occupied by the galvanometer. The deflection of the galvanometer produced by depressing the battery key is nearly annulled by means of a magnet, and the resistances 1, 2, 3 are adjusted until no alteration of the galvanometer deflection takes place when the key in CD is depressed. When this is the case C and D are at the same potential, since the addition of the conductor CD does not disturb the current distribution in the network; and we have for the resistance 4 of the galvanometer

=

1213

r4 11

CHAPTER IX.

THE MEASUREMENT OF ENERGY IN ELECTRIC

CIRCUITS.

WHEN a circuit in which a current of electricity is flowing contains a motor, or machine by which work is done in virtue of electromagnetic action, the whole electrical work done in the circuit consists, as was first shown by Joule, of two parts, work spent in heat in the generator and motor and in the conductors connecting them, and work done in moving the motor against external resistance. The total rate at which electrical energy is given out in the circuit is, as we have seen, E C watts, where E is the total electromotive force of the generator in volts, and C is the number of amperes of current flowing. The rate at which work is spent in heat is in watts, by Joule's law, CR, where R is the total resistance in circuit in ohms; hence, if we call W the rate at which work is done in the motor,* we have,

*We consider here a system in which C is constant, and neglect loss of energy due to local currents, &c., in the motor. For fuller information regarding motors and their action see a paper by Profs. Ayrton and Perry, Proc. Soc. Tel. Engs., 1883, republished in the electrical journals, also Prof. S. P. Thompson's Dynamo-Electric Machinery.

which shows that the current flowing is equal to that which would flow in the circuit if, the resistance remaining the same, the motor were held at rest, and the electromotive force diminished by an amount equal to W/C. This is what is called the back electromotive force of the motor, and is due to the action of the motor in setting up when driven an electromotive force tending to send a current through the circuit in the opposite direction to that of the current by which the motor is driven. We shall denote the back electromotive force by E1. Hence equation (2) becomes,

and the rate at which work is spent in driving the motor is E1C.

To determine E we have simply to measure with a potential galvanometer or voltmeter, the difference of potential between the two terminals of the generator. Calling this V, and R1 the effective resistance of the generator, we have plainly,

Again, since C and

also the total resistance R in the circuit can be found by measurement, we find by (3)

where all the quantities on the right-hand side are known.

The ratio of E1C, the electrical energy spent per unit of time in the circuit otherwise than in heating the conductors, to the whole electrical energy EC spent in the circuit per unit of time, that is the ratio of E, to E, we may call the electrical efficiency of the arrangement.

Denoting this efficiency by e, we find, by equation (4),

Hence the smaller C is made, that is, the slower the energy is given out, the value of the efficiency of the arrangement is the more nearly equal to unity, the value of the efficiency of an arrangement in which the energy in the motor done against external resistance is exactly equal to the whole electrical energy given out in the circuit.

When however energy is spent at the maximum rate in working the motor, E1 C has its greatest value. But by (5)

E1C = EC C2R

This equation may be written,

= W.

Now in order that these values of C may be real, 4 RW cannot be greater than E. Hence the greatest value, W can have is E2/4R. When W has this maximum value, C is equal to E/2R, and therefore E1 equal to E/2. Hence the electrical efficiency is . It is to be very carefully observed that although in this case the arrangement is that of greatest electrical activity, it is not that of greatest electrical efficiency, as it has only about one-half the efficiency of one in which energy is given out at a very slow rate. The case is exactly analogous to that referred to in p. 85, of a battery arranged so as to give maximum current through a given external resistance.

All that has been stated above is applicable to the case of a motor fed by any kind of generator whatever. The generator employed however is generally some form of dynamo- or magneto-electric machine driven by an external motor, such as a steam- or gas-engine or a waterwheel, and a few of the results obtained below apply only to such cases, which will be indicated as they occur.

When the generator and motor are exactly similar machines, and the same current passes through both, the ratio of E1 to E will be that of n Aƒ(C) to n' A ƒ (C); where n and n' are the speeds of the machines, A a constant depending on the form and disposition of the magnets, and ƒ(C) a function of the current. Hence in this case the efficiency is measured simply by the ratio of the speed of the motor to that of the dynamo. The more nearly therefore the speed of the motor approaches to that of the generator, the greater is the efficiency. It is to be observed however that two machines identically alike will not in practice be perfectly similar in their action, even with the same currents flowing in their armatures and field-magnet coils. The armature currents tend to weaken the field in the generator, and to strengthen the field in the motor.

In general, the higher the speed at which the motor is run, the greater is the electrical efficiency of any arrangement, for it is obvious that the higher the speed the more nearly does E1 approach to E, and therefore the value of E1/E, the measure of efficiency, to unity.

For a constant difference E-E1, the ratio of the energy spent in heating the conductors by the current to the whole energy expended in the circuit, may be reduced by increasing the total electromotive force E of the circuit. The energy spent in heat is CR, or (E—E1)2/R, and the