Theory and Calculation of Transient Electric Phenomena and Oscillations — part 14 of 20

ALTERNATING MAGNETIC FLUX DISTRIBUTION 365 hence, substituting for c from equation (9), | L = 10‘ _ 3570 40

EMSinf Via (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, /,, in centimeters for various materials and frequencies are given below. Frequency. 25 60 1000 | 10,000 10° Soft iron, #= 1000, A=10.....] 0.0714 | 0.0460 | 0.0113 | 0.0036 | 0.00036 Cast iron, w= 200, A=10*.....| 0.504 | 0.325 | 0.080 | 0.0252 | 0.0025 Copper, w= 1, A=6X10......] 0.922 | 0.595 | 0.144 | 0.0461 | 0.0046 Resistance alloys, «= 1,A=10* | 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, l,, = “2 , substituting for c, from equa- tion (9), is 31,600 lL, =—=; 41) nf | ( 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 /,, with equa- tion (40) of the depth of penetration J,, it follows that the depth of penetration is about one-ninth of the wave length, or 40 degrees, or, more accurately, since 1 22 L, = evaend l, = ad |

866 TRANSIENT PHENOMENA we have l, 1 1 =~ =—— = 42 L, 2V2n 89’ @) or 40.5 degrees. The speed of propagation is S = fl, A = 31,600 Vf f : (43) , Vie

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.

Soft iron, #= 1000, A= 10°....................] S=15.8¢em, 316 em.

Cast iron, A= 200,4=10................... | 11 em. | 2230 cm

Copper, P= 1, A=6X10°................] 204 em.| 4080 cm.

It is seen that these speeds are extremely low compared with the usual speeds of electromagnetic waves.

57. Since instead of @,, corresponding to the impressed m.m.f. and permeability », the mean flux density in the lamina is B., the effect is the same as if the permeability of the material were changed from y» to

w= we, (44) ~1 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 cl,; ’ appears in complex form thus, w= wl t+ dey; that is, the permeability is reduced and also made lagging. For high values of cl,, that is, thin laminations or high fre- quencies, from (33), we have _ Bt we (1 — j) el, * 4, _#, “de, 7120, (#5)

ALTERNATING MAGNETIC FLUX DISTRIBUTION 367 58. As illustration, for iron of 14 mils thickness, or 1, = 0.018 centimeters, and the constants » = 1000 and A = 10°, that is a = 1.98, the absolute value of the effective permeability is Le and " dl, V2 c=avyf; hence, Le I ie 19,800 VF (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, jy = 1000; at 10,000 cycles, iw = 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 y oscillations by the lag of the mean magnetic flux by 45 degrees. 7 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 J, 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 J, 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.

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 /,; 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,

869

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 371 is copper, 40 per cent space between the strands, the mean conductivity is 60 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 « = 1000, 4.= 10°, and the conductor section is reduced by stranding to 60 per cent, the permeability to one-tenth, the mean values would be

fg = 100 and A, = 0.6 x 10°,

and the factor VAy, in the equation of current distribution, is reduced from Vin = 10,000 to VA,#, = 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 asection of a flat conductor, as shown diagrammatically in Fig. 92, page 356, let 4 = the electric conductivity of conductor material; « = the magnetic permeability of conductor material;

Z = the distance counted from the center line of the conductor, and 21, = the thickness of conductor.

Furthermore, let £, = 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- imductance of the external magnetic field of the conductor; thus, if E, = the total supply voltage per unit length of conductor

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372 TRANSIENT PHENOMENA

and /, = the external reactance voltage, or voltage consumed by the magnetic field outside of the conductor, between the con- ductors, we have

. Hy =f, — EF,

Let

I =1, + jt, = current density in conductor element dl,

® = b, + jb, = magnetic density in conductor element dl,

E = e.m.f. consumed in the conductor element di by the self- inductance due to the magnetic field inside of the conductor; then the current J dl in the conductor element represents the m.m.f. or field intensity,

dx = 0.4 xI dl, (1) which causes an increase of the magnetic density ®@ between the two sides of the conductor element dl by

dos = pda

= 0.4 zl dl. (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 @ 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 jzfe 10-* dl, (3) where f = the frequency of the impressed e.m.f.

