75 8.56 Joop 195 j........ | 7.9 | 8.3

90 9.18 |.........f..... |} 901.7 Foo] 7.55 | — 7.85

105 38 joe ene eenns Peeeenenn neennen beneaaee

The mean value of the rectified current is derived herefrom as 8.92 amp., while without rectification the effective value of

110 alternating current would be -————- = 3.48. 110 volts . VP +x . 22 wa: effective corresponds to ~ - 110 = 99 volts mean, which in Zn 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 (6 — 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.

. | Zei-4 \é = 156 sing | | 15 $— tN tt 1 — 9074+ Py EES (as ER AS ee YE IN] | lOe780 ! Ca (Ey A a Ca Hs UV (i is“ | [Phas i737 758 || ENO OO OA we ue Rit 1A oe | Bt (ea) I) i | Joe a Teal Py ae ee ee = (eae CCC eer Se £ | || bee! Ge AR BS | iF Peet CT ‘EP ARRAS PARR EPAL Gees sift 4 Pe ie A tt} Ltt ECC er Wet NS rrr ray ARRRRESE ARG | 20 #0 60 fo] 100 «6100 «(640 16180 200 Degrees 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., EZ, = ecos @ and EF, = esin 0, are connected to a rectifying commutator, so that while the first E, is in circuit E, is open-circuited. At the moment @,, E, is connected in parallel, as shown diagrammatically in Fig. 58, with Z,, and the rising e.m.f. in £, gradually shifts the current i, away from £, into E,, until at the moment 6,, E, is dis- connected and £, 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 7,. Thus up to the moment 6, the current in Z,ist,,in

|

MECHANICAL RECTIFICATION 248 E,is 0. From @, to 0, the current in F, may be 7; thus in £, it isi, =1, —1. After 6,, the current in £, is 0, in £, it is z,. A change of current occurs only during the time from 6, to 8,, and it is only this time that needs to be considered. ri .

/ 7 N -- ; ( , s % “ 7 -_ Fig. 68. Quarter-phase constant-current rectifying commutator. | Let Z =r — jz = impedance per phase, ‘where z = 2 njL; then at the time ¢ and the corresponding angle 0 = 2 zt the difference of potential in E, is ecos0 — (i, 1) r — Loo = dt ae di | =ecosd-(@, —a)r+aa, (1) the difference of potential in E, is esin @ — ir — Mt, do ? and, since these two potential differences are connected in parallel, they are equal ¢ (Sind — c080) + ig - 2ir — 22% = 0. (2)

244 TRANSIENT PHENOMENA The differential equation (2) is integrated by 1=A + Be“ +C cos (6 — 3); (3) di we one ; thus — = — aBe~™ — Cin (6 — 8), dé . and substituting in (2), e (sin 0 — cos 6) +%,r — 2 Ar — 2 Bre~™ — 2Cr cos (6 — 3)

• 2aBure—-% + 2Czxsin @ — 3) = 0- or, transposed, ((, -2A)r +2 Be-%(ax — r)+ sin 6[e — 2Crsind
• 2 Cz cos 8] — cos 6 [e + 2Crcosd + 2Czsin 3] = 0; thus ; 1,-2A=0, ax —r =0, e — 2Crsind + 2Crcosd = 0, . , and e+2Czrsind + 2Crcosé = 0, and herefrom, letting = = tan oc, we have e= — 2Czsin (o — 0), e = — 2Czcos (7 — 4), 1, A = 9 J r a=-) zx : . tant =, : (4) r+r. C=- —“., V2(2 +7) . tan (¢ — 4) = 1, e and Cc =——-.- V 22 |

MECHANICAL RECTIFICATION 245 These values substituted in (3) give ty ~1e e i . t=24+Be * —-—— cos (6-6 g B® ~ rary ROO z-r tan d = z4r ° (5) At @ = 6,,7 = 0, and we have 1, —26 e 0="°+Be » —————— cos (f, — 9); 2 2@ +?) 6 hence, -74 e t, | Be = = ——————. ¢08 (6, — 0) - 3; 6) . 2 +7) ~~ Fi ' substituting in (5), we have the equations of current in the two coils as follows: , i-2 ‘(— = cos (6, — 8) — ie), se" ] 2 2(2 +7) , 2 e -—=— - os (6 — 3 V2 (2? +77) ( ) r (7) é — -(6-4) — ————— c0s (6-6) — cos (6, —d) ¢ * 4,; Fae e8 C— 2) 0080, —2) a and

