Variable armature leakage reactance in salient-pole synchronous machines

The electrical performance characteristics of a polyphase synchronous machine, that is, its voltage-current relations under load, depend essentially upon the nature and the extent of the magnetomotive forces of the armature currents. Broadly speaking, the effect of these magnetomotive forces is two-fold; i. e., (a) they oppose and distort the field magnetomotive force and (b) they create leakage fields linked with the armature conductors. The first influence is known as the armature reaction, and the second as the armature reactance. More specifically, in a machine with salient poles, the armature reaction may be resolved for purposes of computation into the direct reaction (along the center lines of the poles) and the transverse reaction, midway between the poles. In polyphase machines of usual proportions, the armature leakage reactance, x, usually plays a secondary role, and for most purposes is assumed to be constant and independent of the power factor of the load. The vector of the reactive drop, Ix, is simply drawn in a leading time quadrature with the current I. However, in machines with considerable armature reactance, or where higher accuracy is required, the assumption of a constant x leads to noticeable discrepancies between the computed and observed data. This is of particular importance in problems which involve hunting, instability, etc., and in which the torque (or displacement) angle must be predicted. This angle depends to a considerable degree upon the leakage reactance of the machine. It has been previously proposed by others to use two distinct values of leakage reactance, one when the leakage paths around a group or belt of armature slots are closed through the center of a pole face (maximum reactance), and the other when such slots are midway between the poles (minimum reactance). However, no account has been taken apparently of a gradual change in the reactance between the two extreme positions, nor have the results been properly correlated w- th the rest of the factors which enter in the performance of the machine. In the present paper, the leakage inductance is assumed to consist of two parts, one of which is constant (the average inductance), and the other, varying harmonically at a double frequency, reaches a maximum opposite the centers of the poles. A magnetic linkage equation is written, and its derivative with respect to the time angle is taken to obtain the induced voltage. The result shows that the foregoing assumption leads to two reactive drops, one, the usual average Ix drop and another a supplementary drop, leading Ix by an angle 2 ψ, where ψ is the internal phase angle at which the machine is operating. These quantities are introduced in the usual Blondel diagrams for the generator and the motor, and the relationships among the various quantities are established both graphically and analytically.