# Consider an elementary cylindrical-rotor twophase synchronous machine with uniform air gap, as…

Consider an elementary cylindrical-rotor twophase synchronous machine with uniform air gap, as illustrated in the schematic diagram in Figure P12.4.12. It is similar to that of Figure E12.4.1, except that Figure P12.4.12 has two identical stator windings in quadrature instead of one. The self-inductance of the rotor or field winding is a constant given by Lff H; the selfinductance of each stator winding is a constant given by Laa = Lbb. The mutual inductance between the stator windings is zero since they are in space quadrature; the mutual inductance between a stator winding and the rotor winding depends on the angular position of the rotor,

where L is the maximum value of the mutual inductance, and θ is the angle between the magnetic axes of the stator a-phase winding and the rotor field winding.
(a) Let the instantaneous currents be ia, ib, and if in the respective windings. Obtain a general expression for the electromagnetic torque Te in terms of these currents, angle θ, and L.
(b) Let the stator windings carry balanced twophase currents given by ia = Ia cos ωt, and ib = Ia sin ωt, and let the rotor winding be excited by a constant direct current If. Let the rotor revolve at synchronous speed so that its instantaneous angular position θ is given by θ = ωt +δ. Derive the torque expression under these conditions and describe its nature.
(c) For conditions of part (b), neglect the resistance of the stator windings. Obtain the volt–ampere equations at the terminals of stator phases a and b, and identify the speed– voltage terms.

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