CHAPTER XXXII TRANSFORMATION OF POLYPHASE SYSTEMS 289. In transforming one polyphase system into another poly- phase system, it is obvious that the primary system must have the same flow of energy as the secondary system, neglecting losses in transformation, and that consequently a balanced sys- tem will be transformed again into a balanced system, and an unbalanced system into an unbalanced system of the same bal- ance-factor, since the transformer is not able to store energy, and thereby to change the nature of the flow of energy. The energy stored as magnetism amounts in a well-designed trans- former 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 effi- ciently, as revolving machinery, etc. Since any e.m.f. can be resolved into, or produced by, two components of given directions, the e.m.f. of any polyphase sys- tem can be resolved into components or produced from compon- ents of two given directions. This enables the transformation of any polyphase system into any other polyphase system of the same balance-factor by two transformers only. 290. Let El, E^, E3 .... he the e.m.fs. of the primary sys- tem which shall be transformed into E\, E'2, E'i .... the e.m.fs. of the secondary system. Choosing two magnetic fluxes, 4> and $, of different phases, as magnetic circuits of the two transformers, which generate the e m.fs., e and e, per turn, by the law of parallelogram the e.m.fs., El, E2, .... can be resolved into two components, Ei and El, E2 and E2, .... of the phases, e and e. Theri_ El, Ei, .... are the counter e.m.fs. which have to be gen- _ erated in the primary circuits of the first transformer; El, E2, .... the counter e.m.fs. which have to be generated in the primary circuits of the second transformer. 422 Hence_ El E2 TRANSFORMATION OF POLYPHASE SYSTEMS 423 . . are the numbers of turns of the primary coils of e e the first transformer. _ Analogously -^-> -^ . . . . are the numbers of turns of the primary coils in e e the second transformer. In the same manner as the e.m.fs. of the primary system have been resolved into components in phase with e and e, the e.m.fs. of the secondary system, E\, E\, ._. . . are produced from com- ponents, £"], and E'\, E'2, and E'2 ■ ■ . • in phase with e and e, and give as numbers of secondary turns — -^> -=- ... .in the first transformer; e e -^' -5- ... .in the second transformer. e e That means each of the two transformers, m and m, contains in general primary turns of each of the primarj^ phases, and second- ary turns of each of the secondary phases. Loading now the secondary polyphase system in any desired manner, correspond- ing to the secondary currents, primary currents will exist in such a manner that the total flow of energy in the primary polyphase system is the same as the total flow of energy in the secondary system, plus the loss of power in the transformers. 291. As an instance may be considered the transformation of the symmetrical balanced three-phase system, E sin /3, E sin (^ - 120), E sin (^ - 240), into an unsymmetrical balanced quarter-phase system, E' sin /3, E' sin (/3 - 90). Let the magnetic flux of the two transformers be chosen in quad- rature $ cos iS and $ cos (/S — 90). Then the e.m.fs. generated per turn in the transformers are e sin /3 and e sin (/3 — 90) ; 424 ALTERNATING-CURRENT PHENOMENA hence, in the primary circuit the first phase, E sin /3, will give, in E the first transformer, — primary turns ; m the second transformer, 0 primary turns. The second phase, E sin (j3 — 120), will give, in the first trans- former, -^ — primary turns ; in the second transformer, „ primary turns. The third phase, E sin (|8 — 240), will give, in the first trans- former, -^ — primary turns ; in the second transformer, ^ ■ primary turns. In the secondary circuit the first phase, E' sin /3, will give in E' the first transformer: — secondary turns; in the second trans- e "^ ' former: 0 secondary turns. The second phase: E' sin ((8 — 90) will give in the first trans- E' former: 0 secondary turns; in the second transformer, — second- ary turns. Or, if E = 5000, E' = 100, e = 10. Primary Secondary 1st. 2d. 3d. 1st. 2d. First transformer +500 -250 -250 10 0 Second transformer 0 +433 -433 0 10 turns. Using autotransformer connection in the three-phase primaries of the first transformer, that is, using as coils of the second and the third phase the two halves of the coil of the first phase, this gives the well known T-connection of three-phase-quarter-phase transformation. That means : Any balanced polyphase system can he transformed hy two transformers only, without storage of energy, into any other balanced polyphase system. Or more generally stated: Any polyphase system can be transformed hy two transformers only, without storage of energy, into any other polyphase system of the same balance factor. TRANSFORMATION OF POLYPHASE SYSTEMS 425 292. Some of the more common methods of transformation between polyphase systems are: 1. The delta-Y connection of transformers between three-phase systems, shown in Fig. 210. One side of the transformers is connected in delta, the other in Y. This arrangement becomes necessary for feeding four-wire three-phase secondary distribu- tions. The Y connection of the secondary allows the bringing out of a neutral wire, while the delta connection of the primary maintains the balance, in regard to the voltage between the phases at unequal distribution of load. The delta-Y connection of step-up transformers is frequently used in long-distance transmissions, to allow grounding of the high-potential neutral. Under certain conditions — which there- fore have to be guarded against — it is liable to induce excessive voltages by resonance with the line capacity. J_I_i P^^lIM nm Fig. 210. The reverse thereof, or the Y-delta connection, is undesirable on unbalanced load, since it gives what has been called a "float- ing neutral;" the three primary Y voltages do not remain even approximately constant, at unequal distribution of load on the secondary delta, but the primary voltage corresponding to the heavier loaded secondary, and, therefore, also the corresponding secondary voltage, collapses. Thereby the common connection of the primary shifts toward one corner of the e.m.f. triangle, away from the center of the triangle, and may even fall outside of the triangle. As result thereof the secondary triangle becomes very greatly distorted even at moderate inequality of load, and the system thus loses all ability to maintain constant voltage at unequal distribution of load, that is, becomes inoperative. In high-potential systems in this case excessive voltages may be induced by resonance with