CHAPTER XXI REGULATING POLE CONVERTERS 230. With a sine wave of alternating voltage, and the com- mutator brushes set at the magnetic neutral, that is, at right angles to the resultant magnetic flux, the direct voltage of a syn- chronous converter is constant at constant impressed alternating voltage. It equals the maximum value of the alternating voltaga between two diametrically opposite points of the commutator, or "diametrical voltage," and the diametrical voltage is twice the voltage between alternating lead and neutral, or star or J voltage of the polyphase system. A change of the direct voltage, at constant, impressed alter- nating voltage (or inversely), can be produced: Either by changing the position angle between the eiuimjuia- tor brushes and the resultant magnetic flux, so that the direct voltage between the brushes is not the maximum diametrical alternating voltage but only a part thereof. Or by changing the maximum diametrical alternating voltage, at constant effective impressed voltage, by wave-shape distortion by the superposition of liigher harmonics. In the former case, only a reduction of the direct voltage lx*- low the normal value can lie produced, while in the latter case an increase as well as a reduction can be produced, an increase if the higher harmonies are in phase, and a reduction if the higher harmonics are in opposition to the fundamental wave of the dia- metrical or Y voltage. A. Variable Ratio by a Change of the Position Angle between Commutator Brushes and Resultant Magnetic Flux 231. Let, in the commutating maclane shown diagrammatic- ally in Fig. 195, the potential difference, or alternating voltage between one point, a, of the armature winding and the neutral, 0 (that is, the 1' voltage, or half the diametrical voltage) be repre- sented by the sine wave, Fig. 197. This potential difference is a maximum, e, when a stands at the magnetic neutral, at A or Ji. 422 REGULATING POLE CONVERTERS 423 If, therefore, the brushes are located at the magnetic neutral, A and B, the voltage between the brushes is the potential differ- ence between A and B, or twice the maximum Y voltage, 2 c, as indicated in Fig. 197. If now the brushes are shifted by an angle, r, to position C and D, Fig. 196, the direct voltage between s = r- -N Fig. 195. — Diagram of Fig. 196. — E.m.f. variation com mutating machine by shifting the brushes, with brushes in the mag- netic neutral. the brushes is the potential difference between C and Df or 2 e cos f with a sine wave. Thus, by shifting the brushes from the position A, B, at right angles with the magnetic flux, to the posi- tion E, F, in line with the magnetic flux, any direct voltage be- Fig. 197. — Sine wave of e.m.f. tween 2 e and 0 can be produced, with the same wave of alter- nating volage, a. As seen, this variation of direct voltage between its maximum value and zero, at constant impressed alternating voltage, is in- 424 EL ECTRIC A L A PPA RA T f '8 dependent of the wave shape, and thus run be produced whether the alternating voltage is a sine wave or any other wave. It is obvious that, instead of shifting the brushes on the com- mutator, the magnetic field poles may \k< shifted, in the opposite direction, by the same angle, as shown in Fig. 198, A, B, C. Instead of mechanically shifting the field poles, they can bt shifted electrically, by having each field pole consist of a numUr of sections, and successively reversing the polarity of these sec- tions, as shown in Fig. 199, A, B, C, D. by mechanically shifting llie poles. Instead of having a large number of field pole sections, obvi- ously two sections are sufficient, and the same gradual change can be brought about by not merely reversing the sections but reducing the excitation down to zero and bringing it up again In opposite direction, as shown in Fig. 200, A, B, C, D, E. Fin. 11)9.— E.in.f. by electrically shifting the polos. In this case, when reducing one section in polarity, the othtf section must be increased by approximately the same amount] to maintain the same alternating voltage. When changing the direct voltage by mechanically shifting the brushes, as soon as the brushes come under the field pole faces, self-inductive sparking on the commutator would result if the iron of the field poles were not kepi away from the brush REGULATING POLE CONVERTERS 425 position by having a slot in the field poles, as indicated in dotted line in Fig. 196 and Fig. 198, B. With the arrangement in Figs. 196 and 198, this is not feasible mechanically, and these arrange- ,f. variation by shifting-flux distribution. ments are, therefore, unsuitable. It is feasible, however, as shown in Figs. 199 and 200, that is, when shifting the resultant magnetic flux electrically, to leave a commutating space between Fio. 201. — Variable ratio or split-pole converter. the polar projections of the field at the brushes, as shown in Fig. 200, and thus secure as good commutation as in any other com- mutating machine. 