CHAPTER II INSTANTANEOUS VALUES AND INTEGRAL VALUES 9. In a periodically varying function, as an alternating cur- rent, we have to distinguish between the instantaneous value, which varies constantly as function of the time, and the integral value, which characterizes the wave as a whole. As such integral value, almost exclusively the effective value is used, that is, the square root of the mean square ; and wherever the intensity of an electric wave is mentioned without further reference, the effective value is understood. The maximum value of the wave is of practical interest only in few cases, and may, besides, be different for the two half-waves, as in Fig. 3. As arithmetic mean, or average value, of a wave as in Figs. 4 and 5, the arithmetical average of all the instantaneous values dur- ing one complete period is understood. 0 \ I Fig. 4. — Alternating wave. This arithmetic mean is either = 0, as in Fig. 4, or it differs from 0, as in Fig. 5. In the first case, the wave is called an alternating wave, in the latter a 'pulsating wave. Thus, an alternating wave is a wave whose positive values give the same sum total as the negative values; that is, whose two half-waves have in rectangular coordinates the same area, as shown in Fig. 4. 11 12 ALTERNA TING-C URRENT PHENOMENA A pulsating wave is a wave in which one of the half- waves pre- ponderates, as in Fig. 5. By electromagnetic induction, pulsating waves are produced only by commutating and unipolar machines (or by the super- position of alternating upon direct currents, etc.). All inductive apparatus without commutation give exclusively alternating waves, because, no matter what conditions may exist in the circuit, any line of magnetic force which during a complete period is cut by the circuit, and thereby generates an e.m.f., must during the same period be cut again in the opposite direc- tion, and thereby generate the same total amount of e.m.f. (Ob- viously, this does not apply to circuits consisting of different / p- -^ s^ / ^ ^ N / / \ V 1 / \ 1 f \ t VEF AQI VA .UE 1 / 0 j / 1 f s. / \ / Fig. 5. — Pulsating wave. parts movable with regard to each other, as in unipolar machines.) A direct-current machine without commutator or collector rings, or a coil-wound unipolar machine, thus is an impossibility. Pulsating currents, and therefore pulsating potential differ- ences across parts of a circuit can, however, be produced from an alternating induced e.m.f. by the use of asymmetrical circuits, as arcs, some electrochemical cells, as the aluminum-carbon cell, etc. Most of the alternating-current rectifiers are based on the use of such asymmetrical circuits. In the following we shall almost exclusively consider the alter- nating wave, that is, the wave whose true arithmetic mean value = 0. Frequently, by mean value of an alternating wave, the average of one half-wave only is denoted, or rather the average of all instantaneous values without regard to their sign. This mean value of one half-wave is of importance mainly in the rectifica- INSTANTANEOUS AND INTEGRAL VALUES 13 tion of alternating e.m.fs., since it determines the unidirectional value derived therefrom. 10. In a sine wave, the relation of the mean to the maximum value is found in the following way: Let, in Fig. 6, AOB represent a quadrant of a circle with radius 1. TT Then, while the angle 6 traverses the arc ^ fi'oni A to B, the sine varies from 0 to OB = 1. Hence the average variation of TT the sine bears to that of the corresponding arc the ratio I -^ ^, 2 or — '- 1. The maximum variation of the sine takes place about TT its zero value, where the sine is equal to the arc. Hence the maximum variation of the sine is equal to the variation of the Fig. 6. Fig. 7. corresponding arc, and consequently the maximum variation of the sine bears to its average variation the same ratio as the av- erage variation of the arc to that of the sine , that is, 1 -^ -, and since the variations of a sine function are sinusoidal also, we have Mean value of sine wave -r- maximum value = — ^ 1 = 0.63663. TT The quantities, "current," "e.m.f.," "magnetism," etc., are in reality mathematical fictions only, as the components of the entities, "energy," "power," etc.; that is, they have no inde- pendent existence, but appear only as squares or products. Consequently, the only integral value of an alternating wave which is of practical importance, as directly connected with the me- chanical system of units, is that value which represents the same power or effect as the periodical wave. This is called the effective 14 ALTERNATING-CURRENT PHENOMENA value. Its square is equal to the mean square of the periodic function, that is: The effective value of an alternating wave, or the value repre- senting the same effect an the periodically varying wave, is the square root of the mean square. In a sine wave, its relation to the maximum value is found in the following way: Let, in Fig. 7, AOB represent a quadrant of a circle with radius 1. Then, since the sines of any angle, d, and its complementary angle, 90° — 6, fulfill the condition, sin2 e + sin2 (90 - 0) = 1, the sines in the quadrant, AOB, can be grouped into pairs, so that the sum of the squares of any pair = 1 ; or, in other words, the mean square of the sine = ^i, and the square root of the mean square, or the effective value of the sine, = ~^-' That is: The effective value of a sine function bears to its maximum value the ratio, 1 V2 ^ 1 = 0.70711. Hence, we have for the sine wave the following relations: Max. Eff. Arith. mean Half period Whole period 1 1 V2 2 IT 0 1.0 0.7071 0.63663 0 1.4142 1.0 0 . 90034 0 1 . 5708 1.1107 1.0 0 11. Coming now to the general alternating wave, * = Ai sin 2 irft + A2 sin 4 vr/i + A3 sin 6 x/i + . . . + Bi cos 2 Trft + B2 cos 4 irft + ^3 cos Qirft -\- . . . , we find, by squaring this expression and cancelling all the prod- ucts which give 0 as mean square, the effective value I = VM(Ai2 ^ A2' -\- A^^ -{-... -\- B,' + B2' + 53^.. T The mean value does not give a simple expression, and is of no general interest. INSTANTANEOUS AND INTEGRAL VALUES 15 12. All alternating-current instruments, as ammeter, volt- meter, etc., measure and indicate the effective value. The maxi- mum value and the mean value can be derived from the curve of instantaneous values, as determined by wave-meter or oscillograph. Measurement of the alternating wave after rectification by a unidirectional conductor, as an arc, gives the inean value with direct-current instruments, that is, instruments employing a permanent magnetic field, and the effective value with alternating- current instruments. Voltage determination by spark-gap, that is, by the striking distance, gives a value approaching the maximum, especially with spheres as electrodes of a diameter larger than the spark- gap.