204
Inductance, Capacitance, and Mutual Inductance
where VI and i l are the voltage and current in circuit 1,
and Vz and iz are the voltage and current in circuit 2. For
coils wound on nonmagnetic cores, M 12 = M ZI = M
(See page 190.)
• The relationship between the self-inductance of each
winding and the mutual inductance between windings is
• The dot convention establishes the polarity of mutually
induced voltages:
The coefficient of coupling, k, is a measure of the degree
of magnetic coupling. By definition, 0 :::; k :::; 1. (See
page 197.)
When the reference direction for a current enters
the dotted terminal of a coil, the reference polar­
ity of the voltage that it induces in the other coil
is positive at its dotted terminal.
Or, alternatively,
When the reference direction for a current leaves
the dotted terminal of a coil, the reference polar­
ity of the voltage that it induces in the other coil
is negative at its dotted terminal.
• The energy stored in magnetically coupled coils in a lin­
ear medium is related to the coil currents and induc­
tances by the relationship
(See page 199.)
(See page 190.)
Problems
- - - - - - - _ . _ . _ - _ ..•
Section 6.1
6.1 The current in the 2.5 mH inductor in Fig. P6.1 is
known to be 1 A for l < O. The inductor voltage for
t 2: 0 is given by the expression
VL(t)
=
VL(t)
=
3e-4t mY,
6.3 The voltage at the terminals of the 200 M-H inductor
in Fig. P6.3(a) is shown in Fig. P6.3(b). The inductor
~IUlTIS[f·, current i is known to be zero for t :::; O.
PSP[CE
a) Derive the expressions for i for t
b) Sketch i versus t for 0 :::; t :::;
0+:::; t :::; 2 s
-3e-4('-Z) mY,
2 s :::; t <
Sketch VL(l) and iL(t) for 0 :::; t <
2:
O.
00.
Figure P6.3
00
Vs
i
00.
(mY)
~
v.~ 200~HI
Figure P6.1
51----,
o
(a)
2.5mB
2
t (ms)
(b)
6.4 The triangular current pulse shown in Fig. P6.4 is
applied to a 20 mH inductor.
6.2 The current in a 50 M-H inductor is known to be
PSP[CE
iL
~lUlTIS[M
=
18le-
JOt
A
for t
2:
PSP!CE
1·1ULTISH-1
O.
a) Find the voltage across the inductor for t > O.
(Assume the passive sign convention.)
b) Find the power (in microwatts) at the terminals
of the inductor when t = 200 ms.
c) Is the inductor absorbing or delivering power at
200 ms?
d) Find the energy (in microjoules) stored in the
inductor at 200 ms.
e) Find the maximum energy (in microjoules)
stored in the inductor and the time (in milli­
seconds) when it Occurs.
MUlTIS[~1
O. Assume the passive sign convention.
a) At what instant of time is the voltage across the
inductor maximum?
b) What is the maximum voltage?
for t
2:
6.9 a) Find the inductor cunent in the circuit in
Fig. P6.9 if V = -50 sin 250 t Y, L = 20 mH,
MULTlSm
and i(O) = 10 A.
PSPICE
a) Write the expressions that describe i(t) in
the four intervals t < 0, 0:::; t :::; 5 illS,
5 ms :::; t :::; 10 ms, and t > 10 ms.
b) Derive the expressions for the inductor volt­
age, power, and energy. Use the passive sign
convention.
Figure P6.4
i(~)
250
....
1
o
5
10
­
t (ms)
test at 10 and 30 J.LS.
c) Check the energy expression within each interval
by selecting a time within the interval and seeing
whether the energy equation gives the same
result as ~Cvz. Use 10 and 30 J.Ls as test points.
6.13 Initially there was no energy stored in the 5 H
inductor in the circuit in Fig. P6.13 when it was
placed across the terminals of the voltmeter. At
-
206
Inductance, Capacitance, and l>1utual Inductance
= 0 the inductor was switched instantaneously to
position b where it remained for 1.6 s before returning
instantaneously to position a. The d'Arsonval volt­
meter has a full-scale reading of 20 V and a sensitivity
of 1000 D/V. What will the reading of the voltmeter
be at the instant the switch returns to position a if the
inertia of the d'Arsonval movement is negligible?
t
6.16 The rectangular-shaped current pulse shown in
Fig. P6.16 is applied to a 0.1 fLF capacitor. The ini­
(,lUlTlSII·1
tial voltage on the capacitor is a 15 V drop in the
reference direction of the current. Derive the
expression for the capacitor voltage for the time
intervals in (a)-(d),
PSPICE
a) 0
Figure P6.13
b
:5 I :5
10 Vs;
b) 10 fLs
:5
t :5 20 fLS;
c) 20 fLS
:5
t
:5
40 VS
d) 40 fLs
:5 t
<
00
e) Sketch vet) over the interval -10 fLS
:5 ( :5
50 fLS.
