Example 13.
5 kΩ
a
i
1 kΩ
1 kΩ
20 V
1 µF
t=0
5 kΩ
10 µF
b
The circuit is in the steady state mode before the switch closes at t=0; Determine the
current through terminal a and b i(t) for t>0.
The circuit has two capacitors. Some beginners might try to combine the two capacitors
to one. The two capacitors are not in parallel so we cannot combine the two capacitors.
The circuit is a second order circuit. Here let us see if we can solve it by the methods we
learned for the first order transient response. Let us consider the circuit before switching,
vc1 (0  )  10V vc 2 (0  )  10V
t>0
5 kΩ
a
i
1 kΩ
1 kΩ
20 V
1 µF
5 kΩ
10 µF
b
Immediately after the switching, a and b are shorted, the 20V source will drive a current
through a and b. Also the voltage though each of the capacitors will drive the current
flow from a to b. Let’s divide the problem into 3 sub problems. The current i can be
considered as the superposition of three portions.
t > 0 5 kΩ
a
a
a
i2
i1
i3
1 kΩ
1 kΩ
20 V
10 µF
1 µF
b
b
b
i1 (t )  4mA
i2 (0  )  10mA  2  RC  1000  10 6  0.001 s i2 (t )  i2 (0  )e t /   10e 1000t mA
i3 (0  )  10mA  3  RC  1000  10 5  0.01s i2 (t )  i2 (0  )e t /   10e 100t mA


i  i1 (t )  i2 (t )  i3 (t )  4  10e 1000t  10e 100t mA t  0 