actually constructed and in more or less extensive use, are represented diagrammatically, according to Silv. Thompson; ' and they are so chosen that those diagrams are not given against which serious objections may be raised from a magnetic point of view. The order of the figures A-O is progressive from the simplest to the most complicated magnetic arrangement. The ferromagnetic parts which produce the field are cross-shaded, and the current-producing armatures are simply shaded. The names of the constructors are not given, since many of the arrangements are used in various machines with slight modifications, and this is a matter of less moment for the objects of the present book.

Single magnetic circuits are met with in the machines represented in figures A to E, several of which are extensively used. The arrangements are sufficiently clear from the figures, so that any further description is needless.

Double magnetic circuits are represented in figures F to J. The latter shows an arrangement in which the magnetising coils are wound round the armature instead of round the limbs of the field-magnets. This interesting mode of winding has been recommended by different electricians on the ground that the total flow of induction is thereby utilised, and there is no useless leakage. Armatures thus wound do not, however, allow of sufficient ventilation.

Multiple magnetic circuits are, finally, shown in the diagrams K to O. In K the last-mentioned principle of winding is, in a certain sense, met with in hollow cores of magnets and spherical armatures. The magnetic circuit is closed by a number of iron rods, the top and bottom ones of which are represented in the figure. L and M represent fourfold magnetic circuits; in the latter machine the armature is outside the field-magnets, as has

1 Silv. Thompson, Dynamoelectric Machinery, 4th edition, chapter viii., from which the substance and the figures of these last paragraphs are taken with kind permission of the author. In Kittler, loc. cit. figs. 462–466, over 60 such diagrams of magnetic circuits are depicted.

2 Compare Grotrian, Wied. Ann. vol. 50, p. 737, 1893, and du Bois, ibid. vol. 51, p. 536, 1894, where the magnetic circuit of such machines is subjected to discussion. (See also note 2, p. 263.)

DIAGRAMS OF VARIOUS MAGNETIC CIRCUITS

223

recently been often arranged (§ 140). N, finally, represents a sixfold, and O an eightfold, magnetic circuit.'

The classification of machines according to the number of magnetic circuits is perhaps the most rational from the magnetic point of view; in practice, indeed, we still frequently speak of non-polar, unipolar (compare note, p. 219), bipolar, multipolar, as well as external polar, internal polar, consecutive polar machines, and the like.

224

CHAPTER IX

MAGNETIC CIRCUIT OF VARIOUS KINDS OF ELECTROMAGNETS AND TRANSFORMERS

A. Physical Principles

§ 147. Magnetic Cycles. In the present chapter we shall continue to treat practical examples of the applications of the principles which hold for magnetic circuits; but from the great variety of arrangements we shall only select those which are typical, and have special theoretical interest. We have, however, previously to discuss the results of experimental investigations and theoretical considerations which have come into account, and which refer more particularly to the alteration of magnetic states. We shall first consider the phenomenon of magnetic hysteresis. In § 8 we have defined its general character, but in our subsequent developments we have always disregarded hysteresic processes. We shall limit ourselves, at present, to stating the main features of this important phenomenon, referring for experimental details to works in which ferromagnetic induction is more completely dealt with than is consistent with the object of the present book.'

We will, after Warburg,2 subject a ferromagnetic substance of endless shape to a magnetic cycle, whereby we cause the magnetic intensity to pass through all values from - He to + Ha, and back again too, where may have any chosen limiting value. Hysteresis then asserts itself by the fact that the corresponding successive values of magnetisation do not lie in a single curve, but, with a sufficient number of repetitions. of the process, on a perfectly definite loop. For instance, the

1 See Ewing, Magn. Induction, &c. chap. v.

2 E. Warburg, Wied. Ann. vol. 13, p. 141, 1881; Warburg and Hönig, Wied. Ann. vol. 20, p. 814, 1883. See also E. Cohn, Wied. Ann. vol. 6, p. 388, 1879.

curve, fig. 36, A, represents such a typical hysteresis loop, which corresponds to the behaviour of a specimen of annealed steel

FIG. 36.-ANNEALED STEEL PIANOFORTE WIRE, He = 20 C.G.S. units;

u = 84,000 ergs per cubic cm.

pianoforte wire. The numerical values are given in round numbers for the sake of clearness. The range of abscissæ is restricted to the values (- 100 << + 100), while the corresponding range of magnetisation amounts to (-1000 <3<+1000). The arrows denote the so-called increasing or decreasing branches of the curves corresponding to the ascending or descending values of magnetisation respectively. The curve OG, that with no arrow, is what is called the curve of ascending reversals,' which has always formed the basis of the preceding considerations (§ 85). On sufficiently frequent repetition, a definite loop of greater or less extent, according to the range of magnetisation, corresponds to each such cyclical change of intensity between given limits.

§ 148. Dissipation of Energy by Hysteresis.-The chief property of hysteresis loops lies in the fact, that their area furnishes a measure of the energy u transformed into heat in the cycle in question, per unit volume of the ferromagnetic substance. For, as was shown by Warburg (loc. cit.), and soon afterwards independently by Ewing,

taken between the limiting values of the cycle.' The proof of this fundamental principle, which can be given in various ways, would here lead too far. The heat disengaged produces, in certain circumstances, a rise of temperature of the ferromagnetic substance, which has, however, but a small value for a single cyclebeing of the order of about the thousandth of a degree-but is very appreciable with frequent repetitions of the process.

A complete cycle is, strictly speaking, one which ranges between the limits of intensity - ∞ and ∞, in which the maximum magnetisation obviously represents the range of ordinates. For most purposes, however, it is sufficient to keep in view high finite values of intensity. Neither the limiting value

The second expression for u not only holds with the same approximation with which we can often write B = 43, but is perfectly exact, since the integral fd must necessarily vanish for all closed loops. Strictly speaking, equation (I) holds only under definite assumptions as to interchange of heat; for instance, for isothermal or for isentropic (adiabatic) cycles (see Warburg and Hönig, loc. cit., p. 817; Ewing, Proc. Roy. Soc., vol. 23, p. 22, 1881, and vol. 24, p. 39, 1882). For most purposes, however, this is of small importance.