166

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY

[CHAP. 8

Since 2 > !20 , the circuit is overdamped and from (2) we have qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s1 ¼  þ 2  !20 ¼ 1271 and s2 ¼   2  !20 ¼ 4717  dv  At t ¼ 0; and ¼ s A þ s2 A 2 V0 ¼ A1 þ A2 dt t¼0 1 1 From the nodal equation (1), at t ¼ 0 and with no initial current in the inductance L,  V0 dv dv  V ¼0 or þC ¼ 0 dt dt t¼0 R RC Solving for A1 , A1 ¼

V0 ðs2 þ 1=RCÞ ¼ 155:3 s2  s1

and

A1 ¼ V0  A1 ¼ 50:0  155:3 ¼ 105:3

Substituting into (2) v ¼ 155:3e1271t  105:3e4717t

ðVÞ

See Fig. 8-8.

Fig. 8-8

Underdamped (Oscillatory) Case ð!20 > 2 Þ The oscillatory case for the parallel RLC circuit results in an equation of the same form as that of the underdamped series RLC circuit. Thus, v ¼ et ðA1 cos !d t þ A2 sin !d tÞ

ð3Þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ where  ¼ 1=2RC and !d ¼ !20  2 . !d is a radian frequency just as was the case with sinusoidal circuit analysis. Here it is the frequency of the damped oscillation. It is referred to as the damped radian frequency. EXAMPLE 8.5 A parallel RLC circuit, with R ¼ 200 , L ¼ 0:28 H, and C ¼ 3:57 mF, has an initial voltage V0 ¼ 50:0 V on the capacitor. Obtain the voltage function when the switch is closed at t ¼ 0. ¼

1 1 ¼ ¼ 700 2RC 2ð200Þð3:57  106 Þ

2 ¼ 4:9  105

Since !20 > 2 , the circuit parameters result in an oscillatory response.

!20 ¼

1 1 ¼ ¼ 106 LC ð0:28Þð3:57  106 Þ

Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An
Mahmood_Nahvi_eBook_Schaum_s_Outlines_Theory_An