Pad Rubber - Virginia Tech...using microwaves for the evaluation of tank track pad rubber. In the preliminary study, an 8 GHz signal was transmitted from one side of a 1.37 cm (0.50

Microwave Sensing of Bulk Electrical Properties of Tank Track Pad Rubber

by

Michael W. Lee

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

in

The Bradley Department of Electrical Engineering

APPROVED:

Dr. R. Clark Robertson, Chairman

Dr. W. A. Davis

August, 1988

Blacksburg, Virginia

Dr. I. M. Besieris

Dr. n. A. de Wolf

Microwave Sensing of Bulk Electrical Properties of Tank Track Pad Rubber

by

Michael W. Lee

Dr. R. Clark Robertson, Chairman

The Bradley Department of Electrical Engineering

(ABSTRACT)

A complex permittivity measurement system composed of a network analyzer and a

open-ended coaxial waveguide has been used to evaluate the permittivity of rubber

samples. The conductivity of rubber provides an indication of the dispersion of carbon

black throughout the rubber matrix. The technique is based on the Deschamps antenna

modeling theorem which relates the effective admittance of an antenna in some arbitrary

medium to the effective admittance of the same antenna embedded in free space. This

technique is well suited for material with loss tangents between 0.1 and 1.0. Only

material within a radius on the order of the outer conductor radius of the coaxial

waveguide is interrogated. Inferred permittivity measurements for rubber samples are

presented. An APC-7 connector is used as the transducer which provides a means for

convenient calibration because standard calibration terminations can be used. The

amount of pressure from the sample applied to the waveguide affects reflection

coefficient measurements, preventing consistent results.

A:cknowledgements

I first wish to thank the members of my graduate committee; Dr. R. C. Robertson,

Chairman, Dr. I. M. Besieris, Dr. W. A. Davis, and Dr. D. A. de Wolf, for their guidance

and advice, not only in the course of this study, but throughout all phases of my

education at Virginia Tech.

A very special thanks are due to my very special friends, Sherry A. Bergmann,

Joseph P. Havlicek, and John F. Jockell. Unfailingly, they were always there for

sympathetic ears, wholehearted advice, laughter when all seemed hopeless, and

reprimand when the old noggin got too big to fit the hat.

Last, but far from least, I thank my family for their unending love and supp_ort

which make all my endeavors possible and worthwhile.

Acknowledgements iii

DAMNANT QUOD NO INTELLIGUNT.

NON SUM QUALIS ERAM.

Acknowledgements iv

Table of Contents

CHAPTER 1 Introduction . . • . . . . . • . . . . . . . . . .. . . . • . • • . . . . . • . • . • • . . . . . . . • . • . • . 1

1.1 Tank Track Pads and Carbon Black . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Preliminary Study ................................... ·. . . . . . . . . . . . . . . . . . . 3

1.3 Conductivity Measurement System for Actual Tank Pads . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Overview of This Presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER 2 General Theory Development . . . . . . . . . . • . . • • • • . • . . • • • • • • • . • • . • . . • 11

2.1 Time Harmonic Maxwell's Equations ..................................... 11

2.2 The Concept of Complex Permittivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Transmission Line Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Transmission Line Representation of Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 Coaxial Waveguide Modes ............................................. 26

2.5 Modeling Waveguide Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

CHAPTER 3 Open-Ended Coaxial Waveguide Transducer for Complex Permittivity

Measurement. . .................................... ~ . . .. . . . . . . . . . . . . . . . . . 32

3.1 Theoretical Basis . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . 33

Table of Contents v

3.2 Network Analyzer Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.3 Complex Permittivity Measurement System Description . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

CHAPTER 4 Conclusions and Recommendations for Further Research . . . . . . . . . . . . . . . . 76

4.1 Recommendations for Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Literature Cited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Appendix A. Marcuvitz Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Appendix B. Sizwght Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Vita ................................................................. 96

Table of Contents vi

List of Illustrations

Figure 1. Typical tank track pad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Figure 2. Two~way transmission scheme ................................ 4

Figure 3. One-way transmission scheme ................................ 6

Figure 4. Distributed parameter transmission line model . . . . . . . . . . . . . . . . . . . 17

Figure 5. Terminated transmission line ................................ 19

Figure 6. Uniform waveguide with an arbitrary cross-section ............... 21

Figure 7. Cross-section of a uniform coaxial waveguide ................... 27

Figure 8. _Open-ended coaxial waveguide transducer ...................... 34

Figure 9. Equivalent capacitance of a coaxial waveguide radiating into air 36

Figure 10. Equivalent capacitance of a coaxial waveguide radiating into air ..... 37

Figure 11. Open-ended coaxial waveguide transducer ...................... 39

Figure 12. Equivalent conductance of a coaxial waveguide radiating into air ..... 41

Figure 13. Equivalent conductance of a coaxial waveguide radiating into air ..... 42

Figure 14. Equivalent susceptance of a coaxial waveguide radiating into air ..... 43

Figure 15. Equivalent susceptance of a coaxial waveguide radiating into air ..... 44

Figure 16. Conductance/Susceptance vs. frequency for coaxial waveguide open into air .................................................... 45

Figure 17. Conductance/Susceptance vs. frequency for coaxial waveguide open into air . . . . . . . . . . . . . . . . . . . . . . . . . . . . ·. . . . . . . . . . . . . . . . . . . . . . . . 46

Figure 18. Fringing field at the coaxial waveguide aperture .................. 47

List of Illustrations vii

Figure 19. Schematic of a basic network analyzer system for reflection coefficient measurement ............................................ 49

Figure 20. Error model for network analyzer calibration .................... 51

Figure 21. WR-90 rectangular waveguide radiating into air - reflection coefficient . 53

Figure 22. WR-90 rectangular waveguide radiating into air - equivalent admittance 54

Figure 23. Schematic of the WR-90 rectangular waveguide transducer. . ........ 55

Figure 24. APC-7 connector radiating into air - reflection coefficient .......... 59

Figure 25. APC-7 connector radiating into air - equivalent capacitance ......... 60

Figure 26. APC-7 connector radiating into 15NAT1 rubber - equivalent admittance 63

Figure 27. APC-7 connector radiating into 15NAT1 rubber - loss tangent ....... 64

Figure 28. APC-7 connector radiating into 15NA T22A rubber - equivalent admittance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Figure 29. APC-7 connector radiating into 15NA T22A rubber - loss tangent .... 66

Figure 30. APC-7 connector radiating into 15SBR2 rubber - equivalent admittance 67

Figure 31. APC-7 connector radiating into 15SBR2 rubber - loss tangent ....... 68

Figure 32. APC-7 connector radiating into 15NAT42 rubber - equivalent admittance 69

Figure 33. APC-7 connector radiating into 15NAT42 rubber - loss tangent ...... 70

Figure 34. APC-7 connector radiating into SM02SP rubber - equivalent admittance 71

Figure 35. APC-7 connector radiating into SM02SP rubber - loss tangent ...... 72

Figure 36. Effect of pressure on a 150TR6 rubber sample - equivalent admittance 74

Figure 37. Effect of pressure on a 150TR6 rubber sample - loss tangent ........ 75

List of Illustrations viii

CHAPTER 1 Introduction

1.1 Tank Track Pads and Carbon Black

Tracked military vehicles such as tanks and armored personnel carriers utilize

rubber pads to dampen vibrations, reduce noise, and prevent damage to roadway

surfaces. The rubber pads are oblong pieces of rubber varying in thickness from about

6.35 to 20.3 cm (2.5 - 8.0 in) and are approximately flat on one side while bonded to a

metal piece on the other (see Fig. 1). Carbon black is added during the processing of the

rubber as a reinforcing filler to increase tensile strength, improve durability, and reduce

the tendency of rubber to swell when exposed to oils. The concentration and distributive

uniformity of carbon black within the composition greatly influence the rubber quality.

Because carbon black is a moderate electrical conductor while the other ingredients of

the rubber composition are nonconductors, the degree of dispersion of carbon black in

rubber is reflected in its electrical conductivity. [ 1-4].

CHAPTER I Introduction

.. , I

I I I r--.J 1---,

I I I

I I I I .,....- .. I , 'i

' J ,, '1 "_ ...... I I I I I I I I

I

I I

I I I r·-' r-·i I I I I I r 1 I r I

I I I :

. I I

Figure 1. typical tank track pad

CHAJ>TER I Introduction 2

1.2 Preliminary Study

In 1983, the United States Army Tank and Automotive Command (TACOM)

contracted the Bradley Department of Electrical Engineering to study the feasibility of

using microwaves for the evaluation of tank track pad rubber. In the preliminary study,

an 8 GHz signal was transmitted from one side of a 1.37 cm (0.50 in) thick rubber slab

and measured on the opposite side (hereon referred to as a two-way transmission

scheme) (see Fig. 2) [5]. Both the transmit and receive antennas were placed in close

contact with the sample. The mathematical model assumes the antennas are situated in

the far field and the rubber is homogeneous, isotropic, linear, and nonmagnetic. The

conductivity is inferred from a measurement of the transmission loss through the rubber

whereby transmission loss is approximated by

P - -2r1.dp r- e o (1.2.1)

2ef [JI . 0 J ex = -c- Ime, + J 2neJ' (1.2.2)

where d is the thickness of the sample, P0 is the source power, P1 is the received power,

o: is the absorption coefficient,fis- the frequency, a is the conductivity, e: is the relative

dielectric constant, e0 = 1/36n x 10-9F/m, and c = 3 x 108 m/s. The relative dielectric

constant, e~ , cannot be extracted from this measurement approach and must be assumed

from a priori knowledge.

Absorption loss measurements for rubber compositions with an assumed nominal

value of e: = 5 vary from 10 to 25 dB per L37 cm (0.5 in), implying the conductivity at

CHAPTER 1 Introduction 3

TRANSMIT HORN

RUBBER PAD

d

RECEIVE HORN

Figure 2. Two-way transmission scheme: measurement test configuration used in the preliminary study.


8 GHz is somewhere between l and 3 S/m. Thus, the study concluded that it was indeed

feasible to use microwaves for rubber evaluation.

1.3 Conductivity 1Vleasurement System for Actual Tank

Pads

In 1986, TACOM contracted the Bradley Department of Electrical Engineering to

develop an ac conductivity measurement system for implementation with actual tank

track pads. The measurement system is to be used for quality control purposes.

TACOM expressed the need for a system that did not necessarily yield an accurate

absolute conductivity value but for a system that reliably distinguishes differences in

conductivity, both differences among different rubber samples and differences within a

sample (i.e., homogeneity).

The preliminary study demonstrated the possibility of using microwaves to

distinguish varying degrees of conductivity, but the two-way transmission scheme cannot

be directly applied to actual tank pads because the metal backing prevents microwave

transmission from one side of the rubber to the other. Hence, in a measurement system

initially proposed for study (hereon referred to as the one-sided transmission scheme), the

receiving horn was moved to the same side of the rubber pad as the transmitting horn

and microwave energy at about 8 GHz was launched from an oblique angle of incidence

(see Fig. 3). The receiving horn was oriented to receive the signal that was transmitted

into the tank pad and specularly reflected from the metal backing. The direct signal and

the signal specularly reflected from the front surface were to be blocked by placing a

CHAPTER 1 Introduction s

TX HORN

x .l I I I

METAL BACKING

Figure 3. One-way transmission scheme: measurement system for actual tank track pads initially proposed.


large conducting plate between the transmit and receive horns. Absorption loss was

measured by comparing the received power from the rubber and from a metal plate

alone. Both parallel and perpendicular polarization cases were modeled. Measurements

were to be made in the near field to alleviate free space spreading loss but modeling

assumed far field antenna placement for mathematical simplicity. A Hewlett-Packard

HP8559A/853A spectrum analyzer was acquired for use as a receiver.

In August 1986, the author joined the investigation group and attempted to

implement the one-sided transmission scheme. Measurements were performed on rubber

slabs with a metal plate placed behind the slab to simulate the metal backing on actual

tank pads (the author later learned in November 1986 that the metal backing on actual

tank pads are nonplanar). The angle of incidence was set by using a measuring straight

edge and trigonometric methods. Results from the one-way transmission scheme were

not promising. A disadvantage found with this technique is the extreme sensitivity of

measurement to the placement of the sample, the angle of the horns, and the placement

of the shield. This sensitivity to position is a consequence of near field coupling,

refraction, noise, and the inadequacy of the shield. Another disadvantage with this

system is the range of the conductivity expected is such that the signal that is specularly

reflected from the metal back undergoes at least 40 dB more absorption loss than the

signal that is specularly reflected from the front surface. Therefore, the shield must be

capable of isolating an unwanted signal at least 40 dB greater than the desired signal, a

difficult condition to satisfy. In addition, the back conductor is not flat, and the signal

reflected from the back conductor is only approximately specular.

In December 1986, when the one-sided transmission scheme began to show little

promise of success without substantial and costly modification, the author began a


review of literature and theory to find an alternative measurement system. During this

time of review, it was also felt that the initial specifications given for the project

development were only loosely defined. Through an iterative process of working on the

project until an ambiguity arose, and then raising pertinent questions to resolve the

ambiguity, a more rigorous performance criteria was developed by the author and Dr.

