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# Since all the energy starts out the capacitor

684 C H A P T E R

T W E L V E

t r a n s i e n t s

i n s e c o n d - o r d e r

2. Since there is no drive, and since there is a dissipative element in the circuit, the final value of the capacitor voltage is given by

vC(∞) = 0 V.

4.
α = 1
2RC
ωo = 1
LC

and the ringing frequency is

5.
ωd =
Q =ωo 2α

The form of vC(t) can now be plotted as shown in Figure 12.59.

12.8

vC(V) 1.0 VCM
2

12.8 Two-Capacitor or Two-Inductor Circuits

R1 C H A P T E R

T W E L V E

685
0.5
v1 v2
0.0
-0.5 -VCM C1
R2 C2
-1.0
3 4

0

t (ms)

FIGURE 12.59 Sketching the form of vC for a parallel RLC circuit driven by an impulse current input.

C dv1(t) 1 v(t)1

(12.238)

1 dt + R1

1 + R3

C dv2(t) 1 v(t)1
2 dt + R2

2 + R3

2 −1 =

v2(t) = R3C1 dv1(t) + �1R3
dt + R1

d2v1(t)

1
1 + 1

+ R3C2� dv1(t) 1

dt2 + R1C1
+ R2C2 R3C1
1 +
+ R1R3C1C2

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