The Reluctance of Some Irregular Magnetic Fields

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To be presented at the 32d A nnual Convention of the American Institute of Electrical Engineers, Deer Park, Md., June 29, 1915.

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 equipotential 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. The idea of a magnetic flux is also of gradual growth. At first, magnetic effects were considered as being exerted at a distance and manifested only when iron filings, a compass needle, or other magnetic body was present. However the labors of Gauss and Weber with their charts of magnetic fields served to make the idea of fields of force more tangible. In 1813 Gauss10 proved that the surface integral of magnetic force over any closed surface is proportional to the amount of at tracting matter within it. This he proved directly from the law of inverse squares. The constancy of this integral could not help but suggest the idea of a flux or flow of force, constant in amount, from one magnet pole to another. This so-called solenoidal character of magnetic flux was, however, really im plied in La Place's equation.

If HXy Hy> and Hz are the three components of the magnetic field parallel to the three co-ordinate axes, and if the field is solenoidal,

dHx dHy d Hz _ ~ ,Rv dx dy dz

That electromagnetism was solenoidal followed at once from Ampere's discovery that a current was equivalent to a mag netic shell. In 1831 when Faraday11 discovered magnetic induction, he was inevitably lead by the idea of cutting lines of force to the conception of the physical reality to be ascribed to them. His experiments in 185112 confirmed the solenoidal character of magnetism within the substance of the magnet itself. The concept of tubes of force and of induction followed at once. The belief in a magnetic circuit of flux was of gradual ac- 9. Werner v. Siemens, Wiedeman's Annalen, Vol. 24 (1885), p. 94; Larmor, " Aether and Matter," (1904), p. 108. 10. Werke, Band 5, p. 9. 11. Experimental Researches, First Series, Vol. 1. 12. Ibid, Vol. 3, Sects, 3077 and 3109. 872 DOUGLAS: RELUCTANCE [June 29 ceptance. At first the identity of the tubes of induction out side the magnet with those inside was not clear. The term Magnetic Circuit was first used in 1833 by Ritchie,13 and the term closed magnetic circuit in 1853 by Dub14 and De la Rive.15 The connection between magnetic force and induction in iron and the distinction between them, outside magnetic substances, was all that was then needed before Ohm's law for the magnetic circuit could be formulated. These were supplied by the discoverer. The earlier theory of induction in iron took into account two distinct actions going on at once. First, there was a magnetic field inside the iron but due to outside influences. Second, there was a field due to the polarization of the iron itself. Al though this conception is helpful, it delayed the formulation of Ohm's law for the magnetic circuit. It was on the basis of this theory that the early determinations of the quality of iron was made. In 1854 Kirchofï16 calculated the susceptibility of iron from some of Weber's experiments with an iron ellipsoid. In January 1873 Stoletow17 first used the ring method of testing for determining the quality of iron. We owe the statement of Ohm's law for the magnetic circuit to Rowland,18 who not only determined the experimental re lation between magnetic force and induction within iron, but also distinguished between the two. He originated the con ception of magnetic force as a localized magnetomotive force. That is, magnetic field intensity was the gradient of magneto motive force, rather than the flux density which would be set up in air, were the magnetic substance removed. The name of permeability as the ratio of magnetic induction developed in iron to that produced in other bodies is due to Lord Kelvin,19 who attributed the idea of magnetic conductivity to Faraday. However, this point of view can be traced back as far as 1821, when Cumming20 measured crudely what corresponded to mag netic conductivity rather than susceptibility. This was a rather close approach to the idea underlying Ohm's law for 13. Phil. Mag. Series 3, Vol. 3, p. 122. 14. Poggendorf's Annalen, Vol. 90, p. 436. 15. Treatise on Electricity, Walker's Translation, Vol. 1, p. 293. 16. Crelle's Journal, Vol. 48, p. 374. 17. Phil. Mag., Series 4, Vol. 45, p. 40. 18. Phil. Mag., Series 4, Vol. 46, p. 140, (Aug. 1873). 19. Papers on Electricity and Magnetism, p> 484, (1872). 20. Camb. Phil. Trans., Apr. 2, 1821. 1915] DOUGLAS: RELUCTANCE 873 the magnetic circuit. Permeability as the permeance of a unit cube is of course due to Rowland. Rowland also proposed, in his article, the ampere-turn as the unit of magnetic potential difference. In opposition to this Bosanquet21 later proposed as a unit (10/4π) of an ampere- turn. He also coined the term " magnetomotive force.'' This unit of Bosanquet's has persisted in spite of the priority of Row land, and in spite of the return to the confusion between field in tensity in air and induction in air, apparently because of the great influence of Maxwell. Maxwell22 spoke of the intensity of a magnetic field as defined by the result of excavating a cavity in the iron of a particular shape. This " mouse and cheese" theory has always been a hindrance to electrical engineers, and manufacturing companies have always used magnetization curves in the form developed by Rowland.23 Recent Text books24 still disagree as to the issue thus raised. The term reluctance for magnetic resistance is due to Heavi- side,25 while the term permeance is due to S. P. Thomson.26 The use of reluctance and Ohm's law for the magnetic circuit of dynamos was first brought to the attention of engineers by Hopkinson,27 who however introduced a modification by using curves of induction plotted against field intensity, or B-H curves. The suitability of Ohm's law for parallel as well as series mag netic circuits was pointed out first by Kapp.28 The beginning of the theory of the reluctance of two-dimen sional magnetic fields, was in the mathematical theory of goedetic lines of various kinds. This was next applied to the flow of electric currents in two dimensions and finally extended to magnetic problems. Maxwell attributes the application to the electric current to Prof. W. R. Smith,29 and develops it himself in Chap. XII of his " Treatise." The following points are worthy of notice. (1) His treatment involves the theory 21. Phil. Mag., Series 5, Vol. 15, p. 205, (Mar. 1883). 22. Electricity and Magnetism, Vol. 2, Sect. 398, p. 24, 3rd Ed. 23. Ayrton and Perry, Jour. Soc. Teleg. Eng. & Elee, 1886, p. 518. 24. Karapetoiï, " Magnetic Circuit," 1st Ed. p. 12 and p. 266. Pender, 11 Principles of Electrical Engineering," Chap. II. 25. Electrician, (London), Vol. 21, (1888), p. 27. 26. Dynamo Electric Machinery, 4th Ed. (1892), p. 186. 27. Phil. Trans. Royal Soc, 1886 Part 1, p. 331. 28. Jour. Soc. Teleg. Eng. & Electr., 1886, p. 530. 29. Proc. R. S. Edin., 1869-70, p. 79. See also Treatise on Electricity and Magnetism, 3rd Ed., p. 185. 874 DOUGLAS: RELUCTANCE [June 29

of the conjugate functions of a complex variable. (2) Although this is the usual basis for the theory, his conclusions are inde pendent of this method of treatment. (3) Part of this theory- is essential to an understanding of the graphical and the experi mental methods. (4) His exposition is confusing to those who do not care to go into the mathematical method deeply. (5) A brief outline of this theory is necessary. The reluctance of an irregular two dimensional magnetic field, such as " OAPB" in Fig. 1, is R = R'v/t (6) where v is the reluctivity or reciprocal of the permeability μ, t is the thickness, and R' is a constant which depends only upon the shape of the field. If we always work with a thick ness t = v (7) of the field, we see that Rf is the reluctance of such a slice and is independent of the material. R' may be called the geometric re luctance of the field.30 Each line of flux may be char acterized by a number called the F - flux function. This is defined as the amount of flux passing between this line of flux and an arbitrary point or zero of reference, and within a thickness of the field equal to its reluctivity. The flux function at a point is that belonging to the flux line passing through that point. The geometric reluctance R' is expressible through the flux and potential functions. Thus in Fig. 1, the geometric reluct ance of OAPB is,

i^OAPB = (νρ-νο)/(ΦΡ-φθ) (8) The total reluctance of a slice of any thickness can be obtained by combining equations (6) and (8). A study of the reluctance of two-dimensional irregular magnetic fields is nothing else tnan the study of the graphical and analytical relations between the potential and the flux functions. Let us consider the field in Fig. 1 as divided by lines of force at equal increments of the flux function Δφ, and alsobyequi- 30. Karapetoff, " Magnetic Circuit," Chap. 5, p. 93. 1915] DOUGLAS: RELUCTANCE 875 potential lines at equal intervals Δ V which are equal to Δφ. Let us consider that the field is divided also by parallel planes into laminae of a thickness / = v. The field is divided thus into cells each of which has a reluctance of unity, and all the angles of which are right angles. Moreover if the intervals Δφ and AFare sufficiently small, the curvature of the sides of the cells can be neglected. That each cell is a square, is easily seen from equation (3). Putting A = tXw, the thick ness being / and the width of the cell being w, we get, („) (/) (0 («0 Since / is equal to v, the length of any cell I must be equal to its width w. The dimensions / and w are, however, shown in only one of the squares of Fig. 1. The fact that the flux and potential lines form what may be termed curvilinear squares, is made the basis of the graphical method of determining the reluctance of an irregular magnetic field. If we consider in Fig. 1 an infinitesimal square formed by two lines of flux, φ and φ + Δφ, and by the two equipotential lines Fand V+AV, we can see from the geometry of the figure that,

dV = άφ_. _dV_ = άφ dy dx ' dx dy

Functions satisfying these relations are called conjugate func tions. By differentiating the first of the preceding equations with regard to x and the second with regard to y, we obtain, after combining, &+ f - °

