6. • If a given circuit is not in the form of standard RL/RC circuit. It can be
converted to its thevenin’s equivalent across the inductor/capacitor.
Eg Numerical 6)
• For the circuit shown in figure below, the switch S has been closed for
a long time and then opens at t = 0.
Find,
i. vab(0−
)
ii. ix 0−
iii. iL 0−
iv. ix(0+
)
v. vab(0+
)
vi. ix(t = ∞)
vii. vab(t = ∞)
viii.ix t , for t > 0
7. Solution:
There is a current (I0) flowing in the inductor before switch is opened. To determine
I0, Analyze the circuit before the switch is operated.
𝐢 𝐭 =
𝑽𝒔
𝑹
𝟏 − 𝒆−
𝑹𝒕
𝑳
𝑽𝒔=Vth=12.857
R=Rth=2.142Ω
𝐢 𝐭 =
𝟏𝟐. 𝟖𝟓𝟕𝟏
𝟐. 𝟏𝟒𝟐
𝟏 − 𝒆−
𝟐.𝟏𝟒𝟐𝒕
𝟏 = 𝟔 𝟏 − 𝒆−
𝟐.𝟏𝟒𝟐𝒕
𝟏
1
0
6
5
1
H
2
0
V
a
b
i
x
5
2
0
V
a
b
1
0
i
x Since the switch is closed for a long time.
Inductor behaves as a short circuit carrying a
current 6A.
i. vab 0− ⇒ 0. Behaves as a short circuit
ii. ix 0− ⇒ 6 ∗
10
15
= 4𝐴
iii. iL 0− ⇒ 6𝐴
8. 10
6
5
1H
20V
a
b
ix
𝐢 𝐭 =
𝑽𝒔
𝑹
−
𝑽𝒔
𝑹
−𝑰𝟎 𝒆−
𝑹
𝑳
(𝒕−𝒕𝟎)
t0 = 0
I0=6A
Vs=Vth=20V
R=Rth=5Ω
𝐢 𝐭 =
𝟐𝟎
𝟓
−
𝟐𝟎
𝟓
− 𝟔 𝒆−
𝟓
𝟏
(𝒕−𝟎)
𝐢 𝐭 = 𝟒 − 𝟒 − 𝟔 𝒆−𝟓𝒕
𝐢 𝐭 = 𝟒 + 𝟐𝒆−𝟓𝒕
iv. ix 0
+
⇒ 𝑖𝐿 0
+
= 6𝐴
v. vab(0+
) ⇒ 20 − 5𝑖𝐿 0
+
− 𝑉𝑎𝑏 = 0 ⇒ Vab = −10V
vi. ix(t = ∞) ⇒ 𝑖𝐿 ∞ = 4𝐴
vii. vab t = ∞ = 0V. Inductor behaves as a short
Before the switch is opened the current would reach a maximum of 6A .
Therefore I0=6A
For t>0
9. Numerical 7)
For the circuit shown in figure below, the switch S has been kept open
for a long time and then it is closed at t = 0.
Find,
i. vc(0−
)
ii. vc 0+
iii. ic 0−
iv. ic(0+
)
v.
dvc
dt t=0+
vi. vc t = ∞
vii. find the time constants of the circuit before and after the switch is
closed
10. 1
0
4
0
V
4
F
before the switch is operated
𝐕𝐜 𝐭 = 𝑽𝒔 𝟏 − 𝒆−
𝟏
𝑹𝑪
𝒕
Vc t = 40 1 − 𝑒−
1
40
𝑡
Capacitor is charged to a maximum of 40 V
before the switch is operated. Hence behaves as
an open circuit.
vc 0−
= 40V; ic 0−
= 0A
𝝉1 = 40s
After the switch is operated
𝐕𝐜 𝐭 = 𝑽𝒔 − (𝑽𝑺 − 𝑽𝟎)𝒆−
𝟏
𝑹𝑪
(𝒕−𝒕𝟎)
Vs=Vth=15V
R=Rth= 3.75Ω
Vc t = 15 − 15 − 40 𝑒−
1
15
𝑡−0
Vc t = 15 + 25 𝑒−
1
15𝑡
vc 0+ = 40V; vc ∞ = 15V; 𝝉2 = 15s
1
0
4
0
V
4
F
6
𝑑Vc 0+
𝑑𝑡
= −
5
3
𝐶
𝑑Vc 0+
𝑑𝑡
= −
20
3
11. Numerical 8)
For the circuit shown below, switch S is kept in position ‘1’ for a long
time and then it is moved to position ‘2’ at time 𝑡 = 0 . Find
i. The current expression for 𝐢 𝐭 for 𝑡 ≥ 0
ii. Calculate the time constants of the circuit before and after the
switching phases
