Bird J. Electrical Circuit Theory and Technology
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948 Electrical Circuit Theory and Technology
D
V/L
R
2L
2
1
LC
R
2L
2
1
LC
s C
R
2L
2
R
2L
2
1
LC
2
and current i D L
−1
fI.s/g
=
V
L
R
2L
2
−
1
LC
e
−.Rt=2L/
sinh
R
2L
2
−
1
LC
t
from 15 of Table 45.1
This curve is shown as curve D in Figure 45.28.
Problem 37. An L –R–C series circuit contains a coil of induc-
tance 1 H and resistance 8 and a capacitor of capacitance 50
µF.
Assuming current i D 0 at time t D 0, determine (a) the state of
damping in the circuit, and (b) an expression for the current when
a step voltage of 10 V is applied to the circuit.
(a)
R
2L
2
D
8
21
2
D 16 and
1
LC
D
1
150 ð 10
6
D 20000
Since
R
2L
2
<
1
LC
the circuit is underdamped
(b) When
R
2L
2
<
1
LC
, equation (45.41) applies,
i.e., i D
V/L
1
LC
R
2L
2
e
Rt/2L
sin
1
LC
R
2L
2
t
D
10/1
p
20000 16
e
4t
sin
p
20000 16t
i.e., i D 0.0707e
−4t
sin141.4t A
Problem 38. Values of R, L and C in a series R–L –C circuit are
R D 100 , L D 423 mH and C D 169.2
µF. A step voltage of 2 V
is applied to the circuit. Assuming current i D 0 at the instant of
applying the step, determine (a) the state of damping, and (b) an
expression for current i.
Transients and Laplace transforms 949
(a)
R
2L
2
D
100
20.423
2
D 13972
and
1
LC
D
1
0.423169.2 ð 10
6
D 13972
Since
R
2L
2
D
1
LC
the circuit is critically damped.
(b) From equation (45.42), current
i D
V
L
te
Rt/2L
D
2
0.423
te
100/20.423t
i.e., i D 4.73te
−118.2t
A
45.9 Initial conditions
In an L–R–C c
ircuit it is possible, at time
t D 0, for an inductor to carry
a current or a capacitor to possess a charge.
(a) For an inductor:
v
L
D L
di
dt
The Laplace Transform of this equation is:
v
L
s D L[sIs i0]
If, say, i0 D I, then
v
L
s D sLIs LI
0
45.43
The p.d. across the inductor in the s-domain is given by: (sL)I(s)
Equation (45.43) would appear to comprise two series elements,
i.e.,
v
L
s D (p.d. across L) C (voltage generator of LI
0
An inductor can thus be considered as an impedance sL in series with
an independent voltage source of LI
0
as shown in Figure 45.29.
I(s)
sL
LI
0
+
−
Series equivalent circuit for
an inductor in the s-domain
Figure 45.29
Transposing equation (45.43) for I(s) gives:
Is D
v
L
s C LI
0
sL
D
v
L
s
sL
C
I
0
s
Thus, alternatively, an inductor can be considered to be an impedance sL
in parallel with an independent current source I
0
/s
i.e., Is D current through L C
current source
I
0
s
sL
I(s)
+−
I
0
s
Parallel equivalent circuit for
an inductor in the s-domain
Figure 45.30
as shown in Figure 45.30.
Problem 39. An L –R series circuit has a step voltage V applied
to its input terminals. If after a period of time the step voltage is
removed and replaced by a short-circuit, determine the expression
for the current transient.
The L –R circuit with a step voltage applied to the input is shown in
Figure 45.31.
A
L
V
B
R
i
Figure 45.31
950 Electrical Circuit Theory and Technology
A
L
B
R
Figure 45.32
Using Kirchhoff’s voltage law:
V D iR C L
di
dt
45.44
If the step voltage is removed the circuit of Figure 45.32 results.
The s-domain circuit is shown in Figure 45.33 where the inductor is
considered as an impedance sL in series with a voltage source LI
0
with
it’s direction as shown.
