NisrUNITED STATES

DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

AT L INST. OF STAND & Tfc^H R.I.C.

A111D3 M7flMm

NISI

PUBLICATIONS NIST Technical Note 1341

Transmission / Reflection ii'if

and Short-Circuit Line

^^

Permittivity Measurements

James Baker-Jarvis

Electromagnetic Fields Division

Center for Electronics and Electrical Engineering

National Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

\

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director

Issued July 1990
National Institute of Standards and Technology Technical Note 1341

Natl. Inst. Stand. Technol., Tech. Note 1341, 154 pages (July 1990)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1990

For sale by the Superintendent of Documents, U.S. Government Printing Office, Wasliington, DC 20402-9325
Contents

1 Introduction 2

2 Transmission/Reflection Method 5

2.1 Theory 5

2.2 Numerical Determination of Permittivity 13

2.3 Corrections to Data 16

2.3.1 Gaps Between Sample and Holder 17

2.3.2 Attenuation Due to Imperfect Conductivity of Sample

Holders 17

2.4 Instrumentation 21

2.4.1 ANA Calibration 21

2.4.2 Sample Holder Specifications 23

2.4.3 Sample Preparation 24

2.4.4 System Diagnostics 28

2.5 Measurement Results 28

2.6 Appearance of Higher Order Modes 28

2.7 Uncertainty Analysis 43

2.7.1 Uncertainty in Gap Correction 66

2.7.2 Reference Plane Position Uncertainty 67

3 Short-Circuit Line 68

3.1 Theory 68

3.1.1 Reflection Coefficient Method 69

3.1.2 Slotted Line Technique 74

3.2 Corrections to Data 76

3.3 Instrumentation 77

3.3.1 Sample Holder Specifications 78

111
Contents (cont.)

3.3.2 Sample Preparation 78

3.3.3 System Diagnostics 79

3.3.4 Measurement Procedure 79

3.4 Measurement Results 79

3.5 Uncertainty Analysis 86

4 Discussion and Conclusions 97

5 Acknowledgments and References 99

5.1 Acknowledgments 99

5.2 References . . . 100

6 Appendices 104

6.1 Appendix A: The Scattering Matrix 104

6.1.1 Theory 104

6.1.2 Lossless Network 106

6.1.3 Lossy Networks 107

6.2 Appendix B: Imperfect Waveguide Walls lOS

6.3 .Appendix C: The Gap Correction 110

6.3.1 Frequency Independent .Approaches 110

6.3.2 Frequency Dependent Capacitor Model 117

6.4 Appendix D: Fields in a Transmission Line 119

6.4.1 Theory 119

6.4.2 TE, TM and TEM Modes 120

6.4.3 Green's Dyadic Function 124

6.4.4 Fields of a Coaxial Line 12S

6.5 Appendix E: Numerical Methods 131

6.6 .Appendix F: Kramers- Kronig Relations 132

6.7 .Appendix G: Data Smoothing 133

6.S Appendix H: Software 134

iv
Abstract

The transmission/reflection and short-circuit hne methods for measuring

complex permittivity are examined. Equations for permittivity are developed

from first principles. New robust algorithms that eliminate the ill- behaved

nature of the commonly used transmission/reflection method at frequencies

corresponding to integral multiples of one-half wavelength in the sample are presented. These allow measurements to be made on samples of any length.

An uncertainty analysis is presented which yields estimates of the errors incurred due to the uncertainty in scattering parameters, length measurement

and reference plane position. The equations derived here indicate that the

minimum uncertainty for transmission/reflection measurements of nonmagnetic materials occurs at integral multiples of one-half wavelength in the

material. In addition, new equations for determining complex permittivity

independent of reference plane position and sample length are derived. New

equations are derived for permittivity determination using the short-circuit

line allow positioning the sample arbitrarily in the sample holder.

Key words: cahbration; coaxial line; dielectric constant; loss factor; mi- crowave measurements; permeability measurement; permittivity measurement; reflection method; short- circuit; transmission; uncertainty; wave

guide.
Chapter 1

Introduction

Broadband measurements of complex dielectric constants are required

for a multitude of applications. Due to their relative simplicity, the transmission/reflection (TR) and short-circuit line (SCL) methods are presently

widely used broadband measurement techniques. In these methods a sample

is placed in a section of waveguide or coaxial line and the scattering parameters are measured, preferably by an automatic network analyzer [ANA]. The

relevant scattering equations relate the measured scattering parameters to

the permittivity and permeability of the material.

For the TR measurement method, the system of equations contains as variables the complex permittivity and permeability, the two reference plane

positions, and, in some apphcations, the sample length. In the TR procedure

we have more data at our disposal than in SCL measurements since we have

all four of the scattering parameters. The system of equations is generally

overdetermined and therefore can be solved in various ways.

In SCL measurements the variables are complex permittivity and permeability, sample length, distance from sample to short-circuit and reference

plane positions. However, in most problems we know the sample length, refer- ence plane position, and distance from short-circuit plane position to sample.

In these cases we have four unknown quantities (complex permittivity and

permeability) and therefore require four independent real equations to solve

for these variables. These equations can be generated by taking reflection

coefficient data at two positions in the transmission line, thus yielding the

equivalent of four real equations for the four unknown quantities.
Measurements are said to have a high precision if there is a small random

uncertainty, whereas measurements are said to have a high accuracy if the

systematic uncertainty is small. Measurement uncertainty is a combination

of the random and systematic errors. It is possible to correct for some systematic errors. The uncertainties and corrections of a set of measurements

is of paramount importance. The uncertainties yield an estimate of the deviation of the mean of a set of sample measurements from the true value.

The literature lacks a comprehensive summary of the uncertainties for the

TR and SCL methods.

In addition there are corrections which can be made to measurements.

For example, correction formulas exist that take into account the gap that

exists around a sample in a waveguide. A summary of corrections is presented

in this report.

With the advent of modern ANA systems there is generally no paucity of

data, and thus efficient numerical algorithms for the reduction of the scat- tering data are of paramount importance. To accommodate modern ANA

systems, Nicolson and Ross [1] and Weir [2] introduced procedures for obtaining broadband measurements, both in the time and frequency domains.

In the Nicolson-Ross-Weir (NRW) procedure the equations for the scattering

parameters are combined in such a fashion that the system of equations can be decoupled, so that this procedure yields an explicit equation for the permittivity and permeability as a function of the scattering parameters. This

solution has formed the basis of the commonly used technique for obtaining permittivity and permeabihty from scattering measurements [3], [4], [5]. The compact form of these equations, while elegant, is not well-behaved for

low-loss materials at frequencies corresponding to integral multiples of onehalf wavelength in the sample. In fact the Nicolson-Ross-Weir equations are

divergent, due to large phase uncertainties, for very low-loss materials at integral multiples of one-half wavelength in the material. Many researchers avoid

this problem by measuring samples which are less than one-half wavelength

in length at the highest measurement frequency. However, this approach, as shown later in the report, severely hmits the viability of the TR method since

short samples increase measurement uncertainty. Stuchly and Matuszewski

[6] presented a shghtly different derivation than Nicolson and Ross and obtained two explicit equations for the permittivity.

Ligthardt [7], in a detailed analysis, presented a method for determining permittivity by solving the scattering equations for the permittivity over
a calculated uncertainty region and then averaged the results. Ligthardt's

equations are useful for high-loss materials, but for low-loss materials they

suffer the same pathologies as the Nicolson-Ross [1], Weir [2], and Stuchly

[6] equations at multiples of one-half wavelength. In this report I present a procedure for obtaining complex permittivity from the scattering equations which is stable over the frequency spectrum. This procedure minimizes

instabihties in determination of permittivity by setting the permeability to

unity. This new procedure thus allows measurements to be taken on samples

of arbitrary length.

Another problem encountered is the transformation of 5-parameter measurements at the calibration reference planes to the plane of the sample. This

transformation requires knowledge of the position of the sample in the sample holder. However this information may be limited in many apphcations.

The port extension and gating features of network analyzers are of some help

in determining reference plane position, but does not completely solve the

problem. Equations that are independent of reference plane positions are

desirable. Also, equations that are independent of sample length are useful

for high temperature applications. In the past other authors, for example,

Altschuler [8], Harris [9], Scott [10], addressed the problem of either reference

plane invariance or sample length invariance, but the problem of combined

reference plane and sample length invariance remains to be resolved.

The goal of this report is threefold: first, to examine the scattering equations in detail for the TR and SCL methods and to present an improved

method for solving the equations in an iterative fashion with application to

permittivity measurements; second, to derive scattering equations that are

invariant to the position of the reference planes and sample lengths; and

third, to present an uncertainty analysis for this new solution.
Chapter 2

Transmission/Reflection

Method

2.1 Theory

In the TR measurement, a sample is inserted into either a waveguide or coaxial line and the sample is subjected to an incident electromagnetic field

[see figure 2.1]. The scattering equations are found from an analysis of the

electric field at the sample interfaces. If we assume electric fields Ej, Ejj^

and, Ejii in either the TEM mode in a coaxial line or the TEio mode in a waveguide, (with a time dependence of exp(ja;t)) in the regions I, II, and III, we can write the spatial distribution of the electric field for an incident field

normalized to 1:

El = exp(-7ox) + Ci exp(7ox) , (2.1)

En = C2 exp{--fx) + C3 exp(7x), (2.2)

Em = C4 exp{-jox), (2.3)
oLi

E|ncEpefl.-^

I m

L2- . c I ^ i-Trans. I

X

Port 1

I

I

I

I

I o

I J

I oI

I

I

I

I

I 1 Port 2

Figure 2.1: A dielectric sample in a transmission line and the incident(inc)

and reflected (refl) electric field distributions in the regions I, II, and III. Port

1 and port 2 denote calibration reference plane positions.
'='\

^vac

^27ry

l( "" V

/ \Clab)

R ~ J^rI^o =

^27r\2

where

(2.4)

(2.5)

(2.6)

^ = [/^fl - J>'A]/^o = fi*Rf^o- (2.7)

Here j = >/— 1, c„ac and c/at are the speed of hght in vacuum and laboratory, to is the angular frequency, Ac is the cutoff wavelength, Co and Ho are the permittivity and permeabihty of vacuum, e^ and //|j are the complex

permittivity and permeability relative to a vacuum, and 7o,7 are the propagation constants in vacuum and material respectively. The constants Cj are determined from the boundary conditions. The boundary condition on the

electric field is the continuity of the tangential component at the interfaces.

The tangential component can be calculated from Maxwell's equations given an electric field with only a radial component.

Ei{x = Lr) = Eii{x = Lr), (2.8)

Eii{x = L,^L) = Eni{x = Ii + X), (2.9)

where Lair = Li -\- L2 + L is the length of the air hne, Li and L2 are the

distances from the respective ports to sample faces, and L is the sample

length. The boundary condition on the magnetic field requires the additional

assumption that no surface currents are generated. If this condition holds,

then the tangential component of the magnetic field is continuous across the interface. The tangential component can be calculated from Maxwell's

equations for an electric field with only a radial component.

IdEr IdEjr

-{x = Li}^ —(x = Li), (2.10)

Ho dx fi dx

i^(. = i, + i) = l^(. = £. + i). (2.11)

fJ, OX flo (jx
For a two-port device the expressions for the measured scattering parameters are obtained by solving eqs (2.1)- (2.3) subject to the boundary

conditions. We assume that 512 = 'S'21. The explicit expressions are given by

where

Su = Ri

S22 — R2

S21 = R1R2

r(i -z'Y

1- T^z\

r(i -z')]

1- T^z\

\zOL-r2)

1 - V^z^

Ri = exp(-7oIi),

(2.12)

(2.13)

(2.14)

(2.15)

R2 = exp(-7ol2), (2.16)

where Li and L2 are the distances from the calibration reference planes to the

sample ends, and Ri and R2 are the respective reference plane transformation

expressions. Eqs (2.12)-(2.14) are not new and are derived in detail elsewhere

[1], [11]. We also have an expression for 2, the transmission coefficient,

z = exp(—7Z).

We define the reflection coefficient by

I2. p Mo

I2. _(_ 2

For coaxial line the cutoff frequency approaches 0, {cOc

reduces to

(2.17)

(2.18)

0) and therefore F

Cyac

Clab 1

Cvac iiK _^ 2

(2.19)

Additionally, 52i for the empty sample holder is
521 = R\R2G^v{-loL)- (2.20)

For nonmagnetic materials, eqs (2.12), (2.13), (2.14) contain e^, e^, L, and

the reference plane transformations i?i,i?2 as unknown quantities. Since the

equations for 5i2 and ^21 are equivalent for isotropic materials, we have four

complex equations, eqs (2.12), (2.13), (2.14), (2.20), plus the equation for

the length of the air line, or equivalently, nine real equations for the five un- knowns. Additionally, in many applications we know the sample length. For

magnetic materials we have seven unknowns. Thus, the system of equations

is overdetermined, and it is possible to solve the equations in various combinations. As an example, in nonmagnetic materials if the position of the

reference planes is not known accurately, then Li and L2 can be eliminated

from the equations to obtain equations that are reference plane invariant. A

whole family of reference plane independent equations exists, but only the

most useful are given below as examples:

1^21 i

S\\S22 _

S12S21

r(i -^')

1-

-n

1- ^2^2

(1 - ^Y

5ji smh 7L,

(2.21)

(2.22)

(2.23)

(2.24)

where the vertical bar denotes the magnitude of the complex expression.

Equation (2.23) is valid only for coaxial line.

Nicolson and Ross [1], and Weir [2] combined the equations for 5ii and

^21 and discovered a formula for the permittivity and permeability. This

procedure works well at frequencies where the sample length is not a multiple

of one-half wavelength in the material. At these latter frequencies, however.

^21 , nili:il!l

'21

5*21 5*12 - 5ii522 = exp[{-2jo){Lair - L)]-

-T'
the solution procedure completely breaks down. In the Nicolson- Ross- Weir

algorithm the reflection coefficient

Ti = X ± \/X2-l, (2.26)

is given explicitly in terms of the scattering parameters where

-^ = Tr¥' (2.27)

and

Vi = S2i + S,u (2.28)

V2 = 5'2i - Su. (2.29)

The correct root is chosen in eq (2.26) by requiring [Fi | < 1. The transmission

coefficient, zi, for the Nicolson- Ross-Weir procedure is given by

5"!! + 'S'21 — Fi

and the permeability is given by

where Aq is the free space wavelength and Ac is the cutoff wavelength. In this

report ^z^ = 1 (this removes the ambiguity in the logarithm branch). The

permittivity is given by

-^ = -[:^ln(-)r, (2.32)

A^ ZttL zx 4 = 4[l^-[;^ln(-)n. (2.33)

Equation (2.32) has an infinite number of roots for magnetic materials, since

the logarithm of a complex number is multi-valued. In order to pick out

the correct root it is necessary to compare the measured group delay to the

calculated group delay. The calculated group delay is related to the change

of the wave number, k, with respect to angular frequency

10
oup — '^

The measured group delay is

Tcalc.group — ^ jr \ \2 \ 2

' {Z.64:)

_

\ d(f> 'meas.group — 7^ TF' \Z.OO)

ZTT aj

where (f) is the measured phase. To determine the correct root, the calculated

group delays are found from eq (2.35) for various values of n in the logarithm

term in eq (2.32), using In z = In \z\ + j{6 -\- 27rn), where n — 0, ±1, ±2...,

and compared to the measured value from eq (2.35). The comparison yields

the correct value of n. Many researchers think of the Nicolson-Ross-Weir

solution as an explicit solution; however, due to the phase ambiguity, it is not.In the limit of no loss, the Nicolson-Ross-Weir solution is divergent at integral multiples of one-half wavelength in the sample. This singular behavior can be minimized in cases which permeability is known a priori. An stable

algorithm will be developed in the next section.

In figure 2.2, a typical plot utilizing the NRW [1] equations is displayed.

At points corresponding to one-half wavelength the scattering parameter

I ^ii I gets very small. Also, for small | 5ii | the uncertainty in the measurement of the phase of 5ii on a network analyzer is very large. Thus,

this uncertainty dominates the solution at these frequencies. To bypass this

problem many researchers resort to short sample lengths. Use of short samples lowers the measurement sensitivity, however. In fact, as will be shown

later in the report that when minimizing the uncertainty for low-loss, low

permittivity materials, a relatively long sample is preferred.

Another interesting result can be obtained if we assume that e|j and the

measured value of an 5-parameter are known at a single angular frequency,

Uk. In this case we can solve either (2.15) or (2.16) for the reference positions

and then substitute this length into eqs (2.12)-(2.14) to obtain relations for

the reference plane positions at other frequencies. This length is the equivalent electrical length of the section of line. If we let m denote "measured

value" then we can obtain a relation for the reference plane rotation term at an angular frequency of uji

11
2.5

2.3

^%

1.9 -

1.7 -

1.5

' 1 —^^—

r

—

1

1

\

\ r^ 1\ 1

1 1

1 ~

- Lj , y|l-^^ p4^

7 mm

-

. r^

- coaxial

sample

— 3.279 cm ~

1 I

long

6 9 12

Frequency (GHz)

15 18

Figure 2.2: The determination of the permittivity of a polytetrafluoroethylene sample as a function of frequency using the Nicolson and Ross equations

(soUd line) and the iteration procedure (dashed Une).

12
Rliu;,) = [SnM]m

i_^2r2-|(^)

r(i-z2)

Rl{LOi)R2{LOi) = [S21M]m

^2p2-|($)

(2.36)

(2.37)

2(1 -P)

Thus, we can determine the reference plane positions in terms of e*fi{u}k) and

the measured value of the scattering parameter at u^. Equations (2.36),

(2.37) may be very useful for problems where other methods have produced

an accurate measurement of e|^ at a single frequency.

2.2 Numerical Determination of Permittivity

There are various ways of solving the scattering equations depending on

the information available to the experimenter. In cases where the sample

length and reference plane positions are known to high accuracy taking vari- ous linear combinations of the scattering equations and solving the equations

in an iterative fashion yields a very stable solution on samples of arbitrary

length. A useful combination is

^{[Sn + S2i]i-f3[Sn^S22]}

1 - z^T^

(2.38)

In eq (2.38) the 5-parameters to be used need to be transformed from the

calibration plane to the sample face by use of eq (2.15), (2.16). Here 13 is a constant which varies as a function of the sample length, uncertainty in

scattering parameters and loss characteristics of the material. The constant

/? is a weighting function for the S- parameters. For low-loss materials,

the 5*21 signal is strong and so we can set /? = 0, whereas for materials of

high loss Sii dominates and a large value of f3 is appropriate. A general

relation for ^ is given by the ratio of the uncertainty in 521 divided by the

uncertainty in 5ii. In figure 2.2 the iterative solution of eq (2.38) (in the

dashed line) is compared to the Nicolson-Ross procedure for a sample of

13
2.05

€r 2 045

2.04

_ i

L* ' .

1|W">^ 1 ; fm^fVtKi#^

;.^L^ M :' ^ - •• ^ j^

in' « Hi* I •1 -1

—

P - n P -

- - ,o - 1 P - i - /3—

. ! ! 1 1 . .