R. C. Robertson of Virginia Tech. It was determined that the ideal conductivity

measurement system would meet the following performance requirements:

1. Sufficient sensitivity .. system must reliably indicate if the conductivity is in the abnormal range.

2. Nondestructive measurement - the evaluation must not alter the structure or performance of the tank pad.

' \

3. One-sided measurement - nonremovable metal backing precludes two-sided transmission measurements, i.e., the preliminary study approach.

4. Ample tolerance to sample thickness - system must be capable of evaluating tank pads 6.35 - 20.3 cm (2.5 - 8.0 in) thick.

5. Sufficient immunity to noise - measurements are to be performed in the manufacturing environment

6. Batch processing capability - evaluation duration should be minimal.

7. User-friendliness - measurement system must require minimal interpretive skills from the ultimate user.

The first criterion is still rather nebulous; the value of AC conductivity which demarcates

the boundary of normal and defective rubber is not known and hence, the amount of ,

resolution the measurement system must possess is also unknown. Although -the

preliminary study by de Wolf chose a frequency of 8 GHz for analysis, there are no

restrictions on the frequency chosen for analysis and so it would seem that a system

capable of a wide frequency range of analysis would be advantageous at this time.

In January 1988, a complex permittivity measurement system consisting of a

network analyzer and a open~ended coaxial waveguide transducer was chosen for study.


The real dielectric constant, in addition to the conductivity, are inferred from complex

reflection coefficient measurements in which the transducer is simply placed against the

rubber sample. This noninvasive measurement approach is based on Deschamps

antenna modeling theorem which relates the effective admittance of an antenna

embedded in some arbitrary medium to the effective admittance of the· same antenna

embedded in free space. Relating the dielectric constant and conductivity to the

reflection coefficient yields a set of nonlinear coupled equations. However, if the system

is operated at frequencies sufficiently low such that the radiation conductance of the

transducer in free space is negligible, then an elegant analytical solution can be obtained.

Negligible radiation conductance implies that the constitutive parameters being

measured are primarily those of the rubber material within a radius of the outer

conductor of the transducer since only the near-field will be present to interrogate the

rubber sample. The network analyzer is capable of swept frequency measurements and

automation. Researchers have reported accuracies within 2% of reference data obtained

by other methods [23]. The network analyzer approach to measuring permittivity is best

suited for materials with loss tangents between 0.1 and 1.0 [36].

1.4 Overview of Tliis Presentation

In this study, the application of the network analyzer with an coaxial transducer

for measuring the conductivity of rubber is examined. Chapter 2 contains a general

theory development o_f time harmonic Maxwell's equations in a source free medium, the

concept of complex permittivity, transmission lines, waveguides, and modeling of

discontinuities. The purpose of chapter 2 is not meant to provide a comprehensive

CHAPTER I Introduction 9

survey of electromagnetic theory; rather, the intention of its inclusion is to set forth the

focus of this paper. In chapter 3, the theoretical development of the inference scheme

used to determine the conductivity and dielectric constant is provided. Also a qualitative

description of the network analyzer calibration is described. Furthermore, the two

experiments carried out for this project are presented. The first experiment examines the

coaxial w~veguide radiating into free space and the second experiment examines

permittivities of various rubber samples. Chapter 4 coneludes the presentation with

discussion of aspects of the project which need further study.

CHAPTER I Introduction IO

CHAPTER 2 General Theory Development

The purpose of this chapter is to provide some fundamental background for the

development of the ac conductivity measurement system. The development is not

mathematically rigorous and it is only meant to invoke the notion of constitutive

parameters which depend on frequency, the idea of a transmission line and its relation

to the TEM mode, the idea of the waveguide as a generalized transmission line and the

existence of the TE and TM mode, and finally the notion of evanescent waves and

discontinuities.

2.1 Time Harmonic Maxwell's Equations

The fundamentals of electromagnetic theory, established by James Clerk Maxwell

(1831-1879), are given by four equations

V x E(r t) = - _bB_(_r,r_) ' bl

(2.1.l(a))

CHAPTER 2 General Theory Development 11

t5D(r,t) V x H(r,t) = J + bt

V • D(r,t) = p

V • B(r,t) = 0

(2.1.l(b))

(2.1. l(c))

(2.1.l(d))

where r, represents the spatial coordinates dependence and t the time dependence, E is

the electric field intensity (V/m), D is the electric flux density (electric displacement)

(C/m2), Bis the magnetic flux density (T), His the magnetic field intensity (A/m), and

J is the volume current density (A/m2). These equations are collectively known as

Maxwell's Equations (note that boldface type denotes vector quantities.)

The field quantities D and E, B and H, or J and E, respectively are related by

constitutive parameters in the following way:

D=tE (2.1.2( a))

B=µH (2.1.2(b))

(2.1.2(c))

wheres is the permittivity (F/m), µis the permeability (H/m), and a is the conductivity

(S/m).

If the wave propagation is in a medium such that the constitutive parameters are

independent of the magnitude of the field quantities (linear) and independent of its

spatial position (homogeneous), and the field quantities D and E, or B and H,


respectively, are always parallel (isotropic), then the constitutive parameters B, µ, and a

may be regarded as scalar constants and hence, Maxwell's equations reduce to:

bH VxE=-µ-bt

bE VxH=aE+e-flt

p V·E=-. e

V ·H=O.

(2.1.3(a))

(2.1.3(b))

(2. l.3(c))

(2.1.3(d))

Linear, homogeneous, isotropic media are called simple media. However, if the time

derivative of e, µor a is not negligible, then (2.1.3(a)-(c)) are no longer valid [9]. The

material is still linear in the general sense, and the permittivity, permeability, or

conductivity are linear functions of the frequency and hence, Maxwell's equations

become

bH VxE=-µ(w)-M

bE V x H = a(w)E + e(w) Tr

p V·E=--e(w)

v ·I-I =0.


(2.1.3(e))

(2.1.3(/))

(2.l.3(g))

(2.1.3(h))

13

Field vectors that vary with space coordinates and are sinusoidal functions of time

can be represented by vector phasors that depend on space coordinates but suppress

time dependence. We can write a time-harmonic F field as

F(x,y,z,t) =Re[ F(x,y,z) l 001 ]. (2.1.4)

With the additional constraint that the electromagnetic fields are in a source free region,

the time harmonic Maxwell's equations in phasor notation are

V x E(r) = - jwµ(w )H(r) (2.1.S(a))

V x H(r) = a(w)E(r) + jwe(w)E(r) (2.1.5(b))

V • E(r) = 0 (2.1.5(c))

V • H(r) = 0. (2.1.5(d))

Henceforth, usage of phasor representation and the dependence of e and a on w 1s

implied. In addition, only nonmagnetic material will be considered such thatµ= µ0 •


2.2 The Concept of Complex Permittivity

Maxwell's second equation may be manipulated in the following way:

V x H =(a+ jcm:)E

=F»(e +~ )E JW

= jwe ( ...£_ + -!I-- )E o eo JWeo

(2.2.l(a))

where e0 represents the permittivity of free space; If the notation e. is introduced such

that

e a ec = eo( T + -. - ) o JWE0

= e0(e~ - je;) (2.2.l(b))

= eoer.

We may write

V x H = jwecE = jwe0 erE. (2.2.l(c))

The quantity.e. is called the complex permittivity ande, is called the complex relative

p~rmittivity. The introduction of complex permittivity allows Maxwell's equations for

wave propagation in a conducting media to be manipulated similarly to Maxwell's

equations for wave propagation in a nonconducting media·(a = 0).

CHAPTER 2 General Theory Development IS

2.3 Transmission Line Concepts

Transmission line concepts can be developed either from the point of view of

electromagnetic field theory or essentially from the point of view of electric circuit

theory. In the latter case, the transmission line is considered as a uniform distributed

parameter circuit consisting of certain values of series inductance and resistance and

parallel capacitance and conductance. Energy storage in the magnetic field is accounted

for by the series inductance L per unit length, whereas energy storage in the electric field

is accounted for by the distributed shunt capacitance C per unit length. Power loss in

the conductors is taken into account by a series resistance R per unit length. Finally,

the power loss in the dielectric may be included by introducing a shunt conductance G

per unit length. A uniform transmission line is one for which the primary line constants

L, C, R, and G do not change with distance along the line. The distributed parameters

are necessary for modeling propagation time delay caused when the physical dimensions

of the network are a considerable fraction of a wavelength long.

Consider the distributed parameter model in Fig. 4. Applying Kirchoff s voltage

and current laws and letting {Jz -+ 0, the following transmission line equations are

derived:

Ci V(z) (iz = Rl(z) + jwLI(z) (2.3. l(a))

Ci I(z) -~ = GV(z) + jwCV(z). (2.3.l(b))

Equations (2.3.l(a)) and (2.3.l(b)) can be manipulated into:


1(%) L&z

• •

V(z) GSz

Figure 4. Distributed parameter transmission line model


~-- C~z

l(Z ... ~Z) ~

V(Z+iz)

17

62 V(z) -y2 V(z) = 0 c5z2

2 c5 l(z) -y2 /(z) = 0

()z2

where the complex propagation constant y is given by

y =a+ jfJ = )(R + jwL)(G + jwC) .

(2.3.2(a))

(2.3.2(b))

The term cc is the attenuation constant and fl is the propagation constant. If the line is

assumed lossless (a= 0) then the general solution to (2.3.2(a)) and (2.3.2(b)) is

V(z) = Ae -jf3z + Be"f3z (2.3.3(a))

/( ) A -jf3z B jf3z z =-e --e-Zo Zo

(2.3.3(b))

where the constants A and B are complex quar{tities, Z 0 = J ~ is the characteristic

impedance of the transmission line and fl = wflC. Thus, we see the solution to the

transmission line equations are of the transverse electromagnetic (TEM) form.

Now consider a transmission line of characteristic impedance Z 0 terminated in a

load ZL (See Fig. 5). The reflection coefficient f(z) is defined as

aiJf3z r(z) = ll Ae-1 z

= r e"2f3z 0

(2.3.4(a))

= re""' e"2f3z


Ae-"'61- I I I I

ZL Zo I

~

Be .. 'lSz.

Z=O

Figure 5. Terminated transmission line


where r is the reflection coefficient at z = 0 which can be written in terms of Z0 and

(2.3.4(b))

2.4 Transmission Line Representation of Waveguides

A uniform waveguide is characterized by its cross section which is identical in both

size and shape everywhere along the longitudinal direction of propagation (see Fig. 6).

Within such regions the electromagnetic field may be represented as a superposition of

an infinite number of mutually orthogonal functions -- a generalized Fourier series. The

mathematical representation of the electromagnetic field within a uniform region is in

the form of a superposition of an infinite number of modes or wave types. The electric

and magnetic field components of each mode are factorable into form functions,

depending only on the cross-sectional coordinates transverse to the direction of

propagation, and into amplitude functions depending only on the coordinate in the .

propagation direction. The transverse functional form of each mode is dependent upon

the cross-sectional shape of the given region and, except for the amplitude factor, is

identical at every cross section. As a result the amplitudes of a mode completely

characterize the mode at every cross section. The variation of each amplitude along the

propagation direction is given implicitly as a solution of a one-dimensional wave or

transmission line equation. According to the mode in question, the wave amplitudes

may be either propagating or attenuating [10].


z

Figure 6. Uniform waveguide with an arbitrary cross-section


To stress the independence of the transmission line description upon the

cross-sectional coordinate system, an invariant transverse vector formulation of the

Maxwell field equations is employed. This form of the field equations is obtained by

elimination of the field components along the z-direction and can be written, for a steady

state of radian frequency w, as

(2.4.l(a)}

(2.4.l(b))

where E1 = E1(r) is the electric-field intensity transverse to the z-axis, H, = Hi(r) is the

magnetic-field intensity transverse to the z-axis, ( = ~ = jf is the intrinsic

impedance of the medium, k = w.j;; = 2)..rc is the propagation constant in the medium,

V, = V - z,, :z is the gradient operator transverse to the z-axis, and 8 is a unit dyadic

such that 8 • A= A• 8 = A. The z components of the electric and magnetic fields follow

from the transverse components by the relations

(2.4.2(a))

(2.4.2(b))

The cross-sectional dependence may be integrated out of (2.4.1) by means of a

suitable set of vector orthogonal functions. Functions such that the result of the

operation V 1V 1 on a function is proportional to the function itself are of the desired type

provided they satisfy, in addition, appropriate conditions on the boundary curves of the

cross-section (eigenfunction-eigenvalue problem). Such vector functions are known to

be of two types: the E-mode function (TE and TEM) functions, e; , defined by


(2.4.3)

(2.4.4)

where

(2.4.5)

<Di= 0 on s if k~i ¥:- 0

a<D. --1 = 0 on s if k' · = 0 OS Cl

and the H-mode (TM) functions, e;' defined by

(2.4.6)

(2.4.7)

where

(2.4.8)

o'If Tv=O on s.

The subscript i denotes a double index mn and v is the outward normal to s in the

cross-section plane. For the sake of simplicity, the explicit dependence of

e~ , e;' ,' <D;, and '¥; on the cross-sectional coordinates has been omitted in the writing of

the equations. The constants k:, and k;, are defined as the cutoff wave numbers or

eigenvalues associated with the guide cross section.


The functions e1 possess the vector orthogonality properties

I I I I . { 1 for i = j} ej • ej dS = e'j • e'j dS = . .

0 for 1 i= J

J J ej • e'j dS = 0

with the integration extended over the entire guide cross section.

