18. EQUIVALENT SINE WAVES 87. In the preceding chapters, alternating waves have been assumed and considered as sine waves. EQUIVALENT SINE WAVES 107 The general alternating wave is, however, never completely, frequently not even approximately, a sine wave. A sine wave having the same effective value, that is, the same square root of mean squares of instantaneous values, as a general alternating wave, is called its corresponding "equivalent sine wave." It represents the same effect as the general wave. With two alternating waves of different shapes, the phase relation or angle of lag is indefinite. Their equivalent sine waves, however, have a definite phase relation, that which gives the same effect as the general wave, that is, the same mean (ei). Hence if e = e.m.f. and i = current of a general alternating wave, their equivalent sine waves are defined by e0 = -\Anean (e2), io = A/mean (i2); and the power is Po = eQiQ cos eoiQ = mean (ei)', thus, mean (ei) COS €QIQ = — / - Vmean (e2) v mean (i2) Since by definition the equivalent sine waves of the general alternating waves have the same effective value or intensity and the same power or effect, it follows that in regard to inten- sity and effect the general alternating waves can be represented by their equivalent sine waves. Considering in the preceding the alternating currents as equiva- lent sine waves representing general alternating waves, the investigation becomes applicable to any alternating circuit irrespective of the wave shape. The use of the terms reactance, impedance, etc., implies that a wave is a sine wave or represented by an equivalent sine wave. Practically all measuring instruments of alternating waves (with exception of instantaneous methods) as ammsters, volt- meters, wattmeters, etc., give not general alternating waves but their corresponding equivalent sine waves. EXAMPLES 88. In a 25-cycle alternating-current transformer, at 1000 volts primary impressed e.m.f., of a wave shape as shown in 108 ELEMENTS OF ELECTRICAL ENGINEERING e §M »OCOOI>.C^O5(NCOOOOi'— l i— 1 CO CO CO »H i— 1 • OQ '^ CO CO C^J ^H >O CO iQ CO C^ O O5 CO CO iQ CO O TfitOOOiOiOO ^ rH 00 00 T-( c co" co"co"co'Nco>i-r ^^^ooooooo 1— «O ..» 00 00^2 ^ ^ ^ se ^ oo oo . < r-l CO ?CI> i-H QJ CO EQUIVALENT SINE WAVES 109 Fig. 41 and Table I, the number of primary turns is 500, the length of the magnetic circuit 50 cm., and its section shall be chosen so as to give a maximum density B = 15,000. At this density the hysteretic cycle is as shown in Fig. 42 and Table II. FIG. 41. — Wave-shape of e.m.f. in example 88. What is the shape of current wave, and what the equivalent sine waves of e.m.f., magnetism, and current? The calculation is carried out in attached table. TABLE II / B 0 ±8 ,000 2 + 10,400 - 2,500 4 + 11,700 + 5,800 6 + 12,400 + 9,300 8 + 13,000 + 11,200 10 + 13,500 + 12,400 12 + 13,900 + 13,200 14 + 14,200 + 13,800 16 + 14,500 + 14,300 18 + 14,800 + 14,700 20 + 15,000 In column (1) are given the degrees, in column (2) the relative values of instantaneous e.m.fs., e corresponding thereto, as taken from Fig. 41. Column (3) gives the squares of e. Their sum is 24,939; 24 939 thus the mean square, . ' * - = 1385.5, and the effective value, lo 110 ELEMENTS OF ELECTRICAL ENGINEERING Since the effective value of impressed e.m.f. is = 1000, the 1 000 instantaneous values are eQ = e^-^ as given in column (4). Since the e.m.f. e0 is proportional to the rate of change of magnetic flux, that is, to the differential coefficient of B} B is proportional to the integral of the e.m.f., that is, to Se0 plus an integration constant. 2e0 is given in column (5), and the integration constant follows from the condition that B at 180° FIG. 42. — Hysteretic cycle in example 88. must be equal, but opposite in sign, to B at 0°. The integration constant is, therefore, 1 SO -it n i ^ and by subtracting 7324 from the values in column (5) the values of B' of column (6) are found as the relative instantaneous values of magnetic flux density. Since the maximum magnetic flux density is 15,000 the in- 15 000 stantaneous values are B = B' ' . , plotted in column (7). From the hysteresis cycle in Fig. 42 are taken the values of magnetizing force /, corresponding to magnetic flux density B. They are recorded in column (8), and in column (9) the instan- taneous values of m.m.f. F = If, where I = 50 = length of magnetic circuit. EQUIVALENT SINE WAVES 111 i = — , where n = 500 = number of turns of the electric circuit, gives thus the exciting current in column (10) . Column (11) gives the squares of the exciting current, i2. 25 85 Their sum is 25.85; thus the mean square, ' = 1.436, and lo the effective value of exciting current, i' = Vl.436 = 1.198 amp. Column (12) gives the instantaneous values of power, p = ieo. Their sum is 4766; thus the mean power, p' = 4766 18 = 264.8. FIG. 43. — Waves of exciting current. Power and flux density corresponding to e.m.f . in Fig. 41 and hysteretic cycle in Fig. 42. FIG. 44. — Corresponding sine waves for e.m.f. and exciting current in Fig. 43. Since p' = i'e'Q cos 0, where e'0 and i' are the equivalent sine waves of e.m.f. and of current respectively, and 0 their phase displacement, substitut- ing these numerical values of p', er, and i', we have 264.8 = 1000 X 1.198 cos 6. hence, cos 0 = 0.221, 6 = 77.2°, 112 ELEMENTS OF ELECTRICAL ENGINEERING and the angle of hysteretic advance of phase, a = 90° - 0 = 12.8°. The hysteresis current is then i' cos e = 0.265, and the magnetizing current, i' sin 0 = 1.165. Adding the instantaneous values of e.m.f. eQ in column (4) 14 648 gives 14,648; thus the mean value, — f-r— = 813.8. Since the J-O effective value is 1000, the mean value of a sine wave would be 2 -v/2 1000- - = 904; hence the form factor is 7T 904 7 = Adding the instantaneous values of current i in column (10), irrespective of their sign, gives 17.17; thus the mean value, 17.17 ' = 0.954. Since the effective value = 1.198, the form lo factor is 1.198 2 V2 7 = 0954 — The instantaneous values of e.m.f. e0, current i, flux density B and power p are plotted in Fig. 43, their corresponding sine waves in Fig. 44.