Page 54 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 11 SIGNALS AND SYSTEMS

SYSTEMS AND CLASSIFICATION OF SYSTEMS

1.32. Consider the RC circuit shown in Fig. 1-32. Find the relationship between the input
x(t) and the output y(t)
If ~(t) v,(t) and y(r) = v,( t ).
=
If ~(t) u,(t) and y(t) =i(t).
=
Applying Kirchhofs voltage law to the RC circuit in Fig. 1-32, we obtain

c,(r) = Ri(r) + ~,(t) ( 1.103)

The current i(t ) and voltage or( t ) are related by

Letting u,(t) = x(t) and u,(t) = y(r) and substituting Eq. (1.04) into Eq. (1.1031, we obtain

RC-
dr

Thus, the input-output relationship of the RC circuit is described by a first-order linear
differential equation with constant coefficients.
Integrating Eq. (1.104), we have
1 ,
(t) = - i(i) di (1.106)
C -,
Substituting Eq. (1.106) into Eq. (1.103) and letting c,(t) =x(r) and $11 = y(t), we obtain

R

Fig. 1-32 RC circuit.