9.1 Natural Response of a Series RL circuit

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.

Figure 9.1: Circuit for determining the natural response of a series RL circuit

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.

1. Separation of Variables Method

   Rearranging (9.1), so that the variables i and t are separated, we obtain

   
   

   Next, integrating both sides and using the initial condition, we obtain

   
   

   where ? is a dummy variable. Integration yields

   
   

   or

   
   

   or

   
   

   Recalling that x = lny implies y = e ^x, we obtain

   
   

   Substitution of (9.2) into (9.1) yields 0 = 0 and that at t = 0, i(0) = I ₀. Thus, both the...