Steinmetz General Equations of the Electric Circuit Pt3 1919

8/2/2019 Steinmetz General Equations of the Electric Circuit Pt3 1919

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Pr-W ., 1M , 1 1 1 JlU,""" COII_,iota of1MA_ie.. 1.$1"... of EUefI"ul8.,.,sN . . Yori, F . " ' _ ' 7 :1O,l{)l{).Copyright 111111. By A. I. B. E.

THE GENERAL EQUATIONS OF THE ELECTRICCIRCUIT-IIIVariation of Constants r, L,C, and g, and ita Effects

BY CHARLES P. STEINMETZ

ABSTBACT 01' PAPBRIn the usual theory of transients the assumption ismade, thatresistance, inductance, capacity and conductance are constant.This however, is not correct, and as the result thereof, it wasnot possible to theoretically investigate, and numerically cal-culate the dissipation of high-frequency disturbances, the fiat-tening of the wave fronts of impulses, the rounding off of steepwaves, etc., with the time and the distance of travel, and there-from to determine the distance, to which the danger from suchdisturbances extends, and to investigate the conditions of line

construction, which limit the danger zone of such phenomena tothe smallest local extent.In the foUowingJtwo of the foremost causes of change of theline constants witn the equivalent frequency are investigated,the unequal current distribution in the conductor, and the electricradiation from the conductor, and shown, that within the rangeof frequencieswhich may be met in industrial circuits, the effec-tive resistance of the conductor, and its attenuation constant,may increase more than a million fold.Equations of the line constants as function of the equivalentfrequency are derived, and applications thereof made to a fewproblems:(1) The laws of conduction of high-frequency currents, suchas lroduced by lightning disch.ariresand similar disturbances,an the conclusionsresulting therefrom on the nature of the con-ductor.(2) The decay of high-frequency sine waves in transmiBSionlines.(3) The attenuation of rectangular waves.(4) The fiatte~ of the wave front of stee~ impulses.It is shown that m high-frequency conduction the section ofthe conductor is of little importance, but the circumference isthe determining factor, except at very high frequencies, wheresize, shape, and material-within certain limits-becomes ofsecondary importance. The inductance and the capacity ofthe conductor remain constant up to very high frequencies, butthe effective resistance may increase many thousand fold. Athigh frequencies, not all the energy consumed by the effectiveresistance of the conductor is converted into heat, but the majorpart may be radiated into space and thereby aJfect other circwts.The conditions, which cause a fiattening of steep wave frontsand impulses, and a roun~ of irregular waves, are investigated,and it is shown, that within the range of distances in which

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192 STEINMETZ: GENERAL EQUATIONS [Feb. 20danger may result from a steep wavefront or high frequency, thesteepness of the wave front decreases with the square root ofthe distance traveled by the wave; that size and material ofthe conductor have little influence, but the distance betweenconductor and return conductor is the main factor in flatteningthe wave front and thereby limiting the danger zone.For convenience, the theoretica.lpart has been separated andpla.cedin an appendix, giving in the text the discussion, withnumerous tables derived from jhe theoretical equations, andcurves illustrating these tables.The investigation is not general, but rather limited to specialapplications, as the field is too new to permit fargoing general-ization.

I.-GeneralI1908, in a paper under the above title, the general equationsof the electric circuit were presented and discussed, and ina second paper in 1916, under the title "Outline of a Theoryof Impulse Currents," the application of these general equationsto the special cases of direct-current circuits, alternating-current circuits, etc., was discussed, and more particularly atheory of impulse currents outlined, that is, transient currents.which are non-periodic in time, and either non-periodic orperiodic in space.In both of these papers however, and in all general theoreticalinvestigations on these subjects, as far as I know, the assump-tion is made, that the circuit constants; resistance r, inductanceL, capacity C and shunted conductance g, are constant. Whilethis is true, with sufficient approximation, for the usual machinefrequencies and for moderately high frequencies, experience

shows that it is not even approximately true for very highfrequencies and for very sudden circuit changes, as steep wavefront impulses, etc.If r, L, C and g are assumed as constant, it follows that theattenuation is independent of the frequency, that is, waves ofall frequencies decay at the same rate, and as the result, a

complex wave or an impulse traversing a circuit dies out with-out changing its wave shape or the steepness of its wave front.Experience, however, has conclusively shown that steepwave fronts are dangerous only near their origin, and rapidlylose their destructiveness by the flattening of the wave frontwhen running along the circuit. Experimentally, small in-ductances shunted by a spark gap, inserted in transmissionlines for testing for high frequencies or steep wave fronts, haveshown appreciable spark lengths, that is, hlgh-voltagegradients,only near the origin of the disturbance, but no appreciable


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1919) OF THE ELECTRIC CIRCUIT 193spark length beyond a few miles from the origin of the distur-bance.The rectangular wave of starting a transmission line byconnecting it to a source of voltage, which is given by the theoryunder the assumption of constant T, L, C and g, and thus shownin most text books, has never been shown in any of the oseil-lograms of transmission lines, as far as I know.If T and L are constant, the power factor of the line con-

ductor, r , should with increasing frequency~~+~~) -continuously decrease, and reach extremely low values, atvery high frequencies, so that at these, an oscillatory dis-turbance should be sustained over very many cycles, and showwith increasing frequency an increasing liability to become asustained or cumulative oscillation. Experience,however, showsthat high-frequency oscillations die out much more rapidlythan accounted for by the standard theory, and show at veryhigh frequency practically no tendency to become cumulative.This does not mean that the present theory of transients,which is based on the assumption of constant r, L, C and -g ,is useless by not correctly representing the actual phenomena,but it means that it correctly represents the transient only inits initial stage and near its origin, but does not satisfactorilyrepresent its course after its initial stage and at some distancefrom the origin, at least not for high-frequency transients orsteep wave fronts.As the destructive effect of voltage does not depend only onthe value of the voltage, but also on the duration of its appli-cation, this is a limitation, though not as serious as it mayappear, since it tends to understate the danger of high-fre-quency transients.However, it makes it impossible to determine such importantquestions as; how far the danger zone of a high-frequencydisturbance or steep wave front extends; how the destructive-ness decreases with the distance; how it depends on line con-struction, conductor shape and materials, etc., and to approachthe problem of so designing the circuit as to limit the dangerfrom any such disturbance to the smallest local extent.In the following, therefore, the attempt is made to in-vestigate and calculate theoretically some of the factors, whichcause a change of the line constants r, L, C, and g, with increas-ing frequency or steepness of wave front, and to investigate

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194 STEINMETZ: GENERAL EQUATIONS [Feb. 20theoretically the effect produced on the course of the transientwith regard to duration and wave shape, by the variation of theline constants.As this field is rather new, no very general treatment is yetpossible, but rather, an outline of the problem, and applica-tions to some of the most important phenomena, such as the~onduction of high-frequency currents, the changes of waveshape with time and distance of wave travel, and the flatteningof the wave front, and 'its bearing on circuit construction.. The two most important factors in the variation of the circuitconstants T, L, C, and g, seem to be the u neq ua l cu rren t du-

tT ibution in the conductor and the fin ite velocity of the electricfield.UNEQUAL CURRENT DISTRmUTION IN THE CONDUCTOR

The magnetic field of the current surrounds this currentand fills all the space outside thereof, up to the return current.Some of the magnetic field due to the current in the interiorand in the center of a conductor carrying current, thus isinside of the conductor, while all the magnetic field of thecurrent in the outer layer of the conductor is outside of it.Therefore, more magnetic field surrounds the current in theinterior of the conductor than the current in its outer layer,and the inductance therefore increases from the outer layer ofthe conductor towards its interior, by the "internal magneticfield." In the interior of the conductor the reactance voltagethus is higher than on the outside.At low frequency, with moderate size of conductor, thisdifference is inappreciable in its effect. At higher frequencieshowever, the higher reactance in the interior of the conductor,due to this internal magnetic field, causes the current densityto decrease towards the interior of the conductor, and the cur-rent to lag, until finally the current flows practically onlythrough a thin layer of the conductor surface.As the result thereof, the effective resistance of the conductoris increased, due to the uneconomical use of the conductormaterial caused by the lower current density in the interior,and due to the phase displacement, which results in the sum ofthe currents in the successive layers of the conductor beinglarger than the resultant current. Due to this unequal cur-rent distribution, the internal reactance of the conductor isdecreased, as less current penetrates to the interior of the con-

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1919) OF THE ELEC7'RlC CIRCUIT 195ductor, and thus produces less magnetic field inside of theconductor.The derivation of the equations of the effective resistanceof unequal current distribution in the conductor, TIr and of theinternal reactance Xl under these conditions, is given in appen-dix A. It is interesting to note, that effective resistance andinternal reactance, with increasing frequency, approach thesame limit, and become proportional to the square root of thefrequency, the square root of the permeability, and the squareroot of the resistivity of "the conductor material, while at lowfrequencies the resistance is independent of the frequency anddirectly proportional to the resistivity, and the internal re-actance is independent of the resistivity and directly propor-tional to the frequency.

