CHAPTER II INSTANTANEOUS VALUES AND INTEGRAL VALUES. 8. IN a periodically varying function, as an alternating current, 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 Fig. 4. Alternating Wave. value is used, that is, the square root of the mean squares ; 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 instan- taneous values during one complete period is understood. 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 pttlsating wave. 12 ALTERNA TING-CURRENT PHENOMENA. 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. A pulsating wave is a wave in which one of the half- waves preponderates, as in Fig. 5. By electromagnetic induction, pulsating waves are pro- duced only by commutating and unipolar machines (or by the superposition of alternating upon direct currents, etc.). All inductive apparatus without commutation give ex- clusively alternating waves, because, no matter what con- Fig. 5. Pulsating Wave. ditions may exist in the circuit, any line of magnetic force, which during a complete period is cut by the circuit, and thereby induces an E.M.F., must during the same period be cut again in the opposite direction, and thereby induce the same total amount of E.M.F. (Obviously, this does not apply to circuits consisting of different parts movable with regard to each other, as in unipolar machines.) In the following we shall almost exclusively consider the alternating 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 INSTANTANEOUS AND INTEGRAL VALUES. 13 average of all instantaneous values without regard to their sign. This mean value is of no practical importance, and is, besides, in many cases indefinite. 9. In a sine wave, the relation of the mean to the maxi- mum value is found in the following way : — Fig. 8. Let, in Fig. 6, AOB represent a quadrant of a circle with radius 1. Then, while the angle <£ traverses the arc -n- / 2 from A to B, the sine varies from 0 to OB = 1. Hence the average variation of the sine bears to that of the corresponding arc the ratio 1 -j- 7r/2, or 2 / TT •+- 1. The maximum variation of the sine takes place about 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 corresponding arc, and consequently the maximum variation of the sine bears to its average variation the same ratio as the average variation of the arc to that of the sine ; that is, 1 -f- 2 / 77-, and since the variations of a sine-function are sinusoidal also, we have, o Mean value of sine wave -r- maximum value = • — • -f- 1 7T = .63663. The quantities, "current," "E.M.F.," "magnetism," etc., are in reality mathematical fictions only, as the components 14 AL TERNA TING-CURRENT PHENOMENA. of the entities, "energy," "power," etc. ; that is, they have no independent 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 mechanical system of units, is that value which represents the same power or effect as the periodical wave. This is called the effective value. Its square is equal to the mean square of the periodic function, that is : — TJie effective value of an alternating wave, or tJie value representing the same effect as 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 : Fig. 7. Let, in Fig. 7, AOB represent a quadrant of a circle with radius 1. Then, since the sines of any angle and its complemen- tary angle, 90°— <£, fulfill the condition, — sin2 $ + sin2 (90 — <£) = 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 =1/2, and the square root of the mean square, or the effective value of the sine, = 1/V2. That is: INSTANTANEOUS AND INTEGRAL VALUES. 15 The effective value of a sine function bears to its mum value the ratio, — 1 V2 Hence, we have for the sine curve the following rela- tions : 1 = .70711. MAX. EFF. ARITH. MEAN. Half Period. Whole Period. 1 1 V2 2 7T 0 1 .7071 .63663 0 1.4142 1 .90034 0 1.5708 1.1107 1 0 10. Coming now to the general alternating wave, / = Ai sin 27r Nt + Az sin 4-n- Nt + A3 sin GTT Nt + . . . + BI cos 2-n-Nt + B* cos ±TrNt + £s cos GTT Nt + . . we find, by squaring this expression and canceling all the products which give 0 as mean square, the effective value, — 1= V* W The mean value does not give a simple expression, and is of no general interest. 16 ALTERNATING-CURRENT PHENOMENA,