• 4 ( e =) -2 (6-4) t —-t == —|—————c08 (0, — 8) —=Je = e

• —————. cos (6 — 8). V2(e +7) e-9) J

246 TRANSIENT PHENOMENA At 6 = 6,,1 = 1,; thus 1, ( e to\ — 2-6 ~ — | —————— cos (6 — 6) - se = 2 \W2@+P) ie 2 e

• ————— cos (4 — 0d) = 0; Vier’ or, multiplied by « =" and rearranged, we have the condition connecting moments 0, and 8,, as follows: On (3" Fe) é -124 ~ +e 7 |) +———— Je * cos (6,- 3) 2 V2(F +7) ; — =" e908 (6, - af =0 and t ( an :*) e £6, za + e = ——_—. 4 &® cos (@, — d) 2 V2 (2 +7) , ~ * cos 0,2). (8) Rearranged equation (8) gives f(6,) = enti + 2? ___ o (0 0)| ; 1,2 (2 +7") > : Zo, 2e =er | —__——-— cos (9, — 8 -1], 9 ers 1 — 9) © where tang =~". z+r By approximation, from this equation the value of @,, corre- sponding to a given @,, is derived. .

16. Example: : e = 2000, 7, = 10, and Z = 10 — 407. | Thus 6 = 31° = 0.54 radians -

MECHANICAL RECTIFICATION 247 and i = 5 + [34.3 cos (0, — 31°) — 5) 90 — 34.3 cos (@ — 31°), f (0,) = "(1 + 6.86 cos (0, — 31°)] . ; = °%% [6.86 cos (8, — 31°) — 1]. ae oF ot wo Substituting for @,, 30° = g7 2 = q? and 60° = 37 Tespec- tively, gives: nm . — 096 (9— 5)- Om Fim 5+ 20.36 6) _ 34.3 cos (@—31°) Z,im5+ 28.37 °*(?~ 4)_34 3 cos (6— 31°)

• 6% Z,t=54+25.1e 035 (° ~ 5) _ 34. 3 cos (0—31°) 3 and . cs wv a= => a=5 a=3 | @ ‘ tr i te ‘ ta e a1 25 |......4...00)....0).... |... fe....| 1810 | 850

30. 0 | 10 |.....)....0, dp. 35 ]......)......)......,.....-).00..-fe.0. 0.) 1640 | 1150 40|-0.9| 10.9|...........0)......f.0000)... fe... 45]......[....... 0 | 40 |......]......) 1410 | 1410 50 |—0.6 | 10.6]......]......]......J....0.,...00) B5]......]......, 8] 9.2]......[......] 1150 | 1640 60/+0.6| 9.4|......]......] 0 | 10 |......[...... 65)......[......) 2.5] 7.5 |......[......] 850] 1810 -| 70} 3.0] 7.0]......)......) 22] 7.8].....).0.... 75)......[......) 5.1] 4.9 ]......]......) 520 | 1930 80} 6.0| 4.0]......]......) 5.4] 46].....)...... 85]......[......] 8.6] 2.4 ]......[......1 170 | 1990 90| 9.9] O.1]......[......f 93] O7]...0./000...
31. .....]......] 12.8 |-2.8|......]......J—170 | 1990 100 | 14.3 |—4.3]......]......] 13.8 |-3.8]......[...... 105 |......]......] 17.3 |-7.3 |......[......]—520 | 1930 110} 19.1 |-9.1]......]......] 18.6 |-8.6]......]...... 115 |......)......] 22.2 |—12.9)......]......]—850 | 1810 These values are plotted in Fig. 59, together with e, and e,. It follows then, 6,= 90.2° 88.6° 91.7°

248 TRANSIENT PHENOMENA ' The actual curves of an arc machine differ, however, very greatly from those of Fig. 59. In the arc machine, inherent regu- lation for constant current is produced by opposing a very high ‘armature reaction to the field excitation, so that the resultant m.m.f., or m.m.f. which produces the effective magnetic flux, is | | ea 20 30 40 50 60 70 80 90 100 «6110 120 | BRAD RRR oe | oes SeeeUTUORREREEBE LICE a nace “SSS, ean 8 2000}-r=f ae : 6 1 a SK gf 3000 HN | = et Ct eee Ne pee! Fm El GRE an se ct OO bat ad aoe} g | | 8 HH & ‘a | | ARS SRah wt “8 |_| a pO OS Oe a a a oie seESaECSUUEeeeEEEe eal (25 1G ed Be at TSS 20 30 40 50 ee 90 100 «10 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.

