## Tuesday, August 31, 2021

### THE PRINCIPLE BEHIND ELECTRICAL INDUCTANCE

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.

DECAY OF CURRENT
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.