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Technology Administration, U.S. Department of Commerce

NIST Technical Note 1536

Measuring the Permittivity

and Permeability of Lossy

Materials: Solids, Liquids,

Metals, Building Materials,

and Negative-Index Materials

James Baker-Jarvis

Michael D. Janezic

Bill F. Riddle

Robert T. Johnk

Pavel Kabos

Christopher L. Holloway

Richard G. Geyer

Chriss A. Grosvenor

too

NIST Technical Note 1536

Measuring the Permittivity and Permeabiiity of

Lossy Materials: Solids, Liquids, Metals, Building

Materials, and Negative-Index Materials

James Baker Jarvis

Michael D. Janezic

Bill F. Riddle

Robert T. Johnk

Pavel Kabos

Christopher L. Holloway

Richard G. Geyer

Chriss A. Grosvenor

Electromagnetics Division

National Institute of Standards and Technology

Boulder, CO 80305

February 2005

o

%

U.S. Department of Commerce

Carlos M. Gutierrez, Secretary

Technology Administration

Phillip J. Bond, Under Secretary for Technology

National Institute of Standards and Technology

Hratch G. Semerjian, Acting Director
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an experimental procedure or concept

adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the

best available for the purpose.

National Institute of Standards and Technology Technical Note 1536

Natl. Inst. Stand. Techno!. Tech. Note 1536, 172 pages (February 2005)

CODEN: NTNOEF

U.S. Government Printing Office

Washington: 2005

For sale by the Superintendent of Documents, U.S. Government Printing Office

Internet bookstore: gpo.gov Phone:202-512-1800 Fax:202-512-2250

Mail: Stop SSOP, Washington, DC 20402-0001
Contents

1 Introduction 2

2 Electrical Properties of Lossy Materials 4

2.1 Electromagnetic Concepts for Lossy Materials 4

2.2 Overview of Relevant Circuit Theory 7

2.3 DC and AC Conductivity 11

3 Applied, Macroscopic, Evanescent, £ind Local Fields 14

3.1 Microscopic, Local, Evanescent, and Macroscopic Fields 14

3.2 Averaging 19

3.3 Constitutive Relations 20

3.4 Local Electromagnetic Fields in Materials 22

3.5 Effective Electrical Properties and Mixture Equations 23

3.6 Structures that Exhibit Effective

Negative Permittivity and Permeability 23

4 Instrumentation, Specimen Holders, and Specimen Preparation 27

4.1 Network Analyzers 27

4.2 ANA Calibration 28

4.2.1 Coaxial Line Calibration 28

4.2.2 The Waveguide Calibration Kit 28

4.2.3 On-Wafer Calibration, Measurement,

and Measurement Verification 29

4.3 Specimen-Holder Specifications 30

4.3.1 Specimen Holder 30

4.3.2 Specimen Preparation 31

5 Overview of Measurement Methods for Solids 33

6 Transmission-Line Techniques 40

6.1 Coaxial Line, Rectangular, and Cylindrical Waveguides 41

6.1.1 Overview of Waveguides Used in Transmission/Reflection

Dielectric Measurements 41

6.1.2 sections of Waveguides Used as Specimen Holders 43

6.2 Slots in Waveguide 45

6.3 Microstrip, Striphnes, and Coplanar Waveguide . 45

6.4 Ground-Penetrating Radar 50

6.5 Free-Space Measurements 50

m
7 Coaxial Line and Waveguide Measurement

Algorithms for Permittivity and Permeability 51

7.1 Specimen Geometry and Modal Expansions 52

7.2 Completely Filled Waveguide 55

7.2.1 Materials with Positive Permittivity and Permeability 55

7.2.2 Negative-Index Materials 57

7.3 Methods for the Numerical Determination of Permittivity 59

7.3.1 Iterative Solutions 59

7.3.2 Exphcit Method for Nonmagnetic Materials 60

7.4 Corrections to Data 60

7.4.1 Influence of Gaps Between Specimen and Holder 60

7.4.2 Attenuation Due to Imperfect Conductivity of Specimen Holders ... 61

7.4.3 Appearance of Higher Order Modes 62

7.4.4 Mode Suppression in Waveguides 62

7.4.5 Uncertainty Sources and Analysis 62

7.4.6 Systematic Uncertainties in Permittivity Data Related to Air Gaps . 65

7.5 Permeability and Permittivity Calculation 66

7.5.1 Nicolson-Ross-Weir Solutions (NRW) 66

7.5.2 Modified Nicolson-Ross: Reference-Plane Invariant Algorithm .... 68

7.5.3 Iterative Solution 69

7.5.4 Exphcit Solution 70

7.5.5 NIM Parameter Extraction 70

7.5.6 