CHAPTER VII SHAPING OF WAVES : GENERAL 69. In alternating-current engineering, the sine wave, as shown in Fig. 46, is usually aimed at as the standard. This is not duo to any inherent merit of the sine wave. For all those pm-poses, where the energy developed by the cur- rent in a resistance is the object, as for incandescent lighting, heating, etc., any wave form is equally satisfactory, as the energy of the wave depends only on its effective value, but not on it^ shape. With regards to insulation stress, as in high-voltage systems, a flat-top wave of voltage and current, such as shown in Fig. 47, would be preferable, as it has a higher effective value, with tho same maximimi value and therefore with the same strain on tho insulation, and therefore transmits more energy than the sine wave. Fig. 46. Inversely, a peaked wave of voltage, such as Fig. 48, and such as the common saw-tooth wave of the uni tooth alternator, is superior in transformers and similar devices, as it transform s tho energy with less hysteresis loss. The peaked voltage wave. Fig. 4:8, gives a flat-topped wave of magnetism. Fig. 47, and therciby transforms the voltage with a lesser maximum magnetic flux, than ^ sine wave of the same effective value, that is, the same powc^r. the hysteresis loss depends on the maximum value of t\u) mag- 5tic flux, the reduction of the maximum value of tho magncitic; fl^Jx, due to a peaked voltage wave, results in a lower hyHtorvMiH ioes, and thus higher efficiency of transformation. This reduc- ^i^on of loss may amount to as much as 15 to 25 p(;r cent, of the ^^otal hysteresis loss, in extreme cases. Inversely, a peaked voltage wave like Fig. 48 would be obj(i(j- t-xonable in high- voltage transmission apparatus, by giving an un- necessary high insulation strain, and a flat-top wave of voltage ^vke Fig. 47, when impressed upon a transformer, would give a ^^^ed wave of magnetism and thereby an increased hyHteresis Ill 112 ELECTRIC CIRCUITS The advantage of the sine wave is, that it remains unch&nged in shape under most conditions, while this is not the case with any- other wave shape, and any other wave shape thus introduces the danger, that under certain conditions, or in certain parts of the circuit, it may change to a shape which is undesirable or even Figs. 46 to 49. dangerous. Voltage, e, and current, i, are related to each other \>y proportionality, by differentiation and by integration, with sistance, r, inductance, L, and capacity, C, as factors, e = n, r di e = cl idt, and as the differentials and integrals of sines are sines, as long SB r, L and C are constant — which is mostly the case — sine waves of SHAPING OF WAVES 113 voltage produce sine waves of current and inversely, that is, the sine wave shape of the electrical quantities remains constant. A flat-topped current wave like Fig. 47, however, would by differentiation give a self-inductive voltage wave, which is peakedj like Fig. 48, A voltage wave like Fig. 48, which is more efficient in transformation, may by further distortion, as by intensifica- tion of the triple harmonic by line capacity, assume the shape, Fig. 49, and the latter then would give, when impressed upon a transformer, a double-peaked wave of magnetism, Fig. 50, and such wave of magnetism gives a magnetic cycle with two small i secondary loops at high density, as shown in Fig. 51, and an additional energy loss by hysteresis in these two secondary loops, which is considerable due to the high mean magnetic density, at which the secondary loop is traversed, so that in spite of the reduced maximum dux density, the hysteresis loss may be increased. Therefore, in alternating-current engineering, the aim gener- 114 ELECTRIC CIRCUITS ally is to produce and use a wave which is a sine wave or nearly so. 60. In an alternating-current generator, synchronous or in- duction machine, commutating machine, etc., the wave of voltage induced in a single armature conductor or "face conductor" equals the wave of field flux distribution around the periphery of the magnet field, modified, however, by the reluctance pulsations of the magnetic circuit, where such exist. As the latter produce higher harmonics, they are in general objectionable and to be avoided as far as possible. By properly selecting the length of the pole arc and the length of the air-gap between field and armature, a