FINITE VELOCITY OF THE ELECTRICjFIELDIn calculating the inductance or the capacity of a con-ductor, usuaJIy the assumption is made, that the electric fieldis instantaneous, that is, the magnetic field appears and dis-appears simultaneously with the current, and therefore, is in

phase with the current, and the dielectric field appears anddisappears simultaneously with the voltage. In reality, how-ever, this is not strictly correct, but the electric field moveswith a finite velocity from the conductor outwards. Whilethis velocity is inconceivably high: S = 3 X 1010 cm. per sec.,or 300,000 Jan. per sec., its effect is appreciable at high fre-quencies. Consider, for instance, a circuit representing averagetransmission line conditions with 6 ft. = 182 cm., betweenconductors, traversed by a current of f = loa, or one millioncycles, such as maybe produced by a nearby lightning discharge.The wave length of this current and thus of its magnetic field

S 3 X 1010then would bef = loa = 30,000 em. and the distanceof 182 em. between the line conductors would be 30~~ or1~5 of a-wave length or ~:~ = 2.2 deg. That is, the mag-netic field of the current, when it reaches the return conductor,would not be in phase with the current, but 1~5 of a wave

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196 STEINMETZ: GENERAL EQUATIONS [Feb. 20length or 2.2 deg. behind the current. The voltage inducedby the magnetic field would not be in quadrature with thecurrent, or wattless, but lag 90 + 2.2 = 92.2 deg. behind thecurrent, thus have an energy component equal to cos 92.2deg. = 3.8 per cent, giving rise to an effective resistance Tbequal to 3.8 per cent of the reactance x. Even if at normalfrequencies of 60 cycles the reactance is equal only to theohmic resistance To - usually it is larger-the reactance x to10e cycles would be 1~~ , = 16,700 times the ohmic resistance,and the effective resistance of magnetic radiation, Ta, being3.8 per cent of this reactance, thus would be 630 times theohmic resistance: Ta = 630 To. Thus, the ohmic resistancewould be entirely negligible compared with the effective re-sistance resulting from the finite velocity of the magnetic field.Considering however, a high-frequency: oscillation of 10'cycles, not between the line conductors, but between lineconductor and ground, and assume, under average transmissionline conditions, 30 ft. as the average height of the conductorabove ground. The magnetic field of the conductor then canbe represented as that between the conductor and its imageconductor, 30 ft. below ground, and the distance between con-ductor and return conductor would be 2 X 30 ft. = 1820 em.The lag of the magnetic field, due to the finite velocity of propa-gation, then becomes 22 deg. thus quite appreciable, and theenergy component of the voltage induced by the magnetic fieldis cos (90 + 220) = 37 per cent. This would give rise to aneffective or radiation resistance Ta = 0.37 x. As in this case,the 6O-cycle reactance usually is much larger than the ohmicresistance, assuming it as twice would make the radiationresistance TI = 12,600 To, or more than ten thousand timesthe true ohmic resistance. It is true, that at these high fre-quencies, the ohmic resistance would be very greatly increasedby unequal current distribution in the conductor. But theeffective resistance of unequal current distribution increasesonly proportionally to the square root of the frequency, and,assuming for instance conductor No. 00 B. & S. gage, the effec-tive resistance of unequal current distribution, Tit would at lO tcycles be about 36 times the low-frequency ohmic resistanceTo. Thus the effective resistance of magnetic radiation wouldstill be 123~00- =350 times the effective resistance of unequal

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19191 OF THE ELECTRIC CIRCUIT 197current distribution: r 3 = 350 rio and the latter, therefore,is negligible compared with the former.It is interesting to note, that the effective resistance ofradiation, r " does not represent energy dissipation in the con-ductor by conversion into heat, but energy disslpation byradiation into space, and in distinction from the "radiationresistance," which dissipates energy into space, the effectiveresistance of unequal current distribution may be called a"thermal resistance," as it converts electric energy into heatThus in this instance of a lOS-cyclehigh-frequency dischargebetween transmission line conductor and ground, the heating .of the conductor would be increased 36 fold over that producedby a low-frequency current of the same amperage, by theincrease of resistance by unequal current distribution in theconductor; but the total energy dissipation by the conductorwould be increased by magnetic radiation still 350 times more,so that the total energy dissipation by the 108-cycle currentwould be 12,600 times greater than it would be with a low-fre-quency current of the same value, and the attenuation or rateof decay of the current thus would be increased many thousandtimes, over that calculated on the assumption of constantresistance at all frequencies.The derivation of the equations of the effective resistanceof magnetic radiation, and in general of the effects of. thefinite velocity of the electric field on the line constants, aregiven in appendix B.The magnetic radiation resistance is proportional to thesquare of the frequency (except at extremely high frequencies)..It therefore, is negligible at low and medium frequencies, butbecomes the dominating _factor at high frequencies. It isproportional to the distance of the return conductor, but en-tirely independent of size, shape, or material of the conductor,as is to be expected, since it represents the energy dissipatedinto space. Only at extremely high frequencies, the rise ofradiation resistance becomes less than proportional to thesquare of the frequency. It becomes practically independentof the distance of the return conductor, when the latter becomesof the magnitude of the quarter wave length.The same applies to the capacity. Due to the finite velocityof propagation, the dielectric or electrostatic field lags behindthe voltage which produces it, by the same angle by which themagnetic field lags behind the current, and the capacity current

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198 STEINMETZ: GENERAL EQUATIONS [Feb. 20or charging current thus is not in quadrature with the voltage,or reactive, but displaced in phase by more than 90 deg. thuscontains a negative energy component, which gives rise to ashunted conductance of dielectric radiation g. This gives riseto an energy dissipation by the conductor, at high frequencies,by dielectric radiation -into space, of the same magnitude asthe energy dissipation by magnetic radiation, above considered.The term "shunted conductance" g has been introducedinto the general equations of the electric circuit largely fromtheoretical reasons, as representing the power consumption. proportional to the voltage. Most theoretical investigationsof transmission circuits consider only r, L and C as the circuitconstants, and omit g, since under average transmission lineconditions, at low and moderate frequencies, g usually is negli-gible. In communication circuits, as telegraph and telephone,there is a "leakage," which would be represented by a shuntedconductance, and in underground cables there is a considerableenergy consumption by dielectric losses in the insulation, asthe investigations of the last years have shown, which gives ashunted conductance. In overhead power lines however,energy losses depending on the voltage,-and leading to a termg-have been known only at those high voltages where coronaappears. It is interesting therefore to note, that at highfrequencies, "shunted conductance" g may reach very formid-able values even in transmission lines, due to electrostaticradiation.In investigating the effect of the finite velocity of the electricfield on the inductance L and the capacity C, in appendix B,it is seen, that the equations of external inductance and ofcapacity are not affected, but remain the same as the usualvalues derived by neglecting the velocity of the electric field,except at extremely high frequencies, when the distance of thereturn conductor approaches quarter wave length.EQUATIONS OF ELECTRICAL CONSTANTS, AND NUMERICAL

VALUESIn appendix C are given, compiled from appendices A and B,the equations of the components of the electrical constants, asfunctions of the frequency, for conductors with return con-ductor, and also for conductors without return conductor (asapproximated by lightning strokes or wireless antennae):Resistance: true ohmic resistance or effective resistance ofunequal current distribution, and magnetic radiation resistance.

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1919) OF THE ELECTRIC CIRCUIT 199Reactanu: low frequency internal reactance or internal re-actance of unequal current distribution, and external reactance.lnductanu: low frequency internal inductance or internal in-

ductance of unequal current distribution, and external in-ductance.Shunted conductance and capacity are not so satisfactorilyrepresented, and therefore, instead of representing energystorage and power dissipation depending on the voltage by aconductance g and a capacity C or susceptance b, in shuntwith each other, it is more convenient to represent them by aneffective resistance, the dielectric radiation resistance r 0 , anda capacity reactance Xc, in series with each other.As seen from appendix C, four successive stages may be dis-tinguished in the expressions of the circuit constants as functionsof the frequency.1. Low frequencies, such as the machine frequencies of 25"and 60 cycles. The resistance is the true ohmic resistance,the internal reactance and inductance that corresponding touniform current density throughout the conductor, with con-ductors of moderate size, and of non-magnetic material.2. Medium frequencies, of the magnitude of a thousand toten thousand cycles. Resistance and internal reactance orinductance are those due to unequal current distribution inthe conductor, that is, the resistance is rapidly increasing, and

the internal inductance decreasing. The conductance g isstill negligible, radiation effects still absent, and all the energyloss that of thermal resistance.3. High frequencies, of the magnitude of one hundredthousand to one million cycles. The radiation resistance isappreciable and becomes the dominating factor in the energydissipation. The internal inductance has practically dis-appeared, due to the current penetrating only a thin surfacelayer. A considerable shunted conductance exists due to thedielectric radiation.4. Extremely high frequencies. Of themagnitude of manymillions of cycles, when the quarter wave length has becomeof the same magnitude or less than the distance of the returnconductor. 'Radiation effects entirely dominate, and theusual expressions of inductance and of capacity have ceasedto apply ..This last case is of little industrial importance, as suchextremely high frequencies propagate only over short distances.

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200 STEINMETZ: GENERAL EQUATIONS [Feb. 20It would come into consideration only in calculating the flat-tening of the wave front of a rectangular impulse in the immedi-ate neighborhood of its origin, and similar problems.Thus far, a general investigation does not seem feasible.Substituting the equations of the circuit constants, as functionsof tp.e frequency, into the general equations of the electriccircuit, leads to expressions too complex for general utility,and the investigation thus must largely be made by numericalcalculations.Only when the frequencies which are of importance in theproblem, lie fairly well in one of the four ranges above discussed-as is the case in the investigation of the flattening of a steepwave front in moderate distances from its origin,-a moregeneral theoretical investigation becomes possible.As instances of industrial importance may be considered inthe following:The conduction of high frequency currents such as producedby lightning and similar disturbances.The attenuation of waves in overhead transmission circuits.The rounding of rectangular waves.The flattening of steep wave fronts.

