Machines EPM405A Presentation 06

Dr. Amr AbdAllah 1

Electric Machines IIIA

COURSE EPM 405A

FOR

4th Year Power and Machines

ELECTRICAL DEPARTMENT

Lecture 06

Dr. Amr AbdAllah 2

Synchronous machine

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 (Lr-L’lr.I) is not function of the rotor position.

mθ

fWeT

)qdr

i()r

L(T)qdr

i()2

3(

2

1

)qdr

i()sr

L(T)abcs

(i )abcs

(iI)s

(LT)abcs

(i2

1

lrL

lsLfW

lkdLlfdLlkqLlkqLdiagLlr 2 1

Dr. Amr AbdAllah 3

Synchronous machine

Electromagnetic torque equation: Accordingly and for a P-Pole synchronous machine the

electromagnetic torque is given by:

The resultant torque equation as function of the machine currents and rotor position is:

)qdri()srL(T)abcs(i)abcs(iI)-s(LT)abcs(i2

1-

2 rlsL

r

PTe

)( sin)(2

3)cos()

2

1

2

1()(

)( cos)(2

3)sin()

2

1

2

1()(

-

)2cos()22-22(2

3

)2sin()2--22

1-2

2

1-2(

32

21

rcsbsrcsbsaskdfdmd

rcsbsrcsbsaskqkqmq

e

iiiiiiiL

iiiiiiiL

rcsiasibsiasicsibsi

rcsibsicsiasibsiasicsibsiasimqLmdLPT

Dr. Amr AbdAllah 4

Synchronous machine

Electromagnetic torque equation: In addition to the 6 differential equations representing the three phase

synchronous machine the following mechanical equations are governing the electromechanical transformation

Where Tl : prime-mover Torque, J : moment of inertia, and m is the rotor speed and since

Then

We have also

……………

dt

dJlTeT

m

mP

r 2

dt

d

PJlTeT

r2

dtrd

r

Dr. Amr AbdAllah 5

Torque equation Derivation

)120cos())(3

2()

2

3(

2

1)120cos())(

3

2()

2

3(

2

1

)cos())(3

2()

2

3(

2

1

)2

3( ))(

3

2())(

3

2())

2

3(

2

1)

2

3( )

3

2())(

3

2( ))(

2

3(

2

1

)120cos(2

1

)120cos(2

1)cos(

2

1

2

1

2

1

11

1

212

1

11

11

11

11

11

11

21

11

2121

2

)(

21

211

2

11

11

111121212

111

rcs

i

kqs

kq

Lmq

skqkq

srbs

i

kqs

kq

Lmq

skqkq

s

ras

i

kqs

kq

Lmq

skqkq

s

i

kqs

kq

i

kqs

kq

L

kqkqkqkq

s

i

kqs

kq

L

mkqkq

skq

rcskqskq

rbskqskqraskqskqkqkqkqkqkqmkqkq

iiN

NL

N

Nii

N

NL

N

N

iiN

NL

N

N

iN

Ni

N

NL

NN

Ni

N

NL

N

NW

iiL

iiLiiLiiLiLW

kqkq

kq

kqkqmqkqmq

.....)cos())(3

2()

2

3(

2

1.....))2cos((

2

1

1

11

11

2ras

i

kqs

kq

Lmq

skqkq

sasrBAas ii

N

NL

N

NiLLW

kq

Dr. Amr AbdAllah 6

Transformation to arbitrary reference frame

The voltage equation for the balanced three phase stator windings is similar to that of the induction machine can be transformed to the arbitrary reference frame using the transformation Ks

where

sqdsqdsqdssqd pp 01-

ss01-

ss00 λ)K(KλKKirv

0 )120cos( )120sin(

0 )120cos( )120sin(

0 )cos( )sin(

)K( 1-s

ωp

0 0 0

0 0 1

0 1 0

)KK( 1-ss ωp

dqssqdsqdssqd p λλirv 000

abcsabcssabcs pλirv

Dr. Amr AbdAllah 7


Since the rotor windings are different therefore the change of variables to the arbitrary reference frame will offer no advantage in the analysis of rotor circuits.

Note that the equation is raised to index r to show that the rotor variables are still in the rotor qd frame, while the stator variables are transformed to the stator arbitrary reference frame rotating with speed ; the flux equations for the new machine is given as:

rqdr

rqdrr

rqdr pλirv

rqdr

sqd

rT

sr

srs

rqdr

sqd

i

i-

L )(K)L(3

2

LK )(KLK

λ

λ 0

1-s

s1-

ss0

Dr. Amr AbdAllah 8


It can be shown that all terms on inductance matrix are sinusoidal in nature except L’r

For example:

The sinusoidal terms will be constants only when the = r, that is the rotating frame is fixed in rotor. This shows that the only reference frame that is useful in the analysis of the synchronous machine is the rotor reference frame.

