NBSIR 73-326

WAVELENGTH OF A SLOTTED RECTANGULAR LINE

CONTAINING TWO DIELECTRICS

Howard E. Bussey

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

July 1973

Prepared for

U.S. ARMY MOBILITY EQUIPMENT RESEARCH

AND DEVELOPMENT CENTER

Fort Belvoir, Virginia

NBSIR 73-326

WAVELENGTH OF A SLOTTED RECTANGULAR LINE

CONTAINING TWO DIELECTRICS

Howard E. Bussey

Electromagnetics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

July 1973

Prepared for

U.S. ARMY MOBILITY EQUIPMENT RESEARCH

AND DEVELOPMENT CENTER

Mine Detection Division

Fort Belvoir, Virginia

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director

CONTENTS

Page

1. Introduction 1

2. Propagation Constant as a Function of Line

Capacitance 2

3. Capacitance of the Line 3

4. Comparison with Experiment- 6

5. Error Discussion 7

6. Conclusions 8

7. Acknowledgments 8

References 9

iii

WAVELENGTH OF A SLOTTED RECTANGULAR LINE CONTAINING

TWO DIELECTRICS

The titled electromagnetic wave property is

obtained approximately for a rectangular slab line

with two dielectrics. The perturbing dielectric

is a thin sheet set on the center conductor and

slotted to permit travel of the probe when the line

is used as a slotted line. The purpose is to measure an unknown dielectric filling most of the line,

but perturbed by the thin sheet.

Key words: Capacitance; dielectric measurement;

slab line; slotted line.

1 . In troduct ion

A rectangular line has been designed by Ramon L. Jesch

of the NBS Electromagnetics Division for use as a slotted

section. The center conductor is rectangular shaped (fig. 1) , and a dielectric strip sets on one boundary of the center

conductor to protect the traveling probe when dirt fills the

remainder of the cross section.

The purpose of this work is to find the propagation

constant of the line containing the dielectric strip while

the remainder of the line is filled with another dielectric,

such as dirt, the dielectric constant of which is to be

determined from a weighted average dielectric constant of

the two materials.
Section 2 of this report gives the dependence of the phase

velocity on the capacitance and hence on the dielectric filling.

Section 3 describes the method of finding the capacitance by

computer approximation and gives results for certain dielectric

fillings. Section 4 compares the theoretically computed wavelength with measurements.

2 . Propagation Constant as a Function of Line Capacitance

The lumped circuit representation of transmission line

characteristics uses a series inductance, L, and a shunt

capacitance, C, each per unit length [1]. The characteristic

impedance of a uniform line is

Z = (L/C)^, (1)

and the propagation velocity is

V = 1/(LC)^. (2)

For a TEM mode in vacuum, v = c, the velocity of light. When

a uniform dielectric of relative permittivity e* = e/Eq =

e' - je" fills the line,'00 then C = £*C where C is the vacuum

capacitance, and the velocity v is

V = c/e'^^ (3)

When the dielectric space is inhomogeneously filled, an

exact solution is difficult. Therefore, we approximate by

assuming that eqs. (1) to (3) still apply when the electrostatic capacitance is found with two dielectrics present.

This is a good approximation when the width, 1 inch, multi2
plied by e ' ^ is small compared to the wavelength, i.e.

when the line is below cutoff, transversely, and the volume

of the perturbation is small. (A correct treatment would

involve hybrid modes instead of the assumed TEM mode.)

Losses were not treated.* It is believed that the method

given below, that of using a filling factor to calculate the

perturbation of the velocity by the small strip, is applicable even for damp soil with e"/e' up to 0.2 to 0.3.

3 . Capacitance of the Line

The capacitance C and charge q per unit length of the

cross section, and the voltage V are related by the expression

C = q/V, (4)

and assuming V = 1, C = q. A method o£ solving electrostatic

field problems is given in reference [2], and a computer program

for two dimensional problems is available [3]. The program

was modified by multiplying the charge on each segment of the

boundary by e', thus giving the true charge. Since C = C^e'

the computed charge is also taken as a measure of the average

dielectric constant.

The transmission line (fig. 1) has a plane of symmetry

(fig. 2). The program was run for one half of the total cross

section using this symmetry. Running half the problem introduced a small error that was removed by an approximate method.

*With high loss the complex propagation constant should be

used, whether or not a dielectric perturbation strip is involved.

