**ELECTRICAL
INDUCTANCE**

__MUTUAL INDUCTION__

We have seen that an e.m.f
is induced whenever the magnetic flux linked with an electric circuit is
changed, and that one method of producing such a change is to caus4e a

conductor forming part of the circuit to cut lines of force, either by moving
the conductor itself or the magnet producing the flux.

It is possible, however, for a change influx linkage to
occur without actual movement, consider the two single-turn coils in the
center, one of which can be put in series with an electric cell c by
means of a key k while the other is permanently connected to a galvanometer g.
we will call the coil connected to the cell the primary coil and the
one connected to the galvanometer the secondary. There is no electrical connection between one coil and the other.

When the key is
closed and current flows in the primary coil, the galvanometer gives a
momentary deflection, showing that for a short time current flows in the
secondary coil also. We can explain this effect by considering the magnetic
fieldset up by the current in the primary.

The lines of force form closed paths around the conductor, some of them are represented in the present figure
by dots, although they do not exist, of course, until the key is closed. It
is clear that when the magnetic field has been set up, there are lines of force linked with the secondary coil which was not there before, and it is this change in flux
linkage which produces the momentary current through the galvanometer. When the key is opened, the current in the primary coil
ceases and the magnetic field disappears. Once again there is a change in flux
linkage and momentary current in the secondary, but this time the deflection of
the galvanometer is in the opposite direction. Evidently, the current in the
secondary flows in one direction when the primary circuit is closed and in the
opposite direction when it is opened.

This is in accordance with Lenz's law, which
states that induced currents always tend to oppose the change which produces them.
Thus, when the primary circuit is closed, the current in the secondary tries to
prevent the flux from being set up, but when the primary circuit is opened it
tries to prevent it from dying away.

A similar result would be obtained if the outer coil were
used as the primary and the inner as the secondary, or if two coils of equal
size were placed together. The effect would be more pronounced if each coil
consisted

Of many turns, particularly
if the flux were increased by winding them upon an iron core.

__SELF-INDUCTION__

In dealing with the motor, we said that when the armature
conductors were moving across the
magnetic field, an e.m.f. was induced in them, just as it would have been had
the machine has been a generator, we called this the back e.m.f. because it
opposed the e.m.f. applied to the motor.

A similar action occurs in the present case, for as the
magnetic field builds up, the flux linkage changes not only in the secondary
coil, but also in the primary. Let us, therefore, ignore the secondary and
consider what happens when a single coil, say a solenoid, is connected to a
source of current.

As soon as the current starts to flow, the magnetic field
starts to build up, and the flux linked with the coil changes. The result is to
produce and e.m.f., called the back e.m.f. of self-induction, which opposes the
applied e.m.f. were it not for the back e.m.f., the current would rise
instantaneously to its value as determined by ohm’s law, thus:

Current
=applied e.m.f/resistance

As it is, however, the
back e.m.f. retards the growth of current, and the rise takes place gradually.

A circuit in which
these effects are marked is said to be inductive. The rate at which the current
starts to grow (that is, the slope at the beginning of the curve) is determined
by the applied voltage and by a property of the particular circuit known as its
self-inductance, or often simply as its inductance. As the current grows, the
rate of growth becomes less. Therefore a reduction in the
rate of change of flux linkage and of the back e.m.f. as the back e.m.f. falls,
more and more of the applied e.m.f. is available for forcing the increased
current against the resistance of the circuit.

Theoretically, the current never quite reaches its ohm’s
law value in an inductive circuit, but it does so for all practical purposes
within a comparatively short time. According to the inductance and resistance
of the circuit, this time may be anting from a small fraction of a second
upwards. and increase in resistance, while reducing the value of the final
current, also reduces the effect of inductance in retarding the current growth.

Unit of Inductance

The unit of inductance is the henry (plural, henrys; not
henries), named after joseph henry, the American physicist. A circuit has an
inductance of I henry if a current which is changing at the rate of I ampere
per second produces in it a back e.m.f. of I volt.

The inductance of a coil is dependent upon the linkage
between turns and flux. In a single turn coil, the linkage is equal to the total
flux, since all the flux links with the single turn. In a coil with two turns, there is twice as much flux and also twice as many times as great. Provided
that every line of force links with every turn, we can say that in general, the
inductance is proportional to the square of the number of turns. Actually,
this condition is not always attained in practice, particularly in the case of
a ling coil such as a solenoid, and some of the flux links with less than the a full number of turns.

Note that if iron is present, the flux varies i.e, it is not proportional to the
current flowing. The inductance of an iron-cored coil is therefore dependent in
part upon the amount of current flowing through it, that is, upon the state of magnetization of the core.

