equals the retarding torque of the of the

balances the torque of the load.

It was observed that the rotor runs in the direction of rotating magnetic field.

When the rotor as stationary, the rotor conductor are being cut by the rotating

flux at synchronous speed , at this the frequency of rotor current is

same as that of the supply frequency, but when the rotor starts revolving, the

rate at which the rotor conductors are being cut by the rotating flux depends

upon the relative speed between the rotor and the stator revolving magnetic

field called the slip speed usually express as

Frequency of rotor {f } = = S=

Since slip = S= , Ns - Nr =SNs= . and by away of substitution,

rotor frequency will be given as f = SNs= Sf. So we conclude that

the frequency of rotor emf or rotor current is given a the product of the slip

and the supply frequency. And this is called slip frequency.

2.3.2 ROTOR CURRENT AND POWER FACTOR

If we assume the circuit diagram of an induction motor rotor to be


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At stand still,

Induced emf per phase in rotor equals =

Rotor winding resistance per phase =

Rotor winding reactance per phase = 2 f

Where f =frequency of supply.

Rotor winding reactance impedance per phase

=

Rotor Current = =

Power factor = =


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AT SLIP S

I

nduced emf per phase in rotor winding =s

Rotor winding resistance per phase=

Rotor winding reactance per phase =2 f

Rotor winding impedance per phase =

Rotor Current

Power factor = =

2.3.3 Rotor Torque and Condition

S


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The running torque is therefore is proportional to the square of supply voltage

because rotor induced emf is proportional to the supply voltage

=

Where is a constant and is the starting torque.

2.3.5 Full Load Torque and Maximum

Let be the speed corresponding to full load torque

Dividing both the numerator and the denominator by , we have

Where a = =


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In fact a = = slip corresponding to the maximum torque. In that case , the

relationship becomes

Where = Full load.

In general =

T = -

Fixed supplied voltage will be maximum at

OR OR .

IF = 0 OR S = so that = =

Hence we conclude that maximum torque

1. Is independent of the rotor circuit resistance.

2. The slip of maximum is govern by rotors.

3 . It varies inversely as a standstill reactance of the rotor.


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4. It varies directly as the square of supply voltage [4].

2.4.1 Condiction for Maximium Starting Toque

will be maximium when

=1 or

2.4.2 Effect of Rotor Resistance upon Torque-Slip

From the torque equation, it was observed that for a Constance voltage

supply when rotor resistance is very small compared to , the

torque for a given slip is directly proportional to but at large the

slip for a given torque varies inversely as . Hence variation of rotor

resistance does not changed the value of the magnitude of the

maximum torque but merely changes the value of slip at which it

occurs. Therefore, the larger the rotor resistance the larger the slip at

which maximum torque occurs. Recall that at a given torque , the slip is

proportional to the rotor resistance so additional in the rotor circuit

merely stretches the maximum torque so that the same torque value

occur at a lower speed .


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2.4.3 The Steady State Equivalent Circuit

The key variables in the machine are the air gap power, mechanical and shaft

output power, and electromagnetic torque. These are derived from the

equivalent circuit of the induction machine as follows. The real power

transmitted from the stator. P,. To the air gap, p., is the difference between

total input power to the stator windings and copper losses in the stator and is

given as

.

FIGURE 2.1: SIMPLIFIED PHASOR DIAGRAM OF INDUCTION MOTOR


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they have to be represented as a function of speed for evaluation of the

variable - speed performance of the induction motor. There are also losses due

to stray magnetic fields in the machine; they are covered by the term stray

load losses. The stray - load losses vary from 0.25 to 0.5 percent of the rated

machine output. The stray - load losses are obtained from the measurements on

the machine under load from the remainder of the difference between the

input power and the sum of the known losses such as the stator and rotor

copper and core losses, friction and windage losses, and power output. Note

that the stray - load losses have not been accounted for in the equivalent circuit

of the machine. Various analytical formulae and empirical relationships are in

use, but a precise prediction of the stray losses is very difficult.