1 , . 1 1 ' 1 1 1 1 ! 1 • 10 11 Frequency (GHz)

12 13

Figure 2.3 The real part of the permittivity obtained from eq (2.38) for

various values of /S. The dotted Une is for /? —* co, the soUd Hne for /? =

and the dashed hne for = 1.

14
0.01

-0.01

T—

I

I I I

•I/Wl/VvV^' '''•'

8

/3 ^0

^ =1

J_J. I I I I I 10 11

Frequency (GHz)

J \ L—

12 13

Figure 2.4 The imaginary part of the permittivity obtained from eq (2.38)

for various values of f3. The dotted line is for /? -^ oo, the solid line for /? =

and the dashed hne for /3 = 1.

15
polytetrafluoroethylene (PTFE) in 7 mm coaxial line. The iterative solution

process is summarized in appendix E. A striking contrast is seen between the

solutions. In figure 2.3 and 2.4 data for PTFE in an X-band waveguide is reduced for permittivity while varying the parameter /?. In the case of PTFE,

which has low loss and relatively low e^, the best results are produced by

using only ^21 and S12 data. One requirement for an iterative technique is the selection of the initial guess for the permittivity. As an initial guess we

use the solutions to the Nicolson- Ross-Weir equations as a starting value and

then use the previously obtained permittivity at one frequency as the initial

guess for the next frequency.

For cases when the reference plane positions are uncertain, we find that

eq (2.25) is robust. When we use eq (2.25), no reference plane transformation

need be performed since it has been eliminated by use of the relation Lair =

Li -\- L2 -\- L. Equation (2.25) works well for both low-loss and high-loss

materials. If eq (2.25) is solved in tandem with any one of eqs (2.21)-(2.24)

the measurement can become independent of reference plane position and

sample length. That is, we have four real equations for the four unknowns:

f' f" T T Ri Ri 1 ^aiTFor magnetic materials, four independent real equations are required (or

two independent complex equations). Since in this case there are seven un- knowns and nine real equations, (eqs (2.12)- (2.14), (2.20) and the equation

for length of sample holder) it is possible to use various combinations of the

basic equations. When permeability is included for materials with both lowloss magnetic and dielectric properties, the solutions for both permittivity

and permeability will be unstable at integral multiples of one-half wavelength

in the material.

2.3 Corrections to Data

Once a set of measurements is obtained it is necessary to correct the

data taking into account correctable, systematic uncertainties. These known

uncertainty sources include air gaps around samples, waveguide wall imperfections, imperfect short-circuits, and waveguide losses. Air gaps are particularly important for coaxial samples; in particular, the gap near the center

conductor is important since the electric field is higher in this region. In

order to make corrections it is necessary to obtain precise measurements of

16
both the sample and the sample holder; air gauging equipment is useful for

these measurements.

2.3.1 Gaps Between Sample and Holder

Air gaps around samples affect measured value of permittivity. In waveguide the air gap along the wide side of the guide has the primary influence on the calculated permittivity since this region has a higher electric field. For the same reason in coaxial line the gap near the center conductor contributes more than a gap of the same thickness near the outer conductor.

See appendix C for gap correction formulas [12], [13], [14]. These correction

formulas are approximate and generally undercorrect for the effects of the air

gap. Imperfect waveguide or coaxial hne walls have been studied by Hill [15]

and the results from this work are summarized in appendix C. Waveguide

and coaxial line losses can be probed by measuring the scattering parameters

of the empty waveguide.

2.3.2 Attenuation Due to Imperfect Conductivity of

Sample Holders

Since no waveguide is perfectly conducting, all propagating modes are at- tenuated to some degree. The finite conductivity of waveguide walls promotes

power loss in the guide.

The power flow down the waveguide is given by

P = l f Re[E X H*] dS = f I \Ht\^dS, (2.39) Z Js 2, Js where Z is the wave impedance and (<) denotes the tangential component

[16]. The power loss in the guide is

PL-lY^jyds = ^Jm'ds, (2.40)

where Rs = Jt^J^ is the effective surface resistance, a is the conductivity,

and c is a closed path. The effective attenuation constant can be defined as

17
Table 2.1 Cutoff Frequencies.

EIA WR() Band Cutoff frequency(GHz)

650 L 0.908

430 W 1.372

284 S 2.078

187 c 3.152

90 X 6.557

42 K 14.047

22 Q 26.342

a

2P'

(2.41)

For rectangular waveguide the integrals in eq (2.40) can be performed to

obtain [16]

2R.

OtJEn

hri^l - i^r

KJ^

(i + -)(Tr + (i-(Tr) /c.2^iK(!) + ^')

/ f ' m\^^-rn , (2.42)

where a and h are the guide dimensions, /c is the cutoff frequency, m,n =

0,1,2,. . . and 7] is the impedance of free space. For n — the attenuation

Waveguides can be used over only a band of frequencies, so the selection

of a waveguide depends on the frequencies to be measured. The cutoff fre- quency is a lower bound on frequencies that can be transmitted in the guide

without introducing evanescent modes. A list of cutoff frequencies for various

waveguide sizes is given in table (2.1). The cutoff frequency for either TE or TM waves in rectangular waveguide is given by

^•^^^"^"~ v^r7^'"^^i^'' (2.44)

18
where a is the long dimension of the guide, 6 is the short dimension of the

guide and m,n = 0, 1,2.... For the TEio mode the cutoff frequency is:

1 c_ 2>/J7ea ~ 2a'

yjc/mn — (2.45)

Rectangular waveguide operated in the TEiq mode becomes overmoded at twice the cutoff frequency. Operating frequencies and waveguide dimensions

are given in tables (2.1) and (2.2).

For a circular waveguide of radius a the attenuation is given by [16]

o^TMmn -

R. 1

O^TEr,

Rs

«^yr^

«Vi-(^)''

m^

(2.46)

(2.47)

where p'^^ are the roots of J'^ip'^n) = 0- The cutoff frequencies in circular waveguide are given for TMmn by

Pr.

where pr,

\Jc)mn . r\ '

are the nth root of Jm{Pmn) — 0- For TEmn waves we have

(2.48)

\Jc)mn

P. (2.49)

yqI2Tra

where p'^^ are the nth roots of J^ip'^nn) = 0- Coaxial line has the distinct advantage of having no cutoff frequency,

however, coaxial line becomes multi-moded above a certain frequency. These

multi-modes are due to bead resonance which couples in the TEn and higher

modes. The approximate upper frequency limit due to bead resonance is given in table (2.3).

Also, the propagation of the TE and TM modes in the cable in addition

to the TEM mode is possible at higher frequencies. The cutoff wave numbers

for higher TM waves in coaxial line are given by the roots of the equation

[16],

= 0, (2.50)

19
Table 2.2 Rectangular waveguide dimensions and operating frequencies in

air.EIA WR Band a (cm) h (cm) TExQ Operating Frequency(GHz)

650 L 16.510 8.255 1.12- 1.7

430 W 10.922 5.461 1.7 - 2.6

284 S 7.710 3.403 2.6 - 3.95

187 c 4.754 2.214 3.95 - 5.85

90 X 2.286 1.016 8.2 - 12.4

42 K 1.067 0.432 18 - 26.5

22 Q 0.569 0.284 33- 50

Table 2.3 Air-filled coaxial cable operating frequencies.

Coaxial Cable Dimensions (mm) Useful Operating Frequency (GHz)

3.5 - 34.5

7.0 - 18.2

14.0 - 08.6

and for TE waves in coaxial line by

0, (2.51)

J'n{Kr^) J'AKro)

where J and N denote the Bessel functions of the first and second kind,

and Tj- and Tq are the inner and outer radii, respectively [16]. The cutoff

wavelengths are given approximately by

A, «-(r,-r,) , 9=1,2,3,.... (2.52)

For example, the TM mode cutoff frequency in 7 mm coaxial line for eq (2.52)

is approximately 34 GHz. For headless air line it is possible to exceed the

frequencies given in table 2.3

20
2.4 Instrumentation

The following apparatus is needed for broadband TR measurements:

• Automated network analyzer

• Data acquisition and analysis systems

• Precision waveguide or coaxial line

Network analyzers are rapidly becoming the preferred data acquisition system

for many researchers. Network analyzer systems have various error sources. These include [17]:

• Imperfect matching at connectors

• Imperfect calibration standards

• Nonlinearity of mixers, gain and phase drifts in IF amplifiers, noise

introduced by the analogue to digital converter

• Imperfect tracking in dual channel systems

Generally, the manufacturer furnishes the specification of its own system.

The choice of network analyzer is crucial for good phase data; the ANA

which is best suited for the frequency band of interest should be chosen.

2.4.1 ANA Calibration

Coaxial Line Calibration

Various components of the ANA introduce phase and magnitude uncertainties. Examples of these uncertainty sources are frequency response, various

mismatches and radio frequency (r.f.) leakage. Cahbration of the ANA re- moves the systematic uncertainties in a measurement of a set of standards,

for example, shielded open circuit, reflect or load. Information on the dif- ference between a specific measurement of these standards and the expected

values (stored in ANA system) generates a set of error correction coefficients.

The calibration coefficients are determined from solving a set of simultaneous

21
equations generated from the linear fractional transformation. After calibration, when the system is operated with correction on mode, the measurements

are corrected by the calibration information.

The 7 mm hne calibration kit contains the following standards:

• open-circuit

• short-circuit

• low and high frequency loads

• sliding load

The Waveguide Calibration Kit

For caHbration of a given waveguide it is necessary to construct a calibration kit for the ANA. The TRL calibration consists of measuring a thru, a reflect, and a section of line. The length of hne to be used as the thru

is calculated as follows: the phase delay {(f)) in waveguide is related to line

length [i] and guided wavelength (A^) by

e=p, (2.53)

where the guided wavelength is related to the free-space wavelength by:

K = ,, \.._ - (2.54)

^/^^W

The procedure for the calculation of the line length is to calculate i for a phase delay of 20° at the lowest frequency of interest, and again for a phase delay of 160° at the highest frequency of interest, and then choose a

line length between these extreme values, typically, A/4 at geometric center

frequency [Jcenter = VJminJmax)-

When using waveguide for measurements it is necessary to insert sec- tions of waveguide, each approximately two wavelengths in length, between

the coax-to-waveguide adapter and the sample holder. The function of the

waveguide sections is to damp out any evanescent modes. The X-band caHbration kit contains the following items:

22
• coax to waveguide adapters

• short-circuit

• a section of waveguide line to be used as a sample holder

• a section of calibration "line"

• two lengths of X-band waveguide approximately two wavelengths long

• loads

2.4.2 Sample Holder Specifications

The sample holder for TR measurements should consist of high precision

waveguide or coaxial hne. There should be a length of waveguide between the

waveguide-to- coaxial line adapter and sample holder to damp out evanescent waves generated at these discontinuities. The length of the sample holder

should be measured with a high degree of precision. Nicks and other abrasions

in the sample holder will, of course, degrade performance. When 7 mm

coaxial headless air line is used, APC-7 connectors are usually preferred. The

sample holder should be treated with extreme care and stored in a protected

area.The impedance of coaxial line is given by

Zo =

\

(« + ^"^)

(2.55)

where R = resistance, C = capacitance, L = inductance, and G = conductance (all per unit length). The surface impedance determines the loss in the

line and is given by

!+.;

+ (2.56)

TT [Z)((J^)oc d{(T6)i

where D is inner diameter of outer conductor, d is outer diameter of inner

conductor, a and 6 = l/y/TrJJia are the conductivity and skin depth respectively [18].

23
Table 2.4 Connector directivity repeatability

Connector Type Repeatability at 18 GHz(dB)

7 mm with 6 slot collet 59

Precision Type-N 49

PSC-N 59

Wong has shown that the impedance of a precision 7 mm coaxial air line,

with uniformity of approximately ±2//m, varies shghtly with frequency from

50.25 n at 0.1 GHz to 49.95 ft at 20 GHz [18].

The repeatability of the 7 mm connectors are summarized in tablei 2.4

[18].

2.4.3 Sample Preparation

Samples to be used for measurements must be prepared carefully. Scratches,

nicks and cracks may alter the measured dielectric properties. Minimize any

unnecessary wear and tear on the sample by placing it in a secure area between measurement sessions. The sample length measurement is critical and

should be performed carefully with a precision micrometer at laboratory temperatures.

The following list summarizes the preparation procedure:

• Carefully select a piece of material free of unusual inhomogeneities,

cracks, etc.

• Have the sample machined to fit as tightly as possible in the sample

holder. The machining process should not leave metalhc residue on

the sample. Note that gaps near the center conductor in coaxial line are more serious than gaps near the outer conductor. Samples which

fit very tightly on the outer conductor can be inserted more easily by

prior cooling.

• Measure the sample length with a high degree of precision at a temperature very close to that realized in the laboratory. The resulting strain,

24
Table 2.5 Metrologist tolerance specifications.

Symbol Term

_L perpendicularity

concentricity

cylindricity

circularity

parallelism

flatness

® position

/ runout

AZ//L, from increased temperature can be calculated from the linear

thermal expansion coefficient, a, by using the relation IS.LJL = aAT.

• Keep the sample very clean and store in a secure area. If the sample

requires cleaning, an ultrasonic cleaner will suffice.

• Keep the gap between sample and guide walls to a minimum. We have

found that clearances of (2.5-7) xlO"^ m (0.0001 - 0.0003 in.) are acceptable.

In figures 2.5 an4 2.6 examples of specifications for coaxial and waveguide

samples are shown.

The parameters of interest to the machinist in this example are the cylindricity, concentricity, and perpendicularity. A hst of metrology symbols is given in table (2.5). It is very important to specify precisely your tolerances

to the machinist. Generally the machinist will ask that the sample holder be

available for fitting purposes from time to time. Corrections for the resulting

effect of the air gap on the permittivity is given by the equations in appendix

C.

25
|0| 0.0003 ifT

0.2751 in.0

±0.0003

|@| 0.0003 in~

Glass Rod

MAPL - GLASS

SCALE - 4:1

Figure 2.5 Sample specifications for a coaxial sample

26
0.399 in. ±0.0006 _i *— 0.9 in. ±0.05

•>

Alfl 0.00031 c] ¥

/

1

0.8<

±0.0

k U10.0003 ^^ » in 1006

l-R 1 [ /

//I 0.0003 IaK

-C- -A-i

DIELECTRIC BLO

SCALE 2:1 CK

Isometric View

Figure 2.6 Sample specifications for a rectangular sample in inches

27
2.4.4 System Diagnostics

In order to verify the accuracy of the system before each measurement it

is important to have a verification test. For the TR method our standard

verification procedure consists of measuring the permittivity of air in the

empty sample holder and comparing the results to those of a reference set. An example reference is given in figure 2.7 and 2.8.

2.5 Measurement Results

The accuracy of TR measurements is limited by a number of factors.

These factors include the uncertainties of the network analyzer, dimensional

tolerances of sample holder and sample, and the accuracy of the underlying

field model.

In figure 2.9 through 2.20 the permittivity of coaxial samples of air and

cross- linked polystyrene and waveguide X-band samples of air, PTFE, and

glass are plotted as a function of frequency. All of these data were reduced

with eq (2.25). The PTFE sample, in both waveguide and coaxial line has an effectively a nonexistent air gap. It is possible to smooth the plot of the

calculated permittivity in various ways. We currently use either a loss-pass

filter or a maximum entropy smoothing routine. A smoothing procedure that

utilizes the method of maximum entropy is given in appendix G.

2.6 Appearance of Higher Order Modes

The field model developed in this report assumes a single mode of propagation in the sample. Due to cutoff conditions, higher modes of propagation

become possible in inhomogeneous samples of high dielectric constant. Air

gaps also play an important role in mode conversion. If mode conversion

does occur due to some type of perturbation, then the theoretical model

expounded in this report will break down. The degree of breakdown of the

model depends on the power coupled to the higher modes. Generally, the appearance of higher modes manifests itself as a sudden dip in |5ii|. This dip is a result of resonance of the excited higher mode. We can expect the general

28

I
CALIBRATION VERIFICATION (L = 99.97 mm)

1 1/02/89

-0.16

0.045 3.636 7.227 10.818 14.409

Frequency in GHz (1.7955 GHz per DIv.)

18

Figure 2.7 System diagnostic example for |52i|

29
CALIBRATION VERIFICATION (L = 99.97 mm)

1 1/02/89

Q

CO<

-0.45

0.045 3.636 7.227 10.818 14.409

Frequency in GHz (1.7955 GHz per Div.)

18

Figure 2.8 System diagnostic example for arg(52i)

30
1.005 '

I

'

I

' I

'

I

' I

' I

' I' I'

I I I I

' I' 1'

I

'

I

'

1.003 -

i^w^ I iiig»i^ ^11 *^0^mm^^^0^^^^^m^^0^^^^mm^K^aa^^^

1.001

^ L

0.999

0.997

0.995 I I I I I

. I I I'

I I I

- I I I I 6 9 12

Frequency (GHz)

15 18

Figure 2.9 e^ of air in a coaxial line for empty air line using TR method.

The worst case uncertainty at 10 GHz, Ae'^ = 0.005.

31
0.01 I

I I

I II II

0.008

0.006

0.004

0.002

T—

1

1 II

I 1I I I 1—1 I—I T-l 1I I T rI I , I . I I , I , I , I

'

I '

1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18

Frequency (GHz)

Figure 2.10 e'j^ of air in a coaxial line for empty air line using TR method.

The worst case uncertainty at 10 GHz, Ae'^ = 0.0012.

32
2.1 I '

I

' I' I

I I

'

I

'

I I

'

I I

'

I

' I 1 M '

I

' I'

I

'

2.08

2.06

^ \-

2.04

2.02

I I r I I II I II I II ,r I I . I I I , I , I . I , I I , I

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Frequency (GHz)

Figure 2.11 Calculated e'^ for PTFE in a coaxial line as function of frequency

with TR method. The worst case uncertainty at 10 GHz, Ae^ = 0.009.

33
0.05 I '

I

'

I 'I

I

I M II M I I

'

I

I I

'

I

' I M '

0.04

1 I r^

0.03

0.02

0.01

Ql I I I

'

I

' ' '

I '

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Frequency (GHz)

Figure 2.12 Calculated e^ for PTFE in a coaxial line as function of frequency

with TR method. The worst case uncertainty at 10 GHz, Ae^ = 0.0017.

34
1.005

1.004

1.003

1.002

1.001

0.999

0.998

0.997

0.996

0.995

1—I—I—r-|—I—I—I—1—]—I—I—I

\—

I—

r

I I I I I r

|kg,^,.---Avv^^^

J I I I i \ I I \ I i \ L-J I I I \ \ I I I L 8 9 10 11 12 13

Frequency (GHz)

Figure 2.13 e^ of air in a X-band waveguide for empty air line using TR

method. The worst Ceise uncertainty at 10 GHz, Ae^ = 0.005.

