Theory and Calculation of Transient Electric Phenomena and Oscillations — part 9 of 20

I F = re = armature m.mf., . nn l hence, F =n, — 3 (13) = resultant m.m_f., and E = ap¥ = e.m.f. maximum generated thereby, E and p=—=%s (14) z, z, = armature current, maximum. Substituting (13) in (14) gives al = apni, — ee apn,t apn and I = PN, 0 = PNobo ; (15) n,apn, %,+% 2,474 2 or, by (9), 2n,. I =-— i, —*— n,n, r,+ 2,’ (16)

208 TRANSIENT PHENOMENA where 2 we = k, = transformation ratio of field turns to p 1 . resultant armature turns; hence, =ki—2—. I = ka, a +t, (17) Substituting (11) in (17) thus gives the maximum value of the armature current as _ 10g yy es ed I hides +h, z, , (18) the instantaneous value of the armature current as x (2 + 2. “2° " 1 = kyl Ae tae 0-W)—e *° # 1=kI, z, (@, + %) cos ( )-e cos >, (19) and by equation (10) of Chapter XIII, the armature reaction as NpyN z,(z + 27>") r : _ Mot, tt, + 2; _ ae . po {1 ‘ cos Of, (20) where zx, + z, = x, is the synchronous reactance of the alter- nator. For @ = o, or in permanent condition, equations (18), (19), (20) assume the usual form: = v Il=k, 7,’ . ZL, - t= k,l, 3 cos (6 — 0) (21) Npn x and f= = kl, z . . 117. As an example is shown, in Fig. 53, the instantaneous . value of the transient short-circuit current of a three-phase alternator, with the time angle 0 as abscissas, and for the con- stants: the field turns, n, = 100; the normal field current, I, = 200 amp.; the field impedance, Z, =r, — jz, = 1.28 — 1607 ohms; the armature turns, n, = 25, and the armature

| | | SHORT-CIRCUIT CURRENTS ‘OF ALTERNATORS 209 i impedance, Z, = 7, — jz, = 0.4 —5j7 ohms. For the phase angle, & = 0, the transformation ratio then is 2n, 8 =——=- = 267 he Nn, 3 2.67, and the equivalent impedance of armature reaction is nN 2 = > () Xo = 15, and we have T= 40001+36°°0°), (18) t = 400 (1 4 3 e~ 070086) (cos _ e—0088) (19) and f = 15,000 (1 + 367%") (1 — 2 ° cos 6). (20) 1600 . “AAA AAA AAAS = maivaiy vi UAL AEE AEA Pet Aas [| | Feige mem doe TL “AAAARAAAARA AAAS by SS anes | | pee EY mA ALAIN ALALA LAA aL ol EYE Ve TTS Fig. 58. Short-circuit current of a three-phase alternator. (B) Single-phase alternator. 118. In a single-phase alternator, or in a polyphase alternator with one phase only short-circuited, the armature reaction is pulsating. The m.m.f. of the armature current, t = I cos (6 — @), (22) of a single-phase alternator, is, with regard to the field, f, = n,I cos (8 — 6’) cos (8, — &); hence, for position angle 6, = time angle 0, or synchronous / Totation, f, = GI {1 + cos 2 (0 - 0); (23)

i qT 210 TRANSIENT PHENOMENA that is, of double frequency, with the average value, n $,=5 1, (24) pulsating between 0 and twice the average value. The average value (24) is the same as the value of the poly- phase machine, for n, = 1. Using the same denotations as in (A), we have: (1) €, =nd,, (25) (4) 0 eter 114008 2 (0-0) } = eT 11 + cos 207}. 22, 22, . (26) Denoting the effective reactance of armature reaction thus: | z= 7) 2 and substituting (27) in (26) we obtain | $= =n{1 + 008.2 (0 — #)} = =Fnql, {1+00820}; (28) 1 1 hence, by (6), | Nl y = Nie — 2g! {1 + cos 2 6} 1 and “9 att z; ., a, z, T,jl t+ z, + 7,02 F (29) and the field current, _% pe tte oe _ On z, I,j)1+ z+ 7,08” (0— 4) (30) 119. If J = maximum value of armature current, | . F, = 1 {1 + cos2 (0 - 0} (31) = armature m.m.f.;