CHAPTER THREE

DYNAMIC MODELLING OF THREE PHASE INDUCTION MOTOR

3.0. The Analysis of Induction Machine in the Arbitrary Reference Frame

During start up and other severe motoring operations, the induction

motor draws large currents, produce voltage dips, oscillatory torques and can

even generate harmonics in the power system. [ 4] It is therefore important to


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be able to predict these phenomena. Various models have been developed and

the q d or two axis model for the study of transient behaviours has been tested

and proven to be very reliable and accurate. [ 5 ] This method of modelling has

been applied by several authors. [4,5 ]

3.1.0 Circuit Model of Three Phase Induction Machine

Using the coupled circuit approach and motor notation, the voltage equations

of the magnetically coupled stator and rotor circuits shown in figure 3 .1 can be

written as follows: [ 5 ]


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FIGURE 3.1: The Idealized Circuit of as Three Phase Induction Motor

3.1.1 Stator and Voltage Equations

dt d

r iv

dt d

r iv

dt d

r iv

cs scscs

bs sbsbs

as sasas

P

P

P

!

!

!

V (3 .1)

3.1.2 Rotor Voltage equations

d t d

r iv

d t d

r ivd t

d r iv

cr r cr cr

br r br br

ar r ar ar

P

P

P

!

!

!

V (3 .2)

3.1.3 Flux Linkage Equations

The flux linkage equations of the stator and rotor windings in terms of the

winding inductances and currents may be written in compact form using

matrix notation as:


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!

abcr

abc s

abcrr

abcrs

abc sr

abc ss

abcr

abc s

i

i

L L

L L

P

PWb - turn ( 3 .3 )

Where

t cr br ar

abcr

t

csbsas

abc

s

t cr br ar

abcr

t csbsas

abc s

iiii

iiii

),,(

),,(

),,(

),,(

!

!

!

!

PPPP

PPPP

(3 .4)

and the superscript, t, denotes the transpose of the array

The submatrices of the stator - to -stator and rotor - to - rotor winding inductances

are of the form:

!

ssls sm sm

sm ssls sm

sm sm sslsabs ss

L L L L

L L L L

L L L L

L H (3 .5)

!

rr lr rmrm

rmrr lr rm

rmrmrr lr absrr

L L L L

L L L L

L L L L

L H (3 .6)


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The stator - to - rotor winding inductances are functions of the rotor angle as

shown below:

? A

!!

r r r

r r r

r r r

sr

t abcrs

abc sr

UT

UT

U

T UU

T U

T UT UU

cos)3

2cos ()

32

cos (

)3

2cos (cos)

32

cos (

)3

2cos ()3

2cos (cos

H (3 .7)

Where L ls is the per phase stator winding leakage inductance, L lr is the per

phase rotor winding leakage inductance, L ss is the self - inductance of the stator

winding, L rr is the self inductance of the rotor winding, L sm is the mutual

inductance between stator windings, L rm is the mutual inductance between

rotor windings, and L sr is the peak value of the stator - to - rotor mutual

inductance.

3.2.0 Machine Model in Arbitrary qd0 Reference Frame

Let us first derive the equations of induction machine in arbitrary reference

frame which is rotating at a speed of in the direction of the rotor rotation.

The relationship between the abc quantities and qdo quantities of a reference

frame rotating at an angular speed is shown in figure 3 .2.