426 ELECTRICAL APPARATUS Such a variable-ratio converter, then, comprises an armature A, Fig. 201, with the brushes, H, B', in fixed position and field poles, P,P', separated by inter polar spaces, C, C, of such width as required for commutation. Each field pole consists of two parts, P and Pi, usually of different relative size, separated by a narrow space, DD', and provided with independent windings. By vary- ing, then, the relative excitation of the two polar sections, Pand Pi, an effective shift of the resultant field flux and a corresponding change of the direct voltage is produced. As this method of voltage variation does not depend upon the wave shape, by the design of the field pole faces and the pitch of the armature winding the alternating voltage wave can 1* made as near a sine wave as desired. Usually not much atten- tion is paid hereto, as experience shows that the usual distributed winding of the commutating machine gives a sufficiently close approach to sine shape. Armature Reaction and Commutation 232. With the brushes in quadrature position to the resultant magnetic flux, and at normal voltage ratio, the direct -current generator armature reaction of the converter equals the syn- chronous-motor armature reaction of the power component of the alternating current, and at unity power-faetor the converter thus has no resultant armature reaction, while with a lagging or leading current it has the magnetizing or demagnetizing re- action of the wattless component of the current. If by a sliift of the resultant flux from quadrature position with the brushes, by angle, t, the direct voltage is reduced by factor cos r, the direct current and therewith the direct-current armature reaction are increased, by factor, -. as by the law of conservation of energy the direct-current output must equal the alternating-current input (neglecting losses). The dueet- current armature reaction, ff, therefore ceases to be equal to the armature reaction of the alternating energy current, 5F», but is greater by factor, '■ The alternating-current armature reaction, Su, at no | placement, is in quadrature position with the magnetic flux. REGULATING POLE CONVERTERS 427 The direct-current armature reaction, £, however, appears in the position of the brushes, or shifted against quadrature position by angle t; that is, the direct-current armature reaction is not in opposition to the alternating-current Armature reaction, but differs therefrom by angle t, and so can be resolved into two components, a component in opposition to the alternating-cur- rent armature reaction, £0, that is, in quadrature position with the resultant magnetic flux: £" = $ cos T = $0, that is, equal and opposite to the alternating-current armature reaction, and thus neutralizing the same; and a component in quadrature position with the alternating-current armature reac- tion, $0, or in phase with the resultant magnetic flux, that is, magnetizing or demagnetizing: $' = $ sin t = $0 tan r; that is, in the variable-ratio converter the alternating-current armature reaction at unity power-factor is neutralized by a component of the direct-current armature reaction, but a result- ant armature reaction, 5', remains, in the direction of the resultant magnetic field, that is, shifted by angle (90 — r) against the position of brushes. This armature reaction is magnetizing or demagnetizing, depending on the direction of the shift of the field, t. It can be resolved into two components, one at right angles with the brushes : 5'i = & cos t = $0 sin r, and one, in line with the brushes: $'2 = $' sin t = £ sin2 r = $0 sin t tan t, as shown diagrammatically in Figs. 202 and 203. There exists thus a resultant armature reaction in the direc- tion of the brushes, and thus harmful for commutation, just as in the direct-current generator, except that this armature reac- tion in the direction of the brushes is only $'2 = & sin2 t, that is, sin2 t of the value of that of a direct-current generator. The value of 5'2 can also be derived directly, as the difference between the direct-current armature reaction, (F, and the com- ELECTRICAL APPARATUS Fig. 203. — Diagram of minis, in split-pole converter. REGULATING POLE CONVERTERS IHiuciii of i hi- alternating-current armature reaction, in the direc- tion of the brushes, 50 cos r, that is: ff'j ■ = ff (1 ■ COS! t) ■- -- Jo sin r ton r 233. The shift of the resultant magnetic flux, by angle r, gives ii component of the m.m.f. of field excitation, 5"/ = S/sinr, (where ;T, = m.m.f. of field excitation), in the direction of the commutator brushes, and either in the direction of armature reaction, thus interfering with commutation, or in opposition to the armature reaction, thus improving commutation. If the magnetic flux is slutted in the direction of armature rotation, that is, that section of the field pole weakened toward which the armature moves, as in Fig. 202, the component 5"/ of field excitation at the brushes is in the same direction as the armature reaction, 3'j, thus adds itself thereto and