CHAPTER XXIX. TRANSFORMATION OF POLYPHASE SYSTEMS. 283. In transforming a polyphase system into another polyphase system, it is obvious that the primary system must have the same flow of power as the secondary system, neglecting losses in transformation, and that consequently a balanced system will be transformed again in a balanced system, and an unbalanced system into an unbalanced sys- tem of the same balance factor, since the transformer is an apparatus not able to store energy, and thereby to change the nature of the flow of power. The energy stored as magnetism, amounts in a well-designed transformer only to a very small percentage of the total energy. This shows the futility of producing symmetrical balanced polyphase systems by transformation from the unbalanced single-phase system without additional apparatus able to store energy efficiently, as revolving machinery. Since any E.M.F. can be resolved into, or produced by, two components of given directions, the E.M.Fs. of any polyphase system can be resolved into components or pro- duced from components of two given directions. This en- ables the transformation of any polyphase system into any other polyphase system of the same balance factor by two transformers only. 284. Let Elt E2, Ez . . . . be the E.M.Fs. of the primary system which shall be transformed into — E{, £2', £s' . . . . the E.M.Fs. of the secondary system. Choosing two magnetic fluxes, <£ and <£, of different TRANSFORMATION OF POLYPHASE SYSTEMS, 461 phases, as magnetic circuits of the two transformers, which induce the E.M.Fs., e and ?, per turn, by the law of paral- lelogram the E.M.Fs., Elf E^, . . . . can be dissolved into two components, El and Elt E^ and Ez, .... of the phases* "e and J. Then, - E!, £2, • • ' • are the counter E.M.Fs. which have to be- induced in the primary circuits of the first transformer;. Ev E2, .... the counter E.M.F.'s which have to be in- duced in the primary circuits of the second transformer.. hence EI 1 7, £2 1 J . . . . are the numbers of turns of the primary coils of the first transformer. Analogously EI /T £2 IT . . . . are the number of turns of the primary coils in the second transformer. In the same manner as the E.M.Fs. of the primary system have been resolved into components in phase with J and FJ the E.M.Fs. of the secondary system, E-^> E^, .... are produced from components, E-f and E^, E£ and EJ, .... in phase with ~e and J, and give as numbers of second ary turns, — £il / J, £2l /?»•••• in the first transformer ; EI 1 7, EZ / F, .... in the second transformer. That means each of the two transformers m and m con- tains in general primary turns of each of the primary phases, and secondary turns of each of the secondary phases. Loading now the secondary polyphase system in any desired manner, corresponding to the secondary cur- rents, primary currents will flow in such a manner that the total flow of power in the primary polyphase system is the 4j^ ALTERNATING-CURRENT PHENOMENA. same as the total flow of power in the secondary system, plus the loss of power in the transformers. 285. As an instance may be considered the transforma- tion of the symmetrical balanced three-phase system E sin ft, E sin (ft — 120), E sin (ft — 240), in an unsymmetrical balanced quarter-phase system : E' sin ft, E' sin (ft — 90). Let the magnetic flux of the two transformers be (/> cos £ and cos (ft — 90). Then the E.M.Fs. induced per turn in the transformers e sin ft and e sin (ft — 90) ; hence, in the primary circuit the first phase, E sin ft, will give, in the first transformer, E/e primary turns; in the second transformer, 0 primary turns. The second phase, E sin (ft — 120), will give, in the first transformer, — E / 2 e primary turns; in the second E x ~\/3 transformer, — — primary turns. 2 e The third phase, E sin (ft — 240), will give, in the first transformer, — E /le primary turns; in the second trans- former, — primary turns. 2 e In the secondary circuit the first phase E' sin ft will give in the first transformer: E' / e secondary turns; in the second transformer : 0 secondary turns. The second phase : E' sin (ft — 90) will give in the first transformer : 0 secondary turns ; in the second transformer, E' I e secondary turns. Or, if : E = 5,000 E' = 100, e = 10. TRANSFORMATION OF POLYPHASE SYSTEMS. 463 PRIMARY. 1st. 2d. SECONDARY. 3d. 1st. 2d. Phase. first transformer second transformer + 500 0 - 250 - 250 4- 433 - 433 10 0 0 10 turns. That means : Any balanced polyphase system *.jm be transformed by two transformers only, without storage of energy, into any other balanced polyphase system. 286. Some of the more common methods of transfor- mation between polyphase systems are : Fig. 799. 1. The delta -Y connection of transformers between three-phase systems, shown in Fig. 199. One side of the transformers is connected in delta, the other in Y. This arrangement becomes necessary for feeding four wires rwi nnr V Fig. 200. three-phase secondary distributions. The Y connection of the secondary allows to bring out a neutral wire, while the delta connection of the primary maintains the balance be- tween the phases at unequal distribution of load. 464 ALTERNA TING-CURRENT PHENOMENA. 2. The L connection of transformers between three-phase systems, consisting in using two sides of the triangle only, as shown in Fig. 200. This arrangement has the disadvan- tage of transforming one phase by two transformers in series, hence is less efficient, and is liable to unbalance the system by the internal impedance of the transformers. Fig. 201. 3. The main and teaser, or T connection of trans- formers between three-phase systems, as shown in Fig. 201. V3 One of the two transformers is wound for ~-~- times the voltage of the other (the altitude of the equilateral triangle), and connected with one of its ends to the center of the Fig. 202. other transformer. From the point £ inside of the teaser transformer, a neutral wire can be brought out in this con- nection. 4. The monocyclic connection, transforming between three-phase and inverted three-phase or polyphase mono- cycle, by two transformers, the secondary of one being reversed regarding its primary, as shown in Fig. 202. TRANSFORMATION OF POLYPHASE SYSTEMS. 465 5. The L connection for transformation between quar- ter-phase and three-phase as described in the instance, para- graph 257. 6. The T connection of transformation between quarter- phase and three-phase, as shown in Fig. 203. The quar- ter-phase side of the transformers contains two equal and Fig. 203. independent (or interlinked) coils, the three-phase side two Vs coils with the ratio of turns 1 -=- — ^ connected in T. 7. The double delta connection of transformation from three-phase to six-phase, shown in Fig. 204. Three trans- formers, with two secondary coils each, are used, one set of Fig 204. secondary coils connected in delta, the other set in delta also, but with reversed terminals, so as to give a reversed E.M.F. triangle. These E.M.F.'s thus give topographically a six-cornered star. 466 AL TERN A TING-CURRENT PHENOMENA. 8. The double Y connection of transformation from three-phase to six-phase, shown in Fig. 205. It is analo- gous to (7), the delta connection merely being replaced by the Y connection. The neutrals of the two F's may be connected together and to an external neutral if desired. 9. The double T connection of transformation from Fig. 205. three-phase to six-phase, shown in Fig. 206. Two trans- formers are used with two secondary coils which are T con- nected, but one with reversed terminals. This method allows a secondary neutral also to be brought out. 287. Transformation with a change of the balance factor of the system is possible only by means of apparatus \ \ •/ / y / \ y 2' v ' Fig. 208. able to store energy, since the difference of power between primary and secondary circuit has to be stored at the time when the secondary power is below the primary, and re- turned during the time when the primary power is below TRANSPORMATION OF POLYPHASE SYSTEMS. 467 the secondary. The most efficient storing device of electric energy is mechanical momentum in revolving machinery. It has, however, the disadvantage of requiring attendance ; fairly efficient also are capacities and inductances, but, as a rule, have the disadvantage not to give constant potential. 468 ALTERNATING-CURRENT PHENOMENA.