CHAPTER XIV CONSTANT-POTENTIAL CONSTANT-CURRENT TRANS- FORMATION 127. The generation of alternating-current electric power prac- tically always takes place at constant voltage. For some pur- poses, however, as for operating series arc circuits, and to a lim- ited extent also for electric furnaces, a constant, or approximately constant alternating current is required. While constant alter- nating-current arcs have largely come out of use and their place taken by constant direct-current luminous arc circuits, or incan- descent lamps, the constant direct current is usually derived by rectification of constant alternating-current supply circuits. Such constant alternating currents are usually produced from constant- voltage supply circuits by means of constant or variable inductive reactances, and may be produced by the combination of inductive and condensive reactances; and the investigation of different methods of producing constant alternating current from constant alternating voltage, or inversely, constitutes a good application of the terms "impedance," admittance," etc., and offers a large number of problems or examples for the symbolic method of dealing with alternating-current phenomena. Even outside of arc lighting, such combinations of inductance and capacity which t«nd toward constant-voltage constant-cur- rent transformation are of considerable importance as a poffsiblo source of danger to the system. In a constant-current circuit, the load is taken off by short-circuiting, while opc;n-circuiting causes the voltage to rise to the maximum value pcjnnitted by the power of the generating source. Hence, whrjrrj the circuit constants, with a constant-voltage supply source, are Huch as U) approach constant-voltage constant-current tran.sfonnation, as in for instance the case in very long transmission line«, or>^;n-<:ircuit- ing may lead to dangeroiLs or even destructive voltage rh¥% 128. With an inductive reactance inserted in series to an alt^^r- 245 246 ELECTRIC CIRCUITS nating-current non-inductive circuit, at constant-supply voltage, the current in this circuit is approximately constant, as long as the resistance of the circuit is small compared with the series inductive reactance. Let ^0 = Co = constant impressed alternating voltage; r = resistance of non-inductive receiver circuit; Xo = inductive reactance inserted in series with this circuit. The impedance of this circuit then is Z = r + jxof and, absolute, and thus the current, / = ^* = -^ (1) ^ r + jxo and the absolute value is eo Co the phase angle of the supply circuit is given by (2) and the power factor. tan ^0 = - (3) T cos ^0 = -• (4) z ^ ^ If in this case, r is small compared with Xq, it is ,-^£o _-l (5) Xo ' ^* xM¥" or, expanded by the binomial theorem. V • • • \xj hence, : (6) 6o I = — Xo 2xo2^8xo* -r . . . that is, for small values of r, the current, z, is approximately constant, and is 6o I = — Xo CONSTANT-CURRENT TRANSFORMATION 247 For small values of r, the power-factor cosfl — - is very low, however. Allowing a variation of current of 10 per cent, from short- circuit or no-load, r = 0, to full-load, or r = ri, it is, substituted in (2): No-load current: / -: / ^ y / ^ i < \ s 1 / / \ s ^ c JV N -^ \ s / \ \ ^ y S 1 / ... .„. \ Full-load current: Vn' + a Vr^~+x? and therefore, ■ ri = 0.485 x„, and the power-factor, from (4), is 0.437. That is, even allowing as large a variation of current, i, as 10 per cent., the maximum power-factor only reaches 43.7 per cent., when producing constant-current regulation by series inductance reactance. 248 ELECTRIC CIRCUITS As iUustrations are shown, in Fig. 113, for the constants: Co = 6600 volts applied e.m.f. ; zo = 792 ohms series reactance; the current: 6600 t = \/f« + 792« 8.33 amp.; and the power-factor: cos d = yA^T792^ 792 yl'+imY' with the voltage at the secondary terminals: e = ri as abscissas. 129. If the receiver circuit is inductive, that is, contains, in addition to the resistance, r, an inductive reactance, x, and if this reactance is proportional to the resistance, X = kr, as is commonly the case in arc circuits, due to the inductive reactance of the regulating mechanism of the arc lamp (the effective resistance, r, and the inductive reactance, a:, in this case are both proportional to the number of lamps, hence pro- portional to each other), it is: total impedance: Z = r +j (xo + x) = r +j (xo + kr) ; or the absolute value is z = Vr^ + {xo + xy = Vr2 + {xo + kr)^; thus, the current r + j{xo + krY i *. _J_ A(^. _L hW (•) and the absolute value is i= , '" = = '-^ -, ^ ; (8) 4' Xq Xo2 CONSTANT-CURRENT TRANSFORMATION 249 and the power-factor: cos ^0 = - = J ' (9) 2 Vr^ + (xo + kry By the binomial theorem, it is i _ 1 « ^ _ r^ (2 - fc^) _ V 2 fer r^(l + k^) ^o 4 Xo Xo Xo' Hence, the current .2 +-...) (10) Xo fcr _ r2(2 - A;)' Xo 4 Xo^ that is, the expression of the current, i (10), contains the ratio? T — , in the first power, with k as coefficient, and if therefore A; Xo is not very small, that is, the inductive reactance, x = At, a very small fraction of the resistance, r, the current, i, is not even approximately constant, but begins to fall oflE immediately, even at small values of r. Assuming, for instance, k = 0.4. That is, the inductive reactance, x, of the receiver circuit equals 40 per cent, of its resistance, r, and the power-factor of the receiver circuit accordingly is r cos 6 = r2 + x^ 1 1 + fc2 = 93 per cent.; it is, substituted in (8), eo I = As illustrations are shown, in the same Fig. 113, for the constants: Co = 6600 volts supply e.m.f . ; Xo = 792 ohms series reactance; the current: 8.33 2 = — -. — amp. yliW+V+^-'m) This current is shown by dotted line. In this case, in an inductive circuit, the current, i, has decreased 250 ELECTRIC CIRCUITS by 10 per cent, below the no-load or short-circuit value of 8.33 amp. that is, has fallen to 7.5 amp., at the resistance r = 187 ohms, or at the voltage of the receiving circuit, e = i V r« + x2 = n V 1 + fc« = 1.077 H = 1500 volts; while, in the case of a non-inductive load, the current has fallen off to 7.5 amp., or by 10 per cent, at the resistance r = 395 ohms, or at the voltage of the receiving circuit : e = 2950 volts. 130. As seen, a moderate constant-current regulation can be produced in a non-inductive circuit, by a constant series inductive reactance, at a considerable sacrifice, however, of the power-factor, while in an inductive receiver circuit, the con- stant-current regulation is not even approximate. To produce constant alternating current, from a constant- potential supply, by a series inductive reactance, over a wide range of load and without too great a sacrifice of power-factor, therefore re- quires a variation of the series inductive reactance with the load. That is, with increasing load, or increasing resistance of the receiver circuit, the series induc- tive reactance has to be decreased, so as to maintain the total impedance of the p, . . ^ circuit, and thereby the ciurent, constant. In constant-current apparatus, as trans- formers from constant potential to constant ciurent, or regula- tors, this variation of series inductive reactance with the load is usually accomplished automatically by the mechanical motion caused by the mechanical force exerted by the magnetic field of the current, upon the conductor in which the ciurent exists. For instance, in the constant-current transformer, as shown diagrammatically in Fig. 114, the secondary coils, S, are arranged so that they can move away from the primary coils, P, or in- versely. Primary and secondary currents are proportional to each other, as in any transformer, and the magnetic field between primary and secondary coils, or the magnetic stray field, in which the secondary coils float, is proportional to either current. The magnetic repulsion between primary coils and secondary coils is proportional to the current (or rather its ampere-turns), and to the magnetic stray field, hence is proportional to the square of the current, but independent of the voltage. The secondary CONSTANT-CURRENT TRANSFORMATION 251 coils, Sf are counter-balanced by a weight, W, which is adjusted so that this weight, TF, plus the repulsive thrust between second- •ary coils, S, and primary coils, P (which, as seen above, is propor- tional to the square of the current), just balances the