impairs the commutation, and such a converter is hardly operative. In this case the component of armature reaction, 5', in the direction of the field flux is magnetizing. If the magnetic flux is shifted in opposite direction to the armature reaction, that is, that section of the field pole weakened which the armature conductor leaves, as in Fig. 203, the Com- ponent, it",, of field excitation at the brushes is in opposite direc- tum to the armature reaction, J'i, therefore reverses it, if suffi- ciently large, and gives a commutating or reversing flux, $„ that , improves commutation so that this arrangement is used in such converters. In this case, however, the component of arma- ture reaction, $', in the direction of the field flux is demagnet- izing, and with increasing load the field excitation has to be in- creased by ff* to maintain constant flux. Such a converter thus requires compounding, as by a series field, to take care of the demagnetizing armature reaction. If the alternating current is not in phase with the field, but lags or leads, the armature reaction of the lagging or leading component of current superimposes upon the resultant armature reaction, 5', and increases it — with lagging current in Fig. 202, leading current in Fig. 203 — or decreases it — with lagging cur- rent in Fig. 203, leading current in Fig. 202 — anil with lag of the alternating current, by phase angle, 6 = t, under the conditions of Fig. 203, the total resultant armature reaction vanishes, that is, the lagging component of synchronous-motor armature reaction compensates for the component of the direct -current reaction, 430 ELECTRICAL APPARATUS which is not compensated by the armature reaction of the power component of the alternating current. It is interesting t<> note that in this case, in regard to heating, output based (hereon, etc., the converter equals that of one of normal voltage ratio. B. Variable Ratio by Change of Wave Shape of the Y Voltage 234. If in the converter shown diagranimatieally in Fig. 204 the magnetic flux disposition and the pitch of the armature winding are such that the potential difference between the point, a, of the armature and the neutral 0, or the 1" voltage, is a sine wave, Fig, 205 A, then the voltage ratio is normal. Assume, however, thai the voltage curve, a, differs from sine shape by the superposition of some higher harmonics: the third harmonic in Figs. 205 B and C. the fifth harmonic in Figs. 20.5 D and E. If, then, these higher harmonics are in phase with the fundamental, that is, their maxima coincide, as in Figs. 205 B and D, they increase the maximum of the —Variable ratio con- alternating voltage, and thereby the s shape direct voltagc;andiflhescharmonics are in opposition to the funda- mental, as in Figs. 205 C and E, they decrease the maximum alternating and thereby the direct voltage, without Appreciably affecting the effective value of the alternating voltage. For in- stance, a higher harmonic of 30 per cent, of the fundamental increases or decreases the direct voltage by 30 per cent . bill varies the effective alternating voltage only by -y/i + 0.3' = 1.044, or 4.4 per cent. The superposition of higher harmonics thus offers a DMUM '•> increasing as well as decreasing the direct voltage, at i-urist.mi alternating voltage, and without shifting the angle between the brush position and resultant magnetic flux. Since, however, the terminal voltage of the converter does not only depend on the generated e.m.f. of the converter, but also on that of the generator, and is a resultant of the two e.m.fs. in approximately inverse proportion to the impedances from the converter terminals to the two respective generated e.mJs., hi REGULATING POLE CONVERTERS 431 varying the converter ratio only such higher harmonics can be used which may exist in the Y voltage without appearing in the converter terminal voltage or supply voltage. In general, in an n-phase system an nth harmonic existing in the star or Y voltage does not appear in the ring or delta voltage, Fig. 205. — Superposition of harmonics to change the e.m.f. ratio. as the ring voltage is the combination of two star voltages dis- 180 placed in phase by — degrees for the fundamental, and thus by IV 180°, or in opposition, for the nth harmonic. Thus, in a three-phase system, the third harmonic can be in- troduced into the Y voltage of the converter, as in Figs. 205 B and C, without affecting or appearing in the delta voltage, so can be used for varying the direct-current voltage, while the fifth harmonic can not be used in this way, but would reappear and 432 ELECTRICAL APPARATUS cause a short-circuit current in the supply voltage, hence should be made sufficiently small to be harmless. 