The transverse electric and magnetic fields can be expressed m terms of the

orthogonal functions by means of the representation

(2.4.9(a))

Hr= 2)i (z)hi +,Lr; (z)h'i (2.4.9(b)) l

and inversely

(2.4.IO(a))

(2.4.lO(b))


Ii = f J Ht • hj dS (2.4.lO(c))

(2.4.lO(d))

The longitudinal field components then follow as

(2.4.1 l(a))

)k(Hz =I Vj (z)k~12 'fi (2.4.1 l(b)) i

As long as no discontinuities exist within the waveguide cross-section or on the

guide walls the substitution of (2.4.9) transforms (2.4.1) into an infinite set of

transmission line equations. Explicitly,

(2.4.1 l(a))

(2.4.ll(b))

which completely define the variation with z of the mode amplitudes JI; and I;. The

superscript distinguishing the mode type has been omitted since the equations are of the

same form for both modes. The parameters K; and Z; are however of different form; for

E-modes


(2.4.12(a))

K 1 K~ Zj =(-+= wie (2.4.12(b ))

for H-modes

"-Jk2 k"2 K1 - - cl (2.4.12(c))

(2.4.12(d))

2.4 Coaxial Waveguide Modes

Consider an infinitely long coaxial waveguide (see Fig. 7) where the inner and outer

walls assumed to be perfect conductors. The auxiliary scalar functions for the three

possible modes are as follows:

TE modes E. = 0:

(2.4. l(a))

where


~--2~ __ ....,....

Figure 7. Cross-section of a uniform coaxial waveguide

CH.APTER 2 General Theory Development 27

( ') ~ Zm Xtb = 2 (2.4.l(b)).

m = 0,1,2,3, ... ,

The arguments a,b are the outer, inner conductor radii, c == a/b, r is the cross sectional

radial coordinate, Jm is a Bessel function of the first kind and order n and Nm is a

Neumann function of order n (Bessel function of the second kind and order n).

TM Modes H. = 0:

'I1 = zm( xi ~ ) sin(mc/>) (2.4.2(a))

Jm( xi f )N~ (xi) - Nm( xi f )1~ <xi) (2.4.2(b))

{[ ::(~:i) ]'[i-( crl )']-[I-(~ )']}"'

m = 0,1,2,3, ... ,

where' x: = x~ .. is the nth nonvanishing root of the derivative of the Bessel-Neumann

combination. Zm( cx1).


TEM mode E. = H, = 0:

ln r <Doo = -~===-

) 2n ln ~ (2.4.3)

The modes for a coaxial waveguide are similar to a single conductor circular

waveguide, but single-conductor waveguides cannot support TEM waves. Magnetic flux

lines always close upon themselves. Hence, if a TEM wave were to exist in a waveguide,

the fields of Band H would form closed loops in a transverse plane. However, the

generalized Ampere's circuital law requires that the line integral of the magnetic field

around any closed loop in a transverse plane must equal the sum of the longitudinal

conduction current inside the waveguide. By definition, a TEM wave does not support

an E, component; consequently, there is no longitudinal displacement current. The total

absence of a longitudinal current inside a waveguide leads to the conclusion that there

can be no closed loops of magnetic field lines in any transverse plane. Therefore, TEM

waves cannot exist in a single-conductor hollow (or dielectric-filled) waveguide of any

shape.

The TEM mode is uniquely different from the other modes in the following ways:

1. Both the electric and magnetic fields of TEM wave are perpendicular to the

direction of propagation. Higher order modes also have a field component in the

direction of propagation.


2. A transmission line that is to transmit TEM waves must have two or more

conductors. Higher order modes can propagate on any kind of transmission line,

including single conductor structures such as hollow waveguides.

3. TEM waves propagate at any frequency; higher mode waves can propagate only

above certain cutoff frequencies that depend on the particular mode and cross

section of the transmission line.

4. The phase velocity of TEM waves is independent of frequency, while the phase

velocities of waves belonging to the higher order modes are frequency dependent

(the electric and magnetic fields of a TEM wave are uniquely related to a voltage

and current because the field equations satisfy Laplace's equation)

2 .5 Modeling Waveguide Discontinuities

The dimensions and field excitation of a waveguide usually permit only one mode

to propagate. Because of the transmission line behavior of the mode amplitudes, the

field quantities .of this dominant mode can be described almost everywhere in terms of

the voltage and current. However, if the· uniform structure of the waveguide is

interrupted, an infinity of nonpropagating higher order modes are generated. The '

excited higher order modes give rise to a reflected wave and storage of reactive energy

in the vicinity of the discontinuity. The transmission line description can be extended

to describe the behavior of the higher order modes by introducing mode voltages and

currents as measures of the amplitudes of the transverse electric and magnetic field


intensities of each of the higher modes. Thus, for a complete description of the

discontinuity an infinite number of transmission lines are required. For example, the

electromagnetic field at the opening of a coaxial waveguide opening into some arbitrary

medium consists of a TEM wave superimposed with other field components satisfying

the boundary conditions. Because of the axial symmetry, only TM0M modes are excited

by the discontinuity, and the aperture field is represented as the summation of the TEM

mode and a series of TM0M modes.

If the effect produced by the dominant mode is of most interest and the magnitude

of these higher order modes is small, approximate techniques may be used to describe

the effect at the discontinuity. These mathematical techniques have been classified as

[10]

l. the variational method

2. the integral equation method

3. the equivalent static method

4. the transform method

Once the field quantities have been determined, the effects of the discontinuity fields can

be represented by equivalent lumped parameter circuit models since the nonpropagating

higher order modes restrict the complication in field description to the immediate vicinity

of the discontinuity.


CHAPTER 3 Open-Ended Coaxial Waveguide

Transducer for Complex Permittivity Measurement

The open-ended coaxial transducer approach to measure the complex permittivity

of biological substances is well documented [ 18-35]. This noninvasive measurement

·procedure relates the complex reflection coefficient to the equivalent lumped parameter

conductance and capacitance of an unknown medium. The measurement system

consists of a coaxial waveguide whose transducer surface is faced off flat and a vector

network analyzer. In this section, the theoretical development of this technique is

presented, followed by a qualitative description of a calibration procedure necessary for

accurate reflection coefficient measurement. This section concludes with a presentation

of experimental results.

CHAPTER 3 Open-Ended Coaxial Waveguide Transd11cer for Complex Perinittivity Measurement 32

3.1 Theoretical Basis

The theoretical basis behind this permittivity measurement approach is the Deschamps

antenna modeling theorem which relates the effective admittance of an antenna

embedded in some arbitrary homogeneous medium to the effective admittance of the

same antenna embedded in free space [13]. Mathematically stated, Deschamps theorem

says

(3.1.1)

where Y2 is the admittance of the antenna in the arbitrary nonmagnetic (µ = µ0 ) medium,

Y1 is the admittance of the antenna in free space, and e, is the complex relative

permittivity of the arbitrary medium. The Deschamps antenna modeling theorem is

valid for any probe and can be employed provided an analytical expression for the

terminal impedance of an antenna is known and the dielectric medium extends far

enough to contain the antenna radiation field.

An open-ended coaxial waveguide radiating into free space from an infinite ground

plane can be modeled as a transmission line terminated with an equivalent conductan~e

(representing far field radiation) in parallel with an equivalent capacitance (representing

reactive near field) (see Fig. 8). The equivalent admittance, Y1 = G0 + jB0 = G0 + jwC0 ,

for the coaxial waveguide radiating into free space is given by:

J~ 0

~() e [Jo(ka sin8) - lo(kb sin8)]2 sm

(3.1.2(a))

CHAPTER 3 Open-Ended Coaxial Waveguide Transducer for Complex Permittivity Measurement 33

Figure 8. Open-ended coaxial waveguide transducer: (a) coaxial waveguide radiating into air (b) lumped parameter model.


Ya J0n 2 Si(kJ a 2 + b2 - 2ab cos</J)

n ln .E... b (3.l.2(b))

- Si(2ka sin ~ ) - Si(2kb sin ~ ) d</J

where J0 (x) is a Bessel function of the first kind, order 0, and argument x, Si(x) is the

· · 1 ~ · f k 2n Y l · h h · · d · f sme-mtegra 1unct10n o x, = -.-, 0 = -Z is t e c aractenst1c a mlttance o the A. o

coaxial waveguide, and a and b are the radii of the outer and inner conductor,

respectively. For a = 3.50 mm and b = 1.52 mm (7 mm, 50Q airline), equations

(3.1.2(a)) and (3.1.2(b)) are valid for frequencies less than 75 GHz [10, p. 212]. The

· above circuit parameters are derived from the variational method approach assuming a

dominant TEM mode incident at the aperture and are presumed to be in error by less

than 10 percent over most of the range of validity.

Examination of Fig. 9 shows that the capacitance C0 for a 7mm airline is essentially

constant up to 2 GHz. but beyond that frequency the capacitance C0 varies as a function

of frequency due to the increase of evanescent TM modes being generated at the

junction discontinuity. The magnitude of c. and the frequency at which it begins to rise

depends on the absolute dimensions of the line and the dielectric filling the waveguide.

The admittance given by equations (3.1.2(a)) and (3.1.2(b)) can be approximated

by less complicated functions if the outer radius of the coaxial waveguide is small

compared to a wavelength at the frequency of operation. For a 7 mm airline, the outer

radius is small for frequencies less than 15 GHz (Marcuvitz states that the approximate

equivalent circuit values are within 15% of values yielded by (3.1.2(a)) and (3.1.2(b)))

[10, p. 212]. The approximate admittance is given by


i Ill u ; ~

~

~ ...., Ill u I I ~ e

COAXIAL LINE RADIATING INTO AIR 0.100 I

I

i O.OIO

O.OIO

I i

0.070 j

Q.080

o.oeo

0.°"41

I 1

I l~ -t--~-·~: ~--t~~~:~~~--4~~-.-~1~~1 I

1 i I 0.030

0.020

0.010

0.000 O.D

0.084

G.083

OJJl2

O.Oe1

O.OIO

0.079

0.078

0.077

0.071

~

0.075 0.0

I !

40.0

l'REQUENCY (CHz)

IO.O

COAXIAL LINE RADIATING INTO AIR . UARCINIT.Z UOD£1.: a-.1.!0mm ll-1.521ntn

/ , /

/ /

v /

/ v

_/ /

-----2.0

IO.O

/

7 / /v

.10.0 12.0

Figure 9. Equivalent capacitance of a coaxial waveguide radiating into air: model results; a ... 3.50 mm, b • 1.52 mm, (a) 0-80 GHz (b) 0-12 GHz.


COAXIAL LINE RADIATING INTO AIR UARClMTZ MODEL: a-3.SOmm b-1.52mm

0.07!18 0.0791

0.07514

- / !/

0.071562 / v 0.07560 c

Q. 0.07558 I ! // I ' L /' ......

Id () a u § ..J ~ g

0.07551 0.079'4 0.07582

0.07550 0.07548 0.07!W& 0.075-M

0.07542 0.075-40

I I I ! ! /,,,

I v /

I .,v I ,,v // '

i / ·-1+1 ./ i..-~

I v ' I / I t

~ I

0.07538 0.07536

_,...

-l--" i

0.07534 : ' I

O.& a.a 1.D 1.2 1.4 1.1 1.8 2.D

FREQUENCY (CHz)

Figure 1 O. Equivalent capacitance of a coaxial waveguide radiating into air: model results; a = 3.50 mm, b = 1.52 mm, 0.6-2.2 GHz.

'

2.2


TCµo eo a - b 4 3/2 5/2 [ 2 2 ]2 Go= 12 ln(b/a) w (3.1.2(c))

B -w E -1 -we { 8(a+b)e0 [ ( 2Jab ) ]} o - (ln(b/a))2 a+ b - o

(3.1.2(d))

where E(x) is the complete elliptic integral of the second kind and argument x. [9, p. 112]

[10, p. 216]

Using the Deschamps theorem, the free space equivalent admittance is related to

the equivalent admittance of the coaxial waveguide radiating into an arbitrary medium

with the relation (see Fig. 11)

(3.1.3)

At the plane of discontinuity (boundary between the unknown medium and the

coaxial waveguide, defined as z = 0) the effective admittance is related to the measured

complex reflection coefficient by

( 1-r) - Y2 =Yo 1 + r (3.1.4)

Substituting (3.1.3) into (3.1.4) and comparing real and imaginary parts, we obtain the

following set of equations

+ R [( I - ·_!!__ )5/2] f.oGo = f.oYo(l - r 2) CJ e f.r J Wf. C ( 2

0 o C0 I + r + 2rcos¢) (3.1.5(a))

e' + Im[(e' - ._a_ )st2] _G_o_ - ___ -_2_Y_o_r_si_n_4> __ r r J weo wCo - roC0 (l + r 2 + 2rcos¢)

(3.1.5(b))


Figure 11. Open-ended coaxial waveguide transducer: (a) coaxial waveguide radiating into an unknown sample (b) lumped parameter model.

where


R [( I • (1 )5/2] e e -J-- = r we0

,5/2[ ( (1 )2]5/4 [ 5 -1 ( (1 )] e, 1 + , · . cos -2 tan , . we, e0 we, e0

(3.1.5(c))

and

l [(. I • (1 )5/2] m e -J-- = . . r we0

- e~5 '2[1 + ( ~ ) 2]514sin[ 52 tan-1( ~ ·)] we, e0 . we, e0

(3.1.5(d))

Equations (3.1.5(a)) and (3.1.5(b)) are nonlinear coupled equations that must be.

solved to obtain the. conductivity and relative dielectric constant of the unknown

material. One approach to solve (3.1.5(a)) and (3.1.5(b)) fore~ and a is by iterative

methods. However, if the outer conductor. radius is small compared with the guide

wavelength, then G0 is much smaller than the B. (see Figs. 12-17). Hence, at lower

frequencies G. may be neglected. Stuchly states that G. can be excluded from the model

when a/).< 0.04 [21]. For G0~0, equations (3.l.5a) and (3.l.5b) decouple and provide

an elegant analytic solution for o and e~. Using Stuchly's criteria for 7.0 mm coaxial

airline, G. may be excluded for frequencies below 3.4 GHz.

