TABLE OF CONTENTS This chapter discusses the natural response of electric circuits. The term natural implies that there is no excitation in the circuit, that is, the circuit is source-free, and we seek the circuit's natural response. The natural response is also referred to as the transient response.
Let us find the natural response of the circuit of Figure 9.1 where the desired response is the current i, and it is given that at t = 0, i = I 0, that is, the initial condition is i( 0) = I 0.
Application of KVL yields
or
Here, we seek a value of i which satisfies the differential equation of (9.1), that is, we need to find the natural response which in differential equations terminology is the complementary function. As we know, two common methods are the separation of variables method and the assumed solution method. We will consider both.
Rearranging (9.1), so that the variables i and t are separated, we get
Next, integrating both sides and using the initial condition, we get
where ? is a dummy variable. Integration yields
or
or
Recalling that x = lny implies y = e x, we get
Substitution of (9.2) into (9.1) yields 0 = 0