weight of the secondary coils. Any increase of secondary ciurrent, as, for instance, caused by short-circuiting a part of the secondary load, then increases the repulsion between primary and secondary coils, and the secondary coils move away from the primary; hence more of the magnetic flux produced by the primary coils passes between primary and secondary, as stray field, or self-inductive flux, less passes through the secondary coils, and therefore the second- ary generated voltage decreases with the separation of the coils, and also thereby the secondary current, until it has resumed the same value, and the secondary coil is again at rest, its weight balancing counterweight plus repulsion. Inversely, an increase of load, that is, of secondary impedance, decreases the secondary current, so causes the secondary coils to move nearer the primary, and to receive more of the primary flux; that is, generate higher voltage. In this manner, by the mechanical repulsion caused by the cur- rent, the magnetic stray flux, or, in other words, the series induct- ive reactance of the constant-cmrent transformer, varies auto- matically between a maximum, with the primary and secondary coils at their maximum distance apart, and a minimum with the coils touching each other. Obviously, this automatic action is independent of frequency, impressed voltage, and character of load. If the two coils P and S in Fig. 114 are wound with the same number of turns and connected in series with each other and with the circuit, Fig. 114 is a constant-current regulator, or a regulating reactance, that is, a reactance which varies with the load so as to maintain constant current. If P is primary and S secondary circuit. Fig. 114 is a constant-cmrrent transformer. Assuming then, in the constant-current transformer or regula- tdr-or other apparatus, a device to vary the series inductive reactance so as to maintain the current constant. Let ^0 = 6o = constant = impressed e.m.f., Z = r + jx, = r (1 + jk) the impedance of the load, and let Xo = inductive series reactance, as the self-inductive internal reactance of the constant-current transformer. 252 ELECTRIC CIRCUITS The current in the circuit then is r+j{xo + x) or, the absolute value, 60 I = and, to maintain the current, i, constant (i = to), then requires 60 ^o = or, transposed. or, for Vr^ + {xo + xy' -M'-"- X (11) X = fcr, • xo = y[(^y^^ - *^^ (12) that is, to produce perfectly constant current by means of a variable series inductive reactance, this series reactance must be varied with the load on the circuit, according to equation (11) or (12). For non-inductive load, or a; = 0, it is Xo the maximum load, which can be carried, is given by a^o = and is z = Vr^ + x^ = r Vl + /c2 = ^ (14) to As seen from equation (13), the decrease of inductive reactance, Xo, required to maintain constant current with non-inductive load, is small for small values of resistance, r, when the r^ imder the root is negligible. With inductive load, equation (11), the inductive reactance, rco, has still further to be decreased by the inductive reactance of the load, x. Substituting: __ eo Xoo — '^~~ as the value of the series inductive reactance at no-load or short- circuit, equations (11), (12), (13) assume the form: CONSTANT-CURRENT TRANSFORMATION 253 General inductive load: Xo = Vxoo^ - r2 - z, (14) Inductive load of — = fc: r Xo = Vxoo* - r2 - At (15) Non-inductive load : Xo = Vxoo^ - r2. (16) 131. As seen, a constant series inductive reactance gives an approximately constant-current regulation with non-inductive load, but if the load is inductive this regulation is spoiled. Inversely it can be shown, that condensive reactance, that is, a source of leading current in the load, improves the constant- current regulation. With a non-inductive load, series condensive reactance exerts the same efifect on the current regulation as series inductive re- actance; the equations discussed in the preceding paragraphs re- main the same, except that the sign of x© is reversed and the cur- rent always leading. With series condensive reactance, condensive reactance in the load spoils, inductive reactance in the load improves the constant- current regulation. That is, in general, a constant series reactance gives approxi- mately constant-current regulation in a non-inductive circuit, and with a reactive load this regulation is impaired if the react- ance of the load is of the same sign as the series reactance, and the regulation is improved if the reactance of the load is of opposite sign as the series reactance. Since a constant-current load is usually somewhat inductive, it follows that a constant series condensive reactance gives a better constant-current regulation, in the average case of a some- what inductive arc circuit, than the constant series inductive reactance. Let -Fo = Co = constant = impressed, or supply voltage. Z = r + jx = impedance of the load, or the receiver circuit, and X = fcr, thatisy Z = r(l+jk) 254 ELECTRIC CIRCUITS or, absolute, z = rVl + kK Let now a constant condensive reactance be inserted in series with this circuit, of the reactance, — Xe, then the total impedance of the circuit is Z' = r - i (Xc - kr). (17) The current is / = „ .v,^" ,._x ' (18) r - j(Xe - kr)' 6o Vr^ + (Xc - kry Xe — kr r r (19) or, the absolute value is t = the phase angle is tan ^0 = - ^^^^-—^ (20) and the power-factor is cos ^0 = / (21) -Vr^ + (Xe — kr)^ for A; = 0, or non-inductive load, equations (19) and (21) assume the form: t = — , and cos 6 = r2 + rcc^ V r^ + Xc^ that is, the same as with series inductive reactance. From equation (19) it follows, that with increasing current, t, from no-load: r = 0, hence: 2o = — (22) Xc the current, Zo, first increases, reaches a maximum, and then decreases again. When decreasing, it once more reaches the value, io, for the resistance, ri, of the load, which is given by . eo __ Co , y/ri^ + {Xc - kri)^ "" Xc' hence, expanded, and the maximum value through which i passes between r = and r = ri, is given by - =0 dr "' CONSTANT-CURRENT TRANSFORMATION 256 or hence, ^{r« + {xc - fcr)*} = = 2r - 2k{xc - kr); KXc Ti fckA\ This maximum value is given by substituting (24) in (19), as for = io Vl +k^ (25) k = 0.4, this value is P = 1.077 to, that is, the current rises from no-load to a maximum 7.7 per cent, above the no-load value, and then decreases again. As an example, let 6o = 6600 volts impressed e.m.f. and Xe = 880 ohm condensive reactance, Xe being chosen so as to give for then. 2*0 = — = 7.5 amp.; Xc k = 0.4, 6600 I = cos ^0 = VV2T(880-0.4r)*' r Vr^ + {SSO - OAry' e = zi = 1.077 ri. These values of current and power-factor are plotted, with the receiver voltage as abscissse, in Fig. 115. 132. The conclusions from the preceding are that a constant series reactance, whether condensive or inductive, when inserted in a constant-potential circuit, tends toward a constant-current r^ulation, at least within a certain range of load. That is, at varying resistance, r, and therefore varying load, the current is approximately constant at light load, and drops off only gradu- ally with increasing load. 256 ELECTRIC CIRCUITS This constant-current regulation, and the power-factor of the circuit, are best if the reactance of the receiver circuit is of oppo- site sign to the series reactance, and poorest if of the same sign. That is, series condensive reactance in an inductive circuit, aBd series inductive reactance in a circuit carrying leading current, * y t / 7^ ■— ' — XT' ^ / -\ s s / S 1 <.»; y \ ^ / y / / ""» -" / Fig. 115. give the best regulation; series inductive reactance with an in- ductive, and series condensive reactance with leading current in the circuit, give the poorest regulation. Since the receiver circuit is usually inductive, to get best regula- tion, either a series condensive reactance has to be used, as in Fig. 115, or, if a aeries inductive reactance is used, the current in the receiver cir- cuit is made leading, as, for instajice, . by shunting the receiver circuit by a condensive reactance. Assuming, then, as sketched diagram- matically in Fig. 116, in a circuit of constant impressed e.m.f., fio = co = constant, a constant in- ductive reactance, Xo, inserted in series; and the receiver circuit, of impedance, Z = r + jx = r{l + jk) where k = tangent of the angle of lag = -; H Fig 116. CONSTANT'CURRENT TRANSFORMATION 257 let the receiver circuit be shunted by a constant condensive react- ance, Xe'f let then: ^ = potential difference of receiver circuit or the condenser terminals, / = current in the receiver circuit, or the "secondary current,'' /i = current in the condenser, /o = total supply current, or "primary current.'' Then /o = / + /i (26) and the e.m.f. at receiver circuit is ^ = Z/ (27) at the condenser, ^ = - jxch (28) hence, /i=if/ (29) and, in the main circuit, the impressed e.m.f. is ^0 = eo = ^ + jV« (30) Hence, substituting (26), (27) and (29) in (30), eo-^Zj +jxM+j^l) \ Xe I eo= (z^^+ixo)/ (31) I = -^ZTZ (32) Xc If Xc = Xo, that is, if the shunted condensive reactance equals the series inductive reactance, equations (32) assume the form, ^ = +^=-J? (33) ja:o xq or and and the absolute value is i = -^^ (34) that is, the current, i, is constant, independent of the load and the power-factor. 