3
The error arose because the charge on the center conductor

increased (as given by the program) to a higher than correct

value, at and near the intersections of the symmetry plane

with the center conductor. Within the scope of the present

work the program itself could not be corrected. As shown

in figure 3, the charges on segments 1 to 6 and 55 to 60 were

replaced by an extrapolated curve. The extra charge eliminated

by this method amounted to 4.9% of the total, for the air

filled line. Similar extrapolations were made with the various

dielectrics inserted. The computed charge on the center conductor, proportional to e' and C, is given in table I for

various conditions.

A useful viewpoint is to assume that the capacitance is

proportional to a weighted average dielectric constant given

by

^AV = ^4 '^l-g)^- C5)

With this notation eq. (3) becomes

V = c/e^^^. (6)

In eq. (5) g is the so-called filling factor for e^, where

is a perturbation of the main dielectric filling . One

might expect g to be nearly constant; however, g as computed

from the results in table I and eq. (5) changes from approximately

0.04 when = 1, to 0.067 when = 18. Interpolation of g

will allow estimation of for values of other than those

given in the table.

4
The equation for the wavelength o£ the line with its

dielectrics is

X^^ = v/f = c/(f e^/o. (7)

The equation for the measured dielectric quantity is

^AV = (^/f)'/^AV

We consider two examples of using the results of table I.

Frequency is Measured

If the frequency f is measured then the TEM mode wavelength in vacuum is

Suppose the wavelength with unknown is X^y = ^^Z'^'

Equation (8) gives

e =9

^AV • Interpolating in table I we choose g = .063. Using these

values in eq . (5) gives

= 9.43.

Frequency is Not Measured

If the frequency is not measured then must be obtained

from the wavelength of the empty line with = 2.6, £2 ~ 1-

For this case g = 0.041. From eqs . (7) and (9)

'^AV,air '^o^^Av''

where e^^ = 0.0406 x 2.6 + 0.959 = 1.0646, giving

X = A.,, . X 1.032 , o AV,air ' which is to be used for c/f in eq . (8).
In summary, the measurement of an unknown dielectric

material requires the following:

1. Find the free space wavelength either from the

frequency or from X ... ^^-^^ 1 .032.

A. V f air

: 2. Measure A^^ with the unknown material and find e^y

, , , , from eq . (8)

3. Interpolate a value of g from table I and convert

from e^y to using eq . (5).

4 . Comparison with Experiment ,

The measured A^y of the air filled perturbed line was 3%

less than the free space wavelength [4], i.e.,

A /A.,, .= 1.0 31.

o' AV,air

The predicted value from the computer work above is

,s^' ^ V^AV.air =

The prediction is in good agreement with experiment. If the

effect of the slot in the dielectric strip were taken into

account, the predicted ratio would be slightly lower.

The line was carefully filled with nylon, the dielectric

constant of which had been measured as 3.08 using a coaxial

holder. The present slab line gave e^y = 3.09 and using

g = 0 . 0 56 , £2 is 2 • 12 . An earlier experimental model had a narrower dielectric

strip which with an air sample reduced the wavelength by 21,

compared to a theoretical prediction of 2.11.

6
5 . Error Discussion

We have assumed a quasi static model to relate the

velocity to the effective dielectric constant of the two

materials, and a filling factor model to obtain the effective

dielectric constant of the two materials. It is believed

that these are valid models when the filling factor, g, is

as small as was found here, g - 0.05. The error limits of

either assumption have not been determined, and cannot be

determined theoretically within the scope of this investigation .

The limits may perhaps be estimated. Experimentally

we obtained ^/^^y ~ 1.031 and theoretically we obtained

1.032. The fractional error of the perturbation of X equals

that of g , i . e .

A(AX)/AA = Ag/g,

which is 31 in the above comparison. Thus the experimental

evidence in one case suggests that g and AX are in error

by few percent. The error in the correction, Ac', i.e.,

ACAc') would be twice as many percent.

For = 10, Ac = -0.472 and A(Ae) would be 0.03 x 2 x 0.47 - ± 0.03, out of £* = 10, i.e., a final error in the

measured e' of 0.31 due to a 31 error is g. It is likely

that g is accurate to ± 10%, thus contributing approximately

up to ± 1% errors in the final determination of el.

7
6 . Conclus ions

A quasi-static model has been developed that predicts

the wavelength and average dielectric constant of an inhomogeneous ly filled slab line, to a good approximation. A

method is given for calculating the dielectric constant of an unknown material from the measured wavelength, as perturbed by

the described dielectric strip.

The analysis may also be applied to the characteristic

impedance in eq. (1) .

'-^ ' 7, Acknowledgment

We thank Mr. Ramon Jesch for bringing this problem to

our attention, and for his suggestions, and Dr. J, Brian Davies

for references to the computer program.

