NBS MONOGRAPH 96

Electrical Parameters of Precision,

Coaxial, Air-Dielectric Transmission Lines

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

UNITED STATES DEPARTMENT OF COMMERCE • John T. Connor, Secretary

NATIONAL BUREAU OF STANDARDS • A. V. Astin, Director

Electrical Parameters of Precision,

Coaxial, Air-Dielectric Transmission Lines

Robert E. Nelson and Marlene R. Coryell

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado

National Bureau of Standards Monograph 96

Issued June 30, 1966

For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price $1.25
Library of Congress Catalog Card Number: 66-60053
Contents

Page

1. Introduction 1

2. Description of the air lines 1 3. Computational methods 3

4. Graphs 5

4.1. Standard conditions 6

4.2. Variations with resistivity, p 6

4.3. Variations with relative dielectric constant, — 6

4.4. Variations with diameters a and b 6

5. Summary 6

6. References 6

List of illustrations

Line Size No. 1: (7 mm)

Standard conditions 9

Figure 1 -figure 7 Variations with resistivity, p 17

Figure 8—figure 14.

Variations with relative dielectric constant, — 25

€0 Figure 15-figure 17 Variations with diameters a and b 29

Figure 18-figure 19

Line Size No. 2: (14 mm)

Standard conditions 33

Figure 20-figure 26

Variations with resistivity, p 41

Figure 27-figure 33

Variations with relative dielectric constant, — 49

Figure 34-figure 36

Variations with diameters a and b 53

Figure 37-figure 38

Line Size No. 3: (3/4 in.) Standard conditions 57

Figure 39-figure 45

Variations with resistivity, p 65

Figure 46—figure 52

Variations with relative dielectric constant, — 73

€o Figure 53-figure 55

Variations with diameters a and b 77

Figure 56—figure 57

Line Size No. 4: (21 mm)

Standard conditions 81

Figure 58-figure 64

Variations with resistivity, p 89

Figure 65-figure 71

Variations with relative dielectric constant, — 97

Co Figure 72-figure 74

Variations with diameters a and b 101

Figure 75-figure 76

iii

Electrical Parameters of Precision, Coaxial,

Air-Dielectric Transmission Lines

Robert E. Nelson and Marlene R. Coryell

Since precision coaxial connectors have become commercially available, precision, coaxial, air- dielectric transmission lines are being widely used as radio frequency immittance standards. The

evaluation of seven electrical parameters for four different line sizes, which are commonly used as standards, are presented in graphic form. The seven parameters, evaluated as functions of frequency,

are inductance per inch, resistance per inch, characteristic impedance magnitude, characteristic impedance phase angle, attenuation per inch, phase-shift per inch, and wavelength. Also included

are graphs showing how these parameters vary with changes in the resistivity of the conductors, the

relative dielectric constant of air, and the diameters of the conductors. Key words: Air-dielectric, coaxial transmission lines, electrical parameters, evaluation, functions

of frequency, graphic presentation, and precision.

1. Introduction

For several years, the High Frequency Impedance

Section of the National Bureau of Standards has maintained

a set of fixed lengths of coaxial air lines which are used as

the reference standards for high frequency immittance

measurements [Powell, Jickling, and Hess, 1958; Jones and

Nelson, 1962]. However, before these standards, or any

standards, may be used to calibrate a measuring instru- ment with any degree of accuracy, a precision connector

is needed to establish a definable reference plane between

the standard and the instrument being tested. Recognizing

a need to establish guidelines for commercial development

of precision coaxial connectors, a committee was formed

(later becoming the Institute of Electrical and Electronics

Engineers, Inc., Instrumentation and Measurement Group,

Technical Subcommittee on Precision Coaxial Connectors)

to determine realistic limits on the mechanical and electrical parameters of precision coaxial connectors. Following the guidelines estabhshed by this committee,

several commercially developed precision connectors have

appeared on the market which have led to the wide use of

precision coaxial fines as immittance standards. However, before these fines can be used as standards, the electrical parameters of the fines must be known to a high degree

of accuracy. This has required rather tedious calculations

using long and involved formulas. This paper has been

prepared to alleviate this situation by providing graphs

from which the electrical parameters of the fines may be

determined rapidly and accurately.