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 are 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 electrit 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, ie., 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 flarne 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 forthe 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 are stream. Due to the low temperature of the boiling point of mercury, enclosure in glass is feasible with the mercury arc.

II. Mercury Arc REcrIFIER.

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, b, 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- TY] [-) _ ne cay | a | | & B . ay | ; = | | = git TS af | = | 7 ’ | Fig. 60. Constant-current Fig. 61. Constant-potential | mercury arc rectifier. 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 . ares, ca and cb, and thereby the required continuity of the arc stream atc. 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 ACB and the reactive coils Aa and Ba, generally a compensator cr 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, ot .

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 6 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), hence ave é = mean value, the direct current voltage is

& = v2, — 13. . Zz III. MopE or OperarIoN.

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,b,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 @ to c and from 6 to c, under the impressed e.m.fs. II and ITI, 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. III before the current of the next half wave 2 starts by the e.m.f. IT. 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. IIT until the current of the next half wave 2 has started, i.e., to laa overlap the currents of the successive half ANG) B waves. This is done by inserting reactances ia into the leads from the transformer to the rectifier, i.e., between A and a, B and b respec- i tively, as shown in Fig. 60. The effect of ? | this reactance is that the current of half wave a b t 1, V, continues beyond the zero of its im- Ye i _ pressed e.m.f. II] i.e., until the e.m.f. IIT has VI x died out and reversed, and the current of the l 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 an Constante a period or 180 degrees. are rectifier. my The current waves then are shown in curve VII. The current half wave 1 starts at the zero value of its e.m.f. III, but rises more slowly than it would without react- sey a SEES ENGEEPEREEEESCEE . OANA NCCCONC SA ae ne SSH == x a: N a NLI* sissaaibecaatinssiinaiie misiZ_ Nr NI TT er Ty a CATT NCAT NICE | BRC : TV =a ae P — y{LUIZENN LAK BEES SASS xa eee a ms ea oe t 1 IAL IA 7 | | XIIt vo pt Ppl 2] Dt | ia | 4 = wpe RE EER +H Fig. 63. E.m.f. and current waves of constant-current mercury arc rectifier. ance, following essentially the exponential curve of a starting ° ; current wave, and the energy which is thus consumed by the reactance as counter e.m.f. is returned by maintaining the

254 TRANSIENT PHENOMENA

current half wave 1 beyond the e.m.f. wave, i.e., beyond 180 degrees, by 0, time-degrees, so that it overlaps the next half wave 2 by 0, time-degrees.

Hereby thé rectifier becomes self-exciting, i.e., each half wave of current, by overlapping with the next, maintains the cathode blast until the next half wave is started. .

The successive current half waves added give the rectified or unidirectional current curve VIII.

During a certain period of time in each half wave from the zero value of e.m.f. both arcs ca and cb exist. During the existence of both arcs there can be no potential difference between the rectifier terminals a and b, and the impressed e.m.f. between the rectifier terminals a and b therefore has the form shown in curve IX, Fig. 63, i.e., remains zero for 0, time-degrees, and then with the breaking of the arc of the preceding half wave jumps up to its normal value.

The. generated e.m.f. of the transformer secondary, however, must more or less completely follow the primary impressed e.m.f. wave, that is, has a shape as shown in curve I, and the difference between IX and I must be taken up by the reactance. That is, during the time when both arcs exist in the rectifier, the a. c. reactive coils consume the generated e.m.f. of the transformer secondary, and the voltage across these reactive coils, therefore,

. is as shown in curve X. That is, the reactive coil consumes voltage at the start of the current of each half wave, at x in curve X, and produces voltage near the end of the current, at y. Between these times, the reactive coil has practically no effect and its voltage is low, corresponding to the variation of the rectified alternating current, as shown in curve XI. That is, during this intermediary time the alternating reactive coils merely assist the direct-current reactive coil.

Since the voltage at the alternating terminals of the rectifier, a, b, has two periods of zero value during each cycle, the rectified voltage between c and C must also have the same zero periods, and is indeed the same curve as IX, but reversed, as shown in curve XII.

