Bird J. Electrical Circuit Theory and Technology
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Electromagnetic induction 105
Figure 9.5
Induced e.m.f. in a coil of N turns,
E = −N
d8
dt
volts,
where d is the change in flux in Webers, and dt is the time taken for
the flux to change in seconds(i.e., d/dt is the rate of change of flux).
Induced e.m.f. in a coil of inductance L henrys,
E = −L
dI
dt
volts,
where dI is the change in current in amperes and dt is the time taken
for the current to change in seconds(i.e., dI/dt is the rate of change of
current). The minus sign in each of the above two equations remind us
of its direction (given by Lenz’s law).
Problem 6. Determine the e.m.f. induced in a coil of 200 turns
when there is a change of flux of 25 mWb linking with it in 50 ms
Induced e.m.f. E DN
d8
dt
D200
25 ð 10
3
50 ð 10
3
D −100 volts
Problem 7. A flux of 400 µWb passing through a 150-turn coil is
reversed in 40 ms. Find the average e.m.f. induced.
Since the flux reverses, the flux changes from C400 µWb to 400 µWb,
a total change of flux of 800
µWb
Induced e.m.f. E DN
d
dt
D150
800 ð 10
6
40 ð 10
3
D
150 ð 800 ð 10
3
40 ð 10
6
Hence the average e.m.f. induced E = −3 volts
Problem 8. Calculate the e.m.f. induced in a coil of inductance
12 H by a current changing at the rate of 4 A/s
Induced e.m.f. E DL
dI
dt
D124 D
−48 volts
106 Electrical Circuit Theory and Technology
Figure 9.6
Problem 9. An e.m.f. of 1.5 kV is induced in a coil when a
current o
f 4 A collapses uniformly to zero in 8 ms. Determine the
inductance of the coil.
Change in current, dI D 4 0 D 4A;dt D 8msD 8 ð 10
3
s;
dI
dt
D
4
8 ð 10
3
D
4000
8
D 500 A/s; E D 1.5 kV D 1500 V
Since jEjDL
dI
dt
, inductance, L D
jEj
dI/dt
D
1500
500
D 3H
(Note that jEj means the ‘magnitude of E’, which disregards the minus
sign)
Further problems on inductance may be found in Section 9.8, problems 6
to 9, page 110.
9.4 Inductors
A component called an inductor is used when the property of inductance
is required in a circuit. The basic form of an inductor is simply a coil of
wire.
Factors which affect the inductance of an inductor include:
(i) the number of turns of wire—the more turns the higher the
inductance
(ii) the cross-sectional area of the coil of wire —the greater the cross-
sectional area the higher the inductance
(iii) the presence of a magnetic core— when the coil is wound on an
iron core the same current sets up a more concentrated magnetic
field and the inductance is increased
(iv) the way the turns are arranged— a short thick coil of wire has a
higher inductance than a long thin one.
Two examples of practical inductors are shown in Figure 9.6, and the
standard electrical circuit diagram symbols for air-cored and iron-cored
inductors are shown in Figure 9.7.
Figure 9.7
An iron-cored inductor is often called a choke since, when used in a.c.
circuits, it has a choking effect, limiting the current flowing through it.
Inductance is often undesirable in a circuit. To reduce inductance to a
minimum the wire may be bent back on itself, as shown in Figure 9.8,
so that the magnetizing effect of one conductor is neutralized by that
of the adjacent conductor. The wire may be coiled around an insulator,
as shown, without increasing the inductance. Standard resistors may be
non-inductively wound in this manner.
Figure 9.8
Electromagnetic induction 107
9.5 Energy stored
An inductor possesses an ability to store energy. The energy stored, W,
in the magnetic field of an inductor is given by:
W =
1
2
LI
2
joules
Problem 10. An 8 H inductor has a current of 3 A flowing
through it. How much energy is stored in the magnetic field of
the inductor?
Energy stored, W D
1
2
LI
2
D
1
2
83
2
D 36 joules
Further problems on energy stored may be found in Section 9.8,
problems 10 and 11, page 110.
9.6 Inductance of a coil
If a current changing from 0 to I amperes, produces a flux change from 0
to Webers, then dI D I and d D . Then, from Section 9.3, induced
e.m.f. E D N/t D LI/t, from which
inductance of coil,
L =
N8
I
henrys
Problem 11. Calculate the coil inductance when a current of 4 A
in a coil of 800 turns produces a flux of 5 mWb linking with the
coil.
