CHAPTER III MAGNETISM Reluctivity 29. Considering magnetism as the phenomena of a "magnetic circuit," the foremost differences between the characteristics of the magnetic circuit and the electric circuit are: (a) The maintenance of an electric circuit requires the ex- penditure of energy, while the maintenance of a magnetic circuit does not require the expenditure of energy, though the starting of a magnetic circuit requires energy. A magnetic circuit, there- fore, can remain "remanent" or "permanent." (6) All materials are fairly good carriers of magnetic flux, and the range of magnetic permeabilities is, therefore, narrow, from 1 to a few thousands, while the range of electric conductivi- ties covers a range of 1 to 10^^. The magnetic circuit thus is analogous to an uninsulated electric circuit inunersed in a fairly good conductor, as salt water: the current or flux can not be carried to any distance, or constrained in a "conductor," but divides, "leaks" or "strays." (c) In the electric circuit, current and e.m.f . are proportional, in most cases; that is, the resistance is constant, and the circuit therefore can be calculated theoretically. In the magnetic circuit, in the materials of high permeability, which are the most important carriers of the magoietic flux, the relation between flux, m.m.f. and energy is merely empirical, the "reluctance" or mag- netic resistance is not constant, but varies with the flux density, the previous history, etc. In the absence of rational laws, most of the magnetic calculations thus have to be made by taking numerical values from curves or tables. The only rational law of magnetic relation, which has not been disproven, is Frohlich's (1882) : "jTAe premeability is proportional to the magnetizability^^ M = a(S - B) (1) where B is the magnetic flux density, S the saturation density, 43 44 ELECTRIC CIRCUITS and S — B therefore the magnetizability, that is, the still avail- able increstse of flux density, over that existing. From (1) follows, by substituting, . = 1 (2) and rearranging, where / •— — ' •r. y p / jj p y in 1 J / 1, 1 / ' 1 / ^ 1 v ^ ^ / / y ^ ^ / ■^ \ y ^^ '. ^_, ^■^ ^ / c ■^ ^ H /^ '^ Fia. 23. Space flux, /f, or flux carried by space independent of the material in apace. The best evidence seems to corroborate, that with the excep- tion of very low field intensities (where the customary magneti- zation curve usually has an inward bend, which will be discussed later) in perfectly pure miagnetic materials, iron, nickel, cobalt. 46 ELECTRIC CIRCUITS etc., the linear law of reluctivity (5) and (3) is rigidly obeyed by the metallic induction Bq. In the more or less impure commercial materials, however, the p — H relation, while a straight line, often has one, and occasion- ally two points, where its slope, and thus the values of a and a change. Fig. 23 shows an average magnetization curve, of good standard iron, with field intensity, H, as abscissae, and magnetic induction, B, as ordinates. The total induction is shown in drawn lines, the metallic induction in dotted lines. The ordinates are given in kilolines per cm.^, the abscissae in units for 5i, in tens for B2, and in hundreds for £3. The reluctivity curves, for the three scales of abscissse, are plotted as pi, p2, pa, in tenths of milli-units, in milli-units and in tens jof milli-units. Below ff = 3, p is not a straight line, but curved, due to the in- ward bend of the magnetization curve, B, in this range. The straight-line law is reached at the point Ci, at ff = 3, and the re- luctivity is then expressed by the linear law Pi = 0.102 + 0.059 H (7) for 3 < ff < 18, giving an apparent saturation value. Si = 16,950. At H = 18, a bend occurs in the reluctivity line, marked by point C2, and above this point the reluctivity follows the equation P2 = 0.18 + 0.0548 ff (8) for 18 < ff < 80, giving an apparent saturation value S2 = 18,250. At H = 80, another bend occurs in the reluctivity line, marked by point cs, and above this point, up to saturation, the reluctivity follows the equation P3 = 0.70 + 0.0477 H (9) for H>SO giving the true saturation value, S = 20,960. MAGNETISM 47 Point cs is frequently absent. Fig. 24 gives once more the magnetization curve (metallic in- duction) aa B, and gives as dotted curves Bi, Sj and B» the mag- netization curves calculated from the three linear reluctivity equa- tions (7), (8), (9). As seen, neither of the equations represents ^ .. 1 , . . n , . », Li, n B — _= -IB — — — ^ P' r"' —■ S: ■^ . — 3 ^ — — 17 — ■ / / / '/ A —■ — ■ "^ -'' / B= / B ^ ^^ " •' / -' / / / 6. 1 / / / .-- / / / ^ ^' / .^ '' / // 1 ■-■ ^ ^ ;' / 1 ^ =v y II / / ijj ' y i 3 s i . 