W = φ +jV = F (X +jY) = F (Z) (11) we obtain two functions of X and F, that is φ and V, which are conjugate to each other.32 Thus φ and V are related to each other as a two-dimensional flux function and potential function. The properties of rational algebraic and trigonometric func tions of a complex variable were familiar to Cauchy and Euler but precise ideas of what constituted functionality were lack ing until Rieman.33 In 1799 Gauss34 proved that all functions of a complex number were themselves complex numbers. Thus graphically, «for every point (X, F) in the Z plane there is a corresponding one (φ, V) in the W plane. Equation (11) can be thought of as transforming any figure in the W plane to a corresponding figure in the Z plane. In 1822 Gauss35 proved that the transformations effected by equations of the type of equation (11) were conformai; ix. the angular relations were preserved and the corresponding in- 31. Essai sur une manière de represente les quantities imaginaires dans les constructions géométrique. 32. Rieman, Gessamelte Werke, p. 5, see also Webster, Dynamics of Particles and of Rigid, .Elastic and Fluid Bodies, p. 522. 33. Chrystal, Algebra 4th Ed. 1898, Vol. 1, p. 253. 34. Gessamelte Werke, Vol. 3, p. 1. 35. ibid, Vol. 4, p. 192. 1915] DOUGLAS: RELUCTANCE 877 finitesimal parts of the two diagrams were similar. For instance in the equation, t = i+jr) = F{X +JY) (12) we may regard the function F ( ) as transforming an irregular conductor in the (X, Y) plane into another conductor in the (£, η) plane. These conductors are equivalent since the equipotential and flux lines in one conductor are all changed conformally in the new conductor; that is, without changing the shape of the enclosed cells of reluctance. In particular in equation (11), the lines in the Z plane formed by the lines φ = 0, 1, 2, etc. and V = 0, 1, 2, etc. are small curved squares, and any rectangular conductor bounded by flux and equipoten tial lines is converted into an equivalent irregular conductor. If one tries a few simple functions of (X + j Y), and plots the curves of constant φ and V, he can quickly convince himself of the truth of the above statements. The importance of this principle to the theory of the reluctance of irregular magnetic fields, is that it furnishes a means of converting irregular into rectangular conductors, of which the reluctance is known. In 1825 Gauss36 showed that every conformai change could be represented by equation (12) provided only that the proper functions be found. This was equivalent to proving that every irregular conductor could be converted by this equation into an equivalent rectangular conductor. While it would be more convenient to make this conversion directly, it is generally found necessary to make the conversion in two steps through an intermediate irregular conductor equivalent to both. A conductor whose only boundary is an infinite straight line is found to be a most convenient intermediate form. As a simple example of the use of the above method, let us take the function, φ + jV = (X + jY)2; φ = X2-Y2 ; V = 2 X Y. the lines of flux and equipotential are mutually orthogonal hyperbolas. We can use this function therefore to find the reluctance of any figure bounded by these lines, such as that in Fig. 2. Thus

Va-Vb 2Xa Yg-2Xb Yb 2 2 2 2 φΡ-φο (Xp - Yp ) -{Xo - Yo )

2.5·5-2·5·(-5) 100 = 1 (32-02)-(02-02) 9 9 36. Schumachers Astr. Abh. 1825, Ges. Werke, Vol. 4, pp. 189-216. 878 DOUGLAS: RELUCTANCE [June 29

From the point of view of equivalent conductors, the above equation may be said to convert the irregular conductor in Fig. 2 into the rectangle V = 50, V = - 50, φ = 0, and φ = 9. The method illustrated by this problem has its limitations; for the proper function needed to suit a particular boundary is not always known. The transformation of a polygonal area conformally into a half plane has been known since 1867, and is due to Christ off el37 and Schwartz.38 This is equivalent to converting a polygonal conductor into its equivalent conductor having a straight line boundary. We are then able to find its reluctance easily by converting it a second time by the same theorem into a rectangle. Let it be desired to convert the polygonal conductor shown in Fig. 3 into its equivalent conductor consisting of an infinite

FIG. 2 SCHWARTZ'S THEOREM half plane with a straight line boundary. Let Z = (X + jY) be the co-ordinates of any point in the polygonal conductor, and let / = ί + ]η be the two rectangular co-ordinates of the corresponding point in the equivalent conductor with the straight line boundary. Then Schwartz's and Christoffers theorem states that,

C(t-h)*> (t-h)\ ... (t-tn)*dt (13)

where θι, θ2, etc. are the angles shown in the figure, being plus when measured toward the right and minus when measured to the left. The constants h, h, etc. at the corners of the boundary 37. Annali di Matematica, Vol. 1, (1867), p. 89. 38. Creile, Vol. 70 (1869), p. 105. See Also, J. J. Thomson, " Recent Advances in Electrictiy and Magnetism" p. 208. 1915] DOUGLAS: RELUCTANCE 879 are the co-ordinates of the corresponding points in the straight line boundary of the Kt plane. Three of these constants are arbitrary, and the remainder are determined by the geometry of the polygonal conductor. The constant C determines merely the size and orientation of the equivalent conductor as a whole. There is of course another equation similar to equation (13) which transforms the conductor in the half / plane into a rec tangular conductor in the W plane, whose co-ordinates are φ and V. When the equations of the lines of force and equipotential surfaces are desired, it is necessary to keep the signs and argu ments of the various terms in equation (13) perfectly correct. When, however, the reluctance or resistance only of the irregular field is desired, without regard to the shape of the flux and equi potential lines, a simpler method may be used. In this case we integrate equation (13) without regard to argument, using only the moduli of the various terms, from one corner of the figure to another. In other words, the equation is inte grated from ti to /2, h to h etc. This method makes graphical integration possible. The first use of Schwartz' and Christofïers theorem for an electrical problem was made by Kirchofï39 who used it to de termine the capacity of plate condensers. The first suggestion of using the method in magnetic phenomena is due to F. W. Carter,40 who adapted Kirchoff's and Thomson's results to the pole-tip fringing in an electric machine. J. J. Thomson41 used the method in his book " Recent Advances'' to find the capacity of the guard-ring electrometer and F. W. Carter42 applied the result with some improvements to slot and air-duct fringing. The results are limited theoretically to the case of infinitely broad teeth, and infinitely deep slots. C. H. Lees43 has de termined the effect of a sudden change in the width of a con ductor upon its resistance. He also found the effect of a narrow saw-cut on its resistance.44 The writer has applied the method generally to check his experimental results. Since these com- 39. Monatsberichte der Akad. der Wiss. zu Berlin, Vol. 15, p. 144. See also J.. J. Thomson, " Recent Advances" Chap. 3, Section 238. 40. Inst. Elee. Eng. Jour. Vol.« 29 (1900), pp. 925-933. 41. Chap. 3, Sect. 241, p. 227; and Sect. 243, p. 233. 42. Electrical World and Engineer, Vol. 38 (1901), p. 884. See also Inst. Civil Eng. Proc, Vol. 187 (1912), p. 311. 43. Phil. Mag., 5th Series, Vol. 16 (1908), pp. 734-739. 44. Phys. Soc. Proc, Vol. 23 (Aug. 1911), pp. 361-366. 880 DOUGLAS: RELUCTANCE [June 29 putations are not included, it should be stated that the agree ment was in all cases very close. An application of this theorem to hydrodynamical problem has been made by Mitchell45 and Love46. There are a few other more or less complicated cases where Schwartz' theorem has been applied, but these will not be quoted because they seem to be mostly of academic interest. For irregular conductors bounded by circular arcs, reference may be made to Forsythe Theory of Functions 2nd Ed., Art. 271, p. 644. For the flux and potential problems in thin curved sheets, see Darboux, " Theorie General des Surfaces."47 The reluctance or resistance of irregular fields in three di mensions does not in general admit of solution. Indeed, there is not such thing as a flux function, except in a few cases. One of these is when the bounding surfaces and all the flux and po tential lines are in surfaces of revolution. These cases are fre quently met with in electrostatic problems, and therefore offer scope for future work along these lines. The following refer ences are given to aid further study.48 Closely allied to the analytical method of conjugate func tions is a graphical method due to L. F. Richardson49 and to Dr. Th. Lehman.50 It is quite fully developed by these authors for a two-dimensional field. It depends essentially upon the fact that the small square cells of such a field can be quite ac curately drawn by the judgment of the eye, and also upon the fact that it is easy to adjust them correctly by the use of a pair of dividers. In this way various irregular fields can be drawn, and by counting the number of squares in series and in parallel, the reluctance of the field can be obtained. Mr. Richardson points out how the method may be used in the case of irregular fields bounded by surfaces of revolution, and also to the case of bodies made of a material showing saturation. Use was made of this method in drawing Fig. 26. A very striking experimental method of determining the 45. Phil. Trans. Royal Soc, 1890 part A, p. 389. 46. Proc. Camb. Phil. Soc, Vol. 7 (1891), p. 175. 47. Vol. 1, Book 2, Chap. 4, pp. 170, 176-180. 48. Stokes. " On the Steady Mqtion of Incompressible Fluids" Camb. Phil. Trans. Vol. 7, 1842: Math, and Phys. Papers, Vol. 1, p. 15. Sampson. " On Stokes' Current Function" Phil. Trans. 1891 * Part A. 49. Phil. Mag., Series 6, Vol. 15, Feb. 1908, p. 237. 50. E. T. Z. Vol. 30, (1909) pp. 995-1019. Lumiere Electrique, Vol. 8, No.' 43, p. 403, 1915] DOUGLAS: RELUCTANCE 881 shape of the lines of force in a magnetic field is due to Hele- Shaw, Powell, and Hay51 using the property that a viscous fluid in steady motion flows according to the same laws as an electric current. In order to render the stream lines visible, they.injected thin lines of coloring matter. They controlled the permeability of the medium, by varying the thickness of the layer of fluid. They determined in this way the field re sulting when cylinders, rings, toothed and shuttle armatures, and other variously shaped bodies are introduced into a magnetic field. They checked in this manner the theoretical work done by Carter.52 They were followed by Dr. W. M. Thornton,53 who used the method for determining the flux distribution in the armature cores of electrical machines. These plates are reproduced in a paper by Hansen before the A. I. E. E.54 An application of the same method for determining the electro static fields in a three-phase cable has been made by Thornton and Williams.55 In order to use the above method to determine the actual re luctance of an irregular field of force, measurements were made of the pressure of the liquid supply, the quantity of flow, and the coefficient of viscosity. The most useful experimental method of determining the reluctance of irregular magnetic fields, is to make use of the similarity of the electric and the magnetic circuits, and measure the resistance of the corresponding irregular electric circuit. Kirchoff56 was the first to propose determining the equipotential lines and resistances of an irregular two-dimensional electric current field, by the use of a specially cut sheet of metal and specially fastened electrodes. Application of the method to other cases was made by G. Carey Foster and Sir Oliver Lodge57 in 1875, who gave a good bibliography of this part of the subject. Gaugain58 modified this method by using liquid conductors such as Mercury, Zinc Sulphate, and Copper Sul phate. W. G. Adams59 developed KirchofFs method, making 51. Electrician (London), Vol. 54, Nov. 25, 1904, p. 213. Phil. Trans. Royal Soc. Vol. 195, 1900 Part A, p. 303-327. 52. Electrical World and Engineer, Loc. Cit. 53. Jour. Inst. Elee. Eng., Vol. 37 (Feb. 26, 1906), p. 125. 54. A. I. E. E. TRANS. Vol. 28, June 30, 1909, pp. 993-1001. 55. Engineering, Vol. 88, Aug. 27, 1909, p. 297. 56. Poggendorf's Annalen, Vol. 64 (1845), p. 497. 57. Proc. Phys. Soc. of London, Vol. 1, pp. 113-193. 58. Ann. de Chemie et de Physik, No. 3, Vol. 64, 1862, p. 200. 59. Proc. Royal Soc. 1875, Vol. 23, p. 280. 882 DOUGLAS: RELUCTANCE [June 29 suititable for three dimensions, and using liquid instead of solid conductors. The use of an electrical experiment upon thin sheets of an electrical conductor, for determining the reluctance of the corresponding magnetic fields was proposed by F. W. Carter60 in 1901. He himself, however, used only the mathematical method. He also proposed the use of the conjugate character of the flux and potential functions to determine the shape of the lines of force. Since a millivoltmeter can only be used to trace equipotential surfaces, if we desire to trace lines of flow, we must contrive to interchange the two sets of lines. Mr. Carter proposed cutting a metal strip, to the shape of the de sired irregular magnetic field, but to solder the electrodes to those parts of the boundary which were flux lines: The re maining or desired equipotential portions of the boundary were to be left free. In this way, an interchange of the roles of flux and potential functions is secured upon the boundary, and con sequently throughout the rest of the strip also. The suggestions of Mr. Carter were first carried out by Mr. C. H. Smoot61, who used the method to check experimentally the reluctance of a case solved theoretically by Mr. Carter. He applied the method, also, to get the flux distribution in the air gap of an alternator. Both suggestions of Mr. Carter were used by the writer in the tests described in this paper. An application of the similarity of the electric current field to the electro-static field, has been made in 1913 by Fortescue and Farns worth62, who used a three-dimensional electrical model of various irregular electrostatic fields. Recently R. Forster63 has developed a method of determining the electrostatic field which he applied to a three-phase cable. He used a conduct ing liquid, and to avoid polarization, inverted the functions of the galvanometer and battery circuit of a Wheatstone bridge. The method seems to have points of similarity to one of those used by G. Carey-Foster and Sir Oliver Lodge.64 III. DESCRIPTION OF EXPERIMENTS In order to determine the reluctance of various irregular magnetic fields, use was made of the similarity of the electric 60. Electrical World and Engineer, loc. cit. 61. Jour. Western Soc. Eng. Vol. of 1905, p. 500. 62. A. I. E. E. TRANS., Vol. 32, Part I (1913), p. 893. 63. Archiv für Elektrotechnik, Vol. 2, 1913, No. 5, pp. 175-180. 64. Proc. Phys. Soc. of London. Loc. Cit. 1915] DOUGLAS: RELUCTANCE 883 and the magnetic circuits, and of the conjugate character of the flux and potential lines. The resistances of various irregular sheets of metal were measured upon an apparatus shown in Fig. 4, and the equipotential lines were also traced out. This was done to determine the permeance of the similar magnetic field, and the shape of the lines of force. Measurement of the resistance consisted in setting the stylus attached to the terminals of the mil li voltmeter alternately on the two points between which the resistance was desired, and adjusting the terminal on the slide wire for a null reading. This was easily done, as the millivoltmeter had a zero center, and both a ten and a hundred millivolt scale. Readings were taken of the two ammeters and of the settings upon the slide wire. The main current and the current in the slide wire were maintained constant to suit frequent calibration tests. Since the absolute resistance was not desired, but only the geometric resistance,.calibra tion of the meters was not made. The calibration of the slide wire was effected check ing it against a regular or rectangular sample cut from FIG. 4—DIAGRAM OF CONNECTIONS the same sheet as the irregular USED IN EXPERIMENTS sample, and ruled into exact squares. In this way identical thickness was secured between the irregular sample and the sample for calibration. The current in the slide wire was adjusted so that the drop in twenty centi meters was twenty millivolts. This current was held rigorously constant. The main current was adjusted carefully at each cali bration so that the drop in one square of the calibration sample was equal to that of approximately twenty centimeters upon the slide wire. In any case the exact drop in the calibration sample K was noted. In other words a drop of K centimeters on the slide wire was equivalent to one unit of geometric re sistance or reluctance both in the regular calibration sample, and the irregular sample for testing. Thus to find the resist ance between two points 1 and 2 at which the slide wire settings were Vi and Vi, we divide the difference of the settings by K. Direct deflection of the millivoltmeter was not used except, 884 DOUGLAS: RELUCTANCE [June 29