For a coil, inductance L D
N
I
D
8005 ð 10
3
4
D 1H
Problem 12. A flux of 25 mWb links with a 1500 turn coil when a
current of 3 A passes through the coil. Calculate (a) the inductance
of the coil, (b) the energy stored in the magnetic field, and (c) the
average e.m.f. induced if the current falls to zero in 150 ms.
(a) Inductance, L D
N
I
D
150025 ð 10
3
3
D 12.5H
(b) Energy stored, W D
1
2
LI
2
D
1
2
12.53
2
D 56.25 J
(c) Induced e.m.f., E DL
dI
dt
D12.5
3 0
150 ð 10
3
D −250 V
108 Electrical Circuit Theory and Technology
(Alternatively, E DN
d
dt
D1500
25 ð 10
3
150 ð 10
3
D −250 V
since if the current falls to zero so does the flux)
Problem 13. A 750 turn coil of inductance 3 H carries a current
of 2 A. Calculate the flux linking the coil and the e.m.f. induced in
the coil when the current collapses to zero in 20 ms
Coil inductance, L D
N
I
from which, flux D
LI
N
D
32
750
D 8 ð 10
3
D 8 mWb
Induced e.m.f. E DL
dI
dt
D3
2 0
20 ð 10
3
D −300 V
(Alternatively, E DN
d
dt
D750
8 ð 10
3
20 ð 10
3
D −300 V
Further problems on the inductance of a coil may be found in Section 9.8,
problems 12 to 18, page 110.
9.7 Mutual inductance
Mutually induced e.m.f. in the second coil,
E
2
= −M
dI
1
dt
volts,
where M is the mutual inductance between two coils, in henrys, and
dI
1
/dt is the rate of change of current in the first coil.
The phenomenon of mutual inductance is used in transformers
(see Chapter 20, page 315). Mutual inductance is developed further in
Chapter 43 on magnetically coupled circuits (see page 841).
Problem 14. Calculate the mutual inductance between two coils
when a current changing at 200 A/s in one coil induces an e.m.f.
of 1.5 V in the other.
Induced e.m.f. jE
2
jDM
dI
1
dt
, i.e., 1.5 D M200
Thus mutual inductance, M D
1.5
200
D 0.0075 H or 7.5 mH
Electromagnetic induction 109
Problem 15. The mutual inductance between two coils is 18 mH.
Calculate the steady rate of change of current in one coil to induce
an e.m.f. of 0.72 V in the other.
Induced e.m.f., jE
2
jDM
dI
1
dt
Hence rate of change of current,
dI
1
dt
D
jE
2
j
M
D
0.72
0.018
D 40 A=s
Problem 16. Two coils have a mutual inductance of 0.2 H. If the
current in one coil is changed from 10 A to 4 A in 10 ms, calculate
(a) the average induced e.m.f. in the second coil, (b) the change of
flux linked with the second coil if it is wound with 500 turns.
(a) Induced e.m.f. E
2
DM
dI
1
dt
D0.2
10 4
10 ð 10
3
D −120 V
(b) Induced e.m.f. jE
2
jDN
d
dt
, hence d D
jE
2
jdt
N
Thus the change of flux, d D
12010 ð 10
3
500
D 2.4 mWb
Further problems on mutual inductance may be found in Section 9.8
following, problems 19 to 22, page 111.
9.8 Further problems on
electromagnetic induction
Induced e.m.f.
1 A conductor of length 15 cm is moved at 750 mm/s at right-angles
to a uniform flux density of 1.2 T. Determine the e.m.f. induced in
the conductor. [0.135 V]
2 Find the speed that a conductor of length 120 mm must be moved
at right angles to a magnetic field of flux density 0.6 T to induce in
it an e.m.f. of 1.8 V. [25 m/s]
3 A 25 cm long conductor moves at a uniform speed of 8 m/s through
a uniform magnetic field of flux density 1.2 T. Determine the current
flowing in the conductor when (a) its ends are open-circuited, (b) its
ends are connected to a load of 15 ohms resistance.
[(a) 0 (b) 0.16 A]
4 A car is travelling at 80 km/h. Assuming the back axle of the car is
1.76 m in length and the vertical component of the earth’s magnetic
field is 40
µT, find the e.m.f. generated in the axle due to motion.