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Figure 3.2 : The Relation between abc and Arbitrary qdo

The transformation equation from abc to qdo reference frame is given by

? A

!

c

b

a

qdod

q

f

f

f

T

f

f

f

)(

0

U (3 .8)

Where the variable f can be the phase voltages, currents, or flux linkages of the

machine. The transformation angle, (t), between the q -axis of the reference

frame rotating at a speed of and the a - axis of the stationary stator winding

may be expressed as

!t

dt t t 0

)0()()( U[ U elect. rad. ( 3 .9)

r

ar

q -axis

bs

cr

br

cs

d- axis

as -axisr


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Likewise, the rotor angle, r(t), between the axes of the stator and the rotor a -

phases for a rotor rotating with speed r(t) may be expressed as

!t

r r r dt t t 0 )0()()( U[ U elect. rad. ( 3 .10)

The qd0 transformation matrix, [T qd0 ( )], is

? A

!

21

21

21

)3

2sin ()

32

sin (sin

)3

2cos ()

32

cos (cos

32

)](0T

UT

UU

T U

T UU

Uqd T (3 .11)

And the inverse

? A

!

1)3

2sin ()3

2cos (

1)3

2sin ()

3

2cos (

1sincos

)( 10

T U

T U

T UT UUU

Uqd T (3 .12)

3.2.1 qd0 Voltage Equations

In matrix notation, the stator winding abc voltage equations can be expressed

as

abc s

abc s

abc s

abc s ir pv ! P (3 .13 )

Applying the transformation, [T qd0 ( )], to the voltage, flux linkage and current,

equation 2.1 3 becomes


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qdo s

qdo s

qd s

qd s

qd s ir pv

! 000

000

001

010

PP[ (3 .14)

Wheredt d U

[ ! and

!

100

010001

0 s

qd s r r (3 .15)

Likewise the rotor quantities must be transformed onto the same qd frame.

From figure 2.2, we can see that the transformation angle for the rotor phase

quantities is ( - r). The qd0 voltage equations for the rotor windings are

likewise:

00000

000

001

010

)( qd r qd

r qd r

qd r r

qd r ir pv

! PP[[ (3 .16)

3.2.2 qd0 Flux Linkage Relations

The stator qd0 flux linkages are obtained by applying T qd0 ( ) to the stator fluc

linkages in equations 2. 3 , that is

? A )()(00 abc

r abc sr

abc s

abc ssqd

qd s iiT ! UP (3 .17)

Using the appropriate inverse transformation to replace the abc stator and

rotor currents by their corresponding qd0 currents, equation 2.17 becomes

? A ? A ? A ? A 0100

0100

0 )()()()( qd r r qd abc sr qd

qd sqd

abc ssqd

qd s iT LT iT LT ! UUUUUP (3 .18)


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000

000

023

0

0023

00

023

0

0023

qd r sr

sr

qd s

ls

ssls

ssls

qd s i

i

!P

(3 .19)

Similarly, the qd0 rotor flux linkages are given by

? A ? A ? A ? A 0100

0100

0 )()()()( qd r r qd abcrr r qd

qd sqd

abcrsr qd

qd r iT LT iT LT ! UUUUUUUP

(3 .20 )

000

00

023

0

0023

000

023

0

0023

qd r

lr

rr lr

rr lr

qd s sr

sr

qd r i

i

!P

(3 .21)

The stator and rotor flux linkage relationship in equations 3 .19 and 3 .21 can be

expressed compactly as


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3 5

!

'

0

'

'

0

'

'

'

'

0

'

'

0

000000000

0000

00000

0000

0000

r

dr

qr

s

ds

q s

lr

lr

lr

ls

ls

ls

r

dr

qr

s

ds

q s

ii

i

i

i

i

PP

P

P

P

P

(3 .22)

Where the primed rotor quantities denote referred values to the stator side

according to the following relationship:

qr r

sqr N

N PP !' , d r

r

sd r N

N PP !' (3 .23 )

qr s

r qr i

i !'

, dr s

r dr i

i !'

