CHAPTER XXXIV METERING OF POLYPHASE CIRCUIT 299. The power of a polyphase system or circuit is the sum of the powers of all the individual branch circuits, and the sum of the wattmeter readings of all the branch circuits thus gives the total power. Let, then, in a general polyphase system, ei, e^, e^ . . . e„ = potentials at the n terminals or supply wires of the /?-phase system. These may be represented topographically by points in a plane, as shown in Fig. 218. ,^-'-' ^^ Fig. 218. The voltage between any two terminals e^ and e^ then is: e.7fc = ei — ek ' (1) And this voltage, in any circuit connected between these two terminals, produces a current, %ik, as the current, which flows from e,- to eu through this circuit. n{n — 1) As there are pairs of terminals ei and e*, there are fly 11 — \ ) existing in a general n-phase system ^ different phases, 7l\7l "" 1 ) and there may thus be ^ different circuits, or rather sets 442 METERING OF POLYPHASE CIRCUIT 443 of circuits — since a number of circuits may and usually are con- nected between the n terminals. Consider one of these numerous circuits of the general w-phase system, that of the current /»<.• passing from ei to ek. The power of this circuit is: Pik = [ei - Ck, iik] (2) where the brackets denote the effective power, as discussed in Chapter XVI. Choosing any point ex, which may be one of the terminals, or the neutral point of the system, if such exists, or any other point. Then the voltage e^ — ek can be resolved by the parallelogram (Fig. 218) into the voltages: e< — ex and e^ — Ci, that is: ei — ek = Bi — ex -\- ex — Sk (3) hence, substituted into (2) : Pik = [{ei - xe) + {ex - Bk), iik\ = [ei - Bx, iik] + [ex - Bk, iik] (4) It is, however: [ex - Bk, iik] = [Bk — Bx, iki] (5) where iki is the current flowing from Bk to Bi, that is, the same cur- rent as iik, only considered in the reverse direction. Thus it is, substituting (5) into (4) : Pik = [e. - Bx, iik] + [Bk - Bx, iki] (6) That is, the power of any branch circuit between two terminals, Bi and Bk, is the product of the powers giving by the two potential differences Bi — Bx and Bk — Bx, of any arbitrarily chosen point Bx, with the current flowing into this branch circuit from the two terminals, Bi and Bk, that is, iik and iki, respectively. 300. The total power of the n-phase system, as the sum of the powers of all the branch circuits, then is; n n p = ^i 2^- p., 1 1 n n = Si Sa; [Bi — Bx, iik] (7) 1 1 where the double summation sign indicates that the summation is to be carried out for all values of k, from 1 to n, and for all values of i, from 1 to n. 444 ALTERNATING-CURRENT PHENOMENA As the term Ci — Cx in (7) does not contain the index k, it is the same for all values of k, thus can be taken out from the second summation sign, that is: P = 2i 1 However: e, - Sx, Xk iik 1 (8) 2*^ iik is the sum of all the currents, flowing from the termi- 1 nal gj to all the other terminals Ck (k = 1, 2 . . n), that is, it is the total current issuing from the terminal e,-, or: n ii = 1>k i-^ (9) 1 and, substituting this in 9, gives as the total power of the n-phase system : P = h [ei - €,, ii] (10) 1 That is: "T/ie total power of a general n-phase system, is the sum of the n powers, given hy the n currents ii, which issue from the n terminals Ci, with the n pote^itial differences of these terminals d against any arbitrarily chosen point Ci." "The total power of the system, no matter how many branch cir- cuits it contains, thus is measured by n wattmeters. Choosing as the point, Cx one of the n-phase circuit terminals, that is one of the phase potentials (for instance, the neutral potential of the system, where such exists), as e„, the number of terms in (10) reduces by one: P =i:i[ei- en,ir] (11) 1 That is: "The total power of a general n-phase system is measured by n — \ wattmeters, connected between one terminal e„ and the n — 1 other terminals ex" Thus for instance, a five-wire, four-phase system (Fig. 195), 5X4 in which —^ — =10 different sets of circuits are possible, is metered by 5 — 1 =4 meters. A four-wire, three-phase system is metered by 3 meters. A three-wire, three-phase system is metered by 2 meters. METERING OF POLYPHASE CIRCUIT 445 301. In a three-phase system with ungrounded neutral, that is, a three-wire, three-phase system, the common method of measuring the total power thus is, by (11), as shown in Fig. 219. Often the two meters of Fig. 219 are arranged in one structure. "ImJ -0- FiG. 219. Thus, if Fig. 220 denotes a general three-wire, three-phase sys- tem, with the voltages and currents in the three phases: El, El, Es and h, h, h counting voltages and currents in the direction indicated by the arrows in Fig. 220. •3, '3 Fig. 220. The voltages may be unequal in sizes and under unequal angles, by a distortion of the three-phase triangle, but it must be: ^1 + ^2 + Es = 0 (12) in a closed triangle. Connecting then the current coils of the two wattmeters into the lines a and h, and the voltage coils between a respectively h, and c, the two wattmeter readings are: and: [-Ei,h-h] = [Ei,h] - [Ei,h] [E,, h - h] = [E^, h] - [^3, h] (13) (14) 446 ALTERNATING-CURRENT PHENOMENA and their sum is: P = [E,, h] - [E,, h] - [Es, h] + [^3, /3]^ = [ii, ii] - [E^ + E„ h] + [^3, h] and since by (12) : El -\- Es — — E2, it is: P = [El, I,] + [E2, h] + [Es, Is] that is, the total power of the three-phase system is the sum of the individual powers of the three branch circuits. 302. In the standard polyphase wattmeter connection of the three-wire, three-phase system, Fig. 219, the voltage coils are out of phase with the current coils at non-inductive load, the one lagging, the other leading by 30°. Therefore, even in a balanced W -OD- Fig. 221. system, if the current lags, the two wattmeter coils do not read alike, as the voltmeter coil in the one lags by the angle of lag of the current plus 30°, and in the other by the angle of lag minus 30°. At 60° angle of lag, the voltage coil of the former lags 60 + 30 = 90°, and the reading becomes zero, and at more than 60° lag, the one meter reads negative, but the algebraic sum of the two meter readings still remains the total power of the circuit, the one meter reading more than the total power, while the other meter reads negative. In a balanced, or nearly balanced three-wire, three-phase sys- tem, instead of connecting the potential coils from a and b to c. Figs. 219 and 220, they are often connected from a to b. This interchanges the lagging and the leading coil, but on balanced loads leaves the same total. In this case, one voltage coil only may be used, acted upon by two current coils. That is, a single- phase wattmeter is constructed, similar to the Edison three-wire METERING OF POLYPHASE CIRCUIT 447 meter, with one current coil in the one, the other current coil in the other line, and the voltage coil connected between these two lines, as shown in Fig. 221. If there is considerable unbalancing, this latter connection gives considerable error, and the double meter has to be used. W -0)- -0- FiG. 222. In a four- wire, three-phase system, the connection of the two meters obviously becomes wrong, if current flows in the neutral, and three meters must be used. Most conveniently these are arranged with the three current coils in the three lines, and the voltage coils between these lines and the neutral, as shown in Fig. 222.