8: Chapter Transient Analysis of
Synchronous Machines
FAMU-FSU College of Engineering
Synchronous Machines
Steady state modeling
rotor mmf and stator mmf are stationary with respect to each other
flux linkage with the rotor are invariant with time
no voltages are induced on the rotor circuits
Transient modeling
flux linkage changes
with time
differential equations
have time-varying
coefficients
Parks transformation
dynamic behavior
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IG
XS
emf
RA
VT
sub-transient period, transient period, and steady-state period
Power Systems I
2
Transient Analysis
Transient analysis will be applied in the dynamic study of
generators
Generators experience dynamic behavior during
switching load
faults
Consider the transient behavior of an RL circuit with a
switched voltage source
R
t=0
L
i(t)
V(t)
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Power Systems I
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Transient Analysis
The voltage source is sinusoidal: v (t ) = Vm sin ( t + )
The KVL equation are:
d i (t )
0 = Vm sin ( t + ) R i (t ) L
dt
i (t ) = I m sin ( t + ) I m e t sin ( )
R
Vm
L
where : I m =
, = ,
Z
R
1 L
= tan
,
V(t)
R
t=0
L
i(t)
and Z = R 2 + 2 L2
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Power Systems I
4
Example
Solve for the time-domain solution of the current
for a faulted generator having the following characteristics
R = 0.125
L = 10 mH
v(t) = 151 sin (377 t + )
which will give
(a) zero dc offset current,
(b) maximum dc offset current
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Power Systems I
5
Example
Z = 0.125 + j (377 )(0.01) = 0.125 + j 3.77 = 3.77288.1
151
0.01
=
Im =
= 40 A
= 0.08 s
3.772
0.125
i (t ) = 40 sin(377 t + 88.1) 40 e t 0.08 sin( 88.1)
(a) Let = 88.1 i (t ) = 40 sin (377 t )
(b) Let = (88.1 90) = 1.9
i (t ) = 40 sin (377 t 90) 40 e t 0.08 sin( 90)
= 40 cos(377 t ) + 40 e t 0.08
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Power Systems I
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Example
40
(a) zero dc
offset current
i(t)
20
0
-20
-40
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.2
0.25
0.3
0.35
t, sec
80
60
40
i(t)
(b) maximum dc
offset current
20
0
-20
-40
0
0.05
0.1
0.15
t, sec
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Power Systems I
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Transient Analysis
Synchronous Machines
Models and analysis were previously developed for steady state
behaviors
rotor and stator magnetic fields are stationary with respect to each
other
the flux linkage in the rotor circuit are constant in time
the per phase equivalent circuit becomes a constant generated emf
in series with a simple impedance
Under transient conditions (time varying) the above assumptions
are no longer valid
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changing stator current are reflected in a dynamic flux linkage
changing flux linkage induces transient currents in the rotor
transient rotor currents in turn react with the stator and the induced
voltages
Power Systems I
8
Synchronous Machine Model
The synchronous machine consist of:
three ac stator windings mounted on the stator
one field winding mounted on the rotor
two fictitious windings which model short-circuited paths of the
damper windings
When modeling, the following are assumed:
a synchronously rotating reference frame with a speed of
the reference frame is along the axis of phase a at time zero
For transient analysis of an ideal synchronous machine
The machine is represented as a group of magnetically coupled
rotating circuits with inductances which depend on the angular
position of the rotor
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Power Systems I
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Synchronous Machine Model
Reference Axis
m
Direct Axis
Quadrature Axis
b
Q
a
c
m = t + +
2
D
F
Q
a
F
D
c
b
Physical Layout
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Power Systems I
10
Synchronous Machine Model
r
rF
VF
r
ia
ic
LF
Laa
rD
Lcc
LD
LQ
Lbb
r
in
Schematic of mutually
coupled circuits
ib
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Power Systems I
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Synchronous Machine Model
The KVL equations for the model
r 0
va
0 r
v
b
0 0
vc
= 0 0
vF
0 0
0
0 0
0
v abc
R abc
v = 0
FDQ
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0
0
0
0
0
0
r
0
0
0 rF
00
0
rD
0
0
0
0 ia
a
0 ib
b
0 ic d c
0 iF dt F
D
0 iD
rQ iQ
Q
0 i abc d
R FDQ i FDQ dt
Power Systems I
FDQ
abc
12
Synchronous Machine Model
The magnetic inductance equations
a Laa Lab Lac LaF
L
Lbb Lbc LbF
b ba
c Lca Lcb Lcc LcF
=
F LFa LFb LFc LFF
D LDa LDb LDc LDF
Q LQa LQb LQc LQF
abc L SS L SR i abc
= L
L RR i FDQ
FDQ RS
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LaD
LbD
LcD
LFD
LDD
LQD
LaQ ia
LbQ ib
LcQ ic
LFQ iF
LDQ iD
LQQ iQ
Power Systems I
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Salient Pole Machines
Rotors have two types of construction
Cylindrical
Salient
The cylindrical rotor has an evenly
spaced air gap and a constant
self-inductance
The Salient has a non-uniform air
gap and a self-inductance that varies