II.-Lightning ConductorsIt has been known for many years that currents of highfrequency, very sudden impulses, etc., such as are given bylightning discharges and similar disturbances in electric cir-cuits, do not follow the laws of low-frequency currents. Theconduction of such currents however, is of very great industrial

importance; in the discharge path of lightning arresters andother protective devices, their ground connections, their con-nections with the circuit, etc., it is of fundamental importanceto design -and select the conductor so as to offer minimum im-pedance to the discharge. Inversely, in the connection of thestation to the transmission line, it is of importance to designthe conductor for maximum impedance against such currents.The protection of buildings by the lightning rod depends onthe design of the conductor system, etc.However, the high-frequency impedanee of the conductoris not the only quantity of importance. At high frequencies,the thermal resistance, which dissipates energy in the conductorby conversion into heat, may be a small part only of the totaleffective resistance, and the radiation resistance dominates.Dig!tized byCoogle


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1919) OF THE ELECTRIC CIRCUIT 201That is, most of the power consumed in the conductor is radiatedinto space. Thus it is not sufficient to merely afford a dischargepath for lightning, because while discharging through suchpath, most of the energy of the lightning may be communi-cated by radiation to other bodies. This accounts for the powerexerted upon bodies near the path of a lightning stroke, thedestructive effectsof a "side stroke:" . Hence it is of importancealso, to study the extent of the electric field of the high-fre-quency conductor, to ascertain that nothing is within theelectric field, which might be damaged by intercepting theelectric field, or may lead the power, communicated to it bythe radiation from the discharge path, into other circuits whereit may be destructive. Thus a lightning arrester, while func-tioning perfectly and supposedly giving perfect protection,may be a source of danger, by concentrating the energy oflightning into its discharge path, if this discharge path iscarried W close inductive relation to conductors of the circuitwhich is to be protected.To show the relative magnitude of the different line constantsand their components, in Table I are given their numericalvalues, in ohms per meter length of conductor, for frequenciesfrom one cycle to one hundred million cycles per second, of acopper wire No. 00 B. & S. gage, for the three conditions:a. Return conductor at l' = 6 ft. = 182 em. correspondingto a discharge between transmission line conductors.b. With the ground as return conductor, at 30 ft. distance,that' is, l' = 2 X 30 ft. = 1820 em, corresponding to a dis-charge between transmission line and ground.c. No return conductor, corresponding to the vertical dis-charge path of a lightning arrester ground, a lightning stroke,etc.Some of these values are plotted in Fig. 1.The low frequency values of resistance To and external andinternal reactance Xo + X,o, have no existence at the higherfrequencies. But as they are the values calculated by theusual formulas, they are given in Table I, for comparison withthe true effective high-frequency values. The values T2 andx " though given for all frequencies, have a meaning only forthe very high frequencies, since at lower frequencies the con-dition of a conductor without return conductor can hardly berealized, as any conductor within a quarter wave length wouldact as a more or less effective return conductor.

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[Feb. 20TEINMETZ: GENERAL EQUATIONSW2

TABLE I.ClRCUIT CONSTANTS OF COPPER WIRE NO. 00 B. '" S. GAGE PER METER LENGTH OF

CONDUCTORa. return conductor at 6 ft. - 182 cm. diatance.b. return conductor at 2 X 30 ft. - 1820 em. diatance.c. no return c:ondu~tor

conductivity: .,. - 6.2 X IO'

1 0 ' 1 0 '0 ' 1 0 'o tO '0Cycle. f-1------- I----I---f---I---~---I---I_ --I-.000240 .000Wl. 000240 .000240 .000240 .000Wl. OOO2tO .0000tO .OOOUO.0000003 .000003 .00003 .0003 .003 .03 .3 3 30fI-* . . -

{ .0000075 .000075 .00075 .0075.0000104 .000104 .00104 .0104

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7501,040.751.04 7.510.4.075.104' " -78It 107 7801,070.781.07 7.810.7.000Wl .000252 .000816 .0078.0000tO .000263 .001097 .0107 .078.107. -

.00031 .000031 .000003 .0000003.00022 .000022.000002.0000002.952.912 .293.218 .0308.0225 .0031.00221.0001.000a. COl _,_b.________ ~---I--- -- -- 1---!I---I---l1---I----1

_____I------I---~---r----I----~--I--" _ .000240 .000240.000240 .000273 .000865 .00273 .00865 .0273 .0865"" _ .0000003 .000003 .000030 .000273 .000865 .00273 .00865 .0273 .086S

\( .. .. .. .. .. .. .. .. .0000048.00048.048.. .. .. .. .. .. .. .. .000048 .0048 .48.000002 .000020 .000197 .00197 .0197 .197 1.97

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{ .0000075 .000075 .00075 (J.075.0000104 .000104 .00104 .0104.0000285 .000256 .00227 .0198490505515

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"'t -.753 7.5 75.2 53e1.043 10.4 81.2 Ml1.42 11.4 85.3 551

.000240 .000252 .000816 .00778 .0759

.000240 .000263 .001097 .01068 .1049

.000242 .000367.00234 .0201 . 170

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1.000 .9521.000 .9121.000 .710

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.121c.13.4 236 20,100 937,000.62a.000.000 1.000. " +"b. -r. 31.3 2,030 94,000 808,000.80.14.000.000 1.000

c. rl +r. _r. 832 8,250 82,000 820,000.35 85.5. 008 1. 082 1. 82

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1919) OF THE ELECTRIC CIRCUIT 203As seen from the equations, and illustrated in Table I, thethermal resistance of the conductor, Tlo that is, the resistancewhich converts electrical energy into heat in the conductor, is

the true ohmic resistance at low frequencies, but with increas-ing frequency begins to rise due to unequal current distributionin the conductor (at about 1000 cycles in copper wire No. 00B. & S. gage) and approaches proportionality with the squareroot of the frequency, hence reaches values many times theohmic resistance, at very high frequencies.o 3 4 5 6 e

10 0

a R e t u m r - l u c t O r - k i - . d l J c J . - ~ ( - 2~~ b 30 Ft. a bo v eGn :>Un d a s r e t u m /.0 e No r e t u m C o n d u c t o r ~ /1 I~ /. -I ~ W /1J - -0e~ /0 " '

, '# /. ~III/~ / - -/.~ /. II.- r - -' " I rl'r-~ f8./~ / .. J .[ "( ': ,/Oi _,; ~ 'i I0 I - 7"l'1I . . . . . : : ; ~ /, V I- ' . - I, I0 \ \\ A , , I~ .:,~ I II :1c i r Xl\ r '


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204 STEINMETZ: GENERAL EQUATIONS [Feb. 20many millions of cycles, its rate of increase becomes less again,and it approaches proportionality with the first power of thefrequency, and approaches the value of radiation resistancer 2 , of the conductor without return conductor, at frequenciesof a wave length comparable with the distance of the returnconductor. The radiation resistance r3 of the conductor withreturn conductor is the larger, the greater the distance of thereturn conductor, and is proportional to this distance, withinthe range of which it is proportional to the square of thefrequency. The radiation resistance of the conductor withoutreturn conductor, at the very highest frequencies, is the sameas that of the conductor with return conductor, but, beingproportional to the frequency, with decreasing frequency, itdecreases at a lesser rate, and would even at commercial ma-chine frequencies still be appreciable, if at such frequenciesthe conditions of a conductor without return conductor couldbe realized.The total effective resistance of a conductor under trans-mission line conditions, that is, with return conductor atfinite distance, is at low frequencies constant and is the trueohmic resistance. With increasing frequency, it begins toincrease first slowly-at about 1000 cycles under transmissionline conditions-and approaches proportionality to the squareroot of the frequency, as the result of the screening effect ofthe unequal current distribution in the conductor. Then theincrease becomes more rapid, due to the appearance of theradiation resistance-at about 100,000 cycles under trans-mission line conditions-and reaches proportionality with thesquare of the frequency, at values many thousand times theohmic resistance. Finally, at the very highest frequencies-10 million cycles-the rate of increase becomes less again, andapproaches proportionality with the frequency.It is interesting to note that the external reactance of theconductor with return conductor, or radiation reactance Xa,has up to very high frequencies, millions of cycles, the samevalue Xo as calculated by the low-frequency formula, that is,by neglecting the finite velocity of the field, hence is propor-tional to the frequency. The internal reactance Xl =XIO, andis proportional to the frequency at low frequencies, but drops. behind XIO due to unequal current distribution in the conductor,and approaches proportionality with the square root of thefrequency. As, however, the internal reactance in a small part

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1919) OF THE ELECTRIC CIRCUIT 205of the total reactance, it follows, that the total reactance of theconductor and thus also the absolute value of the impedance(for all higher frequencies, where the reactance preponderates)can be calculated by the usual low frequency reactance formula,which neglects the finite velocity of the -field. Hence, theinductance L of the conductor can be assumed as approxi-mately constant for all frequencies up to millions of cycles; itdecreases only very slowly by the decreasing internal reactanceof unequal current distribution. Only at the very highestfrequencies, where the 'wave length is comparable with thedistance of the returri conductor, the inductance L decreases,and the reactance Xl + X 3 increases less than proportional tothe frequency.In a conductor without return conductor, the reactance atthe very highest frequencies is approximately the same as in aconductor with return conductor. With decreasing frequency,however, Xl + X2 decreases less than proportional to the fre-quency, that is, the inductance L increases-and becomes in-finity for zero frequency, if such were possible.Without considering unequal current distribution in theconductor and the finite velocity of the electric field, the powerfactor cos w would steadily decrease, from unity at very lowfrequencies, to zero at infinite frequency. Due to the increaseof the effective resistance, the power factor cos w first decreases,from unity at low frequency, down to a minimum at some highfrequency, and then increases again to high values at veryhigh frequencies. The minimum value of the power factor isthe lower and occurs at the higher frequencies, the shorter isthe distance of the return conductor. Thus with the return con-ductor at 6 ft. distance, the power factor is 0.43 per cent at100,000 cycles; with the return conductor at 60 ft. distance, itis 0.72 per cent; in the conductor without return conductorthe power factor is 11.2 per cent at 1000 cycles.I t is of interest to determine the effect of size, shape andmaterial on the high-frequency constants of a conductor.These high-frequency constants are, per unit length ofconductor:

In tern al c on sta nts :Thermal resistance and internal reactance:rl = Xl= ~ ~ O.4" (J l . i 10-4 ohms per em, (8) (9)

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206 STEINMETZ: GENERAL EQUATIONS [Feb. 20E xtern al C01 t8 tan t8 :Radiation resistance:

8".2r l' .r a = S 10"'"ohms per cm.E xterna l reacta nce:

2". l'Xa = 4". J log --r;- 10"'"ohms per cm.(16)

(19)These approximations hold for all but the very highest,and very low frequencies, that is, are correct within the fre-quency range with lower limit of about 1000cycles, and upper

limit of about 10million cycles. Thus they apply for all thosehigh frequencies which are of importance in the disturbancesoccuring in industrial circuits with the exception of the lowestharmonics of low-frequency surges.The constants of the conductor material enter the equationsonly as the ratio ~ , permeability to conductivity, in the'Yinternal constants r) and Xl' Thus higher permeability hasthe same effect in increasing the thermal resistance as lowerconductivity, and for instance, a cast silicon rod of permeability~ = 1, and conductivity 'Y = 55, has the same high-frequencyresistance and reactance, as an iron rod of the same size, ofwrought iron, of permeability ~ = 2,000 and conductivity'Y = 1.1 X 1(}6',hat is, of the same ~ = 0.0182,though the'Ylatter has 2000 times the conductivity of the former.Provided however, that size of conductor and frequency aresuch as to fulfill the conditions under which equations (8) and(9) are applicable, which is, that the conductor is large comparedwith the depth of penetration of the current into the conductor:

l" = ". v' 0.4 'Y ~JThus an iron rod of two inches (five cm.) diameter has at

one million cycles the same thermal resistance as a silicon rodof the same size: 0.17 ohms per meter, since the depth ofpenetration is I" = 0.00034 em, for iron, 0.68 cm. for silicon,thus in either case small compared with radius of the con-ductor l r = 2.5 cm.Digitized byCoogle


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1919) Of" THE ELECTRIC CIRCUIT 207At 10,000 cycles, however, the iron rod has the thermalresistance and internal reactance f1= Xl = 0.017 ohms permeter, the penetration being I, . = 0.0034 cm. thus small. For

the silicon rod, however, at 10,000 cycles the penetration isII ' = 6.8 cm. thus at the radius l, = 2.5 cm. formulas (8) and(9) do not apply any more, but it is approximately (that isneglecting unequal current distribution): f1 = 0.093 ohms permeter, or 5.5 times the resistance of the iron rod, while theinternal reactance is Xl = 0.031 ohms per meter, hence 80per cent higher than that of the iron rod of the same size.In the equations of the external constants, the radiationresistance and reactance, the material constants of the conduc-tor do not enter, and the radiation resistance and the externalreactance thus are independent of the conductor material.The dimensional constants of the conductor, size and shape,enter the equation only as the circumference of the conductorII>It, that is, only the circumference of the conductor countsin high-frequency conduction, and all conductors of the samematerial, regardless of size and shape, have the same high-frequency resistances and reactances as long as they have thesame conductor circumference. Thus a solid copper rod or athin copper cylinder of the same outer diameter as the rod, ora flat copper ribbon of a circumference equal to that of the rod,are equally good high-frequency conductors, though the hollowcylinder or the ribbon may contain only a small part of thematerial contained in the solid copper rod. Provided, however;that the thickness or depth of the conductor,( the thickness ofwall of the hollow cylinder, half the thickness of the copperribbon) is larger than the depth of penetration of current intothe conductor, which is

1 0 4I =I' 1(' v 0.4 'Y ~

In the expression of the radiation resistance, fa, neither thematerial nor the dimensions of the conductor enter, that is,theradiation resistance of a conductor is independent of size,shape or material of the conductor and depends only onfrequency and distance of the return conductor.Thus a thin steel wire or a wet string have the same radiationresistance as a large copper bar. Obviously, the thermalresistance of the former is much larger and the total effectiveresistance thus would be larger except at those very highfrequencies, at which the radiation resistance dominates.

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2 08 STEINMETZ: GENERAL EQUATIONS [Feb. 20As illustration, may be calculated, for frequencies from10thousand to 10million cyclesand for 60 cycles,the resistancesand reactances and thus the total impedance, the power factor,

. toe voltage drop per meter at 100 amperes, of various conduc-tors, for the three conditions:a. High frequency between conductors 6 ft. apart:l' = 182 cm.

b. High frequency between conductor and ground 30 ft.below conductor: l' = 1820em.c. High-frequency discharge through vertical conductorwithout return conductor: I' = CDFor the conductors:(1) Copper wire No. 00 B. & S. gage. t,= 0.463 em, 'Y = 6.2 X 106(2) Iron wire of the same size as (1):

p. = 2,000 'Y = 1.1 X 106(3) Copper ribbon of thickness equal to twice the depth ofpenetration at 10,000cycles, and the same amount of materialas (1), that is, 0.128 cm. by 5.52 cm.(4) Iron ribbon of the same size as (3).(5) Two-inch iron pipe, 7 B inch thick,This gives the depth of penetration at 1= 104 cycles:. 6.4for copper; lp = -== 0.064 cm.vIfor iron: lp = 0.34 = 0.0034cm.vITable II gives the values of To, Xo, r i o Ta, T2, XI, Xa, X2, TI + Ta,

TI + T2, XI + X., XI + X2, Z, cos w, e and lp , for! = 10' cycles.Table III gives the values of Tl + Ta, or TI + T2, Z, cos wand e, for I= 60, 104, 106, 108, 107 cycles for the five kinds ofconductors.Itis interesting to compare in Table III, the constants ofthe first four conductors, as they have the same section, repre-senting about average section of transmission conductors, butrepresent two shapes, round wire and thin flat ribbon, and twokinds ofmaterial, copper-high conductivity and non-magnetic,-and iron-magnetic material of medium conductivity.As seen, the effect of conductor shape and conductor rna-

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1919) OF THE ELECTRIC CIRCUIT 209

TABLE II.CIRCUIT CONSTANTS FOR I-1()1- ONE MILLION CYCLES, PER METER LENGTH OFCONDUCTOR.

return conductor at 6 ft. - 182em, distance.b. return conductor at 2 X 30 ft. - 1820cm. distance.c. no return conductor(1) (2) (3) (4) (5)Copper Iron Copper Iron Iron

Wlre: wire: ribbon: ribbon: pipe:---------------O l l" ---- 0.00024 0.00135 0.00024 0.00135 0.~197A . . , ( 2 r II' I. 1 ' ) a. 7.8 635 3.05 173 81 ohm.'" -0.4rl 101---+-- 1O.,- . 104 b. 10.8 638 5.95 176 84100r~ 0.00865 0.00233 0.248 ,--- --10" - 0.92 0.167, . . . ,0.8 ...p,' { . . 0.048 0.048 0.048 0.048 0.048 r. - 10" -S b. 0.48 0.48 0.48 0.48 0.48 '. -0.2"'/10"- c. 1.97 1.97 1.97 1.97 1.97 100 r V fJT'j]

0.0086& 0.92 0.00233 "--- --10-- 0.248 0.167, . . . ,2 r/I' { a. 7.5 7.5 2.97 2.97 2.47 .. -O.4r/log ---10"- b.l0.4 10.4 5.87 5.87 5.37". -0.4r/(10II ,: -0.5772 ) 1 0 " - c.l1.2 11.2 9.5 9.5 8.9

{ a. 0.0567 0.97 0.0503 0.296 0.215 ohm.r-rl+r.- b. 0.489 1.40 0.482 0.728 0.647 -r.+r,- c. 1.98 2.89 1.97 2.22 2.14 -s,+%.- { . . 7.5 8.42 2.97 3.22 2.64 .10.4 11.32 5.87 6.12 5.64 . -'" + r, - c.11.2 12.12 9.50 9.75 9.07 { a. 7.5 8.46 2.97 3.23 2.65 -","+sI- b.10.4 11.4 5.89 6.16 5.58 . 11.4 12.45 9.7 10.0 9.3 r { a. 0.76 11.5 1.7 9.2 8.1 per.fentCOI_- - - - b. 4.7 12.3 8.2 11.8 11.6 c.17.3 23.3 20.3 22.2 23.0 .. { a. 750 846 296 323 265 votta - 100. - b. 1040 1140 589 616 558c. 1140 1245 970 1000 930

j ' p -1()1

.. "0.4..,,./ - 0.0064 0.00034 0.0064 0.00034 0.00034 em.

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210 STEINMETZ: GENERAL EQUATIONSt-Ot-IQN2 i g 2 i g g:l~:; ;~:! : .....-4~"""NNSi!2!g' l :g~~~"""NN ....~ .. g~;ZO:::ogg ....00000

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~ $ 2 " ' ' ' ' 1 N;:::N8~:::.....00000

C'IIt-C'f)t-O~ ~ t i ~ ~('I)C')C"I')NO

~ ~ g ~ ~& 0 , . . . . . .000~~g~~~O&o('I)lt)~~g; t ; f ; j

[Feb. 20

It)C'),...,._cq" ' ; M o a C ' i" ' ;. . . . 1 : ' 1 . ..-I . .. ..

