To be presented at the 32d A nnual Convention of the American Institute of Electrical Engineers, Deer Park, Md., June 29, 1915. Copyright 1915. By A. I. E. E. (Subject to final revision for the Transactions.) THE RELUCTANCE OF SOME IRREGULAR MAGNETIC FIELDS BY JOHN F. H. DOUGLAS ABSTRACT OF PAPER The development of the idea of magnetic reluctance is sketch­ ed, and the mathematical and experimental methods of de­ termining it are first discussed. On the theoretical side it is the -theory of complex quantities which leads to numerical re­ sults. The electric field of an. electric generator is next analyzed and made to depend approximately upon the reluctance of two- dimensional magnetic fields. This study indicates as preferable the one shown in Fig. 6 at the left. The reluctance of these various magnetic fields is determined by experiments made on sheets of high resistance metal cut in suitable forms, and by mathematical computation. These re­ sults are put into charts and formulas convenient for the use of those interested in electrical design. The results of the tests are proved to be more accurate than most of those already published. In particular, present leakage flux calculations are most in error. Of more general interest are the plates of the shape of the magnetic lines of force in the various parts of electrical machinery. In particular, Figs. 42 to 48 include some new results. The flux lines in the corners of transformers and induction motors are there shown, and the exact solution to the much discussed sine-wave alternator problem is there given. The paper is divided into (I) Introduction (II) Historical Development, (III) Description of Experiments, (IV) Accuracy of results, (V) Conclusions. There are two appendices giving some mathematical details, for reference purposes. I. INTRODUCTION F WE apply %a difference of magnetic potential between I two portions of the surface bounding a region of space, a certain amount of magnetic induction will be developed. If no induction crosses the remainder of the surface, we have a region which is a tube of induction, carrying a definite amount of flux and consuming a definite amount of magnetic potential or mangetomotive force. All such magnetic fields possess a definite amount of reluctance, R, which is the ratio of the applied magnetomotive force, M, to the resulting flux, φ, thus, R = Μ/φ (1) Manuscript of this paper was received August 6, 1914. 867 868 DOUGLAS: RELUCTANCE [June 29 If there is no magnetic saturation present this reluctance is a constant. Only fields in media without saturation are con­ sidered in the following paper quantitatively. A few flux dis­ tributions in iron are, however, considered qualitatively. The reciprocal of reluctance is called permeance or magnetic conductance. It is denoted by P; thus, P = \/R = φ/Μ (2) If the lines of flux flow are everywhere parallel and the equi- potential sections are everywhere the same, the magnetic field can be said to be regular. Such a field has a reluctance which can be readily expressed in terms of its dimensions. Thus, R = L/μΑ (3) where L is the length of the magnetic flux path, A the area of its cross-section, and μ the permeability of the medium. The above equation can not be used for an irregular magnetic field, except for an infinitesimal portion of a tube of induction, where infinitesimal values of L and A may be taken. Re­ course must be had, therefore, to the fundamental definition of reluctance in equation (1) by considering the mutual relations between M and φ. In this paper attention will be limited to some irregular two- dimensional magnetic fields; that is, to fields in which the flux lines are all parallel to some plane surface. The results, how­ ever, will be extended to apply to certain three-dimensional fields, by making approximate assumptions. Owing to the fact that there is no magnetic insulator known, the magnetic fields around dynamo electric machinery is very irregular. There is fringing of flux in the air-gap, and leakage flux between the poles and around the slots. As a consequence designers have to use various correction factors, which even yet are more or less empirical. A study of the field of typical electrical machines was made in search of forms of magnetic circuit which needed study. In addition, on the mathematical side, various cases were canvassed systematically. Beginning with those simple shapes which led to simple integrals, study was made of all possible forms up to those which involved in­ tegrals more difficult than the elliptic integrals of the first and second kind. Of all these cases those promising any use were selected for study. It is in the hope that some of the cases which are worked out here might be of use, that the experi- 1915] DOUGLAS: RELUCTANCE 869 ments and the mathematical work were carried out. Some of the results may be of use in analogous electrical