235. The third harmonic thus can be used for varying the direct voltage in the three-phase converter diagrammatically shown in Fig. 206 A, and also in the six-phase converter with Fio. 206. — Transformer connections for varying the e.m.f. ratio by super- position of the third harmonic. double-delta connection, as shown in Fig. 206 B, or double-}' connection, as shown in Fig. 206 C, since this consists of two sepa- rate three-phase triangles of voltage supply, and neither of them contains the third harmonic. In such a six-phase converter with double- Y connection, Fig. 206 C, the two neutrals, however, REGULATING POLE CONVERTERS 433 must not be connected together, as the third harmonic voltage exists between the neutrals. In the six-phase converter with diametrical connections, the third harmonic of the Y voltage ap- pears in the terminal voltage, as the diametrical voltage is twice the Y voltage. In such a converter, if the primaries of the sup- Fig. 207. — Shell-type transformers. ply transformers are connected in delta, as in Fig. 206 D, the third harmonic is short-circuited in the primary voltage triangle, and thus produces excessive currents, which cause heating and interfere with the voltage regulation, therefore, this arrangement ^ > f TV ^i n tt Fig. 208. — Core-type transformer. is not permissible. If, however, the primaries are connected in Y, as in Fig. 206 E, and either three separate single-phase trans- formers, or a three-phase transformer with three independent magnetic circuits, is used, as in Fig. 207, the triple-frequency voltages in the primary are in phase with each other between 28 434 ELECTRICAL APPARATUS the line and the neutral, and thus, with isolated neutral, can not produce any current. With a three-phase transformer as shown in Fig. 208, that is, in which the magnetic circuit of the third harmonic is open, triple- frequency currents can exist in the sec- ondary and this arrangement therefore is not satisfactory. In two-phase converters, lugher harmonics can he used for regulation only if the transformers are connected in such a man- ner that the regulating harmonic, which appears in the converter terminal voltage, does not appear in the transformer terminals, that is, by the connection analogous to Figs. 206 E and 207. Since the direct-voltage regulation of a three-phase or sis- phase converter of this type is produced by the third harmonic, Fig. 209.— V e.m.f. wa' the problem is to design the magnetic circuit of the converter so as to produce the maximum third harmonic, the minimum fifth and seventh harmonics. If q = interpolar space, thus (1 — q) = pole arc, as fraction of pitch, the wave shape of the voltage generated between the point, a, of a full-pitch distributed winding — as generally used for commutating machines — and the neutral, or the induced Y voltage of the system is a triangle with the top cut off for dis- tance q, as shown in Fig. 209, when neglecting magnetic spread at the pole corners. If then Co = voltage generated per armature turn while in front of the field pole (which is proportional to the magnetic den- sity in the air gap), m = series turns from brush to brush, the maximum voltage of the-wave shown in Fig. 209 is: Ett = ?nc0(l - g); developed into a Fourier series, tliis gives, as the equation of the voltage wave a, Fig. 188: (2»- 1)^ *F ^ COS 2 (1 -?)t5 i (2n - 1)' REGULATING POLE CONVERTERS 435 or, substituting for 2?0, and denoting: A 8 tneo A = — ^-i 2n - 1 - cos 2" -& e = A r (2W-D' cos(2ri " x) • f ir 1 ir 1 i* = il | cos q s cos 0 + q cos 3 g _ c<>s 3 0 + j- cos 5 q = cos 5 0 1 ir + T« cos 7 ^ ^ cos 7 0 + Thus the third harmonic is a positive maximum for q = 0, or 100 per cent, pole arc, and a negative maximum for q = %y or 33.3 per cent, pole arc. For maximum direct voltage, q should therefore be made as small, that is, the pole arc as large, as commutation permits. In general, the minimum permissible value of q is about 0.15 to 0.20. The fifth harmonic vanishes for q = 0.20 and q = 0.60, and the seventh harmonic for q = 0.143, 0.429, and 0.714. For small values of q, the sum of the fifth and seventh har- monics is a minimum for about q = 0.18, or 82 per cent, pole arc. Then for q = 0.18, or 82 per cent, pole arc: ei = A {0.960 cos 0 + 0.0736 cos 3 6 + 0.0062 cos 5 6 - 0.0081 cos 7 d + . . . } = 0.960 A {cos 6 + 0.0766 cos 3 0 + 0.0065 cos 5 0 - 0.0084 cos 7 0 + . . . } ; that is, the third harmonic is less than 8 per cent., so that not much voltage rise can be produced in this manner, while the fifth and seventh harmonics together are only 1.3 per cent., thus negligible. 236. Better results are given by reversing or at least lowering the flux in the center of the field pole. Thus, dividing the pole face into three equal sections, the middle section, of 27 per cent, pole arc, gives the voltage curve, q = 0.73, thus: e2 = A {0.411 cos 0 - 0.1062 cos 3 0 + 0.0342 cos 5 0 -0.0035 cos 7 0 . . .} = 0.411 A {cos 0 - 0.258 cos 3 0 + 0.083 cos 5 0 -0.0085 cos 7 0 . . .}• The voltage curves given by reducing the pole center to one- 436 ELECTRICAL APPARATUS hall intensity, to aero, reversing it to half intensity, to full in- tensity, anil to Btich intensity that the fundamental disappear*, then are given 1 >y : (1) full, e = e, = O.96OA|cos0+O.O77cos30 +0.0065 cos 5 0-0.0084 cos 7 0. . ,| (2) 0.5, «■ = <>, -0.5c2 = O.755A(cos0+O.168cos30 -0.0144 cos 5 0-0.0085 cos 7 0. . . | (3) 0, e = e,-e, =0.549 ,4 {cos 0+0.328 cos 3 0 -0.053 cos 5 0-0.084 cos 7 0. . .