Example 2.
a
b
t  0 1 k
1 k
1V
2 k
iL  t 
+
vL
10 m H
_
The switch has been in position for a long time. It moves to b at t=0. Find
(a) v L (t ) for t  0  and
(b) i L (t ) for t  0 .
The natural response of the circuit is determined by the initial condition and the times
constant.
Key steps to calculate the natural response of a first order circuit:
1. Find the initial current or voltage
2. Find the time constant  
3. i L (t )  I 0 e

t

v L (t )  V0 e

L
R
t

First, the initial current cannot change instantaneously,
So we start from the initial current.
I 0  1mA
Second step is to calculate the time constant. After the switch is moved to position b, we
note that the two resistors are in series., they can be receded to one equivalent resistor.
Then we have …
L 0.01
 
 10 5 s
R 1000
5
i L (t )  1e 10 t mA
v L (t )  L
5
diL
 1e 10 tV
dt