SHORT-CIRCUIT CURRENTS OF ALTERNATORS 211 hence, F =n, —F, (82) = resultant m.m.f. Since, however, ® = ps, e=ad = aps, . l= ee ops , (33) qT, z, and, by (27), ap = 2) i} ny » _ we have, by (33) 5-3 1-3 z 1. @) | Substituting (30), (31), and (34) into (32) gives | _ Re 2 es or ar a _ | 2z,/ Mel z, V+ Fy a, 87 @ #) | — FAT {1 + 0082 0 - #)}; or, substituting, | | k, = 2 ne = transformation ratio, (35) 1 | and rearranging, gives . -@ %, @©$t+r7 ™ . I= kil, z,+ 2, zy (36), as the maximum value of the armature current. This is the same expression as found in (18) for the poly- phase machine, except that now the reactances have different | values. t

212 TRANSIENT PHENOMENA Herefrom it follows that the instantaneous value of the armature ; current is . =k slot te * V9 0) oF cos 0 { (37) pee Zz, (t, + 2) , and, by (31), the armature reactionis | g = kl (e+ 2 *')(, + cos 2 (0 - 0°) (38) 12 T° gx, (a, + 2) For 0 = ©, or permanent condition, equations (30), (36), (37), and (38) give

• Z, = 3. 6 — 0 | oe 1} + =e 082 ( if, I= kde 1 : 2 (39) | t= —— 0-6 7 hdler + z, cos ( )» | ms es - , and f= To Fp, {I + 0082 6 @)}. As seen, the field current 7, is pulsating even in permanent condition, the more so the higher the armature reaction z, compared with the armature self-inductive reactance z,.

120. Choosing the same example as in Fig. 52, paragraph 117, but assuming only one phase short-circuited, that is, a single- phase short circuit between two terminals, we have the effective armature series turns, n, = 25 V3 = 43.3; the armature impe- dance, Z, = r, — jz, = 0.8 — 107; # = 0; the transformation

| ratio, k, = 4.62, and the effective reactance of armature reaction

| 3

. 2 = 35 % =15; herefrom,

T = 555 (1 + 1.5 78%), (36)

@ = 555 (1 + 1.5 67°") (cos 6 — 2 9), ~~ (87)

a and S = 12,000 (1 + 1.5679") (1 + cos26); (38)

SHORT-CIRCUIT CURRENTS OF ALTERNATORS 213 . and the field current is i, = 200 (1 + 1.5 67°) (1 + 0.6 cos 2 4). (30) In this case, in the open-circuited phase of the machine, a high third harmonic voltage is generated by the double frequency pulsation of the field, and to some extent also appears in the short-circuit current.

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| , | | SECTION II | | PERIODIC TRANSIENT PHENOMENA | |

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PERIODIC TRANSIENT PHENOMENA CHAPTER I. INTRODUCTION.

1. Whenever in an electric circuit a sudden change of the circuit conditions is produced, a transient term appears in the circuit, that is, at the moment when the change begins, the circuit quantities, as current, voltage, magnetic flux, etc., cor- respond to the circuit conditions existing before the change, but do not, in general, correspond to the circuit conditions brought about by the change, and therefore must pass from the values corresponding to the previous condition to the values corre- sponding to the changed condition. This transient term may be a gradual approach to the final condition, or an approach by a series of oscillations of ‘gradual decreasing intensities. .

Gradually — after indefinite time theoretically, after relatively short time practically —the transient term disappears, and permanent conditions of current, of voltage, of magnetism, etc., are established. The numerical values of current, of voltage, etc., in the permanent state reached after the change of circuit con- ditions, in general, are different from the values of current, voltage, etc., existing in the permanent state before the change, since they correspond to a changed condition of the circuit. They may, however, be the same, or such as can be considered the same, if the change which gives rise to the transient term can be considered as not changing the permanent circuit con- ditions. For instance, if the connection of one part of a circuit, with regard to the other part of the circuit, is reversed, a transient term is produced by this reversal, but the final or permanent condition after the reversal is the same as before, except that the current, voltage, etc., in the part of the circuit which has been

reversed, are now in opposite direction. In this latter case, the same change can be produced again and again after equal 217

‘ 218 TRANSIENT PHENOMENA intervals of time ¢,, and thus the transient term made to recur periodically. The electric quantities 7, e, etc., of the circuit, from time t = 0 to t = t,, have the same values as from time t=t, tot = 2t,, fromt = 2t, tot = 31,, etc., and itis sufficient - to investigate one cycle, from ¢t = 0 tot = 1,.

In this case, the starting values of the electrical quantities

during each period are the end values of the preceding period,

‘ or, in other words, the terminal values at the moment of start of the transient term, t = 0, 1 = 7, and e = e,, are the same as the values at the end of the period t =¢,, 1 =v and e = e; that is, 7, = + 7’, e, = +e’, etc.; where, the plus sign applies for the unchanged, and the minus sign for the reversed part of the circuit.

