Digital Control of Dyanmic Systems by G. Franklin, J. Powell, M. Workman

By G. Franklin, J. Powell, M. Workman

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Thefirstobjective must therefore be to apply a transformation leading to equations which have constant coefficients. e. not having salient poles. This machine is structurally a two-phase induction motor. The Matrix Equation of the Balanced Two-phase Machine with a Uniform Air-gap (Induction Machine) Consider a two-pole machine of induction motor type with balanced two-phase windings on both stator and rotor and a uniform airgap. In Fig. 12 the stator windings are designated D and Q, and the rotor windings a and ß.

Again, the symmetry of the arrangement shows that there will be no linkage with any winding by flux set up by a current in the winding at right angles to it, and consequently the mutual inductances M α β =Μ β α and MOQ = MQO are all zero. The remaining mutual inductances ΜΛΌ = MDot, M a Q = MQx, AfßD = M Dß and M ßQ = M Qß are all functions of Θ and are therefore functions of time. It is apparent that their variation with Θ is cyclic with a period corresponding to one revolution of the rotor. For simplicity it is desirable to assume that they vary sinusoidally, although this may be only approximately true in practice.

2 Transformers 51 The electric circuit analogue1" of the magnetic circuit is shown in Fig. s of windings 1 and 2 respectively. The "mesh" equations of the magnetic circuits are / i = (S L +SA)I+S L 0M /1-/2 = S L 0 I + 2 ( S L + S Y ) 0 M — S L 0 2 h =-SL0M+(SL+SA)02 which in matrix form is 1 1 h 1 SL+SA M h-h M 2 h 2 2 M 1 Φι SL 2(SL+SY) SL -SL -SL M ΦΜ SL+SA 2 Φζ This reluctance matrix can be inverted to give a permeance matrix of the form _1 M 2 l ' Αχι S" 1 = Λ = M ^ΙΜ 2 Λ12 and 1 1 Φι M ΦΜ 2 φι t See réf.

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