CHAPTER XXIII. EFFECTS OF HIGHER HARMONICS. 244. To elucidate the variation in the shape of alternat- ing waves caused by various harmonics, in Figs. 175 and Fig. 175. Effect of Triple Harmonic. 176 are shown the wave-forms produced by the superposi- tion of the triple and the quintuple harmonic upon the fundamental sine wave. EFFECTS OF HIGHER HARMONICS. 399 In Fig. 175 is shown the fundamental sine wave and the complex waves produced by the superposition of a triple harmonic of 30 per cent the amplitude of the fundamental, under the relative phase displacements of 0°, 45°, 90°, 135°, and 180°, represented by the equations : sin ft sin ft — .3 sin 3 ft sin ft- .3 sin (3/3-45°) sin ft — .3 sin (3 ft — 90°) s'm ft - .3 sin (3 ft - 135°) sin ft — .3 sin (3/3 — 180°). • As seen, the effect of the triple harmonic is in the first figure to flatten the zero values and point the maximum values of the wave, giving what is called a peaked wave. With increasing phase displacement of the triple harmonic, the flat zero rises and gradually changes to a second peak, giving ultimately a flat-top or even double-peaked wave with sharp zero. The intermediate positions represent what is called a saw-tooth wave. In Fig. 176 are shown the fundamental sine wave and the complex waves produced by superposition of a quintuple harmonic of 20 per cent the amplitude of the fundamental, under the relative phase displacement of 0°, 45°, 90°, 135°, 180°, represented by the equations : sin ft sin ft — .2 sin 5 ft sin/3- .2 sin (5,8-45°) sin/3- .2 sin (5/3-90°) smft- .2 sin (5/3- 135°) sin/3- .2 sin (5/8- 180°). The quintuple harmonic causes a flat -topped or even double-peaked wave with flat zero. With increasing phase displacement, the wave becomes of the type called saw- tooth wave also. The flat zero rises and becomes a third peak, while of the two former peaks, one rises, the other 400 AL TERN A TING- CURRENT PHENOMENA. decreases, and the wave gradually changes to a triple- peaked wave with one main peak, and a sharp zero. As seen, with the triple harmonic, flat-top or double- peak coincides with sharp zero, while the quintuple har- monic flat-top or double-peak coincides with flat zero. Distortion of Wave Shapa by Quintuple Harmonfc Sin./S-.2sin.(5/?-S5j/ J \J Fig. 176. Effect of Quintuple Harmonic. Sharp peak coincides with flat zero in the triple, with sharp zero in the quintuple harmonic. With the triple har- monic, the saw-tooth shape appearing in case of a phase difference between fundamental and harmonic is single, while with the quintuple harmonic it is double. Thus in general, from simple inspection of the wave shape, the existence of these first harmonics can be discov- ered. Some characteristic shapes are shown in Fig. 177. EFFECTS OF HIGHER HARMONICS. 401 Sin/?-.225 sinf3/?-180) , ""-.05 sin/5/3-180) Sin./?- 15 sm.(3/?-180). Sin./?-. 15' sin 3/?-.1Q sir (5/J-180) f/jjr. 777. So/ne Characteristic Wave Shapes. Flat top with flat zero : sin /3 — .15 sin 3 /3 — .10 sin 5 0. Flat top with sharp zero : sin 0 - .225 sin (3 /3 - 180°) - .05 sin (5 /3 - 180°). Double peak, with sharp zero : sin (3 - .15 sin (30- 180°) - .10 sin 5 /?. Sharp peak with sharp zero : sin {3 — .15 sin 3 0 — .10 sin (5 (3 — 180°). 245. Since the distortion of the wave-shape consists in the superposition of higher harmonics, that is, waves of higher frequency, the phenomena taking place in a circuit 402 ALTERNATING-CURRENT PHENOMENA. supplied by such a wave will be the combined effect of the different waves. Thus in a non-inductive circuit, the current and the potential difference across the different parts of the circuit are of the same shape as the impressed E.M.F. If self- induction is inserted in series to a non-inductive circuit, the self-induction consumes more E.M.F. of the higher harmon- ics, since the reactance is proportional to the frequency, and thus the current and the E.M.F. in the non-inductive part of the circuit shows the higher harmonics in a reduced amplitude. That is, self-induction in series to a non-induc- tive circuit reduces the higher harmonics or smooths out the wave to a closer resemblance with sine shape. In- versely, capacity in series to a non-inductive circuit con- sumes less E.M.F. at higher than at lower frequency, and thus makes the higher harmonics of current and of poten- tial difference in the non-inductive part of the circuit more pronounced — intensifies the harmonics. Self-induction and capacity in series may cause an in- crease of voltage due to complete or partial resonance with higher harmonics, and a discrepancy between volt-amperes and watts, without corresponding phase displacement, as will be shown hereafter. 