3rnV
Figure P6.16
(rnA)
160
-
Section 6.2
100 -
6.14 The current shown in Fig. P6.l4 is applied to a
PSPICE
0.25 fLF capacitor. The initial voltage on the capaci­
1·1UlTISlf.I
tor is zero.
a) Find the charge on the capacitor at t = 15 fLs.
0
-50
b) Finel the voltage on the capacitor at t = 30 fLS.
1,0
I
I
20
30
40
c) How much energy is stored in the capacitor by
this current?
Figure P6.14
i (rnA)
400
6.17 A 20 fLF capacitor is subjected to a voltage pulse
having a duration of 1 s.111e pulse is described by
the following equations:
--o-+---'--------'-------l---'--------L---=-'--- t (f.Ls)
-300 - - - - - ­
vcCt)
6.15 The initial voltage on the 0.5 fLF capacitor shown in
Fig. P6.15(a) is -20 V. The capacitor current has
(·IUln5IM
the waveform shown in Fig. P6.15(b).
PSPICE
a) How much energy, in microjoules, is stored in
the capacitor at t = 500 fLS?
b) Repeat (a) for t =
Figure P6.15
I(
-20V
•
~O(t - 1)2 V,
0,5 s :5 ( :5 1 s:
elsewhere.
6.18 The voltage across the terminals of a 0.2 fLF capaci­
tor is
PSPICE
l·lUlTlSII·\
50e-2000t rnA, ( 2: 0
25
--o+----I..-----L..-----L---l--.L....-
~-
"
(a)
100
200
t :5 0.5 s;
Sketch the current pulse that exists in the capacitor
during the 1 s interval.
(rnA)
50
•
{
o :5
00.
i
0.5 f.LF
=
30(2 V,
300
(b)
400
500
t (f.Ls)
( :5
0;
t
O.
2:
Problems
The initial current in the capacitor is 250 rnA.
Assume the passive sign convention.
a) What is the initial energy stored in the capacitor?
6.23 The three inductors in the circuit in Fig. P6.23 are con­
nected across the terminals of a black box at t = O.
1·IULTlSi/·!
The resulting voltage for t > is known to be
PSP!CE
a
b) Evaluate the coefficients A 1 and A z.
Vo
c) What is the expression for the capacitor current?
6.19 The voltage at the terminals of the capacitor m
PSPICE
v
=
Fig. 6.10 is known to be
= -6 A
If i1 (0)
2000e- JOOI V.
=
and iz(O)
-20 Y,
{ 100 - 40e-20001(3 cos IOOOt + sin IOOOt) Y
t :5 0;
t 2: O.
b) io(t),t
2:
0;
c) i1(t), t
2:
0;
d) iz(t), t
2:
0;
Assume C = 4 fLF.
e) the initial energy stored in the three inductors;
a) Find the current in the capacitor for t < O.
f) the total energy delivered to the black box; and
b) Find the current in the capacitor for t > O.
g) the energy trapped in the ideal inductors.
Figure P6.23
d) Is there an instantaneous change in the current
in the capacitor at t = O?
e) How much energy (in millijoules) is stored in
the capacitor atl = oo?
l
1
lH
._.
[ -= ()
. 1)
h
~4 H
r
Section 6.3
6.20 Assume that the initial energy stored in the induc­
I
tors of Fig. P6.20 is zero. Find the equivalent induc­
tance with respect to the terminals a,b.
Figure P6.20
14 H
5H
a_----f"Y-yy""-----~---"ry-o,"----~-__,
15 H
60H
80H
30H 20H
6.21 Assume that the initial energy stored in the induc­
I"
1'(.
-
)
~mLnSH·l
= 1 A, find
a) io(O);
c) Is there an instantaneous change in the voltage
across the capacitor at 1 = O?
FSPICE
207
..
Black
box
~
~
3.2H
6.24 For the circuit shown in Fig. P6.23, how many milli­
seconds after the switch is opened is the energy
delivered to the black box 80% of the total energy
delivered?