17 268 ELECTRIC CIRCUITS That is, if in a constant-potential circuit, of impressed e.m.f ., 60, an inductive reactance, Xo, and a condensive reactance, Xe, are connected in series with each other, and if Xe = Xo, (35) that is, the two reactances are in resonance condition with each other, any circuit shunting the capacity reactance is a constant- current circuit, and regardless of the impedance of this circuit, Z = r + jXf the current in the circuit is t = — . Xo 133. Such a combination of two equal reactances of opposite sign can be considered as a transforming device from constant potential to constant current. Substituting, therefore, (35) in the preceding equation gives: (33) substituted in (29) : Current in shunted capacity (36) or, absolute, T Z 2:60 (37) and, substituting (33) and (36) in (26) : primary supply current is /o = ^^P eo (38) Xo* or the absolute value is eo io = i^ Vr* + (xo - xy (39) and the power-factor of the supply current is Xo — X T tan ^0 = — , cos ^0 = — p========== (40) r y/r^ + (xo -xY In this case, the higher the inductive rf ftf^^nnnA^ ^^ n(f \\^ receiving circuit the lower is the supplv current, toy at the same resistance, r, and the higher is the power-factor, and if x = x© /"o = — 2 ^^^ ^^® ^ = 1 (41) Xo that is, the primary, or supply circuit is non-inductive, and the primary current is in phase with the supply e.m.f., and the CONSTANT-CURRENT TRANSFORMATION 259 power-factor is unity, while the secondary or receiver current (33) is 90° in phase behind the primary impressed e.m.f., eo. Inserting, therefore, an inductive reactance, Xi = Xo — x, in series in the receiver circuit of impedance, Z ^ r + jx, raises the power-factor of the supply current, io, to imity, and makes this current, io, a minimum. Or, if the inductive reactance, «o, is inserted in the receiver circuit, thus giving a total imped- ance, Z + jxo = r + j {x + Xo) by equation (38), substituting Z + jxo instead of Z, gives the primary supply current as /o = or the absolute value as to = Zeo Xo^ zeo X{? (42) (43) O < Fig. 117. and the tangent of the primary phase angle X tan ^0 = - = tan 6, r that is, the primary power-factor equals that of the secondary. Hence, as shown diagrammatic- 1 ally in Fig. 117, a combination of two equal inductive reactances in series with each other and with the receiver circuit, and shunted midway between the inductive re- actances by a condensive reactance equal to the inductive reactance, transforms constant potential into constant current, and inversely, without any change of power-factor, that is, the primary supply current has the same power-factor as the secondary current. With an inductive secondary circuit, the primary power- factor can in this case be made imity, by reducing the inductive reactance of the secondary side, by the amount of secondary reactance. 134. Shimted condensive reactance, Xc, and series inductive reactance, x©, therefore transforms from constant potential, e6y to constant current, i, and inversely, if their reactances are equal, Xc ^ Xq, and in this case, the main current is leading, with non-inductive load, and the lead of the main current decreases, with increasing inductive reactance, that is, increasing lag, of the r» \ 260 ELECTRIC CIRCUITS receiving circuit. The constant secondary current, i, lags 90° behind the constant primary e.m.f., eo. Inversely, by reversing the signs of x© and Xc in the preceding equations, that is, exchanging inductive and condensive react- ances, it follows that shunted inductive reactance, Xo, and series condensive reactance, Xe, if of equal reactance, Xe = Xo, transform constant potential, co, into constant current, f, and inversely. In this case, the main current lags the more the higher the inductive reactance of the receiving circuit, and the constant secondary current, i, is 90° ahead of the constant primary e.m.f., €o. In general, it follows that, if equal inductive and condensive reactances, Xo = Xc, that is, in resonance conditions, are con- nected in series across a constant-potential circuit of impressed r=2»a -vC^: B IV e.m.f., eo, any circuit connected to the common point between the reactances is a constant-current circuit, and carries the current, i = —. ' Xo Instead of connecting this secondary or constant-current circuit with its other terminal to line. A, so shunting the con- densive reactance with it, and causing the main current to lead (I in Fig. 118), or to line, B, so shunting the inductive reactance with it, and causing the main current to lag (II in Fig. 118), it can be connected to any point intermediate between A and 5, by a auto transformer as in III, Fig. 118. If connected to the mid- dle point between A and S, the main current is neither lagging nor leading, that is, is non-inductive, with non-inductive, load, and with inductive load, has the same power-factor as the load. The two arrangements, I and II, can also be combined, by connecting the constant-current circuit across, as in IV, Fig. 118> and in this case the two inductive reactances and two conden- CONSTANT-CURRENT TRANSFORMATION 261 sive reactances diagrammatically form a square, with the con- stant potential, eo, as one, the constant current, i, as the other diagonal, as shown in Fig. 119. This arrangement has been called the monocyclic square. The insertion of an e.m.f. into the constant-current circuit, in such arrangements, obviously, does not exert any effect on the constancy of the secondary current, i, but merely changes the primary current, io, by the amoimt of power supplied or consumed by the e.m.f. inserted in the secondary circuit. While theoretically the secondary current is absolutely con- stant, at constant primary e.m.f., practically it can not be per- fectly constant, due to the power consumed in the reactances, but f" falls oflf slightly with increase of load, the more, the greater the loss of power in the reactances, that is, the lower the efficiency of the transforming device. Two typical arrangements of such constant-current transform- ing devices are the T-connection or the resonating-drcuity diagram Fig. 117, and the monocyclic square, diagram Fig. 119. From these, a very large number of different combinations of in- ductive and condensive reactances, with addition of autotrans- formers, and of impressed e.m.fs., can be devised to transform from constant potential to constant current and inversely, and by the use of quadrature e.m.fs. taken from a second phase of the polyphase system, the secondary output, for the same amount of reactances, increased. These combinations afford very convenient and instructive examples for accustoming oneself to the use of the symbolic method in the solution of alternating-current problems. Only two typical cases, the T-connection and the monocyclic square will be more fully discussed. Fig. 119. A. T-Connection or Resonating Circuit 136. General. — A combination, in a constant-potential circuit, of an inductive and a condensive reactance in series with each 262 ELECTRIC CIRCUITS other in resonance condition, that is, with the condensive react- ance equal to the inductive reactance, gives constant current in a circuit shunting the capacity. This circuit thus can be called the "secondary circuit'* of the constant potential constant- current transforming device, while the constant-potential supply circuit may be called the "primary circuit." If the total