CHAPTER III. MECHANICAL RECTIFICATION. 9. If an alternating-current circuit is connected, by means of a synchronously operated circuit breaker or rectifier, with a second circuit in such a manner, that the connection between the two circuits is reversed at or near the moment when the alternating voltage passes zero, then in the second circuit current and voltage are more or less unidirectional, although they may not be constant, but pulsating. If i = instantaneous value of alternating current, and i0 = instantaneous value of rectified current, then we have, before reversal, i0 = i, and after reversal, i0 = — i\ that is, during the reversal of the circuit one of the currents must reverse. Since, however, due to the self-inductance of the circuits, neither current can reverse instantly, the reversal occurs gradually, so that for a while during rectification the instantaneous value of the alternating and of the rectified current differ from each other. Thus means have to be provided either to shunt the difference between the two currents through a non-inductive bypath, or, the difference of the two currents exists as arc over the surface of the rectifying commutator.* The general phenomenon of single-phase rectification thus is : The alternating and the rectified circuit are in series. Both circuits are closed upon themselves at the rectifier, by the resistances, r and r0, respectively. The terminals are reversed. The shunt-resistance circuits are opened, leaving the circuits in series in opposite direction. Special cases hereof are: 1. If r = r0 = 0, that is, during rectification both circuits are short circuited. Such short-circuit rectification is feasible only in limited-current circuits, as on arc lighting machines, or in * If the circuit is reversed at the moment when the alternating current passes zero, due to self-inductance of the rectified circuit its current differs from zero, and an arc still appears at the rectifier. 229 230 TRANSIENT PHENOMENA cases where the voltage of the rectified circuit is only a small part of the total voltage, and thus the current not controlled thereby, as when rectifying for the supply of series fields of alternators. 2. r = r0 = oo , or open circuit rectification. This is feasible only if the rectified circuit contains practically no self -inductance, but a constant counter e.m.f., e, (charging storage batteries), so that in the moment when the alternating impressed e.m.f. falls to e, and the current disappears, the circuit is opened, and closed again in opposite direction when after reversal the alter- nating impressed e.m.f. has reached the value, e. In polyphase rectification, the rectified circuit may be fed successively by the successive phases of the system, that is shifted over from a phase of falling e.m.f. to a phase of rising e.m.f., by shunting the two phases with each other during the time the current changes from the one to the next phase. Thus the Thomson-Houston arc machine is a star-connected three- phase constant-current alternator with rectifying commutator. The Brush arc machine is a quarter-phase machine with rectify- ing commutator. In rectification frequently the sine wave term of the current is entirely overshadowed by the transient exponential term, and thus the current in the rectified circuit is essentially of an exponential nature. As examples, three cases will be discussed: 1. Single-phase constant-current rectification; that is, a rectifier is inserted in an alternating-current circuit, and the voltage consumed by the rectified circuit is small compared with the total circuit voltage; the current thus is not noticeably affected by the rectifier. In other words, a sine wave of current is sent over a rectifying commutator. 