2. With such periodically recurrent changes of circuit con- ditions, the period of recurrence t, may be so long, that the transient term produced by a change has died out, the permanent conditions reached, before the next change takes place. Or, at the moment where a change of circuit conditions starts, a transient term, the transient term due to the preceding change, has not yet disappeared, that is, the time, ¢,, of a period is shorter than the duration of the transient term. ,

In the first case, the terminal or starting values, that is, the

values at the moment when the change begins, are the same as the permanent values, and periodic recurrence has no effect on the character of the transient term, but the phenomenon is cal- culated as discussed in Section I, as single transient term, which gradually dies out.

If, however, at the moment of change, the transient term of

the preceding change has not yet vanished, then the starting or terminal values of the electric quantities, as 7, and e,, also contain a transient term, namely, that existing at the end of the preced- ing period. The same term then exists also at the end of the period, or at t = ¢,. Hence in this case, the terminal conditions are given, not as fixed numerical values, but as an equation between the electric quantities at time ¢ = 0 and at time ¢ = ¢,; or, at the beginning and at the end of the period, and the inte- gration constants, thus, are calculated from this equation.

8. In general, the permanent values of electric quantities after a change are not the same as before, and therefore at least two changes are required before the initial condition of the

INTRODUCTION 219 circuit is restored, and the cycle can be repeated. Periodically recurring transient phenomena, thus usually consist of two or more successive changes, at the end of which the original con- dition of the circuit is reproduced, and therefore the series of changes can be repeated. For instance, increasing the resistance of a circuit brings about a change. Decreasing this resistance again to its original value brings about a second change, which restores the condition existing before the first change, and thus completes the cycle. In this case, then, the starting values of the electric quantities during the first part of the period equal the end values during the second part of the period, and the starting values of the second part of the period equal the end values of the first part of the period. That is, if a resistor is ‘ inserted at time ¢ = 0, short circuited at time ¢ = ¢,, and inserted . again at time ¢ = ¢,, and e and 7 are voltage and current respec- tively during the first, e, and 7, during the second part of the period, we have Lefrao = [¢:/t2u3 /¢:/t-4 = [¢/tmuy and ‘ liftmo = [ti/rms [tr/ tm, = [tL tnt

If during the times ¢, and ¢, — ¢, the transient terms have already vanished, and permanent conditions established, so that the transient terms of each part of the period depend only upon the permanent values during the other part of the period, the length of time ¢, and ¢, has no effect on the transient term, that is, each change of circuit conditions takes place and is calculated independently of the other change, or the periodic recurrence. A number of such cases have been discussed in Section I, as for instance, the effect of cutting a resistor in and out of a divided inductive circuit, paragraph 75, Fig. 33. In this case, four successive changes are made before the cycle recurs: a resistor is cut in, in two steps, and cut out again in two steps, but at each change, sufficient time elapses to reach practically permanent condition. !

In general, and especially in those cases of periodic transient | phenomena, which are of engineering importance, successive | changes occur before the permanent condition is reached, or | even approximated after the preceding change, so that frequently |

° , 220 TRANSIENT PHENOMENA the values of the electric quantities are very different throughout the whole cycle from the permanent values which they would gradually assume; that is, the transient term preponderates in the values of current, voltage, etc., and the permanent term occasionally is very small compared with the transient term.

4. Periodic transient phenomena are of engineering impor- tance mainly in three cases: (1) in the control of electric circuits;

• (2) in the production of high frequency currents, and (3) in the rectification of alternating currents.

1. In controlling electric circuits, etc., by some operating mechanism, as a potential magnet increasing and decreasing the resistance of the circuit, or a clutch shifting brushes, etc., the main objections are due to the excess of the friction of rest over the friction while moving. This results in a lack of sensitiveness,

: and an overreaching of the controlling device. To overcome the friction of rest, the deviation of the circuit from normal must become greater than necessary to maintain the motion of the operating mechanism, and when once started, the mechanism overreaches. This objection is eliminated by never allowing

• the operating mechanism to come to rest, but arranging it in unstable equilibrium, as a “floating system,” so that the con- dition of the circuit is never normal, but continuously and periodically varies between the two extremes, and the resultant effect is the average of the transient terms, which rapidly and periodically succeed each other. By changing the relative duration of the successive transient terms, any resultant inter- mediary between the two extremes can thus be produced. On this principle, for instance, operated the controlling solenoid of the Thomson-Houston arc machine, and also numerous auto- matic potential regulators.