when under special circumstances, a range beyond that of the slide wire was required. For this reason a drop of twenty millivolts per twenty centimeters of slide wire was chosen so as to make extrapolation easy. In most of the problems investigated it was desired to de termine the shape of the lines of flux, while the millivoltmeter could only trace equipotential lines. Hence the roles of the flux and equi-potential lines was transposed, by fastening the electrodes to only those parts of the boundary which, in the magnetic problem, are flux lines. In this way the millivolt meter can be used to trace the flux lines. The resistance, as measured in this way, is really the geometric conductance or permeance of the problem which it is desired to study. Thus we have for computing the permeance the equation

P = (yx - Vi)/K (14) In tracing out the equipotential lines upons the test sample, the slide upon the slide wire was set upon a suitable point and the stylus which touched the plate was moved about so as to find, points for which the millivoltmeter deflection was null. The stylus was carried by a pantagraph and thus the lines of force were drawn at once. The slide wire settings were usually varied four centimeters at a time; so that each strip, marked off in this way, could be divided graphically into five cells of square shape. However this was not invariably followed; and always at any singular point, such as at a corner, an extra line was traced. While experiments were in progress, the use of a pantagraph for tracing equipotential surfaces was described by Fortescue and Farnsworth in their A. I. E. E. paper mentioned above. The metal selected for testing was an alloy made by the Driver-Harris Co. by the name of " Therlo." The sheets were approximately 10 mils thick. (-][·. mm.) The specific resistance was approximately 40 michrohms per centimeter cube. This metal was selected because of its high specific resistance, small temperature coefficient, and low thermo-electric power when used with copper. The first tests were made to test for thermal effects which were found to be negligible. The samples were clamped be tween heavy copper terminals, projections having been left upon those edges of the sample which it was desired to maintain at a constant- potential. 1915] DOUGLAS: RELUCTANCE 885

The next tests were made to determine the effect of contact resistance drop between the test sample and the copper clamps. The drop was found to be considerable and steps were taken to reduce it. First the copper terminals were cleaned with emery cloth, and then partly amalgamated. The drop was reduced in this way considerably, so that regular tests were begun. In order not to include the contact resistance drop the potential was read upon the therlo sheet, but very close to the copper electrodes. To facilitate this work, a fine line was ruled upon the metal sheet on which the potential was desired, The copper electrodes were then clamped very near and parallel to this line. It was noted, in cases where the clamps were so placed that the current entered the metal sheet non-uniformly, that the potential along the desired equipotential line varied, because of a varying contact resistance drop. Upon comparing the results with those calculated from the mathematical theory, certain discrepancies were discovered. Search was made for constant errors, and found in excessive contact resistance, and too infrequent calibration samples. Also, in certain cases, thickness measurements were made rather than the equality of thickness between calibration and test samples secured. In such a small thickness as 10 mils, this was of necessity somewhat in error. The figures shown in this paper were mostly taken during this first series of tests. It is believed, however, that the shape of the lines of force are substantially accurate, even if the spacing between them is only approximate. In the second series of tests a rectangle was cut from every irregular shaped template after it was tested; and this rectangle was used for calibration. Owing to the large number of tests made it did not seem desirable to solder the electrodes directly to the therlo sheet in order to secure a

FIG. 5—FLUX DISTRIBUTION AROUND AN ELECTRIC MACHINE

heterogeneous. As a consequence some constant error in these tests is to be expected. Fig. 25 shows the arrangement in this case, together with the method adopted in laying out an irregular template and trimming it to size. Figure (5) shows roughly the general character of the lines of force around a multipolar electric machine. It is apparent that the lines of force are not parallel to any one plane except perhaps in the air-gap. Nevertheless the sketch suggests vari ous irregular two dimensional fields, which are at least approxi mations to the actual field. In Fig. 6 is shown the various sub divisions of the field of an electric machine, which result from imagining the lines of force guided by certain surfaces. At the right is shown the best assumptions as made at present, while at the left is shown another method, which will be shown 1915] DOUGLAS: RELUCTANCE 887 in a later paragraph to be.more accurate. It will be noted that the assumptions made in the figure at the left reduce the various sections of the field to two-dimensional flux distributions, suit able for treatment by the methods outlined. Another point of interest is that the figure at the right shows some obvious sources of error; namely, in two places two kinds .of flux are supposed to occupy the same space at the same time, and also the field is the more cramped of the two. In the experimental work the shapes of the irregular magnetic fields shown in Fig. 6 at the left were systematically taken up and studied. In ad dition a few extra topics of some interest suggested themselves as the work progressed and were investigated. The effect of the slots and teeth upon the value of the air-

