Linearized Analysis of the Synchronous Machine for PSS Chapter 6 ...

ΔE&
'
q
⎛ 1
= −
⎜
⎝ K
3τ
'
⎛ K
ω ⎜
−
Δ & =
⎝ τ
j
Δ & δ =
2
( Δω ) ω
B
do
⎞
⎟ΔE'
⎠
⎞
⎟ΔE'
⎠
q
q
⎛ K
−⎜
⎝ τ
j
⎛ K
−
⎜
⎝τ
'
1
4
do
⎞ ⎛ 1
⎟Δδ
+
⎜
⎠ ⎝τ
'
⎞ ⎛ ⎞
⎟ δ ⎜
1
Δ + ⎟ΔT
τ
⎠ ⎝ j ⎠
do
m
⎞
⎟ΔE
⎠
FD
(6.79)
The LaPlace transform of the above equations results in the
following relations:
K
3
K
3K
4
ΔE'
q
=
ΔE
FD
−
Δδ
1 + K
3τ
'
d 0
s 1 + K
3τ
'
d 0
s
1
Δω
= ( ΔTm
− ΔTe
)
sτ
j
(*)
1
Δδ
= Δω
s
where the variables ΔE’ q , Δω, and Δδ represent LaPlace transforms
of their corresponding time-domain functions. Two additional
relations may be obtained as well, according to the following:
ΔT
e
ΔV
t
=
=
K
K
1
5
Δδ
+
Δδ
+
K
K
2
6
ΔE'
q
ΔE'
q
+ DΔω
(**)
Finally, we note that E FD , the stator EMF produced by the field
current and corresponding to the field voltage v F , is a function of
the voltage regulator. Under linearized conditions, the change in
E FD is proportional to the difference between changes in the
reference voltage and changes in the terminal voltage, i.e.,
Δ EFD
= G
e(
s )( ΔVref
− ΔVt
) (***)
where G e (s) is the transfer function of the voltage regulator.
4