General Solution for Natural and Step
Responses of RL and RC Circuits
x(t )  x f  [ x(t0 )  x f ]e
Final Value

( t  t0 )

Time Constant
Initial Value
Determine the initial and final values of the
variable of interest and the time constant of
the circuit.
Substitute into the given expression.
ECE 201 Circuit Theory I
1
Example 7.7
b
R1
400kOhm
V1
90 V
a
R3
J1
20 Ohm
Key = Space
+
vC(t)
R2
60 Ohm
V2
40 V
C
0.5uF
-
• What is the initial value of vC?
• What is the final value of vC?
• What is the time constant when the switch is in
position b?
• What is the expression for vC(t) when t>=0?
ECE 201 Circuit Theory I
2
Initial Value of vC
b
R1
400kOhm
a
R3
J1
20 Ohm
+
V1
90 V
Key = Space
+
vC(0)
-
V60
C
0.5uF
R2
60 Ohm
V2
40 V
-
• The capacitor looks like an open circuit, so the
voltage @ C is the same as the voltage @ 60Ω.
60
vC (0)  40V
 30V
20  60
ECE 201 Circuit Theory I
3
Final Value of vC
b
R1
400kOhm
V1
90 V
a
R3
J1
20 Ohm
Key = Space
+
vC(∞)
R2
60 Ohm
V2
40 V
C
0.5uF
-
• After the switch is in position b for a long time,
the capacitor will look like an open circuit again,
and the voltage @ C is +90 Volts.
ECE 201 Circuit Theory I
4
The time constant of the circuit when the
switch is in position b
R1
400kOhm
V1
90 V
b
a
R3
J1
20 Ohm
Key = Space
R2
60 Ohm
V2
40 V
C
0.5uF
• The time constant τ = RC = (400kΩ)(0.5μF)
• τ = 0.2 s
ECE 201 Circuit Theory I
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The expression for vC(t) for t>=0
vC (t )  vC ()  [vC (0)  vC ()]e
vC (t )  90  [30  90]e


t

t
0.2
vC (t )  90  120e 5tV
ECE 201 Circuit Theory I
6
The expression for i(t) for t>=0
b
R1
400kOhm
a
R3
J1
20 Ohm
Key = Space
V1
90 V
i(t)
30V
R2
60 Ohm
V2
40 V
C
0.5uF
+
• Initial value of i is (90 - - 30)V/400kΩ = 300μA
• Final value of i is 0 – the capacitor charges to
+90 V and acts as an open circuit
• The time constant is still τ = 0.2 s
ECE 201 Circuit Theory I
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The expression for i(t) (continued)
i(t )  i ()  [i(0 )  i()]e
i(t )  0  [300 106  0]e


t

t
0.2
i(t )  300e5t  A
ECE 201 Circuit Theory I
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How long after the switch is in position b
does the capacitor voltage equal 0?
vC (t )  90  120e 5t  0
5t
 90
90
5t
e 
120
 90 
5t  ln 
  0.28768
 120 
t  0.05754 s  57.54ms
120e
ECE 201 Circuit Theory I
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Plot vC(t)
ECE 201 Circuit Theory I
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Plot i(t)
ECE 201 Circuit Theory I
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