CHAPTER XXIV. SYMMETBICAIi POLYPHASE STSTEMS. 235. If all the E.M.Fs. of a polyphase system are equal in intensity, and differ from each other by the same angle of difference of phase, the system is called a symmetrical polyphase system. Hence, a symmetrical //-phase system is a system of n E.M.Fs. of equal intensity, differing from each other in phase by 1/;/ of a period: ^1 = ^ sin )8 ; €^ = £ s'\n( P — tlJL \ J ^„ = -£■ sin ( )8 — 2(n - l)ir ^) The next E.M.F. is again : ^i = /^ s\n (P — 2 w) = £ sin /S. In the polar diagram the ;/ E.M.Fs. of the symmetrical «-phase system are represented by ;/ equal vectors, follow- ing each other under equal angles. Since in symbolic writing, rotation by 1/// of a period, or angle 2ir/;/, is represented by multiplication with: cos h J sm = c , the E.M.Fs. of the symmetrical polyphase system are: £• §236] SYMMETRICAL POLYPHASE SYSTEMS, 351 E I cos ^^ + J sin ^^ ) = ^ c ; \ u n ) ^1 cos— ^ +ysin — ^ ) = -fi'c^; ^ / 2 (« - 1) TT , . . 2 (« - 1) ^\ ^ ^.1 j& ( cos — ^» ^ hy sm — ^^ ^— ] =E c" \ The next E.M.F. is again : ^ ( cos 2 «• +>sin 2 «•) = ^ c» = ^. Hence, it is ty o _ c = cos h y sin = Vl. n n Or in other words : In a symmetrical «-phase system any E.M.F. of the system is expressed by : f}E\ where : c = Vl. 236. Substituting now for ;/ different values, we get the different symmetrical polyphase systems, represented by eE, O 9 where, c = vl = cos Yj sm — . n • n 1.) « = 1 c = 1 eE = E, the ordinary single-phase system. 2.) n = 2 c = - 1 c'^fi- = ^ and - ^. Since — ^ is the return of E, ;/ = 2 gives again the single-phase system. Q\« o 2'n-,..2?r — l+y V3 6.) n = 3 c = cos h/sm = -^-^ ^ 3 ^^ 3 2 c = 352 AL TERNA TING-CURRENT PHENOMENA, [§ 237 The three E.M.Fs. of the three-phase system are : ^E^E, Z^Lkn^lE, - ^ ->^ E. Consequently the three-phase system is the lowest sym- metrical polyphase system. 4.) « = 4, € = cos ^^ + y sin "-^ =/, e^ = — 1, e* = — / 4 4 The four E.M.Fs. of the four-phase system are: €' = E, jE, — E, —jE. They are in pairs opposite to each other : E and —E\jE and —jE, Hence can be produced by two coils in quadrature with each other, analogous as the two-phase system, or ordinary alternating-current system, can be produced by one coil. Thus the symmetrical quarter-phase system is a four- phase system. Higher systems, as the quarter-phase or four-phase sys- tem, have not been used, and are of little practical interest. 237. A characteristic feature of the symmetrical n- phase system is that under certain conditions it can pro- duce a M.M.F. of constant intensity. If n equal magnetizing coils act upon a point under equal angular displacements in space, and are excited by the n E.M.Fs. of a symmetrical //-phase system, a M.M.F. of constant intensity is produced at this point, whose direction revolves synchronously with uniform velocity. Let, «' = number of turns of each magnetizing coil. E=^ effective value of impressed E.M.F. / = effective value of current. Hence, (F =///= effective M.M.F. of one of the magnetizing coils- $237] SYMMETRICAL POLYPHASE SYSTEMS, 853 Then the instantaneous value of the M.M.F. of the coil acting in the direction 2 tt//;/ is : //= $FV2 sin /'i8-?^'\ = ///V2sin/'i8-?^\ The two rectangular components of this M.M.F. are: and fi' .. 2iri =// cos n = /i'/V2cos 2 7r/ // A" ^. . 27r/ = // Sin -¥) // = ///V2sin?Ji'sin/')8-?^A Hence the M.M.F. of this coil can be expressed by the symbolic formula : /z = n'/^2sin(fi - ?^\ /cos^+ysin^Y \ ^ A '' « y Thus the total or resultant M.M.F. of the ;/ coils dis- placed under the n equal angles is : 1 1 \ « /\ '' « / or, expanded : /=;//V2 \ sin/J^tfcos^ — +ysin?''-*cos?^^- cos P 2lL sm — cos [-jsirr ) J . It is, however : cos"^ f-y sm cos = J [ 1 + COS f- / sm // // // V // // 354 ALTERNATING-CURRENT PHENOMENA. [$237 . 27r/ 27r/ , . . o^tt/ j I ^ Aiiri . . 47r/\ sm COS h / sin-* = ^ ( 1 — cos / sin \ n n n 2\ n n j and, since: as the sum of all the roots of Vl, it is, /= "AI^ (sin p+J cos /3). or, /=!L^(sin)3+ycos/3) V2 = !^(sin)3 +ycos/3); the symbolic expression of the M.M.F. produced by the n circuits of the symmetrical //-phase system, when exciting n equal magnetizing coils displaced in space under equal angles. The absolute value of this M.M.F. is : V'2 V2 ^ Hence constant and equal h/V2 times the effect've M.M.F. of each coil or ;//2 times the maximum M.M.F. of each coil. The phase of the resultant M.M.F. at the time repre- sented by the angle /3 is : tan a> = cot P ; That is, the M.M.F. produced by a symmetrical «-phase system revolves with constant intensity : F = ,— » V2 and constant speed, in synchronism with the frequency of the system ; and, if the reluctance of the magnetic circuit §238] SYMMETRICAL POLYPHASE SYSTEMS. 355 is constant, the magnetism revolves with constant intensity and constant speed also, at the point acted upon symmetri- cally by the ;/ M.M.Fs. of the //-phase system. This is a characteristic feature of the symmetrical poly- phase system. max 238. In the three-phase system, « = 3, F= 1.5 $F where SF^^^ is the maximum M.M.F. of each of the magne- tizing coils. In a symmetrical quarter-phase system, « = 4, F= 2 ^fHajcf where iF^^ is the maximum M.M.F. of each of the four magnetizing coils, or, if only two coils are used, since the four-phase M.M.F. are opposite in phase by two, /^ = ^HMxy where SF,^^ is the maximum M.M.F. of each of the two magnetizing coils of the quarter-phase system. While the quarter-phase system, consisting of two E.M.Fs. displaced by one-quarter of a period, is by its nature an unsymmetrical system, it shares a number of features — as, for instance, the ability of producing a constant result- ant M.M.F. — with the symmetrical system, and may be considered as one-half of a symmetrical four-phase system. Such systems, of an even number of phases, consisting of one-half of a symmetrical system, are called hemisym- metrical systems. 356 AL TERNA TING-CURRENT PHENOMENA. [§ 239