|
t
= 0
t
= 0.12 s
10 V
5
Ω
5
Ω
0.5 H
S
1
S
2
–
Fig. 10.11-8
20. Initial current at t
= -∞
in the inductor in the circuit in Fig. 10.11-9 was zero. Find i
L
(t) and v
L
(t)
for t
≥
0
+
and plot them.
2
u
(–
t
) V
3
u
(
t
) A
3
Ω
6
Ω
0.5 H
+
+
–
–
i
L
v
L
Fig. 10.11-9
10.62
First-Order
RL
Circuits
21. The switch S in Fig. 10.11-10 is a two-position switch and starts at position-1 at t
=
0. It is kept
in that position for 10ms and then thrown to position-2. It is kept at 2
nd
position till the current in
the inductor goes to zero. At that instant, it is thrown back to position-1. Then, the whole cycle
repeats. (i) Calculate and plot two cycles of i
L
(t), i
S1
(t) and i
S2
(t). (ii) What is the frequency of
switching in the circuit? (iii) What are the average currents in the two sources and inductance?
(iv) What are the average power delivered to the 24V source and the average power supplied by
12V source. (v) If the idea was to charge the 24V battery from 12V battery, what is the efficiency
of this charger? (vi) Suggest a method to control the average charging current in the 24V battery.
12 V
1
2
0.1
Ω
5 mH
24 V
S
+
+
–
–
i
L
i
S1
i
S2
Fig. 10.11-10
22. Repeat the steps in Problem 3 with the circuit in Fig. 10.11-11.
12 V
1
2
0.1
Ω
5 mH
24 V
S
+
+
–
–
i
L
i
S1
i
S2
Fig. 10.11-11
23. The switch S in Fig. 10.11-12 starts at position-1 at t
=
0. It is kept in that position for 10ms and
then thrown to position-2. It is kept at 2
nd
position till the current in the inductor goes to zero.
At that instant, it is thrown back to position-1. Then, the whole cycle repeats.
24 V
1
2
0.1
Ω
5 mH
12 V
+
+
–
–
i
L
i
S1
i
S2
Fig. 10.11-12
(i) Calculate and plot two cycles of i
S1
(t) and i
S2
(t). (ii) What is the frequency of switching in the
circuit? (iii) What are the average currents in the two sources? (iv) What are the average power
delivered to the 12V source and the average power supplied by 24V source? (v) If the idea was
to charge the 12V battery from 24V battery, what is the efficiency of this charger? (vi) Suggest
a method to control the average charging current in the 12V battery.
24. The switch in the circuit in Fig. 10.11-13 was closed for a long time and is opened at t
=
0. Find
and plot the current in inductance and voltage across it as functions of time.
Problems
10.63
+
–
12 V
1 A
10 mH
t
= 0
12
Ω
10
Ω
Fig. 10.11-13
25. Find the impulse response of the voltage variable v(t) in the circuit in Fig. 10.11-14.
δ
(
t
)
0.05 H
+
–
+
–
10
Ω
10
Ω
10
Ω
v
5
Ω
Fig. 10.11-14
26. (i) Find the zero-input response, zero-state response and total response for i
L
(t) and v(t) in the
circuit in Fig. 10.11-15. (ii) Obtain i
L
(t) and v(t) if the current source on the right side is made 2 u(t).
i
L
+
–
5
Ω
u
(
t
) A
–
u
(
t
) A
5
Ω
0.1 H
v
25
Ω
Fig. 10.11-15
27. What must be the value of k in the circuit in Fig. 10.11-16 if v(t)
=
0 for t
≥
0
+
?
+
i
L
–
+
–
+
–
5
Ω
k
δ
(
t
)
3
u
(
t
)
5
Ω
0.1 H
v
5
Ω
Fig. 10.11-16
28. The switch S in the circuit in Fig. 10.11-17 operates cyclically at 10kHz spending equal time
at both positions in a cycle. Estimate the average power delivered by the current source and the
average power dissipated in the resistor. Averages are over switching cycles.
S
2 A
1
2
10 mH
1 k
Ω
Fig. 10.11-17
10.64
First-Order
RL
Circuits
29. In Fig. 10.11-18 Vac
=
10sin (314t) V. (i) Find the steady-state output voltage waveform for v
o
(t)
and plot it. (ii) What is the percentage peak-peak ripple with respect to average value in the output
voltage? (iii) Repeat (ii) if inductance is changed to 2H. (iv) Justify the following statement –
“The steady-state output voltage across resistance in an R-L circuit will be more or less constant at
the average value of driving voltage even if the driving voltage has a.c components if the product
w
L >> R where
w
is the angular frequency of AC components.”
+
+
10 V
0.5 H
20
Ω
L
R
v
O
V
ac
–
–
Fig. 10.11-18
30. Let V
s
(t) an arbitrary time varying periodic voltage source with a cycle average value of V
dc
.
This means that V
s
(t) can be written as V
dc
+
V
ac
(t) where V
ac
(t) is a time varying periodic
component with equal positive half-cycle and negative half-cycle areas. Let that area be A V-s.
This waveform V
s
(t) is applied to a series R-L circuit and output voltage is taken across R.
Assume that L/R >> T where T is the period of V
s
(t). Show that under periodic steady-state the
(i) average value of output voltage is Vdc (ii) the peak-to-peak ripple in output voltage
≈
A/
t
V
where
t
=
L/R. (iii) calculate the quantities in (i) and (ii) for the three inputs given in Fig. 10.11-19
if
t
is 25ms.
100 V
t
in ms
0.8
1
1 8 2
–100 V
2ms
1ms
t
in ms
100 V
sine wave
1ms
t
in ms
100 V
Fig. 10.11-19
31. Two series RL circuits are connected in cascade using a unity gain buffer amplifier as in Fig.
10.11-20. A buffer amplifier is an electronic amplifier that presents infinite resistance at its input
and behaves like an ideal voltage at its output. With a unity gain, its output voltage is same as
input voltage. Buffer amplifiers are used to interconnect circuits that will interact with each other
otherwise. The initial current in the inductor of first stage circuit is 0.5 A and that of second stage
is 2 A. Find v
o
(t) for t
≥
0
+
.
Buffer
Amp
0.02 H
20
Ω
10
Ω
v
O
(
t
)
0.02 H
10
u
(
t
)
+
+
–
–
Fig. 10.11-20
11.1
F i r s t - O r d e r
RC
C i r c u i t s
CHAPTEROBJECTIVES
• Impulse,Step and Ramp Response of First-Order RC Circuits
• Series RC Circuit with Real Exponential Input
• Zero-State Response of Parallel RC Circuit for Sinusoidal Input
• The Use of Frequency Response and Linear Distortion
• First-Order RC Circuits as Averaging Circuits
• Capacitor as a Signal Coupling and Signal Bypassing Element
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