problems. The exact significance of equation (1) can best be obtained by reviewing briefly the historical development of the subject, tracing out the gradual unfolding of the conceptions involved. This will form the first natural division of the subject. The various experimental methods, which have been used will be described, and discussed in the next section of the paper. Before describing a set of experiments made with high resist­ ance metal strips, an analysis of the magnetic field surrounding an electrical machine will be made. The simplifying assumptions, made in picking out certain two-dimensional fields as applying to what is in reality a three dimensional problem are sufficiently serious to warrant dis­ cussion. Before passing on, thereforer to the conclusions, the accuracy of the results will be compared with that of the present methods. The detailed mathematical treatment of the problem of the reluctance of irregular magnetic fields, is not included in this paper, because of its complexity and too special interest. Never­ theless the writer went over the ground, both mathematically, and experimentally, and found both methods needful for se­ curing the best results. Those interested in this side of the subject, can consult the references given in the historical section of this paper. Those who prefer to use an analytical formula to a curve of permeance, can consult an appendix to this paper where the mathematical results are given. Another reason for their inclusion is that, while th3 figures were drawn carefully, greater accuracy might in some casas be desirable. II. HISTORICAL DEVELOPMENT Equation (1) is sometimes called Ohm's law for the magnetic circuit, from its similarity to the law of the same name in the electric circuit. This similarity is made the basis of the ex­ periments described in this paper. Although simple in its final formulation, the law was discovered at a comparatively late date, because the ideas implied were of gradual growth. The existence of a potential function, the solenoidal character of magnetic force, the identity between electromagnetism and permanent magnetism, the existence of magnetic induction within iron, the idea of a closed magnetic circuit, were all needed for the final formulation of Ohm's law for the magnetic circuit. 870 DOUGLAS: RELUCTANCE [June 29 The existence of potential, as a function whose space deriva­ tives give the components of the fields of force, is due to La Place1 who used it with reference to the attraction of gravita­ tion. The name itself is due to Green2 who proposed it in his prize essay published in 1828. La Place showed that gravita­ tional potential satisfies the equation dW dW d'V Ί^ + W + ~d* = ° (4) which is known by his name. This equation followed from the law of the inverse square of the distance discovered by Newton.3 The suitability of such a function in the case of magnetism must have been apparent, because magnetic attraction was already recognized as a case of central forces. In 1785 Coulomb4 demonstrated that the inverse square law was true for magnetism. The application of potential to magnetic theory followed at once. In 1813 Poisson5 called attention to a limitation to La Place's equation showing that it did not apply to the interior of the attracting body. In 1827 he derived the correct equation for the space within an attracting mass.6 Owing to this difference, for a time the magnetic field inside a magnet was thought of as entirely different from that outside of the same. In July 1820 Oersted7 discovered electromagnetism, and a very short time after that Ampere8 showed the equivalence of an electric circuit with a magnetic shell. Thus, the existence of a magnetic potential was proved for all points of the mag­ netic field except those within the magnetizing coil. The variation of magnetic potential within the substance of iron, however, was still unknown. These same experiments of Am­ pere's furnished us also with the idea of magnetic potential as 1. Mémoires de ΓAcadémie Royal de Paris, 1782. Mechanique Celeste Book. 3, Chap. 2. 2. Collected Papers p. 25. 3. Principia, (1687) Book 3, Prop. 1-7." See also Thomson and Tait, " Natural Philosophy" Part 2, p. 9. 4. Mémoires de l'Académie Royal de Paris; 1785 p. 583; 1788, pp. 587 and 603. 5. Nouveau Bui. Soc. Philomatique Vol. 3, pp. 3S8-392. 6. Mémoires de l'Académie de Paris Vol. 6, July 10, 1827. 7. Schweigger's Journal Vol. 29, p. 273. 8. Ann. de Chemie et Physik. 1820. Gilbert's Annalen, Vol. 67, (1821). 1915] DOUGLAS: RELUCTANCE 871 related to an electric current. The unit of magnetic potential, however, came later. The concept of electric current as the universal cause of magnetism, even in permanent magnets was the last step in the development of the theory of magnetic potential.9 In magnets re\^olving electrons within the atom constitute the source of magnetic potential.