2. Single-phase constant-potential rectification; that is, a constant-potential alternating e.m.f. is rectified, and the impe- dance between the alternating voltage and the rectifying com- mutator is small, so that the rectified circuit determines the current wave shape. 3. Quarter-phase constant-current rectification as occurring in the Brush arc machine. MECHA NIC A L REG TIFICA TION 231 i. Single-phase constant-current rectification. 10. A sine wave of current, i0 sin 0, derived from an e.m.f. very large compared with the voltage consumed in the recti- fied circuit, feeds, after rectification, a circuit of impedance Z = r — jx. This circuit is permanently shunted by a circuit of resistance rr Rectification takes place over short- circuit from the moment n — 02 to TT + 0jj that is, at n - 02the rectified and the alternating circuit are closed upon themselves at the rectifier, and this short-circuit opened, after rever- sal, at TT + 6lf as shown by the dia- grammatic representation of a two- pole model of such a rectifier in Fig. 54. In this case the space angles TT -f TJ and TT — r2 and the time angles TT -f Ol and TT - 02 are identical. This represents the conditions ex- isting in compound-wound alter- nators, that is, alternators feeding a series field winding through a rectifier. Let, during the period from Ol to n - 02, i = current in impedance Z, and il = current in resistance rlt then: i + i1 = i0 sin 0. However, di Fig. 54. Single-phase current rectifier commutator. (1) ^1r1=^r (2) and substituting (1) in (2) gives the differential equation : di i (r +r1) + x~ - i0rl sin 0 = 0, au (3) (4) which is integrated by the function : i = Ae-ae+ Bsin (6 - 8). Substituting (4) in (3) and arranging, gives : A (r + rl - ax) e~ a& + [B ( [r + rj cos d + x sin 8) - i/J sin 0 - [(r + rj sin d - x cos d] B cos 0 = 0, (5) 232 TRANSIENT PHENOMENA which equation must be an identity, thus : and and herefrom: r + rl — ax = 0, B ( [r + rj cos d + x sin d) - i0rl = 0 (r + PJL) sin d — x cos d = 0, tand = and where hence: r+r, B = i ° V(r+ rj' (6) z = V(r + r,)2 + x2; (7) (8) During the time of short-circuit, from TT — 02 to TT + 0t, if i' = current in impedance Z, we have di' hence: (9) (10) The condition of sparkless rectification is, that no sudden change of current occur anywhere in the system. In consequence hereof we must have : i = i' = i0 sin 0 at the moment 6 = x — 62, and, at the moment 6 = n + 6V i' must have reached the same value as i and i0 sin 0 at the moment 0 = 0r MECHANICAL RECTIFICATION 233 This gives the two double equations : and (11) or, substituting (8) and (9), - r + ri (IT -^ r - - (* - *2) A e * + ^o-1 sin (d + 0a) = A'e * = 10 sin ^2 (12) z and ^ £ - i0 -i sin 0 - OJ = A'e = i0 sin <9r (13) These four equations (12) (13) determine four of the five quantities, A, A' , Ov 02, rv leaving one indeterminate. Thus, one of these five quantities can be chosen. The deter- mination of the four remaining quantities, however, is rather difficult, due to the complex character of equations (12) (13), and is feasible only by approximation, in a numerical example. 