Page 151 - Schaum's Outline of Theory and Problems of Electric Circuits

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For t > 1 ms, FIRST-ORDER CIRCUITS [CHAP. 7
t
v ¼ 0:632e ðt 1Þ ¼ 1:72e , and i ¼ 1:72e t
(b) V 0 ¼ 10 V, T ¼ 0:1 ms.
For 0 < t < 0:1 ms,
t t 0:1
v ¼ 10ð1 e Þ; i ¼ 10e , and V T ¼ 10ð1 e Þ¼ 0:95 V
For t > 0:1 ms,
t
v ¼ 0:95e ðt 0:1Þ ¼ 1:05e , and i ¼ 1:05e t
(c) V 0 ¼ 100 V, T ¼ 0:01 ms.
For 0 < t < 0:01 ms,
t
v ¼ 100ð1 e Þ 100t; i ¼ 100e t 100ð1 tÞ, and V T ¼ 100ð1 e 0:01 Þ¼ 0:995 V
For t > 0:01 ms,
v ¼ 0:995e ðt 0:01Þ ¼ 1:01e t and i ¼ 1:01e t
t
As the input voltage pulse approaches an impulse, the capacitor voltage and current approach v ¼ e uðtÞ (V)
t
and i ¼ ðtÞ e uðtÞ.

7.11 IMPULSE RESPONSE OF RC AND RL CIRCUITS
A narrow pulse can be modeled as an impulse with the area under the pulse indicating its strength.
Impulse response is a useful tool in analysis and synthesis of circuits. It may be derived in several ways:
take the limit of the response to a narrow pulse, to be called limit approach, as illustrated in Examples
7-11 and 7-12; take the derivative of the step response; solve the differential equation directly. The
impulse response is often designated by hðtÞ.

EXAMPLE 7.12 Find the limits of i and v of the circuit Fig. 7-17(a) for a voltage pulse of unit area as the pulse
duration is decreased to zero.
We use the pulse responses in (14) and (15) with V 0 ¼ 1=T and find their limits as T approaches zero. From
(14c) we have
lim V T ¼ lim ð1 e T=RC Þ=T ¼ 1=RC
T!0 T!0
From (15) we have:
For t < 0; h v ¼ 0 and h i ¼ 0
1 1
þ

For 0 < t < 0 ; 0 h v and h i ¼ ðtÞ
RC R
1 1
For t > 0; h v ðtÞ¼ e t=RC and h i ðtÞ¼ e t=RC
2
RC R C
Therefore,
1 1 1
h v ðtÞ¼ e t=RC uðtÞ and h i ðtÞ¼ ðtÞ e t=RC uðtÞ
2
RC R R C
EXAMPLE 7.13 Find the impulse responses of the RC circuit in Fig. 7-17(a) by taking the derivatives of its unit
step responses.
A unit impulse may be considered the derivative of a unit step. Based on the properties of linear differential
equations with constant coefficients, we can take the time derivative of the step response to find the impulse
response. The unit step responses of an RC circuit were found in (6)tobe
vðtÞ¼ ð1 e t=RC ÞuðtÞ and iðtÞ¼ ð1=RÞe t=RC uðtÞ
We find the unit impulse responses by taking the derivatives of the step responses. Thus