CHAPTER XXII. DISTORTION OF WAVE-SHAPE AND ITS CAUSES. 233. In the preceding chapters we have considered the alternating currents and alternating E.M.Fs. as sine waves or as replaced by their equivalent sine waves. While this is sufficiently exact in most cases, under certain circumstances the deviation of the wave from sine shape becomes of importance, and with certain distortions it may not be possible to replace the distorted wave by an equivalent sine wave, since the angle of phase displacement of the equivalent sine wave becomes indefinite. Thus it becomes desirable to investigate the distortion of the wave, its causes and its effects. Since, as stated before, any alternating wave can be represented by a series of sine functions of odd orders, the investigation of distortion of wave-shape resolves itself in the investigation of the higher harmonics of the alternating wave. In general we have to distinguish between higher har- monics of E.M.F. and higher harmonics of current. Both depend upon each other in so far as with a sine wave of impressed E.M.F. a distorting effect will cause distortion of the current wave, while with a sine wave of current passing through the circuit, a distorting effect will cause higher harmonics of E.M.F. 234. In a conductor revolving with uniform velocity through a uniform and constant magnetic field, a sine wave of E.M.F. is induced. In a circuit with constant resistance and constant reactance, this sine wave of E.M.F. produces 384 ALTERNATING-CURRENT PHENOMENA. a sine wave of current. Thus distortion of the wave-shape or higher harmonics may be due to : lack of uniformity of the velocity of the revolving conductor ; lack of uniformity or pulsation of the magnetic field ; pulsation of the resis- tance ; or pulsation of the reactance. The first two cases, lack of uniformity of the rotation or of the magnetic field, cause higher harmonics of E.M.F. at open circuit. The last, pulsation of resistance and reac- tance, causes higher harmonics only with a current flowing in the circuit, that is, under load. Lack of uniformity of the rotation is of no practical in- terest as cause of distortion, since in alternators, due to mechanical momentum, the speed is always very nearly uniform during the period. Thus as causes of higher harmonics remain : 1st. Lack of uniformity and pulsation of the magnetic field, causing a distortion of the induced E.M.F. at open circuit as well as under load. 2d. Pulsation of the reactance, causing higher harmonics under load. 3d. Pulsation of the resistance, causing higher harmonics under load also. Taking up the different causes of higher harmonics we have : — Lack of Uniformity and Pulsation of tJie Magnetic Field. 235. Since most of the alternating-current generators contain definite and sharply defined field poles covering in different types different proportions of the pitch, in general the magnetic flux interlinked with the armature coil will not vary as simply sine wave, of the form : $ cos /?, but as a complex harmonic function, depending on the shape and the pitch of the field poles, and the arrangement of the armature conductors. In this case, the magnetic flux issu- DISTORTION OF WAVE-SHAPE. 385 ing from the field pole of the alternator can be represented by the general equation, 4> = A0 + A, cos /8 + A* cos 2(3 + Az cos 3/8 + . . . + ^ sin £ + -#2 sin 2 0 + .#, sin 3 ft + . . . If the reluctance of the armature is uniform in all directions, so that the distribution of the magnetic flux at the field-pole face does not change by the rotation of the armature, the rate of cutting magnetic flux by an armature conductor is <£, and the E.M.F. induced in the conductor thus equal thereto in wave shape. As a rule A0, Az, At . . . By B± equal zero ; that is, successive field poles are equal in strength and dis- tribution of magnetism, but of opposite polarity. In some types of machines, however, especially induction alternators, this is not the case. The E.M.F. induced in a full-pitch armature turn — that is, armature conductor and return conductor distant from former by the pitch of the armature pole (corresponding to the distance from field pole center to pole center) is, 8 = $0 - 3>180 = 2 \Ai cos /3 + Aa cos 3 (3 + A6 cos 5 0 + . . . + BI sin j3 + Bz sin 3 ft + jB6 sin 5 ft + . . . \ Even with an unsymmetrical distribution of the magnetic flux