If V D 0 in equation (45.44) then 0 D iR C L
di
dt
i.e., 0 D IsR C L[sIs I
0
]
D IsR C sLIs LI
0
which verifies Figure 45.33
In this case I
0
D V/R, the steady state current before the step voltage
was removed.
Hence 0 D IsR C sLIs L
V
R
i.e.,
LV
R
D IsR C sL
and Is D
VL/R
R C sL
D
VL/R
LsC R/L
D
V/R
s C R/L
sL
R
I(s)
LI
0
Figure 45.33
Hence current i D
L
1
V/R
s C R/L
D
V
R
e
−.Rt=L/
(b) For a capacitor: i D C
d
v
dt
The Laplace transform of this equation is:
Is D C[sVs
v0]
If, say,
v0 D V
0
then Is D CsVs CV
0
45.45
Rearranging gives: CsVs D Is C CV
0
from which, Vs D
Is
Cs
C
CV
0
Cs
D
1
sC
Is C
V
0
s
i.e., Vs D (p.d. across capacitor) C
voltage source
V
0
s
(45.46)
as shown in Figure 45.34.
I(s)
+
−
1
SC
V
0
s
Series equivalent circuit for
a capacitor in the s-domain
Figure 45.34
Thus the equivalent circuit in the s-domain for a capacitor with an
initial voltage V
0
is a capacitor with impedance 1/sC in series with an
impedance source V
0
/s Alternatively. From equation (45.45),
Transients and Laplace transforms 951
+
−
CV
0
1
sC
Parallel equivalant circuit for
a capacitor in the s-domain
I
(
s
)
Figure 45.35
Is D sC Vs CV
0
D (current through C) C (a current source of CV
0
as shown in Figure 45.35.
Problem 40. A C–R series circuit is shown in Figure 45.36. The
capacitor C is charged to a p.d. of V
0
when it is suddenly discharged
through the resistor R. Deduce how the current i and the voltage
v
vary with time.
The s-domain equivalent circuit, from equation (45.46), is shown in
Figure 45.37.
C
i
R
v
Figure 45.36
Applying Kirchhoff’s voltage law:
V
0
s
D Is
1
sC
C RIs
D Is
R C
1
sC
1
sC
V
0
s
R
V
(
s
)
I(s)
Figure 45.37
from which,
Is D
V
0
/s
R C
1
sC
D
V
0
s
R C
1
sC
D
V
0
sR C
1
C
D
V
0
R
s C
1
RC
and current i D L
1
fIsgD
V
0
R
L
1
1
s C
1
RC
D
V
0
R
e
t/CR
i.e., i D
V
0
R
e
.−t=CR/
Since v D iR, then v D
V
0
R
e
t
CR
R
i.e.,
v D V
0
e
.−t=CR/
50 V
x
y
i
1 kΩ
2 µF
Figure 45.38
Problem 41. Derive an equation for current i flowing through the
1k resistor in Figure 45.38 when the switch is moved from x to
y. Assume that the switch has been in position x for some time.
The 2 µF capacitor will have become fully charged to 50 V after a period
of time. When the switch is changed from x to y the charged capacitor can
be considered to be a voltage generator of voltage 50/s.Thes-domain
circuit is shown in Figure 45.39.
Applying Kirchhoff’s voltage law in the s-domain gives:
50
s
D Is
10
3
C
1
2 ð 10
6
s
50
s
1
2 × 10
−6
s
I(s)
1 kΩ
Figure 45.39
952 Electrical Circuit Theory and Technology
from which, Is D
50/s
10
3
C
5 ð 10
5
s
D
50
10
3
s C 5 ð 10
5
D
50/10
3
10
3
s
10
3
C
5 ð 10
5
10
3
D
0.05
s C 500
D 0.05
1
s C 500
Hence current,i = 0.05e
−500t
A
Further problems on circuit analysis using Laplace transforms may be
found in Section 45.10 following, Problems 40 to 55, page 955.