(3 .24)

lr r

slr

2'

! rr r s

sr r

s ssm L N

N L N

N L L 2

323

23

!!! (3 .25)

3.2.2 qd0 Torque Equation

The sum of the instantaneous input power to all six windings of the stator and

rotor is given by

''''''

cr cr br br ar ar cscsbsbsa sa sin iviviviviviv p ! w (3 .26)

In terms of the qd0 quantities, the instantaneous input power is


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)22(23 '

0''''

00 r d r d r qr qr s sd sd sq sqd in viviviviviv p ! w (3 .27)

Substituting equations 2.14 and 2.16 into equation 2.27 and retaining only the

i terms which represents the rate of energy converted to mechanical work.

The electromechanical torque developed by the machine is given by the sum of

the i terms divided by the mechanical speed, that is

? A))(()(22

3 ''''d r qr qr d r r d sq sq sd s

r em iiii

P T PP[[PP[

[! N.m ( 3 .28)

Using the flux linkage relationship in equation 22, we can show that

)()( '''''' d sqr q sd r md r qr qr d r d sq sq sd s iiii Liiii !! PPPP (3 .29)

Equation 28 can thus be expressed in the following fo rms:

)(22

3 ''''qr dr dr qr e ii

T PP! N.m ( 3 .3 0a)

)(22

3dsq sq sdse ii

T PP! N.m ( 3 .3 0b)

)(22

3 ''dsqr q sdr e iiii

T ! N.m ( 3 .3 0c)

For simulation purposes, the preference of one form over another is usually

influenced by what variables are availab le in other pats of the simulation


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The speed voltages are defines as follows:

d sq s E [P! , q sd s E [P! V (3 .3 1a)

'' )( d r r qr E P[[ ! ,'' )( qr r d r E P[[ ! V (3 .3 1b)

Sometimes, machine equations are expressed in terms of the flux linkages per

second, s, and reactances, s, instead of s and L s. These are related simply

but the base or rated value of angular frequency, b, that is

P[ ] b! V or per unit ( 3 .3 2)

Lb[ G ! H or per unit ( 3 .33 )

Where rated b f T [ 2! electrical radians per second, f rated being the rated frequency

in Hertz.

A summary of the Arbitrary Reference Equation in s and s is shown below.

3.2.3 Stator and Rotor Voltage Equations

q s sd sb

q sb

q s ir p

v ! ] [[

] [

(3 .3 4a)

d s sq sb

d sb

d s ir p

v ! ] [[

] [

(3 .3 4b)

s s sb

s ir p

v 000 ! ] [

(3 .3 4c)


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'''''

qr r d r b

r qr

bqr ir

pv

! ] [

[[]

[(3 .3 4d)

'''''

d r r qr

b

r d r

b

d r ir p

v

! ]

[

[[]

[

(3 .3 4e)

'

0

''

0

'

0 r r r b

r ir p

v ! ] [

(3 .3 4f)

Where

!

'

0

'

'

0

'

'

'

'

0

'

'

0

00000

0000

0000

00000

0000

0000

r

dr

qr

s

ds

q s

lr

lr

lr

ls

ls

ls

r

dr

qr

s

ds

q s

i

i

i

i

i

i

x

x x x

x x x

x

x x x

x x x

]

]

]

]

]

]

(3 .3 5)

3.2.4 Torque Equations

! )()(22

3 ''''d r qr qr d r

b

r d sq sq sd s

br em iiii

P T ] ]

[

[[] ]

[

[

[(3 .3 6a)

)(22

3 ''''qr d r d r qr

bem ii

P T ] ]

[! (3 .3 6b)

)(22

3d sq sq sd s

bem ii

P T ] ]

[! (3 .3 6c)

)(22

3 ''d sqr q sd r m

bem iiii x

P T !

