Dr. Amr AbdAllah 1

Electric Machines IIIACOURSE EPM 405A

FOR 4th Year Power and MachinesELECTRICAL DEPARTMENT

Lecture 04

Dr. Amr AbdAllah 2

Induction Machine Modeling Electromagnetic torque equation: The electromagnetic torque is change of the stored magnetic

energy with respect to the angular displacement

It is clear that the energy stored in the leakage inductances is not part of the energy stored in the coupling field, thus the energy stored in the coupling field may be written as:

Notice that (Ls-L’ls.I), (Lr-L’lr.I) are not function of the rotor position.

mθfW

eT

)abcri(I)rL(T)abcri(21)abcs(i)rsL(T)abcri(

21

)abcri()srL(T)abcs(i

21 )abcs(iI)s(LT)abcs(i

21

lrL

lsLfW

Dr. Amr AbdAllah 3

Induction Machine Modeling Electromagnetic torque equation, Cont’d: For a P-pole machine where also

Finally we can compute the electromagnetic torque developed from the following equation:

T)abcri()srL(T)abcs(i

21)abcs(i)rsL(T)abcri(

21

rθfWP

eT

2 mP

r 2

)abcri()srL(T)abcs(i

r

PTe 2

Dr. Amr AbdAllah 4

Induction Machine Modeling Electromagnetic torque equation, Cont’d: The torque equation can thus be written in the

expanded form as:

The above equation shows the electromagnetic equation as function of the stator and rotor currents as well as the rotor position.

rbriaricsiaricribsi

cribriasiraribricricsi

criaribribsicribriariasimsLPeT

cos23sin

21

21

21

21

21

21

2

Dr. Amr AbdAllah 5

Induction Machine Modeling Induction Machine Mechanical Equations: In addition to the 6 differential equations representing the three phase

induction machine the following mechanical equations are governing the electromechanical transformation

Where Tl : Load torque, J : moment of inertia, and m is the rotor speed and since

Then ……………………………………………(4.1)

We have also

……………………………………………(4.2)

dtdJlTeT m

mP

r 2

dtd

PJlTeT r2

dtrd

r

Dr. Amr AbdAllah 6

Induction Machine Modeling Induction Machine Mechanical Equations: Equations 4.1 and 4.2 represent the two mechanical

equations that in addition to the 6 electromagnetic represent the dynamic model of the induction machines.

These 8 equations should be solved simultaneously to obtain the instantaneous value of the system variables (for the induction motor as an example these are the stator and rotor currents as well as the rotor position and rotor speed).

These equations can be arranged in the state space model in the form of:

REPORT Where X represent the vector of state variables, U represent

the vector of input variables.UBXAX

Dr. Amr AbdAllah 7

Reference Frame Theory Applied to Induction Machine Equations of Transformation for Stator

circuits:From the previous chapter it has been shown that the transformation of the abc stationary circuits of the stator to an arbitrary reference frame is obtained by mean of the transformation matrix Ks:

To get

21

21

21

)3

2πsin( )3

2πsin( )sin(

)3

2πcos( )3

2πcos( )cos(

32K s θθθ

θθθ

1 )3

2πsin( )

32π

cos(

1 )3

2πsin( )3

2πcos(

1 )sin( )cos(

)K( 1s

θθ

θθ

θθ

abcssqd fKf s0

Dr. Amr AbdAllah 8

Reference Frame Theory Applied to Induction Machine Equations of Transformation for rotor circuits:

It can be shown by analogy that that the transformation of the abc rotor circuits to the arbitrary reference frame is obtained by mean of the transformation matrix Kr:

Where

To obtain

21

21

21

)3

2πsin( )

32π

sin( )sin(

)3

2πcos( )3

2πcos( )cos(

32

K r

1 )3

2πsin( )3

2πcos(

1 )3

2πsin( )

32π

cos(

1 )sin( )cos(

)K( 1r

r dtd

dtd r

r

abcrrrqd K ff 0

Dr. Amr AbdAllah 9

Voltage Equations of the Induction Machine in Arbitrary reference frame The stator and rotor voltage equations of the induction

machine in the abc reference frame was shown to be given as:

Where

Applying the Ks transformation to the stator voltage equation

abcrabcrrabcr

abcsabcssabcs

pp

λirvλirv

)()(

000

000

sqdsqdssqd

sqdsqdssqd

pp

λKKiKrKvλKiKrvK

-1ss

-1ss

-1s

-1s

-1s

abcr

abcs

rrs

srs

abcr

abcs

ii

L LL L

λλ

Dr. Amr AbdAllah 10

Voltage Equations of the Induction Machine in Arbitrary reference frame The stator voltage equation in the arbitrary reference frame

can thus be shown to be given by:

Where

Which leads to:

(4.3)with

sqdsqdsqdssqd pp 0000 )( λKKλKKirv -1ss

-1ss

0 )120cos( )120sin( 0 )120cos( )120sin(0 )cos( )sin(

)K( 1-s

ωp

000001010

)(

KK -1ss ωp

dqssqdsqdssqd p λλirv 000

Dr. Amr AbdAllah 11

Voltage Equations of the Induction Machine in Arbitrary reference frame Similar voltage equation can be written for the rotor circuits in

the arbitrary reference frame by applying the Kr transformation to the abc voltage equation of the rotor:

That is:

It can also be shown that

Thus (4.4)with

)()(

000

000

rqdrqdrrqd

rqdrqdrrqd

pp

λKKiKrKvλKiKrvK

-1rr

-1rr

-1r

-1r

-1r

rqdrqdrqdrrqd pp 0000 )( λKKλirv -1rr

0 0 0 0 0 10 1 0

)()KK( 1-rr rωωp

dqrrrqdrqdrrqd p λλirv )(000

Dr. Amr AbdAllah 12

Voltage Equations of the Induction Machine in Arbitrary reference frame Equations 4.1 and 4.2 represent the induction machine

voltage equations in the arbitrary reference frame as function of currents and flux linkages

The flux linkages of the stator in the abc frame can be also transformed to the arbitrary reference frame as:

dqssqdsqdssqd p λλirv 000

dqrrrqdrqdrrqd p λλirv )(000

rqdsrssqdsssqd

rqdsrsqdssqd

abcrsrabcssabcs

000

000

iKLKiKLKλiKLiKLλK

iLiLλ

-1r

-1s

-1r

-1s

-1s

Dr. Amr AbdAllah 13

Voltage Equations of the Induction Machine in Arbitrary reference frame The flux linkages of the stator in the dq arbitrary

reference frame is given as: (4.5)

with

where

rqdsrqdsqdsqdsqd 00000 iLiLλ

ls

Mls

Mls

sssqd

LLL

LL

0 0 0 0 0 0

KLKL -1s0

0 0 0 0 0 0 0

KLKL 1-r0 M

M

srssrqd LL

msM LL23

Student assignment

Student assignment

Dr. Amr AbdAllah 14

Voltage Equations of the Induction Machine in Arbitrary reference frame The flux linkages of the rotor in the abc frame can be

also transformed to the arbitrary reference frame as:

Thus (4.6)rqdrdqsqdrsdqrqd 00000 iLiLλ

lr

Mlr

Mlr

rrrqd

LLL

LL

0 0 0 0 0 0

KLKL -1r0

0-1s0 LKLKL srqd

Tsrrrsqd

rqdrrsqdrsrrqd

rqdrsqdrsrqd

abcrrabcsrsabcr

000

000

iKLKiKLKλiKLiKLλK

iLiLλ

-1r

-1s

-1r

-1s

-1r

Student assignmentStudent assignment

rlrr

drdsMdrlrdr

qrqsMqrlrqr

slss

drdsMdslsds

qrqsMqslsqs

iLλiiLiLλiiLiLλ

iLλiiLiLλiiLiLλ

00

00

)()(

)()(

Dr. Amr AbdAllah 15

Voltage Equations of the Induction Machine in Arbitrary reference frame The machine voltage equations in the arbitrary reference

frame can thus be written as:

(4.7)

The flux linkages in the arbitrary reference frame can thus be written as:

(4.8)

rrrr

drqrrdrrdr

qrdrrqrrqr

ssss

dsqsdssds

qsdsqssqs

λpirvλpλωωirvλpλωωirv

pλirvpλλωirvpλλωirv

000

000

)()(

Dr. Amr AbdAllah 16

Voltage Equations of the Induction Machine in Arbitrary reference frame Substituting the flux linkages from equation 4.8 in the

voltage equations 4.7 the stator voltage equations as function of the current can be written as:

drMdsMlsqrMqsMlsqss

dsqsqssqs

iLiLLiLpiLLpirpirv

)()()()(

qrMqsMlsdrMdsMlsdss

qsdsdssds

iLiLLiLpiLLpirpirv

)()()()(

slsss

ssss

iLpirpirv

00

000

)(

Dr. Amr AbdAllah 17

Voltage Equations of the Induction Machine in Arbitrary reference frame Similar equations for the rotor voltage equations as

function of the current can be written as:

drMlrrdsMr

qrMlrqsMqrr

drrqrqrrqr

iLLiLiLLpiLpir

pirv

))(())(( )()(

)(

qrMlrrqsMr

drMlrdsMdrr

qrrdrdrrdr

iLLiLiLLpiLpir

pirv

))(())(()()(

)(

rlrrr

rrrr

iLpirpirv

00

000

)(

Dr. Amr AbdAllah 18

Voltage Equations of the Induction Machine in Arbitrary reference frame The machine voltage equations in the arbitrary reference

frame can be written in matrix form as:

(4.9) Notice that is the reference frame speed while r is the

speed of the rotor

r

dr

qr

s

ds

qs

lrr

rrrrMMr

rrrrMrM

lss

MMsss

MMsss

r

dr

qr

s

ds

qs

iiiiii

LprLprLpLLLLprLpL

pLrpLLpLrLLpLLpLr

vvvvvv

0

0

0

0

000000)0)0)0)000000000

(- (-

( (

-

Mlrr

Mlss

LLLLLL

Dr. Amr AbdAllah 19

Dynamic Equivalent circuits of the Induction Machine in Arbitrary reference frame The machine equivalent circuits can be derived from the

above deduced equations

drrqrqsMqrlrqrrdrrqrqrrqr

dsqrqsMqslsqssdsqsqssqs

iipLiLpirpirviipLipLirpirv

)()()()( e.g.

Dr. Amr AbdAllah 20

Voltage Equations of the Induction Machine for commonly used reference frames Commonly used Reference frames 1. Stator reference frame (=0)

(4.10)2.rotor reference frame (= r) 3. Synchronously Rotating reference frame (= e)

sr

sdr

sqr

ss

sds

sqs

lrr

rrrrMMr

rrrrMrM

lss

Mss

Mss

sr

sdr

sqr

ss

sds

sqs

iiiiii

LprLprLpLLLLprLpL

pLrpLpLr

pLpLr

vvvvvv

0

0

0

0

0000000000000000000000

Dr. Amr AbdAllah 21

Steady State Equivalent Circuit of the Induction Machine Steady State equivalent circuit using stator reference frame

(=0) At steady state p=Jrel , where rel is the relative speed of the

rotating field and speed of reference frame (rel = s -)

From which we can write:

(4.11)

qrMsqsMlssqss

drMdsMlsqrMqsMlsqssqs

iLJiLLJiriLiLLiLpiLLpirv

)()()()()()(

drMsdsMlssdss

qrMqsMlsdrMdsMlsdssds

iLJiLLJiriLiLLiLpiLLpirv

)()()()()()(

rMssMlsssss iLJiLLJirv )()(

drqrr

dsqss

dsqss

iiiiiivvv

JJJ

Dr. Amr AbdAllah 22

Voltage Equations of the Induction Machine in Arbitrary reference frame Similar interpretation can be applied to the rotor equations

(4.12)

drMlrsdsMs

qrMlrsqsMsqrrqr

iLLsiLsiLLiLirv)()1()()(1

)(J)(J

qrMlrsqsMs

drMlrsdsMsdrrdr

iLLsiLsiLLiLirv

)()1()()1(

)(J)(J

)J)(()1()J)(()1( )J)((J))((J)J()J(

qrdrMlrsqsdsMs

drqrMlrsdsqsMsdrdrrdrqr

iiLLsiiLsiiLLiiLiirvv

rMlrssMs

rMlrssMsrrr

iLLsiLsiLLiLirv

)()1(J)()1(J )(J)(J

rMlrssMsrrr iLLiLisr

sv

)(J)(J

lrr Lsr ,

Using equations 4.11 and 4.12 the steady state equivalent circuit can be drawn as below:

Dr. Amr AbdAllah 23

Voltage Equations of the Induction Machine in Arbitrary reference frame

rMssMlsssss iLJiLLJirv )()(

rMlrssMsrrr iLLiLisr

sv

)(J)(J

lss Lr ,

MLsv svrsi ri

It has been shown that the torque of the induction machine in the abc frame can be computed as:

Substituting for the stator and rotor current vectors in the dq arbitrary reference frame we get:

This equation Yields the torque expressed in terms of currents as:

Dr. Amr AbdAllah 24

Torque Equation of the Induction Machine in Arbitrary reference Frame

)abcri()srL(T)abcs(i

r

PTe 2

)qd0ri()1-r(K)srL(T)1-

sK(T)qd0s(i2

)qd0ri1-r(K)srL(T)qd0si1-

s(K2

r

Pr

PTe

)(22

3qrdsdrqsMe iiiiLPT REPORT

Other Forms of torque equation:

Dr. Amr AbdAllah 25

Torque Equation of the Induction Machine in Arbitrary reference Frame

)(22

3qrdrdrqre iiPT

)(22

3dsqsqsdse iiPT

REPORT

Dr. Amr AbdAllah 26

NEXT LECTURE

CHAPTER IIISynchronous Machine Dynamic Modeling