Theory and Calculation of Electric Circuits — part 4 of 15

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 Bo.

To conclude, then, it is probable that: . :

In pure homogeneous magnetic materials, the stable relation between field intensity, H, and flux density, B, is expressed, over the entire range from zero to infinity, by the linear equation of reluctivity

. p=a+tolH, 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 tostability 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.

| 1 CHAPTER IV ; . MAGNETISM Hysteresis 36. Unlike the electric current, which requires power for its maintenance, the maintenance of a magnetic flux does not require | energy expenditure (the energy consumed by the magnetizing | current in the ohmic resistance of the magnetizing winding being an electrical and not a magnetic effect), but energy is required to produce a magnetic flux, is then stored as potential energy in the magnetic flux, and is returned at the decrease or disappear- ance of the magnetic flux. However, the amount of energy re- turned at the decrease of magnetic flux is less than the energy consumed at the same increase of magnetic flux, and energy is therefore dissipated by the magnetic change, by conversion into | heat, by what may be called molecular magnetic friction, at least : | in those materials, which have permeabilities materially higher than unity. . Thus, if a magnetic flux is periodically changed, between

• Band — B, or between B; and Bs, as by an alternating or pul- sating current, a dissipation of energy by molecular friction occurs during each magnetic cycle. Experiment shows that the : energy consumed per cycle and cm.? of magnetic material depends | only on the limits of the cycle, B, and Bz, but not on the speed or. | | _wave shape of the change. | ! If the energy which is consumed by molecular friction is sup- | plied by an electric current as magnetizing force, it has the effect : that the relations between the magnetizing current, 7, or magnetic | field intensity, H, and the magnetic flux density, B, is not revers- | ible, but for rising, H, the density, B, is lower than for decreasing H; that is, the magnetism lags behind the magnetizing force, and | the phenomenon thus is called hysteresis, and gives rise to the . | hysteresis loop. However, hysteresis and molecular magnetic friction are not . 56 . | .

‘MAGNETISM 57 the same thing, but the hysteresis loop is the measure of the mo- lecular magnetic friction only in that case, when energy is supplied - to or abstracted from the magnetic circuit only by the magnetiz- ing current, but not otherwise. Thus, if mechanical work is done by the magnetic cycle—as when attracting and dropping an arma- ture—the hysteresis loops enlarge, representing not only the energy dissipated by molecular magnetic friction, but also that . converted into mechanical work. Inversely, if mechanical en- ergy is supplied to the magnetic circuit as by vibrating it mechan- ically, the hysteresis loop collapses or overturns, and its area becomes equal to the molecular magnetic friction minus the mechanical energy absorbed. Thereaction machine, assynchron- ous motor and as generator, is based on this feature. See “Reaction Machine,” “Theory and Calculation of Electrical Apparatus.’ In general, when speaking of hysteresis, molecular magnetic friction is meant, and the hysteresis cycle assumed under the con- dition of no other energy conversion, and this assumption will be made in the following, except where expressly stated otherwise. ; The hysteresis cycle is independent of the frequency within -commercial frequencies and far beyond this range. Even at . frequencies of hundred thousand cycles, experimental evidence ‘seems to show that the hysteresis cycle is not materially changed, except in so far as eddy currents exert a demagnetizing action and thereby require a change of the impressed m.m-f., to get the same resultant m.m.f., and cause a change of the magnetic: flux dis- tribution by their screening effect. A change of the hysteresis cycle occurs only at very slow cycles —cycles of a duration from several minutes to years—and even then to an appreciable extent only at very low magnetic densities. Thus at low values of B—below 1000—hysteresis cycles taken by ballistic galvanometer are liable to become irregular and erratic, by “magnetic creepage.’’ For most practical purposes, however, ; this may be neglected. 37. As the industrially most important varying magnetic fields are the alternating magnetic fields, the hysteresis loss in alternat- ing magnetic fields, that is, in symmetrical cycles, is of most . interest. In general, if a magnetic flux changes from the condition H,, . B,: point P, of Fig. 29, to the condition Hs, By: point Ps, and we assume this magnetic circuit surrounded by an electric circuit of .