35
0.01

0.008

0.006

0.004

0.002 —

1—I—I—I—I—I\—I—I—I—

r

1—

r

><^yA^\yAAA^yAy»wVW|V^

8 10 11

Frequency (GHz)

12 13

Figure 2.14 e'^ of air in a X-band waveguide for empty air line using TR

method. The worst case uncertainty at 10 GHz, Ae^ = 0.0012.

36
2.55

2.54

2.53

2.52

2.51

-!—I—I—I—I—

r

I

'

I I M

Cross -I inked

Polystyrene

Corrected

2.50' I I I I I I

I

>

' L_l L_J L -L L 9 10 11

Frequency (GHz)

12 13

Figure 2.15 Calculated e^ with and without gap correction for cross linked

polystyrene in a X-band waveguide as function of frequency with TR method.

The worst case uncertainty at 10 GHz, Ae'j^ = 0.011.

37
0.01 n—I

\—\—\—r-:

\—\—\—;

\—\—\—\—\—i—I—I

\—\—\—•—

0.008

0.006

0.004

0.002

01—

' '

I

' I I I I L.J 1 I I I I L_J I' ' 8 9 10 11 12 13

Frequency (GHz)

Figure 2.16 Calculated e'^ with and without gap correction for cross-linked

polystyrene in a X-band waveguide as function of frequency with TR method.

The worst ccise uncertainty at 10 GHz, Acj^ = 0.002.

38
Z.l —

—1—1—1—

1—r —r- "r"'"T-"[ I" 1 I r1

' 1 -I —r""

2.08

- —

2.06

-

^d'

- ^ R ~

2.04

- -

2.02

_ —

9 I

1 1 1 1 1 1 1 1 1 1 t 1 1 I ) 1 J \ 10 11 12

Frequency (GHz)

13

Figure 2.17 Calculated e'^ for PTFE in waveguide as function of frequency

with TR method. The worst case uncertainty at 10 GHz, Ae'^ = 0.009.

39
0.01 I

\—\—I—

r

0.008 —

0.006

0.004

0.002

Tr'-r 1 r T—1—

r

ry^rnA-i^''''''^^\^,\/^^^

Ql L_J I I I I I I I I L J_l I L 9 10 11 12

Frequency (GHz)

13

Figure 2.18 Calculated e^ for PTFE in waveguide as function of frequency

with TR method. The worst case uncertainty at 10 GHz, Ae^ = 0.0016.

40
6.2

€r 6.1

6.0

n I

I—I—

1723 Glass

Corrected

'Uncorrected

J I I L J—L J I I L J L 10 11

Frequency (GHz)

12 13

Figure 2.19 e^ with and without gap correction using TR method for 1723

glciss in a X-band waveguide. The worst case uncertainty at 10 GHz, Ae^ =

0.013.

41
0.01 I—I—I—I—I

\—I

\—I—I—1—I\—1—I

\—1—

r

0.08

0.06

0.04

0.02

I 1 I r

I I I I I I I \ I I I ^ \ ^ 1 I I 'I'll

8 9 10 11 12

Frequency (GHz)

13

Figure 2.20 e^ using TR method for 1723 glass in a X-band waveguide. The

worst case uncertainty at 10 GHz, Ae^ = 0.009.

42
transmission/reflection models to break down (including the one developed

in this report) for materials of high dielectric constant and/or inhomogeneous

samples.

In order to minimize the effects of higher modes, shorter samples can be

used. Higher order modes will not appear if the sample length is less than

one-half guided wave length in the material.

2.7 Uncertainty Analysis

In this section the uncertainty incurred when using the equations ex- pounded in this report will be estimated. The sources of error in TR measurement include

• Errors in measuring the magnitude and phase of the scattering parameters.

• Gaps between the sample and sample holder and sample holder dimensional variations.

• Uncertainty in sample length.

• Line losses and connector mismatch.

• Uncertainty in reference plane positions.

Correction for errors arising from gaps around the sample is obtained

from equations available in the Hterature [12], [13], [14]. The formulas given

in the literature generally undercorrect for the real part of the permittivity

and overcorrect for the imaginary part of the permittivity. We assume that

all measurements of permittivity have been corrected for air gaps around

the sample before the following uncertainty analysis is applied. Errors from

precision air-line dimensional variations have been studied by Hill [15], who

showed that these uncertainties are much smaller than the systematic uncertainty introduced by the network analyzer. In order to evaluate the uncertainty introduced by the measured scattering parameters, a differential

uncertainty analysis is applicable with the uncertainty due to ^n and 521

evaluated separately. We assume that the total uncertainty can be written

as

43
Ae',R

-R e'j,\\d\S^\

A|5, + A^J +

dL

AI +

dd

Ad

(2.57)

^^R

'R

( de'l, dt'k

\i [m'^'^^) ^ ii!

A^. + ''-alWC'-

dL dd

Ad

(2.58)

where a = 11 or 21, A^ is the uncertainty in the phase of the scattering

parameter, A|5o| is the uncertainty in the magnitude of the scattering parameter, Ad is the uncertainty in the air gap around the sample, and AL

is the uncertainty in the sample length. The uncertainties used for the Sparameters depend on the specific ANA used for the measurements. The

derivatives of eqs (2.12) - (2.14) can be explicitly calculated and are given

below for a coaxial line:

where

dt R [l-rV]exp(j^)

d\S:21

di

Q

R = j\S2l\

deR

-CQ

(2.59)

(2.60)

(2.61)

Q = 2ATz[S2iz - 1] + B[{1 - r^) + 2S2iTh], (2.62)

A

iO '^lloCl.cil^^^)

1 +

^°/^fl

1 +

7oM,

B^ Lu^H*f^z

2cLc7

'

C = -7^,

(2.63)

(2.64)

(2.65)

44
"" =expO^)^ -—^, (2.66)

and

d\Sn\ "^-^ ' P

P = A[{1 - z^) + 25n2'r] + 2BTz[S^^V - 1]. (2.69)

The measurement bounds for S'-parameter data are obtained froni specifications for a network analyzer. The dominant uncertainty is in the phase

of 5ii as I S\\ |—^ 0. The uncertainty in 521 is relatively constant until ^21

< -40 dB; it then increases abruptly. The various derivatives are plotted in

figures 2.21 through 2.32.

In figures 2.33-2.38 the total uncertainty in e|j computed from 5*21 and S^

is plotted as a function of normalized sample length, for low-loss and highloss materials at 3 GHz with various values of e|j and the guided wavelength

in the material given by

K = I ^^

(2.70)

In figures 2.33 through 2.38 the error due to the gap correction is not

included. We see that the minimum uncertainty for low-loss materials occurs

at multiples of one- half wavelength. The reason for this can be determined

from examination of eqs (2.12), (2.14) in the hmit that 5ii —> 0, ^21 —+ 1 with r 7^ 0. These equations then reduce to

z^-l^O. (2.71)

Generally, we see a decrease in uncertainty as a function of increasing

sample length. In the case of 821 for high-loss as shown in figures 2.35 and

2.36 we see first a general decrease in uncertainty and then an increase in uncertainty. This increase occurs because A52i increases when the transmitted

signal is less than -40 dB from the reference value. For the case of high loss

45
Figure 2.21 The derivative of 4 with respect to |52i| with e^ = (10.0, 0.01).

46
^|S2l|' ^

Figure 2.22 The derivative of e'^ with respect to |52i| with e^ = (10.0, 0.01).

47
Figure 2.23 The derivative of e^j with respect to 9 using 521 with e^

(10.0,0.01).

48
Figure 2.24 The derivative of e^ with respect to d using 821 with t)^ (10.0,0.01).

49
3500

S21

0.5 1.0 1.5 2.0 2.5 3.0 3.5

L/Xm

Figure 2.25 The derivative of e^ with respect to length using S21 with t*n (10.0,0.01).

50
1600

1200-

l|^l 800

Figure 2.26 The derivative of e^ with respect to length using 521 with e^

(10.0,0.01).

51
Figure 2.27 The derivative of e'j^ with respect to |5ii| with t}^ = (5.0,0.01).

52
Figure 2.28 The derivative of e^ with respect to l^ni with e|j = (5.0,0.01).

53
Figure 2.29 The derivative of e'^ with respect to 6 using 5ii with £^

(5.0,0.01).

54
Figure 2.30 The derivative of e^ with respect to 9 using 5ii with ej^

(5.0,0.01).

55
3000

2000-

1^1

<9L

1000-

Figure 2.31 The derivative of e^ with respect to length using 5ii with e^

(5.0,0.01).

56
1000

800-

600 —^1

400

200

Figure 2.32 The derivative of e'^ with respect to length using 5ii with e^

(5.0,0.01).

57
the uncertainty in ^n approaches a constant value. This is so because for

high-loss materials where the wavelength is much smaller than the sample

length, only weak signals penetrate through the sample and thus the front

face reflection dominates the 5ii parameter.

Also, the uncertainties in the 5-parameters have some frequency dependence with higher frequencies having larger uncertainties in phase. In figure

2.39 a measurement and uncertainty bounds (e^ ± Ae^) are shown; note uncertainty bounds are conservative. This is a consequence of the ^-parameter

uncertainties being conservatively estimated by the manufacturer.

58
0.016

0.012

Acr

—f- 0.008

0.004

21

3.5

Figure 2.33 The relative uncertainty in eij(w) for 821 for a low-loss material

as a function of normalized length, with Cj^ = (5.0,0.001) in solid line and

(10.0,0.001) in dashed line.

59
0.08

0.06 -

-^

0.04

0.02 -

1 1

P\^€r = (10.0, 0.001)

r —

Sn

-

_ / 1

/ L-^R^ (5.0, 0.001)

/ A

-y —

1 ^

I / \ ''^ /\/-\

^r \.

Figure 2.34 The relative uncertainty in ^ji{uj) for S\\ for a low-loss material

as a function of normalized length, with e^ =(5.0, 0.001) in soHd Hne and

(10.0, 0.001) in dashed Une.

60
AeiR 0.010 -

Figure 2.35 The relative uncertainty in e^(w) for ^21 for a high-loss material

as a function of normalized length, with e^= (5.0, 2.0) in solid line and (10.0,

2.0) in dcLshed line.

61
0.09

0.07

^4

4

0.05

0.03

0.01

21

^R = (10.0, 2.0)—

Figure 2.36 The relative uncertainty in t"p{uj) for ^21 for a high- loss material

as a function of normaUzed length, for e% =(5.0, 2.0) in sohd line and (10.0,

2.0) in dashed line.

62
2.0,

A4

0.04

'11

,€r = (5.0, 5.0)

^e, = (10.0, 5.0)_

Figure 2.37 The relative uncertainty in €^ft{uj) for Su as a function of frequency for a high-loss material as a function of normahzed length with e^ =

(5.0, 5.0) in solid hne and (10.0, 5.0) in dashed line.

63
4.8

Ae'/

4

0.8

0.08

11

€r= (5.0, 5.0) _

Figure 2.38 The relative uncertainty in €^(a;) for Su for a high- loss material as a function of normalized length with e^ = (5.0,5.0) in soUd Une and

(10.0,5.0) in dashed hne.

64
11.3

11.26

11.22

€'n

11.18

11.14

11.1

Heavy Metal Fluoride Glass

X-BAND, L = 0.01767m

10 11 Frequency (GHz)

Figure 2.39 A plot of e/^(w) from measurements on a heavy metal fluoride

glass and uncertainty bounds.

65
2.7.1 Uncertainty in Gap Correction

The correction for an air gap between the wall of the sample holder and

sample is very important for measurements of high permittivity materials.

In addition, the uncertainty in the gap correction is very important for high

permittivity materials and may actually dominant the uncertainties of the

measurement. In appendix C the gap correction is worked out in detail. In

this section the uncertainty in the gap correction will be worked out to be

used in eqs (2.57), (2.58). The uncertainty for the air gap is related to the

uncertainty in the measurement of the air gap.

Waveguide Gap Uncertainty

The uncertainty due to an air gap between sample and holder can be calculated from the partial derivatives of e^ with respect to gap thicknesses, d. The relevant derivatives for waveguide are given by

d<R -c' \ ^ 1 r'2 ^

(2.72)

dd -^^6-(6-c/)6U^ '^''[h-{h-d)e'^^r

dd ^'"^^^[6-(6-cf)6U]^'

(2.73)

Coaxial Gap Correction

For coaxial line the relevant derivatives are given by

dRs '^'' RsiLs - eUL,) ^ '^'' RsiLs - c'^^L,r ^' '

66
2.7.2 Reference Plane Position Uncertainty

Another important source of error is the uncertainty in the location of the

reference planes. Generally when TR experiments are carried out, the sample

is placed flush with the end of the sample holder and hence coincident with

a calibration reference plane. This placement procedure leaves room for positioning errors, particularly when the sample is loose. The error introduced

by incorrect positioning of the sample can be estimated in terms of the error

in the reference plane transformation terms eqs (2.15), (2.16). If we have an

uncertainty of AL in the sample position then

^-n - RllSiiW' = l^nlexpfi^ - 2^,{L, + AL,)]. (2.78)

The error in the measured angle is given by

AL

A6 ~ 2j-ioAL = 47r-—

.

(2.79)

A

Therefore small reference plane positioning errors can, in principle, introduce

large uncertainties in the ^n phase at high frequencies. One way to minimize

this is to use equations that are invariant to reference plane position.

67
Chapter 3

Short-Circuit Line

3.1 Theory

The short-circuit hne measurement (SCL) was introduced by Roberts and

von Hippel [19] over fifty years ago as an accurate broadband measurement

procedure. The short-circuit Hne measurement is a 1-port measurement made

either with a slotted hne apparatus or an ANA. The short-circuit may either

be fixed in the hne or moveable. The advantage of a moving short-circuit

[3] is that it allows many separate measurements at a given frequency and

allows the sample to be placed in either a strong electric field or strong

magnetic field region. Generally, a high electric field is advantageous for

permittivity measurements, whereas a high magnetic field is advantageous

for permeabihty measurements.

Ligthart [7] developed an analytical method for permittivity measure- ments at microwave frequencies using an averaging procedure. In Ligthardt's

study, a single-mode cylindrical waveguide was filled with a homogeneous dielectric with a moving short- circuit positioned beyond the sample; however,

this study focused primarily on single frequency measurements rather than on broadband measurements. Chao [20] presented results of SCL measure- ments using a slotted line and also an uncertainty analysis for single frequency

measurements. Chao found that accuracy was reduced when the reflection

coefficient was dominated by the front face contribution.

With the introduction of modern ANA systems it is possible to measure

68
the 1-port scattering parameters with high precision, over a broad band of

frequencies. When an ANA is used the sample is positioned in either a waveguide or coaxial line and the reflection coefficient (which in this case is ^n) is measured. The determination of the permittivity proceeds by solving a transcendental equation which involves the sample length, sample position, and

reflection coefficient. With modern computer systems, iterative procedures

are easy to implement; however, they require an initial guess.

Dakin and Work [21] presented a procedure for low-loss materials and

Bowie and Kelleher [22] presented a rapid graphical technique for solving

the scattering equations. The short-circuit fine method is well suited for

high temperature measurements since these devices can be easily inserted

into tube furnaces. This technique was utilized by Brydon [23] for measuring

permittivity of alumina, beryllia and quartz in the temperature range of

20 - 700°C.

In the case of a slotted line, a sample is placed in the transmission line

and the voltage standing wave ratio VSWR is measured. An equation that

relates the VSWR to the dielectric parameters can be developed.

Uncertainties in the SCL method include network analyzer uncertainties, sample gaps, wall and short-circuit losses, and measurement of sample

dimensions. There are also uncertainties in locating the sample reference

planes and uncertainties in distance from sample to the short. The uncertainty in the network analyzer parameters usually are well documented by

the manufacturer [4]. The goal of this second part of this technical report is to summarize the

short-circuit line measuring procedure. In the following sections the ANA and

slotted line measurement processes are summarized and then an uncertainty

analysis for the SCL method are presented.

3.1.1 Reflection Coefficient Method

We now consider the measurement which uses the reflection coefficient

(Sii). This method is generally used when an ANA is available. We begin

with a mathematical analysis of the electric fields in the sample.

Consider a sample in a transmission fine as indicated in figure 3.1. Assuming only the dominant mode in the waveguide we can write down the

following expressions for the electric fields

69
Ej = exp{--fox) + Ci exp{^ox), (3.1)

Eii = C2 exp(-7x) + C3 exp(7x), (3.2)

Em = C4 exp(-7o(a; - I)) + C5 exp(7o(a; - L)), (3.3)

where

1 = J- '^'^f^R^R /27r^2

^| cLc VA

(3.4)

2 /2rx2

In eqs (3.1)-(3.3) we assumed transverse electromagnetic fields and no radial

dependence. We wish to determine the coefficients in eqs (3.1)- (3.3) by

imposing boundary conditions on the system of equations. The boundary

conditions are:

• Tangential component of the electric field is continuous at sample interfaces.

• Tangential component of the magnetic field is continuous at sample

interfaces.

• The electric field is null at the short-circuit position.

Equations for the coefficients in eqs (3.1)- (3.3) can be obtained by application of the boundary conditions at the interfaces ( ar = and x = L):

1 + Ci = C2 + C3, (3.6)

a2exp(-7X) -f C3exp(7i:) = C4 -f C5, (3.7) —[Ci-1]-C3-C2, (3.8)

7/^0 [C2exp(-7l) - C3exp(7L)] = C4 - C5, (3.9)

7o/^

70
x = o X = L

Figure 3.1 A transmission line with a short-circuit termination.

71
c. = -sa4^ (3.10)

where

S = exp{-2jo^L), (3.11)

and AL is the distance from the short-circuit to the sample. Equations

(3.6)-(3.10) yield the following system of equations:

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

Note that 1//? is an effective impedance. In matrix form these equations can be expressed as

Ci — C2 — C3 — — 1,

7/C2 + — + (<5-l)C4 = 0,

V

Ci+/?C2-^C3 = 1,

^7yC2--C3-C4(l+^) = 0,

where

/?-

7/^0

T] = exp(—7L).

/ 1 -1 -1

V 7; 0\ [ c,\ ( -1 \

C2

V C4 /

1 V )

(3.18)

^-1

1/3-/3

Vo ^77 -f -(i + <5)y

Solution of eqs (3.12)-(3.15) yields an equation for the permittivity in

terms of the reflection coefficient, p = S\\ = Ci, with the sample located a distance AX from the short:

Su

-2/36 + [{S + l) + {6- 1)^2] tanh7X

2/?+[(^+l)-(<5-l)y32]tanh7Z '

72

(3.19)
or in terms of hyperbolic functions _ tanh -iL-\- (3 tanh 70 AL -/?(!+/? tanh 7X tanh 70AZ) " ~ tanh 7L + /? tanh 70AL + ^5(1+^ tanh 7L tanh 70AL) ' ^

' ^

In the derivation, it is assumed that the sample plane coincides with the

measurement cahbration plane. This is not the case in general; however, one can transform the reference plane position by a simple procedure. To

accompHsh this we proceed by assuming that the most general expression for

the reflection coefficient is given by

Ptrans = RiR-, (3.21)

where ptrans is the reflection coefficient at the cahbration reference plane

position and

i?i=exp(-7,Li), (3.22)

and Li is the distance from the calibration plane to the sample front face.