FIG. 6—SUBDIVISIONS OF THE FIELD OF AN ELECTRIC MACHINE 1—Pole tip fringe 6—Pole shoe flank fringe 2—Pole tip leakage fringe 7—Armature core 3—Pole side leakage 8—Pole core 4—Pole flank leakage 9—Pole yoke 5—Pole shoe flank leakage 10—Pole shoe and poletip gap reluctance, has been partially investigated by F. W. Carter and reported in the Electrical Worlds. He assumed infinitely deep slots, and infinitely wide teeth, and neglected the effect of the notches cut to receive the wedges. These factors are reported on here. In Figs. 7 and 8 is shown the flux distribution in the air-gap of a slotted armature. The sample of sheet metal for testing was cut in the form shown by the letters α, ό, c, dy e, /, the elec trode being soldered to the strip at the lines ab and de. When the current was passed through the strip, the curved lines of force were traced out by the pantagraph as shown. The mea sured resistance gave the permeance of the flux from the tooth 65. Loc. cit. 888 DOUGLAS: RELUCTANCE [June 29 ends and that in the tooth fringe combined. Since this com- .bined permeance depends upon the tooth width, 2x, we usually desire to find the excess or fringe of this permeance. That is, the fringe permeance is the amount that the total permeances in excess of that which might be expected to pass from the ends of the teeth, under the assumption that the lines of force were all radial. In Fig. 7 the permeance of the fringe is calculated from experimental values by the formula,

Ve ~Vg 2x (15) Pr = K

Here (Ve — Va)/K is the total measured permeance, and 2x/g is the apparent permeance directly between the teeth and the pole surface. In Figs. 7 and 8, g represents the air-gap, 5 the slot width,

Q z R T Ί FIG. 7—TOOTH FRINGE FLUX FOR FIG. 8—TOOTH FRINGE FLUX FOR SMALL AIR GAP LARGE AIR GAP

D the slot depth, Θ the angle between the tooth side and end. In general the symbols used will be clear by a reference to the figure. The dotted lines labeled a, b and c represent three shapes of notches cut to receive the slot wedges. These notches in the test pieces were cut to a depth of 20 per cent of the slot width on each side. Some of the data together with computations are shown in Table 1 ; the results are plotted in Fig. 9. In addition the effect of the depth of the slot, and the inclination of the tooth tip were studied and these results plotted in Figs. 9 and 10. It is to be noted, that the effect of the depth of the slots is negligbile until the depth is less than the width. Also, it is apparent that the presence of a notch has little effect unless the tooth tip forms an angle less than 90 deg. The data for 1915] DOUGLAS: RELUCTANCE 889 the other cases studied were worked up in exactly the same way, and will not be included in this paper.

TABLE 1. DATA AND COMPUTATIONS FOR DETERMINING FRINGE PERMEANCE FROM TEETH. CALIBRATION: Meter Ai read 12.0 amperes, Meter A« read 32 millivolts. Drop in one square of calibration sample equaled 19.95 centimeters drop on slide wire. Thickness of sample 9.6 mils. TEST DATA AND COMPUTATIONS. (1) Effect of shape of armature slot, when D > S and x = 1.

Ve-Va 2x 5 Shape 5 g Va Ve Pf 1 19.95 g g

(a) 10.0 5.0 9.9 44.4 1.71 0.400 1.31 2.0 (b) 10.0 5.0 9:7 45.5 1.79 0.4 1.39 2 (c) 10.0 5.0 9.7 45.7 1.80 0.4 1.40 2 (d) 10.0 5.0 9.6 45.7 1.805 0.4 1.405 2 (a) 7.0 1.0 12.2 105.8 4.69 2.0 2.69 7.0 (b) 7.0 1.0 12.6 110.5 4.90 2.0 2.90 7 (c) 7.0 1.0 13.1 111.3 4.93 2.0 2.92 7 (d) 7.0 1.0 13.8 112.2 4.93 2.0 2.93 7 |

(2) Effect of depth of armature slot, when S/g = 8, 5 = 10, and x = 1.

Ve—Va 2x P D D g Va Ve f 1 19.95 g S

15.0 1.25 12.5 103.2 4.535 1.60 2 .935 1.50 5.0 1.25 12.1 105.2 4.650 1.60 3.050 0.50 1 2.5 1.25 12.3 113.7 5.070 1.60 3.470 0.25

NOTE: Shapes a, b and c refer to the shapes of the tooth-tip shown in Figs. 7 and 8. Shape d is rectangular pole tips without any notch cut for the slot wedge.

The curve in Fig. 9 for infinitely narrow teeth, and in Fig. 10 for various depths of slots, were obtained mathematically and and not experimentally. It is interesting to note that whether the teeth are infinitely wide or infinitely narrow the fringe per meance is nearly the same. Experimental results for moderately wide teeth agreed more closely to the curve for infinitely wide teeth. The practical effect of the slots is to increase the air-gap reluctance, or to decrease its permeance. We may think of the effective area of the flux as decreased, and the ratio of, this decrease as the air-gap factor Ka. The effective area is very readily computed from the cu yes of. permeance in Figs. 9 and 10, by multiplying the permeance 890 DOUGLAS: RELUCTANCE [June 29

Pf by the air gap "g" to get the amount to be added to the width of each tooth for fringing.66 Analytically the tooth fringe At is Δ/· = PfXg (16) On the other hand we may think of the average length of the lines of force in the air-gap as increased in the ratio Ka.

ν *Α υ^ Ö-90°f-a '(2) ^*θ- 45° ^. >c 3 1 J {\> + <*y& *^ l-U^ ^|oo \r^> α) <£ A 2 A % V / (b + c).

1 / / / / 0 / 3 2 l \ - < 3 10 12 14 16 s/g FIG.9—FRINGE PERMEANCE FROM SIDES OF TEETH TO ARMATURE (DEEP SLOTS) 1. Infinitely wide teeth. 2. Infinitely narrow teeth.

This point of view is perhaps the most helpful when the air-gap is long, for two reasons. In the first place with a long, air-gap the ratio of S/g is small and Figs. 9 and 10 are inaccurate. In the second place, a theoretical investigation showed that the width of the tooth had an important effect in this case while the size of the air-gap was unimportant. The results of this 66 For example see Karapetoff " Magnetic Circuit" Chap. 5, p. 93. 1915] DOUGLAS: RELUCTANCE 891 investigation are shown in Fig. 11. The excess length to be added to the air-gap is obtained by taking the reluctance R from the curve and multiplying it by the tooth pitch, λ. Analytically the excess air-gap with A a is,

Δ α = Re\ <17) That the width of the teeth has an important effect upon the

5 1

s -4d 's-3 d ^d s -2d jr+yf d-ac

3 Jr^

31* GÛ

1 ) / / 0 / ( i i \" i i5 10 12 14 16 5 S/% FIG. 10—FRINGE PERMEANCE FROM SIDES OF TEETH TO ARMATURE, FOR T/G = o air-gap reluctance for long air-gaps, is seen by comparing with the approximate curve also shown in Fig. 11. This approxi mate curve was calculated from the formula for Figure 9, curve 2, for a small ratio of S/g. The case of the air-gap of an induction motor is one that proved too difficult for the writer's power of analysis. How ever, it is a case that might well be studied from the experi- 892 DOUGLAS: RELUCTANCE [June 29 mental side. It has been partly solved by, F. W. Carter,67 in a footnote of an article on Magnetic Centering of Electrical Ma chinery. The various approximate formulas for air-gap factor are now of little interest, a short bibliography, however is given.68 Because of desired economy of space a detailed description of each experiment will not be given. In each case the shape of 2.0 71 1 1 1 / / / 1/ f / / (2) / J roL>

X / 1)

0 5 ,

(1) Exact Curve (2) Approximation Λ <£ p f 0.0 ;· 0.2 0.3 0.4 0.5 0.6 07 0.8 0.9 1.0 SA FIG. 11—EXCESS RELUCTANCE OF AIR-GAP FOR EXTRA LONG AIR-GAPS the template cut for testing can be seen by referring to the figure giving the shape of the lines of force. The portion outlined by the letters a, b, c, d, e,f in each case gives the shape of the test 67. Proc. Inst. Civil. Eng. Vol. 187, (1912), p. 311. 68. Guilbert, Ind. Electrique, Vol. 9, (1900), pp. 377-379. Müller, E. T. Z., Vol. 1906, Vol. 27, p. 1103. Wall, Inst. Elee. Eng. Jour., Vol. 40, pp. 550-568. Baillie, Electrician (London), Vol. 62, (1909), p. 494. Rezelman, Lumiere Electrique, Vol, 11, (1910), p. 202, 1915] DOUGLAS: RELUCTANCE 803 piece, the electrodes being soldered to the lines ab and de. In each case data was taken in a form similar to Table 1, and the permeance calculated in the same manner. In all cases the curves of permeance were calculated by^means of the mathemat ical method as well as the experimental method, and in all cases the results were found to agree very closely. Any special features differing from the above are specially mentioned.