inductive reactance in the constant-current cir- cuit is equal to the condensive reactance, the primary supply current is in phase with the impressed e.m.f. Let, as shown diagrammatically in Fig. 117, Xo = value of the inductive and the condensive reactances which are in series with each other. Xi = the additional inductive reactance inserted in the constant- current circuit. Z = r + jXf OT z = Vr^ + x^ = the absolute value of the im- pedance of the constant-current load. Assuming now in the constant-current circuit the inductive reactance and the resistance as proportional to each other, as for instance is approximately the case in a series arc circuit, in which, by varying the number of lamps and therewith the load, reactance and resistance change proportionally. Let, then, A: = - = ratio of inductive reactance to resistance of the load, or tangent of the angle of lag of the constant-current circuit. It is then Z = r( l + jk) and 2= ry/1 + k^ (1) let, then, jPo = «o = constant = primary impressed e.m.f., or sup- ply voltage, ^1 == potential difference at condenser terminals, JP = secondary e.m.f., or voltage at constant-current circuit, /o = primary supply current, /i = condenser current, / = secondary current, then, in the secondary or receiver circuit, ^ = Z/ (2) at the condenser terminals CONSTANT-CURRENT TRANSFORMATION 263 (3) (4) ^1 = ^ + jxa = (Z + jxi) I and, also, ^1 = - jxo/i hence, z + ixi and the primary current is /o = .,+,.. {,^+ hence, expanded, +:)/ the condenser current is or, the absolute value is Xq^ . _ Vr^ + (x + a;i)^ ^1 ^j a:o^ and (5) ^^^, Z-j{x.-x.) j (6) Xo and the primary supply voltage is 6o = ^1 +ia;o/o; hence, substituting (3) and (6), eo =[(Z + jx,)-{Z - j(xo-Xi)}l/, or, expanded, Co = + jxo/ (7) or, the secondary current is / = ~ ^' (8) * Xq and, substituting (8) in (6) and (5) : the primary current is _ Z - jfa - xi) . . „.£±^.. CO) t - f" (11) Xo ,•„ = VW (a:. - X, =^ ^^ (12) Co (13) tan — - — k gives the secondary phase angle (14) r fl*A ""^ Vt —~ 1* tan ^0 = gives the primary phase angle (15) 264 ELECTRIC CIRCUITS This phase angle ^i = 0, that is, the primary supply current is non-inductive, if Xo — Xi — X = Of that is, Xi = Xo —X. (16) The primary supply can in this way be made non-inductive for any desired value of secondary load, by choosing the reactance, Xi, according to equation (16). If X = 0, that is, a non-inductive secondary circuit (series in- candescent lamps for instance), Xi = Xo, that is, with a non-in- ductive secondary circuit, the primary supply current is always non-inductive, if the secondary reactance, Xi, is made equal to the primary reactance, xo. In this case Xi = xq, with an inductive secondary circuit X tan ^0 = - = tan 6; that is, the primary supply current has the same phase angle as the secondary load, if all three reactances (two inductive and one condensive reactance) are made equal. In general, Xi would probably be chosen so as to make /o non- inductive at full-load, or at some average load. 136. Example. — A 100-lamp arc circuit of 7.5 amp. is to be operated from a 6600-volt constant-potential supply eo = 6600 volts, and i = 7.5 amp. Assuming 75 volts per lamp, including line resistance, gives a maximum secondary voltage, for 100 lamps, of e' = 7500 volts. Assuming the power-factor of the arc circuit as 93 per cent. lagging, gives cos e = 0.93, or tan ^ = 0.4; hence, k = - = 0.4, and Z = r(l + 0.4j), or z = 1.077 r at full-load, if e' = 7500 volts, z' = -. = 1000 ohms, t hence and r' = 0.93 z' = 930 ohms, x' = 0.4 r' = 372 ohms, «o eo 6600 oof\ u i = — > or Xo = — = -irr- = 880 ohms. Xo I 7,0 CONSTANT-CURRENT TRANSFORMATION 265 To make the primary current zo non-inductive at full-load, or for x' = 372 ohms, this requires Xi = Xo — x' = 508 ohms. This gives the equations i = 7.5 amp., e = 7.5 2 = 8.08 r volts. 6600 to = yj r2+ (372 - 0.4 r) 2 x 8802 ^ ^ 372 - 0.4 r tan $0 = r = 0.4 - ^^, r hence, leading current below full-load, non-inductive at full-load and lagging current at overload. 137. Apparatus Economy. — Denoting by 2', /, x' the respective full-load values, the volt-ampere output at full-load is volt-ampere input, «. = '^^' = ^ = ' V." (17) Q = t«eo = ^ (18) That is, the volt-ampere input is less than the Volt-ampere output, since the input is non-inductive, while the output is not. The power output is p = i^r' = ?^ (19) Xq which is equal to the volt-ampere input, since the losses of power in the reactances were neglected in the preceding equations. The volt-amperes at the condenser are Q' = iiHo] hence, substituting (13), The volt-ampere consumption of the first, or primary inductive reactance, xo, is 266 ELECTRIC CIRCUITS hence, substituting (12), Q„ ^ r^* + (X, -X' - x.)« ^^, ^ r>»+(xe-fcr'-xO«^^, ^^^^ the volt-ampere consumption of the second, or secondary induct- ive reactance, Xi, is or Q'" = :^, eo^ (22) The total volt-ampere rating of the reactances required for the transformation from constant potential to constant current then is Q = Q' + Q" + Q'" 2 /«(! + fc*) + 2 )fcr'(2xi - Xo) + (xo* - XoXi + 2x,2) Xo* 60^ (23) and the apparatus economy, or the ratio of volt-amperes output to the volt-ampere rating of the apparatus is . ^ Qo ^ /xoVl + k^ ^ Q 2 r'*(l + A:*) + 2 At'(2 Xi - Xo) + (xo« - XoXi + 2 Xi^) (24) this apparatus economy depends upon the load, /, the power- factor or phase angle of the load, A;, and the secondary additional inductive reactance, Xi. To determine the effect of the secondary inductive reactance, Xi: The apparatus economy is a maximum for that value of secondary inductive reactance, Xi, for which V^ = 0. Instead of directly differentiating /, it is preferable to simplify the function / first, by dropping all those factors, terms, etc., which inspection shows do not change the position of the maxi- mum or the minimum value of the function. Thus the numera- tor can be dropped, the denominator made numerator, and its first term dropped, leaving /' = 2 A;/ (2 Xi - Xo) + (xo^ - XoXi + 2 Xi«) as the simplest function, which has an extreme value for the same value of Xi, as /. Then df j^ = 4 A;r' - Xo + 4 xi = 0, ax I and xi = "" ~/ ^ (26) CONSTANT-CURRENT TRANSFORMATION 267 substituting (25) in (24), gives , ^ 8 r'xoVl + fe" .26^ "'* 16r"-8fcr'a;o + 7xo* To determine the effect of the load r": f\ becomes a maximum for that load, r', which makes or, simplified, ., 16 r'* - 8 hr'xo + 7 Xo* J 1 = ~f > hence ^ = r'(32 / - 8 fcxo) - (16 r'^ - 8 fc/xo + 7 Xo^) = 0, hence r' = 5!>V^ (27) 4 and, substituting (27) in (26), /. . ^^ (28) V 7 — fc hence, for fc = 0: /2 = -^ = 0.378, V7 r' = 5^ = 0.662 xo, 4 Xi = "-r = 0.25 Xo, 4 for fc = 0.4: /, = VOL. ^ ^^^3 ■v/7 - 0.4 r' = 5?V^ = 0.662 xo 4 2' = vTl6 5^ = 0.712 xo 4 xi = ^ (1 - 0.4\/7) 0.016 xo = approximately zero. At non-inductive load fc = and with non-inductive primary supply, that is, Xl =Xo, 268 ELECTRIC CIRCUITS by substituting these values in (24), the apparatus economy is ^ " 2(r'* + xo^) ^^^ which is a maximum for r' = xo (30) /o = 1 = 0.25 (31) which is rather low: That is, non-inductive load and supply circuit do not give very high apparatus economy, but inductive reactance of the load, and phase displacement in the supply circuit, gives far higher appa- ratus economy, that is, more output with the same volt-amperes in reactance. By inserting in (23), with the quantities, Q', Q'', and Q"\ coefficients ni, n2, ns, which are proportional respectively to the cost of the reactances per kilovolt-ampere, the expression then represents the commercial economy, that is, the maximum of this expression, derived by analogous considerations as before, gives the arrangement for minimum cost at given output. 