VELOCITY OF PROPAGATION OF ELECTRIC FIELD 391 jf = the frequency of alternating current. It obviously is not permissible in a conductor having no return conductor.

If a conductor conveying an alternating current has no return conductor, its circuit is closed by electrostatic capacity, either the distributed capacity of the conductor or capacity connected to the ends of the conductor. To produce in such a case con- siderable currents, either the conductor must be very long or the frequency and e.m.f. very high.

No conductor extending parallel to the ground, as a telegraph or transmission wire, can be considered as having no return con- ductor, since even if the conductor is isolated from the ground secondary currents produced in the ground (and in the higher regions of the atmosphere) act inductively as return currents.

Hence the case of the conductor without return conductor can physically be realized only by a conductor perpendicular to the ground, as the sending and receiving antennz of a wireless tele- graph station, and even then completely only on the ocean, where there are no other vertical conductors in the space, as trees, mountains, etc., which may act as inductive returns.

Since a vertical conductor is limited in length, very high fre- quencies are required, and therefore the wave is of moderate length, that is, the velocity of propagation of the magnetic (and electrostatic) field must be considered when investigating the self-induction and the mutual induction of such a conductor.

The magnetic field at a distance J from the conductor and at time ¢ corresponds to the current in the conductor at the time t — t’, where ¢’ is the time required for the electric field to travel the distance J, that is, ? = i where S =the speed of light; or, the magnetic field at distance / and time ¢ corresponds to the current in the conductor at the time t — e

71. Representing the time ¢ by angle 6 = 2zft, where f= the frequency of the alternating current in the conductor, and denoting

Qrf 2x a= s = L, ’ @ ) where L,= ; = the wave length of electric field,

392 TRANSIENT PHENOMENA the field at distance / and time angle 6 corresponds to time angle | 6 — al, that is, lags in time behind the current in the conductor by the phase angle al. Let t =Icosé@ = current, absolute units. (8) The magnetic induction at distance / then is @ = 7 og (9 — al); (9) | hence, the total magnetic flux surrounding the conductor, from distance J to infinity is . | =f, 24 cos (0 — al) at | | = 21, feos of) SF a+ sine [2% a}. (10) ; cos al . . . f a dl cannot be integrated in finite form, but represents a new function which in its properties is intermediate between the sine function fecos atat =1 sin al a and the logarithmic function | f La = log l | U i} and thus may be represented by a new symbol, | sine — logarithm = sil. | i | In the same manner, f ea is related to = cos al and ! to log 1. | Introducing therefore for these two new functions the symbols sil al = I ao dl, (11) col al = It ano a, (2) |

VELOCITY OF PROPAGATION OF ELECTRIC FIELD 3938 , gives

© = 21, {cos @sil al + sin ¢ col al}. (18)

The e.m.f. consumed by this magnetic flux, or e.m.f. of induc- tance, then is _ d® db e- a= 2nf °

hence, :

e = 4 xfIl, {cos 6 col al — sin @ sil al} ; (14) and since the current is

t =I cos@,

the e.m.f. consumed by the magnetic field beyond distance 1, or e.m.f. of inductance, contains a component, in phase with the current, or power component,

e, = 42fIl, col al cos 8, (15) and a component in quadrature with the current, or reactive com- ponent,

e, = — 4 xf1k sil al sin 9, (16) which latter leads the current by a quarter period.

The reactive component e, is a true self-induction, that is, rep- resents a surging of energy between the conductor and its electric field, but no power consumption. The effective component e,, however, represents a power consumption \

p=et = 42fPly col al cos0 (17) by the magnetic field of the conductor, due to its finite velocity; that is, it represents the power radiated into space by the conductor. The energy component e, gives rise to an effective resistance, r= + = 4 nfl, col al, (18) and the reactive component gives rise to a reactance, r= []- 4 nflysil al, (19)

394 TRANSIENT PHENOMENA When considering the finite velocity of propagation of the electric field, self-inductance thus is not wattless, but contains an energy component, and so can be represented by an impe- dance, . Z=r-jx = 4 xfl, (col al — j sil al) 10~* ohms. (20) The inductance would be given by _ 14 L= 2 xf = 21, {sil al + j col al}10~* henrys, (21) and the power radiated by the conductor is . p=vr. 72. The functions sil al = J a dl (11) and col al = f ae dl (12) can in general not be expressed in finite form, and so have to be recorded in tables. Close approximations can, however, be derived for the two cases where lis very small and where 1 is very large compared with the wave length l,, of the electric field, and these two cases are of special interest, since the former rep- resents the total magnetic field of the conductor, that is, its self- inductance, and the latter the magnetic field interlinked with a distant receiving conductor, that is, the mutual inductance . between sending and receiving conductor. It is sil0 =o, rea Cc | 0 =} a (22) sil o = 0, colo = 0.