246. In long-distance transmission over lines of notice- able inductance and capacity, rise of voltage due to reso- nance may occur with higher harmonics, as waves of higher frequency, while the fundamental wave is usually of too low a frequency to cause resonance. An approximate estimate of the possible rise by reso- nance with various harmonics can be obtained by the inves- tigation of a numerical instance. Let in a long-distance line, fed by step-up transformers at 60 cycles, The resistance drop in the transformers at full load = 1%. The inductance voltage in the transformers at full load = 5% with the fundamental wave. The resistance drop in the line at full load = 10%. EFFECTS OF HIGHER HARMONICS. 403 The inductance voltage in the line at full load = 20% with the fundamental wave. The capacity or charging current of the line = 20% of the full- load current / at the frequency of the fundamental. The line capacity may approximately be represented by a condenser shunted across the middle of the line. The E.M.F. at the generator terminals E is assumed as main- tained constant. The E.M.F. consumed by the resistance of the circuit from generator terminals to condenser is Ir = .06 £, or, r = .06 -| . The reactance E.M.F. between generator terminals and condenser is, for the fundamental frequency, Ix = .15 £, -IK E or, x = .15 — , thus the reactance corresponding to the frequency (2/£ — 1) N of the higher harmonic is : x(2k- 1) =.15(2£- 1) — . The capacity current at fundamental frequency is : hence, at the frequency : (2 k — 1) N: / = .2(2£-l)/Z, if: e' = E.M.F. of the (2 k — l)th harmonic at the condenser, e = E.M.F. of the (2 k — l)th harmonic at the generator terminals. The E.M.F. at the condenser is : — e' = V*2 — iar2 + ix (2k — V) • 404 AL TERNA TING-CURRENT PHENOMENA. hence, substituted : ' l — .059856 (2 k — I)2 + .0009 (2 k — I)4 the rise of voltage by inductance and capacity. Substituting : k= 1 2 3 4 56 or, 2 £ - 1 = 1 3 5 7 9 11 it is, a = 1.03 1.36 3.76 2.18 .70 .38 That is, the fundamental will be increased at open circuit by 3 per cent, the triple harmonic by 36 per cent, the quintuple harmonic by 276 per cent, the septuple harmonic by 118 per cent, while the still higher harmonics are reduced. The maximum possible rise will take place for : = 0, or, 2,- 1 = 5.77 That is, at a frequency : N = 346, and a = 14.4. That is, complete resonance will appear at a frequency between quintuple and septuple harmonic, and would raise the voltage at this particular frequency 14.4 fold. If the voltage shall not exceed the impressed voltage by more than 100 per cent, even at coincidence of the maximum of the harmonic with the maximum of the fundamental, the triple harmonic must be less than 70 per cent of the fundamental, the quintuple harmonic must be less than 26.5 per cent of the fundamental, the septuple harmonic must be less than 46 per cent of the fundamental. The voltage will not exceed twice the normal, even at a frequency of complete resonance with the higher har- monic, if none of the higher harmonics amounts to more EFFECTS OF HIGHER HARMONICS. 405 than 7 per cent, of the fundamental. Herefrom it follows that the danger of resonance in high potential lines is in general greatly over-estimated, since the conditions assumed in this instance are rather more severe than found in prac- tice, the capacity current of the line very seldom reaching 20% of the main current. 247. The power developed by a complex harmonic wave in a non-inductive circuit is the sum of the powers of the individual harmonics. Thus if upon a sine wave of alter- nating E.M.F. higher harmonic waves are superposed, the effective E.M.F., and the power produced by this wave in a given circuit or with a given effective current, are increased. In consequence hereof alternators and synchronous motors of ironclad unitooth construction — that is, machines giving waves with pronounced higher harmonics — give with the same number of turns on the armature, and the same mag- netic flux per field pole at the same frequency, a higher output than machines built to produce sine waves. 248. This explains an apparent paradox : If in the three-phase star-connected generator with the magnetic field constructed as shown diagrammatically in Fig. 162, the magnetic flux per pole = $, the number of turns in series per circuit = n, the frequency = N, the E.M.F. between any two collector rings is: E= V2~7T^2;z10-8. since 2« armature turns simultaneously interlink with the magnetic flux 3>. The E.M.F. per armature circuit is : hence the E.M.F. between collector rings, as resultant of two E.M.Fs. e displaced by 60° from each other, is : 406 ALTERNATING-CURRENT PHENOMENA. while the same E.M.F. was found by direct calculation from number of turns, magnetic flux, and frequency to be equal to 2e; that is the two values found for the same E.M.F. have the proportion V3 : 2 = 1 : 1.154. Fig. 178. Three-phase Star-connected Alternator. This discrepancy is due to the existence of more pro- nounced higher harmonics in the wave e than in the wave E = e X V3, which have been neglected in the formula : Hence it follows that, while the E.M.F. between two col- lector rings in the machine shown diagrammatically in Fig. 178 is only e x V3, by massing the same number of turns in one slot instead of in two slots, we get the E.M.F. 2 e or 15.4 per cent higher E.M.F., that is, larger output. EFFECTS OF HIGHER HARMONICS. 