58 ELECTRIC CIRCUITS n turns, the change of magnetic flux induces in the electric cir- cuit the voltage, in absolute units, e=n oe (1) it is, however, . ; . ® = 8B (2) where s = section of magnetic circuit. Hence ~ e= ns 98 6) If ¢ = current in the electric circuit, the m.m.f. is F=ni (4) and the magnetizing force ; | fa" (6) where | = length of the magnetic circuit. And the field intensity H = 4af (6) hence, substituting (5) into (6) and transposing, . is the magnetizing current in the electric circuit, which produces the flux density, B. The power consumed by the voltage induced in the electric circuit thus is . slH dB Pain Tet ®) or, per cm.? of the magnetic circuit, that is, for s = 1 and 1 = 11, H dB PT a (9) and the energy consumed by the change from Hi, B, to Hs, Bs, which is transferred from the electric into the magnetic circuit, or inversely, 1 1,3 = ral HaB ergs (10) = A 1,2 ’ 4x s e

MAGNETISM . 59 where A;,2 is the area shown shaded in Fig. 29. The energy consumed during a cycle, from Ho, Bo to — Ho, — Bo and back to Ho, Bo, thus is 1 w= 7, {#48 ergs (11) A = 7, ores (12) where f HdB = A is the area of the hysteresis loop, shown shaded 0 in Fig. 30. As the magnetic condition at the end of the cycle is the same as \ -H,+B +H+B B, Wom P, SSOENV { fo) Hi Hs SS] -H-B +H,-B Fie. 29. Fia. 30. ‘ at the beginning, all this energy, w, is dissipated as heat, that is, is the hysteresis energy which measures the molecular magnetic friction.

38. If in Fig. 30 the shaded area represents the hysteresis loop between + H, + B, and — H, — B, giving with a sinusoidal alternating flux the voltage and current waves, Fig. 31, the maxi- mum area, which the hysteresis loop could theoretically assume, is given by the rectangle between + H, + B; — H, + B; — d, — B;+ H, — B. This would mean, that the magnetic flux does not appreciably decrease with decreasing field intensity, until the field has reversed to full value. It would give the theoretical wave shape shown as Fig. 32. As seen, this is the extreme ex- aggeration of wave shape, Fig. 31.

60 ELECTRIC CIRCUITS ' ‘The total energy of this rectangle, or maximum available magnetic energy, is . ; 4HB HB = (12) . a: B.. or, if « = permeability, thus H = an it is B? . Wo = 7A (18) | Fia. 31. the maximum possible hysteresis loss. . The inefficiency of the magnetic cycle, or percentage loss of : energy in the magnetic cycle, thus is ; t .. L™ _ a | , Fia. 32. | we TW . f Wo B? —H . _ = TR { HdB (14)

• Ab. | . ~ 4B

39. Experiment shows that for medium flux density, that is, thoses values of B which are of the most importance industrially,

MAGNETISM : 61 from B = 1000 to B = 12,000, the hysteresis loss can with suffi- cient accuracy for most practical purposes be approximated by the empirical equation, w = 7B (15) Pi ttt tT ttt tt tt TTA | Lp | {| fsufeot steed | | | | | | VT af a Lt Pt tt tt tt et ET TTA TY RRR RE EERE | | tit it_t_tEt_?Tt ttt yy | pe] Pt tt tT tT tT TE EE TE ALA | FH ep) fA Pt tet tT tt tt TT Sane FETT TTT ETL TAL TT ey anaes Pt ttt tt YE ET A LETT TTT PATH TTT | EEE Pitt tt tear ET TT TT HE EE" — tt tee ttt tet | Et Tt Brae eerrereT ; Fie. 33. where 7, the “coefficient of hysteresis,” is of the magnitude . of 1 X 10-* to 2 X 10°? for annealed soft sheet steel, if B is given in lines of force per cm.?, and w is ergs per cm.’ and cycle. . Very often w is given in joules, or watt-seconds per cycle and per kilogram or pound of iron, and B in lines per square inch, or w is given in watts per kilogram or per pound at 60 cycles.

62 ELECTRIC CIRCUITS In Fig. 33 is shown, with B as abscisse, the hysteresis loss, w, of a sample of silicon steel. The observed values are marked by circles. In dotted lines is given the curve calculated by the equation | w = 0.824 x 10-3 Bi (16) As seen, the agreement the curve of 1.6* power with the test values is good up to B = 10,000, but above this density, the observed values rise above the curve.

• 40. In Fig. 34 is plotted, with field intensity, H, as abscisse, the magnetization curve of ordinary annealed sheet steel, in | | |rerrite ano MAGNETITE] [| | [| | [a] | [- | | #4 | | [| Macnerization | [tty 7 Tt ttt de SERED? ae ptt eT i Tt tT tT TE TT TE TE TE TTT de PiA LT tT Et Tt tT Ty eet BARRE ae FAT IT TT Peer] | et fT} | | fp eee tT I Hat tt Tt Tt tt te Yl iA | | | rr | | Wee ee Ne a ee eee HIE YY | pet tT ET ETE TT PIA“ tet tT tt tt tt TT TT Tt PZEe 4ne eee Awl mio lolololn | » | » | wl | Fig. 34. half-scale, as curve 1, and the magnetization curve of magnetite, - _ Fez0,—which is about the same as the black scale of iron—in double-scale, as curve II. As III then is plotted, in full-scale, a curve taking 0.8 of I and 0.2 of II. This would correspond to the average magnetic density in a material containing 80 per cent. of iron and 20 per cent. (by volume) of scale. Curves I’ and III’ show the initial part of I and III, with ten times the scale of abscisse and the same scale of ordinates. Fig. 35 then shows, with the average magnetic flux density, B, taken from curve III of Fig. 34, as abscisse, the part of the mag- |