. . .. .. . . . .su. g"oU

o'"

U).&OC'I')O~ici~~~~~;!~0"';000g .. I , 1 C8g~aa . . . .00000a C N ~ = M8S ....00000

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0000000t i g ~ ~ ~......'NCIOCC...... .. ~ o o : : :: : :~ = ~ . a oo ~ o c - o i =. . . . .CIOOIQC')O~~~~~

lQOO~"'C"l0"'; '" a c i r - :. . . .~~~!i!~0....00

. . . . .:8 ... ~8:2~~c : :~~00000.=00....o : i . , ; : ! i ~ g.....0 : : : . . . . = 0 0~----u&~~~~~,S 0~0~!: !i~~~~~

J!"0>cc ....o .0. S " ' : 2 i " ; " , . . ;.;, ; 1 & Of)J0a : : n s 2 1'00"";000>----

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1919) OF THE ELECTRIC CIRCUIT 211terial is very great at machine frequencies, 60 cycles, butbecomes small and almost negligible at extremely high fre-quencies. This is rather against the usua l assumption.The reason is that at machine frequencies, the unequal currentdistribution, or the screening effect of the internal magneticfield, is still practically absent in copper conductors, evenin round wires of medium size, while it is practically completein iron conductors, even in ribbon of 1/2o-inch thickness;while at very high frequencies the effect of radiation prepond-erates, which is independent of the material, and the radiationresistance even independent of the shape of the conductor.Thus under transmission line conditions, first and secondsection of Table III, at 60 cycles the impedance, and hencethe voltage drop in the iron conductor is from 7 to 20 times thatof the copper conductor; at 10,000 cycles the voltage drop inthe iron conductor is only from 1.5 to 2.5 times that in thecopper conductor; the difference has decreased to from 14per cent to 44 per cent at 100,000 cycles, 5 per cent to 12.5per cent at one million cycles, while at 10 million cycles thevoltage drop in the iron conductor is only 3 to 5 per cent higher,thus practically the same and the only difference is that dueto the conductor shape.The effective resistance, and thus the power consumptionin the iron conductor at 60 cycles is from 8 to 30 times that ofthe copper conductor, but with increasing frequency thedifference in the effective resistance increases to from 88 to106 times at 10,000 cycles, reaches a maximum, and thendecreases again, and is from 15 to 65 times that of the copper

conductor at 100,000 cycles, only 1% to 17 times at a millioncycles, while at 10 million cycles and above, all the differencesin the effective resistance practically disappear.As the result, the power factor of the conductor,-being thesame, 100 per cent, at extremely low frequency and not muchdifferent, and fairly high at machine frequency-decreaseswith increasing frequency, reaches a minimum and then in-creases again to considerable values at extremely high fre-quency,-where the high radiation resistance comes into play.The difference between iron and copper, however, is that theminimum value of the power factor, at medium high fre-quencies, is very low in copper, a fraction of one per cent, whilein the iron conductor the power factor always retains consider-able values, the minimum being 10 or more times that of the

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212 STEINMETZ: GENERAL EQUATIONS [Feb. 20copper conductor. Thus an oscillation in an iron conductormust die out at a much faster rate then in a copper conductorand the liability of the formation of a continual or cumulativeoscillation may exist in copper conductors but hardly in ironconductors.The effect of the shape of the conductor on the impedanceor voltage drop is fairly uniform throughout the entire fre-quency range, the voltage drop being the smaller, the larger thecircumference.With regards to the effective resistance, however, the effectof the conductor shape is considerable at very .low frequenciesin iron conductors, but absent with copper conductors, dueto the absence of the screening effect in copper at lowfrequency.With increasing frequency, the difference appears in theeffective resistance of the copper conductor also, with theappearance of unequal current distribution, and the ratioof the resistance of the round conductor to that of the flp.tconductor approaches the same value in copper as in iron.With the approach of very high frequency, however, the dif-ference decreases again, with the appearance of radiationeffect, and finally vanishes.Thus to convey currents of extremely high frequency, aniron' conductor is almost as good as a copper conductor of thesame shape and cross section. As iron is very much cheaperthan copper, it follows that in high-frequency conduction aniron conductor under the conditions of Table III should bebetter than a copper conductor of the same cost and the samegeneral shape, due to the larger size or rather circumference ofthe iron conductor.There is, however, a material advantage at extremely highfrequencies as well as at moderately high frequencies, in lowervoltage drop at the same current resulting from such a shapeconductor as gives maximum circumference, such as ribbon orhollow conductor. This advantage of ribbon or hollow tube,over the solid round conductor, exists also in the resistance andthus power consumption at medium high frequencies, but notat extremely high frequencies, but at the latter, in power con-sumption all conductors, regardless of size, shape or material,are practically equal.With the thickness of ribbon conductor considered inTable III, of about 1/20 inch, which is about the smallest me-chanically permissible under usual conditions, the screening

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1919) OF THE ELECTRIC CIRCUIT 213effect even in copper conductors is practically complete at10,000 cycles, that is, the depth of penetration less than onehalf the thickness of the conductor. It follows, that in thedesign of high-frequency conductor, the thickness of the ribbonor hollow cylinder is essentially determined by mechanical andnot by electrical considerations; in other words, the thinnest .mechanically permissible conductor usually is thicker thannecessary for carrying the current. ABiron usually cannot beemployed in as thin ribbon as copper, due to its rusting, aniron conductor would have a larger section than a copperconductor of the same voltage drop and power consumption.Thereby a part of the advantage gained by the employment ofthe cheaper material would be lost.The last section of Table III gives the constants of the con-ductor without return conductor, such as would be representedby the discharge circuit of a lightning arrester, by a wirelesstelegraph antenna, etc., while the first two sections correspondto transmission line conditions; high-frequency currents be -tween line conductors and between line and ground.In the third section of Table III, the 60-cycle values are notgiven, and the values given for the lower high frequencies,10 and even 1()5cycles, usually have little meaning, are rarelyrealizable; they would correspond to a vertical conductor, aslightning arrester ground circuit, under conditions where noother conductor is within quarter wave distance. Even at 10'cycles, however, the quarter wave length is still 750 m. Thusthere will practically always be other conductors within thefield of the discharge conductor, acting as a partial return con-ductor, and the actual values of impedance and resistance, thatis, of voltage drop and power consumption in the conductor,thus will be intermediate between those given in the thirdsection, for a conductor without return conductor, and thosegiven in the first sections, for conductors with return conduc-tors. Except at extremely high frequencies, at which the wavelength gets so short that the condition of a conductor withoutreturn conductor becomes realizable. Itis interesting to note,therefore, that at extremely high frequencies, the constants ofthe conductor without return conductor approach those of theconductor with return conductor. At lower high frequencies,impedance .and resistance, and thus voltage drop and powerconsumption, of the returnless conductor are much higher than.~those of the conductor with return, the more so the lower thefrequency.

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214 STEINMETZ: GENERAL EQUATIONS [Feb. 20However, while the case of the conductor without returnconductor is not realizable at low and medium high frequenciesin industrial circuits, it probably is more or less realized by

the lightning discharge between ground and cloud, and theconstants given in the third section of Table III would probably. approximately represent the conditions met in the con-ductors of lightning rods such as used for the protection ofbuildings against lightning.Itis interesting to note that with such a conductor withoutreturn conductor, the power factor is always fairly high, evenwith copper as conductor material. The impedance and thusthe voltage drop does not differ much from those of the con-ductor with return conductor. The resistance, however, andthus the power consumption are much higher, sometimes, incopper conductors, more than a thousand times as large, show-ing the large amount of energy radiated by the eonductor-which reappears more or less destructively as "induced light-ning stroke" in objects in the neighborhood of the lightningstroke.

1I1.-Wave Decay in Tranamiaaion LineaFrom the equations given in Appendix C, numerical valuesof the line constants are calculated and given in Table IV, foraverage transmission line conditions, that is, a copper wire,No. 00, with 6 ft. = 182 em, between the conductors, and anaverage height of 30 ft. = 910 cm. above ground, for the twoconditions:a. a high-frequency oscillation between two line conductors;b. a high-frequency oscillation between one line conductorand the ground.The table gives:The thermal resistance T., the radiation resistance T" and thetotal resistance T = TI + Ta;The internal reactance Xh the external reactance Xa and thetotal reactance x = XI + Xa.The Magnetic attenuation UI = ~L' the dielectric attenu-

ation Ut = 2~' and the total attenuation U= Ul + Ut.The table also gives the duration of a transient in micro-seconds t, and in cycles N, that is, the time which a high-he-

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~ 10 7.5 X 10' I 10' 2 X 10' II X 10' Cycle.599 5.875 6 6.301 6.699,~.12 7.50 8.65 12.25 19.3 ohms3.12 7.50 8.65 12.25 19.3 .i! 27 4 8 192 1,200 ' .IU 34.5 56.6 20f 1,219" 5,625 7,500 15,000 37,5003 9 5,632 7,509 15,012 37,520l.1 5 1.195 1.195 1.195 1.195 henrys

14,400 23,700 85,600 510,000

11,200 20,000 80,000 500,000) 25,650 43,700 165,600 1,010,0001.099 4.409 4.640 5.219 6.004~.6 39.0 22.9 6.04 1.00 micro-seconds. 8 29.2 22.9 12.1 5.0 cycles. 4 8 .61 .75 1.36 3.25 per cent.32 ,48 .64 1.27 3.18 per cent

270 480 1,920 12,000 ohms~.1 '1 t17.5 488.6 1,932 12,020~ 7.800 10,400 20,800 52,0007,807 10,409 20,812 52,0201.655 1.655 1.655 1.655 1.655 henry.~ 83.800 147,500 583,000 3,630,000

81,000 144,000 576,000 3,600,000) 164,800 291,500 1,159,000 7,230,0001870 5.217 5.465 6.064 6,8593.5 6.07 3.43 .86 .14 micro-seconds5.75 4.55 3.43 1.72 .69 cycles~.42 3.55 4,70 9.24 22.5 per cent~.31 3.46 4.62 9.22 22.5 per cent

[STIi:INMBTZ)

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, r '