Equation (3.21) transforms the reflection coefficient from the calibration

plane to the plane of sample front face. It is of interest in many applications to eliminate the distance Li from eq (3.19). To accomplish this it is necessary to measure S\\ of the empty sample holder,

Pempty = " eXp(-27o[i/l + AL + L]) = exp(-270 Lair), (3.23)

and therefore,

^^^^^ = - exp(27o[AL + L])p. (3.24)

Pempty

If both the permeability and the permittivity are required then measurement

data for two different short-circuit positions are required.

Equations for Two Short-Circuit Positions

It is possible to obtain an exphcit solution to eq (3.19) when measure- ments at two different short-circuit positions are taken. The explicit solution

is obtained by solving eq (3.19) at a given short-circuit position for tanh7iy

and then substituting this expression into eq (3.19) at another short-circuit

73
position. For two different short-circuit positions at the same frequency we

obtain

_

2/9(Si-[(<5, + l) + (6i-l)^^]tanh7X

^' -2/? + Pi - 1)^2 _ ^s, + l)]tanh7X' ^"^^ ^^

^ 2^<S2-[((^2 + l) + (<^2-l)/g^]tanh7L

^' -2/? + P2-l)^2_(^2_^l)]tanh7L' ^^ ^ where ^i, 62 denote the phases calculated from eq (3.11) for ALi, AL2. These

equations yield:

,

, , m^i+pi) ,,„,,

^2 ^ ^l{^2{pl - P2) + />l/>2 + 1 - 2/J2) - {^2{p\{p2 - 2) + 1) + /92 - Pi)

^l{^2{pl - P2) + /?l/02 + 1 + 2/)2) - {S2{pl{P2 + 2) + 1) + />2 - /?l)

'

(3.28)

Thus, when two measurements are made at different short-circuit positions

then we can obtain e^ exphcitly; note that this solution does not contain

sample length exphcitly. Once /? is known then eq (3.27) can be used to find

permeability for magnetic materials.

3.1.2 Slotted Line Technique

In the past, another measurement procedure was commonly used for shortcircuit hne measurements. This measurement was based on the slotted line,

which can measure the position of the VSWR relative to the sample. Consider a sample in a short- circuited transmission line as depicted in figure 3.2. We can write for the input impedance

Zin = Z6tanh72l, (3.29)

Zin = Za — tanh72X, (3.30)

72

where Zf, is the impedance of the dielectric filled section of the line, Za is the

characteristic impedance of the empty transmission line, and

74
To VSWR

Detector

Sample Probe j \ \i Coaxial Line

•4

> -X—

Z.n

Microwave

Oscillator

•max

iwvw

^min

Figure 3.2 A transmission line containing sample with a slotted line measuring device for VSWR.

75
^,^ = [(^)^-^i^]. (3.32)

7? = [( v) - TtI (3-31)

The impedance can be transformed along the hne to a point Xo from the

sample where the voltage is a maximum value. This impedance is equal to

the VSWR {S) times the impedance of the line, or 7_7 [Z,n + ^atanh7iXo] _ ^^ ^ - '^olJT}—, ry . r ^ - ^^a, [6.66)

[Za + A>itanh7iXoJ

where

"

[1 — 5 tanh7i a^o] Therefore, we can obtain a transcendental equation

tanh72l _ , _\_. S - tanh7i3:o

72^ 7iiy 1 — 5tanh7iXo'

for the permittivity. This equation can be solved by general iterative methods, for example, a Newton-Raphson procedure. Iterative solutions require an initial guess of the dielectric parameters. If this initial guess is in the

neighborhood of the correct solution, the iteration will rapidly converge, in a

quartic manner, to the correct root. There are an infinite number of roots of

this equation so care must be taken in selecting the proper root. Generally,

however, some knowledge of the correct solution is known a priori and can be used in the root selection process.

3.2 Corrections to Data

Once a set of measurements has been made, it is necessary to correct the

data for known errors. Known error sources include air gaps around samples,

short-circuit and waveguide wall imperfections, together with waveguide and

short- circuit losses. The wall losses can be taken into account by attenuation

measurements in the guide.

76
Gap correction formulas (see appendix C) which are relatively easy to

implement [12], [13], [14], can be found in the Hterature. Imperfect waveguide

or coaxial line walls can also be modeled. Hill [15] has studied this problem

and his results are summarized in appendix B. Waveguide losses can be

corrected for by measuring the scattering parameters of the empty waveguide

and calculating the appropriate attenuation constant of the guide. Also the

calculated permittivity data can be smoothed. A smoothing routine using

the method of maximum entropy is given in appendix G.

3.3 Instrumentation

The following apparatus is needed for broadband SCL measurements:

• Automated network analyzer or slotted line

• Data acquisition system

• Precision waveguide or coaxial line

• Data analysis system

When using an ANA it is important to know its frequency limitations. Network analyzer systems have various error sources. These errors include

• matching at connectors

• imperfect calibration standards

• non-linearity of mixers, gain and phase drifts in IF amplifiers, noise

introduced by the analogue to digital converter

• imperfect tracking in dual channel systems

Generally the manufacturer furnishes the performance specifications for its system.

77
3.3.1 Sample Holder Specifications

The sample holder should consist of high precision waveguide or coaxial

line. When 7 mm coaxial headless air line is used, APC-7 connectors are generally preferred. Since the cutoff frequency is related to the dimensions of

the guide it is important to measure the guide dimensions precisely and from

these measurements to calculate the cutoff frequency. Slight discrepancies

in the cutoff wavelength or sample holder length manifest themselves in the

calculated permittivity.

3.3.2 Sample Preparation

The samples to be used for SCL measurements should be prepared carefully since scratches, nicks and cracks will alter the dielectric properties. The

sample should be machined with care to minimize any gaps between sample

holder and sample. Also the front and rear sample end faces must be machined to be very perpendicular to the rest of the sample in order to minimize

mode conversion. It is also very important to minimize any unnecessary wear and tear on the sample by placing it in a secure area between measurements.

The sample length measurement is critical and should be performed carefully

with a precision micrometer. The following list summarizes the preparation

procedure:

• Carefully select a piece of material free of unusual inhomogeneities,

cracks, etc.

• Have the sample machined to fit as tightly as possible in the sample

holder (generally gaps should be < 5 x 10~^ m (0.0004 in)).

• Measure the sample length precisely at a temperature very close to that

realized in the laboratory.

• Keep the sample very clean and store in a secure area.

• Carefully store the sample holder as surface scratches or conductor

bends can erode the line characteristics.

78
3.3.3 System Diagnostics

In order to verify the accuracy of the measurement system before each

measurement, it is important to have a verification test. For the SCL method

a standard comparison may consist of measuring the 5-parameters of air in an empty sample holder and comparing the results to a designated standard

measurement.

3.3.4 Measurement Procedure

The measurement depends on whether an ANA system or a slotted line

apparatus is used. For slotted hues it is necessary to measure the VSWR

and the location this VSWR maximum along the line. Then eq (3.35) can be

used to calculate the complex dielectric constants. When an ANA is used the

scattering parameter ^n is measured, over a broad band of frequencies, with

the sample at a given position in the sample holder. The distance from the

sample to the short- circuit must be known accurately. If both permeability

and the permittivity are required, the sample must be moved in the line and

the S'-parameters measured in the second location.

In general it is better to make permittivity measurements when the sample is predominantly in a strong electric field region, whereas it is advantageous to perform permeability measurements when the sample is in a region

of strong magnetic field. When taking broadband measurements on an ANA

it is possible to calculate when the sample is immersed in the various fields

strengths and then one can select the measurements to be used for permittivity and permeability calculations [13].

3.4 Measurement Results

Measurements were made on an ANA for various samples. Using eq (3.19)

the permittivity was calculated and is shown in figures 3.3-3.8 for various

materials. The associated uncertainties are given in figures 3.15-3.16. In

the case of PTFE there was no measurable gap.

79
2.1 rr 1—I—I—I—

r

n I I I r

2.08

2.06

^ MiNta 2.04

2.02

«MMi>M*^«»M m^^^^m

2L_L

8

'III' I I I I

'

I ' I I I l_l L 10 11

Frequency (GHz)

12 13

Figure 3.3 e'^ using SCL in X-band waveguide for PTFE. The worst case

uncertainty, Ae'^ = 0.011, at 10 GHz.

80
0.01

0.0081—

0.006

0.004

0.002

I I I I r r I 1 1

I T

\—\—I

\—\—I—

r

0* ' ' ' ' I

' ' ' ' I' ' ' ^ ' ' ' ' 10 11 12 13

Frequency (GHz)

Figure 3.4 e^ using SCL in X-band waveguide for PTFE. The worst case

uncertainty, Ae'j^ = 0.004, at 10 GHz.

81
2.60

2.59

2.58

2.57

2.56

< 2.55

2.54

2.53

2.52

2.51

2.50

T 1

\

\

1 r

J l__L

Cross-linked Polystyrene

Corrected

-Uncorrected

J I L 10 11

Frequency (GHz)

12 13

Figure 3.5 e^ with and without gap correction using SCL in X-band waveguide for cross-hnked polystyrene. The worst case uncertainty, Ae^ = 0.009,

at 10 GHz.

82
0.01

0.009

0.008

0.007

0.006 ^" 0.005

0.004

0.003

0.002

0.001

I I I I

I III I I

I r r

1 ^^'^'^~v/^->wv.A/vV^-VvA\|i/WV^^

J I I i I I I I I I I l__J I I I I I L 10 11 12 13

Frequency (GHz)

Figure 3.6 e^ using SCL in X-band waveguide for cross-linked polystyrene.

The worst case uncertainty, Ae'^ = 0.003, at 10 GHz.

83
6.5 I I

6.4

6.3

6.2

6.1

6.0'—^—

I I r T T

-Corrected

Uncorrected

J L J L J L 10 11

Frequency (GHz)

1—^—^—

1723 Glass

J L

(

12 13

Figure 3.7 t'j^ with and without gap correction using SCL for 1723 glass.

The worst case uncertainty, Ae'^ = 0.021, at 10 GHz.

84
0.1

0.09 —

0.08 —

0.07

0.06

0.05

0.04

0.03

0.021—

0.01

I r

J L 9

I 10 11

Frequency (GHz)

12 13

Figure 3.8 e'^ with and without gap correction using SCL for 1723 glass.

The worst case uncertainty, Ae'^ = 0.009, at 10 GHz.

85
3.5 Uncertainty Analysis

The independent sources of uncertainty for a short- circuit Hne measure- ment are basically the same as those for the T/R method except for the

additional uncertainty introduced by an imperfect short-circuit.

The calculation of the theoretical uncertainty in the permittivity due to

the uncertainties in the magnitude and phase of the scattering parameters

requires the calculation of the various partial derivatives with respect to

the various independent variables. For the analysis we will assume that the

sample fits tightly to a perfect short-circuit, so the reflection coefficient is given by

7o tanh 7Z — 7

(3.36)

7o tanh 7Z -|- 7

In an attempt to evaluate the uncertainty introduced by the measured

scattering parameters a differential uncertainty analysis is applicable. We

assume that the total uncertainty can be written as

Ac',R

'R ^'r\ [d\Sa\

A\Sa

(3.37)

(

de'k

'k'iKdis.l

^\Sa\] +

di R

dd,_

AOr +

di R

dL AL\ +

d

'^Ad

dd

(3.38)

where a = 11, A^^ is the uncertainty in the phase of the scattering parameter

and A|5a| is the uncertainty in the magnitude of the 5-parameter. This

uncertainty estimate assumes that the independent sources of uncertainty

are small enough for a linear approximation to be valid.

If we define a variable / by

/ =

7o tanh 7Z — 7

7o tanh 7L -t- 7

P. (3.39)

we can calculate

86
oL 7o tanh 7L + 7 7o tanh 7L + 7

^•^ ^ [(7oIsech^7i' - 1) - '^-^^^^P^f^boLsech'^L + 1]], ^7 7o tanh 7Z + 7 70 tanh 7Z/ + 7

^7 Ho^o^^

(3.41)

(3.42)

Since we can write

we obtain

de*n 27

/>=|/>|expj^, (3.43)

5e^ expjO

5«R_ If

(3.44)

(3.45)

(3.46) 5i [filtr

The partial derivatives are given in figures 3.9 - 3.14 and the calculated

uncertainty is shown in figs 3.15 - 3.16.

For high-loss materials it is possible for the reflection coefficient to go to

0, and this results in a high uncertainty. The possibility of zero reflection

coefficient can be understood by letting eq (3.36) go to zero and then finding an e|j which is consistent with this condition. Consider the case when Sn ~ 0; then for coaxial line:

tanh 7! « — ^ ^ /e^. (3.47)

70 ^ This yields the following relations between real and imaginary components:

cosh2S!'fcos2/^I = ^,^|i4^^'S)^^'n^^^"• (3-48)

87
0.04

0.03

M 3 e'a ,

Figure 3.9 The derivative of e'^ with respect to |5n| with e^ = (30.0,0.01).

88
Figure 3.10 The derivative of e'^ with respect to |5ii| with e^ = (30.0, 0.01).

89
0.0004

0.0003 —

I f^l 0.0002

0.0001

Figure 3.11 The derivative of e^ with respect to B with e^ = (30.0,0.01).

90
Figure 3.12 The derivative of ^ with respect to 6 with e^ = (30.0,0.01).

91
1^1

^.OUL-^M- 1 1 1 1 1 I Sn

2.00E+4 — —

1.50E+4

1

— —

1

l.OOE+4 —

1 —

5000 — ^^^_____ —

n 1 1 1 1 1 1 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Figure 3.13 The derivative of e^ with respect to length with e^

(30.0,0.01).

92
Figure 3.14 The derivative of e'^ with respect to length with C/^ (30.0,0.01).

93
A€r

0.045

0.040

0.035

0.030

. 0.025

R_

" 0.020

0.015

0.010

0.005

€r= (30, 0.01)

Su

Figure 3.15 The relative uncertainty in e^ for 5ii using SCL for a low-loss

material, with e)^ =(5.0,0.01) and (30.0,0.01).

94
Figure 3.16 The relative uncertainty in e^ for 5ii for a low-loss material,

with e^ = (5.0,0.01) and (30.0,0.01).

95
sin2/?I = l=^{e'^ + e'li)-e'l,^

cosh 2aL + cos 2/3L ~

yj2

where we have used

7 = a + i/3,

and

I //2

a = u;

^ = u;

^^1^(7)^-1'

2-^ 1 + (-7)^ + 1,

(3.49)

(3.50)

(3.51)

(3.52)

In the case of very low loss, since a —^ 0, there is not an allowable value

of t\ that satisfies these equations, whereas for cases of appreciable loss, a

solution is possible.

96
Chapter 4

Discussion and Conclusions

We have examined TR and SCL measurement procedures in detail. In

this chapter I will summarize the important results obtained in the analysis.

As we have seen, although the Nicolson-Ross-Weir approach is easy to implement numerically, it fails for broadband measurement of low-loss samples

of arbitrary length. The solution presented in this report uses a NewtonRaphson iteration procedure on linear combinations of the scattering parameters. This procedure yields solutions that are stable at integral multiples

of one-half wavelength in the sample and at the same time does not unduly

increase the complexity of the numerical solution. For materials where the

transmitted signal is greater than -40 dB from the reference signal, 52i data

by themselves are sufficient to calculate permittivity. For materials of large

attenuation, Su by itself will produce optimal results. In general, we have

found eq (2.25) to be robust for high-loss and low-loss materials. For magnetic problems it is necessary to use both ^n and 521 data. The problem

of reference plane position has been addressed, and approaches for the minimization of the error have been presented. Equations that are independent of

reference plane position and sample length have been presented. Equations

that are independent of reference plane position should be very useful in el- evated temperature applications. Generally, sample length can be measured

with great accuracy at laboratory temperature, and for these problems it is preferable to use a measured length. However, in temperature dependent

applications it may be better to use equations independent of both sample

length and reference plane position.

97
An uncertainty estimate for the solution expounded in this report has

been presented. The uncertainty analysis presented here differs in some re- spects from that presented in the literature previously. This difference is due

primarily to the fact that the uncertainties in this paper are derived from

5ii and 5*21 in isolation. The trend indicates that for low-loss materials the

uncertainty decreases as a function of increasing sample length. For high-loss

materials the uncertainty in ^21 decreases until the signal reaches -40 to -50

dB, and thereafter the uncertainty increases and thus Ae^ increases.

The theory for short-circuit line measurements was presented and an uncertainty analysis for the procedure was presented. The uncertainty is a function of the sample length, dielectric constant and S'-parameters. For

fixed frequency measurements, samples of length nA/4, n=l, 3, 5, . . ., yield

minimum uncertainty, a result of the fact that at these frequencies the electric

field is a relative maximum over the sample length. In general, for low-loss

materials, samples lengths that are large in relation to wavelength give more

accurate results; thus for broadband measurements it is preferable to use long samples for low-loss materials. However, for the case of lossy materials

very long samples result in only front face reflection information. Thus, for

relatively lossy materials, sample lengths on the order of one attenuation

length are optimum.

98
Chapter 5

Acknowledgments and

References

5.1 Acknowledgments

I would like to thank Eleanor Livingston for her excellent comments and suggestions, Michael Janezic for help with measurements and software and John

Grosvenor for help with measurements; also Richard Geyer, William Kissick

and Howard Bussey for various discussions and as sources of encouragement.

Finally, I would like to thank Ray King, Steve Buckner, and David Blackham

for reading the manuscript and providing excellent suggestions.

99
5 .2 References

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of materials by time domain techniques," IEEE Trans. Instrum. Meas.,

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[2] Weir, W. B. "Automatic measurement of complex dielectric constant

and permeabihty at microwave frequencies," Proc. IEEE, vol.62, pp.33-

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[3] Deschamps, G. A. "Determination of reflection coefficients and insertion loss of a wave-guide junction," J. Appl. Phys., vol.2, pp. 1046-1050,

August 1953.

[4] "Measuring dielectric constant with the HP 8510 network analyzer,"

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[5] Freeman, M. S., Nottenburg, R. N. and DuBow, J. B. "An automated

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[6] Stuchly, S. and Matuszewski, M. "A combined total reflection transmission method in application to dielectric spectroscopy," IEEE Trans.

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[S] Altschuler, H. M. Dielectric Constant, ch.9, p.502. Vol.3, Polytechnic

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100
[9] Harris, F. J. "On the use of windows for harmonic analysis with discrete

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[10] Scott, W. and Smith, G. S. "Dielectric spectroscopy using monopole

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[11] Kerns, D. M. and Beatty, R. Basic Theory of Waveguide Junctions and

Introductory Microwave Network Analysis. NY: Pergamon Press, 1967.