sinusoidal field flux distribution and thereby a sine wave of voltage induced in the armature face conductor could be produced. In this, direction, however, the designer is very greatly limited by economic con- sideration: length of pole arc, gap length, etc., are determined within narrow limits by the requirement of the economic use of the material, questions of commutation, of pole-face losses, of field excitation, etc., so that as a rule the field flux distribution and with it the voltage induced in a face conductor differs materially from sine shape. The voltage induced in a face conductor may contain even har- monics as well as odd harmonics, and often, as in most inductor alternators, a constant term. The constant term cancels in all turn windings, as it is equal and opposite in the conductor and return conductor of each turn. Direct-current induction (continuous, or pulsating current) thus is possible only in half-turn windings, that is, windings in which each face conductor has a collector ring at either end, so-called unipolar machines (see "Theory and Calculation of Electrical Apparatus"). In every winding, which repeats at every pole or 180 electrical degrees, as is almost always the case, the even harmonics cancel, even if they existed in the face conductor. In any machine in which the flux distribution in successive poles is the same, and merely opposite in direction, that is, in which the poles are symmet- rical, no even harmonics are induced, as the field flux distribution contains no even harmonics. Even harmonics would, however, exist in the voltage wave of a machine designed as shown diagram- matically in Fig. 52, as follows: The south poles S have about one-third the width of the north SHAPING OF WAVES 115 poles N, and the armature winding is a unitooth 60 per cent, pitch winding; shown as A in Fig. 52. Assuming sinusoidal field flux distribution in the air-gaps under the poles N and S of Fig. 52, curve I in Fig. 53 shows the field flux distribution and thus the voltage induced in a single-face con- ductor. Curve II shows the voltage wave in a 50 per cent, pitch turn and therewith that of the winding A . As seen, this contains a pronounced second harmonic in addition to the fundamental. If, then, a second 50 per cent, pitch winding is located on the arma- ria. 52. ture, shown as fi in Fig. 52, by connecting B and A in series with each other in such direction that the fundamentals cancel (that is, in opposition for the fundamental wave), we get voltage wave III of Fig. 53, which contains only the even harmonics, that is, is of double frequency. Connecting A and B in series so that the fundamentals add and the second harmonics cancel, gives the wave IV. If the machine is a three-phase F-connected alterna- tor, with ciu^e IV as the voltage per phase, or Y voltage, the delta or terminal voltage, derived by combination of two Y vol- tages under 60**, then is given by the curve V of Fig. 53. Fig. 54 shows the corresponding curves for the flux distribution of uni- form density under the pole and tapering off at the pole corners, curve I, such as would approximately correspond to actual con- 116 ELECTRIC CIRCUITS ditions. As seen, curve III as well as V are approximately sine waves, but the one of twice the frequency of the other. Thus, such a machine, by reversing connections between the two wind- ings A and B, could be made to give two frequencies, one double the other, or as synchronous motor could run at two speeds, one one-half the other. Fig. 53. 61. Distribution of the winding over an arc of the periphery^ o^ the armature eliminates or reduces the higher harmonics, so tti^^t the terminal voltage wave of an alternator with distributed wind- ing is less distorted, or more nearly sine-shaped, than that of ^ single turn of the same winding (or that of a unitooth alternator)- The voltage waves of successive turns are slightly out of pha^^ with each other, and the more rapid variations due to higher hsLMT- monies thus are smoothed out. In two armature turns different-'*' SHAPING OF WAVES 117 in position on the armature circumference by 8 electrical degrees (** electrical degrees" means counting the pitch of two poles as 360®), the fundamental waves are 8 degrees out of phase, the third harmonics 35 degrees, the fifth harmonics 55 degrees, and so on, and their resultants thus get less