given amount of electrical energy in watts, with a given percentage of "drop" or loss of potential in volts, is inversely proportional to the square of the E.M.F. employed: hence it requires a wire of only one-quarter the cross-section and weight if the initial voltage is doubled. The great advantage thus obtained by the use of high tension can be realized either by a saving in the weight of wire required or by transmitting the current to a greater distance with the same weight of copper. In alternating current electric lighting the primary E.M.F. is usually at least 1,000 and often 2,000 to 10,000 volts. Even at a pressure of 1,000 volts an advantage of 100 to 1 is gained over a system operating at 100 volts. This enormous difference enables a given number of lamps to be supplied at a far greater distance, and at the same time the conductors weigh very much less. These facts make it unnecessary to design a system of alternating current conductors with the great care that is required for direct current distribution, since the use of slightly larger conductors will make up for any small differences in the arrangement of the feeders and mains. The result is that ordinary alternating current systems of conductors are less complicated than those used for direct currents. The very elaborate network of mains, for example, often employed for the latter is seldom required for the former except in the low voltage secondary distribution.

On the other hand, the actual uniformity of voltage secured in direct current circuits is usually superior to that obtained on alternating current systems. This is partly due to the fact just stated, that less care is required in designing the circuits, consequently there is a tendency to exercise too little care. The difference also arises from the effects of inductance and capacity, which produce variations in potential as great as, or greater than, those due to resistance alone. The exact influence of these factors under various conditions will be considered later.

The reason that the alternating current can be used at the high pressure of 1,000 volts or more, while the direct current is limited to about 110, 220, or 440 volts for constant potential lighting is due to the greater facility with which the alternating current can be transformed from a higher to a lower pressure, and vice versa. This is accomplished by simple transformers, consisting merely of two or more coils of wire wound upon an iron core. Since there are no moving parts, the attention demanded and the likelihood of

the apparatus getting out of order are small.

This enables the al

ternating current to be generated at or transformed to a high pressure suitable for transmission over long distances with small conductors, the potential being locally transformed to that required by the lamps, usually about 100 volts. In order to convert a direct current from one potential to another it is necessary to employ a motor-dynamo, which is practically a combination of a motor and a dynamo costing considerably more than an alternating current transformer, having a lower efficiency, and being more troublesome to take care of. In almost every other respect the direct current is preferable for electric lighting; and where the distances are not great, as, for example, in isolated plants and central stations. in thickly populated cities, the direct current has been the system most generally and successfully employed.

Two-phase and three-phase alternating current systems are often employed to supply incandescent and arc lights; but they are only advantageous for operating motors or rotary converters, and so far as lamps are concerned, they are more complicated, and possess no compensating superiority over the single-phase system. The latter, on the other hand, is not desirable when there are a number of motors of anything more than small size, such as fan motors. Hence polyphase systems are used in cases where both lamps and motors are to be supplied with alternating currents.

The principles of alternating currents will now be given; but it is not intended to treat the subject exhaustively, as there are several excellent works entirely devoted to it. It is sufficient herein to consider briefly the chief facts, to serve as a basis for study and calculations concerning alternating current lines, transformers, etc.

Principles of Alternating Currents. Each armature coil of a dynamo tends to generate an E.M.F., which rises to a certain maximum value, then falls to zero, then reverses in direction, and again returns to zero. This cycle of changes, which can be represented by a curve (Fig. 77), constitutes a complete period; and since it is repeated indefinitely at each revolution of the armature in a bipolar field, the currents produced by such an E.M.F. are called periodic currents. The number of complete periods in one second is called the frequency of the pressure, or current. In Fig. 77 the period is completed in .01 second, hence the frequency is 100. Since the

current changes its direction at each half-period, it follows that the number of alternations or reversals is twice the frequency.

Various forms of pressure or current waves may be generated, depending upon the arrangement of the armature winding, pole

pieces, etc. It is possible, by having the pole-faces either considerably wider or considerably narrower than the armature coils to produce a flat-topped wave (Fig. 78); or by making the coils exactly the same width as the pole-pieces, a peaked wave (Fig. 79) may be obtained. If the lines of force are excessive at the edges of the poles, extra waves, or upper harmonics JB W, are superimposed upon the main or fundamental wave A C (Fig. 80). The extra

Fig. 80. Alternating E.M.F. with Third Harmonic.

wave, JB W, shown is the third harmonic, being always an odd number; but in some cases the fifth, seventh, or almost any odd harmonic may be present. These harmonics are alternating pressure or current waves of three, five, seven, etc., times the frequency of the fundamental wave AC, which are generated simultaneously

with the latter, and modify its form. In the case represented, D VEG H is the wave of E.M.F. that is actually produced, being the combination of the fundamental AC and the third harmonic, JBW. For example, the voltage at D is the sum of ST and JT. At it is the algebraic sum of A and B, and so on. The ideal

form of wave generated by a coil of wire revolving about an axis in a uniform field is the sine-curve, in which the E.M.F. at any point, P, is proportional to the sine of the angle 0, through which the coil has moved (Fig. 81).

maxi

If the maximum value of the E.M.F. at M is E then the instantaneous value e at any point is

When the current wave is also a sine-curve, a similar expression gives the instantaneous value as follows:

The sine-wave being the ideal form, practically all calculations are based upon it; other forms tend to be converted into the sine curve; and it seems to be the best for general use, so it should be accepted for the same reason that standard weights and measures, screw-threads, etc., are adopted. This is true, even though some other form might have advantages for certain purposes. The question has been much discussed; * but the tendency has been for manufacturers generally to adopt the sine-form, the actual waves of pressure and current in most commercial apparatus being close approximations to the true sine-curve.

If all alternating current apparatus is designed for the sine-wave * Electrical World, Aug. 4, 1894, p. 107, and many other issues to Dec. 1, 1894.

it is possible to operate dynamos, motors, measuring instruments, etc., on any circuit, thus avoiding the endless confusion and difficulties that would arise if a different form of wave were adopted by each manufacturer and for each particular purpose.

Effective Values of Alternating Pressures and Currents. Since the value of an alternating current is continually varying, it is usually more convenient to consider its mean value; but this is not the ordinary arithmetical average. If an alternating current is flowing through a conductor, its heating effect at any instant will be proportional to the square of the current strength; and the average heating effect for the whole time during which it flows will be the average of these squares, that is, the mean square. It follows, therefore, that a direct current, to produce the same heating effect, would have a value equal to the square root of this quantity, that is, mean square. The same is true of alternating current E.M.F., since the heating effect is ; so that, with a constant re

E2

R

sistance, the heating is proportional to the square of the voltage. The square root of the mean square of the voltage or current is called its effective value, and is the quantity which is indicated by alternating current volt- or ampere-meters. For a sine-wave the

effective pressure or current is

1

√2

=

= .707, or about 71 per cent of

=

the maximum value, and conversely, the maximum is √2 1.41 times the effective value, or 41 per cent greater. These relations may be summed up as follows:

In practical cases it is usually sufficient to determine the effective volts and amperes of alternating currents, the instantaneous values being rarely considered except for the purpose of deducing formulas, studying phenomena, and other investigations. The term virtual is sometimes applied to the Vmean values instead of the word effective; but the latter word is now generally adopted.

Inductance is one of the three fundamental quantities which affect the flow of an alternating or other varying current, the other two being resistance and capacity. It is due to inductive action.