The probing depth, fJ,, of the coaxial transducer is limited since the near-field is the

interrogating field. The probing depth is dependent on the coaxial line dimensions and

Tanabe and Joines calculate that fJ, is given by fJ, = 2(a- b) where a, bare the radii of

the outer, inner conductors [18]. Swicord and Davis similarly calculate that the bulk of

the power is absorbed within a radius of the antenna on the order of the radius of the

outer conductor (See. Fig. 18) [39]. Brunfeldt experimentally verified the theoretical

estimates of Swicord and Davis [37]. Thus, for a 7 mm line, fJ, = 0.35 cm.


COAXIAL LlNE RADIATING INTO AIR LtARaMTZ MOOE!.: o-.1.llOmm lt-1.ll2mm

24.00

22.00

20.00

18.00

/ v- ~

/ """ I I "--'I

ii' 115.00 E ..., Ill 14.00 (J g 12.00 :I Q

10.00 ~

I I I

I I I v

c5 LOO

a.oo

4.00

I I I

/ /

2.00

0.00 __/"'

0.0 60.0 llO.O

COAXIAL LINE RADlATlNG INTO AIR 0.-60

0.35

0.30

ii' ! 0.26 tl

~ 0.20

8 0.15

0

I v - I

I I

/ 0.10 /

/ o.o5 /

~ ~ -o.oo

0.0 2.0 LO !0.0 12.0

Figure 12. Equivalent conductance of a coaxial waYeguide radiating into air: model results; a • 3.SO mm, b • 1.52 mm, (a) 0-80 GHz (b) 0-12 GHz.


,... iJ.OOE-o4 -Ill U2J50E-Q4 ~ ,

6 ~2JJOE-G4

8 ci1J50E-o4

1.00E-cM

O.OOE+oo 0.1

COAXIAL LINE RADIATING INTO AIR UARClMTZ MODEL: a-3.50mm b-1.52mm

I -

I I

T I .

I I I/ I ' I

l/ / --· v

! /

I v 7 --f...-""

- 1..--

0.8 1.0 1.2 1.-4 1.1 1.8

FREQUENCY (GHz)

! / v I v

I/

2.0

Figure 13. Equivalent conductance of a coaxial waveguide radiating into air: model results; a = 3.50 mm, b = 1.52 mm, 0.6-2.2 GHz.


t ..., IS

I i i

! fj

~· a i ii

11.00 17.00 11.00 15.00 14.00 ,~

12.GO 11.00 10.00 9.00 a.oo 7.00 l.00 5.00 -4.00 3.00 2.00 1.00 o.oo

7.00

I.DO

S.00

-4.00

;J.QO

2.00

1.00

0.00

COAXIAL LINE RADIATING INTO AIR lolMC\MTZ WCDEL: a-3.80mm Do-1.12nlnl -. ,," ~ .

-1- '\ - \ f--- \ -- \ l i

~-----1-------1--;f--+--- I ±= ~~+==- --= ! ; ~ -!----------- ~- . '

' , I -l--+ , I . i 1 .. ~t- I

I /

II 0.0

--

I I

v 0.0

I I

I I : t

, \ I I ==l I I r-H -----\--I

I I !

·-I I T--t-

----+--

-t-· ---t =t

l

20.0

!

. -40.0

FREQUENCY (GtGr)

~ ·-~~ '-" ........... ~

·--

IO.O IO.O

COAXIAL LINE RADIATING INTO AIR MMCWrTZ MODEL: a-3.llOmm ti-1.52mnt

-

/

/

v-- I v v

I v / v I

/ I I

. - ·- .

v v· ~ v ' ' i1 . --

2.0 LO a.o 10.0 12.0

Figure 14. Equivalent susceptance of a coaxial waveguide radiating into air: model results; a - 3.50 mm, b = 1.52 mm, (a) 0-80 GHz (b) 0-12 GHz.


COAXIAL LJNE RADIATING INTO AIR MARClMTZ MODEL: a-3.SOmm b•1J52mm

1.20 ...,.-........,..-.,----r-1, --r--r---.--.....---T--'!!-~-r-....---.-1. -...--__..-/

1.10 -l--i!---1---1---+-_.__-- I ' , Iv 1.00 -+--1----l-----+--+---1---1j1---.!----l--1!--.- v>+~

I I i/ ·~-.,.,..~~---l'--~~4---1 I I I I /I

0.90

o.eo --+----'-i-1 1 rn-7 A-tt-1 ~,. -~t-tt-T_/ r 1 t-r 1 1 ~1 ~--t O.&O 11/1 11 Tj a.so --+-L-/----1--// ,,,.. ' ·- I ' -+-1. -+-t-t-·---i-1: _.... _ ___.

/ I I 0.'40 / .

0.1 o.a 1.0 1.2 1 .... 1.i 1.8 2.0

FREQUENCY (GHz)

Figure 15. Equivalent susceptance of a coaxial waveguide radiating into air: model results; a = 3.50 mm, b = 1.52 mm, 0.6-2.2 GHz.


2.2

COAXIAL LINE RADIATING INTO AIR tU.RQM1Z UODEL: ~ 11-1.12mnl

7.00

I.GO

Lao

...... ~

I \ I I \

$ 4.00

I "'° 2.GO

1.00

I I I/

/ ,___-~ -o.ao

0.0 llO.O

COAXIAL LINE RADIATING INTO AJR MARCW11% YODEL: ...ulOtnm ll-1.52mm

/ v I v

/ v

/ / v

1.00E-G2 /

/ / ,___. -- --

0.0 u a.o 10.0 12.0

Figure 16. Conductance/Susceptance vs. frequency for coaxial waveguide open into air: model results; a =- 3.50 mm, b = 1.52 mm, (a) 0-80 GHz (b) 0-12 GHz.


COAXIAL LINE RADIATING INTO AIR MAACWITZ MODEL.: a-J.~mm b-1.~2mm

~OOE-04--~~-r~....,.-~~--r~-;--..---r--r~,--"'l~-,-~T""--:r--r--i

3.00E-04 I '

! i I m UOE-04 Lt+-+-' I I I o I l ··2.00E-04 · 0

l I I I 1~E-04 nl. _ __,,,__--1-"

1~-MH-1-t 5.00E-05

O.OOE+OO ...i::::::~~~-+--4--.f--~--+-+-t--t--t--+----t---t-;---; 0.1 O.IS 1.0 1.2 1.4 1.6 2.0

FREQUENCY (GHz)

Figure 17. Conductance/Susceptance vs. frequency for coaxial waveguide open into air: model results; a = 3.50 mm, b = 1.52 mm, 0.6-2.2 GHz.

2.2


s E> FIEL.0

Figuri= 18. Fringing field at the coaxial waveguide aperture: open-ended waveguide radiating into air.


3.2 Network Analyzer Basics

A schematic of a basic network analyzer system for reflection coefficient

measurement is shown in Fig. 19. The network analyzer consists of a source, directional

couplers, and a receiver. The test signal generated by the reflection from the device

under test (OUT) is compared to the reference signal to obtain the reflection coefficient.

However, the reflection coefficient measured by any physical realizable network analyzer

is not the actual reflection coefficient; the resulting measurement is a vector sum of the

actual reflection coefficient plus error terms .

. Measurement error may be classified in one of two groups: random and systematic

(12]. Random errors are nonrepeatable measurement variations caused by noise,

temperature change, worn connectors, and any other physical changes rendered to the

test configuration between calibration and measurement. A calibration procedure

cannot account for random error, which can only be controlled by allowing sufficient

time for the network analyzer to warm-up and minimizing changes to the system after

calibration. Systematic error are repeatable errors which the system can measure and

are the more significant source of measurement error. Systematic errors are introduced

by imperfect directional couplers (the incident traveling wave is not completely isolated

from the reflected traveling wave), source mismatch (the characteristic impedance of the

source is not matched completely·to the transmission line), and frequency tracking error

of the source (the indicated frequency is not the actual frequency). Source match error

is especially problematic in the measurement of high or low impedances (reflection

coefficients with magnitudes close to unity). The reflection from the device under test

(OUT) encounters various discontinuities as it travels back toward the source. These


--SOURCE

..... --~ TEST PORT

DIRECTIONAL COUPLER

~ I ) x

DIRECTIONAL COUPLER

~ I ~ x

REFERENCE SIGNAL ~

i t RECEIVER

t :>

TEST SIGNAL

Figure 19. Schematic of a basic network analyzer system for reflection coefficient measurement


discontinuities cause power to be reflected back toward the device, in tum causing the

incident power to vary as a function of the impedance difference between the

characteristic impedance of the test port and the input impedance of the DUT.

To account for systematic measurement errors, the network analyzer can be

modeled as a system that consists of a perfect network analyzer (one which has none of

the aforementioned measurement errors) connected to the DUT via a two port error

network (see Fig. 20). The measured reflection coefficient f mm is related to the actual

reflection coefficient f.ct by the equation

fact Efreq f meas = Edir + f 1 - Esource act

(3.2.1)

where Edi, is error parameter modeling the imperfect directional couplers, E1,.q is

frequency tracking error, and E,0 .,ce is the source mismatch error. The network analyzer

is calibrated by measuring at least three terminations whose r .. r is known: ideally, a

short, an open, and a load matched to the characteristic impedance of the test port.

However, open and matched load terminations are difficult to realize and actual

calibration schemes are more elaborate, usually consisting of a known capacitive

termination ( an "open") and a sliding load (load position is changed to vary the phase

of the reflection coefficient but maintain same magnitude). All network analyzer

measurements reported are calibrated by a procedure developed by W. A. Davis of

Virginia Tech [41].

The effect of an incomplete calibration is shown in Figs. 21 and 22 for a rectangular

waveguide radiating into air.· Although a standard calibration was applied at the test

port of the network analyzer (see Fig. 23), only calibration with a short was possible at


EFREQ

IDEAL DUT NETWORK

ANALYZER

••EolR ESOURCE

-,

Figure · 20. Error model for network analyzer calibration

CHAPTER 3 Open-Ended Coaxial Waveguide Transducer for Complex Permittivity Measureme_!lt 51

the WR-90 rectangular waveguide aperture (reference plane). The theoretical reflection

coefficient for dominant TE10 mode was calculated from (2.3.4(b)) and

Y0 rrb G=--o .A g

(3.2.2(a))

(3.2.2(b))

where a and b are the long and short dimension of the aperture. The guide wavelength,

Ag, is related to the free space wavelength, .A, by

(3.2.2(c))

and the characteristic admittance, Y0 is given by

(3.2.2(d))

Equations (3.2.2(a)-(d)) are valid for frequencies above 6.55 GHz and below 11 GHz for

WR-90 waveguide [10, p184]. In Fig. 21, the measured reflection coefficient exhibits an

oscillatory behavior while the predicted reflection coefficient is smoothly varying. The

oscillatory behavior is not a characteristic of the material being examined but is a

characteristic of the source mismatch between the coaxial waveguide of the network

analyzer and the rectangular waveguide of the transducer. Thus, the inherent

measurement error obscures the true nature of the material under examination.


WR-90 WAVEGUIDE RADIATING INTO AIR CUD

OM

0...40

' CUI i ! 0.30

"' 0 ~ 0.25

I 0.20 .. .. • 0.11

0.10

o.oa

0.00 a.a 9.0 10.0 11.0 12.0 _,_ ~data

WR-90 WAVEGUIDE RADIATING INTO AIR 70.00 I0.00 50.00 40.00 - --30.00 20.00 ,, 10.00

1 0.00

r -10.00 11 -20.00 ...... Ill -30.00

~ -40.00 -50.00 .. -I0.00 iii -10.00 -eo.oo -110.00

-100.00 -110.00 -120.00 -130.00

-t I I 1 "' I I I'\ A I I I I I I J I I I I I I \

i i i. \ I -\ J I ~I I I I I l I I {

\ I - - J \ - I \ I I \ I J "" I I I i f --~I I r I -~ J

I \ .....,..,. -r---L \ ,, ; --!'-' ...

1.0 !1.0 10.0 11.0 12.0

1 - 11rp1rtm.rial .data

Figure 21. WR-90 rectangular waveguide radiating into air - reflection coefficient: (a) magnitude, (b) phase.

