Page 176 - Schaum's Outline of Theory and Problems of Electric Circuits

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
CHAP. 8]

Fig. 8-6 165

v ¼ A 1 e s 1 t þ A 2 e s 2 t ð2Þ
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 1 1
2
where s 1 ¼ þ ¼ þ ! 2 0
2RC 2RC LC
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 1 2 1 q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
2
s 2 ¼ ¼ ! 0
2RC 2RC LC
p ffiffiffiffiffiffiffi
where ¼ 1=2RC and ! 0 ¼ 1= LC. Note that , the damping factor of the transient, differs from in
the series RLC circuit.

Fig. 8-7

The transient response is easiest to visualize by assuming an initial charge Q 0 on the capacitor and a
switch that closes at t ¼ 0. However, a step function voltage applied to the circuit will initiate the same
transient response.

2 2
Overdamped Case ð >! 0 Þ
In this case, the solution (2) applies.

EXAMPLE 8.4 A parallel RLC circuit, with R ¼ 1000
, C ¼ 0:167 mF, and L ¼ 1:0 H, has an initial voltage
V 0 ¼ 50:0 V on the capacitor. Obtain the voltage vðtÞ when the switch is closed at t ¼ 0.
We have
1 2 6 2 1 6
¼ ¼ 2994 ¼ 8:96 10 ! 0 ¼ ¼ 5:99 10
2RC LC