188 Fundamental Electrical and Electronic Principles

the primary. If the secondary is connected to a load, then it will cause the
secondary current I2 to flow. This results in a secondary terminal voltage,
V2. Figure 5.42 shows the circuit symbol for a transformer.

N1 : N 2

V1 V2

Fig. 5.42

5.24 Transformer Voltage and Current Ratios

Let us consider an ideal transformer. This means that the resistance
of the windings is negligible, and there are no core losses due to
hysteresis and eddy currents. Also, let the secondary be connected to a
purely resistive load, as shown in Fig. 5.43.

I1 N1:N2 I2

V1 E1 E2 V2 RL

Fig. 5.43

Under these conditions, the primary back emf, El, will be of the same
magnitude as the primary applied voltage, V1. The secondary terminal
voltage, V2, will be of the same magnitude as the secondary induced
emf, E2. Finally, the output power will be the same as the input power.
The two emfs are given by
N1d N 2 d
E1  volt, and E2  volt so,
dt dt
 Electromagnetism 189

d E
 1 …………… [1]
dt N1
d E
and  2 …………… [2]
dt N2

Since both equations [1] and [2] refer to the same rate of change of flux
in the core, then [1]  [2]:
E1 E
 2
N1 N2
E1 N
hence,  1
E2 N2

and since E1  V1, and E2  V2, then

V1 N
 1 (5.22)
V2 N2

From this equation, it may be seen that the voltage ratio is the same
as the turns ratio. This is perfectly logical, since the same flux links
both windings, and each induced emf is directly proportional to its
respective number of turns. This is the main purpose of the transformer.
It can therefore be used to ‘step up’ or ‘step down’ a.c. voltages,
depending upon the turns ratio selected.

Worked Example 5.29

Q A transformer is to be used to provide a 60 V output from a 240 V a.c. supply. Calculate (a) the turns
ratio required, and (b) the number of primary turns, if the secondary is wound with 500 turns.

A
V2  60 V; V1  240 V; N2  500
V1 N 240
(a)  1
V2 N2 60
N1 4
so, turns ratio,  or 4 : 1 Ans
N2 1

N1 4
(b) 
500 1
therefore, N1  2000 Ans

Since the load is purely resistive, then the output power, P2, is given by
P2  V2 I 2 watt
and the input power, P1  V1I1 watt
190 Fundamental Electrical and Electronic Principles

Also since the transformer has been considered to be 100% efficient

(no losses), then
P2  P1
therefore, V2 I 2  V1I 2
I1 V V N
 2 but 2  2
I2 V1 V1 N1

I1 N
hence,  2 (5.23)
I2 N1

i.e. The current ratio is the inverse of the turns ratio.

This result is also logical. For example, if the voltage was ‘stepped up’
by the ratio N2/N1, then the current must be ‘stepped down’ by the same
ratio. If this was not the case, then we would get more power out than
was put in! Although this result would be very welcome, it is a physical
impossibility. It would require the machine to be more than 100% efficient.

Worked Example 5.30

Q A 15  resistive load is connected to the secondary of a transformer. The terminal p.d. at the secondary
is 240 V. If the primary is connected to a 600 V a.c. supply, calculate (a) the transformer turns ratio, (b)
the current and power drawn by the load, and (c) the current drawn from the supply. Assume an ideal
transformer.

A
RL  15 ; V2  240 V; V1  600 V

The appropriate circuit diagram is shown in Fig. 5.43.