CHAPTER 3 Open-Ended Coaxial Waveguide Transducer for Complex Permittivity Measurement SJ

2.IOCIOO 2AOCIOO 2.JOOOO 2.20000 J.10000 2.00000

11.10000 1.IOOOO J 1.10000

b 1.10000

i 1.10000

9 t.40000 • t.30000

1.20000 t.10000 t.00000 G.IOOOO G.IOOOO 0.70000

t.80000

1.IOOOO

uoooo 1.20000

.... 1.00000

! o.aoooo I O.IOOOO

~ 0.'60000 ~ i G.20000

ii 0.00000

-0.20000

-o.aoooo -o.aoooo

WR-90 WAVEGUIDE RADIATING INTO AIR N. U-wlz. ·~ llandllaall ,_114 -

( (\ 7 / . /

II ' /\ ' I

l I I

I I

!

... I ...

7 \ . .,. I \ v :~

/ \ I / v

v a.o t.O

2

r I

7 f\ ! I j \ / ··-\ J :Y \ 1 /I\ \~A ! \ I Yi f \ J

"' \ J v \ \I v

I

10.0

FREQUENC'f (OHz)

r\ / \./ I

l[/1 I 1 \ I i

' I -,

\ : I

T \ ! I \ I \ ~

11.0

WR-90 WAVEGUIDE RADIATlNG INTO AIR

I

'

' Joo'"" J !

v

LO

N. Man:wttz. WGYegUicle Handbook, p.1&4

/\ L i.+-- J

\ \ \ J

\ I v

_1

' I\. I\ I \

~ I -· -- I I

' I \ I ' I \

} \ 7

10.0

l'mMNC't (GIV)

\ 1 \]

_1_

\ \ j

I -, \ I i\ I v

11.0

\ \ \ \ \ I

) Ir/ '

12.0

'"-

' \ \ \ \

: I

i 12.0

Figure 22. WR-90 rectangular waveguide radiating into air - equivalent admittance: equivalent admittance; (a) conductance, (b) susceptance.


t TEST PORT

OF NTWK ANLVZR

WR-90

COAXIAL \ WAVEGUIDE J

\ ADAPTER

Figure 23. Schematic of the WR-90 rectangular waveguide transducer.

REF PLANE


3.3 Complex Permittivity Measurenient System

Description

The complex permittivity measurement system used in this study consists of:

• HP 84 IOA network analyzer

• 2 Hewlett-Packard (HP) 8620C Sweep Oscillator mainframes

• HP 8622A (0.01-2.4 GHz) plug-in (source)

• HP 86290A (2.0-18.0 GHz) plug-in (source)

• HP 8746B S parameter test set (directional couplers)

• IBM XT personal computer (system controller and data collection)

• Keithley System analog/digital converter (data communication)

• Bunker-Ramo Corp. APC-7 connector (transducer)

The coaxial transducer predominantly cited in the literature consists of a semi-rigid

dielectric-filled (Teflon) coaxial cable, faced off at the transducer end. The cable

provides flexibility in the placement of the transducer and the Teflon prevents soft

biological tissue from entering inside the waveguide. However, these coaxial cable

transducers do not allow the use of standard coaxial terminations for calibration and

therefore an alternative calibration procedure is performed using liquids of known

permittivity [23-28, 30). Since equipment availability was limited and we did not

anticipate the rubber sample filling the waveguide would be problematic, the APC-7

airline coaxial connector was selected as the transducer for this study. The APC-7

coaxial transducer (7 mm diameter, son, airline) allows convenient calibration using

standard terminations and, in addition, the planar surface of the connector establishes

a well defined reference plane and firm contact with the rubber surface. Ayer states that

the network analyzer configuration for complex permittivity measurement is most

CHAPTER 3 Open-Ended.Coaxial Waveguide Transducer for Complex Permittivity Measurement 56

accurate for samples with loss tangents (tan b) between 0.1 and 1.0 where tan o is defined

as [36]

e; tano=-. e' r

3.4 Experimental Results

(3.3.1)

Two experiments were performed for this study. The first experiment examines the

theoretical validity of the admittance values provided from equations (3.1.2(a) and (b))

as applied to the APC-7 connector radiating into air. In the second experiment,

reflection coefficient measurements are made with the APC-7 transducer placed on

various rubber samples from which the dielectric constant, conductivity, and loss tangent

are inferred.

Five sets of reflection coefficient measurements in the 0.20-12.00 frequency range

in 0.05 GHz increments were collected and averaged in the first experiment. Each set

of data was taken on a different day with the network analyzer recalibrated each time.

Ten points at each measurement frequency were averaged to reduce the effect of random

noise and frequency deviation. An interval of 50 ms was allowed between each

frequency change and 10 ms between each measurement at the same frequency. The

network analyzer was allowed a minimum warm-up period of 24 hours. Measurements

collected below 0.6 GHz and 11.0 GHz were eventually discarded from consideration

because of suspected problems with the calibration procedure in those ranges. The

measured reflection coefficients were curve-fitted by visual inspection and compared to


results obtained from FORTRAN programs developed for this project (see Appendix).

The theoretical reflection coefficients were derived from (2.24(b)) with appropriate

values for substituted for ZL'

Shown in Fig. 24 is the complex reflection coefficient for the APC-7 connector

radiating into air from 1.0 to 11.0 GHz. The onset of radiation loss, brought upon by

increasing G0 , is evidenced by the decreasing reflection coefficient magnitude in the upper

frequency range. The magnitude of the reflection coefficient is mainly dependent on the

radiation conductance G0 while the phase is mainly affected by reactive term B0 • For 7

mm airline, 50 ohm line, (3.1.2(c)) yields an expected value of G0 = 18.7 x 10-9w4, but the

measured data shows better agreement with an empirically derived value of

G0 = 15.0 x 10-9w4• A similar result is achieved for the reflection coefficient phase; the

measured results show better agreement with the results obtained from(3.1.2(b)) if0.010

pF was subtracted from the theoretical capacitive value c.~0.075 pF (see Figs. 9 and

25).

Several possible reasons are offered for the discrepancy. First, the physical

construction of the connector contains discontinuity structures not accounted for by the

model; for example, the dielectric bead supporting the center conductor. Second,

accuracy of the network analyzer configuration for permittivity measurement is poor for

material of loss tangents less than 0.1; difficulty with source matching is encountered

with reflection coefficients close to unity magnitude. Thirdly, the actual dimensions of

the APC-7 connector might vary from the desired dimensions because of manufacturing

uncertainty. Finally, the limited size of the ground on the APC-7 connector affects the

admittance. Bahl and Stuchly experimentally determined that the ground plane radius

must be greater than 0.2 times the wavelength in the free space if the ground plane is to


APC-7 RADIATING INTO AIR Way 215, 1 !188

1.mo

A 1.010

i 1.000 I? ....

~ O.lllO

J .. O.llO .. • o.970

O.llO 1.oe i.00 7.Dt t.OI 11.00

A - -. c - CUl"l9-flt -~ + ) (e:._= as.o ,.o uJ

APC-7 RADIATING INTO AIR Way 215, HIU

-2.ooo-.-~~~~~~~-~~~~~~~~~~~~~~~~-

4.000

-e.ooo -1.000

-10.000 ...... I -12.000

l -1"-000 ...,

j =::: ~ ~ -20.000 ~ • -22.000 ~

==LJ -30.000 ~~-r--~-,-~~.--~-r-~---.~~-.--~-.-~--1

1.00 3.00 5.00 7.00 1.00 11.00 -A l'REQU£NC'\' (GHz) . - ll'M9CMI (-G.01 Opf')

B

Figure 24. APC-7 connector radiating into air - reflection coefficient: experimental results; (a) magnitude, (b) phase.


0.074r,; 0.073 .

o.072 A

0.071 ~ ::::: l 0.018

0.087

0.061

0.015

0.0&4

0.06.'5

0.012 '

APC-7 RADIATING INTO A!R May 25. 1981

0.060 0.011 j l ::l-1~---.----,1r---~---r-1 --.,-:...._-.---,---r--.----;·

1•00 3.ao s.oo 1.00 1.00 11.oa

mea9UNCI. A

F1GUENC'1' (GHz) - mod9I (-0.010pf)

B

Figure 25. · APC-7 connector radiating into air - equivalent capacitance: experimental result


be considered infinite [2 l, 38]. Hence, for f = 1.0 GHz , the radius of the ground plane

must be a minimum of l 00 mm.

Although there exists a discrepancy between theoretical and measured results for

the coaxial waveguide radiating into air, it is not a critical problem since the primary

project goal is the development of a conductivity measurement system capable of

consistently distinguishing the relative difference between rubber samples. A

conductivity measurement system capable of yielding accurate absolute conductivity

values is not a necessity. Therefore, as long as the discrepancies in the system are

well-behaved, they are not problematic.

In the second experiment, the reflection coefficient is measured for various rubber

samples placed against the transducer and the relative dielectric constant, conductivity,

and loss tangents are calculated. Actual tank pads were not used, but 7.6 cm square

rubber samples of thickness 1.3 cm (0.5 inch) and a 2.5 cm (1.0 inch) are substituted.

Since the probing depth op is only about 0.35 cm for the APC-7 transducer, the rubber

samples can be assumed to be infinite half- spaces. Measurements are made from 0.20

to 2.20 GHz in 0.05 GHz steps where G.~o is valid. Ten points at each measurement

frequency were averaged to reduce the effect of noise and frequency deviation. A period

of 50 ms was allowed between each frequency change and 10 ms between each

measurement at the same frequency. The network analyzer was permitted a minimum

warm-up period of 24 hours. Measurements below 0.6 GHz were discarded because of

suspected calibration problems. The rubber samples were cleaned for every measurement

with a freon-based cleaner to remove surface impurities. Samples were firmly placed

against the APC-7 transducer by hand.


Numerous sets of measurements were taken on six different rubber groups. Each

group contained three or four samples except the SM02SP 2.54 cm rubber group

comprised of one sample. Every measurement was made on the same center location

on the pad. If, for a given group, a set of measurements differed radically from the

remaining sets, it was discarded from analysis. The remaining sets (at least three) were

averaged to obtain the following results. Each set of measurements was taken on a

different day with the network analyzer recalibrated each time.

Inferred relative dielectric constants, conductivities, and loss tangents for various

samples are shown in Figs. 26-35. Equations (2.3.4(b)), (3.1.1), (3.1.2(a)), and (3.1.2(b))

along with the measured reflection coefficients are used to infer the relative dielectric

constant, t:;, the conductivity, a , and the loss tangent, tan~. of various samples. The

range of loss tangents in the frequency range of study fall within the desirable range (0.1

- 1.0) and the complex permittivity is a function of frequency as expected. If the

dielectric constant and conductivity are assumed linear functions of frequency, the

extrapolated values at f = 8 GHz are in close agreement with t:~ = 5 and G between 1

and 3 S predicted in the preliminary study by de Wolf (actual extrapolated conductivity

values fall in a range from 0.3 to 1.7 S) [5].

In Fig. 29, the #1 and #3 15NAT1 samples are lower in conductivity than the

remaining samples. This is a particularly interesting result because of the four 15NA T 1

samples, #1 and #3 are visibly cracked and warped (although measurements were made

on an area in which there were no apparent air pockets) while the other two samples

showed no visible defects. This result seems to substantiate the hypothesis that low

conductivity for a given sample is undesirable [ 5].


.... c

1 I 8

15NAT1 0.5" SAMPLES

14.00 -M--+--+--+--r--t--r--i --!--+--+--+---+-+---+-+-~

O.IO o.ao 1.00 1.20 1.-40 1.50 1.80

FREQUENCY (Otiz) ,, - #2 - #3 - 14

15NAT1 0.5" SAMPLES

G.50 -+---;----;--+- _ _._.__.,_ _ _,__

O.IO OJIO 1.00

,, 1.20 1.-40

l'1tEQUENC'( (GHz) -12 -13

1.IO 1.80

- 14

2.00

2.00

Figure 26. APC-7 connector radiating into I SN A Tl rubber - equinlent admittance: experimental results; (a) dielectric constant (b) conductivity.

CHAPTER 3 Open-,Ended Coaxial Waveguide Transducer for Complex Permittivity Measurement 63

15NAT1 0.5" SAMPLES

i ~ a.J0-+---1 ·~-1-~~-+~-+-~+----1'---+--+---+---1~--+~·-t--t---t~-;

~

a.oo -"---'---1--1--+--+--+--~--1--+--+--+--+--i--t--t----i O.IO 0.80 1.00

f 1

1.20 1.-40

FREQUENCY (GHz) -12 -f3

1.60 1.80 2.00

-~.

Figure 27. APC-7 connector radiating into ISNATJ rubber - loss tangent: experimental results.

2.20


111.00

11.00

I t"-00

1;s.oo

g 12.00 I tt.00 i5

~ 10.00

c t.00

I.GO

7.00

15NAT22A 0.5" SAMPLES

~--

~ ~

I i

! i~ I ! l /~~ I i l___L ' i i ! 3 .. I i I I --;-- I ·~L._ ' I I I I I I ~ l I I : ! . w 1~ -I I I I ·;

I i I j

I I

I ! I I ! I .. I

"I~ I I ~-'I : -.../ I '

I O.IO o.ao

O.IO o.ao

' I

1.00

,,

~

y ..., ...

1.20 t."40

F1'EtlUENC't (GHz) - 13

............ ' ,,.._

1.80 1.ao

f4

15NAT22A 0.5" SAMPLES

1.00 ,, 1.20 .1AO

FREQUENC'I' (GHz) - 13

1.IO 1.ao

f4

-

i i

·1 __ ~ T"

I

·.