[(3 .3 6d)


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3 9

The corresponding equivalent circuit of an induction machine in the arbitrary

reference frame is shown in figure 3 .3 .

vdss vdr

s xm

xlsrs xlr rr

- Edr s ++

-

+

-

idss idr

s

+ -

qrs

b

r

[[

d -axis

vqss vqr

s xm

xlsrs xlr rr

- Eqrs +

+

-

+

-

iqss iqr

s

- +

drs

b

r

[

[

q-axis


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40

Figure 3.3 Equivalent Circuit Representation of an Induction Machine In Arbitrary Reference

Frame

3.3.0 Machine Model in Stationary qd0 Reference Frame

Various reference frames are used for the simulation of induction machine but

in transient studies of adjustable speed drives, it is usually more convenient to

simulate an induction machine and its converters on a stationary reference

frame. [5] The rest of this chapter is devoted to the derivation of the

equivalent circuit suitable for simulation of induction machine in the stationary

reference frame.

Since the equations of the induction machine for the general case that is in the

arbitrary reference frame have been derived, the equations of the machine in

the stationary reference frame can simply be obtained by setting the speed of

the arbitrary reference frame, , to zero. Lets distinguish the variables in the

stationary reference frame by an additional superscript, s. The equations of the

rs xls

v0s

+

-

i0s xlr rr

+

-

v0r

i0r

zero - sequence


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41

symmetrical machine in terms of s and s in the stationary reference frame

are summarized below.

3.3.1 Stator and Rotor Equations

sq s s

sq s

b

sq s ir

pv ! ]

[(3 .3 7a)

sds s

sds

b

sds ir

pv ! ]

[ (3 .3 7b)

s s sb

s ir pv 000 ! ] [

(3 .3 7c)

sqr r

sdr

b

r sqr

b

sqr ir

pv

'''''

! ] [

[ ]

[ (3 .3 7d)

sd r r

sqr

b

r sd r

b

sd r ir

pv ''''' ! ]

[[

] [

(3 .3 7e)

'

0''

0'

0 r r r b

r ir p

v ! ] [

(3 .3 7f)

3.3.2 Flux Linkage Equations

!

'

0

'

'

0

'

'

'

'

0

'

'

0

00000

0000

000000000

0000

0000

r

sdr

sqr

s

sds

sq s

lr

lr

lr

ls

ls

ls

r

sdr

sqr

s

sds

sq s

i

i

ii

i

i

x

x x x

x x x x

x x x

x x x

]

]

]

]

]

]

(3 .3 8)

3.3.3 Torque Equations


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42

)(22

3 '''' sqr

sdr

sdr

sqr

be ii

T ] ] [

! (3 .3 9a)

)(

22

3 sds

sq s

sq s

sds

b

e ii!

T ] ] [

! (3 .3 9b)

)(22

3 '' sds

sqr

sq s

sdr "

be" iiii

#

T ! G [

(3 .3 9c)

The corresponding equivalent circuit of an induction machine in the stationary

reference frame is shown in figure 3 .4.

vdss vdr

s xm

xlsrs xlr rr

- Edrs +

+

-

+

-

idss idr

s

+ -

qrs

b

r

[[

d -axis

vqss vqr

s xm

xlsrs xlr rr

- Eqrs +

+

-

+

-

iqss iqr

s

- +

drs

b

r

[[

q-axis


43/68


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45

d t xr

v sqr smq

lr

r sd r

b

r sqr b

sqr

! )( '''

''' ] ] ] [[

[] (4.4)

d t x

r

vs

d r

s

md lr

r s

qr b

r s

d r b

s

d r

! )(

'

'

''''

] ] ] [

[

[] (4.5)

d t r iv x

i r r r lr

br )(

''

0'

0''

0 ! [ (4.6)

)(

)('

'

sd r

sd sm

smd

sqr

sq sm

smq

ii x

ii x

!

!

]

] (4.7)

smq

sq sls

sq s i x ] ] ! Implying that

ls

smq

sq s s

q s xi

] ] ! (4.8)

smd

sd sls

sd s i x ] ] ! Implying that

ls

smd

sd s s

d s xi

] ] ! (4.9)

s% q

sqr lr

sqr i x ] ] !

'''

Implying that'

'

'

lr

smq

sqr s

qr

xi

] ] ! (4.10)

s& d

sdr lr

sdr i x ] ] !