and less, and becomes zero for that harmonic n, where n8 = 180**. Fig. 64. If 6 = ei sin * + 63 sin 3 (*— as) + 65 sin 6 (*— as) + 67 sin 7 (*— ay) + . . . (1) is the voltage wave of a single turn, and the armature winding of 'm turns covers an arc of w electrical degrees on the armature periphery (per phase), the coefficients of the harmonics of the resultant voltage wave are 118 ELECTRIC CIRCUITS En = men avg. cos nu ■^ 2 nu ~2 (2) or, since + no) avg. cos < 2 . ncu = — sm -^r- no) 2 ncu r, 2r/i . ?ia) -Bn = — en Sin -TT- nco 2 (3) and ^ = — { ei sin TT sin + "TT sin -^ sin 3(0 — as) 0) [ 2, 6 2 + -g Sin -2" sin 5(0 - as) + . } (4) Thus, in a three-phase winding like that of the three-phase synchronous converter, in which each phase covers an arc of 120° 2x .^ . 0) TT , = -0-, It is 2 = Q, nence, E = 3m\/3 2x Ci sin ^ = sin 5{ — as) + ^ sin 7(0 — ay) h . . . . | (5) that is, the third harmonic and all its multiples, the ninth, fif- teenth, etc., cancel, all other harmonics are greatly reduced, the more, the higher theh- order. In a three-phase F-connected winding, in which each phase TT covers 60° = « of the periphery, as commonly used in induction and synchronous machines, it is ^ = ^> hence, E = — I ei sin -}- ^ 63 sin 3(0 — as) + -= e^ sin 5(0 — as) 1 2 - = 67 sin 7(0 - ay) — Q eg sin 9(0 - ag) — Tj-eii sin 11(0 - an) + ^^613 sin 13(0 — a^) H • • • } (6) SHAPING OF WAVES 119 Here the third harmonics do not cancel, but are especially large. Thus in a F-connected three-phase machine of the usual 60° winding, the Y voltage may contain pronounced third harmonics, which, however, cancel in the delta voltage. Thus with the distributed armature winding, which is now al- most exclusively used, the wave-shape distortion due to the non- sinusoidal distribution of the field flux is greatly reduced, that is, the higher harmonics in the voltage wave decreased, the more so, the higher their order, and very high harmonics, such as the seven- teenth, thirty-fifth, etc., therefore do not exist in such machines to any appreciable extent, except where produced by other causes. Such are a pulsation of the magnetic reluctance of the field due to the armature slots, or a pulsation of the armature reactance, as discussed in Chapter XXV of ** Theory and Calculation of Alter- nating-current Phenomena,'' or a space resonance of the armature conductors with some of the harmonics. The latter may occur if the field flux distribution contains a harmonic of such order, that the voltages induced by it are in phase in the successive arma- t\ire conductors, and therefore add, that is, when the spacing of the armatiu'e conductors coincides with a harmonic of the field flux, and the armature turn pitch and winding pitch are such that this harmonic does not cancel. Inversely, if two turns are displaced from each other on the 1 TT armatiu'e periphery by - of the pole pitch, or - , and are connected in series, then in the resultant voltage of these two turns, the n^^ . TT harmonics are out of phase by n times - , or by tt = 180°, that is, are in opposition and so cancel. Thus in a unitooth F-connected three-phase alternator, while each phase usually contains a strong third harmonic, the terminal voltage can contain no third harmonic or its multiples: the two phases, which are in series between each pair of terminals, are one-third pole pitch, or 60 electrical degrees displaced on the armatiu'e periphery, and their third harmonic voltages therefore 3 X 60 = 180° displaced, or opposite, that is, cancel, and no third harmonic can appear in the terminal voltage wave, or delta volt- age, but a pronounced third harmonic may exist — and give trouble — in the voltage between each terminal and the neutral, or the Y voltage. 