[12] Westphal, W. P. "Techniques of measuring the permittivity and permeability of liquids and solids in the frequency range 3 c/s to 50 kmc/s,"

Laboratory for Insulation Research Technical Report, MIT, 1950.

[13] Bussey, H. E. "Measurement of rf properties of materials a survey," Proc.

IEEE, vol.55, pp.1046-1053, June 1967.

[14] Bussey, H. E. and Gray, J. E. "Measurement and standardization of

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[15] Hill, D. A. "Reflection coefficient of a waveguide with slightly uneven

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[16] Ramo, S., Whinnery, J. R. and Duzer, T. V. Fields and Waves in Communication Electronics. New York: John Wiley and Sons, 1984.

[17] Bailey, A. E. Microwave Measurement. London: Peter Peregrinus, 1985.

[18] Wong, K. H. "Using precision coaxial air dielectric transmission lines as calibration and verification standards," Microwave Journal, pp.83-90,

December 1988.

[19] Roberts, S. and von Hippel, A. "A new method for measuring dielectric

constant and loss in the range of centimeter waves," J. Appl. Phys.,

vol.7, pp.610-616, July 1946.

[20] Chao, S. "An uncertainty analysis for the measurement of microwave

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IEEE Trans. Instru. and Meas., vol.IM-35, pp.36-41, March 1986.

101
[21] Dakin, T. W. and Works, C. N. "Microwave dielectric measurements,"

J. Appl. Phys., vol.18, pp.789-796, September 1947.

[22] Bowie, D. M. and Kelleher, K. S. "Rapid measurement of dielectric constant and loss tangent," IRE Trans. Microwave Theory Tech., vol.MTT4, pp.137-140, 1956.

[23] Brydon, G. M. and Hepplestone, D. J. "Microwave measurements of

permittivity and tan S over the temperature range 20 — 700°c," Proc.

Inst. Eke. Eng., vol.112, pp.421-425, 1965.

[24] Kurokawa, K. "Power waves and the scattering matrix" IEEE Trans.

Microwave Theory Tech., vol.MTT-13, pp.194-202, March 1965.

[25] Reiter, G. "Generalized telegraphist's equation for waveguides of varying

cross section," Proc. Inst. Elec. Eng., vol.106, Pt. B, Suppl. 13, pp.54-

57, 1959.

[26] ChampHn, K. S. and Glover, G. H. "Influence of waveguide contact

on measured complex permittivity of semiconductors," J. Appl. Phys.,

vol.37, pp.2355-2360. May 1966.

[27] Schwinger, J. and Saxon, D. S. Discontinuities in Waveguides. New

York: Gordon and Breach Science Pubhshers, 1968.

[28] Champlin, K. S. and Glover, G. H. "Gap effect in measurement of large

permittivities," IEEE Trans. Microwave Theory Tech., pp.397-398, August 1966.

[29] King, R. W. P. and Smith, G. Antennas in Matter. Cambridge, Mass.:

MIT Press, 1981.

[30] Morse, P. M. and Feshbach, H. Methods of Theoretical Physics. New

York: McGraw-Hill Book Company, 1953.

[31] Roberts, D. Electromagnetic wave propagation in dissipative dielectrics

such as oil shales. Master's thesis. University of Wyoming, 1983.

[32] Tai, C. Dyadic Green's Functions in Electromagnetic Theory.

Intext Educational Publishers, 1971.

102
[33] Jackson, J. D. Classical Electrodynamics. New York: John Wiley and

Sons, 1975.

[34] Ben-Menahem, A. and Singh, S. J. Seismic Waves and Sources. New

York: Springer-Verlag, 1981.

[35] Baker-Jarvis, J., Racine, M. and Alameddine, J. "Solving differential

equations by a maximum entropy-minimum norm method with applications to Fokker-Planck equations," J. Math. Phys., vol.30, no. 7,

pp.1459-1463, 1989.

103
Chapter 6

Appendices

6.1 Appendix A: The Scattering Matrix

6.1.1 Theory

In this appendix the basic features of the two-port scattering equations

will be reviewed. We consider incident (a,) and reflected (6j) power waves

defined by [24]

y^ + z,h ,.,.

ai = —, , (6.1)

2,/\ReZ,\

b, = ^^^^S, (6.2)

where Zi,Vi,Ii are the impedance, voltage and current associated with a power wave. The voltage and current are given by

V, = ^£^=^[Z:a, + Z,b,], (6.3)

^/\ReZ,\

li = , ^'[«. - k], (6.4)

^\ReZ,\

where

104
^ I

1 if Re Z. >

.g ^. 1—1 otherwise

The impedance matrix Z is

Z = I ;" ;" I. (6.6)

Zu Z\2

^21 -^22 The voltage vector and impedance matrix are related to the current vector

by Ohm's law:

V = Z/. (6.7)

The voltage and current in a waveguide can be represented by voltage

and current waves:

V{z) = Aexp —-jz + B exp^z, (6-8)

Hz) =

>lexp-7^-J?exp7.

^^^^

Zo

The two-port scattering equations relate the incident voltage wave to the

reflected wave. The two-port scattering equations can be written as

6i = 5iiai -I- 5i2a2, (6.10)

62 = S2\ai + ^2202, (6-11)

or in vector notation

where

h = Sa, (6.12)

\ '-'21 '^22 /

A general n-port junction possesses the following properties [24]:

• reciprocity

105
• losslessness

• symmetry

If off-diagonal elements are equal S21 = S12 we then say the n-port is reciprocal; if 5ii = S221 we say the n-port is symmetrical. A junction is lossless

if the impedance is pure reactance. The impedance matrix that satisfies a

reciprocal network is symmetric:

Z = Z\ (6.14)

The scattering matrix under a similarity transform behaves as

where

therefore

and

S' = P*SP, (6.15)

^ -

;J ^ («-i«)

'-^mn VnVm^nmt v" /

\S,,\' = \Sji\\ (6.18)

6.1.2 Lossless Network

The net power fed into a network is given by

i thus for lossless circuits we have:

W = ^pA\a^'-m=0, (6.20)

106
Therefore

or in matrix form

A+(P-S+PS)A = P, (6.21)

where (+) denotes adjoint. This equation is equivalent to

P = S+PS, (6.22)

which yields three independent conditions:

Pi\Sn\' -^ P2\S2i\' = Pu (6.23)

Pl\Su\' + P2\S22\' = P2, (6.24)

PiSnS*n-\-P2S2iS;2 = 0. (6.25)

\S^r\'\Su\' = \S2i\'\S22\\ (6.26)

|5„P = 1^22^ (6.27)

\Su\' = \S2i\'. (6.28)

6.1.3 Lossy Networks

When the material is lossy we obtain

A'(P - S'PS)A > 0. (6.29)

Therefore (P — S*PS) must be positive definite or semi-positive definite. This

relation is basically a statement of increasing entropy.

107
6.2 Appendix B: Imperfect Waveguide Walls

No waveguide or coaxial line is perfectly uniform and the non-uniformity

of the walls will affect the measured .S-parameters. The goal of this section

is to study these effects and estimate the significance of this effect. Hill [15] recently presented a thorough theoretical analysis of waveguides

with slightly uneven walls. What follows is an overview and summary of

Hill's [15] results. The perturbations in the waveguide wall excites spurious

modes in the structure. The existence of spurious modes in the transmission

line will influence the measurement results. Of course spurious modes are also

generated by inhomogeneities in samples and discontinuities at connectors.

The amplitude and phase of the spurious modes can be studied by use of the

telegrapher's equation developed by Reiter [25]:

^= -JP,At - Y^A- + f:[St,A* + S-,A-l (6.30)

p=l

'^-^ = iM--Y^^At + f:[S-,Al + St,A;\, (6.31)

p=l

where A^ are the amplitudes of the forward and backward traveling modes,

Zi are the impedance of the ith mode, Sip are the coupling coefficients, and

fii are the wave numbers and time dependence is exp(ju;i). The boundary

conditions for a waveguide, of length L, fed by a single mode are:

Ai(0) = Ao, AZ,{L) = 0, (6.32)

A+(0) = 0, A-{L) = for i 7^ m. (6.33)

Hill assumes the inner and outer radii for a coaxial line, pi and po are a function of z. The variation from a base radii, pio^ poo is given by Aj(z), and

therefore

Pi{z) = Pio + A,(2), (6.34)

Po{z) = Poo + Ao(2), (6.35)

108
where Ai/pio <C l,Ao//9oo "C 1. Hill has shown that the 5-parameters due

to uneven walls can be approximated by the equation

5n = JjCi^ + Co^]exp{-2jkz)dz,

where k = 27r/A and

d = -[2pio\n{poo/pio]~ ,

Co = -[2poO^T^{poo/pio]~^ •

This can be written as

where

Sui = 2jkCi / Ai{z)exp{—2jkz)dz,

Jo

S\\o = 2jkCol Ao{z)ey.T^{—2jkz)dz.

Jo

Hill has calculated upper bounds on |5ii|

|5iii| < 2kL\Ci\Ai maxy

I'-'llol S: 2KL\iyo\/\Q max6.36)

6.37)

6.38)

6.39)

6.40)

6.41)

6.42)

6.43)

Hill has shown that for a precision coaxial air line the change in j^nl due to uneven walls is < 1 x 10~^. Therefore for precision air line in good condition

the wall unevenness will contribute to a lesser degree to the uncertainty than

the uncertainty of the ANA parameters, since Al^nl w 1 x 10~^ for network

uncertainties alone.

109
6.3 Appendix C: The Gap Correction

6.3.1 Frequency Independent Approaches

Various researchers have approached the gap problem by representing the

sample with air gap as a layered capacitor [12], [13], [26]. This approach

assumes that the gaps between transmission hne and sample are effectively

modeled by a set of capacitors in series. Champhn [26] approached the

problem using as a starting point the perturbation formula developed by

Schwinger [27]. By substituting into the perturbation formula approximations to the field distribution in the various regions they obtain an estimate

for the effective permittivity. Their answer turns out to be fully equivalent to

the capacitor model of Westphal [12]. Champhn showed that Bussey's theory

[13] is the first two terms in an expansion of Westphal [12] and Champlin's

models.

The capacitor model is frequency independent and thus is strictly valid

only at lower frequencies and d.c. We would expect the capacitor model

to break down at higher frequencies because the wavelength decreases with

increasing frequency to a point where multiple scattering dominants. In order

to account for multiple scattering it is necessary to develop a theory that is frequency dependent.

Coaxial Capacitor model

Consider a capacitor consisting of layers of dielectric and layers of air in a coaxial line. The various dimensions are shown in figure 6.1.

We treat the system as capacitors in series, so that

7^ = 7^ + 7^ + ^ (6.44)

We know that for a coaxial line the electric field distribution is given by

V_

ln(!)

and the voltage between the conductors is given by

Er = 7-7Vr^ (6-4^)

V f

E{r)dr. (6.46)

Ja

110
Figure 6.1 A coaxial sample in holder with air gaps near conductors.

Ill
The capacitance of a coaxial line of length L is given by

27reL

hT ^ = ^T^; (6-47)

thus, for a system of three capacitors in series we have

In ^ In ^ In ^ In ^ —^ = —r^ + —y^ + —r^, (6.48)

where e^, e^ are the corrected and measured values of the real part of the

permittivity and e'j, is the real part of the permittivity of the air gap, respectively. Therefore

e'lc'^ln^

^' " e' In ^

+ e' fin ^+ In ^1

^^'^^^ We can then write

< = Cj Vl-' (6-50)

tan^c = tan^j„[l + e'^^—], (6.51)

^2 where

L,=\n^-^\n^, (6.52)

i?l J?;3

L2 = \n-^, (6.53)

i?2

l3 = ln|^. (6.54)

Equation (6.50) breaks down when e^ > j^. An example is plotted in figure

6.2 for a 7 mm coaxial hne.

112
0.4045

Figure 6.2 The gap correction calculated for various values of e^, where

i?2,^i are the radii of the inner conductor and sample respectively.

113
Rectangular Waveguide Model

For the case of a rectangular guide eqs (6.50) and (6.51) are modified

slightly

' ' ^

(6.55)

^ -b-{b-d)e'J

tan 6c = tan Sm t—77 -jr—, (6.56) b- [b-d)t'^

where b and d refer to the waveguide and sample dimensions and (c) and (m)

refer to corrected value and measured value respectively.

Frequency Independent Model of Bussey

Bussey and Gray used a perturbation theory developed by Bethe and

Schwinger [27] for cavities. We will summarize the theory in this section.

The difference in the propagation constant for a waveguide with and

without an airgap is given by

(gap Inogap 7 .\. , . ^, |^|^^|2J5 I'p - llo gap - M' - (4 - l)(-f ^T,;.,.l. , (6.57)

where Ei is the electric field of the dielectric filled hne with no air gap, E2 is the electric field with an air gap present, S is the cross sectional area. The

following boundary conditions must be satisfied at the dielectric interface,

E2,n = tj,E,^n. (6.58)

E2,t = ^i,t, (6.59)

where [n) denotes normal component,

^2 _^2 - A-,^ - ic' lU^)2 /pap[4l-gl,nP + \E,A'']dS

_ Igap lnogap-^1 - \^R ^)\^) r l^^p^^ ' lO.DUJ

thus we have

Ae^ = {e^ - 1) j^j^^

. (6.61)

114 I
Figure 6.3 A rectangular sample in holder with gaps near conductors.

115
Bussey and Gray showed that for small air gaps, this equation reduces to

Ae'p , , Jb-d) , ^

We see that eq (6.62) is equivalent to the first two terms in a series expansion

of the capacitor eqs (6.55).

Perturbation Model of Champlin

Champlin [26], [28] has used a perturbation approach to the air gap problem. Using a perturbation formula of Schwinger [27] for a single mode in an

inhomogeneously filled waveguide:

• IIAe'J,-E,dS

l2 — li — —J<^ =j =5 ^ ^ ^—

,

(6.63)

JJ[Ei xH2-E2xH^]-kdS

where

At'ji = in region(l), (6.64)

and

Ae'n = e'si? - e'lfl in region (2). (6.65)

The following boundary conditions apply

E2 — El in region(l), (6.66)

E2 = —El in region(2), (6.67)

and

H2t = --HitmTeg[on{l), (6.68)

7i

//2t--^^^itmregion(2), (6.69)

71^2

t^"-" - '1 = fer- <'^-TM^

Substituting eqs (6.64)-(6.69) into eq (6.63) and integrating yields an

expression for an effective permittivity which is equivalent to Westphal's

formula in eq (6.55) for rectangular waveguide. Bussey's result in eq (6.62)

is an approximation of Champhn's and Westphal's results.

116
6.3.2 Frequency Dependent Capacitor Model

The capacitor model was independent of frequency. At high frequencies

the capacitor models given in previous sections break down due to multiple

scattering effects at the material interfaces. It is possible to include frequency

in a medium frequency capacitor model by considering the admittance of a material filled capacitor. The model which was originally developed to model

Maxwell- Wagner effects should work for layered materials in waveguides. We

consider capacitors of cross- sectional area A and admittance Y in series such

that [29]:

The admittance of a capacitor is given by

Y = [a + jioe'f,e,]^, (6.72)

where d is the thickness of the sample and therefore for two samples in series

The effective d.c. conductivity can be obtained from the zero frequency limit:

^if + <T2^

We can also solve eq (6.73) for e^:

^R = ^Roo + 1 , . .2 -J 1, ,.,2-.2

' (6-^^)

where

4oo = , ti't, ^ . (6.76)

^R\ d

'^ ^R2 d

"'" h(t) + -2(^)P ' ^'^

'

<7lC?2 + <^2<^1

117
Equation (6.75) reduces in the high frequency limit to the previous capacitor

model and in the low frequency limit to an expression containing the d.c. conductivity. The effective relaxation time r depends both on the conductivity

and permittivity. In this model the layered structure acts as a material with

a single relaxation time. Many materials have a distribution of relaxation

times.

118
6.4 Appendix D: Fields in a Transmission

Line

6.4.1 Theory

In this appendix we wish to review the field equations for a general, arbitrarily shaped waveguides. We will first review the theory behind propagation of TE, TM and TEM waves in waveguide, then generalize the theory by

means of dyadic Green functions. The solution will be presented in terms of

the vector potential in the so called nameless gauge where the scalar potential is zero [30], [31], [32]. In this gauge only the vector potential needs to be

calculated and the electric and magnetic fields can be obtained from it. The

treatment in this appendix is kept sufficiently general to allow application to

various geometries.

The Fourier-transformed Maxwell equations are V xE = -JLoB, (6.79)

V xH = J-\-ju;D, (6.80)

V-D = p, (6.81)

V • B = 0. (6.82)

The boundary conditions at material interfaces are

n x(^2-^i) = 0, (6.83)

nx{H2- Hi) = Js, (6.84)

n -{32-31) = n, (6.85)

n-{B2-Bi) = 0, (6.86)

119
where Jg is the surface current density and Q is the surface charge density.

In eqs (6.83)-(6.86) we have used a Fourier transform (F) defined as follows

/oo

f{t)exp{ju;t)dt. (6.87)

-oo The Fourier- transformed fields in terms of the vector potential A and scalar

potential (f> are E = -JLoA-V(t>, (6.88)

B = V X A, (6.89)

H =T^ -V X A, (6.90)

D =1 -E. (6.91)

The displacement field is related to the electric field by

V{f,t) = S{f,t) -h f X {t-T)-S{f,T)dT, (6.92)

J — oo and _X {t<0) = 0, (6.93)

and hence the Fourier transform yields the dielectric constant — = /"CO —

e (f,w) =1 + X {z)exp{-juz)dz. (6.94)

J — oo

6.4.2 TE, TM and TEM Modes

We assume in the analysis the following:

• There exists in the guide a preferred direction in the guide which we

call z.

• The cross sectional area of the guide is perpendicular to i'and constant

throughout the length of the guide.

120
In this section we assume no sources of electric and magnetic fields (J = 0)

in the guide, so the fields satisfy

V''E^k'^E = Q, (6.95)

V^H-\-k^H ^0. (6.96)

It is possible to spectrally decompose the combined fields in a waveguide

into TE, TM and TEM modes. If we assume a preferred direction z, we can

write

_ rco _ S{r,i)= I dioJ2En{f,u)exp{±jnz)exp{-ju;t), (6.97)

n{f,t)^ / duj^Hn{f,Lo)exp{±^nz)exp{-jut), (6.98)

where r is a transverse vector and En,Hn are the amplitudes of the modes.

The fields separate into a transverse (T) component which is independent of

z and a longitudinal component which is z dependent. Since the TE, TM and

TEM modes form a complete set of functions, we can expand the transverse

Fourier transformed fields as

Erir^u) =

oo Y.i^nTE^^'^ilnZ) + E'TE^^'^i-lnZ)] Et(TE)

n=\

oo + 5I{^nTMexp(7n^) + E-TMexp{--i'^z)]ET(TM)

n=\

N-\

+ H {KtEM exp(7n^) + KtEM e^P{-7nZ)}ET{TEM) (6.99)

n=l

oo 1 Yl V~^^nTE^MlnZ) - E'^^ ^M-lnZ)}{z X Et(TE)) +

n=l ^TE

121
oo 1 Yl ^ {KtM^MIu^) - ^nTMexp(-7n2)}(^X Et(TM)) +

n=l ^TM

J2 \ -{EnTEM^Mln^) "

n=l V ^ EnTEM^M-ln^)]{^ X Et(TEM)), (6.100)

where

^TM = —

,

(6.101)

Zte = ^. (6.102)

7

Also A'^ is the number of disjoint conductors.