VELOCITY OF PROPAGATION OF ELECTRIC FIELD 395 And it can be shown that for small values of al, that is, such values of J as are only a small fraction of a wave length, the approximations hold: sil al = log 5 — 0.05772, . ; (23) col al = 3” | and for large values of J, that is, values of 1 which make al equal to a considerable number of wave lengths, we have ae < sil al < me, ; 1 2 cos al cos al (24) — < colal < ——, n n, . where n, and n, are the two successive quadrants between which al lies. For instance, for al = 40, since 40 = 25.5 x5, n, = 25, n, = 26, . sin al = sin 1.5 X 5 = + 0.707, cos al = cos 1.5 xX 5 = ~ 0.707, and 0.02825 < sil 40 < 0.02420, | — 0.02825 < col 40 < — 0.02420. As seen, for larger values of al sil al has the same sign as the | sine function, col al the same sign as the cosine function. 73. From equations (20) and (21) then follow, for / = l,, the self-inductive impedance and the self-inductance of the con- | ductor, where J, = the radius of sending conductor, and since J,

896 TRANSIENT PHENOMENA is very small compared with the wave length J, the values (23) can be used, and give Self-inductive impedance: Z=4 nfle\ -j (log — 0.5772) 10-° ohms, = (25) and effective self-inductance: L=21, hogs — 0.5772 +j x 10~* henrys. (26) As an example let a current of := 100 amperes be impressed upon a sending antenna of /, = 100 feet = 3 x 10’ centimeters, consisting of a cylindrical conductor of radius J, = 0.4 inch = 1 centimeter, at a frequency of f= 200,000 cycles, then L, = 1.5 x 10° = 0.94 mile, a = 4.19 x 1075; hence, L= (57.2 + 9.4 7) 107° henrys, Z= (11.8 — 71.8 7) ohms, or, absolute, , z = 72.8 ohms. Hence, the voltage required by 7 = 100 amperes is e = 7280 volts, . and the power radiated into space during the oscillation is p =?r = 118 kilowatts. 74. Since the effective resistance of the total électromagnetic radiation, from the conductor surface to infinity, is, by (25), r= 271,107, (27) it follows that the effective resistance of electromagnetic radia- tion of a conductor is proportional to the frequency and to the length of the conductor, but independent of its size or shape, and the radiated power is p = 27 fly” 10, (28) |

| VELOCITY OF PROPAGATION OF ELECTRIC FIELD 397 or proportional to the frequency. Thus while the radiated power is moderate at commercial frequencies, it becomes considerable | at very high frequencies, and then requires consideration. For instance, at 7 = 100 amperes per 100 feet = 3000 centi- meters of conductor, the radiated power is | At 60 cycles .............3.5 watts; . At 10,000 cycles ..........5.9 kilowatts; At 10® cycles ............5900 kilowatts. | The imaginary component of self-inductance L, that is, the term in L which represents the power radiation, is Iz 10° henrys; (29) hence independent of conductor size, shape, and material, of fre- . quency, current, etc. The imaginary or reactive component of the impedance, | zr=4 nfl (log + - 0.5772) 10~* ohms, : | ta 1 is approximately, neglecting 0.5772 against log al’ and substitut- | ing equation (7), ! g ! t = 4xfhlog —— 10-* ohms flog sh, | =4 rfl, (log or —log i) 10-* ohms. (30) . Hence, with increasing frequency f, the reactance 1 increases, | but less than proportional to the frequency, due to the appearance | of the term — log f in equation (30). | For instance, with the constants J, = 100 feet =3 X10, _ | i. = 0.4 inch = 1, at the speed of light, S = 3 x 10”, we have f= 1¢ 10° 10° 10%, x = 0.0667 4.94 319 14,550. B. Mutual inductance of two conductors of finite length at con- siderable distance from each other.