[1.56 mV]
110 Electrical Circuit Theory and Technology
5 A conductor moves with a velocity of 20 m/s at an angle of (a) 90
°
(b) 45
°
(c) 30
°
, to a magnetic field produced between two square-
faced poles of side length 2.5 cm. If the flux on the pole face is
60 mWb, find the magnitude of the induced e.m.f. in each case.
[(a) 48 V (b) 33.9 V (c) 24 V]
Inductance
6 Find the e.m.f. induced in a coil of 200 turns when there is a change
of flux of 30 mWb linking with it in 40 ms. [150 V]
7 An e.m.f. of 25 V is induced in a coil of 300 turns when the flux
linking with it changes by 12 mWb. Find the time, in milliseconds,
in which the flux makes the change. [144 ms]
8 An ignition coil having 10000 turns has an e.m.f. of 8 kV induced
in it. What rate of change of flux is required for this to happen?
[0.8 Wb/s]
9 A flux of 0.35 mWb passing through a 125-turn coil is reversed in
25 ms. Find the magnitude of the average e.m.f. induced. [3.5 V]
Energy stored
10 Calculate the value of the energy stored when a current of 30 mA is
flowing in a coil of inductance 400 mH. [0.18 mJ]
11 The energy stored in the magnetic field of an inductor is 80 J when
the current flowing in the inductor is 2 A. Calculate the inductance
of the coil. [40 H]
Inductance of a coil
12 A flux of 30 mWb links with a 1200 turn coil when a current of
5 A is passing through the coil. Calculate (a) the inductance of the
coil, (b) the energy stored in the magnetic field, and (c) the average
e.m.f. induced if the current is reduced to zero in 0.20 s.
[(a) 7.2 H (b) 90 J (c) 180 V]
13 An e.m.f. of 2 kV is induced in a coil when a current of 5 A collapses
uniformly to zero in 10 ms. Determine the inductance of the coil.
[4 H]
14 An average e.m.f. of 60 V is induced in a coil of inductance 160 mH
when a current of 7.5 A is reversed. Calculate the time taken for the
current to reverse. [40 ms]
15 A coil of 2500 turns has a flux of 10 mWb linking with it when
carrying a current of 2 A. Calculate the coil inductance and the e.m.f.
induced in the coil when the current collapses to zero in 20 ms.
[12.5 H, 1.25 kV]
Electromagnetic induction 111
16 A coil is wound with 600 turns and has a self inductance of 2.5 H.
What current must flow to set up a flux of 20 mWb?
[4.8 A]
17 When a current of 2 A flows in a coil, the flux linking with the coil
is 80
µWb. If the coil inductance is 0.5 H, calculate the number of
turns of the coil. [12500]
18 A steady current of 5 A when flowing in a coil of 1000 turns
produces a magnetic flux of 500
µWb. Calculate the inductance of
the coil. The current of 5 A is then reversed in 12.5 ms. Calculate
the e.m.f. induced in the coil. [0.1 H, 80 V]
Mutual inductance
19 The mutual inductance between two coils is 150 mH. Find the magni-
tude of the e.m.f. induced in one coil when the current in the other
is increasing at a rate of 30 A/s. [4.5 V]
20 Determine the mutual inductance between two coils when a current
changing at 50 A/s in one coil induces an e.m.f. of 80 mV in the
other. [1.6 mH]
21 Two coils have a mutual inductance of 0.75 H. Calculate the magni-
tude of the e.m.f. induced in one coil when a current of 2.5 A in the
other coil is reversed in 15 ms. [250 V]
22 The mutual inductance between two coils is 240 mH. If the current
in one coil changes from 15 A to 6 A in 12 ms, calculate (a) the
average e.m.f. induced in the other coil, (b) the change of flux linked
with the other coil if it is wound with 400 turns.
[(a) 180 V (b) 5.4 mWb]
10 Electrical measuring
instruments and
measurements
At the end of this chapter you should be able to:
ž recognize the importance of testing and measurements in
electric circuits
ž appreciate the essential devices comprising an analogue
instrument
ž explain the operation of an attraction and a repulsion type of
moving-iron instrument
ž explain the operation of a moving coil rectifier instrument
ž compare moving coil, moving iron and moving coil rectifier
instruments
ž calculate values of shunts for ammeters and multipliers for
voltmeters
ž understand the advantages of electronic instruments
ž understand the operation of an ohmmeter/megger
ž appreciate the operation of multimeters/Avometers
ž understand the operation of a wattmeter
ž appreciate instrument ‘loading’ effect
ž understand the operation of a C.R.O. for d.c. and a.c.
measurements
ž calculate periodic time, frequency, peak to peak values from
waveforms on a C.R.O.