138. Power Losses in Reactances. — In the preceding equations, the losses of power in the reactances have been neglected. However small these may be, in accurate investigations, they require consideration as to their effect on the regulation of the transforming device, and on the efficiency. Let a = power-factor of inductive reactance, that is, loss of power, as fraction of total volt-amperes. b = power-factor of condensive reactance, that is, loss of power, as fraction of total volt-amperes. Here a and b are very small quantities, in general 6, the loss in the condensive reactance, being far smaller than the loss in the inductive reactance. Approximately, the inductive reactances are (a + j)xo and i<^ + j)^i respectively, and the condensive reactance is (6 —j)xo. Assuming the same denotations as in the preceding paragraphs, receiver circuit fJ = Zl (33) CONSTANT-CURRENT TRANSFORMATION 269 at condenser terminals ^, = ^ + (a + j)xj = {Z + (a+ j)x,\l (34) and also hence ^1 = (6 - j)xoh (35) Z + (a + i)a;i , and (37) /o = / + /i ^ ^ + (fe - j)xo + (a + j)xi (b - j)xo * ^ ^ — j(xo — Xi) + (fea^o + axi) (6 - j>o and the impressed e.m.f. eo = ^1 + (a + j>o/o; hence, substituting (35) and (37), xo+[Z(a+b) —jxo{a—b)+jxiia+b)] + {xoab+Xia(a+b)] ^ «»= b^j f (38) Since a and b are very small quantities, their products and squares can be neglected, then xo + [ Z{a + b) - ixoCa - b) + jxija + b)} ,- ,oo^ 6o ^ _ • / \9^) J or ^ Xo + {Z(a + 6) - jxo (a - 6) + ia;i(a + 6)} ^^' this can be written J _ _J£o 1 +jb ""' 1 + j|- (a + 6) - j(a - 6) +i§i (a + 6) I Xq Xo hence / = - J5( 1 + ja - j?'-(a + 6) - A (a + 6) 1 (41) a:o I Xo Xo J that is, due to the loss of power in the reactances, the secondary current is less than it would be otherwise, and decreases with increasing load still further. 270 ELECTRIC CIRCUITS Equation (41) can also be written here the imaginary component is very small in the parenthesis, that is, the secondary current remains practically in quadrature with the primary voltage. The absolute value is, neglecting terms of secondary order, eo t = — Xo 1 - - (o + 6) Xo (43) The primary current is, by equation (37) and (40), Z - j{xo — Xi) + (bxo + axi) /o = Xo + Z{a + 6) — jxo(a — b) + jxi{a + b) Xo Z eo Xo (44) Xo "^ \ Xo' \ Xo/ Xo 139. Example. — Considering the same example as before: a constant-potential circuit of eo = 6600 volts supplying a 100-lamp series arc circuit, with i' = 7.5 amp., and e' = 7500 volts at full-load of 93 per cent, power-factor, that is, k = 0.4, andZ = (1 — OAj)r. Assuming now, however, the loss in the inductive reactance as 3 per cent., and in the capacity as 1 per cent., that is, a =0.03 b =0.01, the full-load value of the secondary load impedance is: z' =1000 ohms, r' =930 ohms and x' =372 ohms. To give non-inductive primary supply at full-load, the follow- ing equation must be fulfilled: iCi = iTo — x' = Xo — 372. From equation (43), the secondary current, at full-load, is i' = !• Xo Xo or 7.5 = 6600 Xo 1 - 930 X 0.04 1 Xo hence Xo = 840 ohms, and Xi = 468 ohms. CONSTANT-CURRENT TRANSFORMATION 271 Substituting in (42), (43), (44), / = -7.86i 1 -0.04^ j/o.052 - 0.01 .-7.86(1-0.04^) e = tz = 1.077 ri -8.46.(1 -0.04 4) and herefrom the power-factor, efficiency, etc. u 1 , 1 -^ 100 PO»EF OR / ^ 1 ,^ t' CIE -^ -^ ^ -. ,.n -^ J / / i; -/ / «/F-^ir+(^-^ 2 CONSTANT-CURRENT TRANSFORMATION 279 or, approximately, t = f»(l-ci??),etc. (49) 146. Example. — Considering the same example as before, of a 7.5-amp. 100- lamp arc circuit operated from a 6600-volt constant-potential supply, and assuming again as in paragraph 139: 3 per cent, power-factor of inductive reactance, or a = 0.03. 1 per cent, power-factor of condensive reactance, or 6 = 0.01. It is then, Ci = 0.02, C2 = 0.01, and at full-load, Xo \ Xq/ or, 7.5 = 6«50/i_ 0.02^); Xo \ Xo / hence, Xo = 861, and i = 7.66^1 - 0.02 g^) , and we have, approximately, r r 0.42 ir 1 ' 0.4 ri /o = 7.66 /x = 3.83{^+i(l+^01 /. = 3.83{3^-i(l-^)) e = zi = 1.077 ri. In Fig. 121 are plotted, with the secondary terminal voltage, e, as abscissae, the values of secondary current, i; primary current, to; condenser current, ii; inductive reactance current, Z2, and eflBciency. As seen, with the monocyclic square, the current regulation is closer, and the efficiency higher than with the T connection. This is due to the lesser amount of reactance required with the monocyclic square. The investigation of the effect of a variation of frequency on the current regulation by the monocyclic square, now can be carried out in the analogous manner as in A with the T connection. 280 ELECTRIC CIRCUITS C. General Discussion of Constant-potential Constant- current Transformation 146. In the preceding methods of transformation between constant potential and constant current by reactances, that is, by combinations of inductive and condensive reactances, the constant alternating current is in quadrature with the constant e.m.f. Even in constant-current control by series inductive reactances, the constancy of current is most perfect for light loads, where the reactance voltage is large and thus the constantr current voltage almost in quadrature, and the constant-current control is impaired in direct proportion to the shift of phase of the constant current from quadrature relation. 13 ^ ^ - „ ^ - ID If ^> r 1 i / 1 y ' M-- r / J -TT SE 1 1 - ■'t ^ s ss ^ " ^ / - " ^ S^ 1 — 7" -^ / 11 Fia. 121. The cause hereof is the storage of energy required to change the character of the flow of energy. That is, the energy supplied at constant potential in the primary circuit, is stored in the react- ances, and returned at constant current, in the secondary circuit. The storage of the total transformed energy in the reactances allows a determination of the theoretical minimum of reactive power, that is, of inductive and condensive reactances required for constant-potential to constant-current transformation, since the energy supplied in the constant-current circuit must be stored for a quarter period after being received from the constant>-po- tential circuit. CONSTANT-CURRENT TRANSFORMATION 281 Let p = P(l + cos 2 e). = Power supplied to the constant-current circuit; thus, neglecting losses, po = P(l - cos 2 e) = Power consumed from the constant-potential cir- cuit, and po — p = 2 P cos 2 ^ = Power in the reactances. That is, to produce the constant-current power, P, from a single-phase constant-potential circuit, the apparent power, 2 P, must be used in reactances; or, in other words, per kilowatt con- stant-current power produced from a single-phase constant-po- tential circuit, reactances rated at 2 kv.-amp. as a minimum are required, arranged so as to be shifted 45° against the constant- potential and the constant-current circuit. The reactances used for the constant-potential constant-cur- rent transformation may be divided between inductive and con- densive reactances in any desired proportion. The additional wattless component of constant-potential power is obviously the difference between the wattless volt-am- peres of the inductive and that of the condensive reactances. That is, if the wattless volt-amperes of reactance is one-half of inductive and one-half of condensive, the resultant wattless volt- amperes of the main circuit is zero, and the constant-potential circuit is non-inductive, at non-inductive load, or consumes cur- rent proportional to the load. If A is the condensive and B the inductive volt-amperes, the resultant wattless volt-amperes is B-A ; that is, a lagging watt- less volt-amperes of B-A (or a leading volt-ampere of A-B) exist in the main circuit, in addition to the wattless volt-amperes of the secondary circuit, which reappear in the primary circuit. 147. These theoretical considerations permit the criticism of the different methods of constant-potential to constant-current transformation in regard to what may be called their apparatus economy, that is, the kilovolt-ampere rating of the reactance used, compared with the theoretical minimum rating required. 1. Series inductive reactance, that is, a reactive coil of constant inductive reactance in series with the circuit. This arrangement obviously gives only imperfect constant-current control. Per- 282 ELECTRIC CIRCUITS mitting a variation of 5 per cent, in the value of the current (that is, full-load current in 5 per cent, less than no-load current) and assuming 4 per cent, loss in the reactive coil, a reactance rated at 2.45 kv.-amp. is required per kilowatt constant-current load. This apparatus operates at 87.9 per cent, economy and 30 per cent, power-factor. Assuming 10 per cent, variation in the value of the current, reactance rated at 2.22 kv.-amp. is required per kilowatt constant- current load. This arrangement operates at an economy of 91.8 per cent., and a power-factor of 49.5 per cent. In the first case, the apparatus economy, that is, the ratio of the theoretical minimum kilovolt-ampere rating of the reactance to the actual rating of the reactance is 88 per cent., and in the last case 92 per cent., thus the objection to this method is not the high rating of the reactance and the economy, but the poor constant- current control, and especially the very low power-factor. 