Such an e.m.f. wave cannot satisfactorily operate arcs, since

’ during the zero period of voltage XII the arcs go out. The voltage on the direct-current line must never fall below the “counter e.m.f.” of the arcs, and since the resistance of this

ARC RECTIFICATION 255 . circuit is low, frequently less than 10 per cent, it follows that the total variation of direct-current line voltage must be below . 10 per cent, i.e., the voltage practically constant, as shown by the straight line in curve XII. Hence a high reactance is inserted into the direct-current circuit, which consumes the excess voltage . during that part of curve XII where the rectified voltage is above line voltage, and supplies the line voltage during the period of zero rectified voltage. The voltage across this reactive coil, therefore, is as shown by curve XIII.

IV. Constant-CurRENT RECTIFIER.

19. The angle of overlap 0, of the two arcs is determined by the desired stability of the system. By the angle 0, and the impressed e.m.f. is determined the sum total of e.m.fs. which has to be consumed and returned by the a. c. reactive coil, and herefrom the size of the a. c. reactive coil.

From the angle 0, also follows the wave shape of the rectified voltage, and therefrom the sum total of e.m.f. which has to be given by the d.c. reactive coil, and hereby the size of the d. c. reactive coil required to maintain the d.c. current fluctuation within certain given limits.

The efficiency, power factor, regulation, etc., of such a mercury arc rectifier system are essentially those of the constant-current transformer feeding the rectifier tube.

Let f = frequency of the alternating-current supply system, tw, = mean value of the rectified direct current, and a = the pulsa- tion of the rectified current from the mean value, i.e., 7, (1 + @) the maximum and 7, (1 — a) the minimum value of direct cur- rent. A pulsation from a mean of 20 to 25 per cent is permissible in an arc circuit. The total variation of the rectified current then is 2 a1,, i.e., the alternating component of the direct current has the maximum value az,, hence the effective value i, (or for a = 0.2, 0.141 7,) and the frequency 2, f. Hysteresis and eddy losses in the direct-current reactive coil, therefore, correspond to an alternating current of frequency 2/f and effective value z t,, or about 0.141 7,, i.e., are small even at relatively high densities.

256 TRANSIENT PHENOMENA

In the alternating-current reactive coils the current varies,

unidirectionally, between 0 and 7, (1 + a), i. e., its alternating

. 1 . component has the maximum value = i, and the effec- . . l+a. . oe tive value Wa” (or, for a = + 0.2, 0.425 2,) and the fre- quency f. The hysteresis loss, therefore, corresponds to an . . 1 . alternating current of frequency f and effective value ros Voy or about 0.425 %,. .

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-are rectifier shown dia-

grammatically in Fig. 64, let e sin 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 @ = total secondary

generated e.m.f. of the constant-current

i transformer; Z, =, ~ jt = imped-

ndAy qe 7 ance of the reactive coil in each anode

‘A i _, Gireuit of the rectifier (“alternating-

wes i current reactive coil’’), inclusive of the

“y* 1 internal self-inductive impedance be-

ig, iy tween the two halves of the transformer

Nie nel secondary coil; 7, and 1, = anode cur-

ory My- Jz; . . .

rents, counted in the direction from

. anode to cathode; es = counter e.m.f.

Fig. 64. Constant-current of rectifying arc, which is constant; Z, =

mercury arc rectifier. T, — JZ, = impedance of reactive coil in rectified circuit (‘‘direct-current re- active coil”); Z, = r, — jx, = 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); 6, = angle of overlap of the two rectifying arcs, or overlap of the currents 7, and 7,; 7, = rectified current during the period, 0 < 6 <6,, where both rectifying ares exist, and 7,’ = . rectified current during the period, 0, < @ < z, where only one are or one anode current 2, exists.

Let e, =e, + ea = total counter e.m.f. in the rectified cir- cuitand Z = r — jx = (r, +7) +7,) — 7 (24, + 2% + 2,) = total impedance per circuit; then we have

(a) During the period when both rectifying arcs exist,

0<40<4, 1 = 1%, +14; (1)

In the circuit between the e.m.f. 2 sin 6, the rectifier tube, and the currents 7, and 7,, according to Kirchhoff’s law, it is, Fig. 64,