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1919) OF THE ELECTRIC CIRCUIT 215quency oscillation of the frequency J would last, if continuingWIth its initial intensity, and the number of cycles, which itwould perform. It also gives the power factor, in per cent, ofthe series circuit, as determined by resistance and inductance,and of the shunt circuit, as determined by shunted conductanceand capacity.As seen, the attenuation constant u is constant up to nearlyone thousand cycles. Thus in this range, all the frequenciesdie out at the same rate. From about one thousand cyclesup to about 100,000cycles, the attenuation constant graduallyincreases, and thus oscillations die out the more rapidly, the

higher the frequency, as seen by the gradual decrease of theduration t. However, as the increase of the attenuation con-stant and thus the increase of the rapidity of the decay of thedisturbance, in this range, is smaller than the increase of fre-quency, the number of cycles performed by the oscillationincreases. Thus, at 25 or 60 cycles, the stored energy,whichsupplies the oscillating power, would be expended in less thanone cycle, that is, a real oscillation would hardly materialize(except by other sources of energy, as the stored magneticenergy of a transformer connected to the line). At 1000cycles,the oscillation would last 9 to 12 cycles, and at still higher fre-quencies reach a maximum of 41.4 cycles at 20,000 cyclesfrequency, in the oscillation against ground; 64.9 cycles at100,000 cycles, in the oscillation between line conductors.This represents a fairly well sustained oscillation, in whichthe cumulative effect of successive cycles may be considerable.Above 100,000 cycles the attenuation constant begins to riserapidly, and reaches enormous values, due to the rapidlyincreasing energy dissipation by radiation. As the result, theduration of the oscillation very rapidly decreases, and thenumber of cycles performed by the oscillation decreases, until,beyond a million cycles, the energy dissipation is so rapid, thatpractically no oscillation can occur; the oscillation dying outin a cycle or less, thus being practically harmless.The attenuation constant is plotted, up to 15,000 cycles, inFig. 2, with the frequency as abscissas, and is plotted in Fig. 3in logarithmic scale, as (3).Noteworthy is the great difference between the oscillationagainst ground, and the oscillation between line conductors.The oscillation against ground is more persistent at low fre-quencies, due to the greater amount of stored energy in the

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216 STEINMETZ: GENERAL EQUATIONS [Feb. 20electric field of the conductor, which reaches all the distanceto the ground. When reaching into very high frequencieshowever, the energy dissipation by radiation becomes appre-ciable at lower frequencies in the oscillation against ground,than in the oscillation between line conductors, and reachesmuch higher values, with the result that the decay of an oscil-lation between line and ground is much more rapid at highfrequencies than the decay of an osciIlation between line con-ductors. For instance, at 100,000 cycles, the latter performs

4W,,------~--~----,_,_,,7T~- " '; '( f:~) t V(2) 6 Fl.D i st an ce b etwe e n Conductors . __V(3 ) C onducto r 3 0 F t. abo ve G ro und 7r - 7

360 -- -- - -- - -----~ i 7 ' 1 7320 -- - - -H-++/+-+-+-,A-V-+--I-+--l~-+-+--f-~f-/-A/'-+--+ -jf- --- r--

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1919] OF THE ELECTRIC CIRCUIT 217ductors,corresponding about to average distribution conductors,in 2 with 6 ft. = 182 cm., between conductors, correspondingto about average transmission line conductors with the oscil-lation between two lines, and in 3 with 60 ft. = 1830 cm.between conductor and return conductor, corresponding to anoscillation between line and ground, under average transmissionline conditions, with the conductor 30' ft .. above ground. 4,5 and 6 give the same condition of an oscillation between lineand ground, but in 4 an iron wire of the size of No. 00 B. & s.

C Y C L E S10' 10' 10' 10'

3.

--II N o .oo B..t: s .Gqe Copper , I II I n . Distant -~No.oo B..t: s .Gqe Co p pe r, 6 F LDi ll ta nt3 ) N o.o o B..t: s .eaae Coppe r , 30 Ft a bo ve G ro un d i4) N o.o o B..t:S .Gap I r o n . 3 O Ft. abo ve G ro w>d f-$) No.4 B.& s .Gqe Coppe r , 30 Ft. above Ground- ~~~~!) J t z lV I - r L1-1--- .-~--r- --r r-V, ~-Ir-l/il~ -/. (3 (2),

V I /V(l)_f-./ / / e-f-:'IV) c / ~ VV ,-- ~ -5 ~ ----_.- j f- t-r- ---, , - L - f . 2 . ~ v - -0 : L ~ 0V J _ --1l- f-- --- - _ -5 ~ ~ -- __ I _ - --(6) ~ ;/'o~ ~

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218 STEINMETZ: GENERAL EQUATIONS [Feb. 20accuracy for practical purposes, and the lower limits, wherethe effects become appreciable, in the various conductors.In Fig. 3 the attenuation constants are plotted, in Fig. 4the power factors and in Fig. 5 the duration, in cycles.As such a transient oscillation dies out exponentially, the-oretically it has no definite duration, but lasts forever, thoughpractically it may. have ceased in a few micro-seconds. Thusas duration is defined the time, or the number of cycles, whichthe oscillation would last if maintaining its initial intensity.In reality, in this time the duration has decreased to _1_, orE37 per cent of its initial value. Physically, at 37 per cent ofits initial value, or 0.372 = 0.135 of its initial energy, it hasbecome practically harmless, S o that this measure of durationprobably is the most representative.From Tables V and VI it is seen that there is no markeddifference between the stranded aluminum conductor (6), andthe solid copper wire of the same conductivity (3), and the val-ues of (6) are not plotted in the figures, but may be representedby (3).The attenuation constant u, in Fig. 3, is plotted in logarith-mic scale, with /f as abscissas. In such scale, a difference ofone unit means ten times larger or smaller, and a straight linemeans proportionality to some power of the frequency. Thisfigure well shows the three ranges; the initial horizontal rangeat low frequency, where the attenuation is constant; the ap-proximately straight moderate slope of medium frequency,where the attenuation constant is proportional to the squareroot of the frequency, the unequal current distribution in theconductor predominating, and the steep slope at high fre-quencies, where the radiation resistance predominates, whichis proportional to the square of the frequency.It is interesting to note that at high frequencies the dist-ance of the return conductor is the dominating factor, whilethe effect of conductor size and material vanishes; in copperwire No. 00 the rate of decay is practically the same as in copperwire No.4, though the latter has more than three times theresistance, or in the iron wire, which has nearly s ix times theresistance and 200 times the permeability. The permeabilityof the iron wire has been assumed as ~ = 200, representingload conditions, where by the passage of the low-frequency

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1919) 0[.' THE ELECTRIC CIRCUIT 219power current the iron is magnetically near saturation, and itspermeability thus lowered. However, the decay of the oseil-lation between conductor and ground is s ix to seven timesmore rapid than that between conductors 6 ft. apart, and thatbetween conductors 18 inches apart about three times less.This shows, that to produce quicker damping of high-frequency waves, such as are instrumental in steep wave fronts,the most effective way is to separate the conductors as far aspossible, perhaps even lead them to the station by separatesingle-conductor lines; but the use of high-resistance conductors,or of magnetic material, as iron, offers little or practically noadvantage in damping very high frequencies or flattening steepwave fronts.At medium and low frequencies however, the relation re-verses, and the decay of the wave is the smaller the greater thedistance of the return conductor. The reason is, that in thisrange the effective resistance is still independent of the conduc-tor distance, while the inductance increases with increasingdistance. At medium and low frequencies, the iron conductoroffers an enormously increased attenuation-from 10 to 20times that of non-magnetic conductors.The power factor of the conductor is plotted in Fig. 4. Asseen, it decreases from unity, at very low frequencies, to aminimum, at medium high frequencies, and then increasesagain to very high values at very high frequencies. Theminimum value is a fraction of one per cent except with theiron conductor, where the minimum is very much higher.The power factor is of importance as it indicates the percentageof the oscillating energy, which is dissipated per wave of oscil-lation. This is represented still better by Fig. 5 the durationof the oscillation in cycles, that is, the number of cycles whichan oscillation lasts before dissipating the stored energy whicheausesit.At medium high frequencies, the oscillation is the morepersistent the lower the ohmic resistance of the conductor andthe further away the return conductor, while at very high fre-quencies the reverse is the case, and the oscillation is the morepersistent, the shorter the distance of the return conductor,whilethe size and material of the conductor ceases to have anyeffect.The maximum number of cycles is reached at medium highfrequencies,in the range between 20,000 and 100,000 cycles-

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220 STEINMETZ: GENERAL EQUATIONS [Feb 20depending on conductor size and distance of return conductor.It thus is in this range of frequencies, where an oscillation causedby some disturbance lasts the greatest number of cycles, thatthe possibility, by some energy supply by means of an arc,etc., to form a stationary oscillation or even a cumulative oscil-lation, thus to become continuous or "undamped", is greatest.It would thus appear, that this range of frequencies, of20,000 to 100,000, represents what may be called the "danger

FlO. 4-POWER FACTOR OF SERIES CONSTANT Cos", _ Tv' J1 + (2 ..!L)'(TABLE VI)

frequencies" of transmission systems. It is interesting to note,that experimental investigations have shown that the naturalfrequency of oscillation of the high-voltage windings of largepower transformers usually is within this range of dangerfrequencies. The possibility of the formation of destructivecumulative oscillations or stationary waves in the high-voltagewindings of large power transformers is therefore greater thanprobably with any other class of circuits, so that such high-Digitized byCoogle


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1919J OF THE ELECTRIC CIRCUIT 221potential transformer windings require specially high disruptivestrength and protection. This aeeounts :for the not infre-quent disastrous experience with such transformers, beforethis matter was realized.Fig. 5a lso shows, that the duration of an oscillation in ironwire, in cycles, is very low at all frequencies. Thus the for-mation of a stationary oscillation in an iron conductor ispractically excluded, but such conductors would act as a deadresistance, damping any oscillation by rapid energy dissipation.The duration ofhigh-frequency oscillations, in cycles,increaseswith increasing frequency, to a maximum at medium fre-

FIG. 5-DuRATION OF OSCILLATION IN CYCLES (TABLE VI)quencies. This obviously does not mean that the time duringwhich the oscillation lasts increases; the time, in micro-seconds,naturally decreases with increasing frequency, due to the in-creasing attenuation constant. Thus in conductor (2) forinstance, the oscillation lasts 65 cycles at a frequency of 100,000cycles, but only 9 cycles at 1ooo-cycle frequency. However,the 65 cycles are traversed in 650 micro-seconds, while the 9cycles last 9000 micro-seconds, or 14 times as long. It is notthe total time of oscillation, but the cumulative effect due tothe numerous and only slowly decreasing successive waves,which increases as represented in Fig. 5.