Although the sums for the TE and the TM waves in eq (6.100) go to oo, in many practical problems some of the coefficients in the sums are due to

cutoff conditions. Two or more modes may have the same eigenvalue; the

eigenvectors in these cases are called degenerate. Generally, it is useful to

separate the gradient into longitudinal and transverse components

V-Vr + i'^, (6.103)

where z Vj = 0. The electromagnetic field can also be separated into

transverse and longitudinal components,

E = Et^E,z, (6.104)

H^Ht^HJ. (6.105)

Maxwell's equations in terms of longitudinal and transverse parts are [33]

E, = zE,, (6.106)

Et = {zx E) X2, (6.107)

-^-ju;zxBt = VtE,, (6.108)

oz

122
dB—^

+ juje^z xEt = VtB,, (6.109) ozz-{V X Et) = -JujB,, (6.110)

z-{V X Bt)-^ juenE,, (6.111)

Vt-^t = -^, (6.112)

oz

Vt-Bt^-^. (6.113)

oz

The component E^ is the generator of the TM mode. The z component of the

electric field for TM modes satisfies the following boundary value problem:

{V^ + A:,2}^,(TM) = 0, (6.114)

where k^ are real, positive eigenvalues we determine the cutoff frequency.

The boundary conditions are

E(z{TM))\on conductor = 0, (6.115)

H^iTM) = 0. (6.116)

The other field components are formed by use of eqs (6.108) and (6.109):

Et(tm) = ^^^tE., (6.117)

1 Ht{tm) = ^—{zxEt(tm)), (6.118)

l = J\/k'-kl (6.119)

If the mode H^ exists, it is the generator of the TE mode. The TE modes

satisfy the boundary value problem,

{Vl + kl}H,^TE) = 0, (6.120)

where k^ are real, positive eigenvalues. The boundary conditions are

n • (VH(^;,^TE)))\on conductor = 0, (6.121)

123
E,iTE) = 0. (6.122)

The other field components are then formed from eqs (6.108) and (6.109)

Ht{te) = -^"^tH,, (6.123)

Et(te) — —Zte{z X Ht(te))- (6.124)

There is also the possibility of TEM modes; TEM modes occur if a struc- ture has more than one conductor (A'^ > !)• For A'^ conductors there are

(A'^ — 1) TEM modes. The TEM modes are the solution of an electrostatic

problem with the eigenvalue k^ = 0. Since both the electric and magnetic

fields are transverse in this case, the TEM modes can be constructed from

the conditions that

Vt X Etem = 0, (6.125)

Vt • Etem = 0, (6.126)

and therefore we must have for the potential <;,

^lon conductor = constant, (6.127)

Et(tem) — Vt<;, (6.128)

Ht(tem) = J-izx Et(tem))- (6.129)

A TEM mode requires a minimum of two conductors in order to be supported.

6.4.3 Green's Dyadic Function

We now generalize our review to waves in an arbitrarily shaped fixture. In

this section we relax the restriction on the assumption that the waveguide

has constant cross sectional area. The wave equation for the vector potential in the nameless gauge is

(V^ + k^)A-\- //[/+ -^VV • J] - 0, (6.130)

where we have used the relations,

124
V-i=--^V-J, (6.131)

k^=io^efi. (6.132)

The boundary condition on the conductors is the vanishing of the tangential

electric field

— ^ X /i|o7i conductors — "• ^^O.iOoj

We also require the asymptotic boundary condition for outgoing waves : [A ^ damped outgoing wave for large r). The Green's dyadic function satisfies

(V"' + A;')G(r', r) = -S{r' - r)I, (6.134)

where I is the unit dyadic. We assume that Green's function satisfies the

boundary conditions,

n' X G\s> = 0, (6.135)

V'-G|5/ = 0, (6.136)

where G is reciprocal, G(r, r') = G {r',r), and G is a damped outgoing

wave for large r. The vector potential can then be written as

A{r,Lo)= (6.137)

fi ^ J{r') [G(r', r) +

^V'V • G(r', r)]dV'

- J^[V A{r') + IV • Jir')]n' G(r', r)dS'

+ / [V • A(r') + -iv • J{r')]n' G(r', r)dS'

- J^[n' A{r') + In' • J] V • G(r', r)dS'

+ /

[n'- A{r') + ^n' J{r')]V G(r', r)dS'

J Soo " - f[{V X i(r')) • («' X G(r',r)) - (n' x A(r')) • (V x G(r', r))]</5"

125
+ / [(V X A{r')) • (n' X G) - (n' x A{r')) (V x G)]dS', (6.138)

where (') denotes prime variables. It is assumed that the surface integrals at

infinity vanish and the use of boundary conditions of eqs (6.135) and (6.136)

eliminates other terms. The resultant equation is

A(r, u;) = fij J{r') [G(r', r) +

^V'V • G(r', r)]dV'

+ f[{n' X A{r')) (V x G{r',r))]dS'. (6.139)

J s Examination of eq (6.139) indicates that it is necessary to specify the tangential components of the vector potential on the boundaries or equivalently,

the tangential component of the electric field. In order to evaluate eq (6.139),

it is necessary to solve the related Green's function boundary value problem.

We begin by introducing three vector eigenfunctions: L, M, and A'^, where

L = AV'rp, (6.140)

M = BV X (^<^a), (6.141)

N = Cy' X (V X {qxa)). (6.142)

The q term is a scalar weighting function [30], which depends on the coor- dinate system used. In cyhndrical coordinates, q — 1 with z the preferred

direction. In cartesian coordinates the preferred direction could be t, y, or z with weighting factor q — I. All of the vector eigenfunctions (denoted by Q)

must satisfy on conductors a relation of the form: nxQ{r) = 0. The function

L has zero curl and is called longitudinal, while M,N have zero divergence

and thus are called transverse.

The vector eigenfunction L is formed from:

(V^ -i- k]^ = 0, (6.143)

where we require

V'ls = 0, (6.144)

126
on surfaces. In most electromagnetic problems L is not required since the

fields are transverse. However, in order to keep the analysis very general, the

L vector eigenfunction will be included.

The vector eigenfunction M is formed from the scalar function:

(V^ + kl)<t> = 0. (6.145)

If n • a = ±1, then

</>\s = 0, (6.146) on surfaces, and if n • a = then

n-(V(#)), = 0. (6.147)

The vector eigenfunction N is formed from

(V' + k])x = 0. (6.148)

If n • a == ±1 then the boundary condition is

^•(Vgx).-O, (6.149)

whereas if n • a = then

Xs = 0, (6.150) on the surface. The normahzation constants A, B, and C are determined

—

by requiring all vector eigenfunctions to satisfy a relation of the form / Qn • Qndf= S{an — Q;^); where a are eigenvalues

S{ae-a'^) = \A\^k''^ f

tP*{a'^,f)^P{ae,f)dr= \A\''k'}S{ae - a'^), (6.151)

Jv

^{c^m-c^'J = \B\'jI fr{c^'^,r)<l>{a^,f}df= \B\'jlS{ar^ - a'J, (6.152)

Jv

%n-a'J =. ICl'-yX fx*K.^xi<^n,^df= |C|^^A:^<5(a. -<), (6.153)

Jv

127
where

(6.154)

(6.155)

and ( is the coordinate in the preferred direction and 7^ is the longitudinal

part of the eigenvalue.

Thus, the constants are given by

1^1 =

\C\ =

1^'

1

im

1

"^nln

(6.156)

(6.157)

(6.158)

Green's dyadic function can then be formed from the basis vector eigenfunctions.

G(r',r)

^ LL*

^ MM

jL2 _ jL2 m ^m k^

+ E

NN'

^-2 _ 1.2 (6.159)

The eigenvalue spectrum on a finite domain is discrete; on the infinite domain

the spectrum may be continuous, discrete or banded. Therefore, the sum in

eq (6.159) goes over to an integral for a continuous eigenvalue spectrum.

6.4.4 Fields of a Coaxial Line

As an example we consider an arbitrary sample in a coaxial line with the

principle direction z{q = 1), of length L, outer radius b and inner radius a. In this example the boundary conditions are

V'(OI.=0,L = 0, (6.160)

128
i^{r)\r=a,b = 0, (6.161)

<t>{r)U=o,L = 0. (6.162) n {V[qcf>{r}])\r=a,b = 0, (6.163)

z-V[x{mz=o,L = 0, (6.164)

X{r)\r=a,b = 0. (6.165)

Green's Function

The functions t/'? X ^^^ ^ ihaX satisfy eqs (6.160)-(6.165) are the building

blocks of the Green's function in eqs (6.140)-(6.142), (6.159)

TTJ? t/.(r') = [Jp(Ar') + R^N^{\r')] sm{—z') expi-jpO'), (6.166)

Tj-rj <l>{r') = [M/3r') + S,N,{^r')] sm{—z')exp{-jp0'), (6.167)

TTT? X(r') = [J,{Xr') + r,7V,(Ar')]cos(— 2')exp(-ip^'), (6-168)

where the constants X, ^,Rp^Sp,Tp are determined from the boundary conditions

Jp(Aa) -f RpNp{Xa) = 0, (6.169)

Jp{Xb) + RpNpiXb) = 0, (6.170)

j;(/?a) + 5piV;(^a) = 0, (6.171)

Jim + SpN;m = 0. (6.172)

and

Tp = Rp. (6.173)

The propagation constants are

A:? = A^ + (!^)^ (6.174)

129
ki = 0' + ifr, (6.175)

kl = A^ + {^?. (6.176)

The vector eigenfunctions are formed from eqs (6.140-(6.142.

130
6.5 Appendix E: Numerical Methods

In the solution for permittivity (2.24) or (2.38) is solved for e^ and e^ using a Newton-Raphson root finding technique. It is easiest to treat real and

imaginary components of (2.24) as independent real equations. The Newton-

— Raphson technique for a system of equations F{x) = approximates the root

by assuming a series expansion around a fixed point, x:

f{x)^fix)-\-ix-x)^{x), (6.177)

where x is the approximate root. It is then assumed that f{x) — and thus

5n~5n-i -/i^Iizii. (6.178)

/'(^n-l)

The roots are then obtained by iteration. For systems of equations the

method is generalized in terms of the Jacobian matrix,

Xk = Xk-i- J {xk-i)F{xk-i). (6.179)

where the Jacobian matrix is given by

J=

I dh dh dh dh ^ dxi dx2 dxz 9x4 dh dh dh dh

dxi 9x2 9x3 dxi

dh dh dh dh

9x1 dx2 9x3 dxi

dh dfi dfi dji I ^ 9x1 9xo 9x^ dxA '

(6.180)

9x1 dx2 9x3 dx\

The roots then are iteratively refined until desired convergence is obtained. The derivatives in the Jacobian can either be calculated analytically or calculated numerically. In our present software the derivatives are calculated numerically.

Equation (2.24) does not require the reference plane positions in the sample holder, but only the length of the waveguide and the sample length.

equation (2.38) does require the reference plane positions and thus the scat- tering parameters must contain the rotation terms (eqs (2.15)- (2.16)). For

short-circuited Hne measurements the reference planes must be rotated to

the sample plane.

The cutoff frequency of the sample holder must be known very accurately.

131
6.6 Appendix F: Kramers-Kronig Relations

The real and imaginary components of any causal function are related by a

dispersion relation. The complex permittivity is a causal function and whose

real and imaginary components are related by the Hilbert transform [34]

4(„)-.. = -^/~Kfc^.., (6.181)

-K Jo O^ — UJ^

,;V) = -?^fK(g^4^... (6.182)

TT Jo U^ — UJ^ The following summarizes some of the features of the Kramers-Kronig relations:

• The Hilbert transform relates real and imaginary components of a

causal function.

• Direct solution requires complete data over full spectrum for one component.

• Equation (6.181) can be thought of as an integral equation for the

unknown component when there exists some data for the other component.

The once-subtracted form of the dispersion relations are given by

e;(„) _ eV.) = ---^PT ^^-«-, (6.183)

e,(u;) - t [ujo) = P

/ J-. 77^ 7, 6.184

where P denotes principal value.

132
6.7 Appendix G: Data Smoothing

Due to the inherent noise in the calculated permittivity it is useful to find a curve that interpolates the data. The method of maximum entropy is particularly well suited for this type of problem and the procedure is summarized

in this appendix.

In vector notation let the measured data be expressed as

e"^=[e:(u;i),4(u;2), ], (6.185)

6''=[e>i),<M, ]. (6.186)

We define various moments of the data:

Mo = 1, (6.187)

Mr, = f2<'^?^ (6-188)

for n=l,2, . . .,K. If we define

ak^=ool (6.189)

and

A = K], (6.190)

then the maximum entropy solution to this system is [35]:

e-; = A*[AA*]-'e^ (6.191)

An analogous equation exists for e". This smoothing algorithm is essentially a A:-power least squares fit of the data of the form

e{uy = J2<^^^\ (6-192)

where, i are the least-squares coefficients. When fitting an actual set of data

the number of moments must be adjusted to obtain a fit that adequately

represents the data trend.

133
Table 6.1: TR Software Variable Names

Type of Variable Variable Name

sample length samplelength

air line length Lairhne

TR method newton=l

hne 1 Li

Hne2 L2

frequencies freq(I)

initial guess erguess, eiguess

cut off wavelength Icut

6.8 Appendix H: Software

Software for data reduction of the scattering parameter data are given

below. The routines use a Newton-Raphson iteration process which requires an initial guess. The program furnishes an initial guess from the solution of

the Nicolson- Ross-Weir equations.

134
885 CALL Rot_ref_planes(Ll,L2,Fco)

886 CASE ELSE

887 Beep(3)

888 DISP "Sample holder selected not available!"

889 PAUSE

890 END SELECT

891 RETURN

892 ! 893 ! ////////////////////////////////////////////////////////////

894 ! 895 Use_tr_method: ! 896 COM /Onee/ REAL Epslli (801) , REAL Epsllr(801)

898 ! 903 FOR 1=2 TO Datacount-1

904 DISP "Using Sll & S21 to calculate epsilon. I=";I

905 Sllr=REAL(Sll(I)

906 Slli=IMAG(Sll(I)

907 S21r=REAL(S21(I)

908 S21i=IMAG(S21(I))

909 S22r=REAL(S22(I)

910 S22i=IMAG(S22(I)

911 S12r=REAL(S12 (I) 912 S12i=IMAG(S12(I)

913 CALL Epnewton(Freq(I) ,Sainplelength,Elguess,E2guess,Erf ,Eif

914 Elguess=Erf

915 E2guess=Eif

919 PRINT "UNCORRECTED EPSILON VALUES"

920 PRINT Erf,Eif

921 Epsl(I)=CMPLX(Erf ,Eif)

922 Mul(I)=l

923 Erfl=Erf

924 Eifl=Eif

925 IF Waveguide_type=l THEN

926 CALL Gap_corr(Erfl,Eifl,Erf ,Eif ,Dl,D2,D3,D4,Waveguide_type)

927 ELSE

928 CALL Gap_corr(Erfl,Eifl,Erf ,Eif ,0,0,Wdl,Wd2,Waveguide_type)

929 END IF

930 ! PRINT EPSILON VALUES

931 PRINT "UNSMOOTHED, GAP CORRECTED EPSILON VALUES" , Erf , Eif

932 Eps2(I)=CMPLX(Erf ,Eif)

933 CALL Uncerts21_real(Erf ,Eif ,Mur,Mui,Samplelength,Freq(I) ,Uncertrs21(I

) ,Uncertis21(I) ,Lcut,Lairline)

934 ! CALL Uncertsll_real (Erf ,Eif ,Mur,Mui,Samplelength, Freq(I) ,Uncertrsll (I

) ,Uncertisll(I) ,Lcut,Ll)

936 Eps5(I)=CMPLX(Uncertrs21(I) ,Uncertis21 (I)

937 PRINT Uncertainties

938 PRINT "FREQ=",Freq(I) , "UNCERT=" ,Uncertrs21 (I) ,Uncertis21 (I) 939 NEXT I 940 Npts=Datacount

941 ! 942 IF Method=l THEN

943 Method=2

944 GOSUB Rotate_spanns

945 END IF

946 ! USE NICOLSON-ROSS TO CALCULATE EPSILON

947 CALL Weir(Er(*) ,Ei(*) ,Lcut,Npts)

948 ! PRINT WEIR-NICOI.SON-ROSS RESULTS

949 PRINT "NICOLSON-ROSS EPSILON VALUES"

950 FOR 1=1 TO Datacount

951 PRINT "FREQUENCy=",Freq(I) , "EPSILON"", Er(I) ,Ei(I)

135
952 Eps3(I)=CMPLX(Er(I) ,-Ei(I) ) 953 Mul(I)=l

954 NEXT I 955 ! 956 Nnioin=3 957 ! THERE ARE TWO SMOOTHING OPTIONS: 1) "SMOOTH" USES MAXIMUM ENTROPY TO SMOOT

H, IN THIS ROUTINE IT IS NECESSARY TO INPUT NUMBER OF MOMENTS (Nmom) . 2) A RUNNI

958 ! AVERAGE TECHNIQUE (WIND=NUMBER OF POINTS AVERAGED OVER)

959 Wind=10

960 Npts=Datacount

961 PRINT "ENTERING SMOOTHING ROUTINE for GAP CORRECTED VALUES " 962 ! CALL Smooth (Datacount, Freq(*) , Nmom, Eps4 (*)

)

963 CALL Smoothl(Npts,Freq(*) ,Wind,Eps2(*) ,Eps4(*)

)

964 PRINT "SMOOTHED VALUES"

965 FOR 1=2 TO Datacount-1

966 Mu4(I)=l

967 PRINT Eps4(I)

968 NEXT I 969 RETURN

970 ! 971 ! ////////////////////////////////////////////////////////////

972 ! 973 Use_scl_method : ! 974 FOR 1=2 TO Datacount-1

975 Reflr=REAL(Sll(I)

)

976 Refli=IMAG(Sll(I))

977 DISP "Using Sll & S21 to calculate epsilon. I=";I

978 CALL Epshort (Freq(I) ,Sainplelength,Elguess,E2guess,Erf ,Eif

)

979 Elguess=Erf

980 E2guesS"Eif

981 Epslr(I)=Erf

982 Epsli(I)=Eif

983 Epsl(I)=CMPLX(Erf ,Eif)

984 PRINT "epsilon without gap correction=",Erf ,Eif

985 IF Waveguide_type-1 THEN

986 CALL Gap_corr(Epslr(I) ,Epsli(I) , Erf , Eif , Dl, D2 , D3 , D4 , Waveguide_typ

e)987 ELSE

988 CALL Gap_corr(Epslr(I) ,Epsli(I) ,Erf ,Eif ,0,0,Wdl,Wd2,Waveguide_typ

e)989 END IF

990 ! 991 Eps2(I)=CMPLX(Erf ,Eif)

992 MU2(I)=CMPLX(1,0)

993 PRINT "uncorrected values="

994 PRINT USING "3D, 2X, 2D. 5D, 2X,MD.5D" ; I,REAL(Epsl (I) ), IMAG(Epsl (I)

)

995 PRINT "GAP CORRECTED VALUES"

996 PRINT Erf, Eif

997 ! 998 CALL Uncerts21_real(Erf ,Eif ,Mur,Mui,Samplelength,Freq(I) ,Uncertrs21(I

) ,Uncertis21(I) ,Lcut)

999 ! CALL Uncertsll_real (Erf ,Eif ,Mur,Mui, Samplelength, Freq(I) ,Uncertrsll (I

) ,Uncertisll(I) ,Lcut)

1000 Eps5(I)=»CMPLX(Uncertrs21(I) ,Uncertis21 (I)

)

1001 PRINT Uncertainties

1002 PRINT "FREQ-",Freq(I) , "UNCERT=" ,Uncertrs21 (I) ,Uncertis21 (I) 1003 NEXT I 1004 ! 1005 Wind=10

1006 Npts=Datacount

i

136
2435 DISP " CONTINUE to try again."