45.10 Further problems
on transients and Laplace
transforms
Response of R–C, R–L and L–R–C series circuits
1A5
µF capacitor is connected in series with a 20 k resistor and the
circuit is connected to a 10 V d.c. supply. Determine (a) the initial
value of current flowing, (b) the value of the current 0.4 s after connec-
tion, (c) the value of the capacitor voltage 30 ms after connection, and
(d) the time after connection when the resistor voltage is 4 V.
[(a) 0.5 mA (b) 9.16
µA (c) 2.59 V (d) 91.63 ms]
2 A 100 V d.c. supply is connected across a 400 nF capacitor as shown
in Figure 45.40. When the switch S is opened the capacitor is left
isolated except for a parallel resistor of 50 M. Determine the p.d.
across the capacitor 5 s after opening the switch. [8.21 V]
100 V
S
5 MΩ
400 nF
Figure 45.40
3 A 40 V d.c. supply is connected across a coil of inductance 25 mH
and resistance 5 . Calculate (a) the final value of current, (b) the
value of current after 10 ms, (c) the resistor p.d. across the resistor
after 5 ms, (d) the value of the voltage across the inductance after
2 ms, and (e) the time when the current reaches 3 A.
[(a)8A(b)
6.92 A (c) 25.28 V (d) 26.81 V (e) 2.35 ms]
4 In the circuit shown in Figure 45.41 a current of 2 A flows from the
source. If the switch S is suddenly opened, calculate (a) the time for
the current in the 0.5 H inductor to fall to 0.8 A, and (b) the maximum
voltage across the resistor. [(a) 114.5 ms (b) 8 V]
2 A
S
4 Ω
0.5 H
E
Figure 45.41
Transients and Laplace transforms 953
5 In a series L –R –C circuit the inductance, L D 5 mH and the resis-
tance R D 5k. Determine whether the circuit is over, critical or
under damped when (a) capacitance C D 500 pF, and (b) C D 10
µF.
[(a) underdamped (b) overdamped]
6 For the circuit in Problem 5 calculate the value of capacitance C for
critical damping. [800 pF]
7 A coil having an equivalent circuit of inductance 1 H in series with
resistance 50 is connected across a fully charged 0.4
µF capacitor
at the instant when the capacitor voltage is 20 V. Determine the nature
of the response and obtain an expression for the current in the coil.
[underdamped; i D0.0127e
25t
sin1580.9 t A]
8 If the coil in Problem 7 had a resistance of 500 and the capaci-
tance was 16
µF, determine the nature of the response and obtain an
expression for the current in the coil.
[critically damped; i D 5000t C 20e
250t
A]
Laplace transforms
Determine the Laplace transforms in Problems 9 to 15.
9 (a) 2t 3 (b) 5t
2
C 4t 3
(a)
2
s
2
3
s
(b)
10
s
3
C
4
s
2
3
s
10 (a)
t
3
24
3t C 2(b)
t
5
15
2t
4
C
t
2
2
(a)
1
4s
4
3
s
2
C
2
s
(b)
8
s
6
48
s
5
C
1
s
3
11 (a) 5e
3t
(b) 2e
2t
(a)
5
s 3
(b)
2
s C 2
12 (a) 4 sin 3t (b) 3 cos2t
(a)
12
s
2
C 9
(b)
3s
s
2
C 4
13 (a) 2te
2t
(b) t
2
e
t
(a)
2
s 2
2
(b)
2
s 1
3
14 (a) 4t
3
e
2t
(b)
1
2
t
4
e
3t
(a)
24
s C 2
4
(b)
12
s C 3
5
15 (a) 5e
2t
cos3t (b) 4e
5t
sint
(a)
5s C 2
s
2
C 4s C 13
(b)
4
s
2
C 10s C 26
16 Determine the Laplace transforms of the following waveforms:
(a) a step voltage of 4 V which starts at time t D 0
(b) a step voltage of 5 V which starts at time t D 2s
(c) a ramp voltage which starts at zero and increases at 7 V/s
954 Electrical Circuit Theory and Technology
(d) a ramp voltage which starts at time t D 2 s and increases at
3 V/s.
(a)
4
s
(b)
5
s
e
2s
(c)
7
s
2
(d)
3
s
2
e
2s
17 Determine the Laplace transforms of the following waveforms:
(a) an impulse voltage of 15 V which starts at time t D 0
(b) an impulse voltage o
f 6 V which starts at time
t D 5
(c) a sinusoidal current of 10 A and angular frequency 8 rad/s.
