CALCULATION OF ARMATURE-REACTION

207

With an Edison-Hopkinson armature of m turns it may be shown, from the manner in which the curve of integration is involved with the windings, that this diminution of the lineintegral—that is, in a certain sense, the 'counter-magnetomotive force of the armature-is equal to 04 am I, in which, as has been observed, a is expressed in circular measure. Hence, with an armature producing a current the total electromotive force will be

With a series machine, in which the current of the armatures also flows through the field magnets, and therefore I'1 = I1, we have

(11)

M = 0·4 I' (πn — am)

Besides this diminution of the total magnetomotive force, a displacement of the distribution of the lines of induction over the hollowed-out pole-pieces occurs, owing to the component of induction due to the rest of the windings of the armature. Thus, for instance, the induction at BC is not the same as at GH (fig. 33). For considering the loop BCGI as a path of integration, the parts of the line-integral along CG and BH are to be neglected. The total line-integral will be determined by the ampere-turns of the armature windings encircled by the path of integration and corresponding to CG. This is obviously equal to the difference AM of the magnetomotive forces between B and C on the one hand, and between G and H on the other; for if both these parts were equal, their algebraical sum-that is, the whole line-integral-would be zero, which is not the case. If the angle COG is denoted by B (in circular measure), it may be

It is to be observed that with a forward lead of the brushes the magnetomotive force of the armature coils forms a 'counter-action,' if the machine generates current; but, on the other hand, assists that of the field magnets if it acts as motor; with a backward displacement of the brushes, precisely the reverse is the case. From what has been said about the position of the brushes, as defined by the position of least sparking, it is clear that in practice a counter-magnetomotive force will always occur. (Compare Silv. Thompson, loc. cit. pp. 585-590.)

This disturbance of the distribution remains without essential influence on the total difference of potential between the brushes, but materially influences the distribution of potential round the commutator. This, as we know, is represented in the ideal case of one turn rotating uniformly in a uniform field, by a sine curve, which, however, is more or less distorted, owing to the armature-reaction.

§ 136. Experiments on Reactions of the Armatures.—In a recent Memoir of J. Hopkinson and Wilson' the calculations of the preceding articles have been tested on two machines of Siemens Brothers, which were investigated both as dynamos and as motors. In the first place, the distribution of potential around the commutator was determined by means of an insulated auxiliary pair of brushes, capable of rotation, and the curves plotted. These show the characteristic distorting influence of the reaction of the armature already referred to, which differs according as the machine is used as a dynamo or as a motor. An agreement was observed with the above theory which must be considered as satisfactory, having regard to the numerous sources of error in such experiments.

In the machines investigated, the armature had a larger section S, than the limbs of the magnet, so that the first member of the Hopkinson equation [(I), § 128] could be neglected. Only the air-gap (L, S2), the pole-pieces, limbs, and yoke were taken into account. The latter had the total length L and mean section S3. The equation is thus simplified, and, apart at first from armature-reaction, has the following form: (13)

v

0·4πn I1 = 2 L2
ΑπηΙ

S2

+ Laf (10)2)

S

2

H

which, by introducing the Hopkinson function of § 96, we can abbreviate as follows:

is thus the magnetic characteristic for the armature current I 0. In Drs. Hopkinsons' first paper a general equation

=

1 J. Hopkinson and Wilson, Proc. Roy. Soc. Feb. 1892; Electrician, vol. 28, p. 609, 1892. Reprint, pp. 136-147, New York, 1893.

EMPIRICAL FORMULÆ

209

has already been deduced, in which the armature-reaction was allowed for. By introducing the function H it became (loc. cit., p. 108):

Η

The letters have the same meaning as before. It was remarked at the outset that in this equation 6, appears as a function of the two independent variable currents I and I, and, strictly speaking, can only be represented graphically by a surface. This characteristic surface' was geometrically discussed in the place referred to. From this was more especially deduced the ordinary two-dimensional characteristic of a series machine, for which the two variables II coincide. Among others, it was shown that in certain conditions the flux of induction through the armature could attain a maximum, and then decrease, in accordance with the known fact that in series machines the electromotive force often attains a maximum, and afterwards considerably decreases.