(a) N1 V 600
 1
N2 V2 240
so, turns ratio, N1/N2  2.5 : 1 Ans

V2 240
(b) I2  ohm 
RL 15
so, I 2  16 A Ans
P2  V2 I 2 watt  240  16
therefore, P2  3.84 kW Ans

(c) P1  P2  3.84 kW
and, P1  V1I1 watt
P1 3840
therefore, I1  amp 
V1 600
h ence, I1  6.4 A Ans
 Electromagnetism 191

Alternatively, using the inverse of the turns ratio:

N2 16
I1  I 2  
N1 2.5
so, I1  6.4 A Ans

Summary of Equations

dφ
Self-induced emf: e  N volt
dt
Emf in a straight conductor: e  Bυ sin φ volt
Force on a current carrying conductor: F  BI sin φ volt
Motor principle: T  BANI newton metre

2  107 I1I 2
Force between current carrying conductors: F  newton
d
1
Voltmeter figure of merit: ohm/volt
I fsd
di
Self-inductance: Self-induced emf, e  L volt
dt
dφ N
LN  henry
di I
N2   N2A
L  o r henry
S 

Energy stored: W  0.5LI2 joule

di1
Mutual inductance: Mutually induced emf, e2  M volt
dt
d 2 N 
M  N2  2 2 henry
di1 I1

M  k L1L2 henry

Energy stored: W  0.5L1I12  0.5L2 I 22 MI1I 2 joule

V2 N
Transformer: Voltage ratio,  2
V1 N1
I2 N
Current ratio,  2
I1 N2
192 Fundamental Electrical and Electronic Principles

Assignment Questions
1 The flux linking a 600 turn coil changes uniformly rotation
from 100 mWb to 50 mWb in a time of 85 ms.
Calculate the average emf induced in the coil.
2 An average emf of 350 V is induced in a 1000
turn coil when the flux linking it changes by
200 Wb. Determine the time taken for the S N
flux to change.
3 A flux of 1.5 mWb, linking with a 250 turn
coil, is uniformly reversed in a time of 0.02 s.
Calculate the value of the emf so induced. Fig. 5.44
4 A coil of 2000 turns is linked by a magnetic
flux of 400 Wb. Determine the emf induced in (b) Indicate the direction of induced emf in
the coil when (a) this flux is reversed in 0.05 s, each side of the coil.
and (b) the flux is reduced to zero in 0.15 s. (c) If this arrangement was to be used as a
motor, with the direction of rotation as
5 When a magnetic flux linking a coil changes, shown, indicate the direction of current
an emf is induced in the coil. Explain the flow required through the coil.
factors that determine (a) the magnitude of
the emf, and (b) the direction of the emf. 11 A conductor of effective length 0.5 m is placed
at right angles to a magnetic field of density
6 State Lenz’s law, and hence explain the term 0.45 T. Calculate the force exerted on the
‘back emf ’. conductor if it carries a current of 5 A.
7 A coil of 15 000 turns is required to produce 12 A conductor of effective length 1.2 m is placed
an emf of 15 kV. Determine the rate of change inside a magnetic field of density 250 mT.
of flux that must link with the coil in order to Determine the value of current flowing
provide this emf. through the conductor, if a force of 0.75 N is
8 A straight conductor, 8 cm long, is moved exerted on the conductor.
with a constant velocity at right angles to 13 A conductor, when placed at right angles to a
a magnetic field. If the emf induced in the magnetic field of density 700 mT, experiences
conductor is 40 mV, and its velocity is 10 m/s, a force of 20 mN, when carrying a current of
calculate the flux density of the field. 200 mA. Calculate the effective length of the
9 A conductor of effective length 0.25 m is conductor.
moved at a constant velocity of 5 m/s, through 14 A conductor, 0.4 m long, lies between two
a magnetic field of density 0.4 T. Calculate the pole pieces, with its length parallel to the pole
emf induced when the direction of movement faces. Determine the force exerted on the
relative to the magnetic field is (a) 90°, (b) 60°, conductor, if it carries a current of 30 A, and
and (c) 45°. the flux density is 0.25 T.
10 Figure 5.44 represents two of the armature 15 The coil of a moving coil meter is wound
conductors of a d.c. generator, rotating in a with 75 turns, on a former of effective length
clockwise direction. Copy this diagram and 2.5 cm, and diameter 2 cm. The former rotates
hence: at right angles to the field, which has a flux
(a) Indicate the direction of the field pattern of density of 0.5 T. Determine the deflecting
the magnetic poles. torque when the coil current is 50 A.
16 A moving coil meter has a coil of 60 turns
wound on to a former of effective length
An armature is the rotating part of a d.c. machine. 22.5 mm and diameter 15 mm. If the flux
If the machine is used as a generator, it contains density in the air gap is 0.2 T, and the coil
the coils into which the emf is induced. In the case current is 0.1 mA. Calculate (a) the force acting
of a motor, it contains the coils through which on each side of the coil, and (b) the restoring
current must be passed, to produce the torque torque exerted by the springs for the resulting
deflection of the coil.
 Electromagnetism 193

