Download Inductors

Inductors
1.1 About inductors
An inductor or choke is a coil of wire with a core of either air or a magnetic material such as
iron. Air- and iron-cored types with their symbols are shown in Figs. 1.01 (a) and (b)
respectively. Inductors oppose or 'choke' changing currents and are said to have inductance,
denoted by L.
The circuit of Fig. 1.02 shows the behaviour of an inductor of inductance L. The rheostat is
first adjusted so that the lamps reach the same brightness when the switch is on d.c.; the
resistance R of the rheostat then equals the resistance of the inductor. If the switch is opened
and closed again, the lamp in series with the inductor lights up a second or two after that in
series with the rheostat. The inductor delays the rise of the d.c. to its steady value because of
its inductance.
If the 3 V battery is replaced by a 3 V a.c. supply, the lamp in series with the inductor
never lights because the current is changing all the time. However, it does light if the
inductance is reduced by removing the iron core.
To sum up, an inductor allows d.c. to flow but opposes a.c.; it has the opposite effect to a
capacitor.
The unit of inductance is the henry (H), to be denned in section 1.3. Submultiples are the
millihenry (1 mH = 1/1 000 H = l0-3 H) and the microhenry (1H = l0-6 H).
An inductor with a magnetic core has a much larger inductance than an air-cored one.
Also, the more turns of wire on the coil the greater the inductance.
1.2 Electricity and magnetism
To understand how an inductor works you need to know two important facts connecting
electricity and magnetism.
(a) Magnetism from electricity
The space round a magnet where it produces magnetic effects, such as deflecting a compass,
is called a magnetic field. A wire carrying a current also creates a magnetic field. In Fig.
1.03(a), if there were no current in the wire the small compasses would all point north (due to
the earth's magnetic field). When current flows, they set in concentric circles, with the wire at
their common centre, all pointing either clockwise or anticlockwise depending on the current
direction. The circles are called field lines (also lines of force) and show the shape of the
magnetic field. The arrows on the lines give the directions in which north poles point.
A coil of wire carrying a current produces a magnetic field with a shape like that in Fig.
1.03(a). The field is quite strong inside the coil, shown by the concentration of field lines
there. One end of the coil acts as if it were the north pole of a magnet and the other as the
south. If the coil has a core of easily magnetized material e.g. 'soft' iron, it is called an
electromagnet; the field is then much stronger but only lasts so long as current flows in the
coil, i.e. an electromagnet is a temporary magnet.
(b) Electricity from magnetism
When a conductor is in a changing magnetic field, a voltage is produced in it. This can be
shown by pushing a magnet into a coil, one pole first (Fig. 1.04), holding it motionless inside
the coil and then withdrawing it. A microammeter shows that current flows when the magnet is
moving but not when it is at rest. The current flows in opposite directions when the magnet
enters and leaves the coil, i.e. a.c. is generated. The result is the same if the coil is moved
towards and away from the stationary magnet.
We say that a voltage has been induced in the coil because the magnetic field passing through
it changed as the magnet and coil approached or separated. If the coil is part of a complete
circuit, the induced voltage causes an induced current. Experiments show that:
(i) the greater the rate at which the magnetic field changes (i.e. the faster the
magnet moves), the greater is the induced voltage—a statement known as
Faraday's law of electromagnetic induction;
(ii) the induced voltage (and current) always opposes what causes it—a statement
called Lenz's law of electromagnetic induction (or the law of cussedness)!
Lenz's law is illustrated by Figs. 1.05(a) and (b). In Fig. 6.05(a), the magnet approaches the
coil, north pole first. The induced current flows so that the coil behaves like a magnet with a
north pole at the top. The downward motion of the magnet is then opposed since two north
(like) poles repel. When the magnet is withdrawn, Fig. 1.05(b), the current direction reverses
and makes the top of the coil a south pole. This attracts the north pole of the magnet and
hinders its removal.
The generation of electricity at power stations and by car alternators and dynamos is by
electromagnetic induction.
1.3 How inductors work
Inductors oppose changing currents because of electromagnetic induction.