11. EXAMPLE: Let an alternating current of effective value of 100 amp., that is, of maximum value t'0 = 141.4, be rectified for the supply of a circuit of impedance Z = 0.2 - 2 j, shunted by a non-inductive circuit of resistance rr Let the series connection of the rectified and alternating circuits be established 30 time-degrees after the zero value of alternating current, that is, d1 = 30 deg. = - chosen. Then, from equation (13), we have A'e x = i0 sin Bv hence, substituting r, x, Ov i0, gives A' = 102. From equation (12), and, substituting, sm02 - 0.527 234 TRANSIENT PHENOMENA approximately sin 02 = 0.527 and 62 = 32°; thus sin 62 = 0.527 £3'2° = 0.558, and 62 = 34°; thus sin 62 = 0.527 £3'4° = 0.559, and 02 = 34°. From equations (12) and (13) it follows : As x + iQ — sin (8 + 02) = i0 sin 02, Ae x ' - i0 - sin (^ - 0J = i sin 04; 2 eliminating A gives 2 - rt sin (5 'T* 7* I /y» substituting sin 8 = -, cos 5 = - — *-> ^2 = (r + rj2 + x2, and z ,z substituting for r, x, 6V 62, gives after some changes : 1.5- 1.04 r e-M,. 1.1 - r, calculating by approximation, assuming rx = 0.5, 0.603 = 0.612; assuming rs = 0.51, 0.597 - 0.602; assuming r^ = 0.52, 0.591 - 0.592; hence, rl = 0.52, and z = 2.124, 5 = 70°. MECHANICAL RECTIFICATION 235 Substituting these values in (12) or (13) gives A =- 113; hence, as final equations, we have i = 112 £-°-36' + 34.6 sin (6 - 70°), i' = 102 £-°-10, i0 = 141.4 sin 0, and it = i0 - i', which gives the following results : Effec- Arith- Quantity. Instantaneous Values. tive metic Value. Mean Value. e° = 30 50 70 90 110 130 146 170 190 210 i = i' = 70.8 70.5 72.6 74.9 79.0 80.4 79.0 79.0 75.8 73.2 "7b'.8f 75.2 75.2 ?0 sin 0= 70.8 108 133 141.4 133 108 79.0 24.7 -24.7 -70. 8J 100.0 *i = 0 37.5 60.4 66.5 54.0 27.6 0 (-51.1-48.5) 0 38.2 '27^3 (44.9) Curves of these quantities are plotted in Fig. 55, for iQ = 100 sin 6. The effective value of the rectified current is 75.2 amp., and this current is fairly constant, pulsating only between 70.5 and 80.4 amp., or by 6.6 per cent from the mean; that is, due to the self-inductance, the fluctuations of current are practically suppressed, and taken up by the non-inductive shunt, and the arithmetic mean value of this current is therefore equal to its effective value. The effective value of the shunt current is 38.2 amp., and this current is unidirectional also, but very fluctuating. Its arithmetic mean value is only 27.3 amp.; that is, in this circuit a continuous-current ammeter would record 27.3, an alternating ammeter 38.2 amperes. The effective value of the total difference between alternating and rectified current (shunt plus short-circuit current) is 44.9 amp. The current divides between the inductive rectified circuit and its non-inductive shunt, not in proportion to their respective impedances, but more nearly, though not quite, in proportion 236 TRANSIENT PHENOMENA to the resistances; that is, in a rectified circuit, self-inductance does not greatly affect the intensity of the current, but only its character as regards fluctuations. 1W 90 80 70 60 40 £ 10 0 -10 -20 -30 -40 s ^ Si i 6 ^N, 1 1 / N \ fc IOC 0.5 0.5 si -5 (- / S \ t / s v / / j \ / / x \^ "> s \ / / / ^ \ s / / / '•il s w \ / / V ? \ \ / f / / \ \ / / / \ f \ \ / / \ / / \ ^ / \ / 0 20 40 60 80 100 120 140 160 ISO 200 Degrees — * Fig. 55. Single-phase current rectification. 2. Single-phase constant-potential rectification. 12. Let the alternating e.m.f. eQ sin 0 of the alternating cir- cuit of impedance Z0 = r0 — jx0 .be rectified by connecting it at the moment 01 with the direct-current receiver circuit of impedance Z = r — jx and continuous counter e.m.f. e, dis- connecting it therefrom at the moment TT - 02, and closing during the time from n — 02 to n + Ol the alternating circuit by the resistance rv the direct-current circuit by the resistance r2, then connecting the circuits again in series in opposite direction, at TT + 6V etc., as shown diagrammatically by Fig. 56, where 1 rn = 1 1 rf +r'" r" + r"1 