62. By the use of a fractional-pitch armature winding, higher harmonics can be eliminated. Assume the two sides of the arma- 120 ELECTRIC CIRCUITS ture turn, conductor and return conductor, are not separated from each other by the full pitch of the field pole, or 180 electrical degrees, but by less (or more) ; that is, each armature turn or coil covers not the full pitch of the pole, but the part p less (or more), that is, covers (1 ± p) 180®. The coil then is said to be (1 ± p) fractional pitch, or has the pitch deficiency p. The voltages in- duced in the two sides of the coil then are not equal and in phase, but are out of phase by 180 p for the fimdamental, and by 180 np for the n^^ harmonic. Thus, if np = 1, for this n^ har- monic the voltages in the two sides of the coil are equal and oppo- site, thus cancel, and this harmonic is eliminated. Therefore, two-thirds pitch winding eliminates the third har- monic, four-fifths pitch winding the fifth harmonic, etc. Peripherally displacing half the field poles against the other half by the fraction q of the pole pitch, or by 180 q electrical de- grees, causes the voltages induced by the two sets of field poles to be out of phase by 180 nq for the n*** harmonic, and thereby eliminates that harmonic, for which ng = 1. By these various means, if so desired, a niunber of harmonics can be eliminated. Thus in a F-connected three-phase alternator with the winding of each phase covering 60 electrical degrees, with four-fifths pitch winding and half the field poles offset against the other by one-seventh of the pole pitch, the third, fifth, and seventh harmonic and their multiples are eliminated, that is, the lowest harmonic existing in the terminal voltage of such a ma- chine is the eleventh, and the machine contains only the eleventh, thirteenth, seventeeth, ninteenth, twenty-third, twenty-ninth, thirty-first, thirty-seventh, etc. harmonics. As by the distrib- uted winding these harmonics are greatly decreased, it follows that the terminal voltage wave would be closely a sine, irrespec- tive of the field flux distribution, assuming that no slot harmonics exist. 63. In modern machines, the voltage wave usually is very closely a sine, as the pronounced lower harm'onics, caused by the field flux distribution, which gave the saw-tooth, flat-top, peak or multiple-peak effects in the former unitooth machines, are greatly reduced by the distributed winding and the use of frac- tional pitch. Individual high harmonics, or pairs of high harmon- ics, are occasionally met, such as the seventeenth and ninteenth, or the thirty-fifth and thirty-seventh, etc. They are due to the pulsation of the magnetic field flux caused by the pulsation of the SHAPING OF WAVES 121 field reluctance by the passage of the armature slots, and occa- sionally, under load, by magnetic saturation of the armature self- inductive flux, that is, flux produced by the current in an arma- ture slot and surrounding this slot, in cases where very many ampere conductors are massed in one slot, and the slot opening bridged or nearly so. The low harmonics, third, fifth, seventh, are relatively harm- less, except where very excessive and causing appreciable increase of the maximiun voltage, or the maximum magnetic flux ahd thus hysteresis loss. The very high harmonics as a rule are rela- tively harmless in all circuits containing no capacity, since they are necessarily fairly small and still further suppressed by the inductance of the circuit. They may become serious and even dangerous, however, if capacity is present in the circuit, as the current taken by capacity is proportional to the frequency, and even small voltage harmonics, if of very high order, that is, high frequency, produce very large currents, and these in turn may cause dangerous voltages in inductive devices connected in series into the circuit, such as current transformers, or cause resonance effects in transformers, etc. With the increasing extent of very high-voltage transmission, introducing capacity into the systems, it thus becomes increasingly important to keep the very high harmonics practically out of the voltage wave. Incidentally it follows herefrom, that the specifications of wave shape, that it should be within 5 per cent, of a sine wave, which is still occasionally met, has become irrational: a third harmonic of 5 per cent, is practically negligible, while a thirty-fifth harmonic of 5 per cent., in the voltage wave, would hardly be permissible. This makes it necessary in wave-shape specifications, to discriminate against high harmonics. One way would be, to specify not the wave shape of the voltage, but that of the current taken by a small condenser connected across the voltage. In the condenser current, the voltage harmonics are multiplied by their order. That is, the third harmonic is increased three times, the fifth harmonic five times, the thirty-fifth harmonic 35 times, etc. However, this probably overemphasizes the high harmonics, gives them too much weight, and a better way appears to be, to specify the current wave taken by a small condenser having a specified amount of non-inductive resistance in series. Thus for instance, if x = 1000 ohms = capacity reactance of the condenser, at fundamental frequency, r = 100 ohms = re- 122 ELECTRIC CIRCUITS sistance in series to the condenser, the impedance of this circuit, for the n*^ harmonic, would be rz -^ inrk 1000. .