CHAPTER XI PHASE CONTROL 80. At constant voltage, eo, impressed upon a circuit, as a transmission line, resistance, r, inserted in series with the receiv- ing circuit, causes the voltage, e, at the receiver circuit to decrease with increasing current, /, through the resistance. The decrease of the voltage, e, is greatest if the current, /, is in phase with the voltage, e — less if the current is not in phase. Inductive reactance in series with the receiving circuit, e, at constant impressed e.m.f., eo, causes the voltage, e, to drop less with a unity power-factor current, 7, but far more with a lagging current, and causes the voltage, e, to rise with a leading current. While series resistance always causes a drop of voltage, series inductive reactance, x, may cause a drop of voltage or a rise of voltage, depending on whether the current is lagging or leading. If the supply line contains resistance, r, as well as reactance, x, and the phase of the current, I, can be varied at will, by producing in the receiver circuit lagging or leading currents, the change of voltage, e, with a change of load in the circuit can be controlled. For instance, by changing the current from lagging at no-load to lead at heavy load the reactance, x, can be made to lower the voltage at light load and raise it at overload, and so make up for the increasing drop of voltage with increasing load, caused by the resistance, r, that is, to maintain constant voltage, or even a voltage, e, which rises with the load on the receiving circuit, at constant voltage, Co, at the generator side of the line. Or the wattless component of the current can be varied so as to maintain unity power-factor at the generator end of the line, eo, etc. This method of controlling a circuit supplied over an induc- tive line, by varying the phase relation of the current in the circuit, has been called "phase control," and is used to a great extent, especially in the transmission of three-phase power for conversion to direct current by synchronous converters for 7 97 98 ALTERNATING-CURRENT PHENOMENA railroading, and in the voltage control at the receiving end of very long high voltage transmission lines. It requires a receiving circuit in which, independent of the load, a lagging or leading component of current can be produced at will. Such is the case in synchronous motors or converters: in a synchronous motor a lagging current can be produced by decreasing, a leading current by increasing, the field excitation. 81. If in a direct-current motor, at constant impressed voltage, the field excitation and therefore the field magnetism is decreased, the motor speed increases, as the armature has to revolve faster to consume the impressed e.m.f., and if the field excitation is increased, the motor slows down. A synchronous motor, however, cannot vary in speed, since it must keep in step with the impressed frequency, and if, therefore, at constant impressed voltage the field excitation is decreased below that which gives a field magnetism, that at the synchronous speed consumes the impressed voltage, the field magnetism still must remain the same, and the armature current thus changes in phase in such a manner as to magnetize the field and make up for the deficiency in the field excitation. That is, the armature current becomes lagging. Inversely, if the field excitation of the synchronous motor is increased, the magnetic flux still must remain the same as to correspond to the impressed voltage at synchronous speed, and the armature current so becomes demagnetizing — that is, leading. By varying the field excitation of a synchronous motor or converter, quadrature components of current can be produced at will, proportional to the variation of the field excitation from the value that gives a magnetic flux, which at synchronous speed just consumes the impressed voltage (after allowing for the impedance of the motor). Phase control of transmission lines is especially suited for circuits supplying synchronous motors or converters; since such machines, in addition to their mechanical or electrical load, can with a moderate increase of capacity carry or produce con- siderable values of wattless current. For instance, a quadrature component of current equal to 50 per cent, of the power com- ponent of current consumed by a synchronous motor would increase the total current only to VI 4- 0.5^ = 1.118, or 11.8 per cent., while a quadrature component of current equal to 30 per cent, of the power component of the current would give an PHASE CONTROL 99 increase of 4.4 per cent, only, that