CHAPTER VI. ALTERNATING MAGNETIC FLUX DISTRIBUTION. 48. As carrier of magnetic flux iron is used, as far as possible, since it has the highest permeability or magnetic conductivity. If the magnetic flux is alternating or otherwise changing rapidly, an e.m.f. is generated by the change of magnetic flux in the iron, and to avoid energy losses and demagnetization by the currents produced by these e.m.fs. the iron has to be subdivided in the direction in which the currents would exist, that is, at right angles to the lines of magnetic force. Hence, alternating magnetic fields and magnetic structures desired to respond very quickly to changes of m.m.f. are built of thin wires or thin iron sheets, that is, are laminated. Since the generated e.m.fs. are proportional to the frequency of the alternating magnetism, the laminations must be finer the higher the frequency. To fully utilize the magnetic permeability of the iron, it there- fore has to be laminated so as to give, at the impressed frequency, practically uniform magnetic induction throughout its section, that is, negligible secondary currents. This, however, is no longer the case, even with the thinnest possible laminations, at extremely high frequencies, as oscillating currents, lightning discharges, etc., and under these conditions the magnetic flux distribution in the iron is not uniform, but the magnetic flux density, (B, decreases rapidly, and lags in phase, with increasing depth below the surface of the lamination, so that ultimately hardly any magnetic flux exists in the inside of the laminations, but practically only a surface layer carries magnetic flux. The apparent permeability of the iron thus decreases at very high frequency, and this has led to the opinion that at very high fre- quencies iron cannot follow a magnetic cycle. There is, however, no evidence of such a " viscous hysteresis," but it is probable that iron follows magnetically even at the highest frequencies, traversing practically the same hysteresis cycle irrespective of 355 356 TRANSIENT PHENOMENA the frequency, if the true m.m.f., that is, the resultant of the impressed m.m.f. and the m.m.f. of the secondary currents in the iron, is considered. Since with increasing frequency, at constant impressed m.m.f., the resultant m.m.f. decreases, due to the increase of the demagnetizing secondary currents, this simulates the effect of a viscous hysteresis. Frequently also, for mechanical reasons, iron sheets of greater thickness than would give uniform flux density have to be used in an alternating field. Since rapidly varying magnetic fields usually are alternating, and the subdivision of the iron is usually by lamination, it will be sufficient to consider as illustration of the method the dis- tribution of alternating magnetic flux in iron laminations. 49. Let Fig. 92 represent the section of a lamination. The alternating magnetic flux is assumed to pass in a direction perpendicular to the plane of the paper. Let n = the magnetic permeability, A = the electric conductivity, I = the distance of a layer dl from the center line of the lamination, and 2 10 = the total thickness of the lamination. If then / = the current density in the layer dl, and E = the e.m.f.'per unit length generated in the zone dl by the alternating magnetic flux, we have The magnetic flux density (Bj at the surface I = 10 of the lamination corresponds to the Fig. 92. Alter- jmpresseci or external m.m.f. The' density (B natmg magnetic ,, 77 , ,, fluxdistribution m the zone dl corresponds to the impressed in solid iron. m.m.f. plus the sum of all the m.m.fs. in the zones outside of dl, or from I to Z0. The current in the zone dl is (2) (3) (4) and produces the m.m.f. 3C= 0.4 TrlE/ dZ, which in turn would produce the magnetic flux density ALTERNATING MAGNETIC FLUX DISTRIBUTION 357 that is, the magnetic flux density (B at the two sides of the zone dl differs by the magnetic flux density d& (equation (4)) pro- duced by the m.m.f. in zone dl, and this gives the differential equation between (B, E, and I, = 0.4 nlpE. (5) The e.m.f. generated at distance I from the center of the lamination is due to the magnetic flux in the space from I to 1Q. Thus the e.m.fs. at the two sides of the zone dl differ from each other by the e.m.f. generated by the magnetic flux ($>dl in this zone. Considering now (B, E, and I as complex quantities, the e.m.f. dE, that is, the difference between the e.m.fs. at the two sides of the zone dl, is in quadrature ahead of ($>dl, and thus denoted by dE = - j 2 TT/CB 10-8 dl, (6) where / = the frequency of alternating magnetism. This gives the second differential equation dj[- -j2^/(BlO-8. (7) 50. Differentiating (5) in respect to I, and substituting (7) therein, gives .0-8(B, (8) or, writing c2 = /a2 = 0.4 Tr2/^ 10-8, (9) a2 = 0.4 rfXn 10-8, (10) we have - This differential equation is integrated by . (19) Denoting the flux density at the outside of the lamination, for Z = Z0, that is, the density produced by the external m.m.f., by