CHAPTER XXX BALANCED AND UNBALANCED POLYPHASE SYSTEMS 273. If an alternating e.m.f., e = E\/2 sin ^, produces a current, i = /V2 sin (/3 - &), where Q is the angle of lag, the power is p = ei = 2 EI sin (3 sin (/3 - d) = EI (cos 0 - cos (2 /3 - e)), and the average value of power, P = EI cos e. Substituting this, the instantaneous value of power is found as cos (2/3 - e)^ _ P A _ cos(2)£J- d)\ \ cos 0 / ' Hence the power, or the flow of energy, in an ordinary single- phase, alternating-current circuit is fluctuating, and varies with twice the frequency of e.m.f. and current, unlike the power of a continuous-current circuit, which is constant, p = ei. If the angle of lag, ^ = 9, it is, p - P(l - cos 2/3); hence the flow of energy varies between zero and 2 P, where P is the average flow of energy or the effective power of the circuit. If the current lags or leads the e.m.f. by angle d, the power varies between P(l ^)andP(l +-^), \ cos 6/ \ cos 6/ that is, becomes negative for a certain part of each half-wave. That is, for a time during each half-wave, energy flows back into 405 406 ALTERNATING-CURRENT PHENOMENA the generator, while during the other part of the half-wave the generator sends out energy, and the difference between both is the effective power of the circuit. lid = 90°, it is p = — EI sin 2 (3; that is, the effective power P = 0, and the energy flows to and fro between generator and receiving circuit. Under any circumstances, however, the flow of energy in the single-phase system is fluctuating, at least between zero and a maximum value, frequently even reversing. 274. If in a polyphase system ei, 62, 63, .... = instantaneous values of e.m.f.; ii, *2, is, .... — instantaneous values of current pro- duced thereby, the total power in the system is p = eiii + 6-212 + 63(3 + . . . , The average power is P = EJi cos ^1 + E2I2 cos ^2 + . . . . The polyphase system is called a balanced system, if the flow of energy p = eiii + 62^2 + esis + . . . . is constant, and it is called an unbalanced system if the flow of energy varies periodically, as in the single-phase system; and the ratio of the minimum value to the maximum value of power is called the halance-J actor of the system. Hence in a single-phase system on non-inductive circuit, that is, at no-phase displacement, the balance-factor is zero; and it is negative in a single-phase system with lagging or leading current, and becomes equal to — 1 if the phase displace- ment is 90° — that is, the circuit is wattless. 275. Obviously, in a polyphase system the balance of the system is a function of the distribution of load between the different branch circuits. A balanced system in particular is called a polyphase system, whose flow of energy is constant, if all the circuits are loaded equally with a load of the same character^ that is, the same phase displacement. POLYPHASE SYSTEMS 407 276. All the symmetrical systems from the three-phase system upward are balanced systems. Many unsymmetrical systems are balanced systems also. 1. Three-phase system: Let ei = E v^ sin ^, and ii = I \/2 sin (/? - 6), 62 = ^ \/2 sin {^ - 120), 12 = I V2 sin {(3 - 9 - 12C), 63 = E \/2 sin (/3 - 240), is = I V2 sin (/3 - ^ - 240), be the e.m.fs. of a three-phase system and the currents produced thereby. Then the total power is p = 2 EI {sin 13 sin (j8 - ^) + sin (jS - 120) sin (^ - 6 - 120) + sin (iS - 240) sin (/3 - ^ - 240) } = S EI cos 9 = P, or constant. Hence the symmetrical three-phase system is a balanced system. 2. Quarter-phase system: Let ei = E \/2 sin /S, ii = I \/2 sin (^ - 9), 62 = E \/2 cos /3, ?*2 = I \/2 cos (/3 - 9) be the e.m.fs. of the quarter-phase system, and the currents produced thereby. This is an unsymmetrical system, but the instantaneous value of power is p = 2 EI {sin i8 sin (^ - 9) + cos /S cos (/3 - 0) } = 2 EI cos 9 = P, or constant. Hence the quarter-phase system is an unsymmetrical balanced system. 3. The symmetrical n-phase system, with equal load and equal phase-displacement in all n branches, is a balanced system. For, let e.- = E\/2 sin I (3 ^j = e.m.f . ; ii = /\/2 sin 1^ — 9 j = current; 408 ALTERNATING-CURRENT PHENOMENA the instantaneous value of power is p = Si Biii 1 = 2 EI h sin (p - ^~) sin (i3 - 0 ^) ' " " / 4 7rA 1 = £7 I Si cos e - Si cos (2^-9 -j ; or p = nEI cos ^ = P, or constant. 277. An unbalanced polyphase system is the so-called inverted three-phase system, derived from two branches of a three-phase system by transformation by means of two transformers, whose secondaries are connected in opposite direction with respect to their primaries. Such a system takes an intermediate position between the Edison three-wire system and the three-phase system. It shares with the latter the polyphase feature, and with the Edison three-wire system the feature that the potential differ- ence between the outside wires is higher than between middle wire and outside wire. By such a pair of transformers the two primary e.m.fs. of 120° displacement of phase are transformed into two secondary e.m.fs., differing from each other by 60°. Thus in the secondarycircuit the difference of potential between the outside wires is V3 times the difference of potential between middle wire and outside wire. At equal load on the two branches, the three currents are equal, and differ from each other by 120°, that is, have the same relative proportion as in a three-phase system. If the load on one branch is maintained constant, while the load of the other branch is reduced from equality with that in the first branch down to zero, the current in the middle wire first decreases, reaches a V3 minimum value of -^ = 0.866 of its original value, and then increases again, reaching at no-load the same value as at full-load. The balance factor of the inverted three-phase system on non- inductive load is 0.333. 