Assignment Questions
17 Two long parallel conductors, are spaced 24 Calculate the self-inductance of a 700 turn
12 cm between centres. If they carry 100 A coil, if a current of 5 A flowing through it
and 75 A respectively, calculate the force per produces a flux of 8 mWb.
metre length acting on them. If the currents 25 A coil of 500 turns has an inductance of 2.5 H.
are flowing in opposite directions, will this be What value of current must flow through it in
a force of attraction or repulsion? Justify your order to produce a flux of 20 mWb?
answer by means of a sketch of the magnetic
field pattern produced. 26 When a current of 2.5 A flows through a
0.5 H inductor, the flux produced is 80 Wb.
18 The magnetic flux density at a distance of 1.4 m Determine the number of turns.
from the centre of a current carrying conductor
27 A 1000 turn coil has a flux of 20 mWb linking it
is 0.25 mT. Determine the value of the current.
when carrying a current of 4 A. Calculate the
19 A moving coil meter has a coil resistance of coil inductance, and the emf induced when the
25 , and requires a current of 0.25 mA to current is reduced to zero in a time of 25 ms.
produce full-scale deflection. Determine the 28 A coil has 300 turns and an inductance of
values of shunts required to extend its current 5 mH. How many turns would be required to
reading range to (a) 10 mA, and (b) 1 A. Sketch produce an inductance of 0.8 mH, if the same
the relevant circuit diagram. core material were used?
20 For the meter movement described in question 29 If an emf of 4.5 V is induced in a coil having an
19 above, show how it may be adapted to serve inductance of 200 mH, calculate the rate of
as a voltmeter, with voltage ranges of 3 V and change of current.
10 V. Calculate the values for, and name, any
30 An iron ring having a mean diameter of 300 mm
additional components required to achieve this.
and cross-sectional area of 500 mm2 is wound
Sketch the relevant circuit diagram.
with a 150 turn coil. Calculate the inductance, if
21 Explain what is meant by the term ‘loading effect’. the relative permeability of the ring is 50.

22 A voltmeter, having a figure of merit of 15 k/ 31 An iron ring of mean length 50 cm and csa
volt, has voltage ranges of 0.1 V, 1 V, 3 V and 0.8 cm2 is wound with a coil of 350 turns. A
10 V. If the resistance of the moving coil is 30 , current of 0.5 A through the coil produces a
determine the multiplier values required for flux density of 0.6 T in the ring. Calculate (a)
each range. Sketch a circuit diagram, showing the relative permeability of the ring, (b) the
how the four ranges could be selected. inductance of the coil, and (c) the value of the
induced emf if the current decays to 20% of
23 Figure 5.45 shows a circuit in which the p.d. its original value in 0.01 s, when the current is
across resistor R2 is to be measured. The switched off.
voltmeter available for this measurement has
32 When the current in a coil changes from 2 A to
a figure of merit of 20 k/V, and has voltage
12 A in a time of 150 ms, the emf induced into
ranges of 1 V, 10 V and 100 V. Determine the
an adjacent coil is 8 V. Calculate the mutual
percentage error in the voltmeter reading,
inductance between the two coils.
when used to measure this p.d.
33 The mutual inductance between two coils
90 V is 0.15 H. Determine the emf induced in one
coil when the current in the other decreases
uniformly from 5 A to 3 A, in a time of 10 ms.
R1 5k 34 A coil of 5000 turns is wound on to a non-
magnetic toroid of csa 100 cm2, and mean
circumference 0.5 m. A second coil of 1000
turns is wound over the first coil. If a current
R2 5k of 10 A flows through the first coil, determine
(a) the self-inductance of the first coil, (b)
the mutual inductance, assuming a coupling
0V factor of 0.45, and (c) the average emf induced
in the second coil if interruption of the current
Fig. 5.45 causes the flux to decay to zero in 0.05 s.
194 Fundamental Electrical and Electronic Principles