CBI; substituted in (19), we have — cos cZ0 - y - - sin cZ0 ( , (20) £t Zi ) and substituting (20) in (19), )cLl 2 2 *] (24) so /"«> "^ o c co\r*n<3/'J V (f "^ c ° -4- /? c'o"\ cjiri /»/ U6. ^t t y UVJo Cte+2^ + £-2^o + 2cos2c/0, (26) (B = and 2 cos 2 (28) 52. Where the thickness of lamination, 2 Z0, or the frequency/, is so great as to give cZ0 a value sufficiently high to make e~cl°, or the reflected wave, negligible compared with the main wave e+cl°, the equations can be simplified by dropping s~cl. In this case the flux density, (B, is very small or practically nothing in the interior, and reaches appreciable values only near the surface. It then is preferable to count the distance from the surface of the 360 TRANSIENT PHENOMENA lamination into the interior, that is, substitute the independent variable s = 10 - I. (29) Dropping €~c* and s~cl° in equation (21) gives ecl (cos cl — f sin d) /r> /T> x » / • 1 ecl° (cos clQ — j sin clQ) = (B^-^o-^jcosc (Z0- 1) + ysinc (Z0- Z)}; hence, (B = ®i *~cs (cos cs + y sin cs); (30) or the absolute value is (B =(B1e-c', (31) and at the center of the lamination, (B0 = $1 £~cZ°(cos cZ0 + y sin cZ0), J sm cy , i From equation (24) the mean value of flux density follows when dropping £~cl° as negligible, thus: *- = oArf' (33) or the absolute value is (34) 53. As seen, the preceding equations of the distribution of alternating magnetic flux in a laminated conductor are of the same form as the equations of distribution of current and voltage in a transmission line, but more special in form, that is, the attenuation constant a and the wave length constant /? have the same value, c. As result, the distribution of the alternating magnetic flux in the lamina depends upon one constant only, clQ. The wave length is given by cZ = 2 * ALTERNATING MAGNETIC FLUX DISTRIBUTION 361 i hence and by (9) 10,000 and the attenuation during one wave length, or decrease of intensity of magnetism, per wave length, is £- 2- = 0.0019, and per half-wave length is e- - = 0.043. At the depth -j below the surface, the magnetic flux lags 90 degrees and has considerably decreased; at the depth -^ it lags Zi 180 degrees, that is, is opposite in direction to the flux at the surface of the lamination, but is very small, the intensity being less than 5 per cent of that at the surface, and at the depth lw the flux is again in phase with the surface flux, but its intensity is practically nil, less than 0.2 per cent of the surface intensity; that is, the penetration of alternating flux into the laminated iron is inappreciable at the depth of one wave length. By equations (33) and (34), the total magnetic flux per unit width of lamination is the absolute value is 2 lQ&m = c" that is, the same as would be produced at uniform density in a thickness of lamination 2 or absolute value, 362 TRANSIENT PHENOMENA which means that the resultant alternating magnetism in the lamination lags 45 degrees, or one-eighth wave behind the im- pressed m.m.f., and is equal to a uniform magnetic density penetrating to a depth L-~ (36) lp, therefore, can be called the depth of penetration of the alternating magnetism into the solid iron. Since the only constant entering into the equation is cl0, the distribution of alternating magnetism for all cases can be repre- sented as function of cl0. If clQ is small, and therefore the density in the center of the lamination v which from equation (30) is tan TI = tan cs. (38) The thickness of the equivalent layer is marked in Fig. 94. \ 3 Q / I 1 100 80 GO 40 Ofl \ 1 \ o o Q I \ 1 s k n 2 1 / \ 1 7 \ \ 1 i ' / \ \ / / \ 1 \ / / s ^ N / , t \ ^> ^ 3 0 ^ / \ / S f / \ / \ / X y cl .0 1.6 1.2 0.8 0.4 0.4 0.8 1.2 1.6 2.0 *. 93. Alternating magnetic flux distribution in solid iron. As further illustrations are shown in Fig. 95 the absolute values of magnetic flux density <& throughout a layer of 14 mils thickness, that is, of Z0 = 0.007 inches = 0.018 cm. thickness. For 60 cycles, by Curve I, c = 15.3 cZ0 = 0.275 For 1000 cycles, by Curve II, c = 62.5 c/0 = 1.125 For 10,000 cycles, by Curve III, c = 198 cZ0 = 3.55 It is seen that the density in Curve I is perfectly uniform, while in Curve III practically no flux penetrates to the center. 55. The effective penetration of the alternating magnetism into the iron, or the thickness lp of surface layer which at con- stant induction (Bt would give the same total magnetic flux as exists in the lamination, is L = ,. 1 - > (39) (1 - j)c or the absolute value is v/2' 364 TRANSIENT PHENOMENA 1.0 a 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 \ 180 100 140 120 100 ( 80 GO 40 20 \ \ / \ / \ / \ / V A \ / f \ / r \ t / \ / \ / E< *~Ti uiv. icku cnt ess s / X / \ / \j / ' \ • / x, \ / — , - — / cs ^~ 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 Fig. 94. Alternating magnetic flux distribution in solid iron. cle's 1.0 \ ycles 0.6 0.4 0.2 — I-I-I-l 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 Fig. 95, Alternating magnetic flux distribution in solid iron. ALTERNATING MAGNETIC FLUX DISTRIBUTION 365 hence, substituting for c from equation (9), 104 3570 (40) that is, the penetration of an alternating magnetic flux into a solid conductor is inversely proportional to the square root of the electric conductivity, the magnetic permeability, and the frequency. The values of penetration, lp, in centimeters for various materials and frequencies are given below. Frequency. 