CHAPTER IV. ARC RECTIFICATION. I. THE ARC. 16. The operation of the arc rectifier is based on the charac- teristic of the electric arc to be a good conductor in one direction but a non-conductor in the opposite direction, and so to permit only unidirectional currents. In an electric arc the current is carried across the gap between the terminals by a bridge of conducting vapor consisting of the material of the negative or the cathode, which is produced and constantly replenished by the cathode blast, a high velocity blast issuing from the cathode or negative terminal towards the anode or positive terminal. An electric arc, therefore, cannot spontaneously establish itself. Before current can exist as an arc across the gap between two terminals, the arc flame or vapor bridge must exist, i.e., energy must have been expended in establishing this vapor bridge. This can be done by bringing the terminals into contact and so starting the current, and then by gradually withdrawing the terminals derive the energy of the arc flame by means of the current, from the electric circuit, as is done in practically all arc lamps. Or by increasing the voltage across the gap between the terminals so high that the electrostatic stress in the gap repre- sents sufficient energy to establish a path for the current, i.e., by jumping an electrostatic spark across the gap, this spark is fol- lowed by the arc flame. An arc can also be established between two terminals by supplying the arc flame from another arc, etc. The arc therefore must be continuous at the cathode, but may be shifted from anode to anode. Any interruption of the cathode blast puts out the arc by interrupting the supply of conducting vapor, and a reversal of the arc stream means stopping the cathode blast and producing a reverse cathode blast, which, in general, requires a voltage higher than the electrostatic striking 249 250 TRANSIENT PHENOMENA voltage (at arc temperature) between the electrodes. With an alternating impressed e.m.f. the arc if established goes out at the end of the half wave, or if a cathode blast is maintained continuously by a second arc (excited by direct current or overlapping sufficiently with the first arc), only alternate half waves can pass, those for which that terminal is negative from which the continuous blast issues. The arc, with an alternating impressed voltage, therefore rectifies, and the voltage range of rectification is the range between the arc voltage and the electro- static spark voltage through the arc vapor, or the air or residual gas which may be mixed with it. Hence it is highest with the mercury arc, due to its low temperature. The mercury arc is therefore almost exclusively used for arc rectification. It is enclosed in an evacuated glass vessel, so as to avoid escape of mercury vapor and entrance of air into the arc stream. Due to the low temperature of the boiling point of mercury, enclosure in glass is feasible with the mercury arc. II. MERCURY ARC RECTIFIER. 17. Depending upon the character of the alternating supply, whether a source of constant alternating potential or constant alternating current, the direct-current circuit receives from the rectifier either constant potential or constant current. Depend- ing on the character of the system, thus constant-potential rectifiers and constant-current rectifiers can be distinguished. They differ somewhat from each other in their construction and that of the auxiliary apparatus, since the constant-potential rectifier operates at constant voltage but varying current, while the constant-current rectifier operates at varying voltage. The general character of the phenomenon of arc rectification is, how- ever, the same in either case, so that only the constant-current rectifier will be considered more explicitly in the following paragraphs. The constant-current mercury arc rectifier system, as used for the operation of constant direct-current arc circuits from an alternating constant potential supply of any frequency, is sketched diagrammatically in Fig. 60. It consists of a constant-current transformer with a tap C brought out from the middle of the secondary coil AB. The rectifier tube has two graphite anodes ARC RECTIFICATION 251 a, 6, and a mercury cathode c, and usually two auxiliary mercury