CHAPTER VII. DISTRIBUTION OF ALTERNATING-CURRENT DENSITY IN CONDUCTOR. 59. If the frequency of an alternating or oscillating current is high, or the section of the conductor which carries the current is very large, or its electric conductivity or its magnetic per- meability high, the current density is not uniform throughout the conductor section, but decreases towards the interior of the conductor, due to the higher e.m.f. of self-inductance in the interior of the conductor, caused by the magnetic flux inside of the conductor. The phase of the current inside of the conductor also differs from that on the surface and lags behind it. In consequence of this unequal current distribution in a large conductor traversed by ^alternating currents, the effective resist- ance of the conductor may be far higher than the ohmic resist- ance, and the conductor also contains internal inductance. In the extreme case, where the current density in the interior of the conductor is very much lower than on the surface, or even negligible, due to this "screening effect/' as it has been called, the current can be 'assumed to exist only in a thin surface layer of the conductor, of thickness lp ; that is, in this case the effective resistance of the conductor for alternating currents equals the ohmic resistance of a conductor section equal to the periphery of the conductor times the " thickness of penetration." Where this unequal current distribution throughout the con- ductor section is considerable, the conductor section is not fully utilized, but the material in the interior of the conductor is more or less wasted. It is of importance, therefore, in alternating- current circuits, especially in dealing with very large currents, or with high frequency, or materials of very high permeability, as iron, to investigate this phenomenon. An approximate determination of this effect for the purpose of deciding whether the unequal current distribution is so small as to be negligible in its effect on the resistance of the conductor, 369 370 TRANSIENT PHENOMENA or whether it is sufficiently large to require calculation and methods of avoiding it, is given in " Alternating-Current Phe- nomena," Chapter XIV, paragraph 133. An appreciable increase of the effective resistance over the ohmic resistance may be expected in the following cases : (1) In the low-tension distribution of heavy alternating cur- rents by large conductors. (2) When using iron as conductor, as for instance iron wires in high potential transmissions for branch lines of smaller power, or steel cables for long spans in transmission lines. (3) In the rail return of single-phase railways. (4) When carrying very high frequencies, such as lightning discharges, high frequency oscillations. In the last two cases, which probably are of the greatest impor- tance, the unequal current distribution usually is such that practically no current exists at the conductor center, and the effective resistance of the track rail even for 25-cycle alternating current thus is several times greater than the ohmic resistance, and conductors of low ohmic resistance may offer a very high effective resistance to a lightning stroke. By subdividing the conductor into a number of smaller conductors, separated by some distance from each other, or by the use of a hollow -conductor, or a flat conductor, as a bar or ribbon, the effect is reduced, and for high-frequency discharges, as lightning arrester connections, flat copper ribbon offers a very much smaller effective resistance than a round wire. Strand- ing the conductor, however, has no direct effect on this phenom- enon, since it is due to the magnetic action of the current, and the magnetic field in the stranded conductor is the same as in a solid conductor, other things being equal. That is, while eddy currents in the conductor, due to external magnetic fields, are eliminated by stranding the conductor, this is not the case with the increase of the effective resistance by unequal current dis- tribution. Stranding the conductor, however, may reduce unequal current distribution indirectly, especially with iron as conductor material, by reducing the