Dr. Amr AbdAllah 9

Transformation to rotor reference frame: Park’s Transformation

By setting the arbitrary reference frame speed to the rotor speed (= r) thus the stator and rotor voltage equations can be written as:

The flux linkages will thus be given as:

rr

rrs

rsdqsqdsqdsqd

p λλirv000

rqdr

rqdrr

rqdr pλirv

rqdr

rsqd

rrs

Tsr

srrs

rss

rs

rqdr

rsqd

i

i-

L )(K)L(3

2

LK )(KLK

λ

λ 0

1-

-1

0

2

1

2

1

2

1

)120sin( )120sin( )sin(

)120cos( )120cos( )cos(

3

2K rrr

rrrrs

Dr. Amr AbdAllah 10


The stator inductance matrix in the dq reference frame attached to rotor is computed as:

1-

1-

)(K

)120cos(2 - )cos(2 -2

1- 120)cos(2-

2

1-

)cos(2 -2

1- 120)cos(2 - 120)-cos(2-

2

1-

)120cos(2 -2

1- )120cos(2 -

2

1- )cos(2-

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

3

2)(KLK

21

21

21

rs

rBAlsrBArBA

rBArBAlsrBA

rBArBArBAls

rrr

rrrrss

rs

LLLLLLL

LLLLLLL

LLLLLLL

Dr. Amr AbdAllah 11


Which finally results in a stator inductance matrix in the dq reference frame attached to rotor that is no more function in rotor position as shown below:

ls

mdls

mqls

rr

rr

rr

lslsls

rmdlsrmdlsrmdls

rmqlsrmqlsrmqlsrss

rs

L

LL

LL

LLL

LLLLLL

LLLLLL

0 0

0 )( 0

0 0 )(

1 )120sin( )120cos(

1 )120sin( )120cos(

1 )sin( )cos(

)120)sin(( 120)-)sin(( ))sin((

)120)cos(( 120)-)cos(( ))cos((

3

2)(KLK

2

1

2

1

2

1

1-

Dr. Amr AbdAllah 12


The inductance matrix for the stator mutual coupling with rotor circuit with the stator circuits transformed to the rotor dq frame is computed as:

0 0 0 0

0 0

0 0

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

120)-sin( 120)-sin( 120)-cos( 120)-cos(

)sin( )sin( )cos( )cos(

)120sin( )120sin( )sin(

)120cos( )120cos( )cos(

3

2LK

2

1

2

1

2

1

mdmd

mqmq

rmdrmdrmqrmq

rmdrmdrmqrmq

rmdrmdrmqrmq

rrr

rrr

srrs

LL

LL

LLLL

LLLL

LLLL

Dr. Amr AbdAllah 13


It can be easily shown that:

The set of voltage equations that describe the synchronous machine in the dq reference frame attached to the rotor can thus be written as:

0 0

0 0

0 0

0 0

)L()(K)(K)L(3

2 1

md

md

mq

mq

Tsr

rs

rs

Tsr

L

L

L

L

rs

rss

rs

rds

rqsr

rdssds

rqs

rdsr

rqss

rqs

pλirv

pλλωirv

pλλωirv

000

rkd

rkdkd

rkd

rfd

rfdfd

rfd

rkq

rkqkq

rkq

rkq

rkqkq

rkq

λpirv

λpirv

λpirv

λpirv

2222

1111

Dr. Amr AbdAllah 14


The flux linkage equations obtained from the transformation applied to the stator variables to the rotor circuit will be given as:

)(

)(

)(

)(

)(

)(

21222

21111

00

21

rkd

rfd

rdsmd

rkdls

rkd

rkd

rfd

rdsmd

rfdls

rfd

rkq

rkq

rqsmq

rkqlkq

rkq

rkq

rkq

rqsmq

rkqlkq

rkq

rsls

rs

rkd

rfd

rdsmd

rdsls

rds

rkq

rkq

rqsmq

rqsls

rqs

iiiLiLλ

iiiLiLλ

iiiLiLλ

iiiLiLλ

iLλ

iiiLiLλ

iiiLiLλ

Dr. Amr AbdAllah 15


The synchronous machine voltage current model in the rotor reference frame can thus be deduced by the substitution of the flux linkage equations in the voltage equations which results in:

rkd

rfd

rkq

rkq

rs

rds

rqs

mdkdkdmdmd

mdmdfdfdmd

mqkqkqmqmq

mqmqkqkqmq

lss

mdmdmqrmqrmdssmqsr

mdrmdrmqmqmdsrmqss

rkd

rfd

rkq

rkq

rs

rds

rqs

i

i

i

i

i

i

i

pLrpLpL

pLpLrpL

pLrpLpL

pLpLrpL

pLr

pLpLLLpLrL

LLpLpLLpLr

v

v

v

v

v

v

v

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0 0 0

0 -

0

2

1

0

22

11

2

1

0

mqlkqmqkq

mqlkqmqkq

mqlsmqs

LLL

LLL

LLL

22

11

mdlkdmdkd

mdlfdmdfd

mdlsmds

LLL

LLL

LLL

Dr. Amr AbdAllah 16

Transformation to rotor reference frame: Equivalent circuit

Dr. Amr AbdAllah 17

Transformation to rotor reference frame: Torque Equation

The expression for the electromagnetic torque in terms of rotor frame variables can be given by:

It can be shown that the above equation results in:

Which is equivalent to:

REPORT

Dr. Amr AbdAllah 18

Hope you all success

COURSE COMPLETED Thanks

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Machines EPM405A Presentation 06