FIG. 12—SHAPE OF POLE-TIP FIG. 13—SHAPE OF POLE-TIP FRINGING FLUX , FRINGING FLUX

The magnetic field in the interpolar regions of a dynamo is shown in Figs. 12, 13, 14, and 15. The fluxwhic h leaves the pole b c d is composed of two parts. The one which passes to the armature directly between a and / is the useful flux, while the rest between / and e passes directly from pole to pole and is leakage. Just as the useful flux may be divided into the air- gap flux, and the pole tip fringe, so the leakage flux may be

FIG. 14—SHAPE OF POLE-TIP FIG. 15—SHAPE OF POLE-TIP FRINGING FLUX FRINGING FLUX, PROJECTING POLE-TIPS divided into the pole side leakage (calculated) as if passing di rectly from one pole to the other) and the remainder which may be called the pole tip leakage flux. The pole tip fringe may be calculated from the experimental data by the formula,

Pf={Vf-Va)/K-(x/A) (18) The pole tip leakage fringe may likewise be calculated by the formula, Pi/= (Ve-Vf)/K -'(y/Bf (19) 894 DOUGLAS: RELUCTANCE [June 29

Here Va, Vf, and Ve are the slide wire readings at the points a, / and e as shown in the figures, and K is the calibration con stant. The values of the pole-tip fringe permeance are plotted in Figs. 16 and 17 and that of the pole-tip leakage permeance in Fig. 18. It is interesting to note that the curves for pole-tip

FIG. 16—PERMEANCE OF FRINGE FLUX FROM POLE-TIPS TO ARMATURE fringe, are similar in shape to those for tooth-tip fringe as in Fig. 9 except for a constant difference of 0.88: this is easily proved by the mathematical method given above. An approxi mate curve used frequently at present is that of Hawkins and Wrightman69 given at top of Fig. 16. The effect of pole tip fringing is practically to increase the pole face width W by an 09. Inst. Elee. Eng. Jour., Vol. 29, pp. 43ί>. 1915] DOUGLAS: RELUCTANCE 895 amount Δ W in all, or one half that amount at each pole tip. This fringe is obtained from curves in Figs. 16 and 17 by mul tiplying the ordinates by the air-gap g. Analytically,

AW = Pfg (20) The occasional negative value for the pole tip leakage in

4 — K/k-1.0^

'K/k = l.l^>^ Scale No. (1) > ^^

K/k = 1.4 ^

) ,+' K/ k=(JK / / 1£A 2 h^k K> M y Scale No. (2) a: - JÄZ 1

0 10 15 20 25 30 35 40 No.(l)k/g 2 3 4 5 6 7 8 No.(2)k/g FIG. 17—FRINGE PERMEANCE FROM POLE-TIPS TO ARMATURE, Q - o

Fig. 18 really indicates that the pole tip fringing encroaches so far on the pole tip that the leakage flux is actually decreased thereby. Thus the error which is committed in Fig. 6 at the right, in assuming two different fluxes in the same place, is rendered plain. The amount to be added to the length of each pole to allow for this pole tip leakage is obtained by multiplying 896 DOUGLAS: RELUCTANCE [June 29 the ordinate of Fig. 18 by K, the distance between the pole cores nearest to the air-gap. Or, expressed analytically,

AH = PifK (21) By referring to Fig. 5 it will be seen that some of the leakage flux passes directly from the pole-core to the pole yoke. This

0.5

0.4 \ [ --— \ \ 0.3 \\ A \ Γ M *\ 0.2 \v V X \, \ \> 0.1 v Λ X0.Ò \ !v R = ?lf \ \ -0.1 ^ \K"i: fk \

-0.2 1:s , ^^ K=] .5k

-0.3 K-l. 0k "

-04

() 4> ,Ì t \ i 10 15 20 2 5 30

FIG. 18—EFFECT OF ARMATURE UPON POLE SIDE LEAKAGE PERMEANCE path is shorter than that assumed from the pole core to the neu tral plane. As a consequence the permeance of the leakage flux is somewhat increased. Study of a field of flux some what similar by the method of complex quantities has indi cated that this effect can be allowed for by adding to the length of the pole a fringe Δ H, which is approximately, AH = 0.3A2/H (22) 1915] DOUGLAS: RELUCTANCE 897

Here A is the half distance between the poles, and H is the length of the poles. The distribution of leakage flux between circular poles of a multipolar machine is given in Fig. 19. The dotted lines give

FIG. 19—LEAKAGE FLUX DIS- FIG. 20—LEAKAGE FLUX BE- TRIBUTION BETWEEN ROUND TWEEN SMALL RECTANGULAR POLES POLES

the best previous assumption, that of Dr. R. Pohl70 which consists in imagining the lines of force to be involutes and straight lines. The cramped character of this assumption is readily seen. In Figs. 20 and 21 is shown the distribution between rectangular

FIG. 21—LEAKAGE FLUX BE- FIG. 22—EFFECT OF LEAKAGE TWEEN* LARGE RECTANGULAR UPON FLUX AND M. M. F. IN POLES POLE-CORES

The effect of leakage upon pole saturation is shown graphically in Fig. 22. The free m.m.f. between the poles and the flux density in the interpolar regions, is shown at the left.

70. Jour. lust. Elee. Eng., Jan. 1, 1914f Vol. 52, p. 120. 71. Jour. Soc. Teleg.' Eng. and Elee., Vol. 15, (1886), p. 551. See also Hawkins and Wallace, " The Dynamo", 3rd. Ed. p. 437. 898 DOUGLAS: RELUCTANCE [June 29

Methods allowing for this effect are well known72 and consist of counting only one-half of this leakage permeance as effective. The variation of the flux and ampere-turns in the pole core are shown at the right, both varying greatly. At present it is as sumed that all the leakage enters at the pole tips; or, what is the same thing, the maximum value of ampere-turns in the pole-core is taken throughout its entire length. It would be

15 l /'·.

' I\ / (1) <

£10 / '<2 )/ / )<

< 5

^ <

'^ / 0 [^ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 08 D/T FIG. 23—LEAKAGE PERMEANCE FROM ROUND POLES TO NEUTRAL PLANE 1—Exact solution 2—Approximate solution

better to assume the leakage flux to enter the core at some point nearer the pole-yoke, say one-half way up. The exact location could be determined by a graphical integration similar to that given by W. B. Hird73 for the case of the reluctance of tapered and saturated armature teeth. 72. Hawkins and Wallace "The Dynamo", 3rd. Ed., p. 441. Pohl. Jour. Inst. Elee. Eng., Loc. cit. 73. Jour. Inst. Elee. Eng. Vol. 29, p. 939. 1915] DOUGLAS: RELUCTANCE 899

The values of the leakage permeance from round poles to the neutral plane is plotted in Fig. 23 together with that calculated from Pohl's formula. The flux which leaves rectangular poles, ♦as in Fig. 21, between the points bed may be considered as divided into two parts. The first is that which would pass

vw/T 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0^8 W/T FIG. 24—LEAKAGE PERMEANCE FROM RECTANGULAR POLES TO NEUTRAL PLANE, IN EXCESS OF THAT BETWEEN THE POLES (1) and (2) exact solutions—(3) approximate solution directly between the poles if the lines of force were straight; this is the pole face leakage. The remainder is fringe of the leakage flux, or what is called elsewhere the pole flank leakage. The formula for computing this leakage from the data is similar to equations (15), (18) and (19), and will not be given here. The result is plotted in Fig. 24, together with the customary ap- 900 DOUGLAS: RELUCTANCE [June 29 proximate formula. In order to compute the leakage we multiply the ordinate taken from Fig. 24 by H the length of the poles increased AH for fringing at the roots, but not at the tip. It is to be noted that, except for very short machines axially, the upper of the two curves in Fig. 24 is the nearer the truth. In all cases the experimental data agreed closely with the upper curve. The template used for determining the pole shoe flank per meance is shown in Fig. 25. This figure shows the method of trimming so as to get a number of different proportions with one sample. In Fig. 26 is shown the flux distribution; this

Assumed Flux Line

Sheet (3)

1 Copper Strip No. 1 Noi

32 I o| ; R 2 4 5 4 3 2 1 I.S'hcetß) Sheet 1 (1)

£2 i»;6,u^io ,7

a» 2 2 2 T p7 0 8 5

FIG. 25—METHODS OF MARK FIG. 26—DISTRIBUTION OF FLUX ING AND CUTTING TEMPLATES OF IN THE FRINGE FROM THE POLE- SHEET METAL TO SIZE, SO AS TO FLANKS TO THE ARMATURE—DE ECONOMI? E MATERIAL TERMINED GRAPHICALLY corresponds to the right hand part of Fig. 5, the line d e being an assumed flux line. This figure was obtained graphically by the method of L. F. Richardson, given above. Lines 2, 4, and 6, and 9, 11, 13 etc. were drawn first. Then lines 1, 3, 5, 5, 8, 10, 12 etc. were drawn to test the accuracy of the preceding lines. For instance in the square 4, 6, s, u the line p r was compared with the line 5 / for equality. Considerable readjust ment was necessary, and the reader will no doubt note improve ments. The flux leaving the pole in Fig. 26 between the points bed may be divided in the usual way into air-gap flux, and the fringe flux. The formula for computing this from the data is 1915] DOUGLAS: RELUCTANCE 901 similar to equations ' (15), (18) and (19). In this way the permeance in Figs. 27 and 28 were computed and plotted. It will be seen that this permeance depends on both the ratio of f to g and of h to g. This permeance, it is to be noted, is the combined pole shoe flank flux, both useful and leakage flux; and includes flux on both ends.

FIG. 27—PERMEANCE OF FRINGE FLUX FROM POLE-FLANKS TO ARMATURE SURFACE—FOR AN OVERHANGING ARMATURE, i. e. LONGER THAN POLES

It will be noted that the point v on line of flux No. 9 in Fig. 26, is also on the dotted line which is the projection of the arm ature coils. Thus the point v may be regarded as the trace of the armature coils. The flux between the line 9 vf and the arm ature links with the armature coils, and is useful. The flux between the line of 9 v f and the line d e is really leakage flux. Readings of the slide wire setting Vz were taken at various points 902 DOUGLAS: RELUCTANCE [June 29 shown on the dotted lines in Figs. 25 and 26, located at δ dis tance z from the armature corner. The fraction of the total pole shoe flank flux that is useful was calculated by the formula,

Pu (V -Vg)/K-(x/g) = z (23) P (Ve-Va)/K-(x/g)

·+

]1-1 0 S + -t 2.5 / f + h« 5* S' f h -3g + +" 20 h-2e^ _). y ê^ *\ Q. h-lg^ >4

h -0.5

1 5 ,/ y /

/ h -0^ / r / / / 1 0 / • V i l Y i i5 1 0 15 2 0 Ì5 () f/g FIG. 28—PERMEANCE OF FRINGE FLUX FROM POLE FLANKS TO ARMATURE —FOR OVERHANGING FIELD POLES

This fraction was computed and plotted in Fig. 29. While this fraction theoretically should-be a function of three variables, it was found it did not depend much upon the ratio of / to g. This rather fortunate coincidence permitted the results to be plotted in a single set of curves, Fig, 29 was not checked theoretically. 1915] DOUGLAS: RELUCTANCE 903