. . adi . di | 2esind — r,t, — a Stn, + 2,3 = 0. (2) | In the circuit from the transformer neutral over e.m.f. e sin 0, current 7,, rectifier arc é. and rectified circuit 7,, back to the transformer neutral, we have . . ay . di . di | esin 0—r,i,—z,—! —ea — Toto — 2958 — Tris — 24S —e,/ = 0; or, . . di . . di esind — ri, — 24797 fo +1,) t, —(2, +2) —e,=0. (3) . (b) During the period when only one rectifying arc exists, | 0,<0<3a, | 1, = Uy hence, in this circuit, ,

di di,’

esind —ry’ — a — (r, +7,) 1" —(x,+ 13) ay —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: . . . d. . 2esin é +7 (lo — 24) + Tae (lo — 24) = 0, (5) . di 2e,+(2r—7,)% +22 - 2) =0, (6) d esind —e — rit — tle! 9 (7) an a 0 0 do = Uz. ‘ In these equations, ¢, and 7, apply for the time, 0 < @ < 4, i, for the time, 0, < 9 < =. . 21. These differential equations are integrated by the func- tions i, — 27, = Ac” + A’sin (0 — 8), (8) i, = Be~® + B’, (9) and if = Ce~% 4+ C’ + C/ sin (6 — 7). (10) Substituting (8), (9), and (10) into (5), (6), and (7) gives | three identities: ‘ 2esin 0 +A’ [r, sin (9-8) +2, cos (6—8)]+Ae~” (r,—az,) =0. 2e,+B’(2r—r,) +Be~"((2 r—r,) —b (2 x—2z,)]=0, ; and | esind —e,—C”[rsin (0 —y) +2 cos(@ —)]—C’r —Ce~%(r —cz) =0; . hence, . ‘ r, — ax, = 0, . (2r —7,) —b (2x -—2,) = 0, and r—cx =0. 2e,+ B’ (2r-7,) =0, . e, + C’r = 0, qn 2e +A’ (r, cos 3? + x, sin 8) = 0, A’ (r, sin 8 — zx, cos #) = 0, e—C” (rcosy + xsiny) = 0, . C” (rsiny — xeosy) =0. Jy |

ARC RECTIFICATION 259 Writing _ 2z,=Vre4+2/, 12 tana, = 71 (12) v, and z=Vrt 2, | x (13) tana = ra . Substituting (12) and (13) gives by resolving the 9 equations. (11) the values of the coefficients a, b, c, A’, B’, C’, C”, B, r:* | a= f, q, 2r— 7, b= 2n—2, (14) r c=-) x 8 = ay (15) rua, . a= — 72, . z, yo 2 e, B= Ir 7 (16) Cc’ = fo, . T e

• (4% = +3 (17) and thus the integral equations of the rectifier are t — 2%, = Ac~® — 2° sin (0 — a,), (18) 1 2e ; , Be-be _ _“%o 19 do Be 2 r— r, ( ) . > _o & ,@. and tf =Ce“% — 7 +; sin (@ — a), (20)

260 TRANSIENT PHENOMENA where a, b, c are given by equations (14), a and a, by equations (12) and (13), and A, B, C are integration constants given by the , terminal conditions of the problem. 22. These terminal conditions are: itl o~o = O, [Vol omo = [to lomns (21) _ and [Jones = [Yolomoo = |%o'lomoo-

That is, at 6 = 0 the anode current 7, = 0. After half a period, or z = 180°, the rectified current repeats the same value. At 6 = @,, all three currents 1,, 2, 2,’ are identical.

The four equations (21) determine four constants, A, B, C, 4,.

Substituting these constants in equations (18), (19), (20) gives the equations of the rectified current 7,, 7,’, and of the anode currents 7, and 7, = 1, — 2,, determined by the constants of the system, Z, Z,, e,, 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, 7,, 7,’, is required to remain constant, or rather its average value.

Let then be given as condition of the problem the average value ¢ 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 1,, 7,’ has its mean value 7 at the moment, 0 = 0. This then gives the additional equation

tolomo = 4, (22) and from the five equations (21) and (22) the five constants A, B,C, 6,, e are determined. ; Substituting (22), (18), (19), (20) in equations (21) gives . 2e, . A=i- z sin @,, Bait 7%, (23) 2r-—7, ower Sit &—Ssinal ' r 2 . ; |