2. Inductive and condensive reactances in resonance condition, the condensive reactance being shunted by the constant-ciu'rent circuit. In this case, condensive reactance rated at 1 kv.-amp. and inductive reactance rated at 2 kv.-amp. are required per kilo- watt constant-current load, and the main circuit gives a constant wattless lagging apparent power of 1 kv.-amp. Assuming again 4 per cent, loss in the inductive and 2 per cent, loss in the condens- ive reactances, gives a full-load efficiency of 91 per cent, and a power-factor (lagging) of 74 per cent. The apparatus economy by this method is 66.7 per cent. 3. Inductive and condensive reactances in resonance condition, the inductive reactance shunted by the constant-current circuit. In this case, as a minimum, per kilowatt constant-current load, condensive reactance rated at 2 kv.-amp. and inductive reactance rated at 1 kv.-amp. is required, and the main circuit gives a con- stant wattless leading apparent power of 1 kv.-amp. The effi- ciency of transformation is at full-load 92.5 per cent., the power- factor (leading) 73 per cent., the apparatus economy 66.7 per cent. 4. T'Connection, that is, two equal inductive reactances in se- ries to the constant-current circuit and shunted midway by an equal condensive reactance. In this case per kilowatt constant- current load, condensive reactance rated at 2 kv.-amp. and in- ductive reactance rated at 2 kv.-amp. are required. The main circuit is non-inductive at all non-inductive loads, that is, the power-factor is 100 per cent. CONSTANT-CURRENT TRANSFORMATION 283 The full-load efl&ciency is 89.3 per cent, apparatus economy 50 per cent. 5. The monocyclic square. In this case a condensive reactance rated at 1 kv.-amp. and inductive reactance rated at 1 kv.-amp. are required per kilowatt constant-current load. The main cir- cuit is non-inductive at all non-inductive loads, that is, the power- factor is 100 per cent. The full-load efficiency is 94.3 per cent., the apparatus economy 100 per cent. 6. The monocyclic square in combination with a constant^ potential polyphase system of impressed e.m.f. In this case, per kilowatt constant-current load, condensive reactance rated at 0.5 kv.-amp. and inductive reactance rated at 0.5 kv.-amp. are re- quired. The main circuits are non-inductive at all loads, that is, the power-factor is 100 per cent. The full-load efficiency is over 97 per cent, the apparatus economy 200 per cent. 148. In the preceding, the constant-potential to constant-cur- rent transformation with a single-phase system of constant im- pressed e.m.f., has been discussed; and shown that as a minimum in this case, to produce 1 kw. constant-current output, reactances rated at 2 kv.-amp. are required for energy storage. The con- stant current is in quadrature with the main or impressed e.m.f., but can be either leading or lagging. Thus the total range avail- able is from 1 kw. leading, to zero, to 1 kw. lagging. Hence if a constant-quadrature e.m.f. is available by the use of a poly- phase system, the range of constant current can be doubled, that is, reactance rated at 2 kv.-amp. can be made to control the po- tential for 2 kw. constant-current output in the way shown in Fig. 122 for a three-phase, and Fig. 123 for a quarter-phase system of impressed e.m.f. In this case, one transformer feeds a monocyclic square, the other transformer inserts an equal constant e.m.f. in quadrature with the former, which from no-load to half-load is subtractive, from half-load to full-load is additive, that is, at full-load, both phases are equally loaded; at half-load only one phase is loaded and at no-load one phase transforms energy into the other phase. The monocyclic e.m.f. square in this case, when passing from full-load to no-load, gradually collapses to a straight line at half-load, then overturns and opens again to a square in the opposite direction at no-load. That is, at full-load the trans- formation is from constant potential to constant current, and at 284 ELECTRIC CIRCUITS no-load the transformation is from constant current to constant potential. Obviously with this arrangement the efficiency is greatly increased by the reduction of the losses to one-half, and the con- stant-current control improved. CONSTANT CUB8ENT 8INQLE-PHA8E Fig 122. s- At the same time, the sensitiveness of the arrangement for dis tortion of the wave shape, as will be discussed later, is greatly reduced, due to the insertion of a constant-potential e.m.f. into the constant-current circuit. Obviously the arrangments in Figs. 122 and 123 are not the only ones, but many arrangements of inserting a constant-quad- rature e.m.f. into the monocyclic square or triangle are suitable. IND. IND. ^^nrs-^TRTH < z -lU- z o o OOND. CONSTANT CURRENT 8INQLE-PHA8E Fig. 123. Different arrangements can also be used of the constant-current control, for instance, the inductive and condensive reactances in resonance condition with their common connection connected to the center of an autotransformer or transformer, with the insertion of the constant-potential quadrature e.m.f. in the latter circuit as shown in Fig. 124, or the T-connection, shown applied to a quarter-phase system in Fig. 125. CONSTANT-CURRENT TRANSFORMATION 285 Constant-potential apparatus and constant-current single- phase circuits can also be operated from the same transformer secondaries in a similar manner, as indicated in Fig. 124 for a three-phase secondary system. In Figs. 122 to 125 the arrangement has been shown as applied to step-down transformers, but in the estimate of the eflBciency ^ S o ■O O 1 '^^^• Ill POTENTi; -PHASE o^ o '^^^\ POTE THREE z £ 2 I dfe o< Jo CONSTANT CURRENT ' 8INQIC*PHA8E Fig. 124. the losses in these transformers have not been included, since these transformers are obviously not essential but merely for the convenience of separating electrically the constant-current cir- cuit from the high-potential line. It is evident, for instance, in Fig. 124, that the constant-current and constant-potential cir- g Ul 2 u 5 ^ ^ $-§- f ^'^"^^ o VTX o 5 o _/ o "X ^v.. o ^>' o ^ V \ ^ CONSTANT CURRENT SINOLE-PHASE Fig. 125. cuits instead of being operated from the three-phase secondaries of the step-down transformers can be operated directly from the three-phase primaries by replacing the central connection of the one transformer by the central connection of the auto- transformer. 286 ELECTRIC CIRCUITS D. Problems 149. In the following problems referring to constant-potential to constant-current transformation by reactances, it is recom- mended: (a) To derive the equation of all the currents and e.m.fs., in complex quantities as well as in absolute terms, while neglecting the loss of power in the reactances. (6) To determine the volt-amperes in the different parts of the circuit, as load, reactances, etc., and therefrom derive the apparatus economy, to find its maximum value, and on which condition it depends. (c) To determine the effect of inductive load on the power of the primary supply circuit, to investigate the phase angle of the primary supply circuit, and the conditions under which it becomes a minimum, or the primary supply becomes non- inductive. (d) To redetermine the equations of the problem, while con- sidering the power lost in the reactances, and apply these equa- tions to a numerical example, plotting all the interesting values. (e) To investigate the effect of a change of frequency on the equations, more particularly on the constant-current regulation. (/) To investigate the effect of distortion of wave shape, that is, the existence of higher harmonics in the impressed e.m.f., and their suppression or reappearance in the secondary circuit. (g) To study the reversibility of the problem, that is, apply (a) to (f) to the reversed problem of transformation from constant current to constant potential. Some of the transforming devices between constant potential and constant current are: A. Single-phase. (a) The resonating circuit, or condensive and inductive reactances, of equal values, in series with each other in the con- stant-potential circuit, and the one reactance