898 TRANS[ENT PHENOMENA 76. Let 1, and 1, be the length of the sending and of the receiv- ing conductor respectively. By equation (2), the electric field of a conductor of length 1,, at a considerable distance J, is given by LY. ¢ ~ OP , (31) hence, for current t = Icos0 lJ cos (6 — a @ = Af cos 6-0) (32) is the electromagnetic component of the field at distance I. The magnetic flux intercepted by the receiving conductor of length /,, at distance 1, from the sending conductor, and assumed to be inductively parallel thereto, then is . bm fo Nel cos (0 — a) ly P = Lit Sco 9 f SH a+ cing f Mad. ay i &-P 4 F By partial integration, ” cos al ° 1 cosal J —p_ a= -f cos al d= —>— — a col al, in al 1 sin al @#) fo set ae — [sin at dt = 8 4 og sil t P l l l hence, cos al, . sin aly . © =I) {cos T ns —a col aly) + sin 9 (“> 4 asil al.) ‘d d

• (35) and the mutual inductance is Ly =ll, (a ~ aco! al,) + (34 +asil al,) 10-* henrys; d d (36)

VELOCITY OF PROPAGATION OF ELECTRIC FIELD 899 the mutual impedance, . Z = 2xf1),} (“5 + asil al,) -j (ao ~acol ai,) ‘a da X 10-*° ohms, (37) or, absolute, ’

2 2 2=22fll,V js es sil al,{ + 5S He a-col aly Ci da x 10-° ohms. © (38)

76. As an example, let . l, = 1, = 100 feet = 3 X 10°, ; lz = 100 miles = 15.9 x 108, J = 200,000 cycles; hence, a = 4.19 x 10-5, al, = 666 : nz = 424.5 x5; and | sin al, = sin 0.55 = 0.707, cos al, = cos 0.55 = 0.707, then, by (24), . sin al, _ sil al, > 5 =1.661 x 107° sin al, _ -3 . < “Gog = 1-665 x 10 ’ | cos al, 3 col al, > 25 =1.661 x 10 “cos aly _ “5. < 4 = 1.665 x 107°; ; | | |

400 TRANSIENT PHENOMENA hence, approximétely, sil al; = 0.1663 x 107°, col al; = 0.1663 x 107°, and . Ly = (— 0.227 + 1.026 7) 10~* henrys, Z = (0.387 + 0.086 7) 10-* ohms, or, absolute, | L,, = 1.051 < 10~ henrys, 2 = 0.3964 x 107° ohms. Hence, with an oscillating current of 100 amperes in the sending antenna, the oscillating voltage generated in the receiving an- tenna, 100 miles distant, is e=% = 0.03964 volte. C. Capacity of a sphere in space. . TI. The electrostatic field of a sphere in free space decreases . with the square of the distance /; hence, : el, r= P , (39) where J, = the radius of the sphere and e = the voltage of the sphere. ‘

Therefore, if e, = E cos 0 (40) is the potential or voltage of the sphere, the electrostatic field at distance 1 is

LE cos[@ — a(l — 1] v= oT P or, neglecting J, compared with 1, Ve LE cos (6 — al) (41) . P . |

VELOCITY OF PROPAGATION OF ELECTRIC FIELD 401 and the potential, or voltage at distance J, is e= f Vdl -1E J 29 a, (42) hence, expanded, e=LE feos f= dl +sino f 24a}, (43) l P l P by equations (34) this gives e=LE cos (se _ acol al) -+sin (ta sil al) . (44) . hence, for l=1,e = Ecos#. The inductive capacity thus is ee cos al ) (= al . ; K-55 -1, (2S aol al + (5+ ail al) : (45) hence, at distance /, kK =l1, a —acol al,) + j(% + asil al,)} ; (46) l, I, or the absolute value is” 2 7 2 k=1V jose — a col al, + jan asil al, - (47) 0 0 f 78. For instance, let J, = 10 feet = 300 centimeters; 1, = 100 miles = 15.9 x 10° centimeters, and f = 200,000 cycles per second; then (see example in section B) we have | k = 10.5 x 1078; hence, with e, = 10,000 volts impressed upon the sending sphere, , the voltage induced statically in the receiving sphere, at 100 miles | distance, is - , e = ke, = 0.105 volts. D. Sphere at a distance |, from ground. ;

402 TRANSIENT PHENOMENA

79. If ,is the radius of a sphere, at a distance /, from ground and at a potential difference e from ground, the ground, as zero potential surface, can be replaced by the image of the sphere, that is, by a sphere of radius /,, elevation —/, and potential difference —e.

The electrostatic field of such a system of spheres, at dis- tance / and elevation /,, then is the difference of the fields of the two spheres, thus: | foe

, P+(,-1) P+(,+4)’ .hence, if / is large compared with J, and /,, 41 lle f= FO? (48) and the induced potential, at distance / and elevation /,, is e= 41118 f~ co C= 49) U , The potential difference of the sending sphere at elevation J, is @ = Ecos @. (50)

This voltage e, at considerable distances /, is very small; that is, the purpose of the condenser at the top of wireless sending and receiving antenne seems not so much to send out or receive elec- trostatic fields, as to afford a capacity return for the oscillating current in the conductor, and thus produce a large current, hence a powerful electromagnetic field.