ARC RECTIFICATION 261 a 2e., _ 2e — Ae sf) 3," (a,— 9,) = Be me sy, = Ce — “2 _ 2 sin (@ — 0,). (24) r 2 Substituting (23) in (24) gives . =£S -nan sin a, — sin (a, — 0) =i dena + —7 >t z, Y 26, } 06 = —.—--%. _ 0 9: Iron, l-e (25) and ; CF cir) gi Ug Sgecr— 00) _ 08 72° sina + sin (a — 9) = 2 € 0? 9% 2e, | 0 L, &0 4 erm — 0) ! . rs be € \ TF - lo (26) and eliminating e from these two equations gives e("—%) sin a + sin (a — 8,) e sin a, — sin (a, — 9,) 2e e c(— 09) __ . — b8 _ A“ *o — 2 — 5% -0 ) .c(w—O) __ oat ‘ {trata cmt stern % — aby “| _ 2& _ 0 fe te i@r_r) 1 é : (27) Equation (27) determines angle @,, and by successive substitu- tion in (26), (23), 2, 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 = b =c = 0; a = a, = 90° or 5 and substituting these values in (27) gives cm & ) e™ + cos 0, 22 ( (i+ l-—cos#, 4 2

262 TRANSIENT PHENOMENA and e(e- (tg) cos 6, = STN eNO (28) = (ee 1) (1 + a 1 2, ar, This value of 6, substituted in the exponential terms of equa- tion (27) gives a simple trigonometric equation in 6,, from which follows the second approximation 6,, and, by interpolation, the final value, _ 2 6, = 0, + Ca ON (29) 1 23. For instance, let e, = 950, 7 = 3.8, the constants of the circuit being Z, = 10 — 1857 and Z = 50 — 1000). Herefrom follows a = 0.054, b = 0.050, and ¢ = 0.050, (14) a, = 86.9° and a = 87.1°. (15) From equation (28) follows as first approximation, 6, = 47.8°; as second approximation, 0, = 44.2°. . Hence, by (29), 0 = 44.4°, Substituting a in (26) gives e = 2100, hence, the effective value of transformer secondary voltage, 2e — = 2980 volts V2 and, from (28), A = — 18.94, B = 24.90, C = 24.20. Therefore, the equations of the currents are ty = 24.90 2°? — 21.10, ij = 24.20 >" _ 19.00 + 2.11 sin (6 — 87.19), 4, = 12.45 2°? + 9.47 6° ™9 — 10.58 + 11.35 sin (9 — 86.9°), and . 1, = 1, — 1. |

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 =1t, -1;.

From these equations are calculated the numerical values of rectified current 7,, 2,’, of anode current 7,, and of alternating current 2’, and plotted as curves in Fig. 65.

ptt tt TT er TT TT SREP ae

rt tT Pe TT TTY TT TT

Lt tT Te Tt ET TT TAT TT TT

Piet EE Ae

ptt izT tT tT tT TP TTY TTT, Pt Tye tT ee tT TT

LETATTTITT TTT TRE Ty .

piv Tt tT Ee |e SSP eee

SRR AEN

PT TIA TT AT TT TT

SRP 4R RRA SERRE SRNR SERN REE ; PtP IN TE ET TT TZ T TT SERRE ee

Pitt tet tt te ET TT TT

pitt et Te

PPA ee rT NEF T

Pt Te

SRR Ree See

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 anodecurrents reaches

. | | 264 TRANSIENT PHENOMENA | | 2 bed BC a WR | oh ed a | TTT ry rr rr err ct CE eet u } |} | | eal ‘CEES SCCeccceeee eed , ae | | |g sco cl, 3 a ee okt t+ + | ett wo Se & TT. ol s 4 = = See ato 8 , ett ttt ttt 2 PEC & |_| bnnm ; | 0 100 200 300 400 500 600 700 so 900 100 110 Volt Load Fig. 66. Results from tests made on a constant-current mercury arc rectifier. i PAPAS Fig. 67. Supply e.m.f. to constant-current rectifier. . L LU U | Fig. 68. Secondary terminal e.m.f. of transformer. | Fig. 69. E.m.f. across a,c. reactive coils. vaNVaNVaavin | Fig. 70. Alternating e.m.f. impressed upon rectifier tube.

ARC RECTIFICATION 265 Fig. 71. Unidirectional e.m.f. produced between rectifier neutral and transformer neutral.

Fig. 72. E.m.f. across d.c. reactive coils. EN Fig. 73. Rectified e.m.f. supplied to arc circuit.

Fig. 74. Primary supply current.

Fig. 75. Current in rectifying arcs, LLL TS

Fig. 76. Rectified current in arc circuit.