shunted by the constant-current circuit. (b) T-connection, as partially discussed in (A). (c) The monocyclic square, as partially discussed in (B). (d) The monocyclic triangle: a condensive reactance and an inductive reactance of equal values, in series with each other across the constant-potential circuit, the constant-current CONSTANT'CURRENT TRANSFORMATION 287 circuit connecting between the reactance neutral, or the common connection between the two (opposite) reactances, and the neutral of a compensator or autotransformer connected across the constant-potential circuit. Instead of the compensator neutral, the constant-current circuit can be carried back to the neutral of the transformer connected to the constant-potential circuit. B. Polyphase. (a) In the two-phase system the two phases of e.m.fs., eo and jeoy are connected in series with each other, giving the outside terminals, A and J5, and the neutral or common con- • nection, C. A condensive reactance and an inductive reactance of equal values, in series with each other and with their neutral or common connection, D, are connected either between A and By and the constant-current circuit between C and D, or the reactances are connected between A and C, and the constant- current circuit between B and D. In either case, several ar- rangements are possible, of which only a few have a good appara- tus economy. (6) In a three-phase system, a condensive reactance, an induct- ive reactance equal in value to that of the condensive reactance and the constant-current circuit, are connected in star connec- tion between the three-phase, constant-potential terminals. Here also two arrangements are • possible, of which one only gives good apparatus economy. (c) In a constant-potential three-phase system, each of the three terminals, A, J5, C, connects with a condensive and an inductive reactance, and all these reactances are of equal value, and joined together in pairs to three terminals, a, 6, c, so that each of these terminals, a, 6, c, connects an inductive with a condensive reactance, a, 6, c, then, are constant-current three- phase terminals, that is, the three currents at a, 6, c, are constant and independent of the load or the distribution of load, and displaced from each by one-third of a period. This arrange- ment is especially suitable for rectification of the constant al- ternating-current, to produce constant direct current. 160. Some further problems are: 1. In a single-phase, constant-current transforming device, as the monocyclic square, the constant current, t, is in quadrature with the constant impressed e.m.f., eo. By inserting a constant- potential e.m.f., ^t, into the constant-current circuit, the appa- 288 ELECTRIC CIRCUITS ratus economy can be greatly increased, in the maximum can be doubled; that is, the e.m.f., ^3 gives constant-power output, and from no-load to half-load, the transformation is from con- stant current to constant potential, that is, a part of the power supply, ^8, is transformed into the circuit, of e.m.f., eo, that is, the circuit, eo, receives power. Above half-load the circuit of €0 transforms power from constant potential to constant current, into the circuit of e.m.f. ^3. Since i is in time quadrature with 60, with non-inductive secondary load, that is, the secondary terminal voltage, -F, in phase with the secondary current, i, ^z should also be in phase with i, that is, ^3 = jez- With inductive secondary load, of phase angle, B, ^z should be in phase with ip, that is, leading i by angle d, or should be: ^3 = jez (1 + kj). It is interesting, therefore, to investigate how the equation of the constant-potential constant-current devices are changed by the introduction of such an e.m.f., ^3, at non-inductive as well as at inductive load, if -^^3 = i^a, or ^z = j{ez — ic'3), in either case, and also to determine how such an e.m.f., ^3, of the proper phase relation, can be derived directly or by trans- formation from a two-phase or three-phase system. 2. If in the constant-potential constant-current transform- ing device one of the reactances is gradually changed, increased or decreased from its proper value, then in either case the regula- tion of the system is impaired. That is, the ratio of full-load current to no-load current falls ofif, but at the same time, the no-load current a-lso changes. With increase of load, the frequency of the system decreases, due to the decreasing speed of the prime mover, if the output of the system is an appreciable part of the rated output. If, therefore, the reactances are adjusted for equality of the frequency of full-load, at the higher frequency of no-load, the inductive reactance is increased, and thereby the no-load current decreased below the value which it would have at constant reactance, and in this manner the increase of current from full-load to no-load is reduced. Such a drop of speed and therefore of frequency, s, can there- fore be found, that the current at full-load, with perfect equality between the reactances, equals the current at no-load, where the reactances are not quite equal. That is, the variation of frequency compensates for the incomplete regulation of the CONSTANT-CURRENT TRANSFORMATION 289 current, caused by the energy loss in the reactances. Further- more, with a given variation of frequency, s, from no-load to full- load, the reactances can be chosen so as to be slightly unequal at full-load, and more unequal at no-load; the change of current caused hereby compensates for the incomplete current regu- lation, that is, with a given frequency variation, s (within certain limits), the current regulation can be made perfect from no-load to full-load, by the proper degree of inequality of the reactances. It is interesting to investigate this, and apply to an example, a, to determine the proper s, for perfect equality of reactance at full-load; 6, with a given value of s = 0.04, to determine the in- equality of reactance required. Assuming a = 0.03; b = 0.01. 3. If one point of the constant-current circuit, either a terminal or an intermediate point, connects to a point of the constant-potential circuit, either a terminal or some intermediate point (as inside of a transformer winding), the constant current is not changed hereby, that is, the regulation of the system is not impaired, and no current exists in the cross between the two circuits. The distribution of potential between the reactances, however, may be considerably changed, some reactances re- ceiving a higher, others a lower voltage. It follows herefrom, that a ground on a constant-current system does not act as a ground on the constant-potential system, but electrically the two systems, although connected with each other, are essentially independent, just as if separated from each other by a transformer. So, for instance, in the monocyclic square, one side may be short-circuited without change of current in the secondary, but with an increase of current in the other three sides. It is interesting to investigate how far this independence of the circuits extends. In general, as an example, the following constants may be chosen: In the constant-potential circuit: eo = 6600 volts and i'o = 10 amp. at full-load. In the constant-current circuit: i = 7.5 amp., e' = 7500 volts at fuU-load. Or, especially in polyphase systems, e\ respectively, i'o corresponding to the maximum economy point, and a = 0.03; b = 0.01. 10 290 ELECTRIC CIRCUITS E. Distortion of Voltage Wave 161. It is of interest to investigate what effect the distortion of the voltage wave, that is, the existence of higher harmonics in the wave of supply voltage, has on the regulation of the con- stant-potential constant-current transformation systems dis- cussed in the preceding. Where constant current is produced by inductive reactance only, higher harmonics in the voltage wave naturally are sup- pressed the more, the larger the inductive reactance and the higher the order of the harmonic. An increase of the intensity of the harmonics in the current wave, over that in the voltage wave, and with it an impairment of the constant-current .regulation, could thus be expected only with devices using capacity reactance. As example may be investigated the effect of the distortion of the impressed voltage wave on the T connection, and on the monocyclic square. The symbolic method of treating general alternating waves may be used, as discussed in Chapter XXVII, of "Theory and Calculation of Alternating-current Phenomena," fifth edition, page 379. That is, the voltage wave is represented by 00 1 and the impedance by Z = r + jn {nxm + ^0 + ;f ) where n = order of harmonic. A. T Connection or Resonating Circuit 152. Assuming the same denotation as before, we have, for the nth harmonic: primary inductive reactance, Zo = + jnxo; secondary inductive reactance, Zi = + jnxi; condensive reactance, Z2 — ; n CONSTANT-CURRENT TRANSFORMATION 291 when neglecting the energy losses in the reactances, load Z = r{l+jnk) therefore, also for the nth harmonic. F = r(l+i7ifc)/i ^1 = ^ + za = [r {l+jnk)'\- jnxi] /, and also hence, and ^.