CHAPTER , IX. . HIGH-FREQUENCY CONDUCTORS.

80. As the result of the phenomena discussed in the preceding chapters, conductors intended to convey currents of very high frequency, as lightning discharges, high frequency oscillations of transmission lines, the currents used in wireless telegraphy, etc., cannot be calculated by the use of the constants derived at low frequency, but effective resistance and inductance, and therewith the power consumed by the conductor, and the voltage drop, may be of an entirely different magnitude from the values which would be found by using the usual values of resistance and induc- _ tance. In conductors such as are used in the connections and the discharge path of lightning arresters and surge protectors, the unequal current distribution in the conductor (Chapter VII) and the power and voltage consumed by electric radiation, due to the finite velocity of the electric field (Chapter VIII), require con- sideration.

The true ohmic resistance in high frequency conductors is usually entirely negligible compared with the effective resistance resulting from the unequal current distribution, and still greater may be, at very high frequency, the effective resistance repre- senting the power radiated into space by the conductor. The total effective resistance, or resistance representing the power consumed by the current in the conductor, thus comprises the true ohmic resistance, the effective resistance of unequal current . distribution, and the effective resistance of radiation.

The power consumed by the effective resistance of unequal current distribution in the conductor is converted into heat in the conductor, and this resistance thus may be called the | “thermal resistance”’ of the conductor, to distinguish it from the radiation resistance. The power consumed by the radiation resistance is not converted into heat in the conductor, but is dissipated in the space surrounding the conductor, or in any other conductor on which the electric wave impinges. That is,

408

404 TRANSIENT PHENOMENA

at very high frequency, the total power consumed by the effective resistance of the conductor does not appear as heating of the conductor, but a large part of it may be sent out into space as electric radiation, which accounts for the power exerted upon bodies near the path of a lightning stroke, as “side discharge.”

The inductance is reduced by the unequal current distribution in the conductor, which, by deflecting most of the current into the outer layer of.the conductor, reduces or practically eliminates the magnetic field inside of the conductor. The lag of the mag- netic field in space, behind the current in the conductor, due to the finite velocity of radiation, also reduces the inductance to less than that from the conductor surface to a distance of one- half wave. An exact determination of the inductance is, how- ever, not possible; the inductance is represented by the electro- magnetic field of the conductor, and this depends upon the presence and location of other conductors, etc., in space, on the length of the conductor, and the distance from the return con- ductor. Since very high frequency currents, as lightning dis-

charges, frequently have no return conductor, but the capacity at the end of the discharge path returns the current as “dis- placement current,” the extent and distribution of the magnetic field is indeterminate. If, however, the conductor under con- sideration is a small part of the total discharge -- as the ground connection of a lightning arrester, a small part of the discharge path from cloud to ground — and the frequency very high, so that the wave length is relatively short, and the space covered by the first half wave thus is known to be free of effective return conductors, the magnitude of the inductance can be calculated with fair approximation by assuming the conductor as a finite section of a conductor without return conductor.

Here then, as in many cases, for the two extremes — low fre- quency, where unequal current distribution and radiation are negligible, and very high frequency, where the current traverses only the outer layer and the total effect, contained within one wave length, is within a moderate distance of the conductor — the constants can be calculated; but for the intermediary case, of moderately high frequency, the conductor constants may be anywhere between the two limits, i.e., the low frequency values and the values corresponding to an infinitely long conductor without return conductor.

|

HIGH-FREQUENCY CONDUCTORS 405 Since, however, the magnitude of the conductor constants, as derived from the approximate equations of unequal current distribution and of radiation, are usually very different from the low frequency values, their determination is of interest even in the case of intermediate frequency, as indicating an upper limit of the conductor constants. 81. Using the following symbols, namely, 1, = the length of conductor, A = the sectional area, l,, = the circumference at conductor surface, that is, following all the indentations of the conductor, ; 1, = the shortest circumference of the conductor, that is, cir- cumference without following’ its indentations, 1, = the radius of the conductor, l, = the distance from the return conductor, A = the conductivity of conductor material,

= the permeability of conductor material,

f = the frequency, | S = the speed of light = 3 x 10° cm., and (1) a= 2a = the wave length constant, the true ohmic resistance is l | r, = 74 ohms; (2) the ohmic reactance, low frequency value is | i x = 2afl, }2I0g, 7# + $110- ohms; (3) | r | or, reduced to common logarithms by dividing by log «, L, = 27xfl (46 log # + f) 10~° ohms. (4) Tr The equivalent depth of penetration of the current into the con- ductor, from Chapter VII, (40), is . La 10° = 5030 | 5 P eVO04Anf = Viaf’ (5) |