8
Re fe rences

[1] Chen, T.S., Determination of the capacitance, inductance

and characteristic impedance of rectangular lines, IEEE

Trans, on Microwave Theory Tech., Vol. MTT-9, pp. 510-

519, Sept. 1960.

[2] Harrington, R.F., K. Pontoppidan, P. Abrahamsen, and

N.C. Albertson, Computation of Laplacian potentials by

an equivalent source method, Proc. lEE (London), Vol. 116,

pp. 1715 -1720 , Oct . 1969

.

[3] Pontoppidan, K. , A Fortran IV subroutine for equivalent

source solutions of electrostatic and magnetostat ic

problems, Report P 163, March 1971, Lab of Em Theory,

Technical Univ. of Denmark, Lyngby. (Note: Several

printing errors in this program must be corrected.)

[4] Jesch, R.L., Slab slotted line. Letter Report, June 4,

1973 (unpublished report).

9
Table I. Charge in arbitrary units on the center conductor

held at 1 unit of potential and resulting e^y and g

The dielectric strip has dielectric constant and

the remainder of the space has dielectric constant

"2-

Charge

(Capacitance)

as Computed

Charge

(Capacitance)

corrected^ ^ 'AV

(c)

1.0

2.6

1.0

1.0

0 .7613

0 . 832 8

0 .7239

0 .7710

1.

1.065 0 .041

3.08

2.6

3.08

3.08

2 .345

2 .318

(b) 2.2 30

2 .210

(b) 3.08

3.053 0 .056

5.0

2.6

5.0

5.0

3.806

3.665

3.620

3.515

5.0

4 .856 0 .060

10 .0

2.6

10 .0

10 .0

7.613

7.145

7. 2 39

6 .897

10 .0

9.528 0 .064

18.0

2.6

18.0

18.0

13.703

12 .694

13.021

12. 286

18.0

16.97 0 .066

(a) Corrected for apparent error in program.

(b) Charge with uniform dielectric, = e^* is always z times the charge found for the vacuum case.

(c) = corrected charge divided by 0 . 7239 .

10
Figure Captions

Figure 1. Cross section dimensions o£ the slab slotted line.

Figure 2. Cross section showing the plane of symmetry, the

image space, and the numbering sequence for the

segments on the center conductor.

Figure 3. Histogram plots of the computed charge t e ' on each

segment of the center conductor. Near the first and

sixtieth segments the charge erroneously increased,

as computed, and was replaced by the extrapolated

curves. The increase at the corners is of course

correct. The segments vary in width; smaller segments were used from 1 to 9 , smaller yet from 10

to 12, and other variations were employed.

11
12
IMAGE

REGION

GROUND

PLANE

I--

LlI §1

Q_

Ixl

CO

DIELECTRIC €,

SEGMENT

SEGMENT 13

-CENTER

CONDUCTOR

^SEGMENT 50

SEGMENT 60

GROUND

PLANE

Figure 2. Cross section showing the plane of symmetry, the

image space, and the numbering sequence for the

segments on the center conductor. 13
14
FORM NBS-n4A (l-7t) U.S. DEPT. OF COMM. T. PUBLICATION OR REPORT NO. 2. Gov't Acc«s>t<M BIBLIOGRAPHIC DATA ^x-dcto -7-? -ir>(. SHEET NBSIR 73-326

3. Recipient's Accession .No.

4. TITLE AND SUBTITLE

Wavelength of a Slotted Rectangular

5. Publication Date July 197 3

6. Performing Organization Code

7 ATITH^R('^')

Howard E, Bussey

8. Performing Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

10. Pro)ect/Task/Work Unit No. 2724430

11. Contract/Grant .No.

12. Sponsoring Organization Name and Address

U.S. Army Mobility Equipment Research

and Development Center

Mine Detection Division

FortBelvoir, Virginia

13. Type of Report & Period

Covered

Aug. '71 - Nov. '72

14. Sponsoring Agency Code

15. SUPPLEMENTAKY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) The titled electromagnetic wave property is obtained approximately for a rectangular slab line with two dielectrics. The perturbing dielectric is a thin

sheet set on the center conductor and slotted to permit travel of the probe when

the line is used as a slotted line. The purpose is to measure an unknown

dielectric filling most of the line, but perturbed by the thin sheet.

17. KEY WORDS (Alphabetical order, separated by semicolons)

Capacitance; dielectric measurement; slab line; slotted line,

18. AVAILABILITY STATEMENT

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TO NTIS.

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UNCL ASSIFIED

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USCOMM-DC 66244.P71