25 60 1000 10,000 106 Soft iron, n= 1000, X= 105 Cast iron, /A= 200, X= 10* 0.0714 0 504 0.0460 0 325 0.0113 0 080 0.0036 0 0252 0.00036 0 0025 Copper, A*=l, X=6Xl05 0 922 0.595 0 144 0 0461 0 0046 Resistance alloys, /*= 1,X= 104 7.14 4.60 1.13 0.357 0.036 As seen, even at frequencies as low as 25 cycles alternating magnetism does not penetrate far into solid wrought iron, but penetrates to considerable depth into cast iron. It also is interesting to note that little difference exists in the penetration into copper and into cast iron, the high conductivity of the former compensating for the higher permeability of the latter. 56. The wave length, lw = — , substituting for c, from equa- tion (9), is 31,600 (41) that is, the wave length of the oscillatory transmission of alter- nating magnetism in solid iron is inversely proportional to the square root of the electric conductivity, the magnetic permea- bility, and the frequency. Comparing this equation (41) of the wave length lw with equa- tion (40) of the depth of penetration lp, it follows that the depth of penetration is about one-ninth of the wave length, or 40 degrees, or, more accurately, since 1 IP = 27T and l = — 366 we have TRANSIENT PHENOMENA 1 lw 2 V2n 8.9 or 40.5 degrees. The speed of propagation is 31,600 \/J (42) (43) that is, the speed of propagation is inversely proportional to the square root of the electric conductivity and of the magnetic per- meability, but directly proportional to the square root of the frequency. This gives a curious instance of a speed which increases with the frequency. Numerical values are given below. Frequency. 25 Cycles. 10,000 Cycles. Soft iron, C*— 1000 X— 105 S— 15 8cm 316 cm Cast iron, Copper, M-= 200, A=104 (tx= i A=6 X 105 111 cm. 204 cm 2230 cm. 4080 cm It is seen that these speeds are extremely low compared with the usual speeds of electromagnetic waves. 57. Since instead of o^, corresponding to the impressed m.m.f. and permeability //, the mean flux density in the lamina is Bm, the effect is the same as if the permeability of the material were changed from /* to // - J£®^, (44) and // can be called the effective permeability, which is a function of the thickness of the lamination and of the frequency, that is, a function of cZ0; // appears in complex form thus, that is, the permeability is reduced and also made lagging. For high values of cl0, that is, thin laminations or high fre- quencies, from (33), we have (45) ALTERNATING MAGNETIC FLUX DISTRIBUTION 367 58. As illustration, for iron of 14 mils thickness, or 10 = 0.018 centimeters, and the constants JJL = 1000 and A = 105, that is a = 1.98, the absolute value of the effective permeability is and hence, (46) that is, the effective or apparent permeability at very high frequencies decreases inversely proportional to the square root of the frequency. In the above instance the apparent per- meability is : At low frequency, JJL = 1000; at 10,000 cycles, // = 198; at 1,000,000 cycles, // = 19.8; at 100 million cycles, // = 1.98, and at 392 million cycles, // == 1, or the same as air, and at still higher frequencies the presence of iron reduces the magnetic flux. It is interesting to note that with such a coarse lamination as a 14-mil sheet, even at the highest frequencies of millions of cycles, an appreciable apparent permeability is still left; that is, the magnetic flux is increased by the presence of iron ; and the effect of iron in increasing the magnetic flux disappears only at 400 million cycles, and beyond this frequency iron lowers the magnetic flux. However, even at these frequencies, the presence of iron still exerts a great effect in the rapid damping of the oscillations by the lag of the mean magnetic flux by 45 degrees. Obviously, in large solid pieces of iron, the permeability // falls below that of air even at far lower frequencies. Where the penetration of the magnetic flux lp is small com- pared with the dimensions of the iron, its shape becomes im- material, since only the surface requires consideration, and so 368 TRANSIENT PHENOMENA in this case any solid structure, no matter what shape, can be considered magnetically as its outer shell of thickness lp when dealing with rapidly alternating magnetic fluxes. At very high frequencies, when dealing with alternating magnetic circuits, the outer surface and not the section is, there- fore, the dominating feature. The lag of the apparent permeability represents an energy component of the e.m.f. of self-induction due to the magnetic flux, which increases with increasing frequency, and ultimately becomes equal to the reactive component.