in the air-gap, the E.M.F. wave induced in a full-pitch armature coil is symmetrical ; the positive and negative half waves equal, and correspond to the mean flux distribution of adjacent poles. With fractional pitch windings — that is, windings whose turns cover less than the armature pole pitch — the induced E.M.F. can be unsymmetrical with unsymmetrical magnetic field, but as a rule is symmetrical also. In unitooth alternators the total induced E.M.F. has the same shape as that induced in a single turn. With the conductors more or less distributed over the surface of the armature, the total induced E.M.F. is the resultant of several E.M.Fs. of different phases, and is thus more uniformly varying ; that is, more sinusoidal, approaching 386 ALTERNATING-CURRENT PHENOMENA. sine shape, to within 3% or less, as for instance the curves Fig. 169 and Fig. 170 show, which represent the no-load and full-load wave of E.M.F. of a three-phase multitooth alternator. The principal term of these harmonics is the third harmonic, which consequently appears more or less in all alternator waves. As a rule these harmonics can be considered together with the harmonics due to the varying reluctance of the magnetic circuit. In ironclad alternators with few slots and teeth per pole, the passage of slots across the field poles causes a pulsation of the magnetic reluc- tance, or its reciprocal, the magnetic inductance of the circuit. In consequence thereof the magnetism per field pole, or at least that part of the magnetism passing through the armature, will pulsate with a frequency 2 y if y = num- ber of slots per pole. Thus, in a machine with one slot per pole, the instanta- neous magnetic flux interlinked with the armature con- ductors can be expressed by the equation : <£ = $ cos /? [1 + e cos [2 (3 — o>] j where, ® = average magnetic flux, c = amplitude of pulsation, and to = phase of pulsation. In a machine with y slots per pole, the instantaneous flux interlinked with the armature conductors will be : = & cos /8 { 1 + c cos [2 y ft — o>] | , if the assumption is made that the pulsation of the magnetic flux follows a simple sine law, as first approximation. In general the instantaneous magnetic flux interlinked with the armature conductors will be : ^ = * cos 0 {1 + 6! cos (2 0 - SO + e, cos (4 £ - oV,) + . . . f , where the term ey is predominating if y = number of arma- ture slots per pole. This general equation includes also the effect of lack of uniformity of the magnetic flux. DISTORTION OF WAVE-SHAPE. 387 Nil LoLd ,"14 .5 y, Fig. 169. No-load of E.M.F. of Multitooth Three-phaser. 130 JMtfl I oad 120 '' = 12 7.0 »= 3 ; ^ '-- --- >s, 110 j^ 5 100 / \ 90 j 7 V SO / s 70 / s 60 / ^ 50 / \, 10 // '^ , 30 /' \ 20 / \ 10 // \\ 0 •'/ /- "--v^ r- 1 — ^.^ ^ — V 10 f ' 10 50 30 10 g (50 70 SO 90 100 no 120 13(1 140 150 100 170 ISO Fig. 170. Full-Load Waue of E.M.F. of Multitooth Three-phaser. 388 ALTERNATING-CURRENT PHENOMENA. In case of a pulsation of the magnetic flux with the frequency 2y, due to an existence of y slots per pole in the armature, the instantaneous value of magnetism interlinked with the armature coil is : <£ = $ COS ft {1 + e COS [2 y ft — £]}. Hence the E.M.F. induced thereby : e = — n — — dt d *» And, expanded : e= V27rA^ (sin /3 + 1 sin (0 — £) + ^ sin (3 /? - £)} ; that is : In a unitooth single-phaser a pronounced triple harmonic may be expected, but no pronounced higher harmonics. Fig. 171 shows the wave of E.M.F. of the main coil of a monocyclic alternator at no load, represented by : e = E (sin (3 — .242 sin ( 3 /3 — 6.3) — .046 sin (5/3- 2.6) + .068 sin (7 £ — 3.3) — .027 sin (9 ft — 10.0) — .018 sin (11 /3 - 6.6) + .029 sin (13 ft - 8.2)}; hence giving a pronounced triple harmonic only, as expected. If y = 2, it is : e = V2 TT Nn 4> j sin £ + ^ sin (3 ft - «J) + |f sin (5 ft - Si) DISTORTION OF WAVE-SHAPE. 389 the no-load wave of a unitooth quarter-phase machine, hav- ing pronounced triple and quintuple harmonics. If 7 = 3, it is : in/3+ sin(5j8— fi) + sin (7 ft - S>) I . That is : In a unitooth three-phaser, a pronounced quin- tuple and septuple harmonic may be expected, but no pro- nounced triple harmonic. Fig. 155. No-load Wave of E.M.F. of Unitooth Monocyclic Alternator. Fig. 156 shows the wave of E.M.F. of a unitooth three- phaser at no load, represented by : e = E (sin /3 — .12 sin (3 £ — 2.3) — .23 sin (5 (3 — 1.5) + .134 sin (7 ft _ 6.2) - .002 sin (9 /3 + 27.7) - .046 sin (11 /? — 5.5) +.031 sin (13)8-61.5)}. Thus giving a pronounced quintuple and septuple and a lesser triple harmonic, probably due to the deviation of, the field from uniformity, as explained above, and deviation of the pulsation of reluctance from sine shape. In some especially favorable cases, harmonics as high as the 23d and 25th have been observed, caused by pulsation of the reluc- tance. 