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# Assymetric field revolving

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### Assymetric field revolving

1. 1. Revolving-field analysis of asymmetric three-phase machines and its extension to single- and two-phase machines Bhag Singh Guru Indexing terms: Equivalent circuits, Machine theory, Magnetic flux, Induction motors, Stators Abstract: The revolving-field theory as applied to single-phase induction motors is extended to develop a concise yet comprehensive theory for asymmetric three-phase induction motors. Each phase is represented by its equivalent circuit, not only to allow for uneven direct transformer interactions, due to asymmetric locations of phase windings, but also to account for the different number of turns, wire size, winding factor etc. each phase may have. The accuracy of the theory was confirmed by actual measurements on symmetric and asymmetric three-phase induction motors. Computed and test data on some motors are included for illustration. It is also shown that the general three-phase development can be easily applied to determine the behaviour of two- and single-phase induction machines, a useful feature for unified computer-aided design, by eliminating one and two phase windings, respectively. Not only the procedural details for determining the performance of single- and two-phase induction motors are given, but comparisons of numerical and test results on some output entities are included as well. Though the comparisons among theoretical and test results are made on fractional horsepower motors, the types being built at Universal Electric, the author is certain that the development presented here is equally applicable to all sizes of induction motors.1 Introduction of suitably chosen components of current and voltage. Further adaptation of this technique to three-phaseA symmetric polyphase induction motor, operating on a induction machines with unequal windings and arbitrarybalanced polyphase power supply, is defined as an spatial displacements in addition to unbalanced lineinduction motor whose windings are evenly displaced in voltages may require a large number of voltage and currentspace and excited by the currents having the same dis- transformations. At this time, such a development, if thereplacement in time phase as the windings have in space is any, is not known to the author. Another analyticalphase. Consequently, the internal interactions of such an method would be the direct extension of the revolving-induction motor are simple, and can be easily characterised field approach as applied for capacitor motors withby one primary circuit and a corresponding secondary windings not in quadrature.3 Even though this methodcircuit based on the revolving-field theory. On the other requires only three unknown quantities, the torquehand, the simple crossfield analysis of such motors involves equation by itself is quite involved.three times as many circuits as there are phases. Hence the All the foregoing methods, in the authors opinion,polyphase induction motors are universally analysed on the satisfy only one particular requirement, that is, if thebasis of the revolving-field theory. However, quite often the development is carried out for a three-phase motor withstator core punchings designed for a certain pole configur- asymmetric windings, it would require comprehensiveation are also used for another pole configuration, and in changes to make it useful for an asymmetric two-phasethat case, if the number of stator slots per pole pair is not motor and vice versa. Stated differently, separate computera whole number, the phase windings would have to be programs are needed to analyse different-phase motors.arranged at resultant angles other than the conventional This, in turn, increases the cost of maintaining theseangle for such motors. This results in a spatially unbalanced programs as permanent files on a time-sharing machine. Tomotor even though the windings are identical. The actual circumvent the situation, we can seek refuge in Kronsperformance of an asymmetric motor is bound to differ generalised theory. 4 5 This theory, as Kron states in hisfrom the conventional motor, and therefore necessitates a book,s has to be extended further to encompass the entiredifferent theoretical approach. As far as the author is range of asymmetric machines. Moreover, an extension ofaware, no attempt has ever been made in this direction. Krons theory requires the in depth understanding ofTherefore, the primary aim of this paper is to develop a tensors, symmetrical components, three-phase/two-phasesystematic method for the analysis of asymmetric three- and inverse transformations. An average user, however, mayphase induction motors. In addition, the effects of not be able to satisfy some or all of these demands.unbalanced terminal voltages are also included to make thedevelopment more general. However, note that the well In literature, almost, all the time, a symmetric three-known method of symmetrical components1 may be used phase induction motor is represented by an analyticallyto analyse a symmetric (equal windings) three-phase simple yet tricky equivalent circuit. An equivalent circuit,induction motor running on an unbalanced power supply. in the authors opinion, should be based upon the physicalThis method has also been extended for the analysis of picture as opposed to a trick circuit. In a three-phaseasymmetric two-phase induction machines2 by making use motor, the flux produced by one phase induces voltages by direct transformer action in the remaining phases and vice versa. An exact representation of this machine, wouldPaper T308 P, first received 9th May and in revised form 24th require an equivalent circuit for each of its phases and anNovember 1978 account for the mutual interactions of the fluxes ratherDr. Guru is with the Universal Electric Company, 300 East MainStreet, Owosso, Michigan 48867, USA than a single yet tricky circuit. Although the former mayELECTRIC POWER APPLICATIONS, FEBRUARY 1979, Vol. 2, No. 1 37 0140-1327/79/010037 + 08 \$01-50/0