278. In Figs. 196 to 203 are shown the e.m.fs., as e and currents as i in full hues, and the power as y in dotted lines, for balance- factor, 0; balance-factor,— 0.333; balance-factor, -1- 1; balance- factor, -}- 1 ; balance-factor, + 1 ; balance-factor, + 1 ; balance- factor, -\- 0.333, and balance-factor, 0. POLYPHASE SYSTEMS 409 279. The flow of energy in an alternating-current system is a most important and characteristic feature of the system, and by its nature the systems may be classified into: Monocyclic systems, or systems with a balance-factor zero or negative. Polycyclic systems, with a positive balance-factor. Balance-factor —1 corresponds to a wattless single-phase circuit, balance-factor zero to a non-inductive single-phase circuit, balance-factor +1 to a balanced polyphase system. 280. In polar coordinates the flow of energy of an alternating current system is represented by using the instantaneous value of power as radius vector, with the angle, /3, corresponding to the time as amplitude, one complete period being represented by one revolution. In this way the power of an alternating-current system is represented by a closed symmetrical curve, having the zero point as quadruple point. In the monocyclic systems the zero point is quadruple nodal point; in the polycyclic systems quadruple isolated point. Thus these curves are sextics. Since the flow of energy in any single-phase branch of the alternating-current system can be represented by a sine wave of double frequency, sin (2 0 - ey V ~ ^ \ "^ cos 0 / ' the total flow of energy of the system as derived by the addition of the powers of the branch circuits can be represented in the form p = P(l +6 sin (2/3 - do)). This is a wave of double frequency also, with e as amplitude of fluctuation of power. This is the equation of the power characteristics of the system in polar coordinates. 287. To derive the equation in rectangular coordinates we introduce a substitution which revolves the system of coordinates by an angle, -^ , so as to make the symmetry axes of the power characteristic the coordinate axes. V = Vx^ + yl 410 ALTERNATING-CURRENT PHENOMENA ta„(,-t) = ^; hence, sin (2 ^- 0o) = 2 sin (p - ~) cos (/3 - |") = ^^"2, substituted, or, expanded, (3.2 _^ ^^2)3 _ p2(3.2 _^ ^2 _|_ 2 ea:y)2 = 0, the sextic equation of the power characteristic. Introducing a = (l+e)P = maximum value of power, h = {\ — t) P = minimum value of power; we have a + h P = 2 ' a — h a + 6' hence, substituted, and expanded, (^2 + 1/2)3 _ 1 {ct(a: + yY + h{x - yY]^ = 0, the equation of the power characteristic, with the main power axes, a and 6, and the balance-factor, -• a It is thus: Single-phase, non-inductive circuit, p = P (1 + sin 2 0), fc =0, a = 2P, (^2 4. ,^2)3 _ p2(3. + yY = 0, ^ = 0. Single-phase circuit, 60° lag: p = P (1 + 2 sin 2 ^), 6 = - P, a = + 3 P, (a;2 _|_ ^2)3 _ p2 (3.2 ^ ^^2 _^ 4 ^yY = 0, - = - o* Single-phase circuit, 90° lag: p = EI sin 2 0, h = - EI, a = -]- EI, (x2 + 2/')' - 4 {Eiyx^y^ ^ = - 1. POLYPHASE SYSTEMS Three-phase non-inductive circuit, p = P, b = 1, a = 1, 411 ^2 _^y2 _ po ^ 0, circle. = + 1. a \ e Fig. 196. — Single-phase, non-inductive circuit. Fig. 197.— Single-phase, 60° lag. Fig. 198.— Quarter-phase, non-inductive circuit. Three-phase circuit, 60° lag, p = P, b = I, a = I, a;2 _|_ ^2 _ p2 ^ 0, circle. - = -f 1. 412 ALTERNATING-CURRENT PHENOMENA Quarter-phase non-inductive circuit, 'p = P, h = \, a = \, x^ -f- w^ — P^ = 0, circle. - = + 1. Fig. 199.— Quarter-phase, 60° lag. e p e P e Fig. 200. — Three-phase, non-inductive circuit. Fto. 201.— Three-phase, 60° lag. Quarter-phase circuit, 60° lag, p=P, 6 = l,a = l, a;2 _|_ ^2 _ p2 _ 0 circle. - = -h 1. POLYPHASE SYSTEMS 413 Fig. 202. — Inverted three-phase, non-inductive circuit. Fig. 203. — Inverted three-phase, 60° lag. 414 ALTERNATING-CURRENT PHENOMENA Inverted three-phase non-inductive circuit, / sin 2 e\ , 1 p 3 {x\+ yY - P' {x' + if + xrjr =0. ^— + |- Inverted three-phase circuit 60° lag, p = P(l -f- sin 2 0), a = 2P, & = 0, Figs. 204 and 205.— Power Figs. 206 and 207.— Power characteristic of single-phase characteristic of inverted three- system, at 0° and 60° lag. phase system, at 0°and 60° lag. a and b are called the main power axes of the alternating-cur- rent system, and the ratio, — , is the balance-factor of the system. 282. As seen, the flow of energy of an alternating-current sys- tem is completely characterized by its two main power axes, a and 6. The power characteristics in polar coordinates, corresponding to the Figs. 196, 197, 202 and 203 are shown in Figs. 204, 205, 206 and 207. The balanced quarter-phase and three-phase systems give as polar characteristics concentric circles.