TABLE OF CONTENTS In spite of its simplicity, the illustrative example of the previous section has evidenced important features that hold for circuits with energy-storage elements in general, regardless of their complexity. We now wish to generalize our observations.
Given a network made up of resistances, capacitances, inductances, and possibly dependent sources, the applied signal x( t) and the response y( t) are in general related by a linear differential equation of the type
where a 0 through a m and b 0 through b n are suitable coefficients whose expressions depend on the elements comprising the network and are thus real and time-independent.
With an applied signal of the complex exponential type
the response is also of the complex exponential type
and the value of s in the expression for y( t) is the same as that given in the expression for x( t). Substituting x and y into Equation (14.23) and exploiting the important exponential property
we obtain
Eliminating the common term e st, this can be abbreviated as
where D( s) and N( s) are polynomials in s with real coefficients,
and n and m are their degrees. These polynomials are, respectively, the left and right members of Equation (14.23), but with derivatives replaced by powers of s. Solving Equation (14.27) for the ratio
generally...