-V Z„ = r-j- = 100--^j (7) or, absolute, the impedance. Zn = 1000^^ + 0.01 (8) and, the admittance, _ 0.001 n . . ^'^ "" Vl + 0.01 n^ ^ ^ and therefore, the multiplying factor, ^ _yn _ 1.005 n . J — yi Vl + 0.01 n« this gives, for n f n / 1 1.0 13 8.0 3 2.9 15 8.4 5 4.5 25 9.3 7 5.8 35 9.6 9 6.7 45 9.8 11 7.4 00 10.0 Thus, with this proportion of resistance and capacity, the maxi- mum intensification is tenfold, for very high harmonics. By using a different value of the resistance, it can be made anything desired. A convenient way of judging on the joint effect of all harmonics of a voltage wave is by comparing the current taken by such a condenser and resistance, with that taken by the same condenser and resistance, at a sine wave of impressed voltage, of the same effective value. Thus, if the voltage wave e = 600 + I83 + 125 + 97 + 49 + 2ii + 3i3 + 3O2S + 2426 = 600 { 1 + 0.033 + 0.025 + 0.0157 + 0.00679+ 0.0033ii + 0.005 13 + 0.0523 + 0.0426 } SHAPING OF WAVES 123 (where the indices indicate the order of the harmonics) of ejffect- ive value e = VeOO^ + 182 + 122 + 92 + 42 + 22 + 32 + 302 + 242 = 601.7 is impressed upon the condenser resistance of the admittance, ynj the current wave is i = 0.603 { 1 + 0.0878 + 0.09b + 0.0877 + 0.04459 + 0.0247ii + 0.04,3 + 0.4623 + 0.3726 } = 0.603 X 1.173 = 0.707 while with a sine wave of voltage, of 60 = 601.7, the current would be io = 0.599, giving a ratio ^ = 1.18, u or 18 per cent, increase of current due to wave-shape distortion by- higher harmonics. 64. While usually the sine wave is satisfactory for the purpose for which alternating currents are used, there are numerous cases where waves of different shape are desirable, or even necessary for accomplishing the desired purpose. In other cases, by the internal reactions of apparatus, such as magnetic saturation, a wave-shape distortion may occur and requires consideration to avoid harmful results. Thus in the regulating pole converter (so-called "split-pole converter") variations of the direct-current voltage are produced at constant alternating-current voltage input, by superposing a third harmonic produced by the field flux distribution, as discussed under "Regulating Pole Converter" in "Theory and Calcula- tion of Electrical Apparatus." In this case, the third harmonic must be restricted to the local or converter circuit by proper transformer connections: either three-phase connection of the converter, or Y or double-delta connections of the transformers with a six-phase converter. The appearance of a wave-shape distortion by the third har- monic and its multiples, in the neutral voltage of F-connected transformers, and its intensifications by capacity in the secondary 124 ELECTRIC CIRCUITS circuit, and elimination by delta connection, has been discussed in Chapter XXV of "Theory and Calculation of Alternating- current Phenomena." In the flickering of incandescent lamps, and the steadiness of arc lamps at low frequencies, a difference exists between the flat- top wave of current with steep zero, and the peaked wave with flat zero, the latter showing appreciable flickering already at a somewhat higher frequency, as is to be expected. In general, where special wave shapes are desirable, they are usually produced locally, and not by the generator design, as with the increasing consolidation of all electric power supply in large generating stations, it becomes less permissible to produce a desired wave shape within the generator, as this is called upon to supply power for all purposes, and therefore the sine wave as the standard is preferable. One of the most frequent causes of very pronoimced wave- shape distortion, and therefore a very convenient means of pro- ducing certain characteristic deviations from sine shape, is mag- netic saturation, and as instance of a typical wave-shape distor- tion, its causes and effects, this will be more fully discussed in the following.