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, 

403 


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, 

Z0 = the length of conductor, 

A = the sectional area, 

lCi = the circumference at conductor surface, that is, following 
all the indentations of the conductor, 

ZCj = the shortest circumference of the conductor, that is, cir- 
cumference without following its indentations, 

lr = the radius of the conductor, 

ld = the distance from the return conductor, 
X = the conductivity of conductor material, 

fi. = the permeability of conductor material, 

/ = the frequency, 

S = the speed of light = 3 X 1010 cm., and (1) 

a = — — = the wave length constant, 

o 

the true ohmic resistance is 


the ohmic reactance, low frequency value is 

*o = 2 7r/70 1 2 loge f + ^l 10~9 ohms; (3) 

or, reduced to common logarithms by dividing by log e, 

x0 = 2 TT/Z f4.6 log^ + |) 10~9 ohms. (4) 

\ l>r ** 

The equivalent depth of penetration of the current into the con- 
ductor, from Chapter VII, (40), is 

104 5030 

(5) 


406 TRANSIENT PHENOMENA 

hence, the effective resistance of unequal current distribution, or 
thermal resistance of the conductor, is, approximately, 

(6) 


and the effective reactance of the internal flux is 

10- ohms. (7) 

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 

r2 = 2 l^flO-* « 1.97 IJ1Q-* 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 

z2 = 4 7r/Z0 flog. 4- - 0.5772) 10-9. (9) 

\ aL I 


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 

42 = 2^r (10) 

in equation (9), and also substituting (1), gives 
x2 = 4 Tr/J^log^ - 0.5772) 10-9 

= 1.26 Z0/(loge ^- - 0.5772) 10-8; (11) 

or, reduced to common logarithms by dividing by log e, and 
substituting for S, 

z* - 5 IJ (1 - 0.547 log IJ) 10-» ohms 1 
= 0.547 IJ (9.15 - logZCi/) 10-" ohms. J 


HIGH-FREQUENCY CONDUCTORS 


407 


82. The total impedance of the conductor for high frequencies 
is, therefore, 


rt = kty— 10-*- L98V^1Q- 

v- A v- A 


.jSVy^ 


1.97 Z0/ 10~8, 
1.98^ 


10~4 = 

^-0.5772)10-, 

= 0.547 Z0/ (9.15 -log U-)10-8; 
while the conductor impedance for low frequencies is 


(13) 


(2-88 log lf + 0.314 /<) io-8. 


(14) 


Although the true ohmic resistance r0 is independent of the 
frequency, the thermal resistance rl is proportional to the square 
root and the radiation resistance r2 to the first power of the fre- 
quency. With increasing frequency, the resistance rl is at first 
appreciable, while r2 is still negligible, and appears only at still 
higher frequencies, and ultimately, at the highest frequency, 
becomes the dominant factor. Since r2 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, XQ, is proportional to the fre- 
quency; the internal reactance, xv 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, x2, 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 x2 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, p = 1, X = 6.2 X 10s; 

wrought iron, p = 2000, ^ = 1.1 X 105, 
we have 


A 

lc{ = 1C2 = 

Wire No. 4 B. and S. Gauge. 

3 In. by | In. 
Ribbon. 

2 In. Pipe, J In. 

Copper. 

Iron. 

Copper. 

Iron. 

0.211 
1.63 

0.211 
1.63 

2.40 
15.85 

4.70 

15.85 

lr - 2^T 

0.259 

0.259 

2.53 

2.53 

r = 

7.6X10-° 

43X10-° 

0.67X10-° 

1. 94X1Q-6 

*••_ 

0.00314X10-6/ 

6.28X10-8/ 

0.00314X10-6/ 

6.28X10-6/ 

t° = 

0.0824X10-6/ 

0.0824X10-6/ 

0.0536X10-6/ 

0.0536X10-3/ 

r, _ ^ = 

0.155X10-°V/ 

16.5xlO-°v7 

0.0159xlO-6\/7 

1.68X10-°V7 

J-- 

0.0197X10-6/ 

0.0197X10-6/ 

0.0197X10~°/ 

0.0197X10~6/ 

0.00547xlO-°/(8.94-log/) 

0.00547X 10-° / (7. 95 -log/) 

HIGH-FREQUENCY CONDUCTORS 409 

In x°, the distance from the return conductor has been 
chosen as ld = 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 

z° = 4 7r//0 loge lf X 10~9 + xv (15) 

ir 

where xl is the internal reactance of equation (7). 

