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 + B and — B, or between Bi and Bz, 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, Bi and B2, 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, i, 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 anna* tare — 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. The reaction machine, as synchron- 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 conmiercial 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 occm*s 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 pm*poses, 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 Hi, Br. point Pi of Fig. 29, to the condition H2j B2: point P2, 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, it is, however, f> = sB (2) where s = section of magnetic circuit. Hence e = ns-^ (3) If i = current in the electric circuit, the m.m.f . is F = ni (4) and the magnetizing force / - f (5 where I = length of the magnetic circuit. And the field intensity H = 47r/ hence, substituting (5) into (6) and transposing, i = -^ ( 47rn is the magnetizing current in the electric circuit, which produc the flux density, B. The power consumed by the voltage induced in the electiMric circuit thus is slH dB ^ *•=•,) or, per cm.^ of the magnetic circuit, that is, f or s = 1 and Z =* ^h H dB V = T- ^ir dt ^ and the energy consumed by the change from H\, Bi to Ht, ^^^^ which is transferred from the electric into the magnetic circiJ-i*> or inversely, 1 C^ ti?i,2 = — I HdB ergs (1«) 47r MAGNETISM 59 where j4 1,1 ia the area shown shaded in Fig. 29. The energy consumed during a cycle, from Ho, Bo to — Hn, — Bo and back to Ho, Ba, thus it -fX HdB ergs (11) (12) r HdB = j4 is the area of the hysteresis loop, shown shaded m Fig. 30. As the magnetic condition at the end of the cycle is the same as B, rj"' i i Hi H, at the beginning, all this energy, v>, 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; — H, — B; + H, — B. This would mean, that the magnetic fiux 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 Wq = IT or, if /* = permeability, thus H = — , it is Wo = B^ TfJL (12) (13) Fig. 31. the maximum possible hysteresis loss. The inefficiency of the magnetic cycle, or percentage loss energy in the magnetic cycle, thus is )f Fig. 32. w ItllW Wo B^ 45 ■I!'" Am 4 3' (1 39. Experiment shows that for medium flux density, that m^^f thoses values of B which are of the most importance industrialL^^; { MAGNETISM 6l from B = 1000 to S = 12,000, the hysteresie lose can with suffi- cient accuracy for moat practical purposes be approximated by the empirical equation, w = i)B'» (15) j r j OCHM SIL COI s EEl 1 H ST Btd S / ,' / / ,' / l/ / / / 'l' SOW 1 / // EOW ' / / 3MK> / / 10»» / y rnn« y B ^ ^ 1 ! , 1 uo' where »j, the "coefficient of hysteresis," is of the magnitude of 1 X 10~' to 2 X 10~' for annealed soft sheet steel, if fi is given in lines of force per cm.*, and io is ergs per cm.* and cycle. Very often to 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 ahscissce, the hysteresis loss, v>, 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-^ B^' (16) As seen, the agreement the curve of l.e** 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 abscissae, the magnetization curve of ordinary annealed sheet steel, in FERRITE AND MAGNETITE III 0-'. MAGNETIZATION - — " li. -^ J / 111 _ — — 1 ■" i --' I -r ^ ^ ^ / l' / / ^ ^ p- 1 k -^ " ^ r— -' f 1 / y , / / y // y ^ .^ 3 s Fig. 34. half-scale, as curve 1, and the magnetization curve of magnetite , FeaO^ — which is about the same as the black scale of iron— ic*. double-scale, as curve II. As III then is plotted, in full-seal^, 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 cen^-fc. of iron and 20 per cent, {by volume) of scale. Curves I' and ILT' show the initial part of I and III, with ten times the scale of abscissae 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 abscissae, the part of the mag- MAGNETISM 63 netic flux density wMch is carried by the magnetite, aa curve I, As seen, the magnetite carries practically no flux up to £ = 10, but beyond B = 12, the flux carried by the magnetite rapidly increases. As curve U 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. . 