The graphs were constructed using the formulas derived

by Stratton [1941]. These results were then compared with

results obtained by using the more exact equations derived

by RusseU [1909].

An additional benefit was obtained from the evaluation

of Russell's equations. A computer program was constructed to evaluate the Bessel functions ber, bei, ker, kei,

and their first derivatives for order zero for large arguments

since tables of these values were not available in the

literature.

2. Description of the Air Lines

There are three basic dimensions used in the calculations of the electrical parameters of coaxial transmission

lines. These are the diameter of the center conductor

(dimension a), the inner diameter of the outer conductor

(dimension b), and the outer diameter of the outer conductor

(dimension c). For the purpose of this paper, the authors

chose the four lines sizes which the committee used for connector studies. Line sizes are usually specified according to the inner diameter of the outer conductor (dimension b). Using this convention, the four line sizes are identified in this paper as: fine size #1 (dimension 6 = 0.275591 in.) (figs 1 to 19)

(commonly referred to as 7 mm size) fine size #2 (dimension b = 0.562500 in.) (figs 20 to 38)

(commonly referred to as 14 mm size) line size #3 (dimension 6 = 0.750000 in.) (figs 39 to 57) (commonly referred to as 19 mm or 3/4 inch size) fine size #4 (dimension 6 = 0.826772 in.) (figs 58 to 76)

(commonly referred to as 21 mm size). In the calculations, both conductors were assumed to exhibit the same electrical properties with the center con- ductor being a solid rod. The annular space between

conductors contains only air as the dielectric. With the inner diameter of the outer conductor specified, the diameter of the center conductor (dimension a) was chosen so that the characteristic impedance of the fine
is 50 n at a temperature of 23 degrees Celsius, an atmospheric pressure of 760 mm of mercury, 50 percent relative

humidity, and infinite conductor conductivity. These were

the conditions chosen by the Subcommittee on Precision

Coaxial Connectors for the purpose of establishing uni- formity of hne sizes. Under these conditions, the equation

for the characteristic impedance of the Unes reduces to the form

where

L = (0.0254)^ In fItt \a

c=

(0.0254) (27reo)

In

Co

which allows dimension a to be easily calculated. Actually,

of course, the characteristic impedances of the hues under

the above conditions are never 50 since the ideal of infinite conductor conductivity can not be achieved. However, the

characteristic impedance approaches 50 fi asymptotically

as frequency increases. The outer diameter of the outer conductor (dimension c) is of relatively minor importance

except at very low frequencies.

In performing the calculations for the graphs, it was assumed that only the TEM wave is propagated in the hnes.

3. Computational Methods

The resistance and inductance per unit length of a

coaxial transmission line as a function of frequency and the

physical and electrical properties of the hne were calculated

by means of a computer program using the mathematical

development of Russell. Russell's equations were derived

-Y{mr;){Y(mr;)S{mr2)-Y{mr2)S{mr3))

using a circuit theory approach with the assumptions that the hne is infinitely long and Faraday's law of magnetic

induction is vahd.

Russell's equations, converted from absolute units to MKS units, are:

R = pm

27rriF(mri)

{ber mri bei' mn — bei mr\ ber' mri)

, pm bei' mn . , , »r , . + 7i T ber mr2 + N bei mr2

Zirri i(mri)

+ O ker mr2 + P kei mri)

pm ber' mri

27rri Y{mri) {M bei mr2 — N ber mr2

2tt \rj iTTtarx F(mri)

+ O kei mr2 — P ker mr2),

(ber mri ber' mri

. 1 1 , ^ , pm ber mri , , m , + bei mri bei mri) + 7i 77, : (M ber mr2 + TV bei mrz

iTTOiri Y[mri)