243 6 PAUSE

2437 Diskdrive$=""

2438 CASE 55,64

2439 DISP " This disk is full, insert new floppy and/or";

2440 DISP " select new drive ...CONTINUE " 2441 PAUSE

2442 Diskdrive$=""

2443 CASE ELSE

24 44 CALL Errortrap

2 44 5 IF LEN(Filenaine$)>0 THEN GOTO Send_to_disk

2446 END SELECT

2447 GOTO Selectdrive

2448 ! 2449 SUBEND

2450 ! 2451 ! it***********************************************************

2452 ! 2453 DEF FNRs21 (F, L,E11,E22)

24 54 COM /Two/ M,EpsO,MuO

2455 COM /Three/ Lcut,Shdis

2456 COM /Test_info4/ REAL Samplelength, LI, L2 , Lairline, Fco

2457 COM /Meth/ Method

2458 COM /S/ Sllr,Slli,S21r,S21i,S22r,S22i,S12r,S12i

2459 COM /Environment/ Temperature, Pressure, Humidity ,

C

2460 COMPLEX Gam, J, Epsol , Trans, S21t, Silt, S12t, S22t, Gl , G2 ,RO,R

2461 S21t=CMPLX(S21r,S21i)

2462 Sllt=CMPLX(Sllr,Slli)

2463 S12t=CMPLX(S12r,S12i)

2464 S22t=CMPLX(S22r,S22i)

2465 J=CMPLX(0,1)

2466 Omeg=2*PI*F

2467 Epsol=CMPLX(Ell,-E22)

24 68 Omegcut=2*PI/Lcut/SQRT(EpsO*MuO)

24 69 Delta=SQRT(l-Omegcut^2/Omeg^2)

2470 KO=Omeg*SQRT(EpsO*MuO)*Delta

2471 R0=SQRT((l/C'2)*Omeg'2-(2*PI/Lcut)^2)

2472 R=SQRT(Epsol*Eps0*Mu0*0meg'^2-(2*PI/Lcut)^2)

2473 Gam=(R0-R)/(R0+R)

2474 Trans=EXP(-J*(SQRT(Omeg^2*Epsol*EpsO*MuO-{2*PI/Lcut) ^2) ) *L)

2475 ! 2476 IF Method=»l THEN

2477 RETURN REAL(S21t*S12t-Sllt*S22t-(Trans"2-Gam^2) / (l-Trans^2*Gam'2)

*EXP(-2*J*K0* (Lairline-L) ) ) 2478 END IF

2479 ! 2480 IF Method"2 THEN

2481 RETURN REAL( (S21t+S12t)/2* (l-Trans^2*Gam^2) -Trans* (1-Gam"2) *EXP(-

J*KO* (Lairline-L) )

)

2482 END IF

2483 FNEND

2484 ! 2485 ! it***********************-!!***********************-!,***********

2486 ! 2487 DEF FNIs21 (F, L, E11,E22)

2488 COM /Two/ M,Eps0,Mu0

2489 COM /Three/ Lcut,Shdis

2490 COM /Test_info4/ REAL Samplelength, LI , L2 , Lairline, Fco

2491 COM /Meth/ Method

2492 COMPLEX Gam, J, Epsol , Trans, S21t, Silt, S22t, S12t,Gl,G2 ,R0,R

137
2493

2494

2495

2496

2497

2498

2499

2500

2501

2502

2503

2504

2505

2506

2507

2508

2509

2510

2511

2512

2)*EXP(

2513

2514

2515

2516

(-J*KO*

2517

2518

2519 ! 2520 ! 2521 ! 2522

2523

2524

2525

2526

2527 ! 2528 ! 2529 ! 2530

2531

2532

2533

2534

2535 ! 2536 < 2537 ! 2538

2539

2540

2541

2542

2543 ! 2544 ! 2545 ! 2546

2547

2548

2549

2550

COM /S/ Sllr,Slli,S21r,S21i,S22r,S22i,S12r,S12i

COM /Environment/ Temperature, Pressure, Humidity,

C

S21t=CMPLX{S21r,S21i)

Sllt=CMPLX(Sllr,Slli)

S12t-CMPIJ((S12r,S12i)

S22t=CMPLX(S22r,S22i)

Omeg=2*PI*F

J=CMPLX(0,1)

Omeg=2*PI*F

Epsol=CMPLX(Ell,-E22)

Omegcut=2*PI/Lcut/SQRT(Eps0*Mu0)

Delta=SQRT ( 1-Omegcut^ 2/Omeg* 2

)

K0=Omeg*SQRT(Eps0*Mu0) *Delta

R0=SQRT( (1/0^2) *Omeg^2-(2*PI/Lcut) '2)

R=SQRT(Epsol*EpsO*MuO*Omeg'^2-(2*PI/Lcut)^2)

Gam= (RO-R) / (RO+R)

Trans=EXP(-J*(SQRT(Omeg^2*Epsol*EpsO*MuO-(2*PI/Lcut) "2) ) *L)

IIF Method=l THEN RETURN IMAG(S21t*S12t-(Sllt*S22t)-(Trans"2-Gam'^2)/(l-Gam-2*Trans*

-2*J*K0* (Lairline-L) )

)

END IF

1 IIF Method- 2 THEN

RETURN IMAG( (S21t+S12t) /2* (l-Trans^2*Gam^2) - (Trans* (1-Gam"2) ) *EXP

(Lairline-L) )

)

END IF FNEND

it*******************************************-****************

DEF FNDrs21el(F,L, Ell, E22, Delta)

Xl=FNRs21(F,L,Ell+Delta,E22)

X2=FNRs21(F,L, Ell-Delta, E22)

RETURN (Xl-X2)/Delta/2

FNEND

DEF FNDrs21e2(F,L, El, E2, Delta)

Xl=FNRs21(F,L,El,E2+Delta)

X2=FNRs21(F,L,El,E2-Delta)

RETURN (X1-X2) /Delta/2

FNEND ************************************************************

DEF FNDis21el(F,L, El, E2, Delta)

X1-FNI321 (F , L, El+Delta , E2

)

X2=FNIs21(F,L, El-Delta, E2)

RETURN (X1-X2) /Delta/2

FNEND ************************************************************

DEF FNDis21e2(F,L, El, E2, Delta)

Xl=FNIs21(F,L,El,E2+Delta)

X2-FNIs21(F,L,El,E2-Delta)

RETURN (Xl-X2)/Delta/2

FNEND

138
2551

2552

2553

2554

2555! 2556!

2557!

2558!

2559! 2560

2561

2562

2563

2564

2565

2566

2567

2568

2569

2570

2571

2572

2573

2574

2575

2576

2577

2578

2579

2580

2581

2582

2583

2584

2585

2586

2587

2588

2589

2590

2591

2592

2593

2594

2595

2596

2597

2598

2599

2600

2601

2602

2603

2604

2605

2606

2607

2608

2609

2610

! ************************************************************

SUB Epnewton(F,L,El,E2,Erf ,Eif)

This sub-program calls all the functions needed to calculate

the permittivity of the sample when the transmission/reflection

method is used.

OPTION BASE 1 COM /Two/ M,EpsO,MuO

COM /S/ Sllr,Slli,S21r,S21i,S22r,S22i,S12r,S12i

DIM Alp (2, 2) ,Injab(2,2) ,Beta(2) ,X(2) ,Deltx(2)

Delta=l.E-8

Ntrial=50

Tolf=l.E-8

Tolx=l.E-8

Epnewton:

!

X(1)=E1

X(2)=E2

FOR K=l TO Ntrial

Alp(l,l)=FNDrs21el(F,L,X(l) ,X(2) , Delta)

Alp(l,2)=FNDrs21e2(F,L,X(l) ,X(2) .Delta)

Alp(2,l)=FNDis21el(F,L,X(l) ,X(2) , Delta)

Alp(2,2)=FNDis21e2(F,L,X(l) ,X(2) .Delta)

Beta(l)=-FNks21(F,L,X(l) ,X(2))

Beta(2)=-FNIs21(F,L,X{l) ,X(2)

)

Errf=0

FOR 1=1 TO 2 Errf=Errf+ABS(Beta(I)

)

NEXT I IF Errf<Tolf THEN GOTO Converged

MAT Injab= INV(Alp)

MAT Deltx= Injab*Beta

FOR J=l TO 2 X(J)=X(J)+Deltx(J)

NEXT J

IF Errf<Tolx THEN GOTO Converged

NEXT K Converged:

!

IF K=Ntrial THEN PRINT "TOOK NTRIAL INTERATIONS"

PRINT "NUMBER ITERATIONS=" ;K; "FREQUENCY=" ,

F

PRINT "REAL EPSILON=" ;X(1) PRINT "IMAGINARY EPSILON=" ;X(2)

Erf=X(l)

Eif=X(2)

SUBEND

I

I ************************************************************

I DEF FNShrs21(F,L,Ell,E22)

COM /Two/ M,EpsO,MuO

COM /Three/ Lcut,Shdis

COM /Refl/ Reflr,Refli

COMPLEX Gam , J , Epsol , Trans , Ref1 , RO , R , XI , Delt , Bet COM /Environment/ Temperature, Pressure, Humidity,

C

Refl=CMPLX(Reflr,Refli)

J=CMPLX(0,1)

Omeg=2*PI*F

Epsol=CMPLX(Ell,-E22)

139
2611

2612

2613

2614

2615

2616

2617

t-1)

2618

2619

2620

2621

2622

2623

2624

2625

2626

2627

2628

2629

2630

2631

2632

2633

2634

2635

2636

2637

2638

t-1)

2639

2640

2641

2642

2643

2644

2645

2646

2647

2648

2649

2650

2651

2652

2653

2654

2655

2656

2657

2658

2659

2660

2661

2662

2663 2664

2665

2666

2667

2668

RO=SQRT( (l/C^2)*Oineg^2-(2*PI/Lcut)^2)*J

R=SQRT(Epsol*EpsO*MuO*Omeg"2-(2*PI/Lcut) "2) *J X1=TANH(R*L)

Delt=EXP(-2*R0*(Shdis)

)

Bet=R/R0

X1=TANH(R*L)

X2=REAL(Refl*(-2*Bet+( (Delt-1) *Bet^2- ( 1+Delt) ) *X1) + (-2*Bet*Delt+ ( (Del

'Bet-2+(l+Delt) )*X1)

)

RETURN X2 FNEND

li*****-!,****************************************-)!************

DEF FNShis21(F,L,Elgues,E2gues)

COMPLEX Gam , J , Epsol , Trans , Ref1 , RO , R, XI , Bet , Delt

COM /Two/ M,EpsO,MuO

COM /Three/ Lcut,Shdis

COM /Refl/ Reflr,Refli

COM /Environment/ Temperature, Pressure, Humidity,

C

Refl=CMPLX(Reflr,Refli)

Omeg=2*PI*F

J=CMPLX(0,1)

Epsol=CMPLX(Elgues,-E2gues)

R0=SQRT( (l/C'2) *Omeg^2-(2*PI/Lcut) ^2) *J R=SQRT(Epsol*Eps0*Mu0*0meg'^2-{2*PI/Lcut)^2)*J

Delt=EXP(-2*R0*Shdis)

Bet=R/RO

X1=TANH(R*L)

X2=IMAG(Refl*{-2*Bet+( (Delt-1) *Bet^2- (1+Delt) ) *X1) + (-2*Bet*Delt+ ( (Del

^Bet^2+(1+Delt) ) *X1)

)

RETURN X2 FNEND

I I

! -k-k*****************-)!**-!!*************************************

I DEF FNShdrs21el(F,L,Ell,E22,Delta)

Xl=FNShrs21(F,L,Ell+Delta,E22)

X2=FNShrs21 (F, L, Ell-Delta, E22)

RETURN (X1-X2) /Delta/2

FNEND

I I

I ************************************************************

I DEF FNShdrs21e2(F,L, El, E2, Delta)

Xl=FNShrs21(F,L,El,E2+Delta)

X2=FNShrs21(F,L,El,E2-Delta)

RETURN (Xl-X2)/Delta/2 FNEND ************************************************************

DEF FNShdis21el{F,L, El, E2, Delta)

Xl=FNShis21 (F, L, El+Delta, E2

)

X2=FNShis21(F,L, El-Delta, E2)

RETURN (X1-X2) /Delta/2

FNEND ************************************************************

DEF FNShdis21e2(F,L, El, E2, Delta)

140
2669

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2671

2672

2673

2674

2675

2676

2677

2678

2679

2680

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2682

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2688

2689

2690

2691

2692

2693

2694

2695

2696

2697

2698

2699

2700

2701

2702

2703

2704

2705

2706

2707

2708

2709

2710

2711

2712

2713

2714 2715

2716

2717

2718

2719

2720! 2721

sr(40

2722

2723

2724

2725

2726

2727

Xl=FNShis21(F,L,El,E2+Delta)

X2=FNShis21 (F, L, El, E2-Delta)

RETURN (Xl-X2)/Delta/2 FNEND

I

I ************************************************************

I SUB Epshort(F,L,El,E2,Erf,Eif)

This sub-program calls all the functions needed to calculate

the permittivity of the sample when the short-circuit line

method is employed.

OPTION BASE 1 COM /Refl/ Reflr.Refli

DIM Alp ( 2 , 2 ), Inj ab ( 2 , 2 ), Beta ( 2 ), X ( 2 ), Deltx ( 2

)

Delta=l.E-8

Ntrial=50

Tolf=l.E-8

Tolx=l.E-3

Epshort: ! X(1)=E1

X(2)=E2

FOR K=l TO Ntrial

Alp(l,l)=FNShdrs21el(F,L,X(l) ,X(2) , Delta)

Alp(l,2)=FNShdrs21e2(F,L,X(l) ,X(2) .Delta)

Alp(2,l)=FNShdis21el(F,L,X(l) ,X(2) , Delta)

Alp(2,2)=FNShdis21e2(F,L,X(l) ,X(2) , Delta)

Beta(l)=-FNShrs21(F,L,X(l) ,X(2)

)

Beta(2)=-FNShis21(F,L,X(l) ,X(2))

Errf=0

FOR 1=1 TO 2 Errf=Errf+ABS(Beta(I) ) NEXT I IF Errf<Tolf THEN GOTO Converged

MAT Injab= INV(Alp) MAT Deltx= Injab*Beta

FOR J=l TO 2 X(J)=X(J)+Deltx{J) NEXT J

IF Errf<Tolx THEN GOTO Converged

NEXT K Converged:

Erf=X(l)

Eif=X(2)

IF K>10 THEN PRINT "CAUTION! NTRIALS, NOT CONVERGED" ,

SUBEND

I

! ************************************************************

! SUB Smooth ( Npts, Freq (*) ,Nmom, COMPLEX Eps2(*))

REAL Alp(8,401) ,Delta(8) , Atran(401, 8) ,C(8,8) ,D{8,8) ,E(8) ,Epsi(401) , Ep

1) ,Epslr(401) ,Epsli(401)

COM /Onee/ Epslli (*) ,Epsllr (*) PRINT "NPTS ", Npts, "NMOM",Nmom

FOR 1=1 TO Npts

Epslr(I)=REAL(Eps2(I) ) Epsli(I)=IMAG(Eps2(I) ) FOR K=l TO Nmom

141
2728 Alp(K,I)=(Freq(I)/l.E+9)^(K-l)

2729 NEXT K 2730 NEXT I 2731 FOR K=l TO Nmom

2732 FOR 1=1 TO Npts

2733 Delta(K)=Delta{K)+Alp(K,I)*Epslr(I)

2734 NEXT I 2735 NEXT K 2736 ! ! 2737 MAT Atran= TRN(Alp)

2738 MAT C= Alp*Atran

2739 MAT D= INV(C)

2740 MAT E= D*Delta

2741 MAT Epsr= Atran*E

2742 ! 2743 FOR K=l TO Nmom

2744 Delta(K)=0

2745 FOR 1=1 TO Npts

2746 Delta(K)=Delta(K)+Alp(K,I)*Epsli(I)

2747 NEXT I 2748 NEXT K 2749 ! ! 2750 MAT Atran= TRN(Alp)

2751 MAT C= Alp*Atran

2752 MAT D= INV(C)

2753 MAT E= D*Delta

2754 MAT Epsi- Atran*E

2755 ! 2756 FOR 1=1 TO 401

2757 Epsllr(I)=Epsr(I)

2758 Epslli(I)=Epsi(I)

2759 NEXT I 2760 SUBEND

2761 ! 2762 ! **'*******************'***************4******«*******«********

2763 ! 2764 SUB Gap_corr (Er_uncor, Ei_uncor , Er_cor , Ei_cor , Dl , D2 , D3 , D4 , Guidetyp)

2765!

2766! In the routine, the real and imaginary parts of epsilon are

2767! corrected due to the airgap between the sample and the sample holder.

2768! If the guidetyp equals one, then an airgap correction for a coaxial

2769! sample holder is done, else a airgap correction is done for a waveguide

2770! sample holder.

2771!