(a) Direct current
When a direct current increases from zero to its steady value, the magnetic field it produces
builds up to its final shape. During the process the field passing through the coil changes so
inducing a voltage in the coil itself which (as Lenz's law tells us) opposes the change that has
caused it, i.e. the rising current that is trying to establish the field. This opposition delays the
rise of current.
The time constant T of the circuit in Fig. 1.06(a) is given by L/R, where L is the inductance
of the inductor and R the resistance of the resistor. T is the time for the current I to rise to
0.63 (approximately two-thirds) of its final steady value of V/R, Fig. 1.06(b). If L is in
henries and R is in ohms, T is in seconds. In 5L/R seconds I will have reached its final value
which depends on R but not L. The energy stored in the final magnetic field of the inductor
can be shown to be ½ LI2 joules if L is in henries and / is in amperes.
When the current is switched off, the switch contacts open and the resistance R becomes very
large, making L/R very small and causing the magnetic field to collapse rapidly. As it does so,
the field passing through the coil changes and a large voltage is induced which opposes the
change producing it, i.e. the collapsing field due to the falling current. It therefore tries to keep
the current flowing longer so delaying its fall to zero and producing sparks at the switch
contacts. The energy for these comes from that stored in the magnetic field.
(b) Alternating current
Since an alternating current is changing all the time, the magnetic field it produces is also
changing continuously. There is therefore always an induced voltage in the coil and permanent
opposition to the current. In Fig. 1.02, when the battery is replaced by an a.c. supply the lamp
in series with the inductor of inductance L does not light up (unless L is reduced by removing
the iron core).
You should note that the induced voltage is caused, as in all cases of electromagnetic
induction, by a change in the strength of the magnetic field passing through the coil. But the
change here arises not as a result of a magnet or a coil moving (as in section 1.2(b)), but from a
changing electric current.
The unit of inductance, the henry (H), is defined in terms of the induced voltage. It is the
inductance of a coil in which a current changing at the rate of one ampere per second induces a
voltage of one volt.
1.4 Types of inductor
(a) Iron cores
Materials based on iron are used where a large inductance is required. Iron increases several
hundred times the strength of the magnetic field caused by the current in the coil wound on it.
Silicon steel and nickel-iron alloys such as Mumetal and Stalloy are used at audio frequencies
(up to 20 kHz). Iron cores are laminated, they consist of flat sheets (stampings) coated thinly
on one side by an insulating material. The laminations reduce the conversion of electrical
energy to heat by making it difficult for currents in the coil to induce currents in the core.
These induced currents are called eddy currents which flow in circles through the iron core,
Fig. 1.07(a). If the laminations are at right angles to the plane of the coil windings, the core
offers a large resistance to the eddy currents. E- and I-shaped laminations for a low frequency
choke are shown in Fig. 1.07(b); such inductors are used as 'smoothing' chokes in mains
power supply units, a typical value being 10 H.
(b) Air cores
Inductors with an air core have small inductances and are used for high frequencies, either
in radio tuning circuits or as 'r.f. chokes' to stop radio frequency currents (greater than 20
kHz) taking certain paths in a circuit. Coils for use at high frequencies are made of Litz wire
which consists of several thin copper wires insulated from each other. High frequency
currents tend to flow near the surface of a wire (the 'skin effect') and by providing more
surface area for a given volume of copper the resistance is reduced. This reduces the loss of
electrical energy to heat.
(c) Iron-dust and ferrite cores
Iron-based cores can be used at high frequencies if the material is in the form of a powder
which has been coated with an insulator and pressed together. An inductor with an iron-dust
core is shown (with the core removed) in Fig. 1.08(a) along with its symbol; it is used in radio
frequency tuned circuits.
Ferrite cores may also be used at high frequencies. They consist of ferric oxide combined
with other oxides such as nickel oxide. Fig. 1.08(b) shows the aerial coil of a radio receiver
wound on a ferrite rod as the core. A pot-type ferrite core is shown in Fig. 1.08(c).
Iron-dust and ferrite cores increase the inductance of a coil considerably. For example if an
air-cored coil has an inductance of 1 mH, a ferrite core could increase it to about 400 mH.
They also have a high resistance and so reduce eddy current losses.
1.5 Inductive reactance
(a) Effect on a.c. and d.c.