406 _ TRANSIENT PHENOMENA hence, the effective resistance of unequal current distribution, or thermal resistance of the conductor, is, approximately, — 1, _ ln, 04 nf ,, 1.981,, fof ., rT, = Uy, = L, ”2 10-4 = L, 2 10 ohms, (6) and the effective reactance of the internal flux is

• L,=?T, 1B yet 10~ ohms. 0) The effective resistance resulting from the finite velocity of the electric field, or radiation resistance, by assuming the conductor as a section of an infinitely long conductor without return con- ductor, from Chapter VIII, (25), is 7, = 21 f10-* = 1.971, f 10-* ohms, (8) . and the effective reactance of the external field of finite section of an infinitely long round conductor without return conductor, from Chapter VIII, (25), is z, = 4nfh (toe, 7 - 0.8772) 10-, @) Assuming now that the external magnetic field of a conductor of any shape is equal to that of a round conductor having the same minimum circumference, as is approximately the case, that is, substituting L, = 2zl, (10) in equation (9), and also substituting (1), gives z,=4 nf (log, S 0.572) 10- Lf , | = 1.261,f (toe. S 0.572) 10-; (11) Lf or, reduced to common logarithms by dividing by log «, and substituting for S, 2, = 5lgf (1 — 0.547 log, f) 10- ohms | (12) = 0.547 1, f (9.15 — log 1. f) 10-* ohms.

| HIGH-FREQUENCY CONDUCTORS 407 82. The total impedance of the conductor for high frequencies is, therefore, Z=r—je=(r,+7,) —] (+ 2), _ ha fost, 1.98254 /uf 4 | 1 WV 10 ~ 7 10 ’ | r, = 2lyf10- = 1.971, f 10-8, | _ lyr os pa 198loy/uf 10-1 (13) | i= l, TZ 10 i, 7T 10 ’ 2, =4al f (Io 5’ _0.5772)10-* 2 0 Be l., f ° ) ’ = 0.547 1, f (9.15 —log 1, f)10-; while the conductor impedance for low frequencies is Zo =% — J =% —J (t+ 2,) l, N ly ta 9 | | Gy ~ 12 7S (2 log, 7 + 5) 10 4) | _ l, . la —8 | =< - ils (2.89 log + 0.314 ) 10-, Although the true ohmic resistance r, is independent of the frequency, the thermal resistance r, is proportional to the square | root and the radiation resistance r, to the first power of the fre- quency. With increasing frequency, the resistance r, is at first appreciable, while r, is still negligible, and appears only at still higher frequencies, and ultimately, at the highest frequency, becomes the dominant factor. Since 7, does not contain the conductor dimensions, it follows that at very high frequencies size, shape, and material of the conductor are immaterial in their effect on the total effective resistance r. The low frequency reactance, x,, is proportional to the fre- quency; the internal reactance, z,, is proportional only to the square root of the frequency; that is, with increasing frequency . the internal field of the conductor has less and less effect on its | reactance. The radiation reactance, z,, increases proportionally | with the frequency for moderate frequencies, but for higher fre-

quencies increases at a lesser rate as soon as the negative term

408 TRANSIENT PHENOMENA

in z, becomes appreciable; it ultimately reaches a maximum and then decreases again, but the latter at such high frequencies as to be of no practical importance. Besides, for extremely high frequencies, thousands of millions of cycles, equation (12) does not apply, as it is only the first term of a series, and the further terms begin to become appreciable.

83. As examples, the following may be considered:

(1) A copper wire No. 4 B. and S. gauge of 0.204 inch = 0.518 centimeter diameter.

(2) An iron wire of the same size, d = 0.204 inch = 0.518 centimeter.

(3) A copper ribbon of 3 inches width and one-eighth inch thickness, or the dimensions 7.6 by 0.317 centimeters.

(4) A wrought-iron pipe of 2 inches = 5.06 centimeters exter- nal diameter and one-eighth inch = 0.317 centimeter thickness of walls, that is, of nearly the same circumference as the copper ribbon in (3).