(a) 15 (b) 6e
5s
(c)
80
s
2
C 64
18 State the initial value theorem. Verify the theorem for the functions:
(a) 3 4 sin t
(b) t 4
2
and state their initial values. [(a) 3 (b) 16]
19 State the final value theorem and state a practical application where
it is of use. Verify the theorem for the function 4 C e
2t
sin t C cos t
representing a displacement and state it’s final value. [4]
Inverse Laplace transforms and the solution of differential equations
Determine the inverse Laplace transforms in Problems 20 to 27
20 (a)
7
s
(b)
2
s 5
[(a) 7 (b) 2e
5t
]
21 (a)
3
2s C 1
(b)
2s
s
2
C 4
(a)
3
2
e
1/2t
(b) 2 cos2t
22 (a)
1
s
2
C 25
(b)
4
s
2
C 9
(a)
1
5
sin5t (b)
4
3
sin3t
23 (a)
5s
2s
2
C 18
(b)
6
s
2
(a)
5
2
cos3t (b) 6t
24 (a)
5
s
3
(b)
8
s
4
(a)
5
2
t
2
(b)
4
3
t
3
25 (a)
15
3s
2
27
(b)
4
s 1
3
(a)
5
3
sinh3t (b) 2e
t
t
2
26 (a)
3
s
2
4s C 13
(b)
4
2s
2
8s C 10
[(a) e
2t
sin3t (b) 2e
2t
sint]
27 (a)
s C 1
s
2
C 2s C 10
(b)
3
s
2
C 6s C 13
(a) e
t
cos3t (b)
3
2
e
3t
sin2t
Use partial fractions to find the inverse Laplace transforms of the functions
in Problems 28 to 33.
Transients and Laplace transforms 955
28
11 3s
s
2
C 2s 3
[2e
t
5e
3t
]
29
2s
2
9s 35
s C 1s 2s C 3
[4e
t
3e
2t
C e
3t
]
30
2s C 3
s 2
2
[2e
2t
C 7te
2t
]
31
5s
2
2s 19
s C 3s 1
2
[2e
3t
C 3e
t
4te
t
]
32
3s
2
C 16s C 15
s C 3
3
[e
3t
3 2t 3t
2
]
33
26 s
2
ss
2
C 4s C 13
2 3e
2t
cos3t
2
3
e
2t
sin3t
In Problems 34 to 39, use Laplace transforms to solve the given differ-
ential equations.
34 9
d
2
y
dt
2
24
dy
dt
C 16y D 0, given y0 D 3andy
0
0 D 3
[y D 3 te
4/3t
]
35
d
2
x
dt
2
C 100x D 0, given x0 D 2andx
0
0 D 0[x D 2 cos 10t]
36
d
2
i
dt
2
C 1000
di
dt
C 250000i D 0, given i0 D 0andi
0
0 D 100
[i D 100te
500t
]
37
d
2
x
dt
2
C 6
dx
dt
C 8x D 0, given x0 D 4andx
0
0 D 8
[x D 43e
2t
2e
4t
]
38
d
2
y
dt
2
2
dy
dt
C y D 3e
4t
, given y0 D
2
3
and y
0
0 D 4
1
3
[y D 4t 1e
t
C
1
3
e
4t
]
39
d
2
y
dt
2
C 16y D 10 cos 4t, given y0 D 3andy
0
0 D 4
[y D 3cos4t C sin4t C
5
4
t sin4t]