1. Then, during the time from #t to n — 02, if i1 = current, the differential equation is eQ sin d - e - i, (r + r0) - (x + x0) ^ = 0, (1) MECHANICAL RECTIFICATION which is integrated by i\ = A, + B , sin (0 - 237 (2) Fig. 56. Single-phase constant-potential rectifying commutator. Equation (2) substituted in (1) gives e0 sin 0 - e - (r + r0) [A, + B^~^9 + C, sin (6 - 9,)] - (x +x0)[- a^Bf-*' + C, cos (0 - dj] = 0; or, transposing, - [ + e + (r + r0) AJ + Bfa* [a, (x + x0) - (r + r0)] + sin 6 [e0 - (r + r0) Cl cos ^ - (x + x0) Ct sin ^J + Ci cos 0 [(r + r0) sin 9t - (x + x0) cos dj = 0; herefrom it follows that e + (r +rJAt = 0, a, (x + x0) - (r + r0) = 0, eQ - (r + r0) (71 cos ^ - (x + XQ) C, sin ^ = 0, and (r + r0) sin 9^ - (x + x0) cos ^1 = 0; 238 hence TRANSIENT PHENOMENA and tan dt = x+x( x+x, r +rf and, substituting in (2), (3) r +r. r + r+r0 tan dj = 2 + (x (a: (4) 2. During the time from n - 62 to n + Ov if t2 = current in the direct circuit, ia = current in alternating circuit, we have Alternating-current circuit: di -i3(r0 + r,) - xQ- = 0, (5) which is integrated the same as in (1), by _ r° + ^ Be x° i sn 0 - x0 cos (r (6) MECHANICAL RECTIFICATION 239 Direct-current circuit: -6-;2(r+r2)-z^ = 0, (7) integrated by _r_+_T20 *'. - - r-^72 + B*s ~ <8) At ^ = TT — <92, however, we must have and i'2 at 0 = n + 6l must be equal to t\ at # = 0t, and opposite to i3 at # = it + (9) These terminal conditions represent four equations, which suffice for the determination of the three remaining integration constants, Blf B2, Bs, and one further constant, as 6l or 62, or rt or r2, or e; that is, with the circuit conditions Z9, Z, rv r2, c0, e chosen, the moment ^ depends on i92 and inversely. 13. Special case: Z0 = 0, r2 = 0, e = 0; (10) that is, the alternating e.m.f. eQ sin 6 is connected to the circuit of impedance Z = r - jx during time 6^ to n - 62, and closed by resistance rv while the rectified circuit is short-circuited, during time TT — #2 to n + Or The equations are : 1. Time^ to n - 62: e + -^ — [r sin 6 - x cos 6]. T -j- 37 2. Time TT - (9 to n + e0 sin 0 io= (11) 240 TRANS I EN T PHENOMENA The terminal conditions now assume the following forms : At 0 = 7T - 09f sin S2 -f- x cos 02) = B2£ en at 0 = TT + Ol and 0X respectively (12) These four equations suffice for the determination of the two integration constants B1 and B2, and two of the three rectifica- tion constants, Ov 02J rv so that one of the latter may be chosen. Choosing 02, the moment of beginning reversal, the equations (12) transposed and expanded give, sin 0 sin <92 4- x2 (13) Bj_?>nVi<'-». ^*i and - B, - (r sin 02 + x cos 0 which give 6lt rv B2, Bl : 6l is calculated by approximation. Assuming, as an example, e0 = 156 sin 6 (corresponding to 110 volts effective), Z = 10-30f, and ^ ^ - s = 30°, MECHANICAL RECTIFICATION 241 and by equations (13) we have : log sin 01 = - 0.3765 - 0.1448 0, Ol - 21.7°, r, - 7.63, B2 = 24.4, 5, - 12.8; and thus (15) and which gives: = 12.8 e ~3 + 1.56 (sin 6 - 3 cos 0), _ 0 = 24.4 r 3, = 20.5 sin 0, (16) 0°, »V V V 0°. V V V 21 7 7 55 135 10 27 • 30 45 60 7.47 7.7 8.02 150 165 180 10.20 10.2 9.4 8.6 10.2 5.3 0 75 90 105 8.56 9.18 9 67 195 201.7 7.9 7.55 -5.3 -7.55 120 10 09 The mean value of the rectified current is derived herefrom as 8.92 amp., while without rectification the effective value of alternating current would be — . = 3.48. 110 volts 2v/§ effective corresponds to - - 110 99 volts mean, which in r = 10 would give the current as 9.9 amp. Thus, in a rectified circuit, self-inductance has little effect besides smoothing out the fluctuations of current, which in this case varies between 7.47 and 10.27, with 8.92 as mean, while without self-inductance it would vary between 0 and 15.6, with 9.9 as mean, and without rectification the current would be 4.95 sin (0 - 71.6°). 