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CHAPTER XXXIV.

ALTERNATE-CURRENT MACHINES.

410. General Properties.-The type of all machines generating alternating currents is the coil rotating about a diameter in a uniform field (Fig. 343). The current is sinusoidal and has a

retardation of phase relatively to the electromotive force defined by the equation

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E being the maximum value of the electromotive force (§ 321, equations (5) and (6)).

The work is equal to the product of the effective electromotive force into the effective strength of the current, multiplied by the cosine of the difference of phase (§ 324),

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It may be added, further, that if, in Fig. 344, OB represents the maximum value E of the electromotive force, and the angle BOA the lag 2, OA represents the maximum of the effective electromotive force, and oc the maximum electromotive force of selfinduction (§ 322).

The effective strength of current is

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while the work per

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§ 411.]

Coupled Machines.

485

of the electromotive force of self-induction being always 90° behind the current, the cosine of the difference of phase vanishes, and the corresponding work is nothing. Hence self-induction produces no loss of energy; its only effect is to limit the power of the machine. 411. Coupled Machines.-Let us imagine two machines coupled together, two coils, for instance, mounted on the same axis, and making an angle of a with each other, the wires being joined so as to form a single circuit. Adopting the same mode of representation, the two lines OC and OD (Fig. 345) making the angle a, representing the given electromotive forces, the maximum value of

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the resultant electromotive force is the diagonal OB of the parallelogram, and the construction is completed, as before. As the coefficients of self-induction are simply added together, the difference of phase relatively to the effective electromotive force or the current, that is to say, the angle BOA is independent of a.

The work expended per second in the form of heat in the

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and it is apparent from the figure that the first is in all cases the algebraical sum of the other two.

If the two projections fall on the same side of the point O, the

amounts of work represented by them are of the same sign, and the two machines act as generators absorbing work. If they fall on opposite sides (Fig. 346), the first machine absorbs work and acts as a generator, the second produces work and acts as a motor. The work that must be done upon the combined system, and is dissipated in the form of heat, is the difference between the amounts of work taken in by the first machine and given out by the second.

412. Coupled Alternators continued.-Instead of two machines mounted on the same axle, let us consider two machines each driven by a special motor and not connected with each other otherwise than electrically. The motors are supposed to be regulated so as to drive the machines at very nearly the same speed. If the machines were independent, the smallest difference of speed would make them pass successively through every possible difference of phase; there would be no stable condition, unless they were able to act on each other in such a manner as to maintain their synchronism and their difference of phase.

Let us take the simplest case, that.of two identical machines; the conclusion arrived at will be none the less generally applicable. It follows evidently from the construction (Fig. 345) that the leading machine always makes, with the effective intensity, a greater angle than the following machine. It follows from this that the latter always does more work than the former; moreover, all causes of disturbance have the effect of increasing the lead of the leading machine and the lag of the follower; they therefore tend to increase the difference of phase until the electromotive forces are in opposition, and the current is at its minimum, which minimum is zero when the two machines are identical.

Two alternate-current machines placed in series in the same circuit cannot, therefore, be made to work in the same direction; they can, however, if joined in parallel. For in the closed circuit containing the two machines, at two points of which the external circuit is connected, the two machines will act in opposition; if the external circuit has a relatively high resistance, which is always the case in practice, its influence on the régime may be neglected, and therefore the machines work under conditions of stable equilibrium. Hitherto, however, no general agreement has been arrived at, so far as the machines are concerned, as to the qualities which best suit coupled working.

These conditions are also those met with when one of the

§ 413-]

Various Types of Machines.

487

machines is used as generator and the other as motor. Equilibrium corresponds to that difference of phase for which the motor gives out the work required from it. The efficiency is equal to (Fig. 346)

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are really alternate. Whatever be the nature of the machine, the current can always be rectified in the external circuit by means of a suitable commutator; but the use of commutators always causes a loss of energy in the form of sparks which rapidly destroy the collectors.

Alternate reversal of the current causes no inconvenience in many applications, especially for electric lighting.

The field-magnets of alternators are usually excited by an independent continuous-current dynamo, which is called the exciter.

The armature has usually no iron core. When such a core is used, it ought to be very finely laminated, so as to avoid Foucault currents. Nor should the magnetisation be pushed too far, otherwise there is an unnecessary loss of energy in each magnetic cycle.

The presence of soft iron causes the value of the coefficient of self-induction to vary with the strength of the current, and thus greatly complicates the theory of the machine.

One characteristic of alternate-current machines is that they can

be easily made to produce very high potentials. This is a point of great importance in connection with the transfer of energy to a distance, which has received an immense development since the introduction of transformers (§ 417).

For equal differences of potential, alternating currents seem to require more care on the part of those who work with them than continuous currents.

As regards efficiency, alternating machines do not seem to be inferior to continuous-current machines, and they are not found to heat more when at work.

The frequency of ordinary machines varies from 40 to 140. In the case of very high frequency, regard must be had to the want of uniform distribution of the current in the conductor (§ 327).

The work taken in or given out during any very short interval is given by an elementary portion of the area of a curve analogous to curve B of Fig. 289 (§ 326), the ordinates of which are the products of the ordinates of two sine-curves, one representing the electromotive force, and the other the current. This work is sometimes work absorbed, and sometimes work given out. The lag being always less than 90°, the work taken in predominates, but it decreases in proportion as the difference of phase increases. An alternating machine consequently acts like a fly-wheel in relation to the motor, sometimes absorbing and sometimes giving

FIG. 347.

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back work. The periodic variations of the driving couple produce vibrations which give rise to a humming sound more or less loud.

As examples of alternating machines, we will describe those of Siemens and of Gramme.

414. Siemens' Machine.

-The inductor consists of two sets of electromagnets, each of them arranged with their axes parallel and at equal distances round the circumference of a circle, and presenting alternately opposite poles in the same direction (Fig. 347, SNS). The two circles of magnets are set facing each other a small distance apart, the axis of each magnet on one circle coinciding with that of one on the other circle which has the same direction

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