MAGNETISM 63 netic flux density which is carried by the magnetite, as curve I. As seen, the magnetite carries practically no flux up to B = 10, but beyond B = 12, the flux carried by the magnetite rapidly increases.

As curve II of Fig. 35 is shown the hysteresis loss in this inhomo- geneous material consisting of 80 per cent. ferrite (iron) and 20 per cent. magnetite (scale) calculated from curves I and II of Fig.

pened | ft LA : Pde | TdT | dT PT WP CT CT TT A poe op a a ALY Ph TT TP rT re ree VY my | fp Pt tte eT TE Ye eee A] | | || |remRiTe AND Maqnetire] AT [| jos fe Z| | | Pitt TTT TPA me fot tt fT ET TT TTA TTT Ty ey Ef LitTtitT tT het ttt TA pop pet | itr | | | tt tt PY SPARES eres Fia. 36. 34 under the assumption that either material rigidly follows the 1.6" power law up to the highest densities, by the equation, Iron: " w, = 1.2 B® K 10-8. Scale: ; We = 23.5 Brn* x 10-3, As curve II’ is shown in dotted lines the 1.6 power equation, w= 1.38 B-* x 10%.

| | 64 ELECTRIC CIRCUITS As seen, while either constituent follows the 1.6" power law, the combination deviates therefrom at high densities, and gives an increase of hysteresis loss, of the same general characteristic as shown with the silicon steel in Fig. 33, and with most similar materials. As curve III in Fig. 35 is then shown the increase of the hyste- resis coefficient 7, at high densities, over the value 1.38 X 107°, which it has at medium densities. Thus, the deviation of the hysteresis loss at high densities, from the 1.6" power law, may possibly be only apparent, and . the result of lack of homogeneity of the material. 41. At low magnetic densities, the law of the 1.6 power must ~ cease to represent the hysteresis loss even approximately. The hysteresis loss, as fraction of the available magnetic energy, is, by equation (14), t=" (14) Substituting herein the parabolic equation of the hysteresis loss, | w = nB (17) where n = 1.6, it is ‘ $ = prn Bs-? (18) = xn B+ 7 With decreasing density B, B*~* steadily increases, if n < 2, and : as the permeability u approaches a constant value, ¢, steadily in- creases in this case, thus would become unity at some low density, B, and below this, greater than unity. This, however, is not | possible, as it would imply more energy dissipated, than available, and thus would contradict the law of conservation of energy. Thus, for low magnetic densities, if the parabolic law of hysteresis (17) applies, the exponent must be: n 5 2. | In the case of Fig. 33, for 7 = 0.824 X 10-, assuming the per- | meability for extremely low density as . | u = 1500, $ becomes unity, by equation (18), at . B = 30. If n > 2, B"— steadily decreases with decreasing B, and the per- . centage hysteresis loss becomes less, that is, the cycle approaches reversibility for decreasing density; in other words, the hys- teresis loss vanishes. This is possible, but not probable, and the { | |

MAGNETISM 65 probability is that for very low magnetic densities, the hysteresis losses approach proportionality with the square of the magnetic density, that is, the percentage loss approaches constancy.

From equation (17) follows

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Fia. 36. log w = log » +n log B (19)

That is:

“Tf the hysteresis loss follows a parabolic law, the curve plotted with log w against log B is a straight line, and the slope of this straight line is the exponent, n.”

5

66 ELECTRIC CIRCUITS Thus, to investigate the hysteresis law, log w is plotted against , log B. This is done for the silicon steel, Fig. 33, over the range from B = 30 to B = 16,000, in Fig. 36, as curve I. |

Curve I contains two straight parts, for medium densities, from log B = 3; B = 1000, to log B = 4; B = 10,000, with slope 1.6006, and for low densities, up to log B = 2.6; B = 400, with slope 2.11. Thus it is

For 1000 < B ¥< 10,000:

w = 0.824 BY* x 107 | For B <¢ 400: _ w = 0.00257 B¥"™ x 10-8 However, in this lower range, n = 2 gives a curve: | w = 0.0457 B? x 107? | which still fairly well satisfies the observed values. | As the logarithmic curve for a sample of ordinary annealed | ‘sheet steel, Fig. 37, gives for the lower range the exponent, n = 1.923, and as the difficulties of exact measurements of hysteresis losses increase with decreasing density, it is quite possible that in both, Figs. 36 and 37 the true exponent in the lower range of mag- netic densities is the theoretically most probable one, n = 2, that is, that at about B = 500, in iron the point is reached, below which the hysteresis loss varies with the square of the magnetic density.