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2 2 2 STEINMETZ: GENERAL EQUATIONS [Feb. 20An oscillation between copper wires No.4. B. & S. gage

6 ft. apart, would give a duration curve, which at moderate,frequenciea follows (5) of Fig. 5, but at high frequenciesfollows (2). Thus the average duration and average rate ofdecay would be about the same as (3), an oscillation betweencopper wire No. 00 and ground. However, the oscillationwould be more persistent in such a conductor at high, and lesspersistent at low frequencies. A complex wave, containingall the harmonics from low to very high ones, such as a steepwave front impulse or an approximately rectangular wave,as may be produced by a spark discharge, etc, would haveabout the same average rate of decay in a copper wire No. 4with return at 6 ft., as in a copper wire No. 00 with groundreturn. The wave front would flatten, and the wave round off,approaching more and more a sine shape, due to the more rapiddisappearance of the higher frequencies, while at the same timedecreasing in amplitude. The wave thus would pass throughmany intermediate shapes. But these intermediate shapeswould be materially different with wire No.4 and return at6 ft., as with No. 00 and ground return; in the latter, theflattening of the steep wave front, and rounding of the wave,would be much more rapid at the beginning, due to the shorterduration of the transient, and while such wave would lastabout the same time, that is, pass over the lines to about thesame distance, it would carry steep wave fronts to muchshorter distances, that is, its danger zone would be materiallyless than that of the wave in copper wire No.4 with return at6 ft.It therefore, is of great interest to further investigate theeffect of the changing attenuation constant on complex waves,and more particularly those with steep wave fronts, as therectangular waves of starting or disconnecting lines, etc.

IV.-Attenuation of Rectangular WavesThe destructiveness of high frequencies or steep wavefronts in industrial circuits is rarely due to over-voltage be-tween the circuit conductors or between conductor and ground,

but is due to the piling up of the voltage locally, in inductiveparts of the circuit, such as end turns of transformers orgenerators, current transformers, potential regulators, etc., orinside of inductive windings as the high-potential coils ofpower transformers, by the formation of nodes and wave crests.

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1919) OF THE ELECTRIC CIRCUIT 223Such effect may be produced by high-frequency oscillationssustained over a number of cycles, as discussed in III. asoscillations lasting only a very few cycles or a fraction of a cycle,or due to non-oscillating transients, as single impulses, etc.As the high rate of change of voltage with the time, and thecorrespondingly high voltage gradients along the conductorare the source of danger, to calculate and compare oscillatoryand non-oscillatory effects in this respect, it has become cus-tomary in the last years to speak of an "equivalent frequency"of impulses, wave fronts or other non-oscillatory transients.As "effective" or "equivalent" frequency of an impulse.

wave front etc., is understood the frequency of an oscillation,which has the same maximum amplitude, e or i, and the same. di t de di Th . . ulsemaxnnum gra len (it or ---;]f' us assuming an Imp

which .reaehes a maximum voltage e = 60,000, and has amaximum rate of increase of voltage of ~ = 1 0 1 0 , that is, amaximum voltage rise at the rate of 1010 volts per second, or10,000 volts per micro-second. As the average voltage r iseof a sine wave is _ !_ times the maximum, the average rise of1 1 'an oscillation of t he same maximum gradient as the impulsewould be

2 de 20 ,000~= 1 1 '

volts per microsecond. The total voltage r ise of e = 60,000thus would occur in

e - 6 0 , 0 0 0 1 1 '_ 9 4 . d-2-~ - 20,000 - . mlcro-secon s.7dtA complete cycle of this oscillation thus would last 4 X 9.4~37.6 micro-seconds, and the equivalent frequency of the im-

purse would bef = i 7 O : S =26,600 cycles or 26.6 kilo cycles.

The equivalent frequency of a perfectly rectangular wavefront lf such could exist, obviously would be infinity.Digitized byCoogle


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224 STEINMETZ: GENERAL EQUATIONS [Feb. 20A . Q\1ARTER WAVE CHARGING OR DISCHARGING OsciLLATION

OF LINEConsidering first the theoretically rectangular wave of con-necting a transmission line to a circuit, or disconnecting itfrom the circuit.Suppose a transmission line, open at the distant end, isconnected to a voltage E. At this moment, the voltage of theline is zero. It should be, in permanent conditions, E. Thusthe circuit voltage consists of a permanent voltage E (which isthe instantaneous value of the alternating supply voltage atthis moment Eosin '" and the transient voltage - E. We thus

have a transient voltage, which uniformly = - E all along theline, except at the switching point l = 0, where the transientvoltage is zero.Or, suppose a transmission line, open at the far end, isconnected to a source of voltage, and at the moment wherethis voltage is E, the line short circuits at some point, by aspark discharge, flash-over, etc. Thus at this moment, thevoltage = 0 at the point of short circuit, and is = E every-where between this point and the end of the line. Thus weget a line discharge leading to the same transient, a theoret-ically rectangular wave. In the part of the line betweengenerator and short circuit, we have a different transient, acircuit of voltage e = 0 at one end, e = E throughout the entirelength at time t = 0, and e = E continuously at the otherend, where the generator maintains the voltage. However,this again leads to the same transient, of a theoreticallyrectangular wave. .Assuming thus, as an instance, a transmission line of 100-km.length, of copper wire No. 00 B. &S. gage, 30 ft. above ground,open circuited at the other end l =100 km., and connected toa source of voltage E at the beginning, l = o . Then the beginning of the line, l = 0, is grounded, at thetime t = 0, thus giving a quarter wave oscillation, with theterminal conditions:Voltage along the line constant =E, at time t = 0, exceptat the beginning of the line, l = 0, where the voltage is o .Current along the line = 0 at t = 0, except at l = 0, wherethe current is indefinite.The equation of the quarter wave oscillation of the lineconductor against ground, as usually. given), then is:1. See for instance. "Theory and Calcula.tion of Transient ElectrioPhenomena," Section IV, Chapter VII, equa.tion (57).

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1919) OF THE ELECTRIC CIRCUIT 2250 1 1e = 4 E E - ' ~ " sin (2 n + 1) T cos (2 n + 1) 8 (1)21" ~ n+1o

where:8 is the time angle of the fundamental wave of oscillation, offrequency:S 3 X 101010 = 4 lo = 4 X 100 X 10' = 750 cycles. (2)

T is the distance angle, for lo= 100 km. =90 = 21"/2,that is:8 = 221"10t

2 I " lT = 21; I (3)Equation (1) however assumes that tt, and thus r, L, C andg are constant for all frequencies. As this is not the case, butuis a function of the frequency, and thus of n: E-' cannotbe taken out of the summation sign. Equation (1) thus mustbe written:

e= 4E ~"E.n' sin(2.n+1)Tcos(2n+1)8 (4)21" ~ 2n+1owhere u" is the value of u for the frequency: 1 = (2 n + 1)10.From (4) follows, as the voltage gradient along the line:

0 1 1

~; = 4: ~" E-IIn' cos (2 n + 1) T cos (2 n + 1) 8 (5)o

2E:=o

G O+ ~" E-n' cos (2n - 1) (T - 8)} (6)o

The maximum voltage gradient occurs at the wave front,that is, for 8:= T. Substituting this, and substituting further,from (3):

( 7 )

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226 STEINMETZ: GENERAL EQUATIONS [Feb. 20gives, as the maximum voltage gradient:~ ~G= ~~ = f {~nE -unICOS(2n+1 )2T+ ~n Eopnl}

o oIt is however:

CI O

~" cos (2 n + 1) 2T = 0o

for all values ofT except r = 0 and T = 1(',that is, the beginningof the line, l = o .Thus, approximately, as u" varies gradually:~

~" E-Un I cos (21(' + 1) 2T = 0 (10)o

except for values of T = 0 or very near thereto.Substituting (10) into (8) gives:

oas the approximate expression of the maximum voltage gra-dient, that is, the steepness of the wave front, at time t, that is,at distance from the origin of the wave:

l = S t = 3 X 1010 t (12)If 8 differs materially from T , the term with ( T - 8) in

equation (6) also vanishes, and :: = 0, that is, there is novoltage gradient except at and near the wave front.From (11) are now calculated numerical values of thesteepness of the wavefront G, for various times t after itsorigin, and thus by (12), various distances l from the origin.These numerical values are given in Table VII, for E =60,000volts.At a fundamental frequency of f o = 750 cycles, successiveharmonics differ from each other by 1500 cycles, and for everyvalue of t, values of E -UnIthus have to be calculated for thefrequencies:n = 0 1 2 3 4 5 6 etc.f = 750 2250 3750 5250 6750 8250 9750 cycles etc.,

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1919) OF TilE ELECTRIC CIRCUIT 227until the further terms add no further appreciable amount to2: E-lln I. In calculating, it is found that for instance for t =5microseconds, or l = 1.5 lan., this occurs at f = 5 X 10' cycle,

5 X 106thus beyond the 750 = 6670th harmonic. Thus 6670terms of the series would have to be calculated to get this onepoint of the wave gradient: more terms for shorter, less termsfor longer distances l of wave travel. This obviously is impos-