, 2. Production of high frequency oscillating currents by period- ically recurring condenser discharges has been discussed under ‘oscillating current generator,”’ in Section I, paragraph 44.

Non-sinusoidal high frequency alternating currents are pro- duced by an arc, when made unstable by shunting it with a condenser, as discussed before.

The Ruhmkorff coil or inductorium also represents an appli- cation of periodically recurring transient phenomena, as also does Prof. E. Thomson’s dynamostatic machine.

3. By reversing the connections between a source of alter-

INTRODUCTION 221 nating voltage and the receiver circuit, synchronously with the alternations of the voltage, the current in the receiver circuit is made unidirectional (though more or less pulsating) and there- fore rectified. In rectifying alternating voltages, either both half waves of voltage can be taken from the same source, as the same trans- former coil, and by synchronous reversal of connections sent in the same direction into the receiver circuit, or two sources of voltage, as the two secondary coils of a transformer, may .be used, and the one half wave taken from the one source, and sent into the receiver circuit, the other half wave taken from the other source, and sent into the receiver circuit in the same direction as the first half wave. The latter arrangement has the disadvantage of using the alternating current supply source less economically, but has the advantage that no reversal, but only an opening and closing of connections, is required, and is therefore the method commonly applied in stationary rectify- ing apparatus. 5. In rectifying alternating voltages, the change of connec- tions between the alternating supply and the unidirectional receiving circuit can be carried out as outlined below: (a) By a synchronously moving commutator or contact maker, in mechanical rectification. Such mechanical rectifiers may again be divided, by the character of the alternating supply voltage, into single phase, quarter phase and three phase, and by the character of the electric circuit, into constant potential and constant current rectifiers. Mechanical rectification by a commutator driven by a separate synchronous motor has not yet found any industrial application. Rectification by a commuta- tor driven by the generator of the alternating voltage has found very extended and important industrial use in the excitation of the field, or a part of the field (the series field) of alternators and synchronous motors, and especially in the constant-current arc machine. The Brush arc machine is a quarter-phase alternator connected to a rectifying commutator on the armature shaft, and the Thomson-Houston arc machine is a star-connected three-phase alternator connected to a rectifying commutator on the armature shaft. The reason for using rectification in these | machines, which are intended to produce constant direct current ” | at very high voltage, is that the ordinary commutator of the |

| 222 TRANSIENT PHENOMENA | continuous-current machine cannot safely commutate, even at | limited current, more than 30 to 50 volts per commutator segment, while the rectifying commutator of the constant- | current arc machine can control from 2000 to 3000 volts per | segment, and therefore rectification is superior to commutation for very high voltages at limited current, as explained by the character of this phenomenon, discussed in Chapter ITI. . (b) The synchronous change of circuit connection required _ by the rectification of alternating e.m.fs. can be brought about without any mechanical motion in so-called ‘‘arc rectifiers, ”’ by the characteristic properties of the electric arc, to be a good conductor in one, an insulator in the opposite direction. By thus inserting an are in the path of the alternating circuit, _ current can exist and thus a circuit be established for that half wave of alternating voltage, which sends the current in the same direction as the current in the arc, while for the reversed half wave of voltage the arc acts as open circuit. As seen, the are cannot reverse, but only open and close the circuit, and so can rectify only one half wave, that is, two separate sources of alternating voltage, or two rectifiers with the same source of voltage, are required to rectify both half waves of alternating voltage. (c) Some electrolytic cells, as those containing aluminum as one terminal, offer a low resistance to the passage of current in one direction, but a very high resistance, or practically interrupt the current, in opposite direction, due to the formation of a non- conducting film on the aluminum, when it is the positive terminal. Such electrolytic cells can therefore bé used for rectification in a similar manner as arcs. The three main classes of rectifiers thus are: (a) mechanical rectifiers; (b) arc rectifiers; (c) electrolytic rectifiers. Still other methods of rectification, as by the unidirectional character of vacuum discharges, of the conduction in some crystals, etc., are not yet of industrial importance.

| CHAPTER II. CIRCUIT CONTROL BY PERIODIC TRANSIENT PHENOMENA.

6. As an example of a system of periodic transient phenomena, used for the control of electric circuits, may be considered an automatic potential regulator operating in the field circuit of the exciter of an alternating current system.

Let, r, = 40 ohms = resistance and L = 400 henrys = inductance of the exciter field circuit.

A resistor, having a resistance, r, = 24 ohms, is inserted in series to r,, L in the exciter field, and a potential magnet, con- trolled by the alternating current system, is arranged so as to short circuit resistance, r,, if the alternating potential is below, to throw resistance 7, into circuit again, if the potential is above normal. .