anodes near the cathode c (not shown in diagram, Fig. 60), which are used for excitation, mainly in starting, by establishing between the cathode c and the two auxiliary mercury anodes, from a small low voltage constant-potential transformer, a pair of low current rectifying arcs. In the constant-potential rectifier, generally one auxiliary anode only is used, connected through a resistor r with one of the main anodes, and the constant- Fig. 60. Constant-current mercury arc rectifier. Fig. 61. Constant-potential mercury arc rectifier. current transformer is replaced by a constant-potential trans- former or compensator (auto-transformer) having considerable inductance between the two half coils II and III, as shown in Fig. 61. Two reactive coils are inserted between the outside terminals of the transformer and rectifier tube respectively, for the purpose of producing an overlap between the two rectifying arcs, ca and cb, and thereby the required continuity of the arc stream at c. Or instead of separate reactances, the two half coils II and III may be given sufficient reactance, as in Fig. 61. A reactive coil is inserted into the rectified or arc circuit, which connects between transformer neutral C and rectifier neutral c, for the purpose of reducing the fluctuation of the rectified current to the desired amount. In the constant-potential rectifier, instead of the transformer ACS and the reactive coils A a and Ba, generally a compensator or auto-transformer is used, as shown in Fig. 61, in which the 252 TRANSIENT PHENOMENA two halves of the coil, AC and BC, are made of considerable self-inductance against each other, as by their location on different magnet cores, and the reactive coil at c frequently omitted. The modification of the equations resulting herefrom is obvious. Such auto-transformer also may raise or lower the impressed voltage, as shown in Fig. 61. The rectified or direct voltage of the constant-current rectifier is somewhat less than one-half of the alternating voltage supplied by the transformer secondary AB, the rectified or direct current somewhat more than double the effective alternating current supplied by the transformer. In the constant-potential rectifier, in which the currents are larger, and so a far smaller angle of overlap 0 is permissible, the direct-current voltage therefore is very nearly the mean value of half the alternating voltage, minus the arc voltage, which is about 13 volts. That is, if e = effective value of alternating voltage between rectifier terminals ab of compensator (Fig. 61), 2\/2 hence - — e = mean value, the direct current voltage is e0 = — e.- 13. 7T III. MODE OF OPERATION. 18. Let, in Figs. 62 and 63, the impressed voltage between the secondary terminals AB of an alternating-current trans- former be shown by curve I. Let C be the middle or center of the transformer secondary AB. The voltages from C to A and from C to B then are given by curves II and III. If now A,B,C are connected with the corresponding rectifier terminals a, 6,cand at c a cathode blast maintained, those currents will exist for which c is negative or cathode, i.e., the current through the rectifier from a to c and from b to c, under the impressed e.m.fs. II and III, are given by curves IV and V, and the current derived from c is the sum of IV and V, as shown in curve VI. Such a rectifier as shown diagrammatically in Fig. 62 requires some outside means for maintaining the cathode blast at c, since the current in the half wave 1 in curve VI goes down to zero at ARC RECTIFICATION 253 the zero value of e.m.f. Ill before the current of the next half wave 2 starts by the e.m.f. II. It is therefore necessary to maintain the current of the half wave 1 beyond the zero value of its propel- ling impressed e.m.f. Ill until the current of the next half wave 2 has started, i.e., to overlap the currents of the successive half waves. This is done by inserting reactances into the leads from the transformer to the rectifier, i.e., between A and a, B and b respec- tively, as shown in Fig. 60. The effect of this reactance is that the current of half wave 1, V, continues beyond the zero of its im- pressed e.m.f. Ill i.e., until the e.m.f. Ill has died out and reversed, and the current of the half wave 2, IV, started by e.m.f. II; that