ž recognize harmonics present in complex waveforms
ž determine ratios of powers, currents and voltages in decibels
ž understand null methods of measurement for a Wheatstone
bridge and d.c. potentiometer
ž understand the operation of a.c. bridges
ž understand the operation of a Q-meter
ž appreciate the most likely source of errors in measurements
ž appreciate calibration accuracy of instruments
Electrical measuring instruments and measurements 113
10.1 Introduction
Tests and measurements are important in designing, evaluating, main-
taining and servicing electrical circuits and equipment. In order to detect
electrical quantities such as current, voltage, resistance or power, it is
necessary to transform an electrical quantity or condition into a visible
indication. This is done with the aid of instruments (or meters) that indi-
cate the magnitude of quantities either by the position of a pointer moving
over a graduated scale (called an analogue instrument) or in the form of
a decimal number (called a digital instrument).
10.2 Analogue
instruments
All analogue electrical indicating instruments require three essential
devices:
(a) A deflecting or operating device. A mechanical force is produced
by the current or voltage which causes the pointer to deflect from
its zero position.
(b) A controlling device. The controlling force acts in opposition to the
deflecting force and ensures that the deflection shown on the meter is
always the same for a given measured quantity. It also prevents the
pointer always going to the maximum deflection. There are two main
types of controlling device— spring control and gravity control.
(c) A damping device. The damping force ensures that the pointer
comes to rest in its final position quickly and without undue oscil-
lation. There are three main types of damping used—eddy-current
damping, air-friction damping and fluid-friction damping.
There are basically two types of scale—linear and non-linear.
A linear scale is shown in Figure 10.1(a), where the divisions or grad-
uations are evenly spaced. The voltmeter shown has a range 0–100 V,
i.e. a full-scale deflection (f.s.d.) of 100 V. A non-linear scale is shown
in Figure 10.1(b). The scale is cramped at the beginning and the gradu-
ations are uneven throughout the range. The ammeter shown has a f.s.d.
of 10 A.
Figure 10.1
10.3 Moving-iron
instrument
(a) An attraction type of moving-iron instrument is shown diagram-
matically in Figure 10.2(a). When current flows in the solenoid, a
pivoted soft-iron disc is attracted towards the solenoid and the move-
ment causes a pointer to move across a scale.
(b) In the repulsion type moving-iron instrument shown diagrammat-
ically in Figure 10.2(b), two pieces of iron are placed inside the
solenoid, one being fixed, and the other attached to the spindle
carrying the pointer. When current passes through the solenoid, the
two pieces of iron are magnetized in the same direction and there-
fore repel each other. The pointer thus moves across the scale. The
force moving the pointer is, in each type, proportional to I
2
. Because
114 Electrical Circuit Theory and Technology
Figure 10.2
of this the direction of current does not matter and the moving-iron
instrument can be used on d.c. or a.c. The scale, however, is non-
linear.
10.4 The moving-coil
rectifier instrument
A moving-coil instrument, which measures only d.c., may be used
in conjunction with a bridge rectifier circuit as shown in Figure 10.3
to provide an indication of alternating currents and voltages (see
Chapter 14). The average value of the full wave rectified current is
0.637 I
m
. However, a meter being used to measure a.c. is usually
calibrated in r.m.s. values. For sinusoidal quantities the indication
is 0.707 I
m
/0.637 I
m
i.e. 1.11 times the mean value. Rectifier
instruments have scales calibrated in r.m.s. quantities and it is assumed
by the manufacturer that the a.c. is sinusoidal.
Figure 10.3
10.5 Comparison of
moving-coil, moving-iron
and moving-coil rectifier
instruments
Type of
instrument
Moving-coil Moving-iron Moving-coil
rectifier
Suitable for
measuring
Direct current
and voltage
Direct and alternating
currents and voltage
(reading in rms value)
Alternating current
and voltage (reads
average value but
scale is adjusted
to give rms value
for sinusoidal
waveforms)
Scale Linear Non-linear Linear
Method
of control
Hairsprings Hairsprings Hairsprings
Method of
damping
Eddy current Air Eddy current