With a single resistance step, r,, in the one position of the regulator, with r, short circuited, and only r, as exciter field winding resistance, the alternating potential would be above normal, that is, ‘the regulator cannot remain in this position, but as soon after short circuiting resistance r, as the potential has risen sufficiently, the regulator must change its position and cut resistance r, into the circuit, increasing the exciter field circuit resistance to r, +7,. This resistance now is too high, would lower the alternating potential too much, and the regula- tor thus cuts resistance r, out again. That is, the regulator continuously oscillates between the two positions, corresponding to the exciter field circuit resistances r, and (r, + 7,) respec- tively, at a period depending on the momentum of the moving mass, the force of the magnets, etc., that is, approximately ; constant. The time of contact in each of the two positions, however, varies: when requiring a high field excitation, the regulator remains a longer time in position r,, hence a shorter time in position (r, + 7,), before the rising potential throws it over into the next position; while at light load, requiring low field excitation, the duration of the period of high resistance,

223

224 TRANSIENT PHENOMENA (r, +1,), is greater, and that of the period of low resistance, r,, less.

7. Let, ¢, = the duration of the short circuit of resistance r,; t, = the time during which resistance r, is in circuit, and t, = t, tty During each period ¢,, the resistance of the exciter field, therefore, is r, for the time ¢,, and (r, + 7,) for the time ¢,.

: Furthermore, let, 7, = the current during time t,, and 7, = the current during time ¢,.

During each of the two periods, let the time be counted anew from zero, that is, the transient current 7, exists during the time 0 < ¢ < ¢,, through the resistance 7,, the transient current, 7,, during the time 0 < ¢ < ¢,, through the resistance (ro + 1,). ,

This gives the terminal conditions:

iy/rmo = /Vo/tmty and (1) /tq/ 10 = Wy /rmt3 _ that is, the starting point of the current, 7,, is the end value of the current, 7,, and inversely. —-

If now, e = voltage impressed upon the exciter field circuit,

the differential equations are: . di, - e=7i, + L a . and (2) . di, e= (ry tr )i, +L; or, di, rT, _ e OL a, Ue r. 0 . 3 4% thy ° . e L . t, -———_ rot,

CIRCUIT CONTROL 225 Integrated, ._@ -=! 4.= 7, + Ce and (4) . e ~ met 4 = 7 +7, + CoE . Substituting the terminal conditions (1) in equations (4), gives for the integration constants c, and c, the equations, & + Cc. = _e + c ~ art ° . Ty + %]% +, # and : ° (4 e@ % b — =— L*. \ ur) +T, + “s T,) + af , herefrom, . er, {1 —¢e &L ut a= ne Ce ee ry (ty +1,) {1 —« L L _% : er, {1 —& a = + (tn) rote +7) {L —e L L Substituting (5) in (4), roth i -2)1 nfs} -T! tt ~ Ba Bal € . (r, tr) tl—e # and (6) _” . e r, fh —e€ nf - mtn, = + ey Ff . mtr, rll —e bet If, e = 250 volts; t, = 0.2 sec., or 5 complete cycles per sec.; t, = 0.15, and t, = 0.05 sec.; then 4, = 6.25 {1 — 0.128 e~°t! } , and (7) @, = 3.91 {1 + 0.391 °'*"}. | |

226. TRANSIENT PHENOMENA
227. The mean value of current in the circuit is 1 f a le , te) Pas fia ; 8) t, +t, lJ, * . ( This integrated gives, e e p- TH, @) t, +t, ’ and, if - +p @ . y T . and (10) if = —2— 7 te th, are the two extreme values of permanent current, corresponding respectively to the resistances r, and (r, + 7,), we have ~_ bt! +t! t,t. 1 , ga es ag pag (11) ty + t, to , t, >? that is, the current, 7, varies between 7,’ and 7,’ as linear function of the durations of contact, ¢, and ¢,. . . The maximum variation of current during the periodic change is given by the ratio of maximum current and minimum current; or, . ‘| =4q (12) On tmo , and is _ 1% (1 — e7"7*) +7, (L — €7*) . q= r, (1 _ e7*1— 4) +r,e7# (1 —e7 ) ) (13) where, r s, = L ty . and (14) rotr 5, = Ae t,. | |