is, the two half waves of the current overlap, and each half wave lasts for more than half a period or 180 degrees. The current waves then are shown in curve VII. The current half wave 1 starts at the zero value of its e.m.f. Ill, but rises more slowly than it would without react- Fig. 62. Constant- current mercury arc rectifier. I / \ / \ ^~. s ^- 1 \ / \ s^ \ 1 ^ J \ 2 f 1 / *S ^ *-* \^ -v s— (s, j / / 11 III 1 /\ \\ ^ \ — s v •^ /" \ / \ / / / \ / ^ \ \£ *-> k / or about 0.425 i0. With decreasing load, at constant alternating-current supply, the rectified direct current slightly increases, due to the increas- ing overlap of the rectifying arcs, and to give constant direct current the transformer must therefore be adjusted so as to regulate for a slight decrease of alternating-current output with decrease of load. V. THEORY AND CALCULATION. 20. In the constant-current mercury-arc rectifier shown dia- grammatically in Fig. 64, let e sirr 6 = sine wave of e.m.f. im- pressed between neutral and outside of alternating-current supply to the rec- tifier; that is, 2 e sin 6 = total secondary generated e.m.f. of the constant-current transformer; Z1 = r1 -- jx1 = imped- ance of the reactive coil in each anode circuit of the rectifier (" alternating- current reactive coil")? inclusive of the internal self-inductive impedance be- tween the two halves of the transformer secondary coil; t\ and i2 = anode cur- rents, counted in the direction from anode to cathode; ea = counter e.m.f. Fig. 64. Constant-current of rectifying arc, which is constant; Z0 = mercury arc rectifier. r0 — jx0 = impedance of reactive coil in rectified circuit (" direct-current re- active coil"); Z2 = r2 ~ JX2 = impedance of load or arc-lamp circuit; e/ = counter e.m.f. in rectified circuit, which is con- ARC RECTIFICATION 257 stant (equal to the sum of the counter e.m.fs. of the arcs in the lamp circuit) ; #0 = angle of overlap of the two rectifying arcs, or overlap of the currents it and i2 ; i0 = rectified current during the period, 0 < 0 < 00, where both rectifying arcs exist, and i0' = rectified current during the period, 00 < 6 < n, where only one arc or one anode current i1 exists. Let e0 = eQ' + ea. = total counter e.m.f. in the rectified cir- cuit and Z = r — jx = (rt + r0 + r2) - j (xl + x0 -f x2) = total impedance per circuit; then we have (a) During the period when both rectifying arcs exist, o < e < 00, % = h + v C1) In the circuit between the e.m.f. 2 e sin d, the rectifier tube, and the currents \ and iv according to Kirchhoff's law, it is, Fig. 64, di di 2esin0 - v\ -^-+^2+^1- = 0- (2) In the circuit from the transformer neutral over e.m.f. e sin 6, current iv rectifier arc ea and rectified circuit i'0, back to the transformer neutral, we have di~ . di0 . di0 esmO-r^-x^ -ea - r,i0 - x0-^- r2i0 - x2— - e,' - or, e sin d - rlil - x^ - (r0 + ra) i0 - (x0 + x2) -^ - e0 = 0. (3) (6) During the period when only one rectifying arc exists, 00 < e < x, h = V; hence, in this circuit, di ' di ' e sin 0 -vY - x^ - (r0 + r2) iQ' - (x0 + x2) -± -e. = 0. (4) 258 TRANSIENT PHENOMENA Substituting (1) in (2) and combining the result (5) of this substitution with (3) gives the differential equations of the rec- tifier: 2 e sin 0 + r, (i0 - 2 i,} + X^'fa - 2 ij - 0, (5) and 2 e0 + (2 r - rj i0 + (2x- xj ° - 0, di ' e sin 6 — e — n'' — x —- = 0. (6) (7) In these equations, iQ and ^ apply for the time, 0 < 0 < 00, ij for the time, 00 < d < n. 21. These differential equations are integrated by the func- tions in -2i, = Ae~a9 + A' sin (6 - /?), (8) L=B*-» +B', (9) and (10) Substituting (8), (9), and (10) into (5), (6), and (7) gives three identities : 2 e sin.fl+A7 [r, sin (0-0) +x, cos (6-p)}+Ae-ae (r^-ax^) =0. 2 e0+B'(2 r-rj +5e-M [(2 r-rj -b (2 x-xj] = 0, and esmd-e0-C"[rsm(d-r)+xcos(0-r)]-C'r-Ce-c9(r-cx)=0', hence, and r, - ax, - 0, (2 r - rt) - b (2 x - x,) = 0, r — ex = 0. e0 + C'r = 0, 2 e + A' (r1 cos /? + x1 sin /?) = 0, A7 (rt sin /? - xl cos /?) = 0, e — C" (r cos 7- + £ sin ?