'l III / 1 ii/ /,' u' rnon /, '' FER RIT A ID MAC NE1 ITE / 1VS TER ESIS / / , / / / •" / / / / / / , ^ '/ li^l _ ^-' L-- ! 3 1 Fig. 35. ^ under the assumption that cither material rigidly follows the 1-8 power law up to the highest densities, by the equation, Iron: Wi = 1.2 B,i-« X 10-^ Scale: Wj = 23.5 S„i-» X 10-^ 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 rj, at high densities, over the value 1.38 X 10~', 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), f = ^ (14) Substituting herein the parabolic equation of the hysteresis loss, w == rjB^ (17) where n = 1.6, it is r = MTTT? B^-^ (18) = wfirj B'^ m With decreasingdensity5,B'»~2 steadily increases, if n < 2, and as the permeability ^u approaches a constant value, f , 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 ^2. In the case of Fig. 33, for t? = 0.824 X 10"^, assuming the per- meability for extremely low density as II = 1500, f becomes unity, by equation (18), at B = 30. If n > 2, B** - 2 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. Prom equation (17) follows . SI licAn TE L 1 irs TER .31! -t Jk A 1 <.«, ^ V V ,..,.1 J^ M in /■ . i. 1. U ^ ■* J. *" ^^ i A V ' ^ -s= '/ y 1 / -" / in •/• / 2, LOG B a , log V} = log II + n log B (19) That is: "If 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 Btra^t line is the exponent, n." 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 S = 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 Si« X 10-» For B < 400: w = 0.00257 B^-ii X 10-3 However, in this lower range, n = 2 gives a curve: w = 0.0457 B^ X 10-3 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 magnetia density. 42. As over most of the magnetic range the hysteresis loss carx be expressed by the parabolic law (17), it appears desirable tci adapt this empirical law also to the range where the logarithm^.^ curve. Figs. 36 and 37, is curved, and the parabolic law does ncz^t apply, above B = 10,000, and between B = 500 and B = lOOO, or thereabouts. This can be done either by assuming the coei cient rj as variable, or by assuming the exponent n as variabli (a) Assuming rj as constant, rj = 0.824 X 10-3 fQj. ^i^Q medium range, where n = 1.6 771 = 0.0457 X 10-3 for the low range, where n\ ^ 2 The coefficients n and n\ calculated from the observed valu^ MAGNETISM 67 of to, 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 ij = 0.0824 X 10-' Ml = 2 for the low range, where in = 0.0457 X 10"* III III ORDINARY SHEET S' EEL, ANNEALED ESIB < h^ / IV' 11- ,., A 1 h^ ^ / >- »-1 •^ / X :^ Ul V ^ ^ >- ^ ^ y i ^ / y /■ /" / H ""^i u The variation of ij and jh, from the values in the constant range, ^•len, are best shown in per cent., that is, the loss to calculated from ^tie paraboUc equation and a correction factor applied for values °^ B outside of the range. 68 ELECTRIC CIRCUITS Fig. 37 shows the values of rj and rji, as calculated from the para- bolic equations with n = 1.6 and ui = 2, and Fig. 36 shows the percentual variation of rj and rji. The latter method, (6), 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 = rjB'' (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 = log 7} + n log B, (19) diiferentiating (19), gives, in the general case, where the parabolic law does not strictly apply, d log w = d log rj + nd log B + log Bdn, hence, the slope of the logarithmic curve is dlogw^ A dn d\ogri \ , . If n = constant, and >j = constant, the second term on the right-hand side disappears, and it is that is, the slope of the logarithmic curve is the exponent. If, however, rj and n 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 , d log w ^ Or, ^=-log5 (22) MAGNETISM 69 In this case, n and much more still 77 show a very great varia- tion, and the variation of 77 is so enormous as to make this repre- sentation valueless. As illustration is shown, in Fig. 36, the slope of the curve as n2. As seen, n2 varies very much more than n or ni. To show the three different representations, in the following table the values of n and rj are shown, for a different sample of iron. Table B 10« (a) 17 =3 const. = 1.254 (6) n ■» const. = 1.6 (c) ns - d log w d log B I2 below 10.00 n = 1.6 1; = 1.254X100-3 n2 = 1 . 