42. As over most of the magnetic range the hysteresis loss can be expressed by the parabolic law (17), it appears desirable to adapt this empirical law also to the range where the logarithmic curve, Figs. 36 and 37, is curved, and the parabolic law does not

• apply, above B = 10,000, and between B = 500 and B = 1000, or thereabouts. This can be done either by assuming the coeffi- cient 7 as variable, or by assuming the exponent n as variable.

(a) Assuming » as constant,

n = 0.824 X 10-* for the medium range, where n = 1.6 nm = 0.0457 X 10-* for the low range, where n; = 2

The coefficients n and 7; calculated from the observed values

|

} oO MAGNETISM 67 of w, then, are shown in Fig. 36 by the three-cornered stars in the upper part of the figure. (b) Assuming n as constant, ; n =. 1.6 for the medium range, where 7 = 0.0824 x 10- m, = 2 for the low range, where 7 = 0.0457 x 107? teal | | dT dT cd Td EE dE EET CL pemrnisras eee eee ae — Pit tt tt fe Hee a HE | pti-|-| 4 Pt tT Tt Te | tel | 7 Pt tt tt Te ee Pitt tT ee Ty Piet tT ET TT EE TE fT aie | | | | | ee att ETT Ta dt ee er Meee YOCELLAT L oF | HT INTE Le , AE Pt tT TTT we gf fs ERR” ARR Pt tt bt TE ET TT TE Pt ty ttt et TT ET Se ee | A || ttt ttt tt ttt Pt tt dd EL eet TE Ta TI Fia. 37. The variation of 7 and 7, from the values in the constant range, then, are best shown in per cent., that is, the loss w calculated from . the parabolic equation and a correction factor applied for values of B outside of the range. J :

68 ELECTRIC CIRCUITS ° Fig. 37 shows the values of 7 and m, as calculated from the para- bolic equations with n = 1.6 and n; = 2, and Fig. 36 shows the percentual variation of 7 and m1. . The latter method, (b), is preferable, as it uses only one expo- nent, 1.6, in the industrial range, and uses merely a correction factor.. Furthermore, in the method (a), the variation of the exponent is very small, rising only to 1.64, or by 2.5 per cent., while ; in method (6) the correction factor is 1.46, or 46 per cent., thus a much greater accuracy possible. 43. If the parabolic law applies, w = 7B" ; (17) the slope of the logarithmic curve is the exponent n. If, however, the parabolic law does not rigidly apply, the slope | of the logarithmic curve is not the exponent, and in the range, where the logarithmic curve is not straight, the exponent thus . can not even be approximately derived from the slope. From (17) follows log w = logy + nlog B, (19) - differentiating (19), gives, in the general case, where the parabolic law does not strictly apply, dlogw = dlog 7 + nd log B + log Bdn, hence, the slope of the logarithmic curve is dlogw _ . dn d log » Toe =" +. (log Barco w + dior B) (20) If n = constant, and » = constant, the second term on the . right-hand side dis&ppears, and it is , dlogw _ | dlog B~ (21) that is, the slope of the logarithmic curve is the exponent. If, however, 7 and 7 are not constant, the second term on the right-hand side of equation (20) does not in general disappear, and the slope thus does not give the exponent. . Assuming in this latter case the slope as the exponent, it must be | dn dlogy _ log B tice Bt dlogB | . Or, dlogn _ —h = log B (22)