is, could be carried by the motor armature without any appreciable increase of the motor heating. Phase control depends upon the inductive reactance of the line or circuit between generating and receiving voltage, So and e, and where the inductive reactance of the transmission line is not sufficient, additional reactance may be inserted in the form of reactive coils or high internal reactance transformers. This is usually the case in railway transmissions to synchronous converters. Phase control is extensively used for voltage control in railway power transmission by compounded syn- chronous converters. It is also used to a considerable extent in very long distance transmission, for controlling the voltage and the power-factor; in a distribution system for controlling the power-factor of the system. While, therefore, the resistance, r, of the line is fixed, as it would not be economical to increase it, the reactance, x, can be increased beyond that given by line and transformer, by the insertion of reactive coils, and therefore can be adjusted so as to give best results in phase control, which are usually obtained when the quadrature component of the current is a minimum. 82. Let, then, e = voltage at receiving circuit, chosen as zero vector. I = i — ji' = current in receiving circuit, comprising a power component, t, which depends upon the load in the receiving circuit, and a quadrature component, i' , which can be varied to suit the requirements of regulation, and is considered positive when lagging, negative when leading. E^ = e'a — jcq" = voltage impressed upon the system at the generator end, or supply voltage, and the absolute value is Co = VevTTv. Z = r -\- jx — impedance of the circuit between voltage e and voltage eo, and the absolute value is z = v r^ + x^- If e = terminal voltage of receiving station, eo = terniinal voltage of generating station, Z = impedance of transmission line; if e = nominal induced e.m.f. of receiving synchronous machine, that is, voltage corresponding to its field excitation, and eo = nominal induced e.m.f. of generator, Z also includes the synchronous impedance of both machines, and of step-up and step-down transformers, where used, 100 ALTERNATING-CURRENT PHENOMENA It is Eo = e + ZI, or, ^0 = (e + ri + xi') - j{n' - xi), (l) and in absolute value we have Co'- - (e + ri + xi'Y + {ri' - xi)\ (2) This is the fundamental equation of phase control, giving the relation of the two voltages, e and eo, with the two com- ponents of current, i and i\ and the circuit constants r and x. From equation (2), follows: e = Veo" - {ri' - xi)" - {ri + xi'), (3) expressing the receiver voltage, e, as a function of eo and I. Denoting tan e =y (5) where 0 is the phase angle of the line impedance, we have r = 2 cos d and x — z sin d (6) and ., , /eo^ /e cos 9 , A 2 e sin d ,„. gives the reactive component of the current, i' , required by the power component of the current, i, at the voltages, e and Co- 83. The phase angle of the impressed e.m.f., £"0, is, from (1), ri' — xi • ,_v tan ^0 = — \ ^T — ~'' (8) e -\- ri -\- XI the phase angle of the current i' tan dy = -T> (9) I hence, to bring the current, /, into phase with the impressed e.m.f., Eo, or produce unity power-factor at the generator ter- minal, eo, it must be ^0 = 0i', hence, ri' — xi' i' e -{- ri -\- xi i PHASE CONTROL 101 and herefrom follows: i' -- ± Ve^ - 4 xH^ - 2x ^ (10) hence always negative, or leading, but i' = 0 for i = 0, or at no-load. From equation (10) follows that i' becomes imaginary, if the term under the square root, (e^ — 4 xH''), becomes negative, that is, if e I > 2x that is, the maximum load, or power component of current, at which unity power-factor can still be maintained at the supply voltage, eo, is given by e 2x (11) 1800 1600 UOO t U1200 a. z 450 400 o > . t'u "~H / / Q. / ^ o & < 800 a 1 / / 400 SOO / ^ / 200 4U0 COO 300 1000 1200 1400 1600 1300 2000 AMPERES LOAD « l Fig. 77. and the leading quadrature component of current required to compensate for the line reactance x at maximum current, im, is from equation (10), • ' ' (12) ^m 2x that is, in this case of the maximum load which can be delivered at e, with unity power-factor at eo, the total current, /, leads the receiver voltage, e, by 45°. 