Assuming the following constants: copper, 2=1, A = 6.2 X 105; wrought iron, # = 2000, 2 = 1.1 x 105, we have ro 7

| Wire No. 4 B. and S. Gauge. 2 In. Pipe, } In. | Copper. | Tron. Iron. [a 0.211 | 0.211 2.40 4.70 by = bey = 1.63 1.63 15.85 15.85 he =- x 0.259 | 0.259 2.53 2.53 | = |7.6x10- 43x 10-* o.e7x10-" | 1.94 10-* i. 0.00314x 10 f} 6.28x10-f | 0.003814x10-°f 6.28x 10°F 7. 0.0824x10-° fF | 0.0824x10- f| 0.0536x10-* f | 0.0536 x10 fF rep 0.155x10-° VF | 16.5x10-° VF ed 1.68x 10-* Vf 2 = (| 0.0197 x10-F_|_ 0.0197 x 10-* f} 0.0197x10-F_| 0.0197 x 10- F) is = — |0.00547x10-* f(8.94—log /) | 0.00547 10-* F (7.95—log SF)

HIGH-FREQUENCY CONDUCTORS 409

In 2x°, the distance from the return conductor has been chosen as 1, = 6 feet = 182 centimeters. The values of x,° for iron are not realized; they are due to the excessive field in the conductor, caused by its high*permeability, but can be realized only at extremely low frequency and small currents; at larger currents, magnetic saturation greatly reduces the reactance, so

that in iron conductors the internal reactance is a function of the current and decreases with increase of current. In conductors of the size above discussed, even at 25 cycles the unequal current distribution in the conductor is so great as to make equation (14) inapplicable, and the reactance is given by Y= 4xfl, log, ‘ xX 107° + x, (15) where z, is the internal reactance of equation (7).

84. In Fig. 97 are plotted values of the resistances r,, 7,, 7, and of the reactances z,°, z,°, Z,, 2,, for the four conductors illus- trated, for frequencies from one cycle to 1000 million cycles. As it is not possible to represent directly quantities varying over such a wide range, in Fig. 97 as abscissas are used the logarithms of the frequency, and as ordinates the logarithms of the ohmic resistance or reactance per meter = 100 centimeters length of conductor; that is, a geometric scale is used. This means that each scale section is a change by factor 10, and since, as discussed above, these quantities can be determined only in their general magnitude, a value one scale section below another one, there- fore, is négligible compared with the latter one, being only one- tenth of it.

The following conclusions may be drawn from the curves shown in Fig. 97. ,

(1) In copper wire No. 4, the true ohmic resistance prepon- derates up to 100 cycles. At 100 cycles the reactance 2° rises beyond the resistance, and the true ohmic resistance becomes negligible in the impedance at 1000 cycles. At 3000 cycles the screening effect, or the unequal current distribution in the con- ductor, becomes appreciable and increases its heating at con- stant value of current. The radiation resistance r, would equal the ohmic resistance 7, at about 500 cycles if at this low fre- quency the case of an infinitely long conductor without return

410 TRANSIENT PHENOMENA

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Fig. 97. High-frequency conductors.

HIGH-FREQUENCY CONDUCTORS 411 conductor could be realized. At 500,000 cycles, the radiation resistance r, equals the external reactance z,, and since the internal resistance r, at high frequencies is equal to the internal reactance x, at 500,000 cycles, the total effective resistance r ; equals the total effective reactance z, that is, the current lags 45 degrees; and at still higher frequencies the lag of the current decreases still further, and the radiation resistance preponder- ates.

(2) In iron wire No. 4, the true ohmic resistance r, ceases to be the main term even at frequencies below 10 cycles, and the screening effect or the unequal current distribution in the con- ductor is marked at 10 cycles. The internal reactance z,°, which corresponds to uniform current density, thus ceases to represent the actual conditions at frequencies even below 10 cycles. Up to about one million cycles, the internal resistance 7, and internal reactance z, prependerate. At about one million cycles, all four quantities — internal resistance r,, representing power converted into heat in the conductor, radiation resistance r,, representing power radiated by the conductor, internal reactance z,, representing the magnetic field in the conductor, the external reactance z,, representing the magnetic field outside of the conductor — are approximately equal and the current lags 45 degrees. Above one million cycles, radiation resistance r, and ° external reactance z, preponderate, and as they are independent of the conductor material above one million cycles, the iron wire thus becomes nearly as good — or poor — a conductor as copper ’ wire.

Similar relations exist between the larger conductors (3) and (4). .

(3) Three-inch by one-eighth inch copper ribbon. Above 10 cycles, the reactance is greater than the resistance, and above 2000 cycles unequal current distribution is marked. At 50,000 cycles, the radiation resistance equals the external reactance, and the current lags 45 degrees, and at still higher frequencies the radiation resistance preponderates, and the current lags less than 45 degrees.