, _ , , r> I

• X , pmbei' mri , . -r O ker mr2 + " kei mr2) + — tt. (M bei mr2

ZTTCoriY (mri) — ber mr2 + O kei mr2 — P ker mri),

where

m = p-o CO

L P

1/2

S (mx) = ber' mx ker' mx + bei' mx kei' mx,

T(mx) = bei' mx ker' mx — ber' mx kei' mx,

Y (mx) — ber'"^ mx + bei'"^ mx,

with a: = Ti, r2, or r^, -OS(mr3) + PT(mr3) M=

N=-

Y(mr3)

0T(mrs)-PS(mr3)

F(mr3)

0 =

r2 (bei' mri bei' mr2 + ber' mn ber' mr2)

(Y(mr3)S(mr2) - Y(mr2)S(mr3))'' + (Y(mr2)T(mr3) - Y(mr3)T(mr2))'^

r2 Y(mr3)(Y(mrt)T(mr3) - Y(mr3)T(mr2))

(ber' mri bei' mr2 ~ bei' mri ber' mr2)

(Y(mr3)S(mr2) - Y(mr2)S(mr3))^ + (Y(mr2)T(mr3) - Y(mr3)T(mr2))^

— —Y(mr3)(Y(mr,)T(mrz) — Y(mr3)T(mr2)),, , , , , < •/ . •/ ^ r2 (ber mriber mr2 + oei mribei mry)

(Y(mr3)S(mr2) - Y(mr2)S(mr3))^ + (Y(mr2)T(mr3) - Y(mr3)T(mr2))''

— Y(mr3)(Y(mr3)S(mr2)-Y(mr2)S(mr3))., , , ., l •> ^ k^.' Tj. (ber mri bei mr2 — bei mri ber mrj)

(Y(mr3)S(mr->)-Y(mr2)S(mr3))^ +(Y(mr2)T(mr3)-Y(mr3)T(mr2))''
The Bessel functions ber, bei, and allied functions ^ _ (41^2—1^) 1

,1, . 1- r 1 1 1 A.v(z) ~ i cos 7 77+* •• appear in Russell s equations and it was tound that presently 1 !8z 4

avaUable tables of values for these functions do not extend 1W4^2 _ 12^4^2 _ 32) . . . r^^2_(2r-m

to large arguments. Consequently, computer programs had +^ '-^ '-^ -J——i i LL

to be written to compute the eight functions ber, bei, ker, kei, ^"^

and their first derivatives for order zero before RusseU's / \

equations could be programmed and values calculated. \cos ^f'TT +

Asymptotic expansion formulas were employed to com-

^^^^^^^ ^^^^ j^^.^^ r + 1; and

pute these functions, since the numerical values computed

were not adaptable to normal series expansions. The

^^^2 _ j expansion formulas are: X^W ' ^ISz

sin -77 — • •

•

, - ^ (-ir+W-Wi^'-S^) • • • {4i/-(2r-l)2}

gz/V2 f, , , / 2 1 ,1 .._\ +

f,zlV2 beLz

V(27r2)

ber^z f==\K{z) cos ( —-^77 + - vtt rliSzY

V(27rz) l VV2 8 2 /

/ z 1 1 M

fsin ^r7r+ •

• • j.

the number of the term being r.

\x cos (-^ — 7:TT + ^vTn , e'l^ r / z 1 1 \

V(27rz) I- VV2 8 2 / + M2) sin (-^-^tt + ^i^tt)}, . / 2 ,1 ,1 \V2 8 2 /J sin ^—+ -7r + -i/7r

6ei„2 '

, iiirlz) cos -^+ -77 + 7:1^77 V(27rz) l VV2 8 2 /

+ ijjUz) sin ^

where

where

l//U2) = 2

l^'' + ^^^^"'"'^''-l^-^^J ^ ^ cos - 77 +

^ (-l)W-l')(4»^^-3^) . . {4j/2-(2r-3)2}{4z/2 + (2r-l)(2r+l)}

rl(8zy

cos 7 r77 + . . . 4

the number of the term being r+ 1; and

(-ir+W-P)(4t^-3^). . . {4i^^-(2r-3)2}{4t^^ + (2r-l)(2r+l)}

r!(8z)''

sin 7 r77 + . . 4

the number of the term being r.