MAGNETISM 69 In this case, n and much more still 7 show @ very great varia- tion, and the variation of 7 is so enormous as to make this repre- . sentation valueless. As illustration is shown, in Fig. 36, the slope of the curve as m2 AS seen, ne varies very much more than n or 7.: : To show the three different, representations, in the following table the values of n and 7 are shown, for a different sample of iron. ; TABLE | - | - - | (c) na = | B 103 (a) J eve (b) n * Sonst. clog w 1 below 10.00 |In=1.6 9 = 1.254100? Ing =1.6 | a= 1.254X10-* 10.00 | =1.601 =1.268 =1.79 230.00 11.23 | =1.604 =1.302 =2.23 3.68 12.63 | =1.617 =1.468 =2.66 0.0488 13.30 | =1.624 =1.570 =2.83 0.0133 14.00 | =1.630 =1.668 =2.98 0.0032 14.65 | =1.634 =1.738 =3.15 0.00069 ; . . ‘| 1.738 As seen, to represent an increase of hysteresis loss by 1954 = 1.39, or 39 per cent., under (c), m2 is nearly doubled, and 73 re- 1 re duced to 1,800,000 of its initial value. 44. The equation of the hysteresis loss at medium densities, W = 7B"; n=1.6 is entirely empirical, and no rational reason has yet been found why this approximation should apply. Calculating the coeffi- cient n from test values of B and W, shows usually values close to 1.6, but not infrequently values of n are found, as low as 1.55, and even values below 1.5, and values up to 1.7 and even above 1.9 : In general, however, the more accurate tests give values of n which do not differ very much from 1.6, so that the losses can . still be represented by the curve with the exponent n = 1.6, without serious error. This is desirable, as it permits comparing different materials by comparing the coefficients 7. This would not be the case, if different values of n were used, as even a small change of n makes a very large change of n: a change of n by 1 per cent., at B = 10,000, changes by about 16 per cent.

70 ELECTRIC CIRCUITS Thus in Fig. 37 is represented as I the logarithmic curve of a sample of ordinary annealed sheet steel, which at medium den- sity gives the exponent n = 1.556, at low densities the exponent nm, = 1.923. Assuming, however, n = 1.6 and n,; = 2.0, gives the average values 7 = 1.21 X 10-* and 7 = 0.10 X 10-*, and the PT tT tTtLttttt tt yy yA | | { Jorowary sueet stee.| [| | | | | |/| | | | ANNEALED. HYSTERESIS Te SERRE | BEE EE EEE RRR | Pe} pe RRR Pt ttt tT tT te ETT FT HERE pti tit ttt tt | yt ly KER EEE EEE Sey fe) eR AAT Ft tt yt Hoe He tT ttt tT ttt tA Ty HERRERA Pt tt tt tt tT ey ty SERRA Pt ttt tT tre TT poppe |e Pt tT TT TT ET TT ert 1 bb Ph en bh bu al) | Fra. 38. | individual calculated values of 7 and m are then shown on Fig. 37 by crosses and three-pointed stars, respectively. Fig. 38 then shows the curve of observed loss, in drawn line, and the 1.6" power curve calculated in dotted line, and Fig. 39 the lower range of the calculated curve, with the observations marked by circles. Fig. 40 shows, for the low range, the curve

. MAGNETISM 71 of 7,B?, in two different scales, with the observed values marked by cycles. As seen, although in this case the deviation of n from 1.6 respectively 2 is considerable, the curves drawn with n = 1.6 and n, = 2 still represent the observed values fairly well in

Fae

EEL,

so HR RE Ao Pitt tt tty Vt pt tT ET Tt A tee Pt ET Ty TTT : Pt tt tt | VT de PTL TAT |} A} Pit tt tt eT See ee PTT TTT YET yy Hy -PE se Pt tT TT Td Se} |} Pt tt yt al. : Pt} TAT] Tt | Soe 4s

. LL a +4 LA tt ttt

| PAGER Fig. 39.

° the range of B from 500 to 10,000, and below 500, respectively, so that the 1.6 power equation for the medium, and the quadratic equation for the low values of B can be assumed as sufficiently accurate for most purposes, except in the range of high densities

72 ELECTRIC CIRCUITS

. in those materials, where the increase of hysteresis loss occurs While the measurement of the hysteresis loss appears a very simple matter, and can be carried out fairly accurately over a

Pt tT tt TE ET tT EE COREEEEE Err | pe pitt tet et T/T Ty Pitt TTT TTT TV, P| tt tty tit tT i” Td. pt tT TT tT TT TT tT, SERRE eee Pt TT tT eT TT AT TT HEHE EEE Pt ti tT tT TT TA TT tt ig} | | TTA TT fel VET TTY Tt TT ed A | tht | tart Ty elf | TVET I tf ype |_| Leg | ee] | | fel | H+} eet | LT tT tt hy Fia. 40.

. narrow range of densities, it is one of the most difficult matters to measure the hysteresis loss over a wide range of densities with such accuracy as to definitely determine the exact value of the exponent n, due to varying constant errors, which are beyond con-

MAGNETISM ; 73 trol. While true errors of observations can be eliminated by multiplying data, with a constant error this is not the case, and if the constant error changes with the magnetic density, it results in an apparent change of n. Such constant errors, which increase or decrease, or even reverse with changing B, are in the Ballistic galvanometer method the magnetic creepage at lower B, and at higher B the sharp-pointed shape of the hysteresis loop, which makes the area between rising and decreasing characteristic difficult to determine. In the wattmeter method by alternating current, varying constant errors are the losses in the instruments, the eddy-current losses which change with the changing flux dis- tribution by magnetic screening in the iron, with the temperature, etc., by wave-shape distortion, the unequality of the inner and outer length of the magnetic circuit, etc.