242 TRANSIENT PHENOMENA As seen, in this case the exponential or transient term of current largely preponderates over the permanent or sinusoidal term. 40 60 80 100 : Degrees 140 160 180 Fig. 57. Single-phase e.m.f. rectification. In Fig. 57 is shown the rectified current in drawn line, the value it would have without self-inductance, and the value the alternating current would have, in dotted lines. 3. Quarter-phase constant-current rectification. 14. In the quarter-phase constant-current arc machine, as the Brush machine, two e.m.fs., E1 = e cos 6 and E2 = e sin #, are connected to a rectifying commutator, so that while the first E^ is in circuit E2 is open-circuited. At the moment Ov E2 is connected in parallel, as shown diagrammatically in Fig. 58, with Ev and the rising e.m.f. in E2 gradually shifts the current i0 away from E1 into E2, until at the moment 02, E^ is dis- connected and E2 left in circuit. Assume that, due to the superposition of a number of such quarter-phase e.m.fs., displaced in time-phase from each other, and rectified by a corresponding number of commutators offset against each other, and due to self-inductance in the external circuit, the rectified current is practically steady and has the value i0. Thus up to the moment 6t the current in E1 is i0, in MECHANICAL RECTIFICATION 243 E2 is 0. From 61 to 62 the current in E2 may be i; thus in El it is i'2 = i'0 - t. After 02, the current in El is 0, in E2 it is i0. A change of current occurs only during the time from Ol to 02, and it is only this time that needs to be considered. Fig. 58. Quarter-phase constant-current rectifying commutator. Let Z = r — jx = impedance per phase, where x = 2 njL ; then at the time t and the corresponding angle 0 = 2 njt the difference of potential in El is d (i'0 - t) 1 6 cos I? — (i0 — i) r — L di = e cos 0 - (ia - i) r + x — ; the difference of potential in E2 is di e sin 0 - ir - x — ; (1) and, since these two potential differences are connected in parallel, they are equal e (sin 0 - cos 6) + v -2ir -2x- = 0. (2) at/ 244 TRANSIENT PHENOMENA The differential equation (2) is integrated by i = A + Bs-ae +Ccos(0 -£); (3) thus -Csm(0 - and substituting in. (2), e (sin 0 - cos 0) + i0r - 2 Ar - 2 Bn'^ -2Cr cos (0 - d) + 2 aBx£~ae +2Cx sin (6 - d) = Q- or, transposed, (i0 - 2 A) r + 2 Bs~ °*(ax - r) + sin d [e - 2 Cr sin 8 + 2Cx cos d] - cos 6 [e + 2 Cr cos d + 2 Cx sin d] = 0; thus i0 - 2 A = 0, ax — r = 0, e - 2 Cr sin d + 2 Cx cos 5 = 0, and e + 2 Cx sin d + 2 Cr cos d = 0, and herefrom, letting - = tan - 34.3 cos (^-31°) 6 ~~ - 31°) -£, i= 5+ 28.3 £ , i = 5 + 25 . 1 3 _34 .3 cos (^- 0i = 7T 6 0! = 7T 4 Oi = IT 3 0° t ft i ii t ii € — - — 4 1000 ^ .- •"J **** ^ S. X * d2-2«50n ^, _i- •"" ~- ^X .// 1 B o •<30 f> Q *""" ii .--- — ^ x* -^ § 12 -2 -500 - — —• -^" "*•* ^ -6 N, •v\\ q i < q ^ \ vv ,\v 10 s^ t. -12 \^ ^'\V 30 40 50 60 70 80 80 1QO UO Degrees > Fig. 59. Quarter-phase rectification. small compared with the total field m.m.f. and the armature reaction, and so greatly varies with a small variation of armature current. As result, a very great distortion of the field occurs, and the magnetic flux is concentrated at the pole corner. This gives an e.m.f. wave which has a very sharp and high peak, with very long flat zero, and so cannot be approximated by an equiva- lent sine wave, but the actual e.m.f. curves have to be used in a more exact investigation.