design and implementation of thyristor switched shunt capacitors a ...

with the initial conditions:
d 2 i(t)
+ 1 w ˆV
i(t) =
dt 2 LC L
cos(wt + α), (2.12)
i(0) = 0,
di(t)
dt | t=0 = ˆV
L sin α − V co
L , (2.13)
The solution to the ordinary differential equation in 2.12 is as follows:
i(t) =
wC ˆV
cos(wt + α) − √ [
V
LC co
− ˆV
1−w 2 LC L
−
wC ˆV
1−w 2 LC
cos α cos(
1 √
LC
t)
( ) ]
1
+ w2 C
sin α sin(
L 1−w 2 √ 1
LC
LC
t)
(2.14)
The details of how to derive this result can be found in Appendix A. To simplify
the above equation, let
√
1
n =
w √ LC = XC
,
X L
1
=⇒
1 − w 2 LC = n2
n 2 − 1 ,
B c = wC,
w o = √ 1 ,
LC
=⇒ n = w o
w ,
Î 1 =
n2
n 2 − 1 B c ˆV ,
assign as new variables. Here w o is natural frequency of oscillation as mentioned
before and n is equal to the per-unit natural frequency. Therefore, the final
equation to represent the single phase TSC is shown below:
[
i(t) = Î1 cos(wt + α) − nB c V co −
n2 ˆV
]
sin α sin(w
n 2 −1 o t)
−Î1 cos α cos(w o t),
(2.15)
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