407 It follows herefrom that the distorted E.M.F. wave of a unitooth alternator is produced by lesser magnetic flux per pole — that is, in general, at a lesser hysteretic loss in the armature or at higher efficiency — than the same effective E.M.F. would be produced with the same number of arma- ture turns if the magnetic disposition were such as to pro- duce a sine wave. 249. Inversely, if su<:h a distorted wave of E.M.F. is impressed upon a magnetic circuit, as, for instance, a trans- former, the wave of magnetism in the primary will repeat in shape the wave of magnetism interlinked with the arma- ture coils of the alternator, and consequently, with a lesser maximum magnetic flux, the same effective counter E.M.F. will be produced, that is, the same power converted in the transformer. Since the hysteretic loss in the transformer depends upon the maximum value of magnetism, it follows that the hysteretic loss in a transformer is less with a dis- torted wave of a unitooth alternator than with a sine wave. Thus with the distorted waves of unitooth machines, generators, transformers, and synchronous motors — and induction motors in so far as they are transformers — operate more efficiently. 250. From another side the same problem can be approached. If upon a transformer a sine wave of E.M.F. is im- pressed, the wave of magnetism will be a sine wave also. If now upon the sine wave of E.M.F. higher harmonics, as sine waves of triple, quintuple, etc., frequency are superposed in such a way that the corresponding higher harmonic sine waves of magnetism do not increase the maximum value of magnetism, or even lower it by a coincidence of their negative maxima with the positive maximum of the fundamental, — in this case all the power represented by these higher harmonics of E.M.F. will be 408 ALTERNATING-CURRENT PHENOMENA. transformed without an increase of the hysteretic loss, or even with a decreased hysteretic loss. Obviously, if the maximum of the higher harmonic wave of magnetism coincides with the maximum of the funda- mental, and thereby makes the wave of magnetism more pointed, the hysteretic loss will be increased more than in proportion to the increased power transformed, i.e., the efficiency of the transformer will be lowered. That is : Some distorted waves of E.M.F. are transformed at a lesser, some at a larger, hysteretic loss than the sine wave, if the same effective E.M.F. is impressed upon the transformer. The unitooth alternator wave and the first wave in Fig. 175 belong to the former class ; the waves derived from continuous-current machines, tapped at two equi-distant points of the armature, in general, to the latter class. 251. Regarding the loss of energy by Foucault or eddy currents, this loss is not affected by distortion of wave shape, since the E.M.F. of eddy currents, as induced E.M.F., is proportional to the secondary E.M.F. ; and thus at constant impressed primary E.M.F., the energy consumed by eddy currents bears a constant relation to the output of the secondary circuit, as obvious, since the division of power between the two secondary circuits — the eddy current circuit, and the useful or consumer cir- cuit — is unaffected by wave-shape or intensity of mag- netism. 252. In high potential lines, distorted waves whose maxima are very high above the effective values, as peaked waves, may be objectionable by increasing the strain on the insulation. It is, however, not settled yet beyond doubt whether the striking-distance of a rapidly alternat- ing potential depends upon the maximum value or upon EFFECTS OF HIGHER HARMONICS. 409 some value between effective and maximum. Since dis- ruptive phenomena do not always take place immediately after application of the potential, but the time element plays ari important part, it is possible that insulation-strain and striking-distance is, in a certain range, dependent upon the effective potential, and thus independent of the wave-shape. In this respect it is quite likely that different insulating materials show a different behavior, and homogeneous solid substances, as paraffin, depend in their disruptive strength upon the maximum value of the potential difference, while heterogeneous materials, as mica, laminated organic sub- stances, air, etc., that is substances in which the disruptive strength decreases with the time application of the potential difference, are less affected by very high peaks of E.M.F. of very short duration. In general, as conclusions may be derived that the im- portance of a proper wave-shape is generally greatly over- rated, but that in certain cases sine waves are desirable, in other cases certain distorted waves are preferable. 410 ALTERNATING-CURRENT PHENOMENA.