-) = 0, C" (r sin ^ - x cos ?) = 0. (11) ARC RECTIFICATION 259 Writing ? = v/r 2 4- r 2 z\ ~ ri r *t. i tan <*,=—, and tan a: = - r (12) (13) Substituting (12) and (13) gives by resolving the 9 equations (11) the values of the coefficients a, b, c, A', J5', C", C", ft r: (14) r c = -> x r— a, 2e B'- 2e« 2r-r1 C" = +-, z and thus the integral equations of the rectifier are and io-Zi, =A£~ae - -sin (0 -a,), 2r - r. r 2 (15) (16) (17) (18) (19) (20) 260 TRANSIENT PHENOMENA where a, b, c are given by equations (14), a and al by equations (12) and (13), and A, J3, C are integration constants given by the terminal conditions of the problem. 22. These terminal conditions are : = o, 0 f« and (21) That is, at 0 = 0 the anode current il = 0. After half a period, or n = 180°, the rectified current repeats the same value. At 6 = 00, all three currents iv iw iQ' are identical. The four equations (21) determine four constants, A, B, C, 00. Substituting these constants in equations (18), (19), (20) gives the equations of the rectified current i0, iQ', and of the anode currents i^ and i2 = i0 — iv determined by the constants of the system, Z, Zv e0, and by the impressed e.m.f., e. In the constant-current mercury-arc rectifier system of arc lighting, e, the secondary generated voltage of the constant- current transformer, varies with the load, by the regulation of the transformer, and the rectified current, iQ, i0', is required to remain constant, or rather its average value. Let then be given as condition of the problem the average value i of the rectified current, 4 amperes in a magnetite or mercury arc lamp circuit, 5 or 6.6 or 9.6 amperes in a carbon arc lamp circuit. Assume as fair approximation that the pulsating rectified current i0, iQr has its mean value i at the moment, 6 = 0. This then gives the additional equation Kol.-o - *> (22) and from the five equations (21) and (22) the five constants A, B, (7, 00, e are determined. Substituting (22), (18), (19), (20) inequations (21) gives A • ' A = i --- sin a C =S*i + -°-sma r z (23) ARC RECTIFICATION 261 _ sn a_ zl 2 r - (24) Substituting (23) in (24) gives 2 e( ) • j € ' 9° sin al - sin (al — #0) > = i z 1 \ ) 2r - and -«-*• (25) ^ r — TJ f ) w~*o) sin « + sin (a — 60] 2 r - rt ( ) r < and eliminating e from these two equations gives ) t(2r-r,){ \ (27) Equation (27) determines angle 00, and by successive substitu- tion in (26), (23), e, A, B, C are found. Equation (27) is transcendental, and therefore has to be solved by approximation, which however is very rapid. As first approximation, a = 6 = c = 0; a=a1 = 90° or - 2i and substituting these values in (27) gives cosfl, 2* V W + W 1 - cos dl zl 262 TRANSIENT PHENOMENA and (28) This value of Ol substituted in the exponential terms of equa- tion (27) gives a simple trigonometric equation in 00, from which follows the second approximation 02, and, by interpolation, the final value, 23. For instance, let e0 = 950, i = 3.8, the constants of the circuit being Zl = 10 - 185 / and Z = 50 - 1000 /. Herefrom follows a = 0.054, b = 0.050, and c = 0.050, (14) «A = 86.9° and a = 87.1°. (15) From equation (28) follows as first approximation, #t = 47.8°; as second approximation, #2 = 44.2°. Hence, by (29), 0 = 44.4°. Substituting a in (26) gives e = 2100, hence, the effective value of transformer secondary voltage, 2 e J— = 2980 volts V2 and, from (23), A = -- 18.94, B = 24.90, C = 24.20. Therefore, the equations of the currents are i0 = 24.90s-0-0500- 21.10, V= 24.20 e-0'050* - 19.00 +2.11 sin (6 - 87.1°), i, = 12.45 e-0'050* + 9.47 £-°-0540- 10.58 + 11.35 sin (0 - 86.9°), and ARC RECTIFICATION 263 The effective or equivalent alternating secondary current of the transformer, which corresponds to the primary load current, that is, primary current minus exciting current, is V V-t Vt From these equations are calculated the numerical values of rectified current iot t'0', of anode current iv and of alternating current i' , and plotted as curves in Fig. 65. \ \ Fig. 65. Current waves of constant-current mercury arc rectifier. 