6.25 The two parallel inductors in Fig. P6.25 are con­
nected across the terminals of a black box at t = O.
The resulting voltage v for 1 > 0 is known to be
64e- 41 Y. It is also known that il(O) = -10 A and
iz(O) = 5 A.
tors of Fig. P6.21 is zero. Find the equivalent induc­
tance with respect to the terminals a,b.
a) Replace the original inductors with an equiva­
lent inductor and find i(t) for I 2: O.
Figure P6.21
b) Find i1 (t) for t
::2:
O.
c) Find i 2 (t) for t
2:
O.
3H
d) How much energy is delivered to the black box
in the time interval 0 :5 t < oo?
8H
e) How much energy was initially stored in the par­
allel inductors?
f) How much energy is trapped in the ideal inductors?
b_---~.../V'yy-,~e---~
6.22 Use realistic inductor values from Appendix H to con­
struct series and parallel combinations of inductors to
yield the equivalent inductances specified below. Try
to minimize the number of inductors used. Assume
that no initial energy is stored in any of the inductors.
a) 3 mH
b) 250 fLH
c) 60 fLH
g) Show that your solutions for i, and i z agree with
the answer obtained in (f).
Figure P6.25
..
i( I)
I'
Black
box
208
Inductance, Capacitance, and Mutual Inductance
6.26 Find the equivalent capacitance with respect to the
terminals a,b for the circuit shown in Fig. P6.26.
6.30 For the circuit in Fig. P6.29, calculate
a) the initial energy stored in the capacitors;
b) the final energy stored in the capacitors;
Figure P6.26
5 fJ.F
30 fJ.F
'-----:d t-rjo
c) the total energy delivered to the black box;
V -
d) the percentage of the initial energy stored that is
delivered to the black box; and
_
e) the time, in milliseconds, it takes to deliver
7.5 mJ to the black box.
4 fJ.F
48 fJ.F
b----1
- 30Y +
- OY +
6.27 Find the equivalent capacitance with respect to the
terminals a,b for the circuit shown in Fig. P6.27.
Figure P6.27
6.31 The two series-connected capacitors in Fig. P6.31
are connected to the terminals of a black box at
t = O. The resulting current i(t) for ( > 0 is known
r
to be 800e--~1
/LA.
a) Replace the original capacitors with an equiva­
lent capacitor and find vo(t) for t ~ O.
b) Find VI(t) fort
~
c) Find V2(t) for t ~
O.
O.
d) How much energy is delivered to the black box
in the time interval 0 :5 t < oo?
e) How much energy was initially stored in the
series capacitors?
f) Hoyv much energy is trapped in the ideal capacitors?
g) Show that the solutions for
the answer obtained in (f).
Figure P6.31
6.28 Use realistic capacitor values from Appendix H to
construct series and parallel combinations of capac­
itors to yield the equivalent capacitances specified
below. Try to minimize the number of capacitors
used. Assume that no initial energy is stored in any
of the capacitors.
b) 750 nF
and V2 agree with
..
i(r)
--­
S:r MF
2/Y~8/.LF
a) 330/J.P
VI
[=
0
VI
+
Co
Co
+
c) 150 pF
6.29 The four capacitors in the circuit in Fig. P6.29 are con­
nected across the terminals of a black box at ( = O.
The resulting current i b for t > 0 is known to be
ib =
-
5e- 501 mAo
If
v,,(O) = -20 V,
vc(O) = -30 V,
and
Vd(O) = 250 V, find the following for ( ~ 0: (a) Vb(t),
(b) v,,(t), (c) vc(t), (d) Vd(t), (e) i1(t), and (f) i 2(t).
Figure P6.29
,
(=
0
-
+
Black
box
Vh
I',.
L -_ _---j
+
1.25/.LF
T
6.32 Derive the equivalent circuit for a series connection
of ideal capacitors. Assume that each capacitor has
its own initi.al voltage. Denote these initial voltages
as Vj(to), V2(tO), and so on. (Hint: Sum the voltages
across the string of capacitors, recognizing that the
series connection forces the current in each capaci­
tor to be the same.)
6.33 Derive the equivalent circuit for a parallel connec­
tion of ideal capacitors. Assume that the initial volt­
age across the paralleled capacitors is veto). (Hint:
Sum the currents into the string of capacitors, rec­
ognizing that the parallel connection forces the
voltage across each capacitor to be the same.)