effective or mean per- meability of the conductor, due to the break in the magnetic circuit between the iron strands, and also by the reduction of the mean conductivity of the conductor section. For instance, if in a stranded conductor 60 per cent of the conductor section DISTRIBUTION OF ALTERNATING CURRENT 871 is copper, 40 per cent space between the strands, the mean conductivity is GO per cent of that of copper. If by the sub- division of an iron conductor into strands the reluctance of the magnetic circuit is increased tenfold, this represents a reduction of the mean permeability to one-tenth. Hence, if for the con- ductor material proper n = 1000, A = 105, and the conductor section is reduced by stranding to 60 per cent, the permeability to one-tenth, the mean values would be fjL0 = 100 and ^0 = 0.6 X 105, and the factor V7/T, in the equation of current distribution, is reduced from VT£ == 10,000 to VI^ = 2450, or to 24.5 per cent of its previous value. In this case, however, with iron as conductor material, an investigation must be made on the cur- rent distribution in each individual conductor strand. Since the simplest way of reducing the effect of unequal current distribution is the use of flat conductors, the most important case is the investigation of the alternating-current distribution throughout the section of the flat conductor. This also gives the solution for conductors of any shape when the conductor section is so large that the current penetrates only the surface layer, as is the case with a steel rail of a single-phase railway. Where the alternating current penetrates a short distance only into the conductor, compared with the depth of penetration the curvature of the conductor surface can be neglected, that is, the conductor surface considered as a flat surface penetrated to the same depth all over. Actually on sharp convex surfaces the current penetrates somewhat deeper, somewhat less on sharp concave surfaces, so that the error is more or less compensated. 60. In a section of a flat conductor, as shown diagram matically in Fig. 92, page 356, let A = the electric conductivity of conductor material; u = the magnetic permeability of conductor material; I = the distance counted from the center line of the conductor, and 2 10 = the thickness of conductor. Furthermore, let E0 = the impressed e.m.f. per unit length of conductor, that is, the voltage consumed per unit length in the conductor after subtracting the e.m.f. consumed by the self- inductance of the external magnetic field of the conductor; thus, if El = the total supply voltage per unit length of conductor 372 TRANSIENT PHENOMENA and E2 = the external reactance voltage, or voltage consumed by the magnetic field outside of the conductor, between the con- ductors, we have Let 7=^4- ji2 = current density in conductor element dl, & = ^ + y&2 = magnetic density in conductor element dl, E = e.m.f. consumed in the conductor element dl by the self- inductance due to the magnetic field inside of the conductor; then the current Idl in the conductor element represents the m.m.f. or field intensity, which causes an increase of the magnetic density (B between the two sides of the conductor element dl by (2) The e.m.f. consumed by self-inductance is proportional to the magnetic flux and to the frequency, and is 90 time-degrees ahead of the magnetic flux. The increase of magnetic flux (B dl, in the conductor element dl, therefore, causes an increase in the e.m.f. consumed by self- inductance between the two sides of the conductor element by dE = + 2 j7r/te 10-8 dl, (3) where / = the frequency of the impressed e.m.f. Since the impressed e.m.f. E0 equals the sum of the e.m.f. con- sumed by self-inductance E and the e.m.f. consumed by the resistance of the conductor element -, we have A ?.-?+!