TABLE OF CONTENTS The network function H( s) can be used to predict not only the natural response but also the transient and the steady-state responses and, hence, the complete response. As such, H( s) provides a unified approach, but using simple algebra instead of differential equations. To investigate the various responses of a circuit, we subject it to the complex exponential signal
and obtain the response by taking the product
where H( s) is calculated at the value of s supplied by the applied signal. Let us examine the most important response types.
This is the response to a dc signal of the type
after all transients have died out. As we know, the complex exponential form of this signal is
so the complex exponential form of the response is
But this is the familiar dc steady-state response,
In words, to find the steady-state response to a dc signal of amplitude X m, we calculate H( s) at s = 0, and then multiply X m by H(0) to obtain y ss. This forms the basis of the following rule:
DC Rule: In dc analysis H( s) is calculated at the origin of the s plane.
Letting s = 0 makes Z L = sL = 0 and Z C = 1/
