Figure shows in schematic cross section a salient-pole synchronous machine having two identical stator windings a and b on a laminated steel core. The salient-pole rotor is made of steel and carries a field winding f connected to slip rings. Because of the non-uniform air gap, the self- and mutual inductances are functions of the angular position ?₀ of the rotor. Their variation with ?₀ can be approximated as:

L_aa = L₀ + L₂ cos 2?₀ L_bb = L_0?? L₂ cos 2?₀ M_ab = L₂ sin 2?_0?where L₀ and L₂ are positive constants. The mutual inductance between the rotor and the stator windings are functions of ?_0?M_af = M cos ?₀ M_bf?= M sin ?₀, where M is also a positive constant. The self-inductance of the field winding, L_ff, is constant, independent of ?₀. Consider the operating condition in which the field winding is excited by direct current If and the stator windings are connected to a balanced two-phase voltage source of frequency co. With the rotor revolving at synchronous speed, its angular position will be given by ?₀ = wt. Under this operating condition, the stator currents will be of the form i_a =?2I_a cos (wt + ?) i_b = ?2I_a sin (wt + ?)

a. Derive an expression for the electromagnetic torque acting on the rotor.

b. Can the machine be operated as a motor and/or a generator? Explain.

c. Will the machine continue to run if the field current If is reduced to zero? Support you answer with an expression for the torque and an explanation as to why such operation is or is not possible.

Transcribed Image Text:

Rotor -b Stator