102 ALTERNATING-CURRENT PHENOMENA Substituting the value, i' , of equation (10), which compensates for the Hne reactance, x, and so gives unity power-factor at eo, into equation (2), gives as required supply voltage eo. 62^2 {x — r) {e — 2 ix)-\/e^ — 4:i^x^ .,o\ ^" = 2x^ + 2x ' ^^^^ As illustration are shown, in Fig. 77, with the load current, i, as abscissas, the values of leading quadrature component of current, i', and of generator voltage, eo, for the constants e = 400 volts; r = 0.05 ohm, and x = 0.10 ohm. 84. More frequently than for controlling the power-factor, phase control is used for controlling the voltage, that is, to maintain the receiver voltage, e, constant, or raise it with in- creasing load, i, at constant generator voltage, eo- In this case, equation (4) gives the quadrature component of current, i', required by current, i, at constant receiver vol- tage, e, and constant generator voltage, eo. Since the equation (4) of i' contains a square root, the maxi- mum value of i, that is, the maximum load which can be carried at constant voltage, e and eo, is given by equating the term under the square root to zero as eoz — er eo — e cos d , , . , ^. = ^^ = ^ (14) and the corresponding quadrature component of current, by (4), is ., ex e sin 0 ._, ^, *- = - ^ = - -^'' (15) that is, leading. From equation (14) follows as the impedance, z, which, at constant line-resistance, r, gives the maximum value of im dim _ ^. eo^ dz hence. = 2r-^ (16) eo and for this value of impedance, Zm, substituting in (14) and (15) ''mm /I ' anCl t mm a ' \^' J PHASE CONTROL 103 The maximum load, i, which can be deHvered at constant voltage, e, therefore depends upon the line impedance, and the voltages, e and eo. Since eo and e are not very different from each other, the ratio g — in equation (16) is approximately unity, and the impedance, Co z, which permits maximum load to be transmitted, is approxi- mately twice the line resistance, r, or rather slightly less. 2 ^ 2r, gives X < rV3. A relatively low line-reactance, x, so gives maximum output. In practice, a far higher reactance, x, is used, since it gives sufficient output and a lesser quadrature component of current. By substituting i = 0 in equation (4), the value of the quad- rature component of current at no-load is found as ., Veo^2^ — e^r^ — ex I 0 = 5 Z^ \/eo^ — e^ cos^ d — € sin d z This can be written in the form (18) ., 's/(eo^ — e-) -f e^ sin^ 0 — e sin 0 z and then shows that for e = eo, i'o = 0, or no quadrature com- ponent of current exists at no-load; for e > eo, i'o < 0 or nega- tive, that is, the quadrature component of current is already leading at no-load. For: e < eo, i'o > 0 or lagging, that is, the quadrature component of current i'o is lagging at no-load, be- comes zero at some load, and leading at still higher loads. The latter arrangement, e < eo, is generally used, as the quad- rature component of current passes through zero at some inter- mediate load, and so is less over the range of required load than it would be if i'o were 0 or negative. From (18) follows that the larger z, and at constant resistance r, also X, the smaller the quadrature component of current. That is, increase of the line reactance, x, reduces the quadrature current at no-load, i'o, and in the same way at load, that is, im- proves the power-factor of the circuit, and so is desirable, and the insertion of reactive coils in the line for this reason customary. 104 ALTERNATING-CURRENT PHENOMENA Increase of reactance, however, reduces the maximum output im, and too large a reactance is for this reason objectionable. Let i = ii be the load at which the quadrature component of current vanishes, i' = 0, that is, the receiver circuit has unity power- factor. Substituting i = ii, i' = 0 into equation (2) gives eo2 = (e + rixY + xHi'' (19) and, substituting (19) in (4), (18), (14), gives reactive component of current /e'^sin^g , 2ecosg^. .. , , . „ I e sin ^ ,^^. = V — 'z^~~ 2 ^^^ - ^) + C^i' - ^') 1 — ' (20) and at no-load ., \e^ sin^ 0 , 2 eii cos 0 , ~ e sin 5 ,^^. ^° =^ V~^^ + — z — + '' - "7-' ^^^^ Maximum output current je^~~2ei^~cos~f~, 77 e cos ^ ,r,n\ '- = \J- + —z— + '-^^ - "T- ^^^^ 85. Of importance in phase control for constant voltage, e, at constant eo, are the three currents ix, the power component of current at which the quadra- ture component of current vanishes: i' = 0. i,n, the maximum load which can be transmitted at con- stant voltage, e. «'o, the reactive component of current at no-load. The equation of phase control, (2), however, contains only two quantities which can be chosen: The reactance, x, which can be increased by inserting reactive coils, and the generator vol- tage, eo, which can be made anything desired, even with an existing generating station, since between e and eo practically always transformers are interposed, and their ratio can be chosen so as to correspond to any