Dr. Amr AbdAllah 1 Electric Machines IIIA COURSE EPM 405A FOR 4 th Year Power and Machines ELECTRICAL DEPARTMENT Lecture 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
Transformation to arbitrary reference frame
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
Transformation to arbitrary reference frame
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
Transformation to rotor reference frame: Park’s Transformation
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
Transformation to rotor reference frame: Park’s Transformation
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
Transformation to rotor reference frame: Park’s Transformation
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
Transformation to rotor reference frame: Park’s Transformation
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
Transformation to rotor reference frame: Park’s Transformation
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
Transformation to rotor reference frame: Park’s Transformation
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