Sections 6.1-6.3
6.34 The current in the circuit in Fig. P6.34 is known to be
+-­
If--­
\
Black
box
i u = 5e -20001 (2 cos 4000t
+ sin 4000t) A
for t ~ 0+. Find vJ(O+) and V2(0+).
Problems
Figure P6.34
209
c) Find the expression for the power developed by
the current source.
40,0,
d) How much power is the current source develop­
ing when t is infinite?
_1_
/'c j 10 rnH
;)
e) Calculate the power dissipated in each resistor
when t is infinite.
6.39 There is no energy stored in the circuit in Fig. P6.39
at the time the switch is opened.
6.35 At t = 0, a series-connected capacitor and induc­
tor are placed across the terminals of a black box,
as shown in Fig. P6.35. For t > 0, it is known that
io
If vc(O)
= 1.5e-16,OOOt
- 0.5e-4000t
-50 V find V o for t
=
2::
A.
O.
Figure P6.35
25mH
I
1',
= ()
625 nF
a) Derive the differential equation that governs
the behavior of i z if L 1 = 4 H, L z = 16 H,
M = 2 H, and R o = 32 n.
b) Show that when i g = 8 - 8e-t A, t 2:: 0, the dif­
ferential equation derived in (a) is satisfied
when iz = e- t - e-Zt A, t 2:: O.
c) Find the expression for the voltage Vl across the
current source.
d) What is the initial value of v]? Does this make
sense in terms of known circuit behavior?
+­
Figure P6.39
Black
box
I'"
L,
•
AIL'
•
'---------'---------'
Section 6.4
6.36 a) Show that the differential equations derived in
(a) of Example 6.6 can be rearranged as follows:
di]
.
di
.
4 - + 25, - 8 -z - 20,
dt
]
dt
z
.
dig
= 5,g - 8dt- ''
di]
di
- 8 - - 20i l + 16-z + 80i z
dt
dt
dig
dt
= 16- .
b) Show that the solutions faT ij, and i z given in
(b) of Example 6.6 satisfy the differential
equations given in part (a) of this problem.
6.37 Let va represent the voltage across the 16 H
inductor in the circuit in Fig. 6.25. Assume va is
positive at the dot. As in Example 6.6,
ig = 16 - 16e-5t A.
a) Can you find Va without having to differenti­
ate the expressions for the currents? Explain.
b) Derive the expression for va'
c) Check your answer in (b) using the appropri­
ate current derivatives and inductances.
6.38 Let v g represent the voltage across the current
source in the circuit in Fig. 6.25. The reference for
vg is positive at the upper terminal of the current
source.
a) Find v g as a function of time when
i g = 16 - 16e-5t A.
b) What is the initial value of v g ?
f;;) fRo
6.40 a) Show that the two coupled coils in Fig. P6.40 can
be replaced by a single coil having an inductance
of Lab = L] + L z + 2M. (Hint: Express Vab as a
function of i ab .)
b) Show that if the connections to the terminals
of the coil labeled Lz are reversed,
Lab = L] + L z - 2M.
Figure P6.40
6.41 a) Show that the two magnetically coupled coils in
Fig. P6.41 (see page 210) can be replaced by a
single coil having an inductance of
L
L Lz - M Z
L] + L z - 2M
--~]---ab -
(Hint: Let i] and iz be clockwise mesh currents in
the left and right "windows" of Fig. P6.41, respec­
tively. Sum the voltages around the two meshes.
In mesh 1 let Vab be the unspecified applied volt­
age. Solve for diJidt as a function of Vab')
b) Show that if the magnetic polarity of coil 2 is
reversed, then
L
_ _L
~l Lz
ab -
-
M _Z
L 1 + L z + 2M
210
Inductance, Capacitance, and Mutual Inductance
Figure P6.41
a
•
•
L 1 _ M -... L z
b
_---.--------.J
6.42 The polarity markings on two coils are to be deter­
mined experimentally. The experimental setup is
shown in Fig. P6.42. Assume that the terminal con­
nected to the negative terminal of the battery has
been given a polarity mark as shown. When the
switch is opened, the dc voltmeter kicks upscale.
Where should the polarity mark be placed on the
coil connected to the voltmeter?
Figure P6.42
I
between the coils is K The coupling medium is non­
magnetic. When coil 1 is excited with coil
2 open, the flux linking only coil 1 is 0.2 as large as the
flux linking coil 2.
a) How many turns does coil 2 have?
b) What is the value of \!P2 in nanowebers per
ampere?
c) What is the value of \!P ll
ampere?