TABLE VIIATTENUATION OF WAVE FRONT OF OUARTERWAVE OSCILLATION, OF

100 KM. LINE, 60,000 VOLTS.

(3) Copper wire No. 00 B. & S. gage 30 (t. - 910 cm. above ground.o 0 I 0 00 60,000 00 0.03 .01 .~2 5,800 5,500 8,800 6.575 .1725 .107 10 X 10' 1,370 1,290 2,060 731.15 .345 .215 8 X 10' 1,02:; 967 1.540 982.3 .69 .43 6 X 10' 670 630 1,000 15011.5 3.45 2.15 4 X 10' 300 280 450 33023 6.9 4.3 2 X 10' 218 205 330 46092 27.6 17 2 10' 104 98 156 960230 69 43 .7 X 10' 62 58 92 1,630

2,300 690 430 .2 X 10' 13 12 19 7,90023.000 6,900 4,300 50,000 1. 3 1.2 1.9 79,000(I) Copper wire No. 00 B, & S, gage. return conductor at 18 ins. -45.5 cm distance.2.3 I .69 I .43 120 X 10' 13,450 1 3,240 I 5.160 2923 6.9 4 3 10 X 10' 1,020 960 I 1.530 98

230 I 69 43 4 X 10' 250 234 370 400(5) Copper wire No, 4 B. & S gage, 30 (t. - 910 em. above ground,23 I 6.9 I 4.3 I 2 X 10' I 227 I 213 I 340230 69 43 .7 X lQ1 64 60 94(4) Iron wire No. 00 B, & S. gag, 30 Ct. ~ 910 cm. above Rround,23 I 6.9 I 4.:1 I 2 X I ( ) I I 143 I 135 I 21:;230 69 43 .3 X 1()1 11 10 16

4401.570

Time I, I I I I Gradient I Wave Front:micro- Distance I f. where Voltage G. Equiva-second... km: I miles: 2: E-tli . 2 . . . d . . . ! _ volts per lent kilo- Length,(rom origin of wave. vanishes E dl I meter, cycles. I meters,

7009,400

sible, and some simpler approximation, of sufficient accuracy,thus is required.This may be done as follows:In the range from 5 X 106 to 106 cycles for instance, there10' - 5 X 105 .are 1500 = 333 harmonics. Instead of calculating u

and E-'" for each of these harmonics, calculate E-'" for theaverage value of these 333 harmonics, and multiply by 333.Thus, dividing the entire frequency range (beyond thelowest harmonics, which are calculated separately), into groups,

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228 STEINMETZ: GENERAL EQUATIONS [Feb. 20and calculating one average value for each group, the calcula-tion of 1: e-'" becomes feasible.Since e-' is calculated through / e-tll = - u t / E, as'values of t,multiples of / ! have been chosen, to still furthersimplify the calculation, in deriving a curve of gradients G.*Table VII gives the values of the maximum voltage gradientof the wave front, in volts per meter at 60,000 volts maximuminitial line voltage, the equivalent frequency of the wave front,in kilo-cycles, and the length of the wave front, in meters, forvarious times of wave travel, from 0.03 micro-seconds up, andcorresponding distances of wave travel, from 10 meters fromthe origin as rectangular wave, up to thousands of kilometers,

TABLE VIII.ATTENUATION OF WAVE FRONT OF QUARTERWAVE OSCILLATION. OFlOO-KM. LINE. 60.000 VOLTS.

Conductor: (3) (I) . (6) (4)Size No.: 00 00 " 00Material: Copper Copper Copper Iron---Distance of return conductor. cm.t ... 1820 "6.6 1820 1820After 23 miC1'OHCOnd.,6.9 Inn.:Gradient, volta per meter: ......... 206 960 213 136Length of wave front, meters: ....... o i 6 O 98 " o i O 700Equivalent kilocyclea: .............. 330 1630 3 o i O 216After 230 miC1'OHCOud 69 Inn.:Gradient, volta per meter: ....... 68 2M 60 10Lencth of ...ave front, meters: ....... 1630 o i O O 1670 WOEquivalent kilocycles: ............... 92 370 So l 16for copper wire No. 00 B. & S. gage, 30 ft. = 910 cm. aboveground, with the ground as return.For comparison are given some data of the same conductor,with the return conductor at 18 inches= 45.5 cm., and alsofor a copper wire No.4, and an iron wire No. 00, with theground as return.These data are plotted in Fig. 6, showing the wave front, asit gradually flattens out in its travel over the line, from thevery steep wave at 172 meters from the origin, to the wavewith a front of 1630meters, 69 km. away.A comparison of the data of the four circuit conditions isgiven in Table VIII, and plotted in Fig. 7.Its interesting to note, that there is practically no difference*Here and in the following, the symbol/has been chosen for theoommon logarithm. -

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1919) OF THE ELECTRIC;_CIRCUIT 229in the flattening of the wave front on a lowresistance conductor,No. 00, and a high resistance conductor, No.4. There is,however, an enormous difference due to the effect of the close-

_ L f /II I f J V0 02 0.4 0.6 0.8 1.0 3 .2 3.3 3 .4 3 .5 3.6 3.7" . ~~. . . . . . 1 - " ' "

./ I - - - ' . . . . . .6.6 6.7 6. 8 6.9 711 ~I 27.0 27.2 27.4 27.6 , 27.8 28J) 28.2~ ~ . . . .I .i-- -- - I-- _ K MIi 0 & l L 2 68.4 6 6 ~LI 69.0 69.2 69.4 69.6 69.8Flo. 6--FLA'M'ENINO OJ' STEEP WAVIC FRONT IN TRANSMI88ION LINENo. 00 B. &. S. GAOE COPPICR 30 FT. ABOVE GROUNDness of the return conductor; with the return conductor at 18inches distance, the wave front is still materially steeper at 6.9km. distance, than it is in the conductor with ground return at

0.60.7 0.8 68 6.9 7071 68.6 68.8 690 692 694 69.6 69.811 I I I I1/ i"'" ~IJIS). . . . . f- No.oo 8." 8.Gqe C o w e rJ . . . . ~ . . . . 30 Ft. allove Ground. . , 1.1 I I I I I1 / I I I I I 1 ( 1 1

No .00 8." 8.Gqe C o w e r18 In.Di I Ian \11 I I I I. . . I " d : : : : : L I -I I6)~ i""" No. 4 8... S . Gqe C o w e r~ " " " 30 Ft. allove Ground. . , . I I t 1 I I' " ..r"r I I I 1 ( 4 )No.oo 8." S. Gqe Ir_I - ' " _I'. 30 Ft. above Ground

64 66 68 70 72 74FlO. 7

0.69 km. distance. That is, the flattening of the wave frontin the conductor with ground return, is more than ten times asrapid, than in the same conductor with the return conductorDigitized by Coogle


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230 STEINMETZ: GENERAL EQUATIONS [Feb. 20closely adjacent. Or in other words, the danger zone of steepwave front, extends in a conductor with the return conductorclosely adjacent, to more than ten times the distance than inthe conductor with ground return.This means, where it is desired to transmit a high-frequencyimpulse or steep wave front to the greatest possible distance,it is essential to arrange conductor and return conductor asclosely adjacent as possible. But where it is essential to limitthe harmful effect of very high frequency or steep wave frontas much as possible to theimmediate neighborhood of [its origin, the' return con- 0 1 . . . 1 - ; : : : = = = = =ductor should be separated }as far as possible.The data on iron wire are 1 .....J...._ _very disappointing; there is _ fan enormous increase in the 2flattening of the wave frontat great distances, by the use 3,---r"-'of iron as conductor material, '-so much so that the wave 4. '\front of the iron conductor at '''''' _69 km. distance had to be 5 Lshown (dotted) at one-tenth ~,-,__the scale of the other con- 6 __ ~-ductors, in Fig. 7. But at """-moderate distances, 6.9 Ian. z - =from the origin, the flatten- 1ing of the wave front in theiron conductor is only littlegreater than in a copper con-ductor of the same size: 215 kilo-cycles against 330 kiJo-cycles.At short distances, the difference almost entirely ceases, andwithin one Ian. from the origin, the wave front in an iron con-ductor is nearly as steep as in a copper conductor of the samesize. Thus a short length of iron wire between station and linewould exert practically no protection against very hlgh-Irequen-cy oscillations or steep wave fronts, such as may be producedby lightning strokes in the neighborhood of the station.This was to be expected from the shape of the curve of theattenuation constant, shown in Fig. 3.From the data in Table VII then are constructed and shown

IO--"" ' ./"::""_-ll---""'Z-'\2-S::: :: :: ::: _

13~14--"" =---15---~'::::::""-16~

17~

Q U A R T E R W A V E C H A R G IN G O S C IL L A T I O NO F T RANS MIS SIO N U NEFlo. 8

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1 9 1 9 ) OF THE ELECTRIC CIRCUIT 231in Fig. 8, the successive curves of voltage distribution in theline, as it is discharging (or charging), by the originally rect-:angular wave running over the line, reflecting at the end ofthe line and running back, then reflecting again at the beginningof the line and once more traversing it, etc., until graduallythe transient energy is dissipated and the line voltage reachesits average, zero in discharge, or the supply voltage in charge.The direction of the wave travel in the successive positions isshown by the arrows in the center of the wave front; theexistence and direction of current flow in the line by the arrowsin the (nearly) horizontal part of the diagram. As seen, afterfour to five reflections, the voltage distribution in the line ispractically sinusoidal.However, in these diagrams Figs. 6 to 8, the wave front hasbeen constructed from the maximum voltage gradient derivedby the calculation, assuming as approximation the

Steinmetz General Equations of the Electric Circuit Pt3 1919

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Transcript of Steinmetz General Equations of the Electric Circuit Pt3 1919