3
ker^z ~ \l I — I e ki,(— z) cos [^^^

+ ^ + ^''T

+ X.(-z) sin + +

A:«„2~

A:er„'Z

|l/>J.(— z) cos ~\'^^ \

~^

sm 1-^-^77 + 1.71

keii,'z —1 e-z/V2

2z,

n„(— z) cos ( — ^TT^-^vn] — (//,,(— z) VV2 8 2

where X^(— z), Xy(~2), z), and z) are obtained by

writing — z for z in the forumlas given for K{z), xj^^)-' ^viz), and n„(z) [McLachlan, 1961]. The terms of these as- ymptotic expansions decrease in magnitude until some minimum value is reached and then increase without

limit. The error in the expansion does not exceed the

first term neglected, provided the series is truncated before the minimum term. A discussion of asymptotic

expansions is found in Watson [1944]. It has been deter- mined that the number of usable terms is approximately

equal to twice the value of the argument for an argument

of 15. As the value of the argument increases, the smallest

term in the series appears progressively farther out in the

series. Since the accuracy of the expansion is limited

by the magnitude of the smallest term, each term in the

series was individually tested until the point was reached

where the next term in the series was less than or equal

to 1 X 10~'^^. The series was then truncated at this point.

For an argument of 35, only 20 terms were needed to reach

this point. Thus. 20 terms, the maximum number of

terms which were used in the program, were sufficient for the size of the arguments which were used. In fact.

if the argument is large, the series can be calculated accurately to several decimal places by using only a few terms

of the series. The relationships among the Bessel functions reveal

two equations used as check sums for each argument.

The equations are

berix) ker'(x) + bei'(x) kei{x) — ber'(x) ker{x) — bei(x) kei'(x) + - — 0 x

and

ber{x) kei'(x) + bei(x) ker'ix) - ber'(x) keiix) - bei'(x) ker{x) = 0 [Lowell, 1959].

These equations provided a means to aid in the determination of the number of significant figures computed for the

different series for each argument. The check sums were

of the order of 1 X 10"-' to 1 X 10"-'^ for all values with many

of the check sums exactly zero. The eight Bessel function programs were then incorporated into a larger program which calculated the resist- ance and inductance per unit length of a coaxial air hne

using Russell's equations. This program was written in 22 digit, double precision, floating point mode for max- imum accuracy.

Another computer program, using 16 digits, was written

to compute the resistance and inductance using the approximate formulas derived by Stratton. These formulas are

R =

IT ohm/inch.

(0.0254, gl„@+^j^^

henry/inch.

These formulas were derived using the theory of electro- magnetic fields and waves. Stratton's formulas neglect

those terms which are of concern only at the lower fre- quencies. It is interesting to note that RusseU obtains

these same formulas as approximations for very high

frequency currents. The results of the two programs were then compared to determine the magnitude of the differences obtained using

the two different sets of formulas. At frequencies above

1 MHz, the differences were small enough to be insignificant

when plotted on the graphs. For example, at a frequency

of 1 MHz under standard conditions for line size #3, the

difference in resistance per inch is less than 2 X 10"^ CI when calculated by both equations. The difference in inductance per inch is less than 1 X 10~'^ H under the same

conditions. This being the case, the formulas derived

by Stratton were used to compute the resistance and

inductance in the final program.

4
The final computer program was written to calculate the

values of the electrical parameters which were used to construct the curves on the graphs. This program was

written in 16 digit, double precision, floating point mode.

The equations used in the program to calculate the desired

electrical parameters are given below: (These equations are

variations of the usual equations which may be found in any good textbook on transmission line theory with the

exception of the equations for R and L which are those

derived by Stratton as described above.)