2.00

2.00

-

I

w I

Figure 28. APC-7 connector radiating into 1SNAT22A rubber - equivalent admittance: experimental results; (a) dielectric constant (b) conductivity.


15NAT22A 0.5" SAMPLES 0.80 _,...-~-.....---r--r---~·-----~--~----------~----.

0.60

~ 0 a 0.50 en g

0.40 r-·r-O.JO

a.so 0.1!10 1.00 1.20 1.-ia 1.60 1.80 2.00 2.20

FREQUENCY (CHz) f1 -- f3 f4

Figure 29. APC-7 connector radiating into 15NAT22A rubber - loss tangent: experimental result.


I 5 ()

()

I i5

I

1 5SBR2 0.5" SAMPLES 1o.ao 10.50

10.«I

10.20 10.00

9.IO

9.50

MO

9.20

9.00

a.ao LIO

"'~ . ,-1~

I"~ l\rJ • 1WVV i ' i\UI I I ' I

~i~ ' I ~! A I l ~ i : ~Al\ ,',( i

I i I JV I lAlA~A ""'i I .v \ 'I '

""~ V\ I\ n ~.I ~! y rrl \ r,y '~ /\)I I A~ r'L ~iti.;l ' I

y ~ !~ v~ '\.A W\1 I\(\_ .~ \ ·~ l

""' l \ vvn . ' ' \t-ro \ ''-'V v i I

L40 V\ I"\.,. v--hr \ ~ """'"-~ ~

r -0.50 o.ao 1.00 1.20 1.50 1.110 2.00

FREQUENCY' (GHz) f1 - 12 p

15SBR2 0.5" SAMPLES o.22....---..,.-.--r--,......---r-__,.-.,-_,,-..,r--,---,--,---,---,--r-llr 0~14--+-+1-+--+---lr-4i-...J_-+--:l-,i--r--r-r--;---t~Aft"' o.20+--+-+--+-1---+---+-+--+--1r--+--+-~r---r---i~~htr'--:"'.1 I \ ' I ! i i ,.r v 1 0.11 4---+-+-1 -l--l---t1--l--+-l--f1--+1-H1-~~J'v"·A:::i~9-~1 --t11'1 o.11-+---+---+---+--~1. --t-+--+--rl -..,!:--~:-A--i v lie£' 0.17 1 : 1 ~ 1r1r~l ,.'"Viii o. us -+--+---1-f--T--~-+--+-A-,+-u.~N1""'¥1-.~l'-+-'.: ~·+-Anit;Ir\:Rl°*J· ~t-H/~l---i 0.15 \ [ JV v_ IJJ\j...\ir'\flJ:liW I 0.14 M'\J .~.i~ 1t••"'L-+-11JV,) I O.l 3 i 1lf I M_~p- I I I 0.12 I _i/t" 11..I ,. Ml\ ,, ! I ' i I J.--+----;--0.11 -t : ~ ij .IA If\'\ I i ' ! I i 0.10 r~ n IN'r'V"'v ,• i ! .J__L__I I - I i I

O.OI f ! .L. I~ • : _, I : , .'1. r-, r-= a.oe 1 ~- I J ~ /I I i I i l 1 i: . . -~~ 0.01 YT i !! I I I : i \ I i j -W O.Dll _ __._ _ _,..__--T-j i I l I . ---t- i J

o.m I I I I I I O I ' • I I I

o.ao o.ao 1.00 1.20 1.40 1.10 1.110 2.20

FREQUENCY (GHz) f1 - 12 13

Figure 30. APC-7 connector radiating into 15SBR2 rubber • equivalent admittance: experimental results; (a) dielectric constant (b) conductivity.


15SBR2 0.5" SAMPLES 0.24

0.23

0.22

0.21

~ 0.20

Cl 0.19 a ~

0.18

0.17

0.11

0.15

0.14

1 I A~ L I I

d ~'a'v lA ~ v\\j ---·-~ ' '" ~I ~ N L[

~ I r AA l

~~ r'Y\' fl ·-~ l/ 'J\J\ ~v' '

- ----!~ fM IAJ v j~- . '

J~V\ ~I I J\ l i 1v ~V\11 ' I A.. I

A f\Ji I l\M J..,:, V' " ~ ~I 'V ~ r Vi~ l ,.. ~ -·-· I' v '~ ~

r ' ' '\/1 ~ f . ~ ~ II" ~JI ' I '

j l ·- (' ! -

0.13 O.IO OJIO 1.00 1.20 1."40 1.80 1.IK> 2.00 2.20

FREQUENCY (GHz) 11 - 12 13

Figure 31. APC-7 connector radiating into 1SSBR2 rubber - loss tangent: experimental result.


I I a

!

15NAT42 0.5" SAMPLES 1.AO-----.-.......... --.-....,.....--.-""'T""--ir--""'T""--ir--....,..--r-....,..--r--r-T-"""'1

1..40--.i4--4_;_-+---+--+---t~-1---1r----t-~r---r-~t---;--r--r--r-'""'1

a.20-+-1~-'4-"*-"--+~-+---1~-+---tt----t--r-~-r--i--r--r--r-""""1

1.00-+-*11llh<-T-\7'\-ill-~-f+-:-:-ir---t-----1r--t-~r---r-r--r~t---r~t--1

7,80 -+---'-""tr--'¥'.ll-'f\f~H-'ltt-11-"t -.iw-,---

7..40 ...... --f--+---IS----+-'lf-<'--~ ·-+++r.r±N-t"'i+++-:l"+\.-"'i-:---+--t---1---1

l.IO-ha'+:-iri-"-'+i---if---ft--~f---+--t---+-~t---+--+--+~t--·-t~-t--;

uo -+---+.l-+...6-r!~i-Hl---IJA-.r-1-----1'---+---+ l..40+--4--+-~l-L-+-~--+1,.J-lll~rl1-4-Jl!W-l~f--·!-+-iJ~f-=---t-=¥i---f

o.eo OAO 1.00

11

1.20 1.40

l'1'IEQUENCI' (GHz) -12 -13

1.10 1.10

- ,. 15NAT42 0.5" SAMPLES

0.11 ...... -4--+-+lil~:....4_,.....,.UW"'-'u1rft--CJ'1-- I-_.____, 0.10 +--+--. O.OI _,_,.._...~--+-~f-0.0I *~~--Pf'll:J(J 0.07 ~-----,._. ....

0.03 ~""""""'~=1,,,.,,..--r-·-tt-.:: _.__.___ 0.02----+-0.01 .._-"--+--"-4---4--+---l--+---tf---+---lt---t--t---+--t--1

o ... 0.IO 1.10 1.20 1AQ 1.10 1.IO 2.00 2.20

11 l'1'f.QUINC\' (GHz) 12 - p - ,.

Figure 32. APC-7 connector radiating into 15NAT42 rubber - equivalent admittance: experimental results; (a) dielectric constant. (b) conductivity.


~ " ~ tit fl) s

0.32

0.30

0.28 Y. ,J ~

o.2e 0.24 ~j "' 0.22

0.20

0.119

0.16

0.1'4

0.12

0.10

o.oa

A J-v 0.0&

0.04 0.60

15NAT42 0.5" SAMPLES .

A A \ ~ vv \). y-Vi ~

I ~ ,\~I , .. 'f

~

-

-

" AA

~ 'fv\J ~

0.80

~ j~ iv .... l

~ -1W\ ,f\~ \j ..... \I/' '\Ir A .J

Ji ~.A r\ /\.J '7 iw I~, ~ ~I 'I~ J. , ~ \

~ .

~ \1, !

j A .~ ' I -v~ 'A I A

Al( ·~ """ M'

11

'"' 1

lf\ j ' v

1.00 1.20

FREQUENCY (GHz) -12 --f3

1.60

! . v~ Ff'-... i... h v - ·-- W'\1-..r ~ t I I

~ ~' N.~ ~ ~ ~

"' ~ 1V

I ' '

v, Ml\·~ ~ ,../\

"""' ~ IV' .

1.80 2.00 2.20

- f4

Figure 33. APC-7 connector radiating into 15NAT42 rubber - loss tangent: experimental result.


SM02SP 1.0" SAMPLE 13.IO 13.IO 13.70 13.IO 13.80

I 1MO ,~

13.20

I ,3.10 lj.00

12.IO

a 12.IO

• }\ 11.

1 1 \M. I 'l

. ' (\ I ~ I II r Yw\ I j

.. i v "\ iA l "'4\

I 12.70 12.IO 12.IO 12.AO 12.30 12.20 12.10

i. Ai . I

\ I

VI ~,. f\ I\ ~ "' "kN I

IL... .>. J IV"' ~ ~ IV

I/\. --. 12.ilO

O.IO 1.00 1.- 1.10

SM02SP 1 .0" SAMPLE 0-10

0.211 I 0.21

I

G.24'

l o.zz

• G.20

i 0.11 ..,

i 0.11

0.14 a 0.12 § 0.10

o.ae o.oe G.04

. Al J..I {'-\ w

A lflN r.JV'tJ !jj I l...r N ~ ' ,,,, ? ·-,..; ~

' V"' !

f'ltJ wv '

~/;/ -- I --t I I

I --v o.m I

I

1.00 1.20 1AO 1.IO !AO

Figur~ 34. APC-7 connector radiating into SM02SP rubber· equivalent admittance: experimental results; (a) dielectric constant, (b) conductivity.

-CHAPTER 3 Open-Ended Coaxial Waveguide Transducer for Complex Permittivity Measurement 71

SM02SP 1.0" SAMPLE

o.so o.eo 1.00 1.20 1.""'° 1.60 1.80 2.00

FREQUENCY (GHz)

Figure 35. APC-7 connector radiating into SM 02SP rubber - loss tangent: experimental result.


An interesting result of the second experiment is the dependence of the permittivity

measurement on the pressure of the sample against the transducer (see Figs. 36-37)~

application of light pressure creates a lower inferred loss tangent, conductivity, and

relative dielectric constant than with the application of moderate pressure (the quantity.

of pressure could not be gauged with available equipment and only a general inexact

description of pressure can be used). The pressure effects are suspected to be the main

source of error and may be caused by one or more factors. First, an air gap might exist

when only light pressure is exerted. Second, the rubber sample might deform under

moderate pressure and begin to fill inside the transducer. And lastly, the pressure effect

may actually be induced by the material. In any case, a more sophisticated method to

hold the sample against the transducer and gauge pressure is desirable to further study

this problem. Once the cause of this pressure dependence known, this measurement

approach can probably be considered reliable.

CHAPTER 3 Open-Ended Coaxial Waveguide Transducer for Complex Permittivity Measurement · 73

I 0

I a

I

EFFECT OF PRESSURE ON 150TR6 #4 0.5" 17.aG

17.00

11.eo

.l

v~ dA ."' 11.00

1UO

15.00

rt\. A -'=t• u++ ' . --fcl,+,-. ! ~I ! ' i I I !'..... I I ' I

~ ! I I ~i~I~ N ~ ... ! .-. 1.y(I

I y\:~ l'V\ ~ 14.00

13.ao

12.eo o.eo A

I

o.ao 1.00

II\ i \,.

'vw A.. I ' '\ ~ ~ wr I\.... • ... -v - '•

/ 1.20 1.40 1.10 1.110 2.00

FREQUENCY (OHz) B lrJhl ~

EFFECT OF PRESSURE ON 150TR6 #4 0.5"

·~,

I

0.M"T""""--r-;--r----ir--.,.--.--r---r--i--r--i--r-'-r--i--r--i 0.52+--+--l-_j__--lf--+--+-+--t---t·-+---t-t--t---i-17~ 0.50 -+----+--+--r---r--r--0.4 +---+-+--;----! 0.<145 -r---i--t---r--0.44 +--t--t--i---+ 0.42 ---t--t'-"--r 0.40 +--+----:+---+----:r---r-0.3114--+-+--+--+--i--r--rrt o.38+--~-+--+-~-t--t-~lf¥--'!--t 0.34 -"---+--+--t---r---:!Jir-1 0.32 ---+-;---r--, ,.,_,,_......,__ 0.30 -+----+--.-t-0.211 -t----+-~!"'To.:ze --:&l""----t---r- ·~~+-o.:z4 ...,.._--+--+- _,_"-t-_ 0.22 -+--- -+-:><'1'f'--'"-"5+---r--o.:zo -.~-0.18 -+----1---r--r-

O.IO 0.80 1.00

A - rnoclwat. ~

:Z.00

Figure 36. Effect of pressure on a l 50TR6 rubber sample - equivalent admittance: experimental results; (a) dielectric constant, (b) conductivity.


EFFECT_ OF PRESSURE ON 150TR6 #4 0.5" 0.43 0..42 0.41

'\ .. VVI

0~40

0.31 0.31 0.37 0.31

:5 0.35

" 0.34 ~ 0.33

I 0.32

9 0.31 0..10 0.29 0.211 0.27 0.21 0.28 G.24

.. ~

IT i I

II" '"'-' .A ' . V' 't

II\ 0 a. 11 - " \1 ~ ~J Wt. f ..... V'J A f v ) "' y IJ .., ,{\/\

rvr II\. rV\.. ~ "\... . ' ~ -Al .....

'w"" ~"' ~ - - . '-V" I'\.. .. ..-'\

\I "'\ A. l - ........ ~ -+--

i I LI'' Al\ I YV ~ . IA

r-Y\J " '-11o.. I l A ,AAA

W' •v i . 0.23

0.80 O.IO 1.00 1.20 1.40 1.IO 1.80

A FREQUENCY (GH:) B - ._ ,.._,,,.

