Page 174 - Schaum's Outline of Theory and Problems of Electric Circuits

HIGHER-ORDER CIRCUITS AND COMPLEX FREQUENCY
CHAP. 8]
Overdamped Case ð >! 0 Þ 163
In this case, both and are real positive numbers.
t
i ¼ A 1 e ð þ Þt þ A 2 e ð Þt ¼ e t ðA 1 e þ A 2 e t Þ

EXAMPLE 8.1 A series RLC circuit, with R ¼ 200
, L ¼ 0:10 H, and C ¼ 13:33 mF, has an initial charge on the
capacitor of Q 0 ¼ 2:67 10 3 C. A switch is closed at t ¼ 0, allowing the capacitor to discharge. Obtain the
current transient. (See Fig. 8-4.)
For this circuit,
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R 1
3 1
5 2
2
2
2
¼ ¼ 10 s ; ! 0 ¼ ¼ 7:5 10 s ; and ¼ ! ¼ 500 s 1
0
2L LC
Then, i ¼ e 1000t ðA 1 e 500t þ A 2 e 500t Þ
The values of the constants A 1 and A 2 are obtained from the initial conditions. The inductance requires that

þ
ið0 Þ¼ ið0 Þ. Also the charge and voltage on the capacitor at t ¼ 0 þ must be the same as at t ¼ 0 , and

v C ð0 Þ¼ Q 0 =C ¼ 200 V. Applying these two conditions,
and
0 ¼ A 1 þ A 2 2000 ¼ 500A 1 1500A 2
from which A 1 ¼ 2; A 2 ¼ 2, and, taking A 1 positive,
500t 1500t
i ¼ 2e 2e ðAÞ
If the negative value is taken for A 1 , the function has simply flipped downward but it has the same shape. The signs
of A 1 and A 2 are fixed by the polarity of the initial voltage on the capacitor and its relationship to the assumed
positive direction for the current.

Fig. 8-4

Critically Damped Case ð ¼ ! 0 Þ
With ¼ ! 0 , the differential equation takes on a different form and the two exponential terms
suggested in the preceding will no longer provide a solution. The equation becomes
2
d i di 2
þ 2 þ i ¼ 0
2
dt dt
and the solution takes the form i ¼ e t ðA 1 þ A 2 tÞ.
EXAMPLE 8.2 Repeat Example 8.1 for C ¼ 10 mF, which results in ¼ ! 0 .
þ

As in Example 8.1, the initial conditions are used to determine the constants. Since ið0 Þ¼ ið0 Þ,
0 ¼½A 1 þ A 2 ð0ފ and A 1 ¼ 0. Then,
di d t at t
¼ ðA 2 te Þ¼ A 2 ð te þ e Þ
dt dt
3
10 t
from which A 2 ¼ðdi=dtÞj 0 ¼ 2000. Hence, i ¼ 2000te (A) (see Fig. 8-5).
þ
Once again the polarity is a matter of the choice of direction for the current with respect to the polarity of the
initial voltage on the capacitor.