Consider an elementary two-pole rotating machine with a uniform (or smooth) air gap, as shown in Figure, inwhich the cylindrical rotor ismountedwithin the stator consisting of a hollow cylinder coaxialwith the rotor. The stator and rotorwindings are distributed over a number of slots so that their mmf can be approximated by space sinusoids. As a consequence of a construction of this type, we can fairly assume that the self-inductances Lss and Lrr are constant, but the mutual inductance Lsr is given by

L_sr = L cos θ

where θ is the angle between the magnetic axes of the stator and rotor windings. Let the currents in the two windings be given by

i_s = I_s cos ωs t and ir = I_r cos(ωr t + α)

and let the rotor rotate at an angular velocity

ω_m = θ rad/s

such that the position of the rotor at any instant is given by

θ = ω_mt + θ₀

Assume that the reluctances of the stator and rotor-iron circuits are negligible, and that the stator and rotor are concentric cylinders neglecting the effect of slot openings.

(a) Derive an expression for the instantaneous electromagnetic torque developed by the machine.

(b) Find the condition necessary for the development of an average torque in the machine.