Figure 37. Effect of pressure on a 150TR6 rubber sample - loss tangent: experimental results .

. CHAPTER 3 Open-Ended Coaxial Waveguide Transducer for Complex Permittivity Measurement 75

CHAPTER 4 Conclusions and Recommendations

for Further Research

4.1 Recommendations for Further Research

A useful· enhancement of the ac conductivity measurement would be to provide a

means to investigate the homogeneity, or lack thereof, of a rubber samples. The

investigated coaxial waveguide system operates at frequencies at which G0 o.!O will only

penetrate the rubber on the order of the outer conductor radius. For example, with the

APC-7 transducer, only a depth of 0.35 cm into the rubber is possible and as s~ch, only

qualities of the rubber near the surface are being observed. On the other hand, if a

radiating field is the interrogation field, the frequency can be changed to vary the skin

·. depth "Of the field and allowing the effective aperture admittance to be determined at

different depths of the material.

CHAPTER 4 Conclusions and Recommendations for Further Research 76

To overcome this lack of probing depth, study into the use of coaxial transducers

of larger diameter could be initiated. Standard sizes of 50.Q air-dielectric coaxial

waveguide vary from 7 mm diameter to 22.9 cm diameter [11]. However, the use of

larger coaxial waveguide requires lower frequencies of analysis. For example, the highest

frequency for 22.9 cm airline in which G0 is negligible is 105 MHz ( ~ < 0.04) and the

probing depth would be extended to 11.45 cm.

If, however, the coaxial waveguide approach of this study is pursued, it is of utmost

importance to alleviate the dependence of pressure on the measurements. While it seems

probable that the error is caused by an air gap left between the transducer and the

sample, the error might also be caused by deformation of the sample into the airline or

the effect might be a property of the material itself. A test fixture to hold the sample

against the transducer and gauge the amount of pressure is desirable for study of the

pressure dependence effect.

Since Deschamps antenna modeling theorem is valid for any antenna, better

radiating transducers, such as hollow waveguides, may also be employed for permittivity

measurements. However, as shown with the results for the WR-90 rectangular

waveguide, the lack of a full calibration in a network analyzer scheme makes

measurement difficult. Perhaps a hollow waveguide approach with a slotted line to

measure reflection coefficient would work. The measurement error caused by calibrating

with only a short circuit in a slotted line configuration may be less substantial than \Vi.th

a network analyzer configuration because the coaxial-to-hollow waveguide connector

would be eliminated. The Deschamps antenna modeling theorem still could be used but

an iterative solution would be required to solve the resulting nonlinear coupled

equations because neither G0 or B0 could be neglected since I G0 I~ I B0 I. Also, the


frequency range of analysis would be much narrower since the frequency band of a

hollow waveguide is limited on the lower end by the cutoff frequency and on the higher

end by propagation of higher modes.

A transmission radiating scheme may be employed if criterion 2 is relaxed; a hole

could be cut into the metal plate and a two-way transmission scheme such as the one

used in the preliminary study can be implemented.

There are two accuracy enhancement procedures for the coaxial transducer

approach cited in the literature that have not been examined in this study. The first

enhancement accounts for a fringe field inside the coaxial waveguide at the aperture, a

capacitance C1 is added in parallel to C0 and G0 [25]. The capacitance C1 must be

experimentally determined from measurements of known materials and primarily affects

the measurement accuracy of the real dielectric constant. The second enhancement

introduces the concept of optimum capacitance. The optimum capacitance is derived

from an error equation introduced by Stuchly that describes the total measurement error

[24]. If the error equation is minimized, a value of capacitance for C0 is found at which

the measurement error is minimized. The optimum capacitance is given by

1 Co=---;====-z I ,2 + ,,2 w o'\/ e, . e,

If further accuracy is deemed necessary, the implementation of these enhancement

procedures must be studied.


4.2 Conclusions

The conductivity of rubber provides an indication of the dispersion of carbon black

throughout a rubber matrix. A permittivity measurement scheme composed of a coaxial

waveguide transducer and a network analyzer has been studied for use with rubber. The

transducer is placed on the sample and the reflection coefficient is measured. Both the

permittivity and the conductivity can be inferred by this approach. The interpretation

of the measured reflection coefficient is based on Deschamps theorem which relates the

admittance of an antenna embedded in some arbitrary homogeneous medium to the

effective admittance of the same antenna embedded in free space. An elegant analytical

solution is provided if the system is operated at frequencies at which the radiation

conductance is negligible. Operation at frequencies at which the radiation conductance

is negligible implies that only the near-field interrogates the material. Hence, only the

material within the outer conductor radius of the transducer is examined.

Experimental results of the transducer radiating into air are compared to

theoretical results derived by Marcuvitz. Experimentation data showed close agreement

with theoretical data except for variations by a constant. There are several possible

reasons for the discrepancy: 1.) discontinuities in the APC-7 connector, 2.) network

analyzer has difficulty with material of loss tangents less than 0.1, 3.) manufacturing

deviation from nominal dimensions, and 4.) lack of an infinite ground plane. Since the

error is well-behaved and our ~ontract sponsor does not desire a system which yields an

accurate absolute value of conductivity, the discrepancy is not critical.


The complex permittivity of some rubber samples is examined. The range of

conductivities and real dielectric constants are consistent with the results of the

preliminary study by de Wolf [5]. The loss tangents of the rubber samples fall within the

acceptable 0.1-1.0 range. There is some evidence that abnormally low conductivity can

indicate defective rubber. An observed dependence on the amount of pressure of the

sample against the transducer precluded any examination of the· homogeneity of the

rubber.

To summarize, the goal of this research project was to develop an ac conductivity

measurement system meeting the following performance criteria:

1. Sufficient sensitivity - system must reliably indicate if the conductivity is in the abnormal range.

2. Nondestructive measurement - the evaluation must not alter the structure or performance of the tank pad. ·

3. One-sided measurement - nonremovable metal backing precludes two-sided . transmission measurements.

4. Ample tolerance to sample thickness - system must be capable of evaluating tank pads 6.35 - 20.3 cm (2.5 - 8.0 in) thick.

5. Sufficient immunity to noise - measurements are to be performed in the manufacturing environment

6. Batch processing capability - evaluation duration should be minimal.

7. User-friendliness - measurement system must require minimal interpretive skills from the ultimate user.

The coaxial waveguide system has shown promise in detecting defective rubber by

measuring a discernible difference in cracked and warped rubber from what appeared to

be acceptable rubber. Hence, the coaxial waveguide system may provide sufficient

measurement sensitivity. It will be important to establish a value of ac conductivity

which demarcates the boundary between acceptable and defective rubber in future study.

Criterion 2 and 3 have been met since the transducer is only required to be placed on a


flat surface of the rubber. For criterion 5, it is unknown at this time if this method is

indeed suitable for the manufacturing environment. Only through testing in such an

environment can this be determined. The ability to automate the network analyzer

certainly makes criterion 6 and 7 a strong possibility. Thus, we see that the coaxial

waveguide system potentially satisfies many of the desired performance specifications.

However, a method to satisfy criterion 4 must be obtained.


Literature Cited

1. R. H. Norman, "Background to Conductive Rubbers," RAPRA: Recent Developments in Conductive Rubbers, ed. by D. J. James, Great Britain: Rubber and Plastics Research Association of Great Britain, 1977.

2. R. J. Cembrola, "Resistivity and Surface Roughness Analyses for Evaluating Carbon Black Dispersion in Rubber," Rubber Chemistry and Technology, vol. 56, pp.233-243, 1983.

3. "An Introduction to Carbon Black," Cabot Corporation Technical Service Report, 1983.

4. D. R. Parris, "Electrical Characterization of Carbon Black Filled Rubber," M.S. Thesis, Materials Engineering, VPI&SU, 1986.

5. D. A. de ·wolf, "Microwave Sensing of Bulk Properties of Tank Pad Rubber: A Feasibility Study," Bradley Department of Electrical Engineering, VPI&SU, funded by TACOM under contract #DAAE07-83-K-R009, from July 1, 1983 to June 30, 1984.

6. R. E. Collin, Field Theory for Guided Waves. New York: McGraw-Hill, 1960.

7. R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1966.

8. W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: John Wiley and sons, 1981.

9. R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961.

10. N. Marcuvitz, Waveguide Handbook. London: Peter Peregrinus Ltd., 1986.

Literature Cited 82

11. D. A. Gray, Handbook of Coaxial Microwave Measurements, General Radio Company, West Concord, Massachusetts, 1968.

12. "Automating the HP8410B Microwave Network Analyzer," Hewlett-Packard, Application Note 221A, June 1980.

13. G. A. Deschamps, "Impedance of an Antenna in a Conducting Medium," IEEE Trans. Antennas and Propagation, vol. AP-10, pp. 648-650, 1962.

14. I. J. Bahl, A. Thansandote, and S. S. Stuchly, "Open-Ended Rectangular Waveguides as Antennas for Medical Diagnostics," Journal of Microwave Power, vol. 15, pp. 81-86, June 1980.

15. M. C. Decreton and F. E. Gardiol, "Simple Nondestructive Method for the Measurement of Complex Permittivity." IEEE Trans. Instrumentation and Measurements, vol. IM-23, PP~ 434-438, Dec. 1974.

16. M. C. Decreton and M. S. Ramachandraiah, "Nondestructive Measurement of Complex Permittivity for Dielectric Slabs," IEEE Trans. Microwave Theory and Technique, vol. MTT-23, no.12, pp. 1077-1080, Dec. 1975.

17. J. R. Mosig, J. E. Besson, M. Gex-Fabry and F. E. Gardiol, "Reflection of an Open-Ended Coaxial Line and Application to Nondestructive Measurements of Materials," IEEE Trans. Instrumentation and Measurement, pp. 46-51, vol. IM-30, no. 1, Mar. 1981.

18. E. Tanabe, W. T. Joines, "A Nondestructive Method for Measuring the Complex Permittivity of Dielectric Materials at Microwave Frequencies Using an Open Transmission Line Resonator," IEEE Trans. Instrumentation and Measurements, vol. IM-25, pp. 222-226, Sept. 1976.

19. E. C. Burdette, F. L. Cain, and J. Seals, "In-Vivo Probe Measurement Technique for Determining Dielectric Properties at VHF Through Microwave Frequencies," IEEE Trans. Microwave Theory Tech., vol. MTT-28, pp.414-427, Apr. 1980.

20. G. B. Gajda, "A Method for Measurement of Permittivity at Radio and Microwave Frequencies," IEEE Intl. Electrical, Electronics Conf. & Expo. Conf. Digest, Oct. 1979.

21. M. A. Stuchly and S. S. Stuchly, "Coaxial Line Reflection Methods for Measuring Dielectric Properties of Biological Substances at Radio and Microwave Frequencies - A Review," IEEE Trans. Instrum. Meas., vol. IM-29, pp. 176-183, Sept. 1980.

22. M. · M. Brady, S. A. Symons, and S. S. Stuchly, "Dielectric Behavior of Selected Animal Tissues in vitro at Frequencies from 2 to 4 GHz," IEEE Trans. Biomed Engr., vol BME-28, pp. 305-307, Mar. 1981.

23. A. Kraszewski, M.A. Stuchly, S.S. Stuchly, and A. M. Smith, "In vivo and In vitro Dielectric Properties of Animal Tissues at Radio Frequencies," Bioelectromagnetics, vol. 3, pp. 421-432, 1982;

Literature Cited 83

24. T. W. Athey, M. A. Stuchly, and S. S. Stuchly, "Measurement of Radio Frequency Permittivity of Biological Tissues with an Open- Ended Coaxial Line: Part I," IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp. 82-86, Jan. 1982.

25. M. A. Stuchly, T. W. Athey, G. M. Samaras, and G. E. Taylor, "Measurement of Radio Frequency Permittivity of Biological Tissues with an Open-Ended Coaxial Line: Part II - Experimental Results," IEEE Trans. Microwave Theory Tech., vol. MTT-30, pp. 87-92, Jan. 1982.

26. M. A. Stuchly, M. M. Brady, S. S. Stuchly, and G. Gajda, "Equivalent Circuit of an Open-Ended Coaxial Line in a Lossy Dielectric," IEEE Trans. Instrum. Meas., vol. IM-31, pp. 116-119, June 1982.

27. A. Kraszewski, M. A; Stuchly, S. S. Stuchly, and S. A. Symons, "Network Analyzer Technique for Accurate Measurements of the Tissue Permittivity in vivo," IEEE CPEM Digest, pp. C15-C17, 1982.

28. A. Kraszewski, S. S. Stuchly, M. A. Stuchly, and S. A. Symons, "On the Measurement Accuracy of the Tissue Permittivity in vivo," IEEE Trans. Instrum. Meas., vol. IM-32, pp. 37-42, Mar. 1983.

29. G. B. Gajda and S. S. Stuchly, "Numerical Analysis of Open-ended Coaxial Lines," IEEE Trans. Microwave Theory Tech., voL MTT-31, pp. 380-384, May 1983.

30 A. Kraszewski, M. A. Stuchly, and S. S. Stuchly, "ANA Calibration Method for Measurements of Dielectric Properties," IEEE Trans. Instrum. Meas., vol. IM-32, pp. 385-386, Jun. 1983.