• w / Differentiating (4) gives dE=-\dI, (5) A DISTRIBUTION OF ALTERNATING CURRENT 373 and substituting (5) in (3) gives dl - - 2 ]Vr/U& 10~8 dl (6) The two differential equations (6) and (2) are in ®, 7 , and Z, which by eliminating &, give the differential equation between 7 and I: differentiating (6) and substituting (2) therein gives . (7) or writing c2 = a2/ = 0.4 7T2 10-8 Itf, (8) where a2 = 0.4 Tr2 10-8 1?, (9) gives ^=-•2 ,<•>/. (10) This differential equation (10) is integrated by / - As-1, (11) and substituting (11) in (10) gives *- -2/c2, v =±c(l r:'i); (12) hence, I = Af+c(l-W + ^2£-c<1-^'. (13) Since / gives the same value for +1 and for — Z, 4i = 42 = 4; (14) hence, 7 "-^T^-^-f £~c(l-j}l\. (15) Substituting e±^' = coscZ ± jsincZ (16) gives /= 45(e+^ + e-rf) coscZ - j (e+cl - e~cl) sincZj, (17) and for Z = Z0, or at the conductor surface, Iml = A{(e+cl» + e-c'») cos cZ0 - j (e+cl° - £~cl<>) sincZJ. (18) 374 TRANSIENT PHENOMENA At the conductor surface, however, no e.m.f. of self-inductance due to the internal field exists, and /o - ^o- (19) Substituting (19) in (18) gives the integration constant A, and this substituted in (17) gives the distribution of current density throughout the conductor section as +Cl -oi-+ci-cl • ° (£+c/0 + s cl°) cos cl0— j (s+cl° - £~cl°) sin cl( The absolute value is given as the square root of the sum of squares of real and imaginary terms, The current density in the conductor center, I = 0, is 2 XE j = m _ ^ A^Q (g+cio + e-e^ cog ^ _ j (£+cl0 _ £-cl0j gin c or the absolute value is 2^« 7 _ ' 61. It is seen that the distribution of alternating-current density throughout a solid flat conductor gives the same equa- tion as the distribution of alternating magnetic density through an iron rail, equations of the same character as the equation of the long distance transmission line, but more special in form. The mean value of current density throughout the conductor section, 1 f*ln (24) which is derived in the same manner as in Chapter V, § 51, is AE0 { (£+cl(> -£~c/0) cos clQ -- j (£+cl° + £~cl°) sin d0\ j ___ rsr . . M»._ g-os) sinc/0j (25) DISTRIBUTION OF ALTERNATING CURRENT 375 and the absolute value is ~i - - 2 cL , o o«7 * (26) Therefore, the increase of the effective resistance R of the conductor over the ohmic resistance RQ is cos cl0 - j (g+c*Q- g-c*Q) sin cZ0 0 (1-j) cZ0 (e+c'°- £-^°) cosc/0 - j (£+^+£- (28) or the absolute value is 2 ^ + £-2^0 _ 2 cos 2 cZ0 62. If clQ is so large that £~cl° can be neglected compared with e+cl°, then in the center of the conductor / is negligible, and for values of I near to Z0, or near the surface of the conductor, from equation (20) we have e +cl (cos d — j sin d) ' °£+cl° (cosd0 - j'sincZ0) = IEQ £c(*~/o) {cos c (Z - Z0) - j sin c (Z - Z0)}. Substituting ^ = Z0 - I, (30) where s is the depth below the conductor surface, we have / = AEQe~cs (cos cs + j sin cs), (31) and the absolute value is 7-^.e— ; (32) the mean value of current density is 376 TRANSIENT PHENOMENA and the absolute value is /\±J Q ^ " dQV2* hence, the resistance ratio, since the current density at the sur- face, or density in the absence of a screening effect, is 1 0 = AEQ : — ifl '}d R0 I m = cZ0 - jdw (35) and the absolute value is §- = <*. ^2; . (so) that is, the effective resistance R of the conductor, as given by equation (28), and, for very thick conductors, from equation (35), appears in the form R = RQ K - jwa), (37) which for very thick conductors gives for ml and m2 the values 63. As the result of the unequal current distribution in the conductor, the effective resistance is increased from the ohmic resistance R to the value R = R0mv R = cl0R0, and in addition thereto an effective reactance X = RQm2, or X = d0R0, is produced in the conductor. In the extreme case, where the current does not penetrate much below the surface of the conductor, the effective resistance and the effective reactance of the conductor are equal and are where Rn is the ohmic resistance of the conductor. DISTRIBUTION OF ALTERNATING CURRENT 377 It follows herefrom that only -r of the conductor section is clQ effective; that is, the depth of the effective layer is i -!..!