| (4) -0.5 e=«i-1.5e, =0.344 A (cos 6 +0.680 cos 3 B -0.131 cos 5 0-0.0084 cos 7 0. . \ (5) - full, c = e,-2<>2 =0.138 A | cos 0+2.07 cos 30 -0.45 cos 5 0-0.008 cos 7 0. . | (6) -1.17, e = c,-2.34e2 = 0.322^jcos 3 0-0.227 cos 5 0. | It is interesling tn note that in the last case the fundamental frequency disappears and the machine is a generator of triple frequency, that is, produces or consumes a frequency equal to three times synchronous frequency. In this ease the sevmUl harmonic also disappears, and only the fifth is appreciable. Iiut could be greatly reduced by a different kind of pole inc. From above table follows: (1) (2) (3) (4) (5) (6) normal MilMIIIHII: fuuiln- rocntal alter- 0.960 0.755 0,549 0.344 0.138 0 0.960 n&ting voile . . Direct volte 1.033 0.883 0.743 0.578 0.423 0.322 0.960 237. It is seen that a considerable increase of direel voltage beyond the normal ratio involves a sacrifice of output, due to the decrease or reversal of a part of the magnetic flux, whereby the air-gap section is not fully utilized. Thus it is not advisable to go too far in tliis direction. By the superposition of the third harmonic upon the funda- mental wave of the Y voltage, in a converter with three seetwni per pole, thus an increase of direct voltage over its norma! voltage can be produced by lowering the excitation of the middk section and raising that of the outside sections of the field pole, and also inversely a decrease of the direct voltage l>e!ow its normal value by raising the excitation of the middl REGULATING POLE CONVERTERS 437 and decreasing that of the outside sections of the field poles; that is, in the latter case making the magnetic flux distribution at the armature periphery peaked, in the former case by making the flux distribution flat-topped or even double-peaked. Armature Reaction and Commutation 238. In such a split-pole converter let p equal ratio of direct voltage to that voltage which it would have, with the same alternating impressed voltage, at normal voltage ratio, where p > 1 represents an overnormal, p < 1 a subnormal direct voltage. The direct current, and thereby the direct-current armature reaction, then is changed from the value which it would have at normal voltage ratio, by the factor — , as the product of direct volts and amperes must be the same as at normal voltage ratio, being equal to the alternating power input minus losses. With unity power-factor, the direct-current armature reac- tion, $, in a converter of normal voltage ratio is equal and opposite, and thus neutralized by the alternating-current armature reac- tion, $0, and at a change of voltage ratio from normal, by factor p, and thus change of direct current by factor — The direct- current armature reaction thus is: * = *-• V hence, leaves an uncompensated resultant. As the alternating-current armature reaction at unity power- factor is in quadrature with the magnetic flux, and the direct- current armature reaction in line with the brushes, and with this type of converter the brushes stand at the magnetic neutral, that is, at right angles to the magnetic flux, the two armature reactions are in the same direction in opposition with each other, and thus leave the resultant, in the direction of the commutator brushes: 5' = $ - So -*(H- The converter thus has an armature reaction proportional to the deviation of the voltage ratio from normal. 239. If p > 1, or overnormal direct voltage, the armature 438 ELECTRICAL APPARATUS reaction is negative, or motor reaction, and the magnetic Hux produced by it at the commutator brushes thus a commutfttWfl, flux. If p < I, or subnormal direct voltage, the armature reaction is positive, that is, the same as in a direct-cum-rii gen- erator, but less in intensity, and thus the magnetic flux of arma- ture reaction tends to impair commutation. In a direct-current generator, by shifting the brushes to the edge of the field poles, the field flux is used as reversing flux to give commutation. In this converter, however, decrease of direct voltage is produced by lowering the outside sections of the field poles, and the edge of the field may not have a sufficient flux density to give commuta- d . LHJ . UHLJ . L Fio. 210. — Three-section pole tor variable -ratio tion, with a considerable decrease of voltage l)elow normal, and thus a separate commutating pole is required. Preferably this type of converter should be used only for raising the voltage, for lowering the voltage the other type, which operates by a shift of the resultant flux, and so gives a component of the main field flux as commutating flux, should be used, or a combination of both types. With a polar construction consisting of three sections, thia can be done by having the middle section at low, the nul.sicfe sections at high excitation for maximum voltage, and, to de- crease the voltage, raise the excitation of the center section, but instead of lowering both outside sections, leave the section in the direction of the armature rotation unchanged, while lowering the other outside section twice as much, and thus produce, in addition to the change of wave shape, a shift of the flux, as represented by the scheme Fig, 210. Pole section . . Max. voltage , Min. voltage . Magnetic Density 3 1' .