31 G. Gajda and S. S. Stuchly, "An Equivalent Circuit of an Open- Ended Coaxial Line," IEEE Trans. Instrum. Meas., vol. IM-32, pp. 506-508, Dec. 1983.

32 . A. Kraszewski and S. S. Stuchly, "Capacitance of Open-Ended Dielectric-Filled Coaxial Lines - Experimental Results," IEEE Trans. Instrum. Meas., vol. IM-32, pp. 517-519, Dec. 1983.

33. L. A. Anderson, G. B. Gajda, and S. S. Stuchly, "Analysis of an Open-Ended Coaxial Line Sensor in Layered Dielectrics," IEEE Trans. Instrum and Meas., vol. IM-35, pp. 13-18, Mar. 1986.

34. T. P. Marsland, S. Evans, "Dielectric Measurements with an Open-ended Coaxial Probe," IEE Proceedings - H, vol. 134, pp. 341-349, Aug. 1987.

35. B. R. Epstein, M. A. Gealt, K. R. Foster, "The Use of Coaxial Probes for Precise Dielectric Measurements: A Reevaluation," IEEE MTT-S Digest, June 1987, pp. 255 -258.

36. J. Ayer, "Measurement of Dielectric Constant at RF and Microwave Frequencies," RF Design, pp. 44-47, April 1987.

terature Cited 84

37. D. R. Brunfeldt, "Theory and Design of a Field-Portable Dielectric Measurement Probe," Proceedings of IGARSS 87 Symposium, Ann Arbor, Michigan, pp. 559 -563, May 1987.

38. I. J. Bahl and S.S. Stuchly, "Effect of Finite Size of Ground Plane on the Impedance of a Monopole Immersed in a Lossy Medium," Electronics Letter, vol. 15, pp. 728-729' 1979.

39. M. L. Swicord and C. C. Davis, "Energy Absorption from Small Radiating Coaxial Probes in Lossy Media," IEEE Transaction Microwave Theory Technique, vol. MTT-29, pp.1002-1209, 1981. .

40. G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design. New Jersey: Prentice-Hall, 1984.

41. W. A. Davis, Virginia Tech, Blacksburg, Virginia, private communication, 1988.

Literature Cited 85

Appendix A. Marcuvitz Program

c***** c c**************************!************************************ c May 10, 1988 c MARCVITZ: MAIN PROGRAM c c Calculates the admittance of an open-ended coaxial line c in a half space of air. The inner radius and outer radius of c the coaxial line are bradius = 1.52 and aradius = 3.50 mm. c The characteristic admittance is yO = 1 / 50.0 siemans. c IMSL subprograms DCADRE and MMBSJO are used for c numerical integration and generation of Bessel functions c of the first kind and order zero, respectively. c c REFERENCE: N. Marcuvitz, Waveguide Handbook, Peter Peregrinus c Ltd., 1986, pgs. 213 - 216. c c Michael W. Lee c*************************************************************** c c*****

common frequency, pi, aradius, bradius, k, c, yO, wght(30) double precision frequency, pi, aradius, bradius, k, c, yO, wght

double precision G, B, rerr, halfpi, aerr, cnstl, cnst2 complex Z integer count

pi = 4.0dO*datan( l .OdO) aradius = 3.50d-03 bradius = l .52d-03

c = 3.0d08 read( 14, *) rerr yo = 1.odo ; so.Odo aerr = O.OdO

cnstl = (yO) / ( dlog ( aradius / bradius)) halfpi = pi I 2.0dO

cnst2 = (yO) / ( pi * dlog ( aradius / bradius ) ) call READWGHT

Appendix A. Marcuvitz Program 86

call HEADING frequency = 0.00d9 do 10 count = 1, 201, 1 frequency = frequency + 0.20d9 k = (2.00 * pi * frequency) I c call CONDUCTANCE ( G, cnstl, halfpi, aerr, rerr)

call SUSCEPTANCE ( B, cnst2, aerr, rerr) call GBFILE (frequency, G, B) call IMPEDANCE ( G, B, Z)

call RXFILE (frequency, Z) call CAPACITANCE ( B)

10 continue end

c***** c c*************************************************************** c May 10, 1988 c MARCVITZ: CONDUCTANCE c c Calculates the conductance term of the admittance. c c Michael W. Lee c**************************************************************** c c*****

subroutine CONDUCTANCE (G, cnstl, halfpi, aerr, rerr)

double precision frequency, pi, aradius, bradius, k, c, yO, wght double precision intgndG, DCADRE

double precision G, cnstl, halfpi, error, aerr, rerr common frequency, pi, aradius, bradius, k, c, yO, wght(30) integer ier

external intgndG G=cnstl*DCADRE(intgndG, O.OdO, halfpi, aerr, rerr, error, ier) write(l 9, *) 'im over here'

return end

c***** c c**************~************************************************ c c MARCVITZ: intgndG c

May 10, 1988

c Generates the integrand for use with DCADRE in the c subroutine CONDUCTANCE. c c Michael W. Lee c****************************************************************


c c*****

double precision function intgndG ( theta )

double precision arg4, arg5, theta, MMBSJO double precision frequency, pi, aradius, bradius, k, c, yO, wght

common frequency, pi, aradius, bradius, k, c, yO, wght(30) if (theta.le. 1.0d-76) then intgndG = O.OdO go to 10 else arg4 = MMBSJO ( k * bradius * dsin (theta), ier) arg5 = MMBSJO ( k * aradius * dsin ( theta ), ier)

intgndG = ((arg5 - arg4)**2) / (dsin (theta)) endif

10 continue return

end

c***** c c*************************************************************** c May 10, 1988 c MARCVITZ: SUSCEPTANCE c c Calculates the susceptance term of the admittance. c c Michael W. Lee c**************************************************************** c c*****

subroutine SUSCEPTANCE ( B, cnst2, aerr, rerr)

double precision B, cnst2, error, aerr, rerr, intgndB, DCADRE double precision frequency, pi, aradius, bradius, k, c, yO, wght common frequency, pi, aradius, bradius, k, c, yO, wght(30) external intgndB integer ier

B = cnst2*DCADRE(intgndB, O.OdO, pi, aerr, rerr, error, ier) return

end

c***** c c*************************************************************** c May 10, 1988 c MARCVITZ: intgndB c c Generates the integrand for use with DCAD RE in the c subroutine SUSCEPTANCE. c


c Michael W. Lee c**************************************************************** c c*****

double precision function intgndB ( phi )

double precision argl, arg2, arg3, sil, si2, si3, phi double precision frequency, pi, aradius, bradius, k, c, yO, wght

common frequency, pi, aradius, bradius, k, c, yO, wght(30) if (phi.le. l.Od-76) phi = O.OdO argl = k * dsqrt ( (aradius**2) + (bradius**2) -

# (2.0dO * aradius * bradius * dcos(phi)) ) arg2 = 2.0dO * k * aradius * dsin ( phi / 2.0dO ) arg3 = 2.0dO * k * bradius * dsin ( phi / 2.0dO )

call SIZ ( argl, sil ) call SIZ ( arg2, si2 )

call SIZ ( arg3, si3 ) intgndB = (2.0dO * sil) - (si2 + si3)

return end

c***** c c*************************************************************** c May 10, 1988 c MARCVITZ: IMPEDANCE c c Calculates the inverse of admittance. c c Michael W. Lee c**************************************************************** c c*****

subroutine IMPEDANCE ( G, B , Z)

double precision G, B complex Y, Z, ONE ONE = dcmplx( l .OdO, O.OdO)

Y = dcmplx ( G, B) Z =ONE/ Y

return end

c***** c c*************************************************************** c May 12, 1988 c MARCVITZ: CAPACITANCE c c Calculates the capacitance of air and stores the value c in a file under device 18.


c Michael W. Lee c**************************************************************** c c*****

subroutine CAPACITANCE( B)


double precision Ctotal, B, gajda, eO eO = 1.0dO I (pi * 36.0d9) Ctotal = (B / (2.0dO * pi * frequency)) write(l8,*) frequency*l.Od-9, Ctotal*l.Od12 gajda = Ctotal*l.Od-12 I ( eO * (aradius - bradius)) write(l 9, *) frequency, gajda

return end

c***** c c*************************************************************** c May 9, 1988 c MARCVITZ: SIZ c c Calculates the sine integral function by means of a c a series representation. c c Michael W. Lee c**************************************************************** c c*****

subroutine SIZ ( z, si )

common frequency, pi, aradius, bradius, k, c, yO, wght(30) double precision frequency, pi, aradius, bradius, k, c, yO, wght double precision x, z, si, siB, diff

integer i, n si = O.OdO

do 20 n = 0, 27, 1 siB = si + (wght(n + 1) * ( z ** ((2*n)+ 1)))

diff = dabs( siB - si) if( diff.le.1.0d-15) then si = siB

go to 40 else

si = siB endif

20 continue 40 return

end


c***** c c*************************************************************** c May 12, 1988 c MARCVITZ: HEAD ING c c Creates the headings for GBFILE AND RXFILE c c Michael W. Lee c**************************************************************** c c*****

subroutine HEADING

write(l6, 10) write( 17, 20)

10 format (8x,'frequency',l lx,'conductance',9x,'susceptance') 20 forma t(8x,' frequency', 11 x,' resistance', 1 lx,' reactance')

write(16,30) write(l 7,40)

30 format (9x,'(hertz)',12x,'(siemans)',12x,'(siemans)', /) 40 format(9x,'(hertz)', 14x,'( ohms)', 14x,'( ohms)',/)

return end

c***** c c*************************************************************** c May 10, 1988 c MARCVITZ: GBFILE c c Enters the values for the conductance and susceptance c into a file under device 16. c c Michael W. Lee c*************************************************************** c c*****

subroutine GBFILE (frequency, G, B)

double precision frequency, G, B write( 16, IO) frequency, G, B

10 format(3e20.8) return

end

c***>S!* c c*************************************************************** c May 10, 1988 c MARCVITZ: RXFILE


c c Stores the data from subroutine IMPEDANCE into a file c under device 17. c c Michael W. Lee c*************************************************************** c c*****

subroutine RXFILE (frequency, Z)

double precision frequency, R, X complex Z R = real(Z)

X = aimag(Z) write (17, 10) frequency, R, X

IO format(3e20.8) return

end

c***** c c*************************************************************** c May 9, 1988 c MARCVITZ: READWGHT c c Reads the weights for the sine integral function series c representation. Device 15 contains the weights of c each term and are generated by the program SIZWGHT FORTRAN. c c c Michael W. Lee c*************************************************************** c c*****

subroutine READWGHT


integer n do 10, n= 1, 28, 1

read(l5,*) wght(n) 10 continue

return end


Appendix B. Sizwght Program

c*************************************************************** c May 9, 1988 c Siz: MAIN PROGRAM c c Program to calculate values for the sine integral c function with argument z by means of a truncated series c representation. c c REFERENCE: M. Abramowitz and I. E. Stegun, Handbook of c Mathematical Functions, 10th printing, p. 232, c equation 5.2.14 c c c Michael W. Lee c**************************************************************** c c*****

call WEIGHT end

c***** c c*************************************************************** c May 9, 1988 c Slz: WEIGHT c c Determines the weight of each term of the sine integral c series. c c Michael W. Lee c*************************************************************** c c*****

subroutine WEIGHT

double precision wght, cnst, factor, argmntl double precision top, bottom

integer n · do 10 n = 0, 27, 1

Appendix B. Sizwght Program 93

call POSITIVE ( cnst, n) call FACTORIAL (factor, n) argmntl = 2.0dO * float (n)

top = cnst bottom = (argmntl + l.OdO) *factor

wght = top / bottom call STOREWGHT( wght )

10 continue return

end c***** c c**************************************************************** c May 9, 1988 c Slz: POSITIVE c c Determines if the weight of each term of the sine integral c series is positive or negative, dependent on whether 'n' is c is even or odd. The argument 'cnst' will be either 1 or -L c c Michael W. Lee c*****************************************************************

\

c c*****

subroutine POSITIVE ( cnst, n )

real argmnt2 double precision cnst

integer n, value 1, value2 argmnt2 = float (n) / 2.0

value 1 = int ( argmnt2 ) value2 = int ( argmnt2 + 0.5 )

if ( value I .eq. value2) then cnst = I.OdO

else cnst = -1.0dO

endif return

end c***** c c*************************************************************** c May 9, 1988 c Slz: FACTORIAL c c Calculates the factorial argument for each term of the c sine integral series. c c Michael W. Lee c****************************************************************

Appendix B. Sizwght Program 94

c c*****

subroutine FACTORIAL (factor, n)

double precision argmnt3, factor integer n

if ( n.eq.O) then factor = l .OdO

return else

argmnt3 = 2.0dO * float (n) factor = factor * ( argmnt3 + l .OdO) * argmnt3

endif return

end c***** c c*************************************************************** c May 9, 1988 c Slz: STOREWGHT c c Stores values 'wght' into a unit 6 file c c Michael W. Lee c******************~********************************************* c c*****

subroutine STOREWGHT ( wght)

double precision wght write ( 11, *) wght

. return end

Appendix 8. Sizwght Program 95

The vita has been removed from the scanned document

Documents

Pad Rubber - Virginia Tech...using microwaves for the evaluation of tank track pad rubber. In the preliminary study, an 8 GHz signal was transmitted from one side of a 1.37 cm (0.50