• *~d-~c' or, in other words, the effective resistance of a large conductor carrying an alternating current is the resistance of a surface layer of the depth (39) and in addition thereto an effective reactance equal to the effective resistance results from the internal magnetic field of the conductor. Substituting (8) in (39) gives n VGA or 5030 ~' (40) It follows from the above equations that in such a conductor carrying an alternating current the thickness of the conducting layer, or the depth of penetration of the current into the con- ductor, is directly proportional, and the effective resistance and effective internal inductance inversely proportional, to the square root of the electric conductivity, of the magnetic permeability, and of the frequency. From equation (40) it follows that with a change of conduc- tivity A of the material the apparent conductance, and therewith the apparent resistance of the conductor, varies proportionally to the square root of the true conductivity or resistivity. Curves of distribution of current density throughout the sec- tion of the conductor are identical with the curves of distribution of magnetic flux, as shown by Figs. 93, 94, 95 of Chapter VI. 64. It is interesting to calculate the depth of penetration of alternating current, for different frequencies, in different materials, to indicate what thickness of conductor may be employed. 378 TRANSIENT PHENOMENA Such values may be given for 25 cycles and 60 cycles as the machine frequencies, and for 10,000 cycles and 1,000,000 cycles as the limits of frequency, between which most high frequency oscillations, lightning discharges, etc., are found, and also for 1,000,000,000 cycles as about the highest frequencies which can be produced. The depth of penetration of alternating current in centimeters is given below. Material M A Penetration in cm. at 25 Cycles. 60 Cycles. 10,000 Cycles. 106 Cycles. 10» Cycles. Very soft iron. . . Steel rail 2000 1000 200 1 1 1 1 1 i.ixio5 105 10* 6.2 X 10s 3.7 X105 0.33 X 105 900 80 0.2 10-* 0.068 0.101 0.71 1.28 1.65 5.53 33.5 112.5 2.25 XlO3 100.6 XlO3 0.044 0.065 0.46 0.82 1.07 3.57 21.7 72.7 1.45 XlO3 65 XlO3 3.4 X lO"3 5.0 X 10-3 0.0355 0.064 0.082 0.276 1.67 5.63 112 5030 0.34x10- 0.5 XlO- 3.55X10- 6.4 xlO- 8.2 xlO- 27.6 X10- 0.167 0.563 11.2 503 0.011 X 10- 0.016X 10- 0.113X10- 0.203x10- 0.263 X 10- 0.88 XlO- 5.3 X10~ 17.9 X10- 0.36 16 Cast iron . . . Copper Aluminum German silver . . Graphite Silicon Salt solu., cone. Pure river water It is interesting to note from this table that even at low machine frequencies the depth of penetration in iron is so little as to give a considerable increase of effective resistance, except when using thin iron sheets, while at lightning frequencies the depth of penetration into iron is far less than the thickness of sheets which can be mechanically produced. With copper and aluminum at machine frequencies this screening effect be- comes noticeable only with larger conductors, approaching one inch in thickness, but with lightning frequencies the effect is such as to require the use of copper ribbons as conductor, and the thickness of the ribbon is immaterial; that is, increasing its thickness beyond that required for mechanical strength does not decrease the resistance, but merely wastes material. In general, all metallic conductors, at lightning frequencies give such small penetration as to give more or less increase of effective resistance, and their use for lightning protection therefore is less desirable, since they offer a greater resistance for higher frequencies, while the reverse is desirable. Only pure river water does not show an appreciable increase of resistance even at the highest obtainable frequencies, and electrolytic conductors, as salt solution, give no screening effect within the range of lightning frequencies, while cast silicon can DISTRIBUTION OF ALTERNATING CURRENT 379 even at one million cycles be used in a thickness up to one-half inch without increase of effective resistance. The maximum diameter of conductor which can be used with alternating currents without giving a serious increase of the effective resistance by unequal current distribution is given below. At 25 cycles: Steel wire 0.30 cm. or 0.12 inch Copper 2.6 cm. or 1 inch Aluminum 3.3 cm. or 1.3 inches At 60 cycles : Steel wire 0.20 cm. or 0.08 inch Copper 1.6 cm. or 0.63 inch Aluminum 2.1 cm. or 0.83 inch At lightning frequencies, up to one million cycles : Copper 0.013 cm. or 0.005 inch Aluminum 0.016 cm. or 0.0065 inch German silver 0.055 cm. or 0.022 inch Cast silicon 1.1 cm. or 0.44 inch Salt solution 22 cm. or 8.7 inches River water . . All sizes. APPENDIX Transient Unequal Current Distribution. 