The opposition of an inductor to a.c. is called its inductive reactance, XL, and increases if
either the inductance L increases or the frequency f of the a.c. increases. It can be shown that
it is given by:
XL = 2  f L
XL is in ohms if f is in hertz and L in henries. For example, if f = 50 Hz and L = 10 H, then
XL = 1 000  = 3.1 k.
An inductor also has resistance and its total opposition to a.c. is made up of its resistance R
and its inductive reactance XL.
We can now sum up the basic facts about resistors, capacitors and inductors. All are
components which, in their different ways oppose current flow and are used to control it in a
circuit.
A resistor allows both a.c. and d.c. to flow through it.
A capacitor allows a.c. to flow 'through' it and its reactance decreases as the frequency of
the a.c. increases. With d.c. the flow stops when the capacitor is charged.
An inductor allows a.c. and d.c. to pass but it opposes a.c. (and varying d.c.) more than
steady d.c.; its reactance increases as the frequency of the a.c. increases.
(b) Phase shift
The current 'through' a capacitor in an a.c. circuit leads the voltage across it. In an inductor
the current I lags behind the voltage V, Fig. 1.09. The lag is due to the fact that when the
current starts to flow, although it is small at first, it is increasing at its fastest rate, therefore so
too is the build-up of the magnetic field. As a result the voltage induced in the inductor has its
maximum value.
The fact that currents and voltages are not in step in a.c. circuits has some important results
which seem surprising. They will be considered in the next section.
1.6 Alternating current series circuit
(a) Impedance
If an a.c. supply of r.m.s. voltage V and frequency f is applied to a series circuit containing
resistance R, capacitance C and inductance L, Fig. 1.10(a), each component offers some
opposition to the current. The total opposition is called the impedance (Z) and is measured
(like resistance and reactance) in ohms. It can be shown that:
Z=…
where XL = 2  f L is the inductive reactance and Xc = 1/(2  f C) is the capacitive reactance.
The r.m.s. current I in the circuit is given by:
I=V/Z
This equation is the same as that connecting current and voltage in a d.c. circuit (i.e.
I = V / R) with the impedance Z replacing the resistance R; it is used for a.c. circuit
calculations.
(b) Worked example
In an a.c. series circuit R = 100 , L = 2.0 H, C = 10 F and the supply has a r.m.s. voltage of
24 V and frequency 50 Hz. Calculate (a) the inductive reactance , XL (b) the capacitive
reactance Xc, (c) the impedance Z, and (d) the r.m.s. current I.
a) I= 50 Hz and L = 2.0 H.
XL = 628 
b) f = 50 Hz and C = 10 F.
XC = 318 
c) R = 100  and XL – XC = 628 - 318 = 310 .
Z = 330 
d) V = 24 V and Z = 330 
I = 0.073 A (73 mA)
(c) Voltages
In a d.c. series circuit the voltages ‘add up’. This is not so for a.c.
For example, consider the r.m.s. voltages in the a.c. series circuit of the worked
example in section 1.6(b) above. The same r.m.s. current I passes through each
component and the r.m.s. voltage across each one equals the current multiplied by the
'opposition' of the component. Therefore:
VL =I * XL = 0.073 * 630 = 46 V
VC =I * XC = 0.073 * 320 = 23 V
VR =I * R = 0.073 * 100 = 7.3 V
These are the voltages which would be recorded by an a.c. voltmeter connected across
each component in turn. The surprising fact is that VL + VC + VR, ( = 76 V) is more
than three times the r.m.s. voltage of 24 V applied to the whole circuit (and which an
a.c. voltmeter connected across the whole circuit would read).
This apparent contradiction is due to the fact that whilst VR and I are in phase, VL
leads I by 90° and VC lags behind I by 90° VR VL and VC are therefore acting as it
were in different 'directions' with the phase shift between VL and VC being 180°, i.e.
when one has its maximum positive value the other has its maximum negative value.
The voltages cannot be added by ordinary addition but this can be done by applying
the parallelogram law to the diagram (called phasor diagram) in Fig. 1.10(b) in which
VL is assumed to be greater than VC so that their resultant (VL – VC) is in the direction
of VL.
