SYNCHRONOUS MOTORS AND CONVERTERS
Theory and Methods of Calculation and Testing

Explanation of Single-Phase Synchronous Motors

But we must take into account the voltage at the terminals, with which the induced E.M.F. combines, and also the self-induction of the machine, which throws the current out of phase by a quarter of a period, i.e., half an interpolar space. Therefore the question can only be treated with precision by calculation, as will be seen later. From the qualitative point of view, the result differs but slightly from the preceding result. The form of the curve of torque remains analogous to that of Fig. 9, but it is no longer so symmetrical, and the lags OC and OB, which determine the limits of stability, take a value γ, which is a little lower than π/2, and which is defined by the relation:

ωL and R being, respectively, the reactance and the resistance of the armature circuit.

It will be observed that, when the current is in phase with the induced E.M.F., the magneto-motive force of the armature-reaction has no action on the inducing field, and can only produce a transverse distortion of the field; while, on the other hand, when the current is out of phase one-fourth of a period in advance, or behind, the M.M.F. Is directly opposed to, or coincident with, that of the field. With regard to the sign, it can be easily seen that a phase-difference of π/2 in advance of the induced E.M.F. produces a magnetizing reaction which is the same as in a generator, and that a phase-difference of π/2 behind produces a demagnetizing reaction. It must not be forgotten, however, that the internal E.M.F. is opposed to the external E.M.F. and that the lag and lead are therefore transposed if they are referred to the latter.

Elementary Explanation of Single-Phase Synchronous Motors

The phenomena are more complicated in single-phase motors. The same explanation may nevertheless be retained by means of a simple artifice of reasoning.

The coils of the armature-winding, being excited by a single alternating current (Fig. 10), produce poles which no longer revolve, but are stationary. These poles are alternately positive and negative, and have a magnetic flux which varies periodically like the current that produces it. There is, therefore, no tendency to rotation; and the motor can only be put in operation by external means, as already seen. But we may suppose it brought previously to synchronism.

M. Maurice Leblanc has enunciated a theorem which is an electrical analogue of the following well-known optical theorem: A vibration of luminiferous ether polarized rectilinearly may be replaced by two circularly polarized vibrations of contrary sign having the same frequency and having amplitudes equal to the half amplitude of the rectilinear vibration.

According to M. Leblanc's theorem, an alternating stationary magnetic field may be replaced by two fields revolving in contrary directions, each having a flux half as large, and having equal velocities, such that they advance a distance equal to that of two poles during a single period. The fields turning in the same direction as the inducing poles will have the same angular velocity a, as the latter, and will drag them in very much the same way as in polyphase motors. On the other hand, the fields which turn in the opposite direction will have a relative velocity, 2 α, which is contrary to and double that of the field-poles, so that their attracting or repelling actions, since they succeed each other in inverse directions, will produce no resisting torque. These reversed revolving fields will give rise only to supplemental losses by hysteresis and by eddy currents.

By this simple analysis (which is, in reality, only approximate) the operation of single-phase motors can, it is seen, be discussed and explained in the same way as that of polyphase motors.

It has been supposed, in what precedes, that the armature is stationary and the field movable. In the contrary case the explanation is the same if we consider the relative velocities of the two portions, but the fields displace themselves only with respect to the armature and therefore remain stationary in space the same as the field-poles.

Equations of Synchronous Motors; Analytical Theory

We have just examined the phenomena of synchronous motors from a physical point of view. We shall now represent them analytically, according to the theory first expounded by Dr. J. Hopkinson, but with a few modifications in form. We shall suppose with him that the E.M.F.'s and currents follow the sinusoidal law, and that the reactances of the machine are constant.

Let us suppose, then, a single-phase A.C. generator and motor, defined by their induced E.M.F.'s, their resistances, and their mean inductances, which are all supposed constant.

Let:
  • T=the duration of the period;
  • &omega=2π/T =the speed of pulsation of the currents;
  • e1 and e2 = the instantaneous values of the generator and motor E.M.F.'s respectively, at the instant t;
  • E1 and E2 = the effective values equal to the amplitudes of the sinefunctions, i.e., the maximum value, divided by √2;
  • t0/T = the phase-difference between e1 and e2;
  • Θ = the angle of lag (phase-difference) corresponding to Θ=2πt0/T;
  • R and L = the resistance and inductance, respectively, of the total circuit, of the two machines;
  • i= the instantaneous value of the current;
  • I = the effective value of the current, equal to the maximum value divided by √2.

Let us suppose the conditions of stability to be unknown and let us seek to ascertain how two alternators connected in series will operate. The two sine-functions of the E.M.F. represented by the curves e1 and e2 in Fig. 11, may be formulated by the equations,

In which Θ designates the angular distance between the actual position of e1 and the position of opposition of e2.

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