+
•
6.43 The physical construction of four pairs of magneti­
cally coupled coils is shown in Fig. P6.43. (See
page 211.) Assume that the magnetic flux is confined
to the core material in each structure. Show two possi­
ble locations for the dot markings on each pair of coils.
nanowebers per
d) What is the ratio (cP22I¢j2)?
6.48 Two magnetically coupled coils are wound on a
nonmagnetic core. The self-inductance of coil 1 is
288 mH, the mutual inductance is 90 mH, the coeffi­
cient of coupling is 0.75, and the physical structure
of the coils is such that \!P 1l = \!P n .
a) Find
b) If N j
- ()
R
In
Lz and the turns ratio N 1/N 2.
=
1200, what is the value of \!P j and \!P2?
6.49 The self-inductances of two magnetically coupled coils
are L] = 180,uH and L 2 = 500,uH. The coupling
medium is nonmagnetic. If coil 1 has 300 turns and
coil 2 has 500 turns, find 0'J I and \!P 2t (in nanowebers
per ampere) when the coefficient of coupling is 0.6.
6.50 a) Starting with Eq. 6.59, show that the coefficient
of coupling can also be expressed as
Section 6.5
6.44 The self-inductances of the coils in Fig. 6.30 are
L] = 18 mH and L 2 = 32 mHo If the coefficient of
coupling is 0.85, calculate the energy stored in the
system in millijoules when (a) i 1 = 6 A, i 2 = 9 A;
(b) i 1 = -6 A, i 2 = -9 A; (c) i 1 = -6 A, i2 = 9 A;
and (d) i 1 = 6 A, i2 = -9A.
6.45 The coefficient of coupling in Problem 6.44 is
increased to 1.0.
a) If i 1 equals 6 A, what value of i2 results in zero
stored energy?
b) Is there any physically realizable value of i 2 that
can make the stored energy negative?
6.46 Two magnetically coupled coils have self-inductances
of 60 mH and 9.6 mH, respectively. The mutual induc­
tance between the coils is 22.8 mHo
b) On the basis of the fractions cP21l¢t and
explain why k is less than 1.0.
cP12N2,
Sections 6.1-6.5
6.51 Rework the Practical Perspective example, except
1'~~~~g,7E that this time, put the button on the bottom of the
divider circuit, as shown in Fig. P6.51. Calculate the
output voltage vet) when a finger is present.
Figure P6.51
Fixed
capacitor
v,.(t)
+
1'(11
a) What is the coefficient of coupling?
b) For these two coils, what is the largest value that
M can have?
c) Assume that the physical structure of these cou­
pled coils is such that \!p] = \!P2' What is the turns
ratio N J! N 2 if N 1 is the number of turns on the
60 mH coil?
6.47 The self-inductances of two magnetically coupled
coils are 72 mH and 40.5 mH, respectively. The 72 mH
coil has 250 turns, and the coefficient of coupling
6.52 Some lamps are made to turn on or off when the
is touched. These use a one-terminal variation
of the capacitive switch circuit discussed in the
Practical Perspective. Figure P6.52 shows a circuit
model of such a lamp. Calculate the change in the
voltage vet) when a person touches the lamp.
Assume all capacitors are initially discharged.
P~~~~~~~~Ebase
Problems
211
figure P6.43
t
\
4
(c)
3
,..
,.
"3
4
Figure P6.52
Figure P6.53
10 pF
Lamp Person 10 pF
~ ---1~
vs(t)
+
lOPFf' 100 PF
1
6.53 In the Practical Perspective example, we calculated
PP,:.cTlCAl the output voltage when the elevator button is the
PEP,5rtmvE
upper capacitor in a voltage divider. In
Problem 6.51, we calculated the voltage when the
button is the bottom capacitor in the divider, and we
got the same result! You may wonder if this will be
true for all such voltage dividers. Calculate the volt­
age difference (finger versus no finger) for the cir­
cuits in Figs. P6.53 (a) and (b), which use two
identical voltage sources.
25 pF F'Ixe d
c<p3citor
T
.-WPF
vit)
Button
vit)
1'(
No finger
r
(a)
vit)
25 pF
25 pF Fixed
25 p~
capacitor
V,(I)
1
lButton
~ t--r
(b)
-PF-''''~II
2-5
r
Finger