desired generator voltage, eo, as they usually are supplied with several voltage steps. Of the three quantities, z'l, im and f'o, only two can be chosen, and the constants, x and eo, derived therefrom. The third current then also follows, and if the value found for it does not suit the requirements of the problem, other values have to be tried. For instance, choosing zi as corresponding to three-fourths PHASE CONTROL 105 load, and i'o fairly small, gives very good power-factors over the whole range of load, but a relatively low value of i^, and where very great overload capacities are required, i„i may not be sufficient, and ii may have to be chosen corresponding to full-load and a higher value of i'o permitted, that is, some sacrifice made in the power-factor, in favor of overload capacity. So, for instance, the values may be chosen ii, corresponding to full-load, and required that i'o does not exceed half of full-load current; i'o < 0.5ii, and that the synchronous converter or motor can carry at least 100 per cent, overload, that is, im > 2 ii. We then can put, tm = 2 ii c and i'o = — — , . (23). and substitute (23) in (19), (22) and determine x, Co, c. 86. The variation of the reactive current, i' with the load, i, equation (4), is brought about by varying the field excitation of the receiving synchronous machine. Where the load on the synchronous machine is direct-current output, as in a motor generator and especially a converter, the most convenient way of varying the field excitation with the load is automatically, by a series field-coil traversed by the direct-current output. The field windings of converters intended for phase control — as for the supply of power to electric railways, from substations fed by a high-potential alternating-current transmission line — ■ are compound-wound, and the shunt field is adjusted for under- excitation, so as to produce at no-load the lagging current, i'o, and the series field adjusted so as to make the reactive compo- nent of current, i', disappear at the desired load, ii. In this case, however, the variation of the field excitation by the series field is directly proportional to the load, as is also the variation of i', that is, it varies from i' — i'o for i = 0, to i' = 0 for i = ii, and can be expressed by the equation (24) where , - ^ C^a) 106 ALTERNATING-CURRENT PHENOMENA is the ratio of (reactive) no-load current, i\, to (effective) non- inductive load current, ii. To maintain constant voltage, e, at constant, eo, the required variation of i' is not quite linear, and with a linear variation of i'f as given by a compound field-winding on the synchronous machine, the receiver-voltage, e, at constant impressed voltage does not remain perfectly constant, but when adjusted for the same value at no-load and at full-load, e is slightly high at inter- mediate loads, low at higher loads. It is, however, sufficiently constant for all practical purposes. Choosing then the full-load current, ii, and the no-load current, t'o = qi\, and let the reactive component of current, i' , by a compound field-winding vary as a linear function of the load, i: i' = q{i\ — i). Then, substituting z'l and i'o — qii in the equations (2) for phase control: No-load: i = 0, i' — qii] eo^ = (e + qxiiY + gn'i^. (26) Full load: ii = ii, i' = 0; eo' = (e + ni)2 + xh\ (27) From these equations (26) and (27) then calculate the required reactance, x, and the generator voltage, Bq, as: qe ±^g(i + «')-[| + Ki-«')]^ X = '-1 1^ ^—p -, (28) and from (27) or (26) the voltage, Cq. The terminal voltage at the receiving circuit then is, by equa- tion (3) : e = V^o^ — [qrii — (qr + x)i]^ — ((r — qz)i + qxii). (29) As an example is shown, in Fig. 78, the curve of receiving voltage, e, with the load, i, as abscissas, for the values: e = 400 volts at no-load and at full-load, ii = 500 amp. at full-load, power component of current, i'o = 200 amp., lagging reactive or quadrature component of current at no-load, hence q = 0.4, i' = 200 - 0.4 i, and r = 0.05 ohm. PHASE CONTROL From equation (28) then follows: X = 0.381 ± 0.165 ohm. Choosing the lower value: X = 0.216 ohm. gives, from equation (27): eo = 443.4 volts; hence 107 e = \ 196,420+ 5.76 i -0.0576 i^ - (43.2 - 0.0264 *)• For comparison is shown,, in Fig. 78, the receiving voltage, e', at the same supply voltage, €o = 443.4 volts, but without phase control, that is, with a non-inductive receiver-circuit. eso 500 ^ ^0 „ — - ■— . eo 450 g400 * 350 800 — e e — ■ — e' 200 -100 Ul ^ s < ,^ < i ^ ■^ 0 IC 0 2( X) 3 » 4( X) 5( ^ ^gI K) 7( K) 8 )0 » lo 10 00 ■^ 100 o -■ ^ <