Question

Show all steps Chapter 4, Problem 28P (1 Bookmark) ON Figure 4.41 shows in cross section a machine having a rotor winding f and two identical stator windings a and b whose axes are in quadrature. The self-inductance of each stator winding is Laa and of the rotor is Lff. The air gap is uniform. The mutual inductance between a stator winding depends on the angular position of the rotor and may be assumed to be of the form where M is the maximum value of the mutual inductance. The resistance of each stator winding is R a. Derive a general expression for the torque T in terms of the angle θ0, the inductance parameters, and the instantaneous currents la, lb, and if. Does this expression apply at standstill? When the rotor is revolving? b. Suppose the rotor is stationary and constant direct currents ia-10, 1b-/0, and if-210 are supplied to the windings in the directions indicated by the dots and crosses in Fig. 4.41. If the rotor is allowed to move, will it rotate continuously or will it tend to come to rest? If the latter, at what value of o? c. The rotor winding is now excited by a constant direct current f while the stator windings carry balanced two-phase currents a Sin ot The rotor is revolving at synchronous speed so that its instantaneous angular position is given by -at-δ, where δ is a phase angle describing the position of the rotor at t :0. The machine is an elementary two-phase synchronous machine. Derive an expression for the torque under these conditions d. Under the conditions of part (c), derive an expression for the instantaneous terminal voltages of stator phases a and b. Figure 4.41 Elementary cylindrical-rotor, two-phase synchronous machine for Problem 4.28 -b Rotor -f Stator

Answer 1

Solution Induclance To derfve the general Expressfon ior the orque terms ot angle 6o xpression or rame ters are ia,b a and d e This expression appltes lor al operancondrlons Suppose rotor s stationarq and direct currents

Provided -there ore ane losses at all, the rotor o come to rest tot 90 hichò_Γ.0 and d )Given a la