+ s = direct current; E° = alternating voltage between adjacent collector rings (ring voltage), and J° = alter- nating current between adjacent collector rings- (ring current); then, as seen in the preceding: £0sin E° = n V2 (1) and as by the law of conservation of energy, the output must equal the input, when neglecting losses: hV2 1° = n sm- n (2) ■140 F.LVJ ■ TRIt A L A PPA HA TUS where I* is the power component of the current corresponding In the duvet -current output. The voltage ratio of a converter can be varied: (a) By the superposition of a third harmonic upon the tar voltage, or diametrical voltage, which does not appear in "he ring voltage, or voltage between the collector rings of lbs converter. (6) By shifting the direction of the magnetic flux. (ii) can be used for raising the direct voltage as well as for lowering it, but is used almost, always for the former purpose, since when using this method for lowering the direct voltage Commutation is impaired. (b) can Ik* used only for lowering (he direct voltage. It is possible, by proportioning (he relative amounts by which the two methods contribute to the regulation of the voltage, to maintain a proper commutating field at the brushes for all loads and voltages. Where, however, this is not done, the brushes are shifted to the edge of the next field pole, and into the fringe of its field, thus deriving the commutating field. 241. In such a variable-ratio converter let, then, ( = intensity of the third harmonic, or rather of that component of it which is in line with the direct-current brushes, and thus (hies the voltage regulation, as fraction of the fundamental wave. / fa chosen as positive if the third harmonic increases the maximum of the fundamental wave (wide pole arc) and thus raises the direct voltage, and negative when lowering the maximum of the fundamental and therewith the direct voltage (narrow pole arc). pi = loss of power in the converter, which is supplied by the current (friction and core loss) as fraction of the alternating input (assumed as 4 per cent, in the numerical example). T,, = angle of brush shift on the commutator, counted positive in the direction of rotation. 0i = angle of time lag of the alternating current (thus negative for lead). r„ = angle of shift of the resultant field from the position :>t right angles to the mechanical neutral (or middle between the pole corners of main poles and auxiliary poles), counted positive in the direction opposite to the direction of armature rotation. that is, positive in that direction in which the field flux has been shifted to get good commutation, as discussed in the preceding article. HEGULATIXG POLE COXYERTERS 441 Due to the third harmonic, f, and the angle of shift of the field flux, ra, the voltage ratio differs from the normal by the factor: (1 + t) COST«, and the ring voltage of the converter thus is: E = r—-r~ — : (3) (1 + /) COST* hence, by (1): E = £0sin n (4) V2(l +/)eosr0 and the power component of the ring current corresponding to the direct -current output thus is, when neglecting losses, from (2): J' = Jo(! +t) COSTa = IoV2(l+t)cosTa. (5) T n sin - n Due to the loss, pi, in the converter, this current is increased by (1 + pi) in a direct converter, or decreased by the factor (1 — pi) in an inverted converter. The power camponent of the alternating current thus is: /, = /'(!+ Vl) T \/2(l+0 (1+P/) COS Ta = '0 t (0) n sin - n where pi may be considered as negative in an inverted converter. With the angle of lag 0i, the reactive component of the current is: J2 = I\ tan 0i, and the total alternating ring current is: z = _iv cos 0t _ JoV2(l+0 (l+p()cosTa (?) n sin - cos 0i n a t 442 ELECTRICAL APPARATUS or, introducing for simplicity the abbreviation: t - (1 + 00 + PJWT. (8) (9) 242. Let, in Fig. 212, Il'OA represent the center line of the magnetic field structure. The resultant magnetic field flux, 0*, then leads OA by angle *Oi = ra. The resultant m.m.f.of the alternating power current,/], isO/i, it f \ \ 1 v\ / \1 h * is ^L s /^--*« Fio. 212.