INTRODUCTION OF MAGNETIC RELUCTANCE 199 layer of zinc Z is interposed, plays an unfavourable part, as it partly forms a magnetic short circuit. 2 Similar very complete measurements have been made by Lahmayer. Experiments on leakage have further been published by Hering, Carhart, Trotter, Esson, Corsepius, Wedding, &c., into the details of which we cannot enter here, since they can hardly claim a general theoretical interest, but hold rather for special cases-that is, special types of machine-and are therefore chiefly valuable from the practical point of view.3 - § 131. Introduction of Magnetic Reluctance. We will now consider the magnetic circuit of dynamo machines from the point of view of the considerations developed in Chapter VII. For this we introduce magnetic reluctance, and again divide the magnetic circuit into five parts, as above (§ 125). The total magnetomotive force is thus the sum of five parts, each of which may be regarded as the product of the flux of induction in question, into the corresponding magnetic reluctance, and therefore takes a form defined by equation I (§ 119). The sum of the five terms-that is, the right side of the Hopkinson's equation (I) § 128-thereby at once becomes susceptible of another interpretation. This transformation of the Hopkinson's equation, which we alluded to in § 122, is a purely formal one, which does not at all affect the essence of the matter, but is, in a certain sense, already contained in it. There is no contradiction between Hopkinson's treatment and that to be now discussed. In order to explain this analytically, we observe that each of the five terms of equation (I), § 128-for instance, the first—by remembering the definitions of § 119, may be transformed as follows: (1) L1ƒ1 (8);) = L, §, = M, = X, 6, Lahmeyer, Elektrotechn. Zeitschrift, vol. 9, p. 283, 1887. 2 2 C. Hering, Electr. Review, vol. 21, pp. 186, 205, 1887; Carhart, Electrician, vol. 23, p. 644, 1889; Trotter, Journ. Inst. Electr. Engineers, vol. 19, p. 243, 1890; Esson, ibid. p. 122, 1880; Corsepius, Theoretische und praktische Untersuchungen zur Konstruktion magnetischer Maschinen, p. 85 et sqq., Berlin, 1891; Wedding, Elektrotech. Zeitschrift, vol. 13, p. 67, 1892. Kittler, Handbuch der Elektrotechnik, 2nd edition, vol. 1, p. 650, Stuttgart, 1892, gives a very complete synoptical table of the results of experiments on leakage. By transforming each individual term of equation (1) in a similar manner, the complete expression of the magnetic characteristic becomes M = X12 + 2 X1⁄2 61⁄2 + 2 X2 v 61⁄2 + X1 v&2 = 2 2 3 2 2 The flux of induction through the air-gap is therefore equal to the total magnetomotive force, divided by the sum of the magnetic reluctances of the parts of the magnetic circuit. The leakage is allowed for by multiplying the resistances of the limbs and of the yoke by the leakage-coefficient. The fraction of the magnetomotive force requisite for magnetising these parts is, in fact, not greater because its resistance had increased; but of course this is only the case because a greater flux of induction 6, v 6, must pass through it. In order to explain this, we may write equation (II) differently, by introducing the total flow of induction &,, instead of the useful flux 2: = 2 by which the influence of leakage becomes clear enough, since it apparently diminishes the reluctances X1, X2, and X ̧. Nothing can be objected to equation (II) or to (IIa) as long as we keep in view that all reluctances are variable except X2, the constant reluctance of the air-gap. Although, theoretically, the mode of writing equation (II), as compared with the original form of equation (I), offers scarcely any advantage, there is, in practice, one circumstance which places the question in an essentially different light. The constant reluctance X, of the air, and of the copper between the armature and the pole-pieces, is, especially in dynamos, almost always considerably greater than all the other variable reluctances together-at any rate, within the small range of induction (B< 15,000 C.G.S.) which occurs in practice so that this portion of the magnetic circuit requires CALCULATION OF AIR-RELUCTANCES 201 by far the greater part of the whole magnetomotive force. We mentioned this already in § 129 in considering fig. 29, in which the straight air-line (B) mainly determines the form of the characteristic (D). We may therefore assume somewhat arbitrary values for the permeability of the main portions of the magnetic circuit, without the result being materially affected, provided the reluctance of the air is correctly brought into the calculation. It is just this, however, which presents special difficulties, and, according to the judgment of experienced electrical engineers, is frequently a weak point in the application of the foregoing theory. We will now enter more closely into the theoretical and empirical determinations of airreluctances. § 132. Calculation of Air-reluctances.