=

The somewhat complex equation (15) was tested by Hopkinson and Wilson on the above machines, and found to be confirmed within certain limits. In this one machine was used as a dynamo and the other as a motor, the field magnets being separately excited by a battery. For more details we must refer to the original, as well as for further investigations on armature-reaction, which are of less interest for the theory of the magnetic circuit; we must be content with a reference to the works cited in the note on p. 189.

§ 137. Empirical Formula.--We have shown in the preceding how from the normal curves of magnetisation or of induction (i.e. for those which hold for endless shapes) we can obtain in a rational, though only approximate manner, the relation which exists between the useful flux of induction through the air-gap, and the total magnetomotive force of a dynamo. The methods developed may be best exhibited and worked out graphically; we have hitherto made no attempt to obtain an analytical expression for the curves.

Р

Such expressions for magnetic curves have long been sought, though it is not easy to see why an approximately correct empirical formula should better satisfy the human tendency to enquiry, or be better fitted for solving practical problems, than a curve which is also empirical, but which at least represents the actual processes exactly as they have been observed. In the course of time a great number of the most varied empirical formulæ have been proposed, often without any regard as to whether they held for perfect or for imperfect magnetic circuits, with more or less extensive air-spaces intervening. The latter circumstance, however, constitutes a fundamental difference, and, from what has been said, needs no further comment.

In § 118 we have already mentioned the tan 'formula which Kapp proposed, after the precedent of J. Müller and von Waltenhofen, and applied to the magnetic circuit of a dynamo. For a critical historical account of all the formulæ proposed, we must be content with a reference to special works, as they can only claim historical interest.' One formula, already frequently mentioned (§§ 33, 121), we will, however, consider more closely; this corresponds to O. Frölich's so-called 'curve of active magnetism.'

§ 138. Frölich's Formula. This formula was established empirically by its inventor, as giving the simplest analytical representation of the curves observed with dynamo machines. It has certain points of connection with a first approximation formerly given by Lamont to the expansion in series of an exponential formula. Frölich's formula forms the basis of his own theory of dynamo machines, as well as of that of Clausius.*

2

This formula, according to Frölich (loc. cit., p. 11), is 'the same for all machines, and the only individual feature is the scale on which the abscissæ are plotted'; and we may, there

1 G. Wiedemann, Elektricität, vol. 3, § 450, et seq. 1883; Silv. Thompson, Dynamo-Elect. Machinery, 3rd edition, pp. 302-311, 1888.

2 Lamont, Handbuch des Magnetismus, p. 41, 1867.

3 O. Frölich, Die dynamoelektrische Maschine, 1886.

4 Clausius, Zur Theorie der dynamoelektrischen Maschinen,' Wied. Ann. vol. 20, p. 353, vol. 21, p. 385, vol. 31, p. 302.

FROLICH'S FORMULE

211

fore, represent it by as general an equation as possible, as

In this a is proportional to the 'ampere-turns' of the field magnet, y to the useful or effective' flux of induction through the air-gap. The latter would from the above formula attain an asymptotic maximum value for x, which, strictly speaking, does not hold for induction (fig. 3, p. 19); of course, the formula only claims approximate validity within the range used in practice. That assumed maximum value is at the same time chosen as unit of ordinates, for it appears from the formula that for x = ∞ y = 1

On the other hand,

for x = Q, y =

1

Q is therefore that value of x for which the iron is half saturated,' or, as Silv. Thompson has appropriately expressed it, has attained its' diacritical point.'

Frölich's empirical formula, as is at once seen, represents a hyperbola, one asymptote of which runs parallel to the x-axis at

a distance 1.

§ 139. Relation of Frölich's to other Formulæ.-There is a certain interest in comparing this curve of active magnetism' 2 with the curve of magnetisation deduced by the author in a rational, though only approximate, manner for bodies with a high factor of demagnetisation. A hyperbola was also here found, one asymptote of which was identical with the preceding; its equation [§ 33, equation (18)] was

Here N is the factor of demagnetisation, P a second constant which depends on the nature of the ferromagnetic

' Silv. Thompson, Dynamoelectric Machinery, 3rd edition, p. 307, 1888. Du Bois, Verhand. der Sections-Sitz. des Elektrotechn. Kongress, Frankfurt, p. 75, 1891.