Assignment Questions
35 Two air-cored coils, A and B, are wound with the core made from laminations? Is the core
100 and 500 turns respectively. A current of 5 A material a ‘hard’ or a ‘soft’ magnetic material?
in A produces a flux of 15 Wb. Calculate (a) Give the reason for this.
the self-inductance of coil A, (b) the mutual
inductance, if 75% of the flux links with B, and 40 A transformer with a turns ratio of 20:1 has
(c) the emf induced in each of the coils, when 240 V applied to its primary. Calculate the
the current in A is reversed in a time of 10 ms. secondary voltage.

36 Two coils, of self-inductance 50 mH and 85 mH 41 A 4:1 voltage ‘step-down’ transformer is

respectively, are placed parallel to each other. connected to a 110 V a.c. supply. If the current
If the coupling coefficient is 0.9, calculate their drawn from this supply is 100 mA, calculate the
mutual inductance. secondary voltage, current and power.

37 The mutual inductance between two coils 42 A transformer has 450 primary turns and 80
is 250 mH. If the current in one coil changes secondary turns. It is connected to a 240 V
from 14 A to 5 A in 15 ms, calculate (a) the emf a.c. supply. Calculate (a) the secondary
induced in the other coil, and (b) the change of voltage, and (b) the primary current when the
flux linked with this coil if it is wound with 400 transformer is supplying a 20 A load.
turns.
43 A coil of self-inductance 0.04 H has a resistance
38 The mutual inductance between the two of 15 . Calculate the energy stored when it is
windings of a car ignition coil is 5 H. Calculate connected to a 24 V d.c. supply.
the average emf induced in the high tension
winding, when a current of 2.5 A, in the low 44 The energy stored in the magnetic field of an
tension winding, is reduced to zero in 1 ms. You inductor is 68 mJ, when it carries a current of
may assume 100% flux linkage between the 1.5 A. Calculate the value of self-inductance.
two windings. 45 What value of current must flow through a 20 H
39 Sketch the circuit symbol for a transformer, inductor, if the energy stored in its magnetic
and explain its principle of operation. Why is field, under this condition, is 60 J?
 Electromagnetism 195

Suggested Practical Assignments

Note: The majority of these assignments are only qualitative in nature.

Assignment 1
To investigate Faraday’s laws of electromagnetic induction.
Apparatus:
Several coils, having different numbers of turns
2  permanent bar magnets
1  galvanometer
Method:
1 Carry out the procedures outlined in section 5.1 at the beginning of this
chapter.
2 Write an assignment report, explaining the procedures carried out, and
stating the conclusions that you could draw from the observed results.

Assignment 2
Force on a current carrying conductor.
Apparatus:
1  current balance
1  variable d.c. psu
1  ammeter
Method:
1 Assemble the current balance apparatus.
2 Adjust the balance weight to obtain the balanced condition, prior to
connecting the psu.
3 With maximum length of conductor, and all the magnets in place, vary
the conductor current in steps. For each current setting, re-balance the
apparatus, and note the setting of the balance weight.
4 Repeat the balancing procedure with a constant current, and maximum
magnets, but varying the effective length of the conductor.
5 Repeat once more, this time varying the number of magnets. The current
must be maintained constant, as must the conductor length.
6 Tabulate all results obtained, and plot the three resulting graphs.
7 Write an assignment report. This should include a description of the
procedures carried out, and conclusions drawn, regarding the relationships
between the force produced and I, , and B.