— Diagram of variable ratio converter. at right angles to 0$, and the resultant m.m.f . of the alternating reactive current, h, is Olt, in opposition to 0*f while the total alternating current, I, is 01, lagging by angle 6\ behind <)/,. The m.m.f. of direct-current armature reaction is in the direc- tion of the brushes, thus lagging by angle r» behind the position OB, where BOA = 90°, and given by 0~lo- The angle by which the direct-current m.m.f., O/0, lags in space behind the total alternating m.m.f., 01, thus is, by Pig. 212: r„ = Si - r„ - rh. (10) If the alternating m.m.f. in a converter coincides with the direct-current m.m.f., the alternating current and the direct cur- rent are in phase with each other in the armature coil midway REGULATING POLE CONVERTERS 443 between adjacent collector rings, and the current heating thus a minimum in this coil. Due to the lag in space, by angle t0, of the direct-current m.m.f. behind the alternating current m.m.f., the reversal of the direct current is reached in time before the reversal of the alter- nating current in the armature coil; that is, the alternating current lags behind the direct current by angle, 60 — t0, in the Fig. 213. — Alternating and direct current in a coil midway between adjacent collector leads. armature coil midway between adjacent collector leads, as shown by Fig. 213, and in an armature coil displaced by angle, t, from the middle position between adjacent collector leads the alternating current thus lags behind the direct current by angle (r + 0O), where t is counted positive in the direction of armature rotation (Fig. 214). Fig. 214. — Alternating and direct current in a coil at the angle t from the middle position. The alternating current in armature coil, t, thus can be ex- pressed by: i = JV2sin(0 - r - 0«); (11) hence, substituting (9): i = sin(0-r-0o), (12) nsin- n and as the direct current in this armature coil is -~>and opposite 444 ELECTRICAL APPARATUS to the alternating current, i, the resultant current in the arma- ture coil, r, is: io to = l — 2 4fc - sin (0 - t - 0O) - 1 7T n sin - n (13) and the ratio of heating, of the resultant current, io, compared with the current, ^, of the same machine as direct-current gen- erator of the same output, thus is: io2 h\2 © 4A: w sin sin(0 - t - 0O) - 1 n •. (14) sin (0 - r - 0O) - 1 1 d$. (15) n sin Averaging (14) over one half wave gives the relative heating of the armature coil, r, as: Integrated, this gives: 8fc2 n2 sin2 - n n 16 k cos (r + 0O) (16) wn sm n 243. Herefrom follows the local heating in any armature coil, t, in the coils adjacent to the leads by substituting t = ± - , and also follows the average armature heating by averaging 7Tfromr = — - to t = H — • n n The average armature heating of the n-phase converter there- fore is : + ~ r - V f > n or, integrated: r = - Sk' + 1 - 16 k cos 0o n2 sin2 - n (17) REGULATING POLE CONVERTERS 445 This is the same expression as found for the average armature heating of a converter of normal voltage ratio, when operating with an angle of lag, 0O, of the alternating current, where k denotes the ratio of the total alternating current to the alternating power current corresponding to the direct-current output. In an n-phase variable ratio converter (split-pole converter), the average armature heating thus is given by: 8fc2 . . 16fccos0o r = n2 sin2 - n + 1 - (18) where h _ (1 + 0 (1 + yi) cos t„ cos 0i ' (8) 0o = 0i - t« - r6; (10) and / = ratio of third harmonic to fundamental alternating voltage wave; pt = ratio of loss to output; 0i = angle of lag of alternating current; r0 = angle of shift of the resultant mag- netic field in opposition to the armature rotation, and n = angle of shift of the brushes in the direction of the armature rotation. 244. For a three-phase converter, equation (18) gives (n = 3): qo k2 1 T = 27 + 1 - 1-621 k cos 0„ = 1.185 A* + 1 - 1.621 k cos 0O. J For a six-phase converter, equation (18) gives (n = 6): 8fc2 r = 9 + 1 - 1.621 k cos 0O = 0.889 k2 + 1 - 1.621 k cos 0„. For a converter of normal voltage ratio: t = 0, r0 = 0, using no brush shift : (20) n = 0; when neglecting the losses: Pi = 0, it is: 1 :- y COS 0i 00 — 01, 446 ELECTRICAL APPARATUS and equations (19) and (20) assume the form: Three-phase: Six-phase : r = i^f -0.621. COS2 0i r = **» - 0.621. COS2 0i The equation (18) is the most general equation of the relative heating of the synchronous converter, including phase displace- ment, 0i, losses, pi} shift of brushes, ny shift of the resultant mag- netic flux, t0, and the third harmonic, t. While in a converter of standard or normal ratio the armature heating is a minimum for unity power-factor, this is not in gen- eral the case, but the heating may be considerably less at same lagging current, more at leading current, than at unity power- factor, and inversely. 245. It is interesting therefore to determine under which con- ditions of phase displacement the armature heating is a minimum so as to use these conditions as far as possible and avoid con- ditions differing very greatly therefrom, as in the latter case the armature heating may become excessive. Substituting for A; and 0O from equations (8) and (10) into equation (18) gives: _ t ,8(1 +02(1 + ?><)2cos2t0 n2 sin2 - cos2 0j n 16 (1 + 0 (1 + pi) cos rtt cos (0i - Ta - n) . . __«>s_ (19) Substituting: - sin - = mt (20) which is a constant of the converter type, and is for a three- phase converter, w3 = 0.744; for a six-phase converter, w6 = 0.955; and rearranging, gives: 8(1+ 02 (1 + Pi)2 COS2 Ta r = i + TT2 W2 1 P - -2 (1 + t) (1 + pt) COS Ta COS (t0 + Tb) REGULATING POLE CONVERTERS 447 8 (1 + !)