CIRCUIT CONTROL 227 Substituting 7? Y l-e a ar or (15) by using only term of first order; l-—¢«7=2, gives } (16) q=1; that is, the primary terms eliminate, and the difference between i, and 7, is due to terms of secondary order only, hence very small Substituting -f ind l-e =2-5; (17) that is, using also terms of second order, gives q= {T. (3, +8,) + 7,8,} -3 {T (s, +8) + 787} . | {r, (8, +8,) +7,8,} — 4 {r, (8, +8)? +7,87 + 27,8,8,}” (18) or, approximately, r,8,8 =1 +——142__ 19 q 7 (s, + 8,) + 7,8, , and, substituting (14), rte = 1 ——thb_ . 20 re 1 2) that is, the percentage variation of current is rt,t -y,=—%_. (21) q LG +t) Equation (21) is a maximum for t Lata 3 , (22) and, then, is rit q-1= IL ; (23)

228 TRANSIENT PHENOMENA or, in the above example, (r, = 24; L = 400; ¢t, = 0.2); q — 1 = 0.003; that is, 0.3 per cent. The time ¢, of a cycle, which gives 1 per cent variation of current, g — 1 = 0.01, is 4L . t, =— (q- 1), (24) r = $sec. The pulsation of current, 0.3 per cent respectively 1 per cent, thus is very small compared with the pulsation of the resistance, . r, = 24 ohms, which is 46 per cent of the average resistance . r, +3 = 52 ohms. | | | .

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 7 = instantaneous value of alternating current, and 1, = instantaneous value of rectified current, then we have, before reversal, 2, = 7, and after reversal, 1, = — 7; 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 r,, respectively. The terminals are reversed. The shunt-resistance circuits are opened, leaving the circuits in series in opposite direction.

Special cases hereof are:

1. Ifr =r, = 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 pasees zero, due to self-inductance of the rectified circuit its current differs from zero, and an arc still appears at the rectifier.

229

280 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 =r, = ©, 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. oo

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.

MECHANICAL RECTIFICATION 231

1. Single-phase constant-current rectification.

2. A sine wave of current, 7, 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 — jz. This circuit is permanently shunted NY f, by a circuit of resistance r,. By i

Rectification takes place over short- s >" circuit from the moment z — 6, to = zx + 0,; that is, at x — 0, the rectified and the alternating circuit are closed upon themselves at the rectifier, and () = }() this short-circuit opened, after rever- sal, at z + 0,, 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 x +7, and z — t, and the time angles x +46, and z — @, are identical. ‘

This represents the conditions ex- 8 54. Single-phase curxent «4. : rectifier commutator. isting in compound-wound §alter- nators, that is, alternators feeding a series field winding through a rectifier.

Let, during the period from 6, to z — 6,, 1 = current in impedance Z, and 7, = current in resistance r,, then:

7 +1, = 1, sin 0. (1)

However,

. dv . aur, =i + 2S (2) and substituting (1) in (2) gives the differential equation: . i@ tn) +25 — igsing =0, (3) which is integrated by the function: t= Ae” + Bain (6 — 2). (4)

Substituting (4) in (8) and arranging, gives:

A(r+r, —az)e-%+(B([r +1r,]cosd + xsind) — i,7,)sin 0 —[(r +7r,) sind — xcosd]Bcos6 =0, (5)

232 TRANSIENT PHENOMENA which equation must be an identity, thus: r+r,—az=0, B ({r + 7,]cosd + xsin dé) — ir, = of and (r + r,) sind — zcosd = 0, ‘ and herefrom: r+r, a= —)> x . x 6 =>S>lUl tan r+r, | (6) r . : and B =i, ——_————-. = 4M , "Virtry+e 2 where z2=V(r+ry +27; (7) hence: ttt, 7 i=Ae 7 + 4, Fain (0 — 8). (8) During the time of short-circuit, from z — 6, to x + 0,, if v = current in impedance Z, we have a’ ur + 2 = 0, (9) hence: -re6 . v=A’e * + (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=? = 17, sin 6 at the moment 0 = z — 0, and, at the moment 6 = z + @,, 7’ must have reached the same value as 7 and 7, sin 6 at the moment 6 = 6,.

MECHANICAL RECTIFICATION 283

This gives the two double equations:

te 0, = v —6, = 1, sin (x - 6,) and (11) do = Ve 40, = 2,91 9;

or, substituting (8) and (9), .

— TE - -lq@- 4) Ae * + ig7tsin (040, =A’e* — =i,sin8, (12) and :

~ rit, - lo 6,) Ae * " ~i,ttsin@-0,) =A’e r= i,sind, (13)

These four equations (12) (13) determine four of the five

| quantities, A, A’, 0,, 9,,7,, 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. Exampue: Let an alternating current of effective value of 100 amp., that is, of maximum value 7, = 141.4, be rectified for the supply of a circuit of impedance Z = 0.2 — 27, shunted by a non-inductive circuit of resistance r,.