-So long as it is only a question of the layer of air between two parallel ferromagnetic surfaces of area S, at a distance d, the magnetic reluctance, in accordance with the theoretical definition, is simply since the permeability of air is equal to unity. In many cases, however, the calculation is not so simple, and the magnetic permeance of air-spaces has to be determined between surfaces of iron of arbitrary shape. Forbes has approximately calculated the permeance for a few special cases of most frequent occurrence by making simple assumptions as to the path of the lines of induction in the air, and has given the following solutions :— I. Magnetic permeance of the gap between two parallel surfaces S, and S2, unequal, but not very much so (fig. 30, p 202). If it be assumed that the lines of induction are straight and uniformly distributed over the two surfaces, then obviously, 1 Forbes, Journ. Soc. Tel. Engineers, vol. 15, p. 551, 1886. II. Magnetic permeance of the space between two equal rectangular ferromagnetic surfaces, which lie near each other in one plane, as represented in fig. 31. Assuming that the lines of induction are semicircles, as shown in the figure by the dotted lines, Forbes finds: III. Magnetic permeance in the case represented by fig. 32, where the ferromagnetic surfaces are further apart. Assuming that the path of the lines of induction is as represented in the figure, Forbes finds : as in If the two surfaces do not enclose the angle π, case II (fig. 31), but an angle a, in which case they are supposed to be rotated about the right line CC', then the value of a in circular measure must be substituted for π in equation (8). In these calculations there is much of an arbitrary nature in the simple assumptions made, that induction lines consist of arcs of circles and straight lines; in such instances, however, success alone can decide. In practical application the above formulas have in many cases been approximately confirmed, inasmuch as with their aid it is often possible to obtain an OTHER DETERMINATIONS OF AIR-RELUCTANCES 203 almost exact estimate of the leakage of a machine before its construction. Forbes had already calculated the leakage of the Edison-Hopkinson dynamo described above (§§ 125-130) from the leakage-diagram represented in fig. 28, p. 189; from this he calculated a coefficient of leakage v1.40, independent of the value of the induction. He showed, further, that if this value is inserted in equation (I), § 128, the curve D (fig. 29, p. 197) agrees still better with the points observed than by introducing the value v = 1·32, as measured by Drs. Hopkinson (§ 120). The cause of this discrepancy is probably partly to be sought in the fact that they placed the exploring coil around the middle instead of around the top of the limbs of the magnet (compare note, p. 197). On the other hand, it is not to be denied that in many other cases the theoretical computation of magnetic air-reluctances leaves much to be desired, since very often considerably smaller reluctances were observed than had been calculated. This is due, in great part, to the fact that at the edge of the air-gap the lines of induction, owing to leakage, bend outwards. The section of the air-gap to be considered in the calculation is thereby greater than the geometrical one, and to an extent which can scarcely be correctly estimated otherwise than empirically. In their fundamental research the Hopkinsons point out this circumstance, and allow for it by putting the section of the air-gap S2 = 1600 sq. cm. (as given in § 128), while it amounted, in fact, only to 1410 sq. cm. From § 133. Other Determinations of Air-reluctances. the special reason, so frequently insisted on, that in dynamo machines the constant magnetic reluctances of the air-spaces play the chief part, the analogies discussed in the previous chapter may be used to determine or to approximately estimate them. The permeability of the air-space is unity, while that of the ferromagnetic substance may be regarded as infinitely great in comparison, since, under the conditions existing in a dynamo it has always very high values (§ 126). Let us now consider the dielectric analogy (Table VII, column IV, § 124). The magnetic permeance of an air-space of any given form is, in consequence of this analogy, there shown to be proportional to its electrostatic capacity, in so far |