45. Symmetrical magnetic cycles, that is, cycles performed be- tween equal but opposite magnetic flux densities, +B and —B, are industrially the most important, as they occur in practically all alternating-current apparatus. Unsymmetrical cycles, that is, cycles between two different values of magnetic flux density, ‘ B, and Bs, which may be of different, or may be of the same sign, are of lesser industrial importance, and therefore have been | little investigated until recently. :

However, unsymmetrical cycles are met in many cases in al- ternating- and direct-current apparatus, and therefore are of importance also.

In most inductor alternators the magnetic flux in the armature does not reverse, but pulsates between a high and a low value in the same direction, and the hysteresis loss thus is that of an unsymmetrical non-reversing cycle.

Unsymmetrical cycles occur in transformers and reactors by the . superposition of a direct current upon the alternating current, as discussed in the chapter “Shaping of Waves,” or by the equiva- lent thereof, such as the suppression of one-half wave of the alter- nating current. Thus, in the transformers and reactors of many types of rectifiers, as the mercury-arc rectifier, the magnetic cycle is unsymmetrical.

Unsymmetrical cycles occur in certain connections of trans- formers (three-phase star-connection) feeding three-wire syn- chronous converters, if the direct-current neutral of the converter is connected to the transformer neutral.

They may occur and cause serious heating, if several trans-

74 ; ELECTRIC CIRCUITS formers with grounded neutrals feed the same three-wire distri- bution circuit, by stray railway return current entering the three- ' wire a ternating distribution circuit over one neutral and leaving it over another one. Two smaller unsymmetrical cycles often are superimposed on ~ . an alternating cycle, and then increase the hysteresis loss. Such occurs in transformers or reactors by wave shapes of impressed voltage having more than two zero values per cycle, such as that ' shown in Fig. 51 of the chapter on “Shaping of Waves.”” They also occur sometimes in the armatures of direct-current motors at high armature reaction and low field excitation, due to the flux’ distortion, and under certain conditions in the armatures of regulating pole converters. _ | A large number of small unsymmetrical cycles are sometimes superimposed upon the alternating cycle by high-frequency pul- sation of the alternating flux due to the rotor and stator teeth, and then may produce high losses. Such, for instance, is the ° case in induction machines, if the stator and rotor teeth are not proportioned so as to maintain uniform reluctance, or in alterna- tors or direct-current machines, in which the pole faces are slotted to receive damping windings, or compensating windings, etc., if the proportion of armature and pole-piece slots is not carefully . designed. 46. The hysteresis loss in an unsymmetrical cycle, between limits B, and Ba, that is, with the amplitude of magnetic variation B= i) follows the same approximate law of the 1.6% power, Wo = noB-6 as long as the average value of the magnetic flux variation, is constant. . With changing Bo, however, the coefficient mo changes, and in- creases with increasing average flux density, Bo. ; John D. Ball has shown, that the hysteresis coefficient of the unsymmetrical cycle increases -with increasing average density, Bo, and approximately proportional to a power of Bo. That is, no = 7 + Bn Bol’.

MAGNETISM 75 Thus, in an unsymmetrical cycle between limits B, and B: of magnetic flux density, it is . _ B, + By"*) /B, — By ™8 w= [a+ 8 ( x) }(+3>) (23)

• where 7 is the coefficient of hysteresis of the alternating-current cycle, and for Bs = —B,, equation (23) changes to that of the symmetrical cycle. ; Or, if we substitute, By = 143s (24) : = average value of flux density, that . is, average of maximum and mini- mum. : B= Pi (25). ‘= amplitude of unsymmetrical cycle, . it is , w= (n+ BBo*)B" (26) or, w = noB-6 (27) where . ; tm = 7 + BBo'® (28) or, more general, . w = 0B" (29) no = 7 + BBo™ (30) . For a good sample of ordinary annealed sheet steel, it was found, . 9 = 1.06 X 10-3 (31) B = 0.344 x 10-"° For a sample of annealed medium silicon steel, n=105X10? - (32) 8 =,0.32 X 10-1 Fig. 41 shows, with Bo as abscisse, the values of n,, by equa- tions (30) and (32). As seen, in a moderately unsymmetrical cycle, such as between B, = +12,000 and B, = —4000, the increase of the hysteresis

