Example 7.
R
t=0
+
V
C
vc
_
In the circuit, the switch closes at t=0. The capacitor is suddenly connected to a power
supply, the voltage source can be modeled as a step function, and the response is called
the step response.
Consider the RC circuit after the switch closes. The source voltage is a constant V. We
want to determine the capacitor voltage.
We assume the initial voltage of the capacitor is V0 .
Since the voltage of a capacitor cannot change instantaneously.
vc (0  )  vc (0  )  V0
Using the KVL law,
RC
dvc
dvc
1
V
 vc  V  0

vc 
0
dt
dt RC
RC
Rearrange the terms:
dvc V  v c
dv
v v
or c   c

dt
RC
dt
RC
dv c
1

dt
RC
vc  V
vc ( t )

dvc
vc ( 0 ) v c  V
t
 
0
1
dt
RC
integrating on both sides:
ln(v c  V )
v c (t )
[v (t )  V ]
1
1 t
ln c

t

t
v c (0)
[v c (0)  V ]
RC
RC 0
taking the exponential of both sides

t
[v c (t )  V ]
 e RC
[v c (0)  V ]
1
v c (t )  V  [v c (0)  V ]e

1
t
RC
 V f  (V0  V f )e

t

, here   RC
Key steps to calculate the step response of a first order circuit:
1. Find the initial voltage V0  vc (0  ) at t  0 
and the final voltage V f  vc () as t  
2. Find the time constant   RC
3. vc (t )  V f  (V0  V f )e

t