84. In Fig. 97 are plotted values of the resistances r0, rv r2 
and of the reactances x*, x2°, xv x2, 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 negligible 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 x° 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 r2 would equal 
the ohmic resistance r0 at about 500 cycles if at this low fre- 
quency the case of an infinitely long conductor without return 


410 


TRANSIENT PHENOMENA 


-10*- 


•10-8 


S7 


Ml 


Copper Wjire 


2. Iron Wi,re 


No. 4 B. & 


Eg 


S. 


b. 4 B. & S. 


Cyc ea 


Cyc 


& 


MM 

-10-g 


x 


1— 


0 


-10— 


4. Ir6n .Pi>e 
2 in. 


Fig. 97. High-frequency conductors. 


HIGH-FREQUENCY CONDUCTORS 411 

conductor could be realized. At 500,000 cycles, the radiation 
resistance r2 equals the external reactance x2, and since the 
internal resistance rl at high frequencies is equal to the internal 
reactance x1 at 500,000 cycles, the total effective resistance r 
equals the total effective reactance x, 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 r0 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 x*, 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 rt and internal 
reactance xl preponderate. At about one million cycles, all 
four quantities — internal resistance rv representing power 
converted into heat in the conductor, radiation resistance r2, 
representing power radiated by the conductor, internal reactance 
xv representing the magnetic field in the conductor, the external 
reactance xv representing the magnetic field outside of the 
conductor — are approximately equal and the current lags 45 
degrees. Above one million cycles, radiation resistance r2 and 
external reactance x2 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 xf here has no meaning, as it would correspond to a 
2-inch iron rod. The ohmic resistance r0 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 
r1 and xl are small compared with r2 and xv 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 r1 and x1 are 
equal, and in the most important range of high frequencies, 
from 10,000 to 1,000,000 cycles, the external constants r2 and 
x2 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 S. Gauge. 

Copper. 

Iron. 

Frequency 

104 
0.0212 
0.0286 
0.0356 
0.59 
3.6 

105 
0.202 
0.221 
0.299 
0.67 
30 

10" 
1.986 
1.626 
2.57 
0.77 
257 

104 
0.185 
0.168 
0.250 
0.74 
25 

105 
0.717 
0.736 
1.028 
0.70 
103 

10" 
3.62 
3.26 

4.87 
0.75 
487 

Resistance r 

Reactance, x 

Impedance z 

Power factor 
Voltage drop at 100 amperes 

Copper Ribbon, 3 In. 
by J In. 

Wrought-iron Pipe, 
2 In. by | In. 

Frequency 

104 
0.0199 
0.0219 
0.0296 

105 
0.198 
0.162 
0.256 
0.77 
26 

106 
1.972 
1.072 
2.24 
0.88 
224 

104 
0.0365 
0.0385 
0.0530 
0.69 
5.3 

105 
0.250 
0.214 
0.326 
0.76 
33 

106 
2.14 
1.24 
2.47 
0.87 
247 

Resistance Y 

Reactance x 

Impedance z 

Power factor 
Voltage drop at 100 amperes 

0.67 
3.0 

As seen herefrom, within the range of frequencies from 104 to 
106 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 104 cycles the iron wire has 
seven times the voltage drop of the same size of copper wire, 
at 106 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 104 cycles is only 80 per cent greater than in the cop- 
per ribbon, while at 106 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 105 
cycles, and very little inferior at 104 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. 


SECTION IV 
TRANSIENTS IN TIME AND SPACE 


TRANSIENTS IN TIME AND SPACE