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[CHAP. 7

For t > 1 ms, v ¼ 0:632eðt1Þ ¼ 1:72et , and i ¼ 1:72et (b) V0 ¼ 10 V, T ¼ 0:1 ms. For 0 < t < 0:1 ms, v ¼ 10ð1  et Þ; i ¼ 10et , and VT ¼ 10ð1  e0:1 Þ ¼ 0:95 V For t > 0:1 ms, v ¼ 0:95eðt0:1Þ ¼ 1:05et , and i ¼ 1:05et (c)

V0 ¼ 100 V, T ¼ 0:01 ms. For 0 < t < 0:01 ms, v ¼ 100ð1  et Þ  100t; i ¼ 100et  100ð1  tÞ, and VT ¼ 100ð1  e0:01 Þ ¼ 0:995 V For t > 0:01 ms, v ¼ 0:995eðt0:01Þ ¼ 1:01et and i ¼ 1:01et

As the input voltage pulse approaches an impulse, the capacitor voltage and current approach v ¼ et uðtÞ (V) and i ¼ ðtÞ  et uðtÞ.



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 V0 ¼ 1=T and find their limits as T approaches zero. From (14c) we have lim VT ¼ lim ð1  eT=RC Þ=T ¼ 1=RC



From (15) we have: For t < 0; 

hv ¼ 0

and hi ¼ 0 1 1 and hi ¼ ðtÞ 0  hv  RC R 1 t=RC 1 hv ðtÞ ¼ and hi ðtÞ ¼  2 et=RC e RC R C


For 0 < t < 0 ; For t > 0; Therefore, hv ðtÞ ¼

1 t=RC uðtÞ e RC


hi ðtÞ ¼

1 1 ðtÞ  2 et=RC uðtÞ 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) to be vðtÞ ¼ ð1  et=RC ÞuðtÞ


iðtÞ ¼ ð1=RÞet=RC uðtÞ

We find the unit impulse responses by taking the derivatives of the step responses.