A synchronous motor can be defined as being, merely, an alternator used as a motor. The transmission of power between an A.C. generator and an A.C. motor is, therefore, nothing more than a particular case of the coupling of two alternators in synchronous operation. Indeed, it is precisely through the study of the features of the coupling of alternators in parallel that the occasion presented itself of noting the phenomenon of the reversibility of alternators, that is to say, the possibility of using the same machine both as a motor and a generator, provided that it shall have been previously brought to a speed absolutely equal to that of the generator which supplies it with current.

We can easily understand the possibility of operating such a motor by comparing it to a motor with commutated current. It is known that if the current of a shuttle armature of the Siemens ("H") type is commutated at each half revolution, the motor-couple is always in the same direction when the machine is supplied by direct current. In an A.C. system, the same result is obtained without a commutator, because the direction of the supply-current changes at each half revolution, and this effect occurs only when the motion of the motor is synchronous, that is to say when the armature advances the distance of one pole during one alternation of the supply-current.

Although this property was noted as early as 1869 by Wilde, it passed unnoticed during more than ten years, and it has really been known only since the experiments of J. Hopkinson and Grylls-Adams, at the South Foreland Lighthouse, in 1883. The Memoir of Hopkinson (in which, without knowing the work of Wilde, he gives the explanation to which reference will be made later) , was epoch-making in the history of alternating currents.

In the South Foreland experiments, the alternators used were three similar de Meritens singlephase alternating current machines, all belt-driven from a common source of power. These machines could be easily coupled in parallel, as generators, by bringing them to the same speed before coupling them. The belt being then removed from one of them, it was observed that it continued to run synchronously by the action of the current of its neighbors, and that it could even develop a considerable amount of power, as measured by a friction brake, before losing its synchronism. These experiments were repeated a few years later by Mordey, on a much larger scale, with machines of low inductance presenting a much greater stability of operation and driven by independent prime movers. He was thus able to demonstrate the synchronizing power of the alternators on the motors or engines driving them, and even to cause one of the latter, with the power shut off, to be dragged by one of the alternators which it was driving. This gives the key to the principles involved in parallel working. He also showed, later, the possibility of accomplishing this coupling with machines connected by means of long lines of high resistance.

## Synchronous Motors and Alternators

Synchronous motors have the same construction as alternators. The few special features relative to the production of the direct current necessary for their excitation will be treated separately, later. It will be assumed that the reader is already familiar with the general details of construction of alternators.

There are motors having movable armatures and stationary fields, or vice versa, and also motors with revolving iron masses in which all the windings are stationary. These machines are similar to the generators of the same types; for example, Fig. 1 indicates, diagrammatically, the principle of construction of a two-phase synchronous motor, with a ring armature and movable fields, receiving an exciting current through the brushes b_{1} and b_{2} .

These motors are designed like generators, the essential condition to be fulfilled being to have a low armature-reaction and powerful inducing fields, in order to obtain good stability.

## Number of Poles

Although it is more difficult to increase the number of poles for small powers than for large powers, the construction of small synchronous motors for ordinary frequencies (40 to 60 cycles) presents no special difficulties, if the speeds corresponding to these frequencies are not objectionable, because these speeds are perfectly allowable so far as centrifugal force is concerned.

## Synchronous Motors at low Speeds

On the other hand, in the construction of small synchronous motors to run at low angular velocities, it is extremely difficult to find space for the numerous conductors and for the exciting or field coils, which must produce as many ampere-turns as in the case of large motors. For this reason non-synchronous motors are more convenient for low rotative speeds.

The author has been able, however, to produce motors of low power (a few hundred watts) which have moving iron and have a very high number of poles (as many as 50 for example), by utilizing inductiontype excitation, the magnetic circuit being closed exteriorly, as shown in Fig. 2, in such a way as to allow all the space needed for the exciting coils.

These coils can then be replaced by permanent magnets, thus producing motors which run without excitation, at speeds sufficiently low to be synchronized by hand, and which can render useful service, in certain applications, such as for oscillographs. For this purpose the author preferably employs a small horseshoe magnet that is made to revolve around a stationary armature having a number of poles which is a multiple of 6. It is possible, in this way, to obtain very stable synchronous rotation of a revolving mirror without expending more than 1.5 to 2 watts.

Several firms made a specialty of synchronous motors, at an early date, among which we may mention La Societe 1'Eclairage Electrique in France, and the Fort Wayne Company in America.

One form of motor constructed in France by the Societe 1'Eclairage Electrique (Figs. 3 and 4), is constructed for polyphase currents or for single-phase currents, for powers ranging from 1 to 130 H.P. Table 1 gives the principal data referring to these matters.

## Efficiency Synchronous Motors

The efficiencies of the three-phase motors are a little higher than those given for the single-phase motors. The horse-powers given in this table correspond to a frequency of 42 periods, but these motors can be also used at frequencies between 40 and 60 periods, and their power then increases with the frequency.