Let the series connection of the rectified and alternating

circuits be established 30 time-degrees after the zero value of alternating current, that is, 0, = 30 deg. = x chosen. . Then, from equation (13), we have -i(8 4) , Ae?” = isin 8, hence, substituting r, 2, 0,, t,, gives A’ = 102. From equation (12), A’e =” — 4 gin 8,, and, substituting, sin 0, = 0.527 e1; |

234 TRANSIENT PHENOMENA approximately sin 0, = 0.527 and 4, = 32°; thus sin 0, = 0.527 2”) = 0.558, and 0, = 34°; thus sin 6, = 0.5272 = 0.559, and 06, = 34°. From equations (12) and (13) it follows: ren, . . Ae = OO 4 i, sin (o + 6,) = i, sin 8,, -ftt, nr, Lo Ae * — t 7 sin (6 — 0,) = 2, sin 4,; eliminating A gives gE nD Fee _ zsin@, — r,sin (6 + 6,) . zsiné, +r,sin(@—96,)’ © substituting sind = cos 6 = os 2? =(r+r,)? +2, and substituting for r, x, 0,, 6,, gives after some changes: jnserge _ 1S — L047 ll-r, ’ calculating by approximation, assuming r, = 0.5, 0.603 = 0.612; assuming r, = 0.51, 0.597 = 0.602; | assuming r, = 0.52, | | 0.591 = 0.592; hence, r, = 0.52, and z = 2.124, é = 70°.

MECHANICAL RECTIFICATION 235 Substituting these values in (12) or (13) gives A = 113; hence, as final equations, we have , 4 = 11267°"" + 34.6 sin (6 — 70°), 7 = 102 e701 - i = 141.4 sind, and 1, = 1, — 1; which gives the following results: ae! Bite. | etic Quantity. Tnstantaneous Values. we ue ean a P ° = 30 alo sce ; 90 110 {130 46 | 170 | 190] 210 |......]..... is 79.0| 75.8] 73.2 io. af 75.2) 75.2 +, 8in 0=|70.8/108 {133 |141. 4/133 |108 |79.0| 24.7|-24.7;-70.8 (100.0 |..... = | 0 [37.5]60.4) 66.5|54.0|27.6| 0 |(—51.1—48.5)' 0 | 38.2 | 27.3 ‘ Il : ‘ Curves of these quantities are plotted in Fig. 55, for % = 100 sin 0. ; . 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. 100 et rijeih ON] | | TT TT tT em a NO BLE EACEEEE EEN EEE bd haere ly eH Sa As DY 40 A tT et oh tn A] én 171 1 TA MD *s que aan SS ahah aaer HH oe i\ 4 “~LLY CHEE 1 A Hn |_| ry | nmr i a a aaeke nan ’ | ae
ae! a Sk I CO | PAA eRe 0 20 40 60 80 100 «120 «©6100 «(160 (180 Degrees —» Fig. 55. Single-phase current rectification. 2. Single-phase constant-potential rectification.

12. Let the alternating e.m.f. e, sin @ of the alternating cir- cuit of impedance Z, = r, — jx, be rectified by connecting it at the moment 6, 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 = — 0,, and closing during the time from = — 0, to z + 6, the alternating circuit by the resistance r,, the direct-current circuit by the resistance r,, then connecting the circuits again in series in opposite direction,

at x + 0,, etc., as shown diagrammatically by Fig. 56, where . _ 1 | my i 7 tom

r + 7’ yl" 47" ? , Tr —_ 1

71 1

7m + a mn

r +r r’ +7

1. Then, during the time from 0, to z — 8,, if 1, = current, the differential equation is

. . a , égsind —e — 1, (7 +1) — (2 +2) 5 = 0, (1)

MECHANICAL RECTIFICATION 237 which is integrated by 1, =A, + Be-™? + C,sin @ — 4,) (2) r VE - rm Fig. 56. Single-phase constant-potential rectifying commutator. Equation (2) substituted in (1) gives e,sind —e — (r + 17,) (A, + Be? + C, sin @ — 4,)] — (x +2,)[— a,Bye~%* + C, cos (0 — 6,)] = 0; or, transposing, —([tet(r+r,) A.) + Be~® [a, (x +2,) —-(r +7,)]

