Page 152 - Schaum's Outline of Theory and Problems of Electric Circuits

CHAP. 7]
1
1 t=RC FIRST-ORDER CIRCUITS 1 t=RC 141
h v ðtÞ¼ e uðtÞ and h i ðtÞ¼ ðtÞ 2 e uðtÞ
RC R R C

EXAMPLE 7.14 Find the impulse responses h i ðtÞ; h v ðtÞ; and h i1 ðtÞ of the RL circuit of Fig. 7-11(a) by taking the
derivatives of its unit step responses.
The responses of the circuit to a step of amplitude 9 were already found in Example 7.5. Taking their
derivatives and scaling them down by 1/9, we find the unit impulse responses to be
1 d 800t 200 800t
h i ðtÞ¼ ½0:75ð1 e Þuðtފ ¼ e uðtÞ
9 dt 3
1 d 800t 800 800t 1
h v ðtÞ¼ ½3e uðtފ ¼ e uðtÞþ ðtÞ
9 dt 3 3
1 d 1 800t 200 800t 1
h i1 ðtÞ¼ ð3 e ÞuðtÞ ¼ e uðtÞþ ðtÞ
9 dt 4 9 18

7.12 SUMMARY OF STEP AND IMPULSE RESPONSES IN RC AND RL CIRCUITS
Responses of RL and RC circuits to step and impulse inputs are summarized in Table 7-1. Some of
the entries in this table have been derived in the previous sections. The remaining entries will be derived
in the solved problems.

7.13 RESPONSE OF RC AND RL CIRCUITS TO SUDDEN EXPONENTIAL EXCITATIONS
Consider the first-order differential equation which is derived from an RL combination in series with
st
a sudden exponential voltage source v s ¼ V 0 e uðtÞ as in the circuit of Fig. 7-18. The circuit is at rest for
t < 0. By applying KVL, we get
di st
Ri þ L ¼ V 0 e uðtÞ ð16Þ
dt
For t > 0, the solution is
þ
iðtÞ¼ i h ðtÞþ i p ðtÞ and ið0 Þ¼ 0 ð17aÞ

Table 7-1(a) Step and Impulse Responses in RC Circuits

RC circuit Unit Step Response Unit Impulse Response

v s ¼ uðtÞ v s ¼ ðtÞ
( (
v ¼ð1 e t=Rc ÞuðtÞ h v ¼ð1=RCÞe t=RC uðtÞ
t=Rc 2 t=RC
i ¼ð1=RÞe uðtÞ h i ¼ ð1=R CÞe uðtÞþð1=RÞ ðtÞ

i s ¼ ðtÞ
(
i s ¼ uðtÞ t=RC
( h v ¼ð1=CÞe uðtÞ
v ¼ Rð1 e t=RC ÞuðtÞ h i ¼ ð1=RCÞe t=RC uðtÞþ ðtÞ
i ¼ e t=RC uðtÞ