6 1,2 = 1. 254 XlO-« 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 As seen, to represent an increase of hysteresis loss by 1.738 1.254 1.39, or 39 per cent., under (c), n2 is nearly doubled, and 772 re- duced to 1 QAA AAA of its initial value. 44. The equation of the hysteresis loss at medium densities, W = 97B"; 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 TF, 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 77. 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 97: a change of n by 1 per cent., at B = 10,000, changes 77 by about 16 per cent. 70 ELECTRIC CIRCUITS Thus in Fig, 37 is represented as 1 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 rti = 1.923. Assuming, however, ti = 1.6 and Wi = 2.0, gives the average values i, = 1.21 X 10"' and vi = 010 X 10"', and the 1 1 1 1 1 J ORDINARY SHEET STEEL, ANNEALED. HYSTERESIS ; !• eooQ i ^ j 1 1 / a'' 1 / 1 ' EOOO I / / / / / '/ 3«» ^' ) / BOW y / y ^ / ^ ^ u 1 ^xlO individual calculated values of jj and iji 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 »|iB', in two different scales, with the obeerved 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 Wi = 2 still represent the observed values fairly well in 1 M M 1 - ORDINARY SHEET STEEL, ANNEALED. HYSTERESIS MEDIUM DENSITIES - 1 1 1 1 1 1 / m / > / ^ / / r ^ 3 .... FiQ. : the range of B from 500 to 10,000, and below 500, respectively, so that the \.^ 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 ia those materials, where the increase of hysteresis loss occurs there. While the measurement of the hysteresis loss appears a very simple matter, and can be carried out fairly accurately over a / ORDINARY SHEET STEEL. ANNEALED. HYSTERESIS LOW DENSITIES / / / r, / / / / / «i / / / / j / 1 / / 1 , { T / 1 / ? 1 y B I u ^ J . Fia. 40. narrow range of densities, it is one of the most difficult matters "•;• measure the hysteresis loss over a wide range of densities wit-I such accuracy as to definitely determine the exact value of tTse exponent n, due to varying constant errors, which are beyond cod- 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. 46. 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, Bi and B2, 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-cmrent 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 Si and B2, that is, with the amplitude of magnetic variation B = 2 — ) follows the same approximate law of the 1.6*** power, Wo = TjoB^-^ as long as the average value of the magnetic flux variation, ^ B1 + B2 Bo = 2 ' is constant. With changing Bo, however, the coefiicient 170 changes, and in- creases with increasing average flux density, Bo. John D. Ball has shown, that the hysteresis coeflScient of the unsymmetrical cycle increases with increasing average density, -Bo, and approximately proportional to a power of Bo. That is, MAGNETISM 75 Thus, in an unsymmetrical cycle between limits Bi and Bt of magnetic flux density, it is 'Bi + B2\''1 iBi -Bs\i« w -|,+^(^ •)}{ ') (23) where tj is the coefficient of hysteresis of the alternating-current cycle, and for B2 = —Bi, equation (23) changes to that of the symmetrical cycle. Or, if we substitute. Bo = ^L+A* (24) = average value of flux density, that is, average of maximum and mini- mum. B = Bi — B2 (25) it is or, where or, more general. = amplitude of unsymmetrical cycle, w = {ri+ pBo'')B^'^ (26) w = rioB^'^ (27) 710 = ri + PBo'-' (28) w = 770B" (29) 770 = 17 + PBo^ (30) For a good sample of ordinary annealed sheet steel, it was found, 17 = 1.06 X 10-3 /3 = 0.344 X 10-i« For a sample of annealed medium silicon steel, 17 = 1.05 X 10-' (31) (32) p = 0.32 X 10 -10 Fig. 41 shows, with Bo as abscissas, the values of 77^, by equa- tions (30) and (32). As seen, in a moderately unsymmetrical cycle, such as between Bi = -|- 12,000 and -Bg = —4000, the increase of the hysteresis ELECTRIC CIRCUITS loss over that in a symmetrical cycle of the same amplitude, moderate, but the increase of hysteresis loss becomes very last "i L UNSyMMETRICAL CYCLE ,„-1.OSx10'V32B''%1(J^ / / /' / / / / / A / ' / y ^ ^ — ^ , 1 1 , 1 .,»' in highly unsymmetrical cycles, such as between B\ — 16,0C and Si = 12,000.