'''

Implying that ''

'

lr

s' d

sdr s

dr x

i] ]

! (4.11)

Where

'

1111

lr ls( M x x x x ! (4.12)

and


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46

! '

'

lr

sqr

ls

sq s

)

smq x x

x] ]

] (4.1 3 )

! '

'

lr

sd r

ls

sd s

0

s

md x x x] ]

] (4.14)

Wr= Rotor speed

The equations of the of the machine is obtained by equating the inertia torque

to the accelerating torque, that is

da 1 p1 eche 1 r 1 T T T

dt d

J ![ (4.15)

Where T mech is the externally applied mechanical torque in the direction of the

rotor speed and T damp is the damping torque in the direction opposite to

rotation.

When used in conjunction with equations 4.1 to 4.6, the per unit speed, r/ b

needed for building the speed voltage terms in the rotor voltage equations,

can be obtained by integrating

dampmechembr b T T T

d t d

P J

!)/(2 [[[ (4.16)


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47

Often, the above equation of motion is written in terms of the inertia constan t,

H, defined as the ration of the kinetic energy of the rotating mass at base

speed to the rated power, that is

b

b2

S J

H 2

2[

! (4.17)

Expressed in per unit values of the machine s own base power and voltage,

equation 2.56 becomes

da mpmechembr T T T

d t d

H !)/(

2[[ per unit (4.18)

4.1a Introduction to Matlab/ Simulink

MATLAB MATRIX LABOURATORY was invented in the late

1970 s by Cleve Moler, then chairman of the computer science Department at

the University of mexico [7]. He designed it to give his student access to

LINPACK without having to learn FORTRAN. Later it spread to other universities

and found a strong audience within maths community.

In 2000, the program was modified to include a newer set of library

for matrix manipulation. The control designers were the first Engineers to

adopt the use of matlab. Today, the processes have found application in most

other field such as education particularly in the teaching of linear algebra,


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48

numerical analysis and very popular amongst scientist involve with image

processing.

Matlab program has gone a long way in reducing the stress posse by

higher order differential equation, trigonometrically functions and generally in

the calculation. It is a high - level language which performs technical computing

language and interactive environment for algorithm development, data

visualization, data analysis and numeric computation

4.1b : Modelling

A model is a representation of a theory that can be used for prediction andcontrol, it is the process of analysis and synthesis to arrive at a suitablemathematical description that encompasses the relevant dynamiccharacteristics of a component, preferably in terms of parameters that can beeasily be determine in pact ice. Model must be realistic and yet simple tounderstand and easy to realistic models cannot be simple. A model should

possess a certain characteristics of the actual condition []. It must be verifiedand validated.

4.1.2 TEMPLATE OF SIMULINK SOURCES

CLOCK Provide and display system time

Constant

Inject constantFROM FILE

Read data from fileFROM WORK SPACE

Read data from a matrix in WorkspaceSIGNAL GENERATOR


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49

Generate various wave formsSINE WAVE

Generate a sine waveSTEP FCN

Generate a step functionREPEATING SEQUENCE

Repeat an arbitrary signalWHITE NOISE

Generate random noise

SINKS

SCOPE

Display signal during simulation

TO FILE

Write data to file

TO WORKSPACE

Write data to a matrix in workspace

CONNECTIONS

INPORT Input port to a masked block

OUTPORT

Output port of a masked blockMUX

Multiplex several scalar input into a vector inputDEMUX

Demultiplex vector input into scalar component input


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50

LINEAR

DERIVATIVE Output a time derivative of the input

GAIN Multiply an input by a constant

INTEGRATOR

Integrate input signal(s)