As table 1 shows, types Nos. 14 to 30 are made with 4 poles, self-exciting. For higher powers, the number of poles increases, and the excitation is obtained by means of a small direct current exciter mounted on the same base. Above type 90 the armature is stationary and the fields turn inside. The fields are of mild cast steel, the armatures being slotted.

As an example of these large motors may be cited several from 50 to 100 H.P., giving the best of results on the power-transmission system around Grenoble, notably at Voiron, a distance of 30 kilometers from the generating station. Their efficiency is from 90 to 92 per cent. One of these motors even works in parallel with a steam-engine of the same power, and it compensates for the variation of angular velocity of the engine as it passes the dead centers.

All these motors are provided with a clutch and with an idle pulley for starting, as will be explained later. When running, they can undergo considerable variations of load without falling out of step.

Attention should also be called to another interesting type of synchronous motor, the Maurice Leblanc type, which is characterized by the addition of closed circuits in the pole-pieces to insure a perfect damping of oscillations, as will be seen later.

## Case of Equal Electromotive Forces

Let us suppose that the electromotive forces of the generator and motor are equal, and, to simplify matters, let us take, as generator and motor, two machines whose excitations are regulated to approximately the same value. Let the two machines be driven by belts (Fig. 5) ; and, when they have attained the same speed, let them be coupled together (experiment of Hopkinson and Grylls-Adams). Let us, moreover, make use of an apparatus of the kind described in Chapter VII, whereby the difference of phase between the two machines may be determained.

It will be noted, in the first place, that as soon as the two machines are brought to the same speed, the current which passes from the one to the other practically disappears. Moreover, the "phases are identical," i.e., the poles of like polarity pass at the same time in front of the corresponding portions of the two armatures.

## The induced E.M.F. Synchronous Motor

The induced E.M.F.'s between the corresponding terminals a, b, and A, B, are therefore in unison. If we measure them, on the contrary, in the direction in which they appear, by following the circuit ab, BA, it will be found that they are exactly opposed to each other.

Let us now suppose the belt of one of the two machines to be removed. This machine will continue to turn at the same speed, but it gives indication of a certain very slight delay or falling behind, technically termed "lag" with respect to the other machine. Moreover, the current in the circuit now becomes appreciable.

## Stalling of a synchronous motor

If a brake be placed on the pulley and if the load be gradually increased in such a way as to increase the mechanical power produced by the motor, the "lag" of the motor will be seen to increase at the same time as the current.

When this lag approaches a quarter of a period, i.e., half an interpolar space, the machine slows up all at once and stops as if held fast by the brake. We then say that it is "stalled," or " out of synchronism", or "out of step." The current in the circuit rises to a very high value as soon as the machine falls out of synchronism; and it becomes approximately equal to the short-circuit current in the circuit when the machine is stopped. In order to avoid accidents, it is necessary to introduce fuses in the circuit, or to provide some automatic disconnecting device, which will prevent the excessive load.

It is seen that what characterizes the synchronous motor is the increase of phase-lag with the load and the "stalling" of the motor or its falling out of step beyond a certain maximum load.

In a good motor, the limiting load should amount to at least 1.5 times, or, better, to twice the normal load. This limit is guaranteed by most makers of synchronous motors.

On the other hand, if the motor is run by a belt in such a way as to give it a "lead in phase" with respect to the machine or the circuit which supplies it with current, it can be found, by wattmeter measurements, that this power changes in sign, i.e., the motor acts as a brake and returns energy to the circuit instead of receiving it therefrom.

The phenomena become more complicated still on varying the E.M.F. of the motor or of the generator.

## Case of Unequal Electromotive Forces

An interesting and characteristic property of synchronous alternating current motors, and which distinguishes them absolutely from direct current motors or from alternating current motors having commutators, is that they can be excited so as to give a voltage greater than that of the supply-circuit. For example, it is possible to feed, from a no-volt circuit, a motor which, driven by belt at the same speed, produces an E.M.F. of 120 to 150 volts at its terminals. But, if the E.M.F.'s are thus unequal, the current passing between the generator and the motor, when the latter is running without load, can, instead of being inappreciable, attain a considerable value.

Likewise, when the motor is running with load, the current is greater than that which corresponds to the work to be done. The same effects are produced when, instead of giving to the motor an excessive excitation, it is given an insufficient induced E.M.F. It is then observed, if the machines are alike, that the potential difference at the terminals assumes a third value, which is the mean of the two E.M.F.'s involved.