. Since the impressed e.m.f. E, equals the sum of the e.m.f. con- sumed by self-inductance # and the e.m.f. consumed by the resistance of the conductor element i, we have

I Ey=E + 2 (4) . Differentiating (4) gives db =—Zal, 6)

DISTRIBUTION OF ALTERNATING CURRENT 3873 . and substituting (5) in (3) gives dI =— 2 jxfa@ 10-* dl. ~ (6)

The two differential equations (6) and (2) are in ®, J, and J, . which by eliminating ®, give the differential equation between ! I and 1: differentiating (6) and substituting (2) therein gives |

= = — 0.8 jx°f 10-8 Apl ; (7) | or writing , | C= af = 047 10-8 dps, (8) where a = 04 7 10-8 Ap, (9) gives , @] . 77 2je1. (10) This differential equation (10) is integrated by I = Ae~, * (11) and substituting (11) in (10) gives ; ? =—2je, , v =tc(1 — 9); (12) hence, l= Ag tea-a! + Ay 9A": (13) Since J gives the same value for +/ and for —I, A, = A, = A; (14) hence, ° TD =A feteG-ae yg eed ary (15) Substituting et! — eos cl + j sin el (16) gives ‘ | I = Af (e+? + e-%) cosel — j (e+ — e-%) sincl}, (17) | and for / = 1,, or at the conductor surface, . | I, = A{(e + ¢-%) cos cl, — j (ete — e~%) sin cl,}. (18) | | |

374 TRANSIENT PHENOMENA At the conductor surface, however, no e.m.f. of self-inductance due to the internal field exists, and I, = ABy. (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 | _ (e+ + 6%) cosel — 7 (e+ — e-“) sine , [= 48, (e+e + @%) cos cl,— j (e+* — e—“) sin cl, (20) . The absolute value is given as the square root of the sum of | squares of real and imaginary terms, : sre + oF + 20s 2 P= MN SIE Th 2 cos 2 chy 21) The current density in the conductor center, J = 0, is 24E, lo- GREE cosd, jem ana,’ ™) or the absolute value is a 93 0 Vette 4g =Fele +2 cos 2 cl, (23) | 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 Im => fT, (24) : 1, . : which is derived in the same manner as in Chapter V, § 51, is y= —AEe {(e* =e7%) cos oly = j (e+ + 67) sin clo} “™ =p, (E*+ e™) cos cl, — j (¢**— e~*) sin cl,} . (25)

DISTRIBUTION OF ALTERNATING CURRENT 87d and the absolute value is dE e tel 4 ——2e _ 2 eos 2 cl In = —=V =" cl, V2 Ve + e2eh + 2 cos 2cl, (26) Therefore, the increase of the effective resistance R of the ° conductor over the ohmic resistance R, is ae | | Re Tn’ @ | R_ 1 (ete + @-%) cos cl, — 7 (e+ — e-%) sin cl, R, (1-9) el, (e— e%) cos cl, — j (c++ e~%) sin cl,’ . (28) or the absolute value is R_ 1 \ [et?eo + ¢-2eb + 2 cos 2cl, (29) . Ry cl, V2 ¥ et# 46-2 — 2 cos 2cl,

62. If cl, is so large that e~°* can be neglected compared with . e+e then in the center of the conductor J is negligible, and for . values of J near to l,, or near the surface of the conductor, from equation (20) we have

e+ (eos cl — 7 sin cl) = a —_—— "nm nn — _ c= P= Ml ra (cos cl, — j sin cl,) = AB, “- {cos c (i — 1) — jsine (I — 1,)}. Substituting s=1,-1, (30) where s is the depth below the conductor surface, we have | I = 4E,¢~® (cos cs + 7 sincs), (31) and the absolute value is [ = AE, e; (82) the mean value of current density is 4E In => 33 , im Ge 8)