(4) Two-inch iron pipe, one-eighth inch walls. The internal reactance r,° here has no meaning, as it would correspond toa | 2-inch iron rod. The ohmic resistance r, ceases to be applicable, and unequal current distribution begins already at one cycle per.

| | |

412 TRANSIENT PHENOMENA

second. At about 30 cycles the external reactance rises beyond the ohmic resistance ; at 5000 cycles beyond the internal reactance and resistance. At 30,000 cycles the current lags 45 degrees, and less than 45 degrees at higher frequencies. At 100,000 cycles r, and x, are small compared with r, and x,, and the conductor material thus ceases to have an effect on the voltage drop; that is, above 100,000 cycles a large iron conductor gives practically the same voltage drop, at the same current, as a copper conductor of the same circumference; that is, iron is nearly as good a con- ductor as copper, when considering a finite section of an infinitely long conductor without return conductor, that is, approximately, when dealing with oscillatory high frequency discharges, as lightning.

It is interesting to note the high power component of impe- dance existing at high frequencies and mainly due to the radia- tion resistance, which causes a rapid decay of the oscillation, due to the high power factor. The internal constants r, and x, are equal, and in the most important range of high frequencies, from 10,000 to 1,000,000 cycles, the external constants 7, and x, are not very different from each other and their plotted curves intersect at some certain frequency. That is, at high frequen- cies, the power radiated into space increases at such a rapid

• rate that the circuit never becomes highly reactive. This, however, applies only under the consideration assumed here; under different conditions the radiation power may be sufficiently

‘ limited to give a large angle of lag of the current and therefore a slower decay of the oscillating discharge, that is, a more sustained oscillation. In general, however, these results show that even at high frequencies and in iron conductors the angle of lag may be moderate.

It is interesting to note that with increasing frequency the conductor material decreases in importance, and even soft iron becomes as good as copper in the voltage drop in the conductor, and at still much higher frequencies even the size and shape of the conductor become less important, and ultimately all con- ductors act practically alike.

85. From the data of the preceding table and Fig. 97 the |

. total effective resistance, reactance, impedance, and the power factor per meter length of conductor for high frequency dis- | charge are given on p. 413.

|

HIGH-FREQUENCY CONDUCTORS 413 Wire No. 4 B. and 8. Gauge. Copper. Iron. Frequeney......0..00....005.....[. 108 10° 10° | 104 10 10° Resistance, r...................-{0.0212] 0.202] 1.986] 0.185] 0.717] 3.62 Reactance, z....................{0.0286] 0.221] 1.626] 0.168] 0.736] 3.26 Impedance, z....................|0.0356| 0.299} 2.57 | 0.250| 1.028] 4.87 Power factor....................0.59 | 0.67 | 0.77 | 0.74 | 0.70 | 0.75 Voltage drop at 100 amperes......|3.6 30 | 257 «| «25 103 | 487 Copper Ribbon, 3 In. | Wrought-iron Pipe, by 3 In. 2 In. by $ In. Frequency.......................| 104 | 105 | 10% | 104 10° | 10° Resistance, r....................0.0199] 0.198} 1.97210.0365] 0.250] 2.14 Reactance, z....................|0.0219] 0. 162] 1.07210.0385| 0.214] 1.24 Impedance, z................--..|0.0296| 0.256] 2.24 |0.0530| 0.326] 2.47 Power factor....................]0.67 | 0.77 | 0.88 (0.69 | 0.76 | 0.87 Voltage drop at 100 amperes....../3.0 26 224 15.3 33 247 As seen herefrom, within the range of frequencies from 10‘ to 10° cycles the power factor varies from 59 per cent to 88 per cent, | being higher at the higher frequencies, but there is little differ- ence in the power factor between iron and copper. The difference between the different conductors decreases with increase of frequency; while at 10* cycles the iron wire has seven times the voltage drop of the same size of copper wire, at 10° cycles it has only 90 per cent more voltage drop. | With the large conductors the difference in the voltage drop between iron and copper is very small; the voltage drop in the . iron pipe at 10‘ cycles is only 80 per cent greater than in the cop- per ribbon, while at 10° cycles the difference has decreased to 10 per cent, that is, has practically become negligible, and the iron pipe is practically as good a conductor and has also practically the same power factor as the copper ribbon, so that | a small increase in the size of the iron pipe, to 2.5 inches in diam- eter, would make it superior to the 3-inch copper ribbon at frequencies of a million cycles and over, about the same at 10° cycles, and very little inferior at 10‘ cycles. This is rather against the usual expectations, but is due to the preponderance of the radiation resistance, and, therefore, does not apply, or at least not to the same extent, where the radiation resistance is smaller. |

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SECTION IV. TRANSIENT PHENOMENA IN TIME AND SPACE |

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TRANSIENT PHENOMENA IN TIME AND SPACE CHAPTER I. GENERAL EQUATIONS.

