DYNAMO
580
gives the flat-topped curve of Fig. 4, will be the upper wavy line of E.M.F. obtained by adding together two of the resultant curves of Fig. 14, with a relative displacement of 45°. The amount of fluctuation for a given number of commutator sectors depends upon the shape of the curve of E.jVI.F. yielded by the separate small sections of the armature winding; the greater the polar arc, the less the fluctuation. In practice, with a polar arc equal to about 0-75 of the pitch, any number of sectors over 32 yields an E.M.F. which is sensibly constant throughout one or any number of revolutions. If the armatures of Fig. 16 be imagined to be cut across to the centre from any point such as X and opened out, as was also done in the case of alternators, any number of pairs of poles, with a corresponding increase in the number
Volts
Fig. 16a
Fig. 16b.
of inductors, may be added, and multipolar continuouscurrent dynamos are obtained. But whereas in the alternator there may in each phase be only one circuit through the armature, in the continuous-current closed-coil armature there can never be less than two parallel circuits, while if there are more than two, there must be as many as there are poles.1 The two conditions coincide in the simple bipolar armature, since it has two parallel paths through the armature, or as many as there are poles. The fundamental electromotive force equation can now be gien a more definite form, suitable lor both the continuous-current dynamo and the alternator. Let Za=the number of C.G.S. lines or the total flux, which issuing from any one pole flows through the armature core, to leave it by another pole of opposite sign. Since each inductor cuts these lines, first, as they enter into the armature core from one pole, and then as they emerge from it to enter another pole, the total number of lines cut in one revolution by any one inductor is 2pZa- The time in seconds taken to peiform one complete revolution is where N is the number of revolutions per minute. The average E.M.F. induced in each inductor in one revolution, being proportional to the total number of lines which it cuts, divided by the time taken to cut them, is therefore 2 x 10'8 volts. Let r be the total number of 60 . , inductors on the surface of the armature, whether ring- or arumwound, and let q be the number of parallel circuits into which the armature winding of a continuous-current dynamo is divided, whence the number of inductors in series is p. The average E.M.F.
The case of the discoidal armature may best be treated by reducing it to the analogous simple ring ; each pair of opposing poles, since they are of the same sign, forms iif effect a single divided pole, whence a total number of lines Za pass into the armature core; r must then be reckoned as equal to the number of loops in the ring, or the number of inductors on one only of its two faces. The equation of E.M.F. thus becomes exactly the same as that of the simple ring machine. The disc winding is analogous to the drum, so that in all cases the E.M.F. of the continuous-current closed-coil armature is E(j = g-• 60 . r. Za. x 10 8 volts, • • • (1>a) where q must be an even number, and can never be less than 2. In the continuous-current dynamo the instantaneous and average values of the E.M.F. from the armature as a whole are the same ; but when we pass to the alternator, its “effective” E.M.F. is equal to the square root of the mean square of the instantaneous values of the E.M.F., since this is the value of the equivalent unidirected and unvarying E.M.F., which, when applied to a given resistance or to a glow lamp, would give the same amount of heat or of light as the alternating E.M.F. produces (see Electricity, iii. Electric Current). The shape of the curve of instantaneous E.M.F. therefore becomes a matter of the greatest importance, for upon it will depend the effective voltage of the alternator, even though its average E.M.F. may be the same with some different shape of E.M.F. curve. Assuming, as we may, that the density of the lines is the same along the length of an inductor for any given position of it during rotation, its instantaneous E.M.F., =B0LV xlO '8 volts, will vary as Bg varies, according to the position of the inductor under or between the poles. Hence the addition of more inductors in series may not give a proportionate addition of E.M.F., and in fact will not do so, unless they are so close beside each other that their difference of spacial position or of phase can be neglected ; the nearest practical approach to this will be in the case of a toothed or tunnel armature, when the inductors of a coil are wound in the same slot or hole. Differential action, which has been already discussed, is indeed only an aggravated case of the same effect as must take place when a coil is gradually coming into or out of action, and is not entirely under a pole. Further, it is evident that the curve of E.M.F. of the coil may be given any shape that we please, by so shaping the pole-piece as to give the required variation in density of the lines in the air-gap. Thus the length of gap between pole-face and armature core might be so varied as to give a curve of E.M.F. varying after a sine law (cp. Fig. 3—for the case of a single inductor), and the ratio of the effective E.M.F. to the average E.M.F. would then have the particular value of =1T1. Two points have, then, to be considered in an alternator, namely, what is the ratio of the square root of the mean square to the average value of its E.M.F., and what is the ratio which either the instantaneous or the average E.M.F. of a coil bears to the instantaneous or average E.M.F. of a single inductor. Both depend on the ratio of the width of coil and pole, and on the law which governs the variation in density of the field. If ee = tve effective E.M.F. of one coil, and the total number of coils in one phase, i.e., 2p, be divided into q parallels, the effective E.M.F. of the phase is Eae = -|-.. ee. Let h' — Ccie ratio of the square root of the mean square to the average
■v/g_ then the effective value of the E.M.F. of the coil, or= JYedt f~ 2p value of the E.M.F. of the phase is ^. E. eav, where T is the time of a period and the net average value of the E.M.F. of a coil. Let k" — the ratio which the net average E.M.F. of the coil bears to the gross average E. M. F. on the supposition that there is differential action, and let e = the average E.M.F. of a single induced in the armature is then Ea=2.^^ .^. ZaX 10 8 volts, and no inductor ; then if t be the number of inductors m a coil, the reducof the E.M.F. of a coil as a whole may be taken into account in the continuous-current machine, with a sufficient number of tion multiplying the product te by a breadth coefficient k , which inductors and commutator sectors, this is the same as the instan- by taneous E.M.F. ; for the sum total of the several E.M.F. s set up must be less than unity if the coil have any width. The effective .k' .k". te. Ifm=the numbei of in the different inductors of one circuit, spread uniformly over the E.M.F. per phase is therefore armature core, will at any moment be equal to the average E.M.l. separate phases into which the armature winding is divided, set up in one inductor multiplied by the number of inductors m series i e it is independent of the distribution of the flux in the t— T - further, the average E.M.F. of the single inductors m. 2p air-crap It should be observed that Za, or the number of lines flowing through the armature, is taken, since in the ring machine is 2 . . Z x 10-8 volts ; whence the effective E.M.F. of one phase 60 a this will differ from the number that cross the air-gap or /g by ttie small percentage which leak across the interior of the ring, and so of the alternator is „, , phT " Za X 10 are cut in the opposite sense by the internal connecting wires. ^ ^ Eae- ^ • £ • £V' • m• 60 ' 7 a xiq-8=2-^" m.q. lio' 1 Except in the special case of multiple-circuit windings, which will volts. The two factors k' and k" may be united into one constant be referred to later.