65. The distribution of a continuous current in a large con- ductor is uniform, as the magnetic field of the current inside of the conductor has no effect on the current distribution, being constant. In the moment of starting, stopping, or in any way changing a direct current in a solid conductor, the correspond- ing change of its internal magnetic field produces an unequal current distribution, which, however, is transient. As in this case the distribution of current is transient in time as well as in space, the problem properly belongs in Section IV, 380 TRANSIENT PHENOMENA but may be discussed here, due to its close relation to the permanent alternating-current distribution in a solid conductor. Choosing the same denotation as in the preceding paragraphs, but denoting current and e.m.f. by small letters as instantaneous values, equations (1), (2), and (4) of paragraph 61 remain the same: dW = OAxidl, (1) , (2) e0=e+-, (4) where e0 = voltage impressed upon the conductor (exclusive of its external magnetic field) per unit length, e = voltage con- sumed by the change of internal magnetic field, i = current density in conductor element dl at distance I from center line of flat conductor, p = the magnetic permeability of the con- ductor, and \ = the electric conductivity of the conductor. Equation (3), however, dE = 2 JTcfB 1(T8 dl, changes to de = -^W-»dl (3) at when introducing the instantaneous values; that is, the integral or effective value of the e.m.f. E consumed by the magnetic flux density ® is proportional and lags 90 time-degrees behind (B, while the instantaneous value i is proportional to the rate of change of (B, that is, to its differential quotient. Differentiating (3) with respect to dl gives and substituting herein equation (2) gives § =-0.4 „„* 10-. (6) Differentiating (4) twice with respect to dl gives ffe 1 tfi 0 -*• (7) DISTRIBUTION OF ALTERNATING CURRENT 381 and substituting (7) into (6) gives ' T° (8) as the differential equation of the current density i in the con- ductor. Substituting c2 = 0.4 n^ 1(T8 (9) gves This equation (10) is integrated by i =A + Be-a2t~bl, (11) and substituting (11) in (10) gives the relation hence, b = ± jca, (12) and substituting (12) in (11), and introducing the trigonometric expressions for the exponential functions with complex imagi- nary exponents, i = A + e-a*' (C1 cos cal + C2 sin cal), (13) where Cl=Bl + B2 and C2 - j (B, - 5a). Assuming the current distribution as symmetrical with the axis of the conductor, that is, i the same for + I and for — I, gives C2=0; hence, i = A + C£~att cos cal (13) as the equation of the current distribution in the conductor. It is, however, for t = <*> , or for uniform current distribution, 382 TRANSIENT PHENOMENA hence, substituting in (13), and A = e (14) At the surface of the conductor, or for I = Z0, no induction by the internal magnetic field exists, but the current has from the beginning the final value corresponding to the impressed e.m.f. e0, that is, for I = 10, and substituting this value in (14) gives eQX = eQ\ + Ce~aH cos cat hence, 0 and or calf cos caL = 0 (2 K - 1) 7T a = (2 K - 1) 7T 2cL "~J (15) (16) where K is any integer. There exists thus an infinite series of transient terms, exponen- tial in the time, t, and trigonometric in the distance, I, one of fundamental frequency, and with it all the odd harmonics, and the equation of current density, from (14), thus is ; (2 * - 1) catZ (2 /c - 1) ^ where 2d, (17) (18) The values of the integration constants CK are determined by the terminal conditions, that is, by the distribution of current DISTRIBUTION OF ALTERNATING CURRENT 383 density at the moment of start of the transient phenomenon, or t = 0. For i = 0, , ^ ^ (2 K - 2l» n . (19) Assuming that the current density i0 was uniform throughout the conductor section before the change of the circuit con- ditions which led to the transient phenomena — as would be expected in a direct-current circuit, — from (19) we have 00 (^ If 1 ^ T/ 2}* CK cos — — — — - = — (eQX — i'0) = constant, (20) and the coefficients CK of this Fourier series are derived in the usual manner of such series, thus: C. = 2 avgf- L (e0A — i0) cos 4 . -;o), (21) 7T where avg [!£(*)]**** denotes the average value of the function F(x) between the limits x=x^ and x = x2 and equation (17) then assumes the form (22) This then is the final equation of the distribution of the current density in the conductor. If now Zj = width of the conductor, then the total current in the conductor, of thickness 2 1Q, is idl or (23) 384 TRANSIENT PHENOMENA For the starting of current, that is, if the current is zero, t*0 = 0, in the conductor before the transient phenomenon, this gives (24) While the true ohmic resistance, r0, per unit length of the conductor is r0 = m> (25) the apparent or effective resistance per unit length of the con- ductor during the transient phenomenon is £ V c 1 fi - (2 < - i)»oi* (26) and in the first moment, for t = 0, is since the sum is 1 ' 7T2 (2* - I)2 .8 The effective resistance of the conductor thus decreases from oo at the first moment, with very great rapidity — due to the rapid convergence of the series — to its normal value. 66. As an example may be considered the apparent resist- ance of the rail ret irn of a direct-current railway during the passage of a car over the track. Assume the car moving in the direction away from the station, and the current returning through the rail, then the part of the rail behind the car carries the full current, that ahead of the car carries no current, and at the moment where the car wheel touches the rail the transient phenomenon starts in this part of the rail. The successive rail sections from the wheel contact backwards thus represent all the successive stages of the transient phenomenon from its start at the wheel contact to permanent conditions some distance back from the car. DISTRIBUTION OF ALTERNATING CURRENT 385 Assume the rail section as equivalent to a conductor of 8 cm. width and 8 cm. height, or Zt =8, Z0 = 4, and the car speed as 40 miles per hour, or 1800 cm. per second. Assume a steel rail and let the permeability p. = 1000 and the electric conductivity X = 105. Then c - v/0.4 npl 10~8 = N/1.2566 = 1.121, ax2 - 0.122. Since i0 = 0, the current distribution in the conductor, by (22), is + \ X" S7T1 ^°>122(2't~1)2' cos 0.393 (2* - = e0A {1- 1.27 [e-0-122* cos 0.393Z-Je-uo'cosl.l8i + t cos 1.96 Z - + ...]}, the ohmic resistance per unit length of rail is r0= Q =0.156 X 10"6 ohms per cm. * V/ and the effective resistance per unit length of rail, by (26), is 0.156 X 1Q~6 At a velocity of 1800 cm. per second, the distance from the wheel contact to any point p of the rail, Z', is given as function of the time t elapsed since the starting of the transient phenom- enon at point p by the passage of the car wheel over it, by the expression V = 1800 t, and substituting this in the equation of the effective resistance r gives this resistance as function of the distance from the car, after passage, = _ 0.156 X IP"6 _ = 1 - 0.81 [fi-**10"1' + l£-612xlO-«*' + ^£-1700xlO-r+ ^ ^ ] ohms per cm. As illustration is plotted in Fig. 96 the ratio of the effective resistance of the rail to the true ohmic resistance, —, and with 386 TRANSIENT PHENOMENA the distance from the car wheel, in meters, as abscissas, from the equation r 1 r0 1-0.81 ,- 0.0068 1' - 0.0612 /' - 0.34 I' As seen from the curve, Fig. 96, the effective resistance of the rail appreciably exceeds the true resistance even at a consider- able distance behind the car wheel. Integrating the excess of effective resistance over the ohmic resistance shows that 7.0 6.0 5.0 .4.0 3.0 2.0 100 200 300 Distance from Car, Meters Fig. 96. Transient resistance of a direct-current railway rail return. Car speed 18 meters per second. the excess of the effective or transient resistance over the ohmic resistance is equal to the resistance of a length of rail of about 300 meters, under the assumption made in this instance, and at a car speed of 40 miles per hour. This excess of the transient rail resistance is proportional to the car speed, thus less at lower speeds.