■ (1 + p,)» COB' r. tanl it2 ra2 2 (1 + 0 (1 + pi) cos r0 sin (t0 + r6) tan B\. (21) r is a minimum for the value, 0i, of the phase displacement given by: dr =0 d tan 0i ' and this gives, differentiated: *« fb = ^-J™ (J ^-^ (22) (1 + 0 (1 +Pl) C0STo Equation (22) gives the phase angle, 02, for which, at given r0> T6, J and pi, the armature heating becomes a minimum. Neglecting the losses, p/, if the brushes are not shifted, n = 0, and no third harmonic exists, t = 0: tan 0'2 = ra2 tan t «l where m2 = 0.544 for a three-phase, 0.912 for a six-phase converter. For a six-phase converter it thus is approximately B\ — ra, that is, the heating of the armature is a minimum if the alter- nating current lags by the same angle (or nearly the same angle) as the magnetic flux is shifted for voltage regulation. From equation (22) it follows that energy losses in the con- verter reduce the lag, 02, required for minimum heating; brush shift increases the required lag; a third harmonic, ty decreases the required lag if additional, and increases it if subtractive. Substituting (22) into (21) gives the minimum armature heat- ing of the converter, which can be produced by choosing the proper phase angle, 02, for the alternating current. It is then, after some transpositions: r. - i + £ { [^-+ ° (1 + p,) c°8 T°]'- 2(1 + 0(1 + pj • = 1- cos ra cos (r0 + n) — m2 sin2 (r« + n) 8m2/, r(l +0 (1 + pi) cost, 7T2 nl + t) (1 + pi) COS Ta , , ,1s ) /rtrtX 1 " L m2 C0S (r° + T6)J |(23) The term To contains the constants /, p/, t0, n only in the square under the bracket and thus becomes a minimum if this 448 ELECTRICAL APPARATUS square vanishes, that is, if between the quantities (, ph i relations exist thai : CM) 246. Of the quantities I, p,, ra, rb; p, and t„ are determined by the machine design. ( and r„, however, are equivalent lo each other, that is, the voltage regulation can be accomplished either by the flux shift, r„, or by the third harmonic, (, or by both, and in the latter ease can be divided between tu and / so as to give any desired relations between them. Equation (24) gives: !■- cos |r„ + n.) J -■ (25) (1 + pt COS T„) and by choosing the third harmonic, (, as function of the angle of flux shift ra, by equation (25), the converter heating becomes a minimum, and is: ■ 1 8 m* l2n henci i2M Tu" — 0.551 for a three-phase converter, IV = 0.261 for a six-phase converter. Substituting (25) into (22) gives: tan 02 = tan (r„ + n); hence: fl2 = u + n; (29) or, in other words, the converter gives minimum heating IV if the angle of lag, 02, equals the sum of the angle of flux shift, r„ and of brush shift, rt. It follows herefrom that, regardless of the losses, /tj, of the brush shift, t,„ and of the amount of voltage regulation required, that is, at normal voltage ratio as well as any other ratio, the same minimum converter heating IV can be secured by dividing the voltage regulation between the angle of flux shift, r#, and the third harmonic, 1, in the manner as given by equation [jMQi and operating at a phase angle between alternating current and voltage equal to the sum of the angles of flux shift, r„ and of bru-h shift, n; that, is, the heating of the split-pole Converter OSS !■<■ made the same as that of the standard converter of normal voltage ratio. (31) REGULATING POLE CONVERTERS 449 Choosing pt = 0.04, or 4 per cent, loss of current, equation (25) gives, for the three-phase and for the six-phase converter: (a) no brush shift (n = 0) : /3° = 0.467, 1 (30) /6° = 0.123; J that is, in the three-phase converter this would require a third harmonic of 46.7 per cent., which is hardly feasible; in the six- phase converter it requires a third harmonic of 12.3 per cent., which is quite feasible. (6) 20° brush shift (r6 = 20) : #0 1 ft-QQ COS (T« + T*) cos Ta <6<»=l-0.877CO8(T,,-+Tfc); COS Ta for ra = 0, or no flux shift, this gives: *300 = 0.500, ) ( h00 = 0.176./ K ' Since " — -- < 1 for brush shift in the direction of cos ra armature rotation, it follows that shifting the brushes increases the third harmonic required to carry out the voltage regulation without increase of converter heating, and thus is undesirable. It is seen that the third harmonic, t, does not change much with the flux shift, r0, but remains approximately constant, and positive, that is, voltage xaising. It follows herefrom that the most economical arrangement regarding converter heating is to use in the six-phase converter a third harmonic of about 17 to 18 per cent, for raising the vol- tage (that is, a very large pole arc), and then do the regulation by shifting the flux, by the angle, ra, without greatly reducing the third harmonic, that is, keep a wide pole arc excited. As in a three-phase converter the required third harmonic is impracticably high, it follows that for variable voltage ratio the six-phase converter is preferable, because its armature heating can be maintained nearer the theoretical minimum by propor- tioning t and ra. 29