Example 3.
t  0
R
L
V
i L (t )
Consider the RL circuit, the switch closes at t=0. Assume the initial current through the
inductor is I 0 . Our goal is to find the inductor current i L (t ) , as t>0
We can apply the Kirchoff’s voltage law to solve the problem.
 V  Ri L  L
di L
0
dt
The circuit is described by the first order differential equation. The equations can be
written as:
di L V R
  iL
dt
L L
We first separate the variable i and t. multiply both sides by –R/L
di L
  R / Ldt
LV

 iL
R L
di L
 dt
V R
 iL
L L
Taking the integration and we get
Consequently we get,
ln
i(t )  V / R
R
 t
i (0)  V / R
L
R
i L (t ) 
V
V  t
 ( I 0  )e L
R
R
i L (t )  I f  ( I 0  I f )e

t

,
L
R
di L
  R / Ldt
V
  iL
R
Key steps to calculate the step response of a first order circuit:
1. Find the initial current I 0 at t  0  and
the final current I f as t  
2. Find the time constant  
3. i L (t )  I f  ( I 0  I f )e

L
R
t

If
I0
τ
2τ
3τ
t