. ; _}0 76 ELECTRIC CIRCUITS So loss over that in a symmetrical cycle of the sayhe amplitude, is , moderate, but the increase of hysteresis loss pecomes very large Pi ttt ett ty Ty Pe Sacceeeeeeneees | | | unsymmetricatcyciey | | | | | /|, nee ea PTT TEE TT ett yt eH 4 | epee pene} by pe Pt tt tT TTT tt | Yt tt ty, SE REEERE AGRE Ft | | eee PEF ETT Tt TAT Tt th. Pt tt tt tte 4 ELTTt prt tT ETT tt eee ae eee pes ETT TTT TET ET TE ET SRR Piet TE TTT EET TTT PT eT tt ety ty et tt Seer sad Fia. 41, in highly unsymmetrical cycles, such as between B, = 16,000 and B, = 12,000. t

CHAPTER V " MAGNETISM : Magnetic Constants

47. With the exception of a few ferromagnetic substances, the magnetic permeability of all materials, conductors and dielectrics, gases, liquids and solids, is practically unity for all industrial purposes. Even liquid oxygen, which has the highest permea- bility, differs only by a fraction of a per cent. from non-magnetic materials.

Thus the permeability of neodymium, which is one of the most paramagnetic metals, is ~« = 1.003; the permeability of bismuth, which is very strongly diamagnetic, is 1 = 1 — 0.00017 = 0.99983. . The magnetic elements are iron, cobalt, nickel, manganese and chromium. It is interesting to note that they are in atomic weight adjoining each other, in the latter part of the first half of

° the first large series of the periodic system: Ti V Cr Mn Fe Co Ni Cu Zn Atomic weight.................. 48 61 52 55 56 58 59 61 65 .

The most characteristic, because relatively most constant, is the metallic magnetic saturation, S, or its reciprocal, the satura- : tion coefficient, c, in the reluctivity equation. The saturation density seems to be little if any affected by the physical condition of the material. By the chemical composition, such as by the presence of impurities, it is affected only in so far as it is reduced approximately in proportion to the volume occupied by the non- magnetic materials, except in those cases where new compounds result. .

It seems, that the saturation value is an absolute limit of the element, and in any mixture, alloy or compound, the saturation value reduced to the volume of the magnetic metal contained therein, can not exceed that of the magnetic metal, but may be lower, if the magnetic metal partly or wholly enters a compound of lower intrinsic saturation value. Thus, if S = 21 x 10? is

. the saturation value of iron, an alloy or compound containing 77

73 ELECTRIC CIRCUITS

72 per cent. by volume of iron can have a maximum saturation value of S = 0.72 X 21 K 10° = 15.1 X 10° only, or a still lower saturation value.

The only known exception herefrom seems to be an iron-cobalt alloy, which is alleged to have a saturation value about 10 per cent. higher than that of iron, though cobalt is lower than iron.

. The coefficient of magnetic hardness, a, however, and the co-

efficient of hysteresis, 7, vary with the chemical, and more still

. with the physical characteristic of the magnetic material, over an enormous range.

Thus, a special high-silicon steel, and the chilled glass hard tool steel in the following tables, have about the same percentage of non-magnetic constituents, 4 per cent., and about the same saturation value, S = 19.2 X 10°, but the coefficient of hardness of chilled tool steel, a = 8 X 10-°, is 200 times that of the special silicon steel, a = 0.04 X 10-*, and the coefficient of hysteresis of

: the chilled tool steel, » = 75 X 10-’, is 125 times that of the sili- con steel, 7 = 0.6 X 10-°. Hardness and hysteresis loss seem to depend in general on the physical characteristics of the material, and on the chemical constitution only as far as it affects the phys- ical characteristics.

Chemical compounds of magnetic metals are in general not ferromagnetic, except a few compounds as magnetite, which are ferromagnetic.

With increasing temperature, the magnetic hardness «, decreases, that is, the material becomes magnetically softer, and the satura-

. tion density, S, also slowly decreases, until a certain critical temperature is reached (about 760°C. with iron), at which the material suddenly ceases to be magnetizable or ferromagnetic, but usually remains slightly paramagnetic.

As the result of the increasing magnetic softness and decreasing | saturation density, with increasing temperature the density, | B, at low field intensities, H, increases, at high field intensities

decreases. Such B-temperature curves at constant H, however, have little significance, as they combine the effect of two changes, the increase of softness, which predominates at low H, and the decrease of saturation, which predominates at high H.

Heat treatment, such as annealing, cooling, etc., very greatly changes the magnetic constants, especially a and 7—more or less in correspondence with the change of the physical constants brought about by the heat treatment. ,

MAGNETISM 79

Very extended exposure to moderate temperature—100 to 200°C.— increases hardness and hysteresis loss with some mate- , rials, by what is called ageing, while other materials are almost free of ageing.