R

1/2 ^1/2 ('lU{poMco)'/2 (I

a/in.

L = (0.0254)^ In

ZTT (167r'y) 1/2

fJ-ci

eo

e

eo

R

L

C

G

e

a

X

magnetic permeability of inner conductor in henry/

meter,

magnetic permeability of outer conductor in henry/

meter,

dielectric constant of free space in farad/meter,

'- relative dielectric constant of dielectric (air), and the parameters calculated were:

resistance in ohms/inch,

= inductance in henrys/inch,

=capacitance in farads/inch,

conductance in mhos/inch {= zero for air dielectric),

characteristic impedance magnitude in ohms,

characteristic impedance phase angle in radians,

phase-shift constant in radians/inch,

- attenuation in nepers/inch,

' wavelength in inches.

^V(poAtco)'/^(|

- F/in.,

(0.0254) (27reo) /€

G = 0 (for air dielectric).

\Zo\=

1/ R'

0 = - tan-i

1/4

rad.,

H/in.

/3 =

[I [( [R^ + (o^L^] [G^ + a>2C2] )i/2-7?G + <o'LC] |

'''rad

The graphs for C and G are not included since C is easily

calculable and not a function of frequency and G is essentially zero for air dielectric. Since \Zo\ is one of the more

important parameters, curves were drawn as deviations

from 50 Q (i.e., AZ = (|Zo|- 50.000) X 10^ O) to aUow more

precise values to be obtained from the graphs. The curves

for /3 and \ were drawn as percent deviations from free

space values (i.e., A/3 X 102 k-ko

ko

X 10^^ for the same reason. The free space values were

calculated using the formulas

ko- r

where, n 1 « = ^ [( + w^L^] [G2 + aj2C2] yl^RG - co^LC] Np/in.,

A. = -o~ mch.

1/0= 1.180285 X 101" in./sec,

which is the velocity of propagation of electromagnetic waves in free space.

The output of the computer program described above was printed in the form of tables and the final graphs were drawn using the values from these tables.

where:

a = outer diameter of inner conductor in inches,

b — inner diameter of outer conductor in inches,

/ = frequency in hertz,

=277/,

Pi = resistivity of inner conductor in ohm-meters,

Po = resistivity of outer conductor in ohm-meters,

fid = magnetic permeability of the dielectric (air) between

conductors in henry/meter.

4. Graphs

The graphs are divided into four main groups corre- sponding to the four jlifferent hne sizes. The scales for the graphs were chosen to allow maximum practical accuracy in obtaining values directly from the curves and still keep the graphs of reasonable physical

size for publication purposes. These considerations also restricted the frequency range to a certain degree. The

upper frequency limit for each line size was chosen to be

5
that frequency below which only the TEM wave can be

propagated. This frequency was determined by the ap- proximate relationship

2v

fc = "T—n [Johnson, 1950],

where v is the velocity of propagation of electromagnetic waves in the line. The lower frequency limit was deter- mined primarily by the physical size of the graphs and re- sulted in values of 10 MHz for line size #1 and 1 MHz for

line sizes #2, #3, and #4.

After the graphs were reduced to pubhcation size, it was

determined that values could be read directly from the curves to within one-half the smallest scale division. In most cases, the uncertainty in reading the curves is much

less than the uncertainties in the original constants (e.g., uncertainties in determining resistivity, p) so that the graphs

are more than adequate considering the present state of

the art in constructing coaxial line immittance standards.

Each line size is subdivided into four general categories

which are discussed below. These subdivisions are (1) standard conditions, (2) variations with resistivity, p, (3) variations with relative dielectric constant, —, and (4) variations with diameters a and b.

4.1. Standard Conditions

The first seven graphs for each line size were drawn

using values obtained from a set of "standard" conditions.

