Page 138 - Schaum's Outline of Theory and Problems of Electric Circuits

First-Order Circuits

7.1 INTRODUCTION
Whenever a circuit is switched from one condition to another, either by a change in the applied
source or a change in the circuit elements, there is a transitional period during which the branch currents
and element voltages change from their former values to new ones. This period is called the transient.
After the transient has passed, the circuit is said to be in the steady state. Now, the linear differential
equation that describes the circuit will have two parts to its solution, the complementary function (or the
homogeneous solution) and the particular solution. The complementary function corresponds to the
transient, and the particular solution to the steady state.
In this chapter we will find the response of first-order circuits, given various initial conditions and
sources. We will then develop an intuitive approach which can lead us to the same response without
going through the formal solution of differential equations. We will also present and solve important
issues relating to natural, force, step, and impulse responses, along with the dc steady state and the
switching behavior of inductors and capacitors.

7.2 CAPACITOR DISCHARGE IN A RESISTOR
Assume a capacitor has a voltage difference V 0 between its plates. When a conducting path R is
provided, the stored charge travels through the capacitor from one plate to the other, establishing a
current i. Thus, the capacitor voltage v is gradually reduced to zero, at which time the current also
becomes zero. In the RC circuit of Fig. 7-1(a), Ri ¼ v and i ¼ Cdv=dt. Eliminating i in both
equations gives
dv 1
þ v ¼ 0 ð1Þ
dt RC
The only function whose linear combination with its derivative can be zero is an exponential
st
st
st
function of the form Ae . Replacing v by Ae and dv=dt by sAe in (1), we get

1 1
st st st
sAe þ Ae ¼ As þ e ¼ 0
RC RC
1 1
from which s þ ¼ 0 or s ¼ (2)
RC RC
Given vð0Þ¼ A ¼ V 0 , vðtÞ and iðtÞ are found to be
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