\ | | | | . 266 TRANSIENT PHENOMENA . di . . zero and stops, and so its L a abruptly changes; that is, a sud- den change of voltage takes place in the circuit aACDec or | bBCDc. Since this circuit contains distributed capacity, that of the transformer coil ACBC respectively, the line, etc., and _ inductance, an oscillation results 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.mf.; ; di . ; ; | that is, the LS of the circuit. An increase of inductance L increases the angle of overlap and so decreases the .. hence does , not greatly affect the amplitude, but decreases the frequency of — this oscillation. An increase of a at constant L, as resulting | . 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 | , | i } ! : | | i | | Fig. 77. E.m.f. between rectifler anodes. | oscillation may reach considerable values, as shown in oscillo- _ gram, Fig. 77, of the potential difference ab. In such cases, if the second half wave of the oscillation reaches below the zero value of the e.m.f. wave ab, the rectifying are 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-are rectifier system, as calculated in the preceding from the con- stants of the circuit, consist of successive sections of exponential or of exponential and trigonometric character. 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 actyal 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 7, = the mean value of direct current; e¢, = the mean value of direct or rectified voltage; 7 = 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, 5 = the effective equivalent sine wave of generated e.m.f. per secondary trans- former coil, and 6, = 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 V2 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, ¢,, 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 @,, when e, is zero. Due to the impedance of the reactive coils in the angde 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, e,, is then given by curve III, Fig. 78, with a maximum value of v2 = we and zero value during the angle of overlap @,, or rather a value = e,, the e.m.f. con- sumed by the rectifying arc (13 to 18 volts).

pit | EAT TT TT PAY

ARENA

PL TTT TTX TAT TT

1 ee AL E71 oT EK

EEA

P| PP tt TE Ty tT

pet ee

PN | eA ENT | EM

Im] NUT TAIT TN ft Ted

PTT | POWELL AEN |

wh TE LIN A fal

TA AEN OLA ENN

wi [ x TT KT Te TT

SO EROS

mE ECCLES

PT] | PreENLLETE EE AK

ee

SEE NE

Mt | AN SET -ERY |

NCE

Pt Ey tt AL YT TT

P71] tt Et eT I I I | Fig. 78. E.m.f. and current curves in a mercury arc rectifier system. |

The direct voltage e,, when neglecting the effective resistance of the reactive coils, is then the mean value of the rectified voltage, ¢,, of curve III, hence is

c= Se ff sin 009 | |

ARC RECTIFICATION 269 _e (1 + cos 9) | . Van 7 or e =e, — v2 , — °1 + cos 8, If e, = the mercury arc voltage, r, = the effective resistance of reactive coils and 1, = the direct current, more correctly it is e=(e + €a + 15%) nVv2 - eee "1 + 0088, The effective alternating voltage between the rectifier anodes is the mean square of e,, curve II, hence is 1". é, = evEV! [sino a 1fé@ sin2 . i=l evi x12 4 —6, sin26 =eV2V ma Mo , WOM eva 2% + 4n _ eV _ 20) — sin 20, 25 and the drop of voltage in the reactive coils in the anode leads, caused by the overlap of the arcs, thus is e-e,- ef1- Vi _ 26, - sin 20s}. 22 26. Let 2’ = the maximum variation of direct current from mean value 2,, hence, 7, = 7, +7’ = the maximum value of rectified current, and therefore also the maximum value of anode current. The anode current thus has a maximum value 7,, and each half wave has a duration z + 0,, as shown by curve IV, Fig. 78. The direct current, 7,, 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 z + 0,, or somewhat more than one half wave. . Denoting, however, each anode wave by 2, that is, considering each anode impulse as one half wave (which corresponds to a lower frequency <5), then, referred to the anode impulse 0' . as half wave, the angle of overlap is ri The direct current, 7,, is the mean value of the anode current curves V, VI, and, assuming the latter as equivalent sine waves of maximum value 7, = 7, + 7’, the direct current, 2,, is . . dt ™ y=, x6, J sin & de’

• 2h x—, . _ 2% +4) % , a . .2z—-80 and 1, = wz ; or 5-7 7 7 2(7 + 4) and the pulsation of the direct current, 7’ = 7, — 1, is . a a _ 1 . . ‘ i) 2 (x + 8,) The effective value of the secondary current, as equivalent sine wave in one transformer coil, is the V/ mean square of curves VII, VIII, or, assuming this current as existing in both trans- former secondary coils in series — actually it alternates, one half