It is to be noted that the line d e is an assumed flux line. There is no natural boundary between the leakage lines and the fringe lines, and hence the point d has to be assumed in a some what arbitrary manner. Several· rough trials have convinced the writer that the best location for the point d is at the junction of the pole shoe with the core, or at one quarter to one third

0 2 4 6 8 10 12 14 16 Zìi FIG. 29—CHART SHOWING THE PERCENTAGE OF THE POLE-FLANK FRINGE THAT is USEFUL, FOR VARIOUS LOCATION OF THE ARMATURE COILS of the width of the pole waist from, the armature. Rough as this method is, it is hoped that it will be useful; a more accurate assumption is discussed in a later paragraph. In order to use Figs. 26, 27 and 28 we first estimate the width /of the pole shoe flux by the method of the preceding paragraph. From the working drawing of the machine we compute the ratios h/g and//g, and from either Fig. 27 or Fig. 28 obtain the 904 DOUGLAS: RELUCTANCE [June 29

flank permeance. This we divide in two parts by means of Fig. 29, into the pole shoe flank fringe Pu and leakage, Pip The effect of the fringe is to add to the effective axial length of the air gap by an amount AL which is given by the equation,

AL = Pu (24) 0.20 v A\ W 0.1 \\ \\ vv 0.09 \\ Y W WË \ - 10 A \\ v\ 0.08 0.15 V \ 10 V \ \ \\ \ \- A 0.07 >V \\ ^ V \ \\ 3 5 k\ V ^0 Ë \ 2 0.06 \ o \\ 1 \ \ y\ \ 2 - > l\ 4>0 ^ '3fo,1 0 r \ >N Ë l ,\ 1 V V \ \\ 2 0 \ 1 •3 s^ ^ 0.04 -1 o 1 [\ \ ^ ^ \\ N ^ ^ 1 -2 \ ^ NS $5 \ 1 ^ ^ ^ 0.03 -3 0,05 ^ sV -5 "—; ^5 ^ 0.02

0 01

• 2 i I t ( 1' 7 5 1Ô 1D io V l FIG. 30—CHART FOR DETERMINING WIDTH OF POLE FLANK FRINGE

The effect of the leakage fringe is to add to the leakage perme ance by an amount,

P = Pif(W+ AW), (25) where W + AW is the corrected width of the pole shoe next the armature. The principle of " maximum effect " can be applied to the results given in Figs. 27 and 28, enabling one to pick out the 1915] DOUGLAS: RELUCTANCE 905 most favorable value of the width σf the pole shoe flank flux,/. This principle states that of two possible assumptions as to flux, that is nearer the truth which is the larger. Fig. 30 is the result of an attempt to apply this principle to the problem. For a maximum total permeance, the derivative of the pole shoe flank permeance with respect to / should be equal to that of the leakage permeance, taken with the minus sign. In symbols,

w_ dP Pi = (26) g d (f/g) where P is given by Figs. 27 and 28, and Pi is the leakage fringe or pole flank leakage given by Fig. 24 for a definite pole waist width W. We may solve the above formula for the derivative, and by referring to Fig. 30 where this quantity is plotted, we can pick out the proper value of f/g at once. The results given in Figs. 27-30 are somewhat complicated, and good service could be done by inventing an empirical formula which would fit the results fairly closely. The only previous attempt to FIG. 31—FLUX DIS- deal with the matter adequately is an TRIBUTION BETWEEN . , , , , 7.1 r 1 artlclA e Μ LeT ros whose £ MAIN AND INTER-POLES ^ · g °™^ are too complicated to give here. The flux distribution between the poles of an interpolar machine (as produced by the m.m.f. upon the main 'poles) is shown in Fig. 31. There is a fringe of permeance over that computed on the assumption of straight lines of force between the poles. This permeance is computed from the data in the usual manner and plotted in Fig. 32. The permeance is that in excess of computed permeance, up to the neutral plane only, for both ends of the interpoles and for two interpoles. The leakage flux distribution inside a slot is shown in Figs. 33 and 34. The template cut in this case is similar to that cut for the preceding case. Both cases are similar to an electrical problem solved by C. H. Lees.75 The fringe permeance is defined in the usual way as the excess of the total permeance over that computed between the parallel parts of the slot. The formula for computing the fringe of permeance is from the experimental 74. Eclairage Electrique, Vol. 30, p. 437; Vol. 51, p. 12. 75. Phil'. Mag. Loc. cit. 906 DOUGLAS: RELUCTANCE [June 29 data in similar to those previously mentioned. The permeance as plotted in Fig. 35 is that for the whole slot, per centimeter of axial length. Various approximate formulae have been pro posed76 most of which are based upon assuming the lines across the slot to be straight. These formulas are compared with the correct formula in Fig. 36 for the tooth tip angle of 45 deg, Ü 1 / 9 / / 8 11 7 / / 1/ / / g •

•

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

FIG. 32—FRINGE OF LEAKAGE PERMEANCE FROM MAIN TO INTERPOLES

All of these formulas give null result when the pole tip angle is zero, and hence are in considerable error. In Fig. 37 we have the field of flux produced by one armature coil embedded in nearly closed slots. At a distance x = n λ from the first coil there is a second coil with a line of flux entering 76. E. Arnold, Wechselstromtechnik, Vol. 4, 1st Ed., p. 44. C. A. Adams, A. I. E. E. TRANS., Vol. 24, (1905), p. 335. J. Rezelman, Lu~ miere Electrique, Vol. 10, (1910), p. 293. 1915] DOUGLAS: RELUCTANCE 907

ÑÖÎÉ FIG. 33—DISTRIBUTION OF FIG. 34—DISTRIBUTION OF LEAKAGE FLUX IN SEMI-CLOSED LEAKAGE FLUX IN SEMI-CLOSED SLOTS SLOTS

2.0

Θ-46 H 1 S Y •i' /

•- / 6-2 2^, CL 1.0 (1 ) ' \r/

Vf È-Ï^ (1) Ö-C ° 0.5 ^ i) . / V? ^ y S y I/H (1) Assuming Wide Slot Opening r^ (2) Assuming Narrow Slot Open ing, i. e. a Line of Flux

0.0 () ;> L ( iI 10 1 2 14 16 S/s FIG. 35—PERMEANCE OF INTERNAL FRINGE OF SLOT LEAKAGE FLUX 908 DOUGLAS: RELUCTANCE [June 29 the center of its slot. The flux inside this line links with only one coil but that between this line and the center of the pole is flux of mutual induction. The permeance of this flux of mutual induction was evaluated theoretically by applying the method outlined in the historical section and is plotted in Fig. 39. The flux of self-induction is the whole flux up to the mouth of the slot. In Fig. 38 is shown a detail of the armature leakage

FIG. 36—COMPARISON OF VARIOUS FORMULAE FOR SLOT LEAKAGE

flux near the slot opening. It will be observed that close to the slot the lines of flux change gradually from a semi-circular form to straight lines. * The fringe of the flux of self-induction outside of the slot is defined in the usual way as the excess over what would pass inside the slot opening, were the lines of force straight. The permeance of the flux was calculated and plotted in Fig. 39 against x which is half of the slot opening, namely 1915] DO UGLAS: RELUCT A NCE 909

(i/2).. The result is correct only provided the slot width is small as compared to the pole pitch. In Fig. 39 is also plotted Arnold's77 formula for self and mutual induction. This formula is based upon the assumption

FIG. 37—LEAKAGE FLUX DISTRIBUTION DUE TO A CONCENTRATED ARAMATURE WINDING that the leakage lines are arcs of circles and straight lines. His results are in error theoretically, in that he integrates over the whole pole pitch. This error is corrected in the second edition. Fig. 39 can be used to deter mine self and mutual induction for the flux outside the slots, provided the field poles are re moved from the armature. This is the case when the re __ M t«+œ actance of an alternator is de ^-S-2h-^ termined from experiment. FIG. 38—LEAKAGE FLUX DISTRI However, it is another question BUTION NEAR THE MOUTH OF A SLOT whether the reactance so de termined has much relation to that of the machine completely assembled. No correction should be made for the curvature of 77. Wechselstromtechnik, 1st Ed. Vol. 4, (1904), p. 44. See also Rezelman, Lumiere Electrique, Vol. 10, p. 259. 910 DOUGLAS: RELUCTANCE [June 29 the armature surface, since a short theoretical investigation proved that it is not necessary. This is in contradiction to Arnold's and Rezelman's recommendation. When there are a number of slots per phase and per pole, the application of Fig. 39 becomes difficult, and further simplification becomes desirable.

FIG. 39—PERMEANCE OF ARMATURE LEAKAGE FLUX—FIELD REMOVED (1) and (2)—Mutual induction (3) and (4)—Self induction (1) and (4)—Exact Formulas (2) an d (3)—Arnold's formulas

The leakage flux set up by an armature winding in the region outside the slots, links in part with each slot individually, and in part with more than one slot. It is convenient therefore, to calculate the inductance due to the leakage flux as if it linked with the whole of the group of conductors belonging to one phase under one pole. This inductance should include the effect of mutual induction between the phases. On the other hand, the 1915] DOUGLAS: RELUCTANCE 911 leakage flux Set up inside the slots is most conveniently calculated separately. Using Fig. 39, and taking into account the factors mentioned, the writer worked out the equivalent leakage permeance of the flux set up in the region outside the slots. It was found that, between the limits of one to twelve slots per phase per pole, the equivalent leakage permeance was practically a con stant ; providing that a certain amount be added to the leakage permeance of the slots, for the flux just outside the slot opening. The equivalent leakage (geometric) permeance was found to be

Peq = 0.725 and 1.19 (27) for two and three phase machines respectively. The fringe of permeance to be added to that inside the slots was found and is recorded in table 2. The best pre vious method for computing the equivalent leakage permeance was k. that proposed by Schenkel78 Ap plying this method the . writer obtained the values 0.51 and 0.86 for two-phase and three-phase FIG. 40—EQUIPOTENTIAL SUR- machines respectively. FACES AROUND A SQUARE WIRE

TABLE 2—VALUES OF EXTERNAL FRINGE OF SLOT LEAKAGE PERMEANCE.

