most likely to effect the desired object, but, being based on alternating currents, is not suitable for general purposes of motive power. By the use of self-regulating motors, such as the one described above, we can, however, not only transmit the electrical energy of a high tension current over a long line of consuming centres, but we can also transform it into the energy of low tension currents at these centres, which currents would have to be distributed over a limited area only, and thus not require any excessive amount of copper for mains. Such a transformator, for continuous currents would consist of a selfregulating motor with auxiliary dynamo and a self-regulating dynamo wound for constant electro-motive force. The three armatures could be mounted on the same spindle, and the whole apparatus could be self-contained. The co-efficient of efficiency of a transformator of this type would be slightly lower than that of an alternating current transformator, because on account of having a revolving part, certain mechanical losses would be incurred. But the efficiency would still be fairly satisfactory; probably about 75 per cent. of the energy of the high tension current could be recovered in energy of the low tension current, and the saving of coal effected by producing the power originally by large engines at the central station would of course be very considerable.

According to the classification made in the beginning of this chapter we have now to consider

System 3), which comprises the transmission of energy between two distant points by means of one generator and

one motor.

Let E, E, and E, represent respectively the electromotive force in the armature, at the brushes and at the terminals of the generator, and let ea, es, and e, repre

LONG DISTANCE TRANSMISSION.

161

sent the same for the motor. Let Ra, Rm, represent the resistance of the armature, and magnetizing coils of the generator, and rasm represent the same for the motor, then we have, according to the equations 15) to 22), if both machines are series-wound, the following relations:

C being the current sent by the generator into the line, and c being the current received by the motor. If the insulation of the line were perfect, these two currents would be equal; but in practice some small leak of current from the positive to the negative circuit, when the line extends over several miles, can hardly be avoided, and therefore we must assume

C > c.

The loss of current C c represents, as far as the generator is concerned, a waste of energy expressed by the product

E, (Ce) watts.

As far as the motor is concerned, this leak not only reduces the current which is available at the receiving station, but it has also the effect of reducing the available electro-motive force e, beyond the value corresponding to the current c. It will be clear that unless the leak occurs close to the generator, part of the line will have to carry a current larger than c, and thus the loss of electro-motive force due to the resistance of the line must be greater than the product of that resistance and the motor-current C. If the line is throughout its entire length equally well insulated, each unit of its length will have the same insulation resistance, which should be very high in com

M

parison to the conducting resistance itself. In a perfect line it should be infinite, but, as remarked above, this is not obtainable in an overhead circuit going many miles across country. Let s represent the conducting resistance of the line, and let i denote the insulation resistance between the positive and negative lead for unit length, then, if the distance from the generator to the motor be 7, the total insulation resistance as measured on a Wheatstone bridge would be. Knowing this from actual measurement, it might be thought that by the application of Ohm's law we could easily find the leak, Cc, by simply dividing the electro-motive force between the wires by the insulation resistance. This would, however, not be correct, for the simple reason, that the electro-motive force between the wires is not a constant, but diminishes in a certain ratio as we approach the distant end of the line, the actual law by which this diminution takes place depending not only on the resistance of the line and the current, but also on the insulation resistance itself. The question is therefore not so simple as it at first sight appears. An approximate solution sufficiently accurate for practical purposes is the following:

Let & represent the electro-motive force between the leads at the distance x from the generator; let the distance be increased to x + dx and the leak of current corresponding to length dx be dc, the drop in electro-motive force corresponding to that length being de. Then the following equations evidently obtain :

LOSS OF CURRENT BY LEAKAGE.

From these equations we obtain

163

To find the constant we apply the formula to the home end of the line, where ɛ = Et, c = C, and obtain between that and the far end the relation

This gives the current arriving at the motor, but in a somewhat inconvenient form. To simplify the expression we develop the square root into a series, and since the second term is very small in comparison to the first we can neglect the second and subsequent powers.

Now E.

2

e ̧2=(E, + e,) (E,— e,) and
e) and ī

total insulation resistance of the line. Hence

= J, the

E,+e,

Now

is the average electro-motive force between

2

1

the out and home lead ; and (E) represents the

J

current which under that average electro-motive force would flow through J, the total insulation resistance. This E gives the actual leak. It

current, multiplied by

C

will be observed that C, being the product of a resistance and a current, represents a difference of potential, and in this case it represents the electro-motive force which would in a line of perfect insulation be required to drive the full initial current C through the circuit, supposing the far ends were in metallic contact. s C represents, therefore, the loss of electro-motive force if there were no leak. The actual loss, E, e,, is naturally somewhat greater, and thus the quotient between the two must always be greater than unity. From this it follows that the loss of current due to leakage along the line is slightly greater than the figure we obtain by dividing the average electromotive force by the total insulation resistance. Where the insulation resistance is very high, and the conducting resistance very low, the leak will with sufficient accuracy be expressed by

but when the conditions are less favourable formula 30) should be used.

It is necessary in this place to briefly consider the influence of the leak on the total efficiency of a system of electric transmission, especially with reference to the most economical speed of the motor. In text books, and in