Assignment 3
To investigate the loading effect of a moving coil meter.
Apparatus:
1  10 k rotary potentiometer, complete with circular scale
1  d.c. psu
1  Heavy Duty AVO
1  DVM (digital voltmeter)
Method:
1 Connect the circuit as shown in Fig. 5.46, with the psu set to 10 V d.c.
2 Start with the potentiometer moving contact at the ‘zero’ end. Measure the
p.d. indicated, in turn, by both the DVM and the AVO, for every 30° rotation
of the moving contact.
Note: Do NOT connect both meters at the same time; connect them IN TURN.
196 Fundamental Electrical and Electronic Principles

Suggested Practical Assignments

10 V

p.s.u

0V

Fig. 5.46

3 Tabulate both voltmeter readings. Plot graphs (on the same axes) for the
voltage readings versus angular displacement.
4 Determine the percentage loading error of the AVO, for displacements of 0°,
180°, and 270°. Write an assignment report, and include comment regarding
the variation of loading error found.

Assignment 4
To demonstrate mutual inductance and coupling coefficient.
Apparatus:
Several coils, having different numbers of turns.
Ferromagnet core
1  galvo
1  d.c. psu
Method:
1 Place the two coils as close together as possible. Connect the galvo to one
coil, and connect the other coil to the psu via a switch.
2 Close the switch, and note the deflection obtained on the galvo.
3 Repeat this procedure for increasing distances of separation, and for
different coils.
4 Mount two of the coils on a common magnetic core, and repeat the procedure.
5 Write an assignment report, explaining the results observed.

Assignment 5
To determine the relationship between turns ratio and voltage ratio for a simple
transformer.
Apparatus:
Either 1  single-phase transformer with tappings on both windings;
or Several different coils with a ferromagnetic core.
Either a low voltage a.c. supply;
or 1  a.c. signal generator.
1  DVM (a.c. voltage ranges)
Method:
1 Connect the primary to the a.c. source.
2 Measure both primary and secondary voltages, and note the corresponding
number of turns on each winding.
3 Vary the number of turns on each winding, and note the corresponding
values of the primary and secondary voltages.
4 Tabulate all results. Write a brief report, explaining your findings.
 Chapter 6
Alternating Quantities

Learning Outcomes
This chapter deals with the concepts, terms and definitions associated with alternating
quantities. The term alternating quantities refers to any quantity (current, voltage, flux, etc.),
whose polarity is reversed alternately with time. For convenience, they are commonly referred
to as a.c. quantities. Although an a.c. can have any waveshape, the most common waveform
is a sinewave. For this reason, unless specified otherwise, you may assume that sinusoidal
waveforms are implied.
On completion of this chapter you should be able to:
1 Explain the method of producing an a.c. waveform.
2 Define all of the terms relevant to a.c. waveforms.
3 Obtain values for an a.c., both from graphical information and when expressed in
mathematical form.
4 Understand and use the concept of phase angle.
5 Use both graphical and phasor techniques to determine the sum of alternating quantities.

6.1 Production of an Alternating Waveform

From electromagnetic induction theory, we know that the average emf
induced in a conductor, moving through a magnetic field, is given by

e  Bv sin  volt  [1]

Where B is the flux density of the field (in tesla)

 is the effective length of conductor (in metre)
v is the velocity of the conductor (in metre/s)
u is the angle at which the conductor ‘cuts’ the lines of
magnetic flux (in degrees or radians)
i.e. v sin u is the component of velocity at right angles to the flux. 197