These conditions are that the conductors be made of copper

with an air dielectric, and that the lines be used under

ambient conditions of 23 °C, 50 percent relative humidity

and an atmospheric pressure of 760 mm of mercury. With

the basic constants of the Hne determined by the above conditions, seven parameters were calculated and plotted as functions of frequency. The seven parameters are (1) re- sistance per inch, (2) inductance per inch, (3) characteristic

impedance magnitude, (4) characteristic impedance phase

angle, (5) attenuation per inch, (6) phase-shift constant,

and (7) wavelength in the line.

4.2. Variations with Resistivity, p

The second set of graphs for each hne size shows how

the seven parameters listed in section 4.1 vary with changes

in resistivity. Each graph is a family of curves with each

curve representing a different value of resistivity. The

resistivity was varied from a value of 0.1 X 10"'' fi-m to a value of 0.1 X lO"'' H-m with several values between for purposes of interpolation.

4.3. Variations with Relative Dielectric Constant,—

Co The third set of graphs for each line size shows how

three of the seven parameters vary with changes in relative

dielectric constant. The three parameters which . vary with

relative dielectric constant are characteristic jmpedance

magnitude, phase-shift constant, and wavelength^ The

other four parameters do not vary significantly with changes

in relative dielectric constant, and consequently were not included in the graphs. The relative dielectric constant was varied from 1.000400 to 1.000800 with an occasional

value in between when necessary for more accurate interpolation.

4.4. Variations with Diameters a and b

The fourth and final set of graphs for each line size shows how inductance per inch and characteristic imped- ance magnitude vary with changes in the diameters a and b. The rest of the seven parameters showed no significant

variations with changes in the diameters. The diameter,

b, was allowed to vary ±0.0002 in. and the diameter, a, was allowed to vary ±0.0001 in. from nominal. Only the

two curves showing maximum variations on either side

of nominal for the parameters were drawn since interpolation is not feasible between curves. Consequently, the

graphs indicating variations with changes in a and b are more qualitative than quantitative and indicate only maximum possible variations from the nominal values of the

parameters.

5. Summary

In conclusion, then, the authors have tried to provide

the user with a rapid and accurate means of determining

the electrical parameters of precision, air-dielectric, coaxial

transmission lines. In addition, graphs have been provided

showing how the electrical parameters of the lines are affected by variations in the resistivity of the conductors,

the relative dielectric constant of air, and the diameters

of the conductors.

The evaluation of the Bessel functions ber, bei, ker, kei,

and their first derivatives for large arguments may be of

considerable value. To the best of the authors' knowledge,

these values are not available in the literature at this time.

If there is a need for these values, the results of the computer program may be published either in tabular or graphic

form at some future time.

The evaluation of resistance and inductance per unit

length by two different methods showed no significant

differences at frequencies above 1 MHz considering the

present state of the art in constructing precision lines. At lower frequencies, however, the differences become

greater and RusseU's equations must be used to obtain

the necessary accuracies in the calculated values.

6. References

Johnson, W. C- (1950), Transmission lines and networks, p. 89 (McGrawHill Book Co., Inc., New York, N.Y.). Jones, R. N., and R. E. Nelson (1962), The role of capacitance in the national reference standards for high frequency impedance. Proceedings

of the 17th Annual I.S.A. Conference and Exhibit, Paper No. 18.1.62.

6
Lowell, H. H. (1959), Tables of the Bessel-Kelvin functions ber, bei, ker,

kei, and their derivatives for the argument range 0(0.01)107.50, National Aeronautics and Space Administration, Technical Report R-32, pp.

6, 7.

McLachlan, N. W. (1961), Bessel Functions for engineers, second ed., cor- rected printing, pp. 137-152, 209-210, 213 (Oxford University Press, London).

Powell, R. C, R. M. Jickling, and A. E. Hess (1958), High-frequency im- pedance standards at the National Bureau of Standards, IRE Trans.

Instr., 1-7, Numbers 3 and 4, 270-274.

Russell, A. (1909), The effective resistance and inductance of a concentric main, and methods of computing the ber and bei and allied functions, Philosophical Magazine 1 7, 524-552.