the line capacity. For instance, if only one phase of the secondary triangle is 426 ALTERNATING-CURRENT PHENOMENA loaded, the other two unloaded, the primary current of the loaded phase must return over the other two transformers, which, at open secondaries, act as very high reactances, thus limiting the current and consuming practically all the voltage, and the loaded primary, and thus its secondary, receive practically no voltage. Y-delta connection is satisfactory if the secondary load is balanced, as induction — or synchronous motors, or if the primary neutral is connected with the generator neutral or the secondary neutral of step-up transformers in which the primaries are con- nected in delta, and the unbalanced current can return over the neutral. If with Y-delta connection, in addition to an un- balanced load, the secondary carries polyphase motors, the motors take different currents in the different phases, so that the total current is approximately the same in all three phases. That is, the motors act as phase converters, and so partially restore the balance of the system. 2. The delta-delta connection of transformers between three- phase systems, in which primaries as well as secondaries are con- nected in the same manner as the primaries in Fig. 210. Since in this system each phase is transformed by a separate transformer, the voltages of the system remain balanced even at unbalanced load, within the limits of voltage variation due to the internal self-inductive impedance (or short-circuit impedance) of the transformers — which is small, while the exciting impedance (or open-circuit impedance) of the transformers, which causes the unbalancing in the Y-delta connection above discussed is enormous. 3. Y-Y connection of transformers between three-phase sys- tems. Primaries and secondaries connected as the secondaries in Fig. 210. In this case, if the neutral is not fixed by connection with a fixed neutral, either directly or by grounding it, the neutral also is floating, and so abnormal voltages may be produced between the lines and the neutral, without appearing in the voltages be- tween the lines, and may lead to disruptive effects, or to over- heating of the transformers, so that this connection is not an entirely safe one. Where in transformer connections in polyphase sj^stems, a neutral or common connection of the transformers exists, care must, therefore, be taken to have this neutral a fixed voltage TRANSFORMATION OF POLYPHASE SYSTEMS 427 point, irrespective of the variation of the load or its distribution, which may occur; otherwise harmful phenomena may result from a "floating" or "unstable" neutral. In connections (2) and (3), the secondary-e.m.f. triangle is in phase with the primary-e.m.f. triangle, while in (1) it is displaced therefrom by 30°. Therefore, even if the voltages are equal, con- nection (1) cannot be operated in parallel with (2) or (3), but (2) 1 [mjim} fW] rm T Fig. 211. and (3) can be operated in parallel with each other, and with the connections (4) and (5), provided that the voltages are correct. 4. The V connection or open delta connection of transformers between three-phase systems, consists in using two sides of the triangle only, as shown in Fig. 211. This arrangement has the disadvantage of transforming one phase by two transformers in series, hence is less efficient, and is liable to unbalance the system mm) mu Fig. 212. by the internal impedance of the transformers. It is convenient for small powers at moderate voltage, since it requires only two transformers, but is dangerous in high potential circuits, being liable to produce destructive voltages by its electrostatic un- balancing. 5. The main and teaser, or T connection of transformers be- tween three-phase systems, is shown in Fig. 212. One of the 428 ALTERNATING-CURRENT PHENOMENA two transformers is wound for V3 2 times the voltage of the other (the altitude of the equilateral triangle), and connected with one of its ends to the center of the other transformer. From the point one-third inside of the teaser transformer, a neutral wire can be brought out in this connection. 6. The monocyclic connection, transforming between three- 1 Fig. 213. phase and inverted three-phase or polyphase monocychc, by two transformers, the secondary of one being reversed regarding its primary, as shown in Fig. 213. 7. The L connection for transformation between quarter-phase and three-phase as described in the example, §291. 8. The T connection of transformation between quarter-phase and three-phase, as shown in Fig. 214. The quarter-phase sides CMT\ rWFl Fig. 214. of the transformers contain two equal and independent (or inter- linked) coils, the three-phase sides two coils with the ratio of V3 turns, 1 -. ^, connected m T. 9. The double delta connection of transformation from three- phase to six-phase, shown in Fig. 215. Three transformers, with two secondary coils each, are used, one set of secondary coils connected in delta, the other set in delta also, but with reversed TRANSFORMATION OF POLYPHASE SYSTEMS 429 terminals, so as to give a reversed e.m.f. triangle. These e.m.fs. thus give topographically a six-cornered star. T — I T\ — r '^q^ I 2 Fig. 215. '3 '2 '3 10. The double Y connection or diametrical connection of trans- formation from three-phase to six-phase, shown in Fig. 216. It Fig. 216. is analogous to (7), the delta connection merely being replaced by the Y connection. The neutrals of the two F's may be con- nected together and to an external neutral if desired. V 3' / V i^ 2' V '■ 3 ^^^FjJW I I Fig. 217. The primaries in 9 and 10 may be connected either delta or F, and in the latter case a floating neutral must be guarded against. 430 ALTERNATING-CURRENT PHENOMENA 11. The double T connection of transformation from three- phase to six-phase, shown in Fig. 216. Two transformers are used with two secondary coils which are T-connected, but one with reversed terminals. This method also allows a secondary neutral to be brought out. 293. Transformation with a change of the balance-factor of the system is possible only by means of apparatus able to store energy, since the difference of energy between primary and secondary circuit has to be stored at the time when the secondary power is below the primary, and returned during the time when the primary power is below the secondary. The most efficient storing device of electric energy is mechanical momentum in re- volving machinery. It has, however, the disadvantage of re- quiring attendance; fairly efficient also are condensive and in- ductive reactances, but, as a rule, they have the disadvantage of not giving constant potential.