Laplace transform analysis of circuit diagrams
40 Determine the impedance of a 2000 pF capacitor in the s-domain.
1
2 ð 10
9
s
or
5 ð 10
8
s
41 Determine the impedance of a 0.4 H inductor in series with a 50
resistor in the s-domain. [50 C0.4s ]
956 Electrical Circuit Theory and Technology
42 Determine the circuit impedance in the s-domain for the following:
(a) a resistor of 100 in series with a 1
µF capacitor
(b) an inductance of 10 mH, a resistance of 500 and a capaci-
tance of 400 nF in series
(c) a 10 resistance in parallel with a 10 mH inductor
(d) a 10 mH inductor in parallel with a 1
µF capacitor
(a)
100 C
10
6
s
(b)
500 C 0.01s C
10
7
4s
(c)
0.1s
10 C 0.01s
or
10s
s C 1000
(d)
10
4
s
0.01s
2
C 10
6
or
10
2
s
1 C 10
8
s
2
43 An L –R–C network comprises a 20 resistor, a 20 mH inductor
anda20
µF capacitor. Determine in the s-domain (a) the impedance
when the components are connected in series, and (b) the admittance
when the components are connected in parallel.
(a)
20 C 0.02s C
1
2 ð 10
5
s
(b) 0.05 C
50
s
C 2 ð 10
5
sS
44 A circuit consists of a 0.5 M resistor in series with a 0.5 µF capac-
itor. Determine how the voltage across the capacitor varies with time
when there is a step voltage input of 5 V. Assume the initial condi-
tions are zero. [
v
C
D 51 e
4t
volts]
45 An exponential voltage, V D 20e
50t
volts is applied to a series R –L
circuit, where R D 10 and L D 0.1 H. If the initial conditions are
zero, find the resulting current. [i D 4e
50t
4e
100t
A]
46 If in Problem 45 a supply of 20.2sin10t C , volts is applied to the
circuit find the resulting current. Assume the circuit is switched on
when , D 0. [i D 2 sin 10t 0.2 cos10t C 0.2e
100t
]
47 An R–C series network has (a) a step input voltage E volts, and (b) a
ramp voltage E volts/s, applied to the input. Use Laplace transforms
to determine expressions for the current flowing in each case. Assume
the capacitor is initially uncharged.
(a)
E
R
e
t/CR
(b) EC1 e
t/CR
48 An R–L series network has (a) a step input of E volts, (b) a ramp
input of 1 V/s, applied across it. Use Laplace transforms to develop
expressions for the voltage across the inductance L in each case.
Transients and Laplace transforms 957
Assume that at time t D 0, current i D 0.
(a) Ee
Rt/L
(b)
L
R
1 e
Rt/L
49 A sinusoidal voltage 5sint volts is applied to a series R –L circuit.
Assuming that at time t D 0, current i D 0 derive an expression for
the current flowing.
i D
5
R
2
C L
2
/
Le
Rt/L
L cost C R sint
0
50 Derive an expression for the growth of current through an inductive
coil of resistance 20 and inductance 2 H using Laplace transforms
when a d.c. voltage of 30 V is suddenly applied to the coil.
[i D 1.5 1.5e
10t
]
51 For the circuit shown in Figure 45.42, derive an equation to represent
the current i flowing. Assume zero conditions when the switch is
closed. Is the circuit over, critical or under damped?
[i D 4.47e
10t
sin447t mA; underdamped]
10 Ω 0.5 H 10 µF
i
1 V
Figure 45.42
52 If for the circuit of Problem 51, R D 100 , L D 0.5HandC D
200
µF, derive an equation to represent current.
[i D 2te
100t
A]
53 If for the circuit of Problem 51, R D 1k,LD 0.5H and C D
200
µF derive an equation to represent current.
[2.01e
1000t
sinh 995t A]
54 In a C–R series circuit the 5
µF capacitor is charged to a p.d. of
100 V. It is then suddenly discharged through a 1 k resistor. Deter-
mine, after 10 ms (a) the value of the current, and (b) the voltage
across the resistor.
[(a) 13.53 mA (b) 13.53 V]
55 In the circuit shown in Figure 45.43 the switch has been connected
to point a for some time. It is then suddenly switched to point b.
Derive an expression for current i flowing through the 20 resistor.
[i D 5e
10
6
t
A]
100 V
ab
50 nF
20 Ω
Figure 45.43