(s/X) = 0.60 0.50 0.40 0.30 0.20 0.15 0.10 0.05 0.025

APS = -0.20 —0.10 —0.00 +0.10 +0.30 +0.40 +0.55 +0.85 +1.200

In order to use the above results to obtain the inductance per phase of a machine, we first add to the fringe of permeance belonging to the slot and obtain the total slot permeance Ps. Then, if Cs is the number of conductors per slot and CPP are the number of conductors per pole, and per phase, P the number of coles, and S the number of slots per phase, and L the effective length of the machine; the inductance of the machine, exclusive of that of the end connections, is

8 L = (4V/10) (S C\ PsL + P Cpp> Peq L) 10~ (28) The electrostatic equipotential surfaces around a square conductor, are shown in Fig. 40. The capacity of a rectangular 78. Electrotechnik und Maschinenbau, Vol. fi, (Feb. 1901), p. 54. 912 DOUGLAS: RELUCTANCE [June 29 wire of dimensions a and b was investigated mathematically and found to be given by the following equation.

7Ã KaL C = (29) Ln (KD /perimeter) where C is the capacity between two wires D cm apart. K{

X 7.0f

ou öl "öS " ä!3 04 05 oe äô áâ a/b FIG. 41—FACTOR FOR DETERMINING CAPACITY OF RECTANGULAR WIRE is the dielectric constant for air, L is the length of the wire in cm. and K is a factor plotted in Fig. 41, the perimeter of the wire being 2 (a + b). The above cases comprise all those of which the writer is prepared to report the permeance. However, the flux distribu tion in the.cores of various machines was studied, also that in the air-gap of alternators. Owing to saturation, the reluctance 1915] DOUGLAS: RELUCTANCE 913 as measured by experiment in these cases would not be of great value; and for this reason it was not calculated. The reluctance of the air gap of alternators was omitted because the assumed shape of the pole is not used. The flux distribution in the core of a transformer is shown in Figs. 42 and 43. While the flux lines could have been obtained by soldering electrodes to the broken lines efmn and g h k I this would have been awkward. By soldering electrodes to the edges e g and / n and passing current through the strip, the equipotential lines/h and m k were traced out. The L-shaped strip was then cut into three lengths e f g h, f h k m, and k l m n. The ftux lines were £hen very easily obtained by soldering electrodes to the lines g h, h k, k l, e f< f m, and m n, and passing current through the three strips in the usual way. The writer is in-

£ι-·;ΒΓ7Ί«

FIG. 42—FLUX DISTRIBUTION FIG. 43—FLUX DISTRIBUTION IN CORNER OF A TRANSFORMER IN THE CORNER OF A TRANS FORMER

debted to Mr. E. M. Shepard, Cornell' 13, for the experimental work involved in this problem. The flux within the core of an armature is shown in Fig. 44; this was suggested by test results published by Dr. Thornton.79 The template has to be laid out with a calculated curve, the equation of which is obtained by the theory outlined earlier in 'the paper. This equation is, (x/R)[(x/R)*-Z(y/R)*] = 1 (30) This figure was determined from experiments and compared with the curves published by Dr. Thornton and found to agree com pletely. This work was also performed by Mr. E. M. Shepard. The flux distributions shown in Figs. 45, 46 and 47 was determined by'calculation only, but the excellent agreement 79. /. E. E. Jour. Loc. cit. 914 DOUGLAS: RELUCTANCE [June 29 between experiment and theory in all other cases indicates that they should be reliable. The theoretical shape of pole shoe required to give an ab solutely sine wave of voltage in an alternator was calculated by the mathematical theory outlined in the Historical Section, for a two-pole alternator. The equation of this pole shoe was, (y/R) - [yR/(x> + f) ] = 0.19 (31)

FIG. 45—FLUX DISTRIBUTION IN A FOUR-POLE ARMATURE WITH FIG. 44—FLUX DISTRIBUTION UNIFORM FLUX DENSITY IN THE IN THE ARMATURE CORE OF A AIR-GAP SIX-POLE MACHINE

FIG. 47—FLUX DISTRIBUTION IN A TWO-POLE INDUCTION MOTOR

FIG. 46—LEAKAGE FLUX DIS TRIBUTION DUE TO A TWO-POLE ALTERNATORARMATUREREMOVED FIG. 48—POLE-SHOES SHAPED TO FROM ITS FIELD GIVE SINE-WAVE E.M.F.

A template was cut to this form and the field distribution shown in Fig. 48 measured. The flux density distribution on the arm ature corresponding electrically to the volts drop per cm. of circumference of the circle, c /, was measured experimentally. This drop is plotted in Fig. 49. The experiment was repeated with a template shaped like a b e df e f in Fig. 48 with the pole piece cut away in a convenient manner. This result also is 1915] DOUGLAS: RELUCTANCE 915 plotted in Fig. 49. The first curve is an exact sine wave. The shape of the pole in a four-pole machine is similar to that in Fig. 48. The mathematical theory is readily adapted to finding the correct pole shape for any shaped wave of flux desired. The application is not difficult as the writer solved a particular case of this sort in a short time.

^ \ \ N^ , 4U \ \ ^N \ y>

ot\ (2)S

0 W 20° W W W W W W 90* FIG. 49—FLUX DENSITY DISTRIBUTION UNDER POLES OF ALTERNATOR (1; Poles shaped for sine wave (2) Same pole tips cut away at convenient angle

It is believed that the most convenient way to use the results of this paper is by means of the curves shown. One no longer uses Frohlich's equation for saturation of iron, but the magnetiza tion curve itself. It is more convenient to use curves of core loss than to calculate it from its co-efficients. Likewise a set of curves like those in this paper for magnetic circuits of various forms, would facilitate magnetic calculations. (Those who pre- 916 DOUGLAS: RELUCTANCE [June 29 fer a formula, however, can use the equations for the curves which will be found in an appendix for reference.)

IV. ACCURACY OF RESULTS The accuracy of the assumptions made in the preceding two-dimensional magnetic fields, when used to compute the total reluctance of a dynamo is the next topic to discuss. This we may do by using a theorem due to Lord Rayleigh.80 If in a conducting medium we insert thin laminae of perfectly in sulating material, the total resistance of the field will be increased. The only exception to this rule is when we place these insulating surfaces so as to co-incide with the natural lines of flow, when the total resistance is unchanged. When we assume arbitrary flux lines, as we do for simplicity when computing the permeance, we are in effect imagining the course of the lines of force restrained by guiding insulating surfaces. Thus any formula which is derived upon this basis gives too small a permeance. Of two different assumptions as to the course of the lines of force, the one yielding the larger total permeance, is the most nearly correct. The closer the assumed lines of force agree with the actual lines of force, the less the error which is made. Of two possible constrained distributions, the one subject to the most restraint is the most in error. The results shown in this paper are of necessity too small, by the extent that the assumptions made in Fig. 6 do violence to the natural course of the lines of force. Where the guiding surfaces merely separate one field from another, as the pole flank leakage from the pole shoe flank fringe and the pole shoe leakage, the error should be very small. The only place where the error could be considerable, would be where the surfaces assumed prevented the flux from reaching the other side. Thus in Fig. 6 the leakage flux is prevented from expanding beyond the pole yoke. Lord Rayleigh also gave a method for obtaining a superior limit to the permeance, but this was not applied by the writer. It will be found, with three apparent exceptions that the results shown in this paper are larger than those obtained by any of the approximations used hitherto. It may be concluded, therefore, that the results here given are closer to the truth. The three exceptions are Figs. 16, 36, and 39. In Fig. 16 the 80. Theory of Sound, Vol. 2, Sect. 305, p. 175. See also V. Karpen, Comptes Rendus, Vol. 134, pp* 88-90. 1915] DOUGLAS: RELUCTANCE 917 approximate curve given by Hawkins and Wrightman is based upon an assumption of the lines of force which encroaches upon the leakage field. Thus, although the permeance of the flux from the pole tip to the armature seems larger than it really is, the total permeance will be found to be less. In Fig. 36, curve No. 3, for a tooth tip angle of 0 = 45 deg., the approximation is larger than the curve here given. It is to be noted, however, that this is a purely empirical curve and does not correspond to any possible assumption of the lines of force. In Fig. 39 the formula as given by Arnold has an error in the integration. When this integration is performed correctly this apparent discrepancy with the principle disappears.

V. CONCLUSIONS Most of the results in this paper are apparent, but they may be briefly summarized. (1) In magnetic circuit calculations a considerable portion of the magnetic field is irregular. (2) Many of the methods in vogue for computing the reluc tance of irregular magnetic fields are very inaccurate. (3) Accuracy of an equal order of magnitude is desirable in all parts of a given problem. In particular it is believed that greater accuracy in leakage flux calculations would be desirable. (4) The reluctance of irregular magnetic fields occurring in electrical machinery, can be estimated fairly well by studying various two-dimensional flux distributions. (5) A knowledge of the properties of the functions of complex numbers, enables one to compute the reluctance of many two- dimensional magnetic fields. (6) Particular cases on the reluctance of irregular magnetic fields, are easily studied experimentally by the vise of templates of the proper shape cut from sheets of high resistance metal. (7) The entire field around a dynamo electric machine has been divided into sections, and the reluctance of each section determined to a greater degree of approximation than before. Many factors have been shown to have negligible effect, and in many cases present methods have been shown to be greatly in error. (8) A number of other magnetic fields have been studied; in particular, the field around a square wire, the flux distribution in transformer cores, dynamo cores, and the air gap of alterna tors. 918 DOUGLAS: RELUCTANCE [June 29

(9) It is of advantage to use graphical charts of permeance rather than empirical formulae. (10) There are other cases of interest and value in the re luctance of irregular magnetic fields yet to be solved. The writer wishes to acknowledge obligation to Cornell University in furnishing facilities for the experimental work, To Professors J. Macmahon, V. Karapetoff, and F. Bedell, and to Mr. F. W. Carter for suggestions, and to Mr. E. M. Shepard for valuable assistance with the experiments.