• sin 6[e, — (r + 17,) C, cos d, — (x + x,) C, sin 4,]
• C, cos O[(r + r,) sind, — (x + z,) cos d,] = 0; . herefrom it follows that e+(r+r,) A, =0, a, (x +2,) — (r +7.) = 9, & — (r +7,) C,cosd, — (x + z,) C, sind, = 0, and . (r +7,) sind, — (x + z,) cos d, = 0;

238 TRANSIENT PHENOMENA hence A, = -—— rt+r, r+r, a, = ——) r+, (3) tan 0, = t+ To, r+r, and Cl,= 5 oe __ “y Vir +r)? + (x + 2,)? and, substituting in (2), r+ : a e + Be F™ 4 e,sin (6 — 3,) r+r, Vir +7)? + (© +2,)? rt, ing — — 24 Bae Fre! , llr trodsind = (e+,)c084], + (4) T+, (r +7r,)? + (x + 2,) tan by = r+%, . rt+r,

2. During the time from z — 0, to z + 6,, if 1, = current in the direct circuit, 7, = current in alternating circuit, we have Alternating-current circuit:

. . di e,sin@ — 4, (r, +1r,) - a! = 0, (5) which is integrated the same as in (1), by mt » 7 (_ iyo Bg wy Soin = 8) Vig tr + ae

• . _ Be mite 4% ((r, +17,) sin @ - Zy cos 6), (6) (r, + r,) + Zo : tan é, = wo, T, + r, | . ~

MECHANICAL RECTIFICATION 239 Direct-current circuit: . . - a —e—%, (rt 7,)— 27% = 0, (7) integrated by e ttn, ee z 1, +r, + By (8) At @ = x — 6,, however, we must have 1 = t, = ty and t,at 0 = x +0, must be equal to o ) 7, at 0 = 0,, and opposite to 1, at 0 = x + 4,; 4,(6 =0,) =7,(0 =x +0)=-1,[(0=2+4+ 4).

These terminal conditions represent four equations, which suffice for the determination of the three remaining integration constants, B,, B,, B,, and one further constant, as 0, or @,, or r, or7r,, or e; that is, with the circuit conditions Z,, Z, 7,, 73, &,) é chosen, the moment 0, depends on @, and inversely.

13. Special case:

Z,=0, 7,=0, e=0; (10) that is, the alternating e.m.f. e, sin 6 is connected to the circuit of impedance Z = r — jx during time 0, to z — 0,, and closed by resistance r,, while the rectified circuit is short-circuited, during time z — 0, to z + 6.

The equations are:

1. Time 6, to x — @,: . . -te e, . 1,= Be + alsin @ — xc0s 6).
2. Time z — 6, toz +90,: 3 x 1 cf ’ (11) 1,= By * i,- e, sin 6 ‘"

240 TRANSIENT PHENOMENA , The terminal conditions now assume the following forms: At G=xnr-4,, 1,=1%1,=1, ae) & . -2@—&) Bye * + sp sin 0, + 2008 6,) = By = ey . = 7 sin 6,; at 6 = x + 6, and 0, respectively (12) re e . — 2 (6+H) Be * tape tain 4, —2zcos0,) = By * ’ = ®osin 6, r,

These four equations suffice for the determination of the two integration constants B, and B,, and two of the three rectifica- tion constants, @,, 0,, 7,, so that one of the latter may be chosen.

Choosing 6,, the moment of beginning reversal, the equations . (12) transposed and expanded give

-Z@+6) sin 0, e* =~ sin 6, m rd rm cot 6, + €* cot 0, = (t= - *) (« -1), Ua 2 0 (13) B, - e, sin Dy +50

d "

an r e . (9-6) B, = B,- z+e (r sin 0, + x cos 0,)e* ~_ which give 0,,7,, B,, B,: 9, is calculated by approximation. Assuming, as an example, ,

€, = 156 sin 6 (corresponding to 110 volts effective), and x |

4, = 6 = 30°,

MECHANICAL RECTIFICATION 241 by equations (13) we have: log sin 0, = — 0.3765 — 0.1448 @,, and 6, = 21.7°, r, = 7.63, B, = 24, (15) and B, = 12.8; thus 6 1, = 12.8¢ 3+ 1.56 (sin @ — 3 cos 8), e . . 1,=%W4de 3, (16) and i, = 20.5sin 8, which gives:

ev. 4. &. 4. | ev. 4. «. i.

21.7 7.55 |........4......|) 135 1

30 7.47 [o..ceee fee |] 150 10.20 | 10.2 10.2

45 i a eres ees | | Sn 5.3

60 8.02 |.........J......]) 180 foo. | 86 0