376 TRANSIENT PHENOMENA and the absolute value is AE Im = —s} (34 cl, V2’ ) . hence, the resistance ratio, since the current density at the sur- face, or density in the absence of a screening effect, is J, = AE: Rk I, — =—° = (1 — 7) dl R, Im ( ] ) c 0 = cl, — jel,, (35) and the absolute value is R —_— = el V2: € 6 R, 0 ? (3 ) 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= R, (m, _ jm), (37) which for very thick conductors gives for m, and m, the values R = R, (el, — jel,). (38)

63. As the result of the unequal current distribution in the conductor, the effective resistance is increased from the ohmic resistance FR, to the value

k = kym,, R= cl,R,, and in addition thereto an effective reactance X = Rym,, or X =a,R,, 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

cl, | where R, is the ohmic resistance of the conductor.

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| DISTRIBUTION OF ALTERNATING CURRENT 377 1 oo, It follows herefrom that only a. of the conductor section is ; 0 effective; that is, the depth of the effective layer is 1, p= cd, ¢’ 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 ; . 1 p= a (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 10‘ lL=--—. ==, . x V0 An or (40) L = 5030 | Ving

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 4 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 materils, 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. | | Penetration in om. at Material Bb A 2 cyaee. | cyclen. | Sa | cyden. | Cheon Very soft fron... .|2000) 1.1105 | 0.068 0.044 3.4x 10-3) 0.34 xX10-3) 0.011 x 10-3 Steel rail........;1000} 10° 0.101 0.065 5.0 X 10-5} 0.5 x10-%| 0.016 x 10-3 Copper 2220] i] "ea ros} 123 | ose; One| et a 0.203 x 10-2 Aluminum ..... 1} 3.7X10°| 1.65 1.07 , 0.082 8.2 x10-3 0.263 x 10-7 German silver .. 1) 0.33 x10 5.53 3.57 0.276 27.6 X10-5) 0.88 x10-* Graphite... ....| 1/900 | 33.5 21.7 | 1.87 0.167 53 Xx10-3 Salt eola., “sone. 1 Oa NI les 108 mw” we 2 x10 Pure river water| 1) 10-4 {100.6 X10" 6 x 105 i038 503 16 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 whiclt 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......................... 0.80 em. or 0.12 inch Copper........................4.+.-- 2.6. em. or 1 inch Aluminum ........................ 33 em. or 1.3 inches At 60 cycles: . Steel wire......................... 0.20 cm. or 0.08 inch Copper. ..................5....... 16 em. or 0.63 inch Aluminum. ....................... 31 em. or 0.83 inch At lightning frequencies, up to one million cycles: | Copper. ............ eee eee eee eee + 0.013 cm. or 0.005 inch Aluminum. ....................... 0.016 em. or 0.0065 inch German silver..................... 0.055 em. or 0.022 inch | Cast silicon........................ 1.1. em. or 0.44 inch Salt solution...................... 22 cm. or 8.7 inches | River water. ................0005. 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: dX =(047idi, (1) dB =0.4xmdl, (2) a é& =e + 7’ (4) where e, = 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, 7 = current density in conductor element di at distance J from center line | of flat conductor, “ = the magnetic permeability of the con- ductor, and A = the electric conductivity of the conductor. Equation (8), however, dE = 2 jnfB 10~* dl, changes to | an . de = — 7 dl (3) 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 ®, while the instantaneous value 7 is proportional to the rate of change of ®, that is, to its differential quotient. Differentiating (3) with respect to dl gives @e d/d®_, : ~~ aa) ©) and substituting herein equation (2) gives @e di.» qe = 7 OF HG 10 . (6) Differentiating (4) twice with respect to dl gives Pe 1d 0 = et ae? 7) |

