CHAPTER XXV. BAIiANCED AND UNBAXiANCBD POLYPHASE SYSTEMa 239. If an alternating E.M.F. : ^ = ^ V2 sin j8, produces a current : /• = /V2sin()8-^), where w is the angle of lag, the power is : / = ^/ = 2 Elsm )8 sin 03 - u») = -£'/(cos w — sin (2 /3 — w)), and the average value of power : 7^= EI cos w. Substituting this, the instantaneous value of power is found as : \^ cos w J 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 -= €t. If the angle of lag w = it is : / = /^ (1 - sin 2 )3) ; hence the flow of power varies between zero and 2/*, where P is the average flow of energy or the effective power of the circuit. 1240} BALANCED POLYPHASE SYSTEMS, 857 If the current lags or leads the E.M.F. by angle a> the power varies between p(\--±J\ and /Yl+-l_V y cos w y y cos w j 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 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. If ci = 90°, it is : / = EIcos2p\ that is, the effective power : /* = 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. 240. If in a polyphase system ^i« ^2> ^8> • . . . = instantaneous values of E.M.F. ; hi h} fsj • • • • = instantaneous values of current pro- duced thereby ; the total flow of power in the system is : The average flow of power is : /* = iSi/i cos oil + E2A cos 0)2 + ... . The polyphase system is called a balanced system, if the flow of energy : / = ^ih + ^2h + ^zH + . . . . is constant, and it is called an unbalanced system if the flow of energy varies periodically, as in the single-phase sys- tem ; and the ratio of the minimum value to the maximum value of power is called the balance factor of ttie system. 358 ALTERNATIXG-CURRENT PHENOMENA. [§§241,242 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 = — 1, if the phase displace- ment is 90° — that is, the circuit is wattless. 241. Obviously, in a polyphase systeiji 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. 242. 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 ^i=^V2sin)3, and /i = /V2 sin ()8 - u») ; e^ = E V2 sin ()3 - 120), i^ = I V2 sin (^ - = nE I = I\ or constant. 243. 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 interrriediate 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- 860 ALTERNATING-CURRENT PHENOMENA. [§ wire system the feature that the potential difference be- tween 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 GO*'. Thus in the secondary circuit 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 minimum value of 87 per cent 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 .333. 244. In Figs. 167 to 174 are shown the E.M.Fs. as e and currents as i in drawn lines, and the power as / in dotted lines, for : Fig* 167. SinglB'phaaB System on Non^inductloe Load. §244] BALANCED POLYPHASE SYSTEMS, 861 Fig. 168. &ngl9-i>lHU9 Syttem on inductlue Load of 90* Lag. Fig. 169. Quart€r-ph€is9 System on Non^inductioe Loot. Fig. 170. Quarter-phase System on inductive Load of 60' Lag, AL TEKNA nXC-CUI^RENT PHENOMENA. [ % 244 « 245, 246] BALANCED POL YPIIASE SYSTEMS. 245. The flow of power 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 circuit, balance factor zero to a non-inductive single-phase circuit, balance factor + 1 to a balanced polyphase system. 246. In polar coordinates, the flow of power of an alternating-current system is represented by using the in- stantaneous flow of power as radius vector, with the angle p 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 cun'e, having the zero point as quadruple point. In the monocyclic systems the zero point is quadruple nodal point ; in the polycyclic system quadruple isolated point. Thus these curves are sextics. 364 ALTERNATING-CURRENT PHENOMENA, [§247 Since the flow of power in any single-phase branch of the alternating-current system can be represented by a sine wave of double frequency : V cos w J the total flow of power of the system as derived by the addition of the powers of the branch circuits can be rep- resented in the form : / = /'(l + esin(2j8-^)) This is a wave of double frequency also, with c as ampli- tude of fluctuation of power. This is the equation of the power characteristics of the system in polar coordinates. 247. To derive the equation in rectangular coordinates we introduce a substitution which revolves the system of coordinates by an angle w/2, so as to make the symmetry axes of the power characteristic of the coordinate axes. ••"(^-f)-f hence, «ii(3^ - ») - 2sin^^ - f Vosp - |"]_ Jiij, substituted, ■\/lF^f = P\ 1 + 2ixy or, expanded : (^' + yy - J''' {^' + f + 2 € xyf = 0, the sextic equation of the power characteristic. Introducing : tf = (1 4- c) y = maximum value of power, ^ = (1 — c) y = minimum value of power ; §247] BALANCED POLYPHASE SYSTEMS, 365 It IS F ^, m hence, substituted, and expanded : the equation of the power characteristic, with the main power axes a and ^, and the balance factor: bfa. It is thus : Single-phase non-inductive circuit ; / = -P (1 -f- sin 2 ^), Single-phase circuit, 60° lag ; / = /* (1 + 2 sin 2 <^), ^ = - /> a = + 3P Single-phase circuit, 90° lag : / = ^ / sin 2 <^, b •= — E ly a = J^ EI — ix" + /)» - (^ /)» {x' - /)S ^ / ^ = - 1. Three-phase non-inductive circuit : p = Py b = 1, a = 1 x^ +/-/>•-» = (): circle, b / a = + 1, Three-phase circuit, 60°^lag :/ = y, b = 1, a = 1 x^ + / _ >2 = : circle, b / a = + 1. Quarter-phase non-inductive circuit :/ = /*, /^ = 1, a = 1 x'+y- P^ = 0: circle. /^/« = + 1. Quarter-phase circuit, 60° lag : p = p^ b = 1, a = 1 jt-2 +/_/>» = : circle, b/a = + 1. 366 ALTERNA TING-CURRENT PHENOMENA. [§248 Inverted three-phase non-inductive circuit : = i/> a^\P p-p{^^f^y ^=i Inverted three-phase circuit 60° lag :/ = /^(l + sin 2 ^), ^ = 0, a = 2P a and b are called the main power axes of the alternating- current system, and the ratio b / a is the balance factor of the system. •-*««««»*i FIga. 175 and 176. 248. As seen, the flow of power of an alternating-cur- rent system is completely characterized by its two main power axes a and bJ The power characteristics in polar coordinates, corre- §248] BALANCED POLYPHASE SYSTEM, 367 spending to the Figs. 167, 188, 173, and 174 are shown in Figs. 175, 176, 177, and 178. f7g«. 777 onrf 778. The balanced quarter-phase and three-phase systems give as polar characteristics concentric circles. 368 ALTERNATING-CURRENT PHENOMENA, [§§240, 26Q