STATESPACE

Linear state space system

SUM

Sum input

TRANSFER FCN Linear transfer function in a domain

ZERO POLE

Linear systems specified in poles and zeros

NONLINEA

Abs Absolute value of input

Black lash Model hysteresis

Dead zone

Zero output dead zone


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51

Fcn Any legal C function of input

look up table Perform piece -wise linear mapping

MATLAB Fcn

Apply a MATLAB function to an input

Product

Multiply input together

Rate Linear

Limit the rate of change of an input

Relay

Switch an output between two values

Saturation

Limit the execution of a signal

S-function Make an S -function into a block

Switch Swi tch bet w een t w o i nputs

Transport Delay Delay an input signal by a given amount of time


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52

Below Are Some Data Of A few Sample Machine Under Study

Table 4. Machine data

RATED VOLTAGE 220v

WINDING CONNECTION

Star

RATED FREQUENCY SAMPLE

60Hz

NUMBER OF POLES 4

RATED SPEED

960rpm

STATOR RESISTANCE 0.4 3 5

ROTOR REFERRED RESISTANCE

0.18

STATOR REACTANCE 0.754

ROTOR REFERRED REACTANCE

0.754

MAGNETISING REACTANCE 26.1 3

MOMENT OF INERTIA

0.089kg.

POWER RATING 3 Hp


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53

4.2 Simulink Implementation of the Induction Machine Model.

Figure 4.1 : Complete Simulink Model of Induction Machine in Stationary Reference Frame

Figure 4.2 abc 2dqs block


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54

Figure 4.3 q-axis block

Figure 4.3 d-axis block


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Figure 4.5 : ROTOR block

Figure 4.6 Zero- sequence block.


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56

Figure 4.7 qds2abc block

4.2.2 The MATLAB processes

%Dynamic Simulation of a 3- hp Induction Motor in the Stationary

%Refrence Frame.

clear

rs=0.4 3 5;%Stator resistance

rpr=0.816;%Refered value o f rotor resistance

xls=0.754;%Stator leakage reactance

xplr=0.754;%Refered value of rotor leakage reactance

xm=26.1 3 ;%magnetizing reactance

J=0.089;%Moment of inertia

P=4;%Number of Poles

fb=60;%Base frequency

wb=2*pi*fb;


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57

xM=1/(1/xls+1/xm+1/xplr);

Vrated=220;

Vm=(sqrt(2/ 3 )*Vrated);

Sb=223 8;%Rate power

Ib=(2*Sb/ 3 *Vm);

Zb=Vm/Ib;

wbm=(2*wb)/P;

Tb=Sb/wbm;

H=(J*wbm^2)/(2*Sb);

disp('run simulation, type ''return'' when ready')

keyboard

figure(1)

plot(y(:,1),y(:,2))

ylabel('Phase Voltage, Vag [V]')

xlabel('Time[Sec]')

grid on

figure(2)

plot(y(:,1),y(:,5))

ylabel('Electromechanical Torque, Tem [N-m]')

axis([0 1 - 25 120])

xlabel('Time[Sec]')

grid on


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figure( 3 )

plot(y(:,1),y(:,6))

ylabel('Per -Unit Speed, wr/wb')

xlabel('Time[Sec]')

grid on

figure(4)

plot(y(:,1),y(:,7))

ylabel('Motor Speed, wr [Rad/Sec]')

xlabel('Time[Sec]')

grid on

figure(5)

plot(y(:,1),y(:,8))

ylabel('Mechanical Load, Tmech [N.m]')

axis([0 1 - 25 0])

xlabel('Time[Sec]')

grid on

figure(6)

plot(y(:,1),y(:,9),' - .')

ylabel('Phase Currents,[A]')

hold on

plot(y(:,1),y(:,10),':')hold on

plot(y(:,1),y(:,11),' -- ')

xlabel('Time[Sec]')

legend('Phase A Current','Phase B Current','Phase C Current')