7i 1 l> 1 < B even approximately over the entire rango, but <;a';h repr»««nt« it very accurately within it« range. Th^ first, wjuation ^T), prol>- ably covers practically the entire induittrially imfx^rtant ra-im*:- 37. Aa these critical points Cj and c, do not fif^m Uj <;xi»t in p"^- fectly pure matcnals, and as the change of direction of ttus n^ 48 ELECTRIC CIRCUITS iuctivity line is in general the greater, the more impure the mate- rial, the cause seems to be lack of homogeneity of the material; that is, the presence, either on the surface as scale, or in the body, as inglomerate, of materials of different magnetic characteristics: magnetite, cementite, silicide. Such materials have a much greater hardness, that is, higher value of a, and thereby would give the observed effect. At low field intensities, H, the harder material carries practically no flux, and all the flux is carried by the soft material. The flux density therefore rises rapidly, giving low «, but tends toward an apparent low saturation value, as the flux-carrying material fills only part of the space. At higher field intensities, the harder material begins to carry flux, and while in the softer material the flux increases less, the increase of flux in the harder material gives a greater increase of total flux density and a greater saturation value, but also a greater hard- ness, as the resultant of both materials. Thus, if the magnetic material is a conglomerate of fraction p of soft material of reluctivity pi (ferrite) and g = 1 — p of hard material of reluctivity, p2 (cementite, silicide, magnetite), Pi = Qi + friH\ P2 = a2 + fT^H J (10) at low values of H, the part p of the section carries flux by pi, the part q carries flux by p2, but as p2 is very high compared with pi, the latter flux is negligible, and it is '>' = ? = ^' + ^'^ (11) V V V At high values of H, the flux goes through both materials, more or less in series, and it thus is p" = VPi + ^P^ = (P«i + Qoii) + {P<^i + QfT2)H (12) if we assume the same saturation value, ; ' ■^C p" > ^ -^ '', /' 'y^ / 1 i 1 f- , 'a K^' />■ / i / / ' '// - / ! 1^' > i/l H - 1 . ' / / / 1 / y / / ._ --' " / -' / ^ » " ^ — r- f \ •in* \ 1 ^ ^ -^ V ^ / \p ^- -'■ ^^\ -' -^ ^v ;^ ^ - ' H _ _. « 1 . The alternating characteristiCjBj, is not a branch of any hystere- sis cycle. It is reproducible and independent of the previous history of the magnetic circuit, except perhaps at extremely low values of H, and in view of its engineering importance as repre- MAGNETISM 53 senting the conditions in the alternating magnetic field, it would appear the most representative magnetic characteristic, and is commonly used as such. It has, however, the disadvantage that it represents an un- stable condition. Thus in Fig. 27, an alternating field H = 1 gives an alternating flux density, B2 = 2.6. If, however, this field strength ff = 1 is left on the magnetic circuit, the flux does not remain at B2 = 2.6, but gradually creeps up to higher values, especially in the presence of mechanical vibrations or slight pulsations of the magnetizing current. To a lesser extent, the same occurs with the values of curve, J5i, to a greater extent with J? 3. At very low densities, this creepage due to instability of the B-H relation may amount to hundreds of per cent, and continue to an appreciable extent for minutes, and with magnetically hard materials for many years. Thus steel structures in the terrestrial magnetic field show immediately after erection only a small part of the magnetization, which they finally assume, after many years. Thus the alternating characteristic, £2, however important in electrical engineering, can, due to its instability, not be considered as representing the true physical relation between B and H any more than the branches of hysteresis cycles Bi and Bz» 34. Correctly, the relation between B and H thus can not be expressed by a curve, but by an area. Suppose a hysteresis cycle is performed between infinite values of field intensity: H = ± ozy that is, practically, between very high values such as are given for instance by the isthmus method of magnetic testing (where values of H of over 40,000 have been reached. Very much lower values probably give practically the same curve). This gives a magnetic cycle shown in Fig. 5 as J5', B". Any point, H, B, within the area of this loop between J5' and B" of Fig. 27 then represents a possible condition of the magnetic circuit, and can be reached by starting from any other point, Hqj Boy such as the zero point, by gradual change of H. Thus, for instance, from point Po, the points Pi, P2, P3, etc., are reached on the curves shown in the dotted lines in Fig. 27. As seen from Fig. 27, a given