In both cases, the greater the inequality between the two E.M.F.'s the more the current measured will increase, by the change of excitation. If we plot a diagram, taking, as abscissae, the values of the excitation of one of the machines, and, as ordinates, the current passing through the circuit, the curve of variation of the latter, as a function of the former, has the form of a V more or less rounded at the bottom (Fig. 6). This form persists, although it may be less marked, when a constant load is placed on the brake. At the same time that the current increases, by reason of an inequality of the E.M.F.'s, it can be noted, by means of an apparatus for indicating phase-difference, that the current undergoes a change of phase, either forward or backward, with respect to the E.M.F. of the motor. This can be expressed in another way by saying that the machine consumes or produces wattless current, i.e., current which is "out of phase" being pi/2 behind or ahead of the E.M.F. This "wattless" current, which has the effect of increasing the "apparent" current, is thus named because it produces no work, the load on the brake remaining constant, by hypothesis.

## 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.

*T*=the duration of the period;- ω=
^{2π}/_{T}=the speed of pulsation of the currents; - e
_{1}and e_{2}= the instantaneous values of the generator and motor E.M.F.'s respectively, at the instant t; - E
_{1}and E_{2}= the effective values equal to the amplitudes of the sinefunctions, i.e., the maximum value, divided by √2; ^{t0}/_{T}= the phase-difference between e_{1}and e_{2};- Θ = the angle of lag (phase-difference) corresponding to Θ=2π
^{t0}/_{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 e_{1} and e_{2} in Fig. 11, may be formulated by the equations,

In which Θ designates the angular distance between the actual position of e_{1} and the position of opposition of e_{2}.

# Equations of Synchronous Motors; Analytical Theory

The E.M.F. which is acting in the circuit is equal to the algebraical sum of the opposing E.M.F.'s.

From this the current, i, may be deduced, by the well-known differential equation,

In this equation let i=X sin ωt+Y cos(ωt).

If this value be substituted in the equation, the values of X and Y can be determined by making the coefficients of the sine-terms and of the cosine-terms successively equal to zero. We can then obtain, by differentiation, substitution, etc., the following value for i:

This may also be written,

_{1}= E

_{2}

this expression reduces to

_{1}and e

_{2}. This result is easily interpreted in Fig. 11, by drawing the resultant curve e

_{1}+ e

_{2}obtained by taking the difference of the ordinates of the first two curves. It will be seen that the curve has actually a phase-difference equal to with regard to

^{π}/

_{2}the mean of e

_{1}and e

_{2}, and that it increases with the phase-difference of e

_{2}with respect to e

_{1}. In consequence of the lag, γ, of the current, measured with respect to this re.sultant E.M.F. (when γ is near

^{π}/

_{2}in value), it will be seen that the current is approximately in phase with this mean value; it would be completely so if there were no resistance-losses.

The power-outputs of the two machines will be obtained by multiplying the instantaneous current i by the E.M.F.'s. e_{1} and e_{2}. For example, in the case where E_{1} =E_{2} , we have

_{2}, we will have

These equations show that the power is not constant, in either case, but pulsating, i.e., it presents variations of frequency= 2 T, as represented in Fig. 11.

These variations constitute sine-functions having pulsations twice as rapid as those of the current, which have for their axes the horizontal lines (P_{1}, P_{2}) corresponding to the *mean powers* given by the first terms within the brackets in the following equations:

The very small difference between P_{1} and P_{2} represents the loss by resistance (Joule effect). The axis of the curve P_{1} is therefore a little more above the axis of zero power than the axis of symmetry of the curve P_{2} is below it.

The torque could be obtained, in each case, by dividing the power by the angular velocity. These expressions show that the current increases with Θ until Θ equals π, but the torques, which equal zero so long as the lag Θ=zero, increase with Θ only until the value Θ = γ; and they then decrease.

Stability will, therefore, exist only with Θ < γ having for its axis the exact opposition of E.M.F.'s.

The solution, in the case where E_{1} is different from E_{2}, will be obtained in an analogous manner, by forming the products e_{1}i - e_{2}i; and it would still give pulsating values for p_{1} and p_{2} ; but, since these caculations are uselessly complicated, we will pass them by and turn to more simple methods.

## Synchronism

In proportion as the armature-speed increases, the magnetic pulsations produced by it in the pole-pieces become less numerous, as can be seen on connecting a lamp to the field-circuit and noting its variations of brightness. It is well to provide a centrifugal regulator which connects this lamp in circuit only when the speed approaches synchronism, because, at lower speeds, it would be subjected to excessive voltage. When synchronism is almost attained (the speed generally remains slightly lower,) the fields are excited when the phases come into opposition, the motor falls into step, and the current immediately diminishes, owing to the disappearance of the reactive current which was absorbed up to that time. The more carefully the time has been selected for closing the excitation-circuit, the more easily the motor will fall into step. It is well, as a rule, to include, in the circuit, a variable self-inductance, which has the effect of preventing excess of current and of damping objectionable harmonics.

When once the motor is in synchronism, it can be loaded progressively, by shifting the belt from the idler to the driving pulley.