1. The energy relations of an electric circuit can be charac- terized, as discussed in Section III, by the four constants, namely: -

r = effective resistance, representing the power or rate of energy consumption depending upon the current, 7’r; or the power component of the e.m.f. consumed in the circuit, that is, with an alternating current, the voltage, ir, in phase with the . current. |

L = effective inductance, representing the energy storage

. PL .

depending upon the current, 3 8s electromagnetic component of the electric field; or the voltage generated due to the change of the current, Le , that is, with an alternating current, the reactive voltage consumed in the circuit — jzi, where z = 2 2fL and f = frequency. ,

g = effective (shunted) conductance, representing the power or rate of energy consumption depending upon the voltage, eg; or the power component of the current consumed in the circuit, that is, with an alternating voltage, the current, eg, in phase with the voltage.

C = effective capacity, representing the energy storage depending upon the voltage, =, as electrostatic component of ; the electric field; or the current consumed by a change of the voltage, C’ e, that is, with an alternating voltage, the (leading) reactive current consumed in the circuit — 7be, where b = 2 xfC and f = frequency. |

417

418 TRANSIENT PHENOMENA In the investigation of electric circuits, these four constants, r, L, g, C, usually are assumed as located separately from each other, or localized. Although this assumption can never be per- fectly correct, — for instance, every resistor has some inductance __ and every reactor has some resistance, — nevertheless in most cases it is permissible and necessary, and only in some classesof phenomena, and in some kinds of circuits, such as high-frequency __ phenomena, voltage and current distribution in long-distance, 'high-potential circuits, cables, telephone circuits, etc., this assumption is not permissible, but 7, L, g, C must be treated as distributed throughout the circuit. In the case of a circuit with distributed resistance, inductance, __ conductance, and capacity, as 7, L, g, C, are denoted the effec tive resistance, inductance, conductance, and capacity, respec- tively, per unit length of circuit. The unit of length of the circuit may be chosen as is convenient, thus: a couple of centimetersin the high-frequency oscillation over the multigap lightning arrester circuit, or a mile in a long-distance transmission circuit or high- potential cable. , |

• The permanent values of current and e.m.f. in such circuits of distributed constants have, for alternating-current circuits, been investigated in Section III, where it was shown that they can be treated as transient phenomena in space, of the complex variables, current J and e.m.f. E. ' Transient phenomena in circuits with distributed constants, | and, therefore, the general investigation of such circuits, leads to | transient phenomena of two independent variables, time ¢ and space or distance /; that is, these phenomena are transient in | time and in space. ™ The difficulty met in studying .such phenomena is that they are not alternating functions of time, and therefore can no longer . : be represented by the complex quantity. . It is possible, however, to derive from the constants of the circuit, r, L, g, C, and without any assumption whatever regand- | ing current, voltage, etc., general equations of the electric cir- | cuits, and to derive some results and conclusions from such equations. | _ These general equations of the electric circuit are based on the | single assumption that the constants r, L, g, C remain constant | with the time ¢ and distance /, that is, are the same for every unit |

| GENERAL EQUATIONS 419 length of the circuit or of the section of the circuit to which the equations apply. Where the circuit constants change, as where | another circuit joins the circuit in question, the integration con- | stants in the equations also change correspondingly.

Special cases of these general equations then are all the phe- nomena of direct currents, alternating currents, discharges of reactive coils, high-frequency oscillations, etc., and the difference between these different circuits is due merely to different values of the integration constants.

2. Ina circuit or a section of a circuit containing distributed resistance, inductance, conductance, and capacity, as a trans- mission line, cable, high-potential coil of a transformer, telephone . or telegraph circuit, etc., let r = the effective resistance per unit . length of circuit; ZL = the effective inductance per unit length of circuit; g = the effective shunted conductance per unit length of circuit; C = the effective capacity per unit length of circuit; ¢ = the time, / = the distance, from some starting point; e = the voltage, and 7 = the current at any point J and at any time ¢; then e and 7 are functions of the time ¢ and the dis- tance J.

In an element di of the circuit, the voltage e changes, by de, by the voltage consumed by the resistance of the circuit element, ri dl, and by the voltage consumed by the inductance of the cir-

. dy cuit element, L Gil. Hence,

de, dt In this circuit element di the current 7 changes, by di, by the | current consumed by the conductance of the circuit element, | gedl, and by the current consumed by the capacity of the circuit element, C’ “dl Hence, . di de.

Differentiating (1) with respect to ¢ and (2) with respect to 1,

and substituting then (1) into (2), gives Cia . di ig ge 7 + OC + gh) t+ We (3)