24. As illustrations of the above phenomena are shown in Fig. 66 the performance curves of a small constant-current rec- tifier, and in Figs. 67 to 76 oscillograms of this rectifier. Interesting to note is the high frequency oscillation at the ter- mination of the jump of the potential difference cC (Fig. 60) which represents the transient term resulting from the electro- static capacity of the transformer. At the end of the period of overlap of the two rectifying arcs one of the anode currents reaches 264 TRANSIENT PHENOMENA =£ 24 20 GO | 12 o 40 n 8 100 200 300 400 500 600 700 800 900 1000 1100 Volt Load Fig. 66. Results from tests made on a constant-current mercury arc rectifier. \ Fig. 67. Supply e.m.f. to constant-current rectifier. A A A , / V V V Fig. 68. Secondary terminal e.in.f. of transformer. Fig. 69. E.m.f. across a.c. reactive coils. r\ . , i/ Fig. 70. Alternating e.m.f. impressed upon rectifier tube. ARC RECTIFICATION 265 w\r\r\r\ Fig. 71. Unidirectional e.m.f. produced between rectifier neutral and transformer neutral. V Fig. 72. E.m.f. across d.c. reactive coils. Fig. 73. Rectified e.m.f. supplied to arc circuit. Fig. 74. Primary supply current. Fig. 75. Current in rectifying arcs. Fig. 76. Rectified current in arc circuit. 266 TRANSIENT PHENOMENA zero and stops, and so its L — abruptly changes; that is, asud- ctt den change of voltage takes place in the circuit aACDc or bBCDc. Since this 'cuit contains distributed capacity, that of the transformer C' ^BC respectively, the line, etc., and inductance, an oscillatk ^.-.esults of a frequency depending upon the capacity and inductance, usually a few thousand cycles per second, and of a voltage depending upon the impressed e.m.f.; that is, the L — of the circuit. An increase of inductance L dt di increases the angle of overlap and so decreases the — , hence does CLL not greatly affect the amplitude, but decreases the frequency of this oscillation. An increase of — at constant L, as resulting dt from a decrease of the angle of overlap by delayed starting of the arc, caused by a defective rectifier, however increases the amplitude of this oscillation, and if the electrostatic capacity is high, and therefore the damping out of the oscillation slow, the Fig. 77. E.m.f. between rectifier anodes. oscillation may reach considerable values, as shown in oscillo- gram, Fig. 77, of the potential difference db. In such cases, if the second half wave of the oscillation reaches below the zero value of the e.m.f. wave db, the rectifying arc is blown out and a disruptive discharge may result. ARC RECTIFICATION 267 VI. EQUIVALENT SINE WAVES. 25. The curves of voltage and current in the mercury-arc rectifier system, as calculated in the p!neding from the con- stants of the circuit, consist of succe, , Sections of exponential or of exponential and trigonometric c In general, such wave structures, built up of successive sections of different character, are less suited for further calculation. For most purposes, they can be replaced by their equivalent sine waves, that is, sine waves of equal effective value and equal power. The actual current and e.m.f. waves of the arc rectifier thus may be replaced by their equivalent sine waves, for general calculation, except when investigating the phenomena resulting from the discontinuity in the change of current, as the high frequency oscillation at the end and to a lesser extent at the beginning of the period of overlap of the rectifying arcs, and similar phenomena. In a constant-current mercury arc rectifier system, of which the exact equations or rather groups of equations of currents and of e.m.fs. were given in the preceding, let % = the mean value of direct current; eQ = the mean value of direct or rectified voltage; i = the effective value of equivalent sine wave of secondary current of transformer feeding the rectifier; e = the effective value of equivalent sine wave of total e.m.f. generated ^> in the transformer secondary coils, hence, - = the effective ft equivalent sine wave of generated e.m.f. per secondary trans- former coil, and #0 = the angle of overlap of rectifying arcs. The secondary generated e.m.f., e, is then represented by a sine wave curve I, Fig. 78, with e \/2 as maximum value. Neglecting the impedance voltage of the secondary circuit during the time when only one arc exists and the current changes are very gradual, the terminal voltage between the rectifier anodes, ev is given by curve II, Fig. 78, with e V2 as maximum value. This curve is identical with e, except during the angle of overlap #0, when e^ is zero. Due to the impedance of the) reactive coils in the anode leads, curve II differs slightly from I, but the difference is so small that it can be neglected in deriving 268 TRANSIENT PHENOMENA the equivalent sine wave, and this impedance