Stratton, J. A. (1941), Electromagnetic theory, p. 553 (McGraw-HiU Book

Co., Inc., New York, N.Y.).

Watson, G. N. (1944), A treatise on the theory of Bessel Functions, second

ed., pp. 194-225 (Cambridge University Press, London).

Supplemental References

Gray, A., G. B. Mathews, and T. M. Mac Robert (1922), A treatise on Bessel Functions and their applications to physics (MacMillan and Co., Ltd., London).

Harris, I. A., and R. E. Spinney (1964), The realization of high-frequency

impedance standards using air-spaced coaxial lines, IEEE Trans. Instr. Meas. IM-13, No. 4, 265-272.

Post, E. J. (1963), Phase, attenuation, and impedance characteristics of coaxial transmission lines with thin tubular conductors, IEEE Trans. Microwave Theory and Techniques MTT-11, No. 2, 129-136.

Schelkunoflf, S. A. (1942), Electromagnetic Waves (D. Van Nostrand Co.,

Inc., New York).

7

Line size #1

Standard conditions

Constants used in calculations:

0 = 0.119670 inch

6 = 0.275591 inch

Pi = po = 0.17241 X 10-^ ohm-meters

A^ci = fJ-co = /i-d = 477 X lO"'^ henry/meter

-=1.000649

Co

9

Cm

> ^

rr ^ 2 ^

UJ I- ID

O e3

U- !Jc

o >

2 c < -2 I— ~

IS '~>

Zld3H Nl A0N3n03ad

11
o> OO

Zia3H Nl A0N3n03yj

12
(/3 5

I

.2 _i ^

UJ

<I

Q.

UJ °

Q. S

P 8

I- J

tr<

X

o

Zld3H Nl A0N3n03Hd

13

218-445 0-66—

2

Zld3H Nl A0N3n03dd

15
16
Line size #1

Variations with resistivity, p

Constants used in calculations are the same as those used under standard conditions except that

Pi {= po) varies from 0.1 X IQ-^ to 0.1 X IQ-^ ohmmeters.
Zld3H Nl A0N3n03dd

18
>-' >

hto

CC

U.o

o

CO<

llJ o

<

I-o

3Q

Z

3

~0

Z1U3H Nl ADN3n03«d

19
>>

(/)

C/J UJ

cc:

Ll(J _c o

(-oz :3 Li. ~S < 3e <

COX

(Jc o %ia

_l

_J c2

(Jc

CO .a.a X a o G qo

If)

fco^ o e

,2 a: • 2

Zld3H Nl A0N3n03ad

20
UJ

cr

u.o

O a O 2

<

2 ^

a: 5

6c UJ I

< -

LlI H < g

UJ -c Q. ^

I- 5

q:

UJ

h-o

<

<X

o

Z183H Nl ADN3n03ijd

21

Ziy3H Nl ADN3n03aj

23
Zia3H Nl A0N3nO3dd

24
Line size #1

Variations with relative dielectric

constant, —

Co Constants used in calculations are the same as those used under standard conditions except that — varies from 1.000400 to 1.000800.
tn2O

O

— N ^o ^

liJ -I 1. UJ ^

26

ZliJ3H Nl ADN3n03HJ
Z1H3H Nl ADN3n03«d

27
c\

1-" z<

ISzoo «J •S o 1^ tr <^

CTUJ

_jUJ to o

UJ> a

.o < a

RE

cy

fo

u. c o 3 zo

1-

C o •2 z (Jc

u.< 0

toa < 3

"a UJ

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28
Line size #1

Variations with diameters a and b

Constants used in calculations are the same

those used under standard conditions except that varies ±0.0001 inch from standard conditions, and

varies ±0.0002 inch from standard conditions.
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30
m

o

< *

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UJ

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UJ ^

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o

Zld3H Nl ADN3n03dd

31

Line size #2

Standard conditions

Constants used in calculations:

0.244255 inch

0.562500 inch

po = 0.17241 X 10-^ ohm-meters

^tco = M^d = 477 X 10" henry/meter

1.000649
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Q.