(d) Power
In an a.c. circuit, power is only dissipated (i.e. electrical energy changed to heat) in
parts having resistance. In a perfect capacitor (no leakage current) no power is used
because the energy taken from the supply to charge the capacitor on the first quartercycle of current is returned to the supply on the second quarter-cycle when it
discharges.
In a pure inductor (no resistance) the energy obtained during the first quarter-cycle
of current supply and which is stored in the magnetic field round the inductor, is
returned to the supply when the field collapses during the second quarter-cycle. In
practice an inductor has some resistance and energy is 'lost' on this account.
1.7 About transformers
(a) Action
A transformer changes (transforms) an alternating voltage from one value to another. It
consists of two coils, called the primary and secondary windings, which are not
connected electrically. The windings are either one on top of the other or are side by
side on an iron, iron-dust or air core. Transformer symbols are given in Fig. 1.11.
A transformer works by electromagnetic induction: a.c. is supplied to the primary and
produces a changing magnetic field which passes through the secondary, thereby
inducing a changing (alternating) voltage in the secondary. It is important that as much
as possible of the magnetic field produced by the primary passes through the secondary.
A practical arrangement designed to achieve this in an iron-cored transformer is shown
in Fig. 1.12 in which the secondary is wound on top of the primary. You should also
notice that the induced voltage in the secondary is always of opposite polarity to the
primary voltage (Lenz's law), as shown in Fig. 1.13.
Too large a current in a transformer causes magnetic saturation of the core, i.e. the
magnetization of the core is a maximum and it is no longer able to follow changes of
magnetizing current. Particular care is required when there is a d.c. component.
(b) Equations
It can be shown that if a transformer is 100% efficient at transferring electrical energy from
primary to secondary (and many are nearly so), then:
(secondary a.c. voltage / primary a.c. voltage) = (turns on secondary) / (turns on primary)
In symbols
Vs / Vp = ns / np
For example, if ns is twice np , the transformer is a step-up one, and Vs will be twice Vp. In a
step-down transformer there are fewer turns on the secondary than on the primary and Vs is
less than Vp. The ratio ns / np is called the turns ratio.
A transformer with two secondaries is represented in Fig. 1.14; it both steps-up and stepsdown the primary voltage. The step-down secondary voltage is 6 V a.c. What is the step-up
one?
If the voltage is stepped-up by a transformer the current is stepped-down in proportion and
vice versa. This must be so if we assume that all the electrical energy given to the primary
appears in the secondary. We can then say:
power in primary == power in secondary
or
Vp * Ip = VS * IS
where Ip and Is are the primary and secondary currents respectively.
Therefore in a perfect transformer, if VS is double VP , IS is half IP . In practice IS is less than
half IP because of small energy losses in the transformer arising from
(i) the resistance of the windings of copper wire, causing the current in them to produce heat,
(ii) eddy currents in iron and iron-dust cored transformers, (reduced by laminations), and
(iii) the magnetic field of the primary not passing entirely through the secondary.
1.8 Types of transformer (a) Mains
Mains transformers—one is shown in Fig. 1.15(a)—are used at a.c. mains frequency (50 Hz
in Britain), their primary coil being connected to the 230 V a.c. supply. Their secondary
windings may be step-up or step-down or they may have one or more of each. They have
laminated iron cores and are used in power supply units. Sometimes the secondary has a
centre-tap.
Step-down toroidal types. Fig. 6.15(b), are becoming popular. They have virtually no external
magnetic field and a screen between primary and secondary windings gives safety and
electrostatic screening. Their pin connections are brought out to a 0.1 inch grid which makes
them ideal for printed circuit board (p.c.b.) mounting.
Isolating transformers have a one-to-one turns ratio (i.e. nS / nS =1/1) and are safety devices
for separating a piece of equipment from the mains supply. They do not change the voltage.
(b) Audio frequency
Audio frequency transformers, as illustrated in Fig. 1.15(c), also have laminated iron cores and
are used as output matching transformers to ensure the maximum transfer of power from the a.f.
output stage to the loudspeaker in, for example, a radio set or amplifier.
(c) Radio frequency
Radio frequency transformers, as in Fig. 1.15(d), usually have adjustable iron-dust cores and
form part of the tuning circuits in a radio. They are enclosed in a small aluminium 'screening'
can to stop them radiating energy to other parts of the circuit.