When one coil acts upon another, they are said to have mutual inductance, this, too, is measured in henrys. Two
coils have a mutual inductance of I henry when a current changing at the rate
of I ampere per second in one of them produces and e.m.f. I volt in the other.

The henry is an inconveniently large unit for many
purposes, and a subdivision, the microhenry, is commonly employed. One henry
equals one million microhenrys.

Suppose that, after the
current has grown to its final value, the coil is suddenly short-circuited, we
need not concern ourselves with what happens to the source of current, but we
will suppose that it is disconnected at the moment that the short circuit is
applied.

If a resistance
without self-inductance is a short circuit, the current in it immediately drops
to zero. In the present case, however, as soon as the current starts to fall,
the magnetic field starts to collapse, and in so doing induces an e.m.f. in
the coil. Since the coil is short-circuited, this e.m.f. can cause a current to
flow, and direction which tends to maintain the flux, i.e., in the same
direction as the original current.

The result is that the current dies away gradually, as
indicated by curve f. as it dies away, the rate of fall becomes less, but in a
short time, it has dropped practically to zero. As in the case of the rising
current, an increase in resistance reduces the effect of inductance.

__STORAGE OF ENERGY IN THE FIELD__

While the current
is dying away, the electrical energy it represents is being converted into heat
in the coil. This energy cannot come from nowhere, and it is clearly not coming
from the original source, which we have supposed to be disconnected. Actually,
it has been stored in the magnetic field. When the circuit was first closed,
energy was required to establish the
field; that is why the current did not rise instantaneously to its full value.
Now, as the field collapses, this energy is restored again to the circuit.

We can compare the
effect of inductance with that of mechanical inertia, which is the tendency of
a body to go on doing what it is doing already, whether moving or standing
still. When a train starts, energy is absorbed in getting it going but once it
is running at a required speed it can be
kept going, but once it is running at the required speed it can be kept going
by supplying just sufficient energy to make good the frictional losses. When
the power f the engine I cut off, the train continues to move until the energy
which it acquired at the start has been exhausted.

Suppose that instead of short-circuiting the coil, we
simply opened its circuit. It would be like stopping the train by putting and
obstacle on the line, for in neither case would there be any opportunity for
the harmless dissipation of the stored energy.

In the electrical case, the collapsing field would induce
the usual e.m.f., and this would force a current to flow by arcing across the
switch contacts. For the present we need note only
that damage to switches and dangerously high voltages can be caused by suddenly
breaking a highly inductive circuit.

__INDUCTANCE IN GENERATOR AND MOTOR COILS__

In a multi-coil armature, the current is generated
by a number of coils connected in series. when the commutator segments
to which a coil is connected pass a brush, this current has to be suddenly
reversed. In the brief time during which the coil is short-circuited by the
brush, therefore, the current must die away in one direction and build up again
the other. Owing to the self-inductance of the coil these operations would
normally require more time that is available, and it is in order to assist the
current to reverse by inducing and e.m.f. opposed to the e.m.f. of
self-induction that interpoles are fitted.

The use of carbon brushes assists towards the same end,
owing to the contact resistance included in series with the coil in which the
current is being reversed. Carbon brushes have therefore replaced the copper
brushes which were used in early machines.

__NON-INDUCTIVE WINDINGS__

We have spoken particularly of the inductance of solenoids
and similar coils, because, owing to the flux from one turn threading others as
well, this is the case in which inductance is most marked. Nearly all conductors,
however, possess inductance in some degree.

When a strictly non-inductive resistance is required, a conductor may be doubled back on itself. It can then be wound in a coil. Since the current flows around the coil in one direction in
the first half of the conductor and in the other direction in the second half,
the two magnetic fields cancel out and there is no inductive effect.

**SERIES AND PARALLEL INDUCTANCES**

Inductances, like resistance, can be connected in series in parallel provided that

They have no mutual
inductance, their joint value may be found by the methods used for resistances.

Example: what is the joint the inductance of two coils, of 2 henrys and 4 henrys inductance respectively,
connected (a) in series and (b)in parallel, there is no mutual inductance
between them?

In series, joint inductance
=(2+4)henrys =6 henry

In parallel, joint
inductance =1/0.5+0.25 =1.33henrys.

If there is mutual
inductance, i.e., if any of the flux of one coil links with any of the turns of
the other, the joint inductance may be either more or less than that obtained
above. It will be more it the two inductances assist each other (as when the two
coils are placed end to end and the current flows in the same direction round
each of them), and less if they oppose.

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