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3

C

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38
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40

21ti3H Nl A0N3n03ad
Line size #2

Variations with resistivity, p

Constants used in calculations are the same as those

used under standard conditions except that p,(=Po) varies

from 0.1 X 10-'^ to 0.1 X 10"^ ohm-meters.

41
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48 Zld3H Nl ADN3n03dd
Line size #2

Variations with relative dielectric constant,—

Constants used in calculations are the same as those

used under standard conditions except that — varies from

Co 1.000400 to 1.000800.
50

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51
52 ZldBH Nl A3N3n03dd
Line size #2

Variations with diameters a and b

Constants used in calculations are the same as those

used under standard conditions except that a varies ± 0.0001

inch from standard conditions, and b varies ± 0.0002 inch

from standard conditions.
ZliJ3H Nl A0N3n03yd

54
Zld3H Nl A3N3n03bd

55

Line size #3

Variations with resistivity, p

Constants used in calculations are the same as those

used under standard conditions except that pi{— po) varies

from 0.1 X 10-^ to 0.1 X lO"*' ohm-meters.


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63

Line size #3

Standard conditions

Constants used in calculations: a = 0.325673 inch

6 = 0.750000 inch

Pi = po = 0.17241 XlO-7 ohm-meters

IJk:i = f^co — l^d = 477 X 10~^ henry/meter

-=1.000649
Zld3H Nl A0N3n03dJ

66
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72 Zld3H Nl ADN3n03dd
Line size #3

Variations with relative dielectric constant,—

Constants used in calculations are the same as those

used under standard conditions except that — varies from

Co 1.000400 to 1.000800.
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o

z

o

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(T

h- <^ o S

UJ ~

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UJ 5 o ^

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75
Zld3H Nl A0N3nO3dd
Line size #3

Variations with diameters a and b

Constants used in calculations are the same as those

used under standard conditions except that a varies ± 0.0001

inch from standard conditions, and b varies ±0.0002 inch

from standard conditions.
78

Zlid3H Nl ADN3n03iJd
Zia3H Nl A0N3n03yj

79

Line size #4

Standard conditions

Constants used in calculations: a = 0.359010 inch

6 = 0.826772 inch

p, = po = 0.17241 X 10-7 ohm-meters

ju.ci = /Xco = iw-d = 477 X IQ-^ henry/meter —=1.000649

Co

LlI c O 5^ o— I,^

< a

I- <o

Ziy3H Nl A0N3nO3Hd

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a. e — e

a. O S — ^

LlI S o ~

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85

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87
Zia3H Nl A0N3n03«d
Line size #4

Variations with resistivity,p

Constants used in calculations are the same as those

used under standard conditions except that p,(=po) varies

from 0.1 X 10-7 to 0.1 X lO-e ohm-meters.
Zia3H Nl A0N3n03dd

90
Zld3H Nl A0N3n03yd

91
Zlb3H Nl A0N3n03ad

92
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96

Zia3H Nl A0N3n03yd i
Line size #4

Variations with relative dielectric constant,—

Go Constants used in calculations are the same as those

used under standard conditions except that — varies from

€0 1.000400 to 1.000800.
Zld3H Nl ADN3n03dd
Zia3H Nl A0N3n03HJ

99
Z1H3H Nl ADN3n03ad

100
Line size #4

Variations with diameters a and b

Constants used in calculations are the same as those

used under standard conditions except that a varies ±0.0001

inch from standard conditions, and b varies ± 0.0002 inch

from standard conditions.
102 ZldBH Nl ADN3n03ai
Zlld3H Nl A0N3n03yd

103

U.S. GOVERNMENT PRINTING OFFICE : 1966 OL— 218-445


U.S. DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20230

POSTAGE AND FEES PAID

U.S. DEPARTMENT OF COMMERCE

OFFICIAL BUSINESS