390 ALTERNATING-CURRENT PHENOMENA. V 100 50 60 70 80 90 1 00 30 140 150 160 170 180 Fig. 172. No-load Wave of E.M.F. of Unitooth Three-phase Alternator. In general, if the pulsation of the magnetic inductance is denoted by the general expression : l + ^"cYcos(2yj8-aY), 1 the instantaneous magnetic flux is : 00 = $ cos 13 ey cos (2 y ff - cos((2y+l) hence, the E.M.F. 2 ; sm(P — DISTORTION OF WAVE-SHAPE. 391 Pulsation of Reactance. 237. The main causes of a pulsation of reactance are : magnetic saturation and hysteresis, and synchronous motion. Since in an ironclad magnetic circuit the magnetism is not proportional to the M.M.F., the wave of magnetism and thus the wave of E.M.F. will differ from the wave of cur- rent. As far as this distortion is due to the variation of permeability, the distortion is symmetrical and the wave of induced E.M.F. 'represents no power. The distortion caused by hysteresis, or the lag of the magnetism behind the M.M.F., causes an unsymmetrical distortion of the wave which makes the wave of induced E.M.F. differ by more than 90° from the current wave and thereby represents power, — the power consumed by hysteresis. In practice both effects are always superimposed ; that is, in a ferric inductance, a distortion of wave-shape takes place due to the lack of proportionality between magnetism and M.M.F. as expressed by the variation in the hysteretic cycle. This pulsation of reactance gives rise to a distortion consisting mainly of a triple harmonic. Such current waves distorted by hysteresis, with a sine wave of impressed E.M.F., are shown in Figs. 66 to 69, Chapter X., on Hy- steresis. Inversely, if the current is a sine wave, the mag- netism and the E.M.F. will differ from sine shape. For further discussion of this distortion of wave-shape by hysteresis, Chapter X. may be consulted. 238. Distortion of wave-shape takes place also by the pulsation of reactance due to synchronous rotation, as dis- cussed in chapter on Reaction Machines. In Figs. 148 and 149, at a sine wave of impressed E.M.F., the distorted current waves have been constructed. Inversely, if a sine wave of current, / = / cos B, 392 ALTERNATING-CURRENT PHENOMENA. passes through a circuit of synchronously varying reac- tance ; as for instance, the armature of a unitooth alterna- tor or synchronous motor — or, more general, an alternator whose armature reluctance is different in different positions with regard to the field poles — and the reactance is ex- pressed by or, more general, X = the wave of magnetism is X = x 1 + yr ^ cos (2 y ft- & l hence the wave of induced E.M.F. = *sin/3 + sin ()8 - fflO + [e, sin ((2 y + 1) sin ((2y+ l)/8 -«,+!)]} ; that is, the pulsation of reactance of frequency, 2y, intro- duces two higher harmonics of the order (2y — 1), and (2y + l\ If ^T=^l , =*{sin0 + |sinG8-a) + .|l sin (3/J-o,)^ Since the pulsation of reactance due to magnetic satu- ration and hysteresis is essentially of the frequency, 21V, DISTORTION OF WAVE-SHAPE. 393 — that is, describes a complete cycle for each half -wave of current, — this shows why the distortion of wave-shape by hysteresis consists essentially of a triple harmonic. The phase displacement between e and i, and thus the power consumed or produced in the electric circuit, depend \ipon the angle, o>, as discussed before. 239. In case of a distortion of the wave-shape by reactance, the distorted waves can be replaced by their equivalent sine waves, and the investigation with suffi- cient exactness for most cases be carried out under the assumption of sine waves, as done in the preceding chapters. Similar phenomena take place in circuits containing polarization cells, leaky condensers, or other apparatus representing a synchronously varying negative reactance. Possibly dielectric hysteresis in condensers causes a dis- tortion similar to that due to magnetic hysteresis. Pulsation of Resistance. 