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59

grid on

figure(7)

plot(y(:,1),y(:,12))

ylabel('q - axis Stator Voltage[V]')

xlabel('Time[Sec]')

grid on

figure(8)

plot(y(:,1),y(:,1 3 ))

ylabel('Zero Sequence Stator Voltage[V]')

xlabel('Time[Sec]')

grid on


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CHAPTER FIVE

5.0 RESULT, OBSERVATION AND CONCLUSION

The simulation yield the following curves

Figure 5.1 : Phase Voltages [V] Against Time [Sec]

Figure 5.2 : The Graph Of Voltages [V] Against Time [Sec]


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Figure 5.3 : Per-Unit Speed ( ) Against Time [Sec]

Figure 5.4 : Motor Speed ] Against Time [Sec]


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Figure 5.5 : Mechanical Load, ] Against Time [Sec]

Figure 5.6 : Phase Current (Ia, Ib, Ia) Against Time [Sec]


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figure 5.7 : q-axis stator voltage against time [sec]

Figure 5.8 : Zero-Sequence Voltage Against Time [Sec]


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5.2 OBSERVATION AND CONCLUSION

At stand still, the input impedance of the induction machine is essential

the stator resistance and the leakage reactance in series with the rotor

resistance and leakage reactance. Consequently, when the rated voltage is

applied, the starting current is large and in some cases may rise up to 10times

the rated value. This is observed in the graph of phase current versus time

(figure 5.6) as the major limitation of direct on line starting of motor. It

therefore recommended that reduced voltage starting method such as starting

in star and ending in delta (star/delta), the use of auto transformer and soft

start methods should be employed to reduce the excess starting current.

The graph of rotor speed against time (figure 5.4) shows that rotor

accelerates from start with zero mechanical torque and since friction and

windage losses are not taken in consideration, the machine accelerates with

synchronous speed. Figure 5.5 shows that the application of 20N -m mechanical

load for say 3 second leads to a sharp drop in the motor speed, this applied

mechanical loading is in the negative sense since the machine operates as a

motor. Also, an increase in electromechanical torque in sympathy with the

applied loading from [20 10 20]N -m in the time sequence [0. 3- 0.4] seconds,

[0.5 - 0.6] seconds respectively as indicated in figures 5.2 and 5. 3 . and these also

show the per unit speed used to compare the actual motor speed and the


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rated speed. The simulated motor is symmetrical and the windage and

frictional losses are assumed negligible.

Finally, this work is concluded by saying that it off ers the user the

opportunity to have access to the basic parameters that makes for the best

operation of three phase induction motor.


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8 Leonard, W. 1995. controlled AC Drives, A successful Transfer From

Ideas Induction practice . CETTT 95. Brasil, ppl -12.

9 MacDonald, M.L and P.C Sen. 1979. Control loop study of induction

Motor Drive Using D - Q Model . IEEE Transaction on Industrial

Electronics and Control Instrumentation. 26(4):2 3 7 -241.

10 Okoro, O.I. 2005. Steady and Transient states Thermal Analysis of a

7.5kw Squirrel - cage Induction Machine at Rated - Load operation . IEEE

Transaction on Energy Conversion. 20(4):7 3 0 - 73 6.

11 Krause,P.O. Wasynrzuk and S.D. Sudhoff 1986. Analysis of Electric

Machinery . McGraw - Hill ine : New York,NY

12 Vas, P. 1992. Electrical machine and Drives - A Space Vector Theory

Approach . Oxford Clarendoon Press: London, Uk

13 Krause,P.C and C.H Thomas. 1965. Simulink of Symmetrical Induction

Machinery . IEEE Transaction PAS -84, 11:105 3 .

14 Mulay, S.P . and M.V. Aware. 2008. V/F Control of an Induction

Machine prediction Inverter Machine interaction . Internation Journal

of Innovation in Energy system and Power. 3 (1): 27 -3 1.

15 Pillay, p and R.G Harley. 199 3 . Comparison of Model for Predicting

Disturbances Caused by Induction Motor Starting . SAIEE Symposium on

Power System Disturbances. Pretoria, Suoth Africa.


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