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 DYNAMO K, and the equation then takes its final form of 2 K 8 Eoe=—‘ mq ^r-. 60 r. Z0 x 10^ volts .... v(I.&)' In contrast to the continuous-current dynamo, q may be unity. The analogy of this equation to that of the continuous-current machine is at once apparent; the process by which it is arrived at really consists in stating the E.M.F. of the alternator as that of a similarly wound continuous-current dynamo multiplied by a certain constant, and only because this is the most convenient method of expressing it are we interested in its average E.M.F. The second term-^A in both equations gives the periodicity of the alternating E.M.F. in the inductors; and even though the current be commuted, as in the continuous-current machine, it is of importance as determining the loss by hysteresis in the core, and by eddy currents in the armature as a whole. The fundamental equation of the electromotive force of the dynamo in its fully developed forms (I.a) and (1.6) may be compared with its previous simple statement (I.). The three variable terms still find their equivalents, but are differently expressed, the density being replaced by the total flux of one field Za, the length L of the single inductor by the total number of inductors r, and the velocity of movement V by the number of revolutions per second. Even when the speed is fixed, an endless number of changes may be rung by altering the relative values of the remaining two factors ; and in successful practice these may be varied between fairly wide limits without detriment to the working or economy of the machine. While it may be said that the equation of the E.M.F. was implicitly known from Faraday’s time onwards, the difficulty under which designers laboured in early days was the problem of choosing the correct relation of Za and r for the required output; this, again, was due chiefly to the difficulty of predetermining the total flux before ■the machine was constructed.1 The general error lay in employing too weak a field and too many inductors, and credit must here be given to the American inventors, Weston and Edison, for their early appreciation of the superiority in practical working of the drum armature, with comparatively few inductors rotating in a strong field. The equation of mechanical force now requires to be given, in its turn, a more fully developed form correlative to that of the electromotive force. Let the inductors of a continuous-current dynamo be carrying a current of c amperes ; then each, so long as it is moving under a pole-face, is subjected to a mechanical drag of BgLc x 10_1 dynes, or if the length and force be expressed in English units of inches and pounds, of 57 B;?L"c x 10 8 pounds. This force is gradually removed and re-established 2p times in each revolution as the inductor passes the interpolar gaps, but its direction is always tangential to the path of rotation, and opposed to it, so that the torque resisting the movement of a single inductor is BgLc.rxlO-1 dyne-centimetres, where r is the radius in cm. of the mean path of the inductor. The total torque acting on the armature is due to the combined effect of the several inductors with which it is wound ; and since the number of inductors within one polar face is r. A where  T—^ r'7TC~ ^ax 10

1

dyne-centimetres, or

10’8 pound-feet • • • • (H.a). In the above statement the lines are treated as if confined strictly to the area of the pole-face—an assumption which is not strictly true, since the fringe surrounding the pole-edges has been neglected. The correction for this slightly decreases the value of B^ for a given value of Za, but correspondingly increases the number of inductors which are subjected to the torque, so that the final form of the equation is strictly true for the continuous-current armature as a whole. This may be proved if the torque be multiplied by the angular velocity in radians per sec., and the rate of absorption of mechanical energy as thus obtained in ergs per sec. or footpounds per sec. be equated to the rate of development of electrical energy. The latter is equal to E C watts, or since Q = qc, the electrical horse-power is 2 r.Zq.lO 8 2gg c r z - 33 x 10-n ' 2-60 746 ^ 6 .c.r. ZaXl. ddx 10. On the other hand, the angular velocity in radians per sec. is equal to the number of revolutions per sec. multiplied by 27r, so 1

Cp. Kapp, Journ. Soc. Tel. Eng. vol. xv. p. 5901, 886.

581

that the mechanical horse-power absorbed is equal to C -A x 10 8 x 2^N x A = ^. c. r. Za x 1 -33 x 10 n. 4-26 Since the expression for the drag on any single inductor involves the term B?, it is evident that it is dependent on the distribution of the flux in the air-gap, i.e., on the shape of the curve of B9, and if the average value of B9 is alone considered, the equation (II.a), although giving the average pull on the inductor, does not express its instantaneous values. As will be explained subsequently, the value of the actual B9 in the continuous-current dynamo when at work is not uniform even under the pole-face. Hence the instantaneous pull on any inductor varies on either side of the average value, and in practice its maximum value at full load may be as much as 20 per cent, greater than the average value. Lastly, in the alternator the current c varies as well as the density Bff, so that the problem becomes much more complicated and no simple equation can express it. If the current in the armature were in phase with the induced E. M. F., and so with the flux-density which causes it, the maxima values of B9 and c would occur simultaneously. The maximum drag thereby produced will be far greater than the average value, and may considerably exceed the force acting on the equivalent continuous-current dynamo. But, more than this, if there be inductance in the circuit, the current in the armature will lag in phase behind the induced E.M.F. and behind the B9 curve. The pull on the inductors now becomes zero not only when Bff is zero, but also when c is zero, and, further, its direction relatively to the rotation changes each time that either the one or the other passes through zero. Thus the alternator is subjected to a racking action which is much more trying than the simple torque of the continuous-current machine. The armature is alternately dragged back when the directions of field and current coincide, and driven forward as a motor when they are opposed, the backward drags being more powerful and lasting longer than the forward pushes, so that on the whole the machine absorbs mechanical energy and acts as a generator. If the inductors, instead of being wound on the surface of a smooth armature core, are sunk into slots or passed through holes close to the edges of the core-discs, the force which would otherwise be expended on them is largely transferred to the iron teeth which project between the slots or tunnels. The lines of the field, after passing through the air-gap proper, divide between the teeth and the slots in proportion to their relative permeances. Fig. 17. Hence at any moment the inductors are situated in a weak field, and for a given armature current the force on them is only proportional to this weak field.2 This surprising result is connected with the fact that when the armature is giving current the distribution of the lines over the face of each tooth is distorted, so that they become denser on the “trailing” side than on the “ leading ” 3 side (Fig. 17) ; the lines tend to straighten and shorten themselves, and the result of the unsymmetrical distribution acting on all the teeth is to produce a magnetic drag on the armature core proportional to the current taken out of the inductors, so that the total resisting force remains the same as if the armature had a smooth core. The amount by which the stress on the inductors is reduced entirely depends upon the degree to which the teeth are saturated, but since the relative permeabilities of iron even at a flux density of 20,000 lines per square centimetre is to that of air as 33 : 1, the embedded inductors are very largely relieved of the driving stress. CONTINUOUS-CURKENT DYNAMOS. On passing to the separate consideration of continuouscurrent dynamos, the winding of the armature may first be taken in greater detail, and certain results drawn from 2 Electrician, vol. xxxiv. p. 757 (Sayers), and vol. xli. p. 108 (Du Bois); Swinburne, Industries and Iron, February 2, 1894 ; Mordey, Journ. Inst. Elec. Eng. vol. xxvi. p. 564 ; Houston and Kennelly, Elec. Eng. vol. xviii. p. 208. 3 By the “leading” side is to be understood that side which first enters under a pole after passing through the gap between the poles, and the edge of the pole under which it enters is here termed the “leading” edge, as opposed to the “trailing” edge or comer from under which a tooth or coil emerges into the interpolar gap ; cp. Mg. 30, where the leading and trailing edges are marked ll and tt.