= -i5/.: _ . n[r (1 + jnk) + jnxi] Xo h-l+l _ j^o — j'^^^1 — nr(l + jnfc) hence, [r (1 + jnk) + jnxi] + n [jxo —jn^Xi — nr (1 + jnfcjj j/ — (n2 — l)r(l +infc) — jnxi (n^ - 1) + jnxo}/; hence. / = -j^o nxa — nxi(n^ — 1) + j (n* — 1) r (1 + jnfc) nxo — (n* — l)[n (ari + fcr) + jr]' hence, approximately, for higher values of n, ^ n^{xi + kr)' that is, for larger values of n, / = 0, or the higher harmonics in the current wave disappear. Herefrom, by substituting in the preceding equations, the supply current, /o, the condenser current, /i, their respective ejn.fs., etc., are derived. It is then, in general expression : If 5. - JC. - i....) = imprest e.m.f. 292 ELECTRIC CIRCUITS / = ^nxo- (n* - 1) [n{xi + kr) + jr] xo ^n^(xi + kr)' the equation of the secondary current. For instance, let Eo = 6600 {li - O.2O3 - 0.155 + O.O67 - 0.25 js} = constant-impressed e.m.f. or, absolute, eo = 6600 \r+ 0.202 + O.I52 + 0,062 + 0.25^ = 6600 X 1.062 = 7010 volts, and choosing the same values as before, in paragraph 143, Xo = 880 ohms, Xi = 508 ohms, r' = 930 ohms, k = 0.4; it is, substituting, . __ _7- . 60.08 -48.8i3- 8.0 is + L2i7 / - /.oji- 508 + 0.4 r or, absolute. i = /7 52 -f ^^ + ^^-^^ + ^'^^ + ^'^^ f = ^ (508 + 0.4 r) 2 = ^7.52+ '^'^^'^ (508 + 0.4 r) 2' hence, at no-load, i = 7.5 X 1.00021 and, at full-load, r = 930, i = 7.5 X 1.00003. That is, the current wave is as perfect a sine wave as possible, regardless of the distortion of the impressed e.m.f., which, for instance, in the above example, contains a third harmonic of 32 per cent. Or in other words, in the T connection or the resonat- ing circuit, all harmonics of e.m.f. are wiped out in the current wave, and this method indeed ofifers the best and most conven- ient means of producing perfect sine waves of current from any shape of e.m.f. waves. CONSTANT-CURRENT TRANSFORMATION 293 1S3. B. Monocyclic Square Assuming the same denotation as before, we have for the nth harmonic: inductive reactance, Z2 = + jnxo; condensive reactance. Zi = -J?; n load, Z = r(l +jnk); currents, and *i - 2 ' e.m.fs., ^0 = Zili + Z2I2, ZJ = Ziji — Z2I2] hence, substituting, we have ^0 = - jxo (^' - xj^ , r(l+jnfc)/= -ja;o(^' + n/2); thus, then, combining, we obtain = +2ixo/, and 2 xo + jr(l + jnA-) ( n - - j = -J^o(n^+l) 2 nxo + jV(n2 — 1)(1 + jnk) and herefrom /i, /i, /2, etc. 294 ELECTRIC CIRCUITS Approximately, for higher values of n, and for high loads, r, / = m. nkr That is, the higher harmonies of current decrease proportion- ally to their order, at heavy loads — that is, large values of r. For light loads, however, or small values of r, and in the extreme case, at no-load, or r = 0, it is _ j^o (n' + 1) 2 nxo and, approximately, * 2x0* That is, the current is increased proportional to the order of the harmonics, or in other words, at no-load, in the monocyclic square, the higher harmonics of impressed e.m.fs. produce increased values of the higher harmonics of current, that is, the wave-shape distortion is increased the more, the higher the harmonics. In general expression: If 00 ^0 = 2^(en — jne'n) = Imprcsscd e.m.f., 1 ^2nxo+ jnr{n^ - 1)(1 + junk)' and herefrom /©, /i, /2, etc. For instance, let Eo = 6600 {li - O.2O3 - 0.25 jz - 0.156 + O.O67} = constant-impressed e.m.f., or, absolute, eo = 7010 volts, and, choosing the same values as before, Xq = 880 ohms, / = 930 ohms, k = 0.4; it is, substituted, (2.5 - 2 jz) 6600 25,740 / = 7.5 - 5280 - (9.6 - Sj)r 8800 - (48 - 24i5)r 19,800 + 12,320 - (134.4 - 48i)r' CONSTANT-CURRENT TRANSFORMATION 295 herefrom follows, at no-load, r = 0, / = 7.5 - (3.12 - 2.5^3) - 2.92 + I.6I7. That is, at no-load, the secondary current contains excessive higher harmonics, for instance, a third harmonic, V3.I22 + 2.52 = 4.0, or 53.3 per cent, of the fundamental. Absolute, the no-load current is i = V7.52 + 3.122 + 2.52 + 2.922 + 1.612 = 9.13 amp. At full-load, or r = 930, it is / = 7.5 + (2.18 4- 1.07 is) + (0.51 + 0.32 jb) - (0.14 + 0.06 jt); that is, at full-load, the harmonics, while still intensified, are less than at no-load, and decrease with their order, n, more rapidly. The absolute value is i = V7.52 + 2.182 + 1.072 + 0.512 _|_ 0.322+ 0.142 + 0.06^ = 7.91 amp. Instead of 7.5 amp., the value which the current would have at all loads if no higher harmonics were present, the higher har- monics of impressed e.m.f. raise the current to 9.13 amp., or by 21.7 per cent, at no-load, and to 7.91 amp., or by 5.5 per cent, at full-load, while the impressed e.m.f. is increased by 6.2 per cent, by its higher harmonics. It follows also that the constant-current regulation of the sys- tem is seriously impaired, and between no-load and full-load the current decreases from 9.13 to 7.91 amp., or by 15.4 per cent., which as a rule is too much for an arc circuit. 164. It follows herefrom: While the T connection of transformation from constant poten- tial to constant current suppresses the higher harmonics of im- pressed e.m.f. and makes the constant current a perfect sine wave, the monocyclic square intensifies the higher harmonics so that the higher harmonics of impressed e.m.f. appear at greatly increased intensity in the constant-current wave. The increase of the higher harmonics is different for the different harmonics and for different loads, and the distortion of wave shape produced hereby is far greater at no-load, and the constant-current regulation of the system is thereby greatly impaired, and at load the dis- 296 ELECTRIC CIRCUITS tortion is less, and very high harmonics are fairly well sup- pressed, and the operation of an arc circuit so feasible. Assuming, then, that in the monocyclic square of constant- potential constant-current transformation, with a distorted wave of impressed e.m.f., we insert in series to the monocyclic square into the main circuit, /o, two reactances of opposite sign, which are equal to each other for the fundamental frequency, that is, a condensive reactance, Z3 = — j— , and an inductive 7* reactance, Z4 = + jnxz. Then for the fundamental, these two reactances together offer no resultant impedance, but neutralize each other, and the only drop of voltage produced by them is that due to the small loss of power in them. At the nth harmonic, however, the resultant reactance is Z3 + Z4 = +jx,(n + ^, or, approximately, = +jxznj and two such impedances so obstruct the higher harmonics, the more, the higher their order while passing the fundamental sine wave. Such a pair of equal reactances of opposite sign so can be called a ''wave screen. ^^ Further problems for investigation by the student then are: 1. The investigation of the effect of the distortion of the wave of impressed e.m.f. on the constant current, with other trans- forming devices, and also the reverse problem, the investigation of the effect of the distortion of the constant-current wave, as caused by an arc, on the system of transformation. 2. What must be the value, Xi, of the reactance of a wave screen, to reduce the wave-shape distortion of the secondary current in the monocyclic square to the same percentage as the distortion of the impressed e.m.f. wave, or to any desired per- centage, or to reduce the variation of the constant current with the load, as due to the wave-shape distortion, below a given percentage? 3. Determination of efficiency and regulation in the mono- cyclic square with interposed wave screen, Xi, assuming again 3 per cent, loss in the inductances, 1 per cent, loss in the capacities and choosing X4 so as to fill given conditions, regarding wave- shape distortion, or regulation, or efficiency, etc.