48. The most important, and therefore most completely in- vestigated magnetic metal is iron. -

Its saturation value is probably between S = 21.0 X 10? and S = 21.5 X 103, the saturation coefficient thusa¢ = 0.047...

As all industrially used iron contains some impurities, carbon, silicon, manganese, phosphorus, sulphur, etc., usually saturation values between 20 X 10’ and 21 X 10° are found on sheet steel or cast steel, etc., lower values, 19 to 19.5 X 10°, in silicon steels containing several per cent. of Si, and still much lower values, 12 to 15 X 10%, in very impure materials, such as cast iron.

Two types of iron alloys seem to exist:

1. Those in which the alloying material does not directly affect : the magnetic qualities, but only indirectly, by reducing the vol- ume of the iron and thereby the saturation value, and by chang- , ing the physical characteristics and thereby the hardness and hysteresis loss.

Such apparently are the alloys with carbon, silicon, titanium, chromium, molybdenum and tungsten, etc., as cast iron, silicon steel, magnet steel, etc.

2. Those in which the alloying material changes the magnetic, ; characteristics. ,

Such apparently are the alloys with nickel, manganese, mercury, copper, cobalt, ete.

In this class also belong the chemical compounds of the mag- netic materials.

Thus, a manganese content of 10 to 15 per cent. makes the iron practically non-magnetic, lowers the permeability to 4 = 1.4. However, even here it is not certain whether this is not an- extreme case of magnetic hardness, and at extremely high magnetic fields the normal saturation value of the iron would be approached.

Some nickel steels (25 per cent. Ni) may be either magnetic, or non-magnetic. However, pure iron, when heated to high incan- descence, becomes non-magnetic at a certain definite temperature, and when cooling down, becomes magnetizable again at another definite, though lower temperature, and between these two tem-

80 ELECTRIC CIRCUITS peratures, iron may be magnetic or unmagnetic, depending whether it has reached this temperature from lower, or from higher temperatures. Apparently, for these nickel steels, the critical temperature range, within which they can be magnetic or un- | magnetic, is within the range of atmospheric temperature, and . _ thus, after heating, they become non-magnetic, after cooling to sufficiently low temperature, they become magnetizable again. | Thus, a steel containing 17 per cent. nickel, 4.5 per cent. chro- | mium, 3 per cent. manganese, bas permeability 1.004, that is, is almost completely unmagnetic.

Heterogeneous mixtures, such as powdered iron incorporated in resin, or iron filings in air, seem to give saturation densities not far different from those corresponding to their volume per- centage of iron, but give an enormous increase of hardness, a, and hysteresis, 7, as is to be expected.

Most chemical compeunds of iron are non-magnetic. Fer- romagnetic is only magnetite, which is the intermediate oxide and may be considered as ferrous ferrite. There also is an alleged magnetic sulphide of iron, though I have never seen it, magnetkies, Fe7Ss or FesSy.

As magnetite, Fe;0,, contains 72 per cent. of Fe, by weight, and has the specific weight 5.1, its volume per cent. of iron would be 48 per cent., and the saturation density S = 10 X 10°.

Observations on the magnetic constants of magnetite give a saturation density of 4.7 <X 10? to 5.91 X 10*, so that magnet- ite would fall in the second class of iron compounds, those in which the saturation density is affected, and lowered, by the

composition.

Not only magnetite, which may be considered as ferrous ferrite, but numerous other ferrites, that is, salts of the acid Fe,0,H:, are to some extent ferromagnetic, such as copper and cobalt fer-

. rite, calcium ferrite, ete.

49. Cobalt, next adjoining to iron in the periodic system of ele-

ments, is the magnetic metal which has been least investigated.

. Its saturation value probably is between S = 12 X 10? and S = 14 x 108, and its magnetic characteristic looks very similar to that of cast iron. Partly this is due to the similar saturation value, partly probably due to the feature that most of the available data were taken on cast cobalt.

It is interesting to note that Cobalt retains its magnetizability

: MAGNETISM . 81 up to much higher temperatures than iron or any other material, so that above 800 degrees C., Cobalt is the only magnetic material.

More information is available on nickel, the metal next ad- joining to cobalt in the periodic system of elements. Its satura-

. tion density is the lowest of the magnetic metals, probably be- tween S = 6 X 10° and S = 7 X 10%.

Some data on nickel and nickel alloys are given in the following table. In general, nickel seems to show characteristics very simi- lar to those of iron, except that all the magnetic densities are re- duced in proportion to the lower saturation density; but the effect of the physical characteristics on the magnetic constants appears ; to be the same. Interesting is, that nickel seems to be least sen- sitive to impurities in their effect on the reluctivity curve.

Nickel ceases to be magnetizable already below red heat. _ .