2772 Gap_corr: ! 277 3 IF Guidetyp=l THEN

2774 L1=LGT(D2/D1)+LGT(D4/D3)

2775 L2=LGT(D3/D2)

2776 L3=LGT(D4/D1)

2777 Er_cor=L2/ (L3-Er_uncor*Ll) *Er_uncor

2778 IF Ei_uncor<=0 THEN

2779 Ei_cor=0

2780 ELSE

2781 Ei_cor=Er_cor*Ei_uncor/Er_uncor*L3/ (L3-L1* (Er_uncor* (l+(Ei_un

cor/Er_uncor) ^2) )

2782 END IF 2783 ! 2784 ELSE

2785 Samplethick=D3

2786 Guidethick=D4

142
2787 Er_cor=Er_uncor* (Samplethick/ (Guidethick- (Guidethick-Samplethick)

*Er_uncor)

2788 IF Ei_uncor>0 THEN

2789 Ei_cor=Er_cor*Ei_uncor/Er_uncor*Guidethick/ (Guidethick- (Guide

thick-Samplethick) *Er_uncor)

2790

2791

2792

2793

2794

2795

2796

2797

2798

2799

2800

2801

2802

2803

2804

2805

2806

2807

2808

2809

2810

2811

2812

2813 2814

2815

2816

2817

2818

2819

2820

2821

2822

2823

2824

2825

2826

2827

2828

2829

2830

2831

2832

2833

2834

2835

2836

2837

2838

2839

2840

2841

2842

2843

2844

ELSE

Ei_cor=0

END IF END IF SUBEND

SUB Edit_data( Prompts, Variable, OPTIONAL Multiplier, Uvariable)

;

! Edit_data is a 'boiler plate' routine which allows easy

! editing of variables throughout the program.

I Edit_data:OFF KEY

IF NPAR>2 THEN Test=Variable*Multiplier

IF NPAR=4 THEN Uvariable=»Uvariable*Multiplier

ELSE

Test=Variable

END IF ON ERROR GOTO Test_again

Test_again:

OUTPUT 2 USING "K,#";Test

DISP "Please enter the value of ";Prompt$;

INPUT Variable

OFF ERROR

IF NPAR=4 THEN Utest=Uvariable

ON ERROR GOTO Utest_again

Utest_again:

OUTPUT 2 USING "K,#";Utest

DISP "Enter the uncertainty in ";Prompt$;

INPUT Uvariable

OFF ERROR

END IF

IF NPAR>2 THEN Variable=Variable/Multiplier

IF NPAR=4 THEN Uvariable=Uvariable/Multiplier

END IF SUBEND

SUB Speed_of_light

This sub-program calculates the speed of light in the laboratory

given the temperature, relative humidity, and pressure.

Init_com: ! COM /Environment/ Temperature, Pressure, Humdity,

Speed_of_light: ! Theta=300/ (273 . 15+Temperature)

Esaturated=Theta"5/(. 4151*10-^ (9. 834*Theta-10)

E=Humidity*Esaturated/100

145
2845

2846

2847

2848

2849

2850

2851

2852

2853

2854

2855

2856

2857

2858

2859

2860

2861

2862

2863

2864

2865

2866

2867

2868

2869

2870

2871

2872

2873

2874

2875

2876

2879

2880

2881

2882

2883

2884

2886

2887

2888

2890

2891

2892

2894

2895

2896

2897

2898

2899

2900

2901

2902

2903

2904

2905

2906

2907

2908

2909

Kpa=Pressure/10

Pdryair=Kpa-E

N=l+( ( ( (41.6*Theta)+2.39)*E*Theta)+(2.588*Pdryair*Theta) ) *10' (-6)

C=2. 99792458*10' 8/N

SUBEND

! NICOLSON-ROSS-WEIR SOLUTION TECHNIQUE FOR NON-MAGNETIC MATERIALS

SUB Weir(Er(*) ,Ei(*) ,Lcut,Npt3)

COMPLEX Gg,Xg, Results

COM /Sparms/ REAL Freq (*), COMPLEX Sll (*) ,S21 (*) , S12 (*) , S22 (*) COM /Sparms/ Mag_sll_id$,Ang_sll_id$,Mag_s21_id$, Ang_s21_id$

COM /Sparms/ Mag_s22_id$,Ang_s22_id$,Mag_sl2_id$,Ang_sl2_id$

COM /Test_info4/ REAL Samplelength, LI, L2 , Lairline, Fco

I FOR Ii=l TO Npts

Xg=(Sll(Ii)'2-S21(Ii)'2+l)/2/Sll{Ii)

Gg=Xg+SQRT(Xg'2-l)

IF ABS(Gg)>l THEN

Gg=Xg-SQRT(Xg''2-l) END IF Lam=2.9972E+8/Freq(Ii)

Results= ( 1-Lam' 2/Lcut' 2

) * ( 1-Gg) ' 2/ ( 1+Gg) ' 2+Lam' 2/Lcut '

Er ( I i

) =REAL ( Results

)

Ei(Ii)=-IMAG(Results) NEXT li

SUBEND

RUNNING AVERAGE SMOOTHING

WIND = NITMBER OF POINTS AVERGED OVER

SUB Smoothl( Npts, REAL Freq (*), Wind, COMPLEX Eps2 (*) , Eps4 (*)

)

I FOR 1=2 TO Npts-1

Remp=0

ltemp=0

IF I<8 THEN

FOR N=I TO 1+5

Remp=Remp+REAL(Eps2 (N)

)

Itemp=Itemp+IMAG(Eps2 (N)

)

NEXT N Eps4 ( I

) =CMPLX (Remp/6 , Itemp/6)

GOTO 2907 END IF

IF I>Npts-6 THEN

FOR N=I-5 TO I Remp=Remp+REAL(Eps2 (N)

)

Itemp=Itemp+IMAG(Eps2 (N)

)

NEXT N Eps4 ( I

) =CMPLX (Remp/ 6 , Itemp/6

)

GOTO 2907 END IF FOR N=I-5 TO 1+5

Remp=Remp+REAL(Eps2 (N) ) Itemp=Itemp+IMAG(Eps2(N) ) NEXT N Eps4(I)=CMPLX(Remp/ll,Itemp/ll)

NEXT I

I SUBEND

i

144
2910 ! 2911 SUB Uncerts21_real (Epsolr,Epsoli,Mur,Mui, Length, Freq,Uncertr,Uncerti, Lcut

,Lairline)

2912 COMPLEX J, A, B, C, Z , X1,X2 ,Gaimna,Ganunaa

2913 COMPLEX Epsol ,Mu,S, Ders

2914 ! 2915 PRINT "LCUT=",Lcut

2916 Uncerts_real : ! 2917 RAD

2918 EpsO=l/36/PI*l.E-9

2919 MuO=4*PI*l.E-7

2920 Epsol=CMPLX(Epsolr,-Epsoli)

2921 Mu=CMPLX(l,0)

2922 Omega=2*PI*Freq

2923 Cv=2.9972E+8

2924 J=CMPLX(0,1)

2925 Ganmia=J*SQRT(Omega^2*Epsol/Cv^2-(2*PI/L<;ut) "2)

2926 Gammaa=J*SQRT(Oniega^2/Cv^2-(2*PI/Lcut)^2)

2927 Xl=FNGain(Epsolr, Epsol i,Mur,Mui, Omega, Lcut)

2928 Z=EXP(-Gainma*Length)

2929 S=(1-X1-^2)*Z/(1-X1"2*Z^2)

2930 Ang=ARG(S)

2931 Mag=ABS(S)

2932 A=OTnega' 2/ 2/Gamnia/Cv'^ 2/Gaiimiaa/ ( l+Ganuna/Gammaa ) * ( 1+ ( l-Gamina/Ganunaa ) / (

1

+Gain]na/Gaminaa)

)

2933 B=Length*Oinega^2*Z/2/Cv^2/Ganmia

2934 C=-Gaituna*Z 2935 Ders=EXP(J*Ang) *(1-X1^2*Z"2)/{2*A*(S*X1*Z^2-X1*Z)+B*( (1-X1"2)+2*S*X1"

2*Z) ) 293 6 Sreal=REAL(Ders)

293 7 Simag=IMAG(Ders)

2938 Agreal=REAL(Ders*J*Mag)

2939 Agimag=IMAG(Ders*J*Mag)

2940 Deltth=FNDelths21(Freq/l.E+9,ABS(S)

)

2941 Delts21=FNDelts21(ABS(S)

)

2942 Deltlen=5.E-6

2943 Ltiinag=IMAG( (-C* (1-X1^2) -2*S*C*X1^2*Z) / (2*A* (S*X1*Z^2-X1*Z) +B* ( (1-Xl"

2)+2*S*Xl-2*Z) )

)

2944 Ltreal=REAL((-C*(l-Xl"2)-2*S*C*X1^2*Z)/(2*A*(S*Xl*Z^2-Xl*Z)+B*( (1-Xl'

2)+2*S*X1^2*Z) )

)

2945 ! 2946 Uncertr=(SQRT((Sreal*Delts21)"2+(Agreal*Deltth)-2+(Ltreal*Del

tlen)^2))

2947 Uncerti=(SQRT( (Simag*Delts21)^2+(Agiinag*Deltth)^2+(Ltimag*Del

tlen)^2)

)

2948 < 2949 SUBEND

2950 ! *******************ii*-k-k**-Hl******-k**-»**ifk********-li**********

2951 SUB Uncertsll_real{Epsolr,Epsoli,Mur,Mui, Length, Freq,Uncertr,Uncerti, Lcut

,L1)

2952 COMPLEX J, A, B, C, Z ,X1,X2 , Devi, Devs,Gamina, Gammaa

2953 COMPLEX Epsol ,Mu,S, Gam

2954 ! 2955 Uncerts_real : I 2956 RAD

2957 Epsol=CMPLX(Epsolr,-Epsoli)

2958 Mu=CMPLX(Mur,-Mui)

2959 Omega=2*PI*Freq

2960 Cv=2.9972E+8

2961 J=CMPLX(0,1)

145
2962 Gaiiuna=J*SQRT(Oinega^2*Epsol/Cv^2-(2*PI/Lcut) '2)

2963 Gaminaa=J*SQRT(Omega"2/Cv'2-(2*PI/Lcut) "2)

2964 Xl=FNGain(Epsolr,Epsoli, 1 , , Omega , Lcut

)

2965 Z=EXP( -Gamma* Length)

2966 A=Omega"2/2/Gamma/Cv^2/Gammaa/ ( 1+Gamma/Gammaa) * ( l+( 1-Gamma/Gammaa) / (1 +Gamma/Gammaa)

)

2 967 B=Length*Omega^2*Z/2/Cv'2/Gamma

2968 C=-Gamma*Z

2969 J=CMPLX(0,1)

2970 S=(l-Z'2) *Xl/(l-X1^2*Z^2)*EXP(-2*Ll*Gammaa)

2971 Ang=ARG(S)

2972 Mag=ABS(S)

2973 ! 2974 Devs=EXP(J*Ang)*(l-Xl'2*Z"2)/(A*(2*S*Xl*Z^2+(l-Z"2) ) +B* (2*S*Xl'2*Z-2*

Z*X1)

)

2975 Sreal=REAL(Devs)

2976 Simag=IMAG(Devs)

2977 Agreal=REAL(Devs*Mag*J)

2978 Agimag=IMAG(Devs*Mag*J)

2979 Devl=2*C*(Z*Xl-S*Xl"2*Z)/(A*(l-Z'2 + 2*S*Xl*Z'2)+2*B*(S*Xl'2*Z-Z*Xl) ) 2980 Ltreal=REAL(Devl)

2 981 Ltimag=IMAG(Devl)

2982 ! 2983 Deltlen=5.E-6

2984 Delttsll=FNDelthsll(Freq/l.E+9,ABS(S) )*.5

2985 Deltsll=(.008*ABS(S)+.002)*(l/17*Freq/l.E+9+16/17)

2986 ! 2987 Uncertr=(SQRT((Sreal*Deltsll)^2+(Agreal*Delttsll)^2+(Ltreal*D

eltlen)'2)

)

2988 Uncerti=(SQRT( (Simag*Deltsll) '2+ (Agimag*Delttsll) -2+(Ltimag*D

eltlen) ^2)

)

2989 ! 2990 SUBEND

2991 I itii-k*******************-)!************************-!!************

2992 SUB Uncertsh_real(Epsolr,Epsoli,Mur,Mui, L, Freq,Uncertr,Uncerti)

2993 COMPLEX J, A, B, C, Z , XI , X2 ,Th, Sh,G, GO , Dedl , Dedr, Dedth, Dfdl, Dfdg, Dgde, RhO

2994 COMPLEX Epsol,Mu,Sll

2995 COM /Two/ Mmx,EpsO,MuO

299 6 COM /Three/ Lcut,shdis

2997 ! 2998 Uncerts_real: ! 2999 RAD

3000 Epsol=CMPLX(Epsolr,-Epsoli)

3001 Mu=CMPLX(Mur,-Mui)

3002 Omega=2*PI*Freq

3003 J=CMPLX(0,1)

3004 G0=SQRT(Omega'2*Eps0*Mu0-(2*PI/Lcut) ^2) *J 3005 G=J*SQRT(0mega^2*Eps0*Epsol*Mu*Mu0-(2*PI/Lcut) "2)

3006 ! 3007 Th=TANH(G*L)

3008 Sh=l./COSH(G*L)

3009 ! 3010 RhO-(GO*Th-G)/(GO*Th+G)

3011 Mag=ABS(RhO)

3 012 Ang=ARG(RhO)

3013 ! 3 014 Dfdl=(l./(GO*Th+G) ) * (G0*G*Sh"2- (GO*Th-G) / (G0*Th+G) *G0*G*Sh'2)

3 015 Dfdg=(l./(G0*Th+G) ) *( (G0*L*Sh"2-l) - (G0*Th-G) / (G0*Th+G) * (G0*L*Sh"2+l) ) 3016 Dgde=-(MuO*Omega"2*EpsO)/G/2

3017 !

i

146
3018 Dedl=-Dfdl/Dfdg/Dgde

3019 Dedr=EXP(J*Ang)/Dfdg/Dgde

3020 Dedth=J*EXP(J*Ang) *Mag/Dfdg/Dgde

3021 ! 3022 Rdedl=REAL(Dedl)

3023 Idedl=IMAG(Dedl)

3024 Rdedr=REAL(Dedr)

3 025 Idedr=IMAG(Dedr)

3 026 Rdedth=REAL(Dedth)

3027 Idedth=IMAG(Dedth)

3028 ! 3029 S11=(G0*TANH(G*L)-G)/(G0*TANH(G*L)+G)

3030 Deltlen=5.E-6

3031 Deltth=FNDelthsll(Freq/l.E+9,ABS(Sll)

3032 Deltsll=( .008*ABS(S11)+.002) * ( l/17*Freq/l .E+9+16/17)

3033 ! 3034 Uncertr=( ( (Rdedr*Deltsll) ^2+ (Rdedth*Deltth) "2+ (Rdedl*Deltlen) "2) -.5)

3035 Uncerti=( ( (Idedr*Deltsll) ^2+ (Idedth*Deltth) "2+ (Idedl*Deltlen) '2)^.5)

3036 ! 3037 ! 3038 SUBEND

3039 [iHiiHHi-k********-!!***********************************************

3040 DEF FNDelthsll(F,Sll)

3041 ! 3042 IF F<18.1 THEN

3043 IF SlK.l THEN

3044 RETURN MIN(6 . 28 , . 029*EXP( . 0246/Sll) * ( . 11138*F+. 9949) ) 3045 ELSE

3046 RETURN MIN(6 . 28 ,(-. 01938*311+ . 03683 )*(. 11138*F+ . 9949 )

3047 END IF

3048 ELSE

3049 PRINT "OUT OF RANGE IN FNDELTTH"

3050 END IF

3051 ! 3052 FNEND

3053 ! 3054 DEF FNGani(Epsolr,Epsoli,Mur,Mui, Omega, Lcut)

3055 COMPLEX Epsol,Mu, Gamma, Gammaa

3056 ! 3057 Eps0=l/36/PI*l.E-9

3058 Mu0=4*PI*l.E-7

3059 Gam: ! 3060 Epsol=CMPLX(Epsolr,-Epsoli)

3061 Mu=CMPLX(Mur,-Mui)

3062 Cv=2.9972E+8

3063 Gamma=SQRT(Omega^2*Epsol/Cv^2-(2*PI/Lcut)^2)

3064 Gammaa=SQRT(Omega^2/Cv^2-(2*PI/Lcut)^2)

3065 ! 3066 RETURN (1-Gamma/Gammaa)/ (1+Gamma/Gammaa)

3067 ! 3068 FNEND

3069 ! ************************************************************

3160 DEF FNDelts21(S21)

3161 Db»20*LGT(S21)

3162 ! ASSUME 0.03 DB UNCERT

3163 IF 20*LGT{S21)>-40 THEN

3164 RETURN .0034

3165 ELSE

3166 A=-. 01113

3167 B=.56+A*90

147
3168 RETURN A*Db+B

3169 END IF

3170 FNEND

3190 DEF FNDelths21(F,Mag)

3191 ! 3192 Db=20*LGT(Mag)

3193 IF Db>-40 THEN

3194 RETURN . 00205*F+. 00113

3195 ELSE

3196 Dsl=2.552E-4*EXP(-.0828*Db)

3197 Ds2=.00437*EXP(-.0575*Db)

3198 A=(Ds2-Dsl)/16.

3199 B=Dsl-2*A

3200 RETURN A*F+B

3201 END IF

3202 FNEND

3203 ! 3204 ! A********-***************************************************

148
BL-114A

(5-90) U.S. DEPARTMENT OF COMMERCENATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1 . PUBUCATION OR REPORT NUMBER

NIST/TN-13A1

2. PERFORMING ORGANIZATION REPORT NUMBER

3. PUBUCATION DATE

TITLE AND SUBTITLE Transmission/Reflection and Short-Clrcuit Line Permittivity Measurements

!. AUTHOR(S)

James Baker-Jarvis

[ PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BOULDER, COLORADO 80303-3328

7. CONTRACT/GRANT NUMBER

8. TYPE OF REPORT AND PERIOD COVERED

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)

SUPPLEMENTARY NOTES

. ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOQRAPHY OR

UTERATURE SURVEY, MENTION IT HERE.)

lie transmission/reflection and short-circuit line methods for measuring complex permittivity

ire examined. Equations for permittivity are developed from first principles. New robust

Igorithms that eliminate the ill-behaved nature of the commonly used transmission/reflection

lethod at frequencies corresponding to Integral multiples of one-half wavelength in the

ample are presented. These allow measurements to be made on samples of any length. An

ncertalnty analysis is presented which yields estimates of the errors incurred due to the

ncertainty in scattering parameters, length measurement and reference plane position. The

quatlons derived here indicate that the minimum uncertainty for transmission/reflection

easurements of nonmagnetic materials occurs at Integral multiples of one -half wavelength in

he material. In addition, new equations for determining complex permittivity independent

f reference plane position and sample length are derived. New equations are derived for

ermlttlvlty determination using the short-circuit line allow positioning the sample

rbitrarily in the sample holder.

KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITAUZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

illbratlon; coaxial line; dielectric constant; loss factor; microwave measurements;

irmeability measurement; permittivity measurement; reflection method; short- circuit;

:ansmlsslon; uncertainty; waveguide

AVAILABIUTY UNUMITED

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS). ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE. WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

154

15. PRICE

ICTRONIC FORM



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durability and safety characteristics of building elements and systems.

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understanding of the characteristics of the products. NIST administers this program as a supplement

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