value of field intensity, such as J? = 1, may give any value of flux density between B = —4.6 and B = +13.6, and a given value of flux density, such as B = 10, may result from any value of field intensity, between H = - 0.25 to ff = + 3.4 54 ELECTRIC CIRCUITS The different values of B, corresponding to the same value of H in the magnetic area, Fig. 27, are not equally stable, but the val- ues near the limits 5' and 5" are very unstable, and become more stable toward the interior of the area. Thus, the relation of point Pi, Fig. 27: H == 2j B = 13, would rapidly change, by the flux density decreasing, to Po, slower to P2 and then still slower, while from point P3 the flux density would gradually creep up. If thus follows, that somewhere between the extremes B' and B", which are most unstable, there must be a value of B, which is stable, that is, represents the stationary and permanent relation between B and Hj and toward this stable value. Bo, all other val- ues would gradually approach. This, then, would give the true magnetic characteristic: the stable physical relation between B and H. At higher field intensities, beyond the first critical point, Ci, this stable condition is rapidly reached, and therefore is given by all the methods of determining magnetic characteristics. Hence, the curves Bi, B2, Bo coincide there, and the linear law of re- luctivity applies. Below Ci, however, the range of possible, B, values is so large, and the final approach to the stable value so slow, as to make it difficult of determination. 36. For H = 0, the magnetic range is from —Ro = —11.2 to +jBo = 11.2; the permanent value is zero. The method of reach- ing the permanent value, whatever may be the remanent mag- netism, is well known; it is by ** demagnetizing" that is, placing the material into a powerful alternating field, a demagnetizing coil, and gradually reducing this field to zero. That is, describ- ing a large number of cycles with gradually decreasing amplitude. The same can be applied to any other point of the magnetiza- tion curve. Thus for ff = 1, to reach permanent condition, an alternating m.m.f. is superimposed upon H = 1, and gradually decreased to zero, and during these successive cycles of decreas- ing amplitude, with ff = 1, as mean value, the flux density gradu- ally approaches its permanent or stable value. (The only re- quirement is, that the initial alternating field must be higher than any unidirectional field to which the magnetic circuit had been exposed.) This seems to be the value given by curve Bo, that is, by the straight-line law of reluctivity. In other words, it is probable that: Frohlich's equation, or Kennelly's linear law of reluctivity MAGNETISM 55 represent the permanent or stable relation between B and H, that is, the true magnetic characteristic of the material, over the entire range down to H = 0, and the inward bend of the magnetic characteristic for low field intensities, and corresponding increase of reluctivity p, is the persistence of a condition of magnetic instability, just as remanent and permanent magnetism are. In approaching stable conditions by the superposition of an alternating field, this field can be applied at right angles to the unidirectional field, as by passing an alternating current length- wise, that is, in the direction of the lines of magnetic force, through the material of the magnetic circuit. This superimposes a cir- cular alternating flux upon the continuous-length flux, and per- mits observations while the circular alternating flux exists, since the latter does not induce in the exploring circuit of the former. Some 20 years ago Ewing has already shown, that under these conditions the hysteresis loop collapses, the inward bend of the magnetic characteristic practically vanishes, and the magnetic characteristic assumes a shape like curve Bq, To conclude, then, it is probable that: In pure homogeneous magnetic materials, the stable relation between field intensity, H, and fiux density, 5, is expressed, over the entire range from zero to infinity, by the linear equation of reluctivity p = a -{- aHy where p applies to the metallic magnetic induction, B ■— H. In unhomogeneous materials, the slope of the reluctivity line changes at one or more critical points, at which the flux path changes, by a material of greater magnetic hardness beginning to carry flux. At low field intensities, the range of unstable values of B is very great, and the approach to stability so slow, that considerable deviation of B from its stable value can persist, sometimes for years, in the form of remanent or permanent magnetism, the inward bend of the magnetic characteristic, etc.