considered after- wards as inserted into the equivalent sine-wave circuit. The rectified voltage, e2, is then given by curve III, Fig. 78, e /— e with a maximum value of - v 2 = —/=- and zero value during the angle of overlap 00, or rather a value = ea, the e.m.f. con- sumed by the rectifying arc (13 to 18 volts). ii in IV VI VII viii Fig. 78. E.m.f. and current curves in a mercury arc rectifier system. The direct voltage e0, when neglecting the effective resistance of the reactive coils, is then the mean value of the rectified voltage, e2, of curve III, hence is en = elf. = — : - / SI v/2 *V* sin d dd ARC RECTIFICATION _ e (1 + cos 00) m y> — — 0 fy C/. 269 If ea = the mercury arc voltage, r0 = the effective resistance of reactive coils and i0 = the direct current, more correctly it is The effective alternating voltage between the rectifier anodes is the Vmean square of ev curve II, hence is 2 00 - sin 2 0 27T and the drop of voltage in the reactive coils in the anode leads, caused by the overlap of the arcs, thus is - sin 2 0( 2x 26. Let iv = the maximum variation of direct current from mean value i'0, hence, i2 = i0 + if =' the maximum value of rectified current, and therefore also the maximum value of anode current. The anode current thus has a maximum value iv and each half wave has a duration TT + 00, as shown by curve IV, Fig. 78. The direct current, i0, is then given by the superposition or addition of the two anode currents shown in curves V, and is given in curve VI. 270 TRANSIENT PHENOMENA The effective value of the equivalent alternating secondary current of the transformer is derived by the subtraction of the two anode currents, or their superposition in reverse direction, as shown by curves VII, and is given by curve VIII. Each impulse of anode current covers an angle n + 00, or somewhat more than one half wave. Denoting, however, each anode wave by n, that is, considering each anode impulse as one half wave (which corresponds to a Ifower frequency— -), then, referred to the anode impulse x + V as half wave, the angle of overlap is TT 0, = 1 ^ , 71 -j- The direct current, i0, is the mean value of the anode current curves V, VI, and, assuming the latter as equivalent sine waves of maximum value i2 = i0 + i' ', the direct current, i0, is sin d' dd' u1 i/o 2L - 0, 2 (n + 00) L and or, and the pulsation of the direct current, if = i2 — i0, is 7T2 ) The effective value of the secondary current, as equivalent sine wave in one transformer coil, is the \/ mean squiare of curves VII, VIII, or, assuming this current as existing in both trans- former secondary coils in series — actually it alternates, one half ARC RECTIFICATION 271 wave in one, the other in the other transformer coil — is half this value, or = 7? V — ~ ; or, substituting ^2 = *0 2 -0,) + 0, cos 0, - sin 0, or, substituting i< — ~~ ~ IT v -L H~ cos sin 2 2V2 where — = ratio of effective value to mean value of sine wave. 2V2 27. An approximate representation by equivalent sine waves, if e0 = the mean value of direct terminal voltage, i0.= the mean value of direct current, is therefore as follows: The secondary generated e.m.f. of the transformer is 272 TRANSIENT PHENOMENA the secondary current of the transformer is 2 < the pulsation of the direct current is 7T2 cos sin x + 60 Ti + 6, the anode voltage of the rectifier is 2 #0 - sin 2 #c and herefrom follows the apparent efficiency of rectification, -V 61 the power factor, the efficiency, etc. 70° 80° Fig. 79. E.m.f. and current ratio and secondary power factor of constant- current mercury arc rectifier. From the equivalent sine waves, e and i, of the transformer secondary, and their phase angle, the primary impressed e.m.f. and the primary current of the transformer, and thereby the ARC RECTIFICATION 273 power factor, the efficiency, and the apparent efficiency of the system, are calculated in the usual manner. In the secondary circuit, the power factor is below unity essentially due to wave shape distortion, less due to lag of cur- rent. As example are shown, in Fig. 79, with the angle of overlap 00 e 2i as abscissas, the ratio of voltages, - — ; the ratio of currents, — ; 2eo % i' the current pulsation, -> and the power factor of the secondary to circuit. SECTION III TRANSIENTS IN SPACE TRANSIENTS IN SPACE