APPENDIX I. MATHEMATICAL RESULTS The fringe of tooth permeance for wide teeth and deep slots as shown in Fig. 9 is

Pf = -g cot"' (S/2K) + 1 Ln [1 + (5/2g)«l (32) ITg 7Ã when the pole tip angle È is 90 deg. The symbol Ln indicates the natural logarithm. When the pole tip angle is 0 deg.,

Pf=±-Ln^±± +—/ί (33) 7Γ Va - 1 7Γ (Va + 1) where the parameter a is determined from the equation,

5 1 Vä+ 1 2 Vä f — = - Ln —j= + —7 rr (34) 2g 7Γ Va - 1 ð («-!) When the pole tip angle È is 45 deg.,

0 . (VT + 7-(I-J)è/ ·" (35) JV-J-Z»(4/a) + :τ*δy . iV.Ë^l + 1 * where the integration is performed graphically and a determined from the following equation. J +l/ /\0.75

-s a / Vi - /

p(l+-yvrTi-(,-i-)«^ + Jo / Vrr? " (36) 1915] DOUGLAS: RELUCTANCE 919

For infinitely narrow teeth, we have with a pole tip angle of 90 deg.,

Ff - 2 —; ^ôô (37) K (tanh^O where K is the complete elliptic integral of the first kind tabu lated in Pierces' Tables of Integrals page 117. For large values of the air gap, and finite width of the teeth, the excess reluct ance of the air gap; plotted in Fig. 11 is *-H1+iMl-+4K(l-4M'-*) (38) When the slots are shallow, and the teeth are infinitely narrow, the fringe permeance plotted in Fig. 10 is given by the equation

K(Vl-a2) where a is a parameter which has to be determined, together with another parameter b from the following equations.

* - K^ (40) d + g X (Vi - b*)

F (VïQ>> V a2~b2 ) 2 2 (41) _ \ ' a (l-ς )/ d + i K (Vl-b2)

The permeance from the two pole tips of a machine to the armature, is less than that from tooth tips of the same shape by an amount

Pf = 2 ^Li*> (42) 7Ã

Thus the permeance curves in Fig. 16 are similar in shape to those in Fig. 9, and can be plotted by applying the above cor rection Pf to equations (32), (33), and (35). 920 DOUGLAS: RELUCTANCE [June 29

The permeance from the pole tips of a machine with pro jecting pole tips, as shown in Fig. 15 and plotted in Fig. 17 is given by the equation

p 1 j /l + V\ , K _, I l-&2\ LK _ p* = T Ln \TW) + 2ïg cos \T+-b>) -Kj (43) where b is a root of the following equation.

2g Z, 1 8-1 x = "2ÏF cos" \V li+ + y /

** L (1 + ό2) (2g/K) (44)

The permeance of the pole tip leakage fringe shown in Fig. 18 is given by,

2 2g . (b - 1\ 1 T 4 Plf = J, cos- (ςTT1) - - Ln ãô¥ (45) the parameter b being the same as in the previous equation. The leakage permeance from round poles to the neutral plane as plotted in Fig. 23, is given by the following equation:

*{ cot^ff +coth*g, I

2 j 7iT j .irr ^ irr T , 7ir | [ coth2 — Lra cot — - cot2 2jT Ln coth -^- j (46) The fringe of leakage permeance from both ends of long rec tangular poles to the neutral plane between the poles of a multipolar machine as plotted in Fig. 24, is

(47) 1915] DOUGLAS: RELUCTANCE 921

In machines of very short axial length, the same permeance, as plotted in Fig. 24, is, tf(cos^) Pi .,--. m x ( Sin 2T ) The total pole shoe flank permeance, to the armature end, which is plotted in Figs. 27 and 28 is

P = -ø^ (49) where the two parameters "6" and "c" are determined from the equations following, when the armature projects beyond the field pole. Note that E is the elliptic integral of the sec ond kind, found in Pierces' tables pp. 117 and 119.

g _ E (e, b) 7 ~ * " E(c,90°) (50)

h E (VT^c2) - <*K (VT^~c2) 7 = £W (51) When the field, however, projects beyond the armature, the parameters "ς" and "c" are given by the following equations,

cos-16 , Vsec20-c2 sec è Üè / i = . o h Ve Vï=Tc2 E (νΤ^?) -Va VW2 K (Vi- e2) (62)

f-g _ (1/c) E (e) - (1-c*) K (e) 2 2 V ; h VT-? E (V\=?) - e* Vì-c K (Vl-C )

The fringe of permeance from the main poles to the interpoles of a dynamo, which is plotted in Fig. 32, is

Ñß= Óç ô {ô) {Ê^Ø-^ºÚÃ)

2 K J_ / . K-W\ (2Ê_Ë + T\K-W+ K )Ln\W ) (54) 922 DOUGLAS: RELUCTANCE [June 29

The permeance of the fringe of flux on the inside of an armature slot shown in Figs. 33, 34 and 35 is,

^^Hf-iKMt+iMi^) (55) when the angle of the tooth tip is 0°: when the angle is 0, where È is (ð/ç), we have the equation, '-b»™ + Tffi*^?-

(56)

The permeance of the armature leakage flux of mutual in duction, outside the slot is plotted in Fig. 39 and is given by the equation,

P = ~rLn cot -g- (57) where the distance x = n\. The same formula will give the self-induction provided we take x = (5/2), and add to the permeance a constant C. The constant C has a value 0.241 as given by

c = 1 Lnhrß) 7Ã 7Ã

The capacity between wire of rectangular cross section is given by equation (20) where the constant K is given by,

(K/8)=c VF? E (Vï^c2)+E(c) -c" V~c2 K Vï^~c2 - (l-åç K(c) (59)

The constant K and the elliptic function K are not to be confused. The parameter c may be determined from equation (53) pro viding that we read (b/a) in place of (f-%)/h. 1915] DOUGLAS: RELUCTANCE 923

APPENDIX II METHODS OF COMPUTING RELUCTANCE THEORETICALLY The method of determining the reluctance of irregular mag netic fields mathematically is best understood by working out two typical cases. The two most representative cases are; the fringe permeance from the sides of infinitely narrow teeth, as plotted in Fig. 9, and the fringe of leakage permeance from rectangular poles, as plotted in Fig. 24. If, in Fig. 7, we continue the sides of the teeth across the air gap, as is shown by the lines n m and o p, these lines will be flux lines. We will convert this conductor by Schwartz* theorem into a half plane with straight line boundary, and then into a rectangle. The point n corresponds in the t plane to the point t = — a, m to the point t = - 1, o to the point / = + 1, and p to the point t = + a. If the slots are of infinite depth the point / = corresponds to the deepest part of the slot. In the W plane, the line n m corresponds to the line ö; = 0, m o to the line V = 0, o p to the line ö = m, and both nQ and Rp in part to the equipotential line V = M. By Schwartz's theorem we have, dt ■/■ C \Ë-Â 7TTT] (60)

W=■!■ \K , dt = (61) 1 V(/2-l) (*2-a2)

In equation (60) when / passes through infinity, there is a dis continuity of jCw in the integral and a discontinuity in Z of the slot width 5. This enables us to determine the constant C Performing this substitution and integrating, equation (60), we obtain,

Z = j | Ln ( t + V/2 - 1 \ = j | cosh"1/ (62)

When, in particular, / = a, we have that Z = + jg) whence by substitution o = cosh(-^M(-?). (63) 924 DOUGLAS: RELUCTANCE [June 29

The permeance is obtained directly from equation (61), for, by definition, the permeance is the ratio of 4>m to M\ and these quantities are obtained by integrating equation (61) between the limits / = - 1 and t = + 1, and / = + 1 and t = + a respectively. Thus dW >r K = 2 }l!±^i (64) M fdWCanW ■ K (Vl- (l/a)2)

By substituting the value for a in this last equation, the result given in Appendix I, equation (37) is obtained. The leakage field shown in Fig. 21 can be adapted for mathe matical treatment by drawing the line K-J prolonging the poles indefinitely in an axial direction. We will convert this case in a similar manner as the preceding, into first an infinite half plane, and then into a rectangle in the W plane. The point h corresponds to the point / = - in the / plane, c to the point t = - a, the point K to t = - 1, J to the point t = 0, and / to / = + °° . In the W plane the line h-C corresponds to the line ö = 0, the line C K J to the line V = 0, and the line J-f to the line V = + M. By Schwartz's theorem we have, Vt + 1 dt ■'!■ (65) t Vt + a dt W == CC ÇI ~^== -- (66) 222 J / Vt + a The constant C in equation (65) and the constant a can both be determined from the discontinuities in the integral when t = 0 and when t = », in a manner similar to the preceding case. This transformation was given in a different problem by C. H. Lees.81 The result is that the constant C = - jT/2w and a = (T/K)\ Also,

z=+j~^rLn\ Vr+1 Vt + {T/KY + * + ^^F]

jK Ln jyr+ i yt + (T/Ky+ (T/K) + r+K> ) 27Γ ( t 2 TK ) (67) 81. Phil. Mag., Loc. cit. 1915] DOUGLAS: RELUCTANCE 925

If the flux passed directly between the pole K J and the neutral plane, the permeance apparently in this region, say between the points / = - 1 and / ^= - e, a small value, near zero, would be

Papp = -Tßr[Z(/=-l) — Z(/ e)]

= i Ln øß^à) - h Ln (-iSr) (68)

On the other hand the constant Ci is determined from equation (66) by the fact that there is a discontinuity in the integral of the amount JC2 V(l/a) when t passes through zero, and W changes suddenly by the amount jM. The permeance of the leakage flux can be determined from equation (66); namely, when / changes from -a to -e, the flux passing is the definite integral of equation (66) taken between these limits. Hence by substitution and integration

P = ^^iLn(iSHKy) (69) M 7Ã \ e I

The value of the fringe permeance is the excess of the total permeance over the apparent permeance. Combining equa tions (68) and (69) and multiplying the permeance by four, we get the final result, which is given in Appendix I, equation (47). The factor four is to give the fringe permeance from all four corners of the pole.