periodically
Cylindrical
maximum when inductance the direct axis
coincides with the stator coil magnetic axis
minimum inductance when the quadrature axis
coincides with the stator coil magnetic axis
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Power Systems I
Salient
14
Salient Pole Machines
The salient pole machine can be represented by cosines
of second harmonics
Stator quantities
Laa = LS + Lm cos 2
Lbb = LS + Lm cos 2( 2 )
3
Lcc = LS + Lm cos 2( + 2 )
3
Lab = Lba = M S Lm cos 2( + 1 )
6
Lbc = Lcb = M S Lm cos 2( 1 )
2
Lca = Lac = M S Lm cos 2( + 5 )
6
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Power Systems I
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Salient Pole Machines
Rotor quantities - All the rotor self inductances are constant since
the effects of the stator slots are negligible
LFF = LF
LDD = LD
LQQ = LQ
LFD = LDF = M R
LFQ = LQF = 0
LDQ = LQD = 0
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Power Systems I
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Salient Pole Machines
Mutual inductance between the stator and rotor circuits
LaF = M F cos 2
LbF = M F cos 2( 2 )
3
LcF = M F cos 2( + 2 )
3
LaD = M D cos 2
LaQ = M Q sin 2
LbD = M D cos 2( 2 ) LbQ = M Q sin 2( 2 )
3
3
LcD = M D cos 2( + 2 ) LcQ = M Q sin 2( + 2 )
3
3
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Power Systems I
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Park Transformation
Changes the abc frame of reference to the dq0 frame of
reference
Voltages and currents on the stator are changed to equivalent
values based on the rotors frame of reference
The transformation is based on the two-axis theory
the electrical quantities are projections onto three new axes:
direct axis - along the direct axis of the rotor field winding
quadrature axis - tangent to the direct axis of the rotor field winding
zero axis - a stationary axis
1 2
2
P=
cos
3
sin
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1
2
cos( 2 )
3
sin ( 2 )
3
2
cos( + 3 )
sin ( + 2 )
3
Power Systems I
1
2
18
Park Transformation
The Park transformation for current
ia
2
2
cos( 3 ) cos( + 3 ) ib
2
2
sin ( 3 ) sin ( + 3 ) ic
Similarly applied to all electrical quantities
i 0 dq = P i abc
i0
id =
iq
1 2
2
cos
3
sin
v 0 dq = P v abc
0 dq
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=P
1
2
1
2
in matrix notation
abc
Power Systems I
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Park Transformation
The Park transformation matix is orthogonal:
1
2
1
T
P =P =
1
3
1
2
2
2
cos
sin
2
2
cos( 3 ) sin ( 3 )
cos( + 2 ) sin ( + 2 )
3
3
Applying the Park transformation to the generator
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Power Systems I
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Park Transformation
Transforming the time-varying inductance to obtain a
rotor frame of reference
P 0
= 0 I
FDQ
33
0 dq
FDQ
abc
P 1 0 0 dq
or
=
0 I 33 FDQ
FDQ
P 1 0 0 dq L SS L SR P 1 0 i 0 dq
= L
0 I 33 FDQ RS L RR 0 I 33 i FDQ
0 dq P 0 L SS L SR P 1 0 i 0 dq
= 0 I L
L RR 0 I 33 i FDQ
FDQ
33 RS
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abc
Power Systems I
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Park Transformation
Resulting inductance matrix
0
0
Ls 2 M s
3
0
Ls + M s + 2 Lm
0
3
0
0
Ls + M s 2 Lm
3
0
0
2 MF
3
0
0
2 MD
3
0
0
2 MQ
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Power Systems I
0
3
2
MF
0
0
3
2
MD
0
LF
MR
MR
LD
0
0
0
3
2 MQ
0
0
LQ
0
22
Park Transformation
Applying the transformation to the machine model KVL
vabc Rabc 0 iabc d abc
v = 0 R i
dt FDQ
FDQ FDQ
FDQ
P1 0 v0dq Rabc 0 P1 0 i0dq d P1 0
v = 0 R
i
0 I33 FDQ
0 I33 FDQ dt 0 I33
FDQ
FDQ
0dq
v0dq P 0 Rabc 0 P1 0 i0dq P 0 d P1 0
i 0 I
v = 0 I 0 R
0 I33 FDQ 33 dt 0 I33
FDQ
FDQ 33
d
v0dq Rabc 0 i0dq P dt P1 0 0dq d 0dq
v = 0 R i
0 I33 FDQ dt FDQ
FDQ
FDQ FDQ
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Power Systems I
FDQ
0dq
23
Park Transformation
Evaluate the expression
d
P dt P 1
d
P dt P 1
1 2
12
1 2 0
sin
cos
= 2 cos cos( 23 ) cos( + 23 ) 0 sin ( 23 ) cos( 23 )
3
sin sin ( 23 ) sin ( + 23 ) 0 sin ( + 23 ) cos( + 23 )
0 0 0
= 0 0 1
0 1 0
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Power Systems I
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Park Transformation
Substituting the original terms into the transformation
0
0
r 0
v0
0 r Lq
0
vd
3
v
q
0 Ld r 2 M F
=
0
rF
0 0
vF
0 0
0
0
0
0 0
0
0
0
0
0
0
L0
0
3
Ld
0
2 MF
0
Lq
0
0
3
0 2MF
0
LF
3
0
MR
0 2 MD
3
0
0
0
2 MQ
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0
0
3
2
MD
0
rD
0
0
3
2MD
0
MR
LD
0
Power Systems I
i0
3
2 M Q id
0 iq
0 iF
0 iD
rQ iQ
0
0
i
0 i0
d
3
i
2 MQ d
q
i
0
dt F
0 iD
iQ
LQ
25
Park Transformation
Observations
The transformation has constant coefficients provided that the
speed is assumed to be constant
The first equation (the zero sequence) is not coupled to the other
equations, and it can be treated separately
While the transformation technique is a mathematical process, it
gives insight into the internal phenomena of the rotor and the
effects of transients
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Power Systems I
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