TABLE OF CONTENTS In this section we illustrate the use of the network function H( s) to derive the natural response of a circuit. This is the response provided by the circuit with no applied signals,
Also called the source-free response, it stems from the ability of the reactive elements in the circuit to store energy. Substituting Equation (14.44) into Equation (14.23) yields
For a circuit to yield a response y( t) ? 0, the characteristic equation associated with this differential equation must vanish,
But, the values of s satisfying this equation are precisely the poles p 1 through p n of H( s), indicating that the network function contains all the information needed to predict the functional form of the natural response. This is the real function
where p 1 through p n are the poles of H( s), and A 1 through A n are suitable time-independent coefficients reflecting the initial conditions in the circuit. We identify the following important cases:
Real poles. If a pole p k is real,
its contribution to the natural response is
As we know, this is an exponential decay if the pole is negative, a diverging exponential if the pole is positive, and a constant function if the pole lies right at the origin of the s