240. To a certain extent the investigation of the effect of synchronous pulsation of the resistance coincides with that of reactance ; since a pulsation of reactance, when unsymmetrical with regard to the current wave, introduces an energy component which can be represented by an " effective resistance." Inversely, an unsymmetrical pulsation of the ohmic resistance introduces a wattless component, to be denoted by "effective reactance." A typical case of a synchronously pulsating resistance is represented in the alternating arc. The apparent resistance of an arc depends upon the current passing through the arc ; that is, the apparent resistance Of the arc = Potential difference^between electrodes jg high for small currents, low for large currents. Thus in an alternating arc the apparent resistance will vary during 304 ALTERNATING-CURRENT PHENOMENA. every half-wave of current between a maximum value at zero current and a minimum value at maximum current, thereby describing a complete cycle per half-wave of cur- rent. Let the effective value of current passing through the arc be represented by /. Then the instantaneous value of current, assuming the current wave as sine wave, is represented by / = 7V2sin/3; and the apparent resistance of the arc, in first approxima- tion, by R = r (1 + e cos 2 j8) ; thus the potential difference at the arc is e = iR = /V2Vsin/3(l -f e cos 2/3) Hence the effective value of potential difference, and the apparent resistance of the arc, r.-f-ry/t-. + f The instantaneous power consumed in the arc is, Hence the effective power, DISTORTION OF WAVE-SHAPE. 395 The apparent power, or volt amperes consumed by the arc, is, thus the power factor of the arc, that is, less than unity. 241. We find here a case of a circuit in which the power factor — that is, the ratio of watts to volt amperes — differs from unity without any displacement of phase ; that is, while current and E.M.F. are in phase with each other, but are distorted, the alternating wave cannot be replaced by an equivalent sine wave ; since the assumption of equivalent sine wave would introduce a phase displace- ment, cos w =/ of an angle, w, whose sign is indefinite. As an instance are shown, in Fig. 173 for the constants, 1= 12 r= 3 £ =.9 the resistance, R = 3 {I + .9 cos 2 /3) ; the current, * = 17 sin /3 ; tha potential difference, e = 28 (sin ft + .82 sin 3 £). In this case the effective E.M.F. is £=25.5; 396 ALTERNATING-CURRENT PHENOMENA. the apparent resistance, the power, the apparent power, the power factor, r0 = 2.13 ; P = 244 ; El =307; / = .796. Fig. 173. Periodically Varying Resistance. As seen, with a sine wave of current the E.M.F. wave in an alternating arc will become double-peaked, and rise very abruptly near the zero values of current. Inversely, with a sine wave of E.M.F. the current wave in an alter- nating arc will become peaked, and very flat near the zero values of E.M.F. 242. In reality the distortion is of more complex nature ; since the pulsation of resistance in the arc does not follow DISTORTION OF WAVE-SHAPE. 397 a simple sine law of double frequency, but varies much more abruptly near the zero value of current, making thereby the variation of E.M.F. near the zero value of current much more abruptly, or, inversely, the variation of current more flat. A typical wave of potential difference, with a sine wave of current passing through the arc, is given in Fig. 174.* 1 13 13 1 15 ONE PAIR CARBONS EG U LATE D BY HAND A. C. dynamo e. m. f •' " " current*. " " " watts. 7 18 19 20 S Fig. 174. Electric Arc. 243. The value of e, the amplitude of the resistance pulsation, largely depends upon the nature of the electrodes and the steadiness of the arc, and with soft carbons and a steady arc is small, and the power factor f of the arc near unity. With hard carbons and an unsteady arc, e rises greatly, higher harmonics appear in the pulsation of resis- tance, and the power factor f falls, being in extreme cases even as low as .6. The conclusion to be drawn herefrom is, that photo- metric tests of alternating arcs are of little value, if, besides current and voltage, the power is not determined also by means of electro-dynamometers. * From American Institute of Electrical Engineers, Transactions, 1890, p- 376. Tobey and Walbridge, on the Stanley Alternate Arc Dynamo. 398 A L TERN A TING-CURRENT PHENOMENA .