| DISTRIBUTION OF ALTERNATING CURRENT 881 and substituting (7) into (6) gives a _,u Be 7 + 0-4 eHA LO (8) as the differential equation of the current density 7 in the con- ductor. Substituting ce = 0.474 1078 (9) gives . ig di aE = C a (10) This equation (10) is integrated by 7=A + Be~ th (11) and substituting (11) in (10) gives the relation = -— ca’; ; hence, ; | b = + jea, (12) and substituting (12) in (11), and introducing the trigonometric : expressions for the exponential functions with complex imagi- ; nary exponents, t=A +e-"(C, cos cal + C, sin cal), (13) | where ' | C, = B, + B, andC, =] (B, — B,). Assuming the current distribution as symmetrical with the axis of the conductor, that is, 7 the same for + / and for — J, gives C, =0; hence, 1 =A + Ce~™ cos cal (13) as the equation of the current distribution in the conductor. It is, however, for ¢ = 0, or for uniform current distribution, t =e,A;

882 TRANSIENT PHENOMENA |

hence, substituting in (13), |

A =e,d |

and | i =e,d + Ce-™ cos cal. (14)

At the surface of the conductor, or for /=J,, 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.

e,, that is, for 7 = 1,,

• 4 =e,A, and substituting this value in (14) gives €,A =e,d + Ce—™ cos cal,; hence, cos cal, = 0 and : (2«-1)x . cal, = Ces ? (15) or _ (2K - l)zx a= 5) cl, ? (16) where « is any integer.

There exists thus an infinite series of transient terms, exponen- tial in the time, ¢, and trigonometric in the distance, /, one of fundamental frequency, and with it all the odd harmonics, and the equation of current density, from (14), thus is

4 =ed t+ DeCe~ Or- 1” cog (2 — 1) ca, 1 2 2 1) al a) =e,a + Sie yen Ox BO ogg (2 — 1) at e—l)z ’ T 21, where a, == (18) 1 2,

The values of the integration constants C’, 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, ort =0.

For t =0, , . < (2«—1)al Vy = yA + mC. cos an (19) Assuming that the current density 7, 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

. C,, cos @e= al =-— (e,d — 1,) = constant, (20) 1 2 l, . and the coefficients C, of this Fourier series are derived in the usual manner of such series, thus: ~ 1) alqr* C, = 2 ave| - (e,A — 2,) cos | 21, imo 4 ; =(-1) = (eyd — 44), (21) where avg [F(x)]?=* denotes the average value of the function F(z) between the limits r=x, and r=2z, and equation (17) then assumes the form a 4 ves (= 1) ge -tyae (2 — Dal t=eA+ ~(€9A %) > deci! eos —— i, (22) This then is the final equation of the distribution of the current density in the conductor. If now /, = width of the conductor, then the total current in the conductor, of thickness 2 J,, is r+ly L=1,f ia nh , 1610, LC Sy eT Ge Dia Deadlds— ae (Cd — to) Be Gap ‘or , 8 t\ < 1 = a __ _ 2 ee DNC 1=21)e fi 3(I a“) oop (23)

384 TRANSIENT PHENOMENA For the starting of current, that is, if the current is zero, t, = 0, in the conductor before the transient phenomenon, this

• gives T=2tded $1 —5 Sen Genel. (24 or1%0 = : (2« — 1) While the true ohmic resistance, r,, per unit length of the conductor is 1 => 25 ro TTR (5) the apparent or effective resistance per unit length of the con- ductor during the transient phenomenon is —~ 2S, ge eat I-35 p> (2 — 1)*° and in the first moment, for ¢ = 0, is r = 0, since the sum is $1 7 (2-1) 8 The effective resistance of the conductor thus decreases from o 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 return 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 em. height, or /,= 8, J,=4, and the car speed as 40 miles per hour, or 1800 cm. per second. Assume a steel rail and let the permeability » = 1000 and the electric conductivity 4 = 10°. Then c = V0.4 741078 = V'1.2566 = 1.121, Rn a,= 2d, = 0.35, a,? = 0.122. Since 7,= 0, the current distribution in the conductor, by (22), is . AQ C= 1)* _oseee—a2e ‘ } ined f+ BS cos 0.393 (2« — 1)1 . = ed {1— 1.27 [e~°* cos 0.393 1-4 e-"* cos 1.181 + $73! | cos 1.961 -+ ... }}, | the ohmic resistance per unit length of rail is 1 -6 r= DTA = 0.156 X 10~* ohms per cm. and the effective resistance per unit length of rail, by (26), is 0.156 x 107° At a velocity of 1800 cm. per second, the distance from the | wheel contact to any point p of the rail, U’, is given as function of the time ¢ elapsed since the starting of the transient phenom- enon at point p by the passage of the car wheel over it, by the expression l’ = 1800 ¢, 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 107° | t= 1 — 0.81 [2 8x10 pf Saxo a eTouxto FF 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 0

386 TRANSIENT PHENOMENA the distance from the car wheel, in meters, as abscissas, from the equation re 1 T, 1 —0.81 [e~ 90088? 4 A 0.08120" a5 eT Os g OMY +o, J 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 the effective resistance over the ohmic resistance shows that TET TT TTT TTT Tt ttt tt tt tt eee VETTE TTT ee tee rit tt eat | | PT TTT TTT r tt te tt tt tt EE soti tt tT Ty tt yy tt Pe TT ET TT ey ey TE TT wtN TT TTT TTT tty te rt et tt BRUREE ERR a MLE tf fd PINT TT TET ETT t ttt tt tt te wet Nitti rey ee ty ee tt tt ob eH rol [rte Obmie| | [> “Lireisenre | TT TTT TE EET tt iL 100 200 300 wo 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. .

CHAPTER VIII. . VELOCITY OF PROPAGATION OF ELECTRIC FIELD.

67. In the theoretical investigation of electric circuits the velocity of propagation of the electric field through space is usually not considered, but the electric field assumed as instan- taneous throughout space; that is, the electromagnetic com- ponent of the field is considered as in phase with the current, the electrostatic component as in phase with the voltage. In reality, however, the electric field starts at the conductor and propa- gates from there through space with a finite though very high velocity, the velocity of light; that is, at any point in space the electric field at any moment corresponds not to the condi- tion of the electric energy flow at that moment but to that at a moment earlier by the time of propagation from the conductor to the point under consideration, or,in other words, the electric field lags the more, the greater the distance from the conductor.

Since the velocity of propagation is very high — about 3 x 10” centimeters per second — the wave of an alternating or oscillating

. current even of very high frequency is of considerable length; at 60 cycles the wave length is 0.5 x 10° centimeters, and even at a million cycles the wave length is 30,000 centimeters, or about 1000 feet, that is, very great compared with the distance to which electric fields usually extend.

The important part of the electric field of a conductor extends to the return conductor, which usually is only a few feet distant; beyond this, the field is the differential field of conductor and return conductor. Hence, the intensity of the electric field has usually already become inappreciable at a distance very small compared with the wave length, so that within the range in which an appreciable field exists this field is practically in phase with the flow of energy in the conductor, that is, the velocity of propagation has no appreciable effect. ,

Thus, the finite velocity of propagation of the electric field requires consideration only:

887

| 388 TRANSIENT PHENOMENA

(a) At extremely high frequencies, hundreds of millions of cycles per second, as given by Hertzian resonators.

(b) In high frequency discharges having no return circuit or no well defined return circuit, as lightning discharges. In this case the effective resistance of radiation may be so large com- pared with the ohmic resistance, even when considering the unequal current distribution in the conductor (Chapter VII), that the effect of the conductor material practically disappears. In the conductors forming the discharge path of lightning arresters this phenomenon therefore requires serious consideration.

(c) With high frequencies, in the case where the field at a considerable distance from the conductor is of importance as in wireless telegraphy.

In wireless telegraphy the electric field of the sending antenne propagating through space impinges upon the receiving antennse | and there is observed by its electromagnetic and electrostatic | effect.

68. The electric field of an infinitely long conductor without return conductor decreases inversely proportionally to the dis- tance, and therefore is represented by .

v | ¢ = T , (1) where Y is the intensity of the electric field at unit distance from the conductor.

The electric field of a finite conductor of length J, decreases inversely proportionally to the distance / and also proportionally to the angle subtended by the conductor J, from the distance J, and since this angle, for great distances, is inversely proportional to the distance 1, the electric field of a finite conductor of length J, without return conductor is represented by

= LY 2 t- OF @)

Since the electric field of the return conductor is opposite to that of the conductor, it follows that the electric field of an infinitely long conductor, with the return conductor at distance L,, by equation (1) is .

. Vv Vv . $= ——7 - —7 (3) (-=~ I+ 2 2 | |

VELOCITY OF PROPAGATION OF ELECTRIC FIELD 389 where l’ = I, cos z is the projection of the distance 1, between the conductors upon the direction 1, that is, l’ is the difference in the distance of the two conductors from the point /.

For large distances 1, equation (3), becomes

y- @) =F:

In the same manner, from equation (2) it follows that the decrease of the electric field of the conductor of finite length 4, with its return conductor at the distance J,, that is, of a recti- linear circuit of the dimensions of J, and l,:

L¥ LW y’ so lV 2 ~ Re? 9 25 hence, 1’ =. 5 $= "5 (5)

69. Since infinitely long conductors, (1) and (4), are of theoretical interest only, practically available are the cases (2) and (5).

The electric field of a closed circuit decreases with the cube of the distance, hence much more rapidly than that of a con- ductor without a return conductor, which decreases only with the square of the distance. Hence, where, as in wireless teleg- raphy, action at great distance is required, only conductors without return conductor can be used. To establish consider- able currents in such open conductors requires high frequen- cies, so that the current is absorbed by the capacity of the conductor or the capacity attached to its end. No conductor parallel to the ground can be treated as conductor without return conductor, since secondary currents in the ground and also in the higher strata of the atmosphere act as return con- ductor with regard to the electric field. The practical reali- zation of a conductor without return thus requires a vertical position of the conductor, and for this reason in wireless teleg- raphy the vertical sending and receiving antenne are necessary, and the transmission is far more successful across the ocean than across the land, since in the latter case every tree, moun-

390 TRANSIENT PHENOMENA tain, etc., acts inductively as return conductor, and thus increases the rapidity of the decrease of the electric field.

In such a case the use of high frequency and of conductors without return conductor, hence with electric fields decreasing relatively slowly with the distance, requires an introduction of the velocity of propagation into the circuit equations.

As illustrations will be discussed:

(A) The inductance of a finite section of an infinitely long con-

ductor without return conductor.

(B) The mutual inductance between two finite. conductors without return conductors, at considerable distance from each other.

(C) The capacity of a sphere in free space.

' (D) The capacity of a sphere against ground, in space.

Cases A and B deal with the electromagnetic, C and D with the electrostatic component of the electric field.

A. Inductance of a length | of an infinitely long conductor without return conductor.

70. The inductance of a length / of a straight conductor is usually given by the equation

L=2l logs x 10~, (6) where /’ = the distance of return conductor, J,= the radius of the conductor, and the total length of the conductor is assumed as infinitely great compared with / and ’. This is approximately the case with the conductors of a long distance transmission line.

For infinite distance /’ of the return conductor, that is, a conductor without return conductor, equation (6) gives L = o; that is, a finite length of an infinitely long conductor without return conductor has an infinite inductance Z and inversely, zero capacity C. .

In equation (6) the magnetic field is assumed as instantaneous, that is, the velocity of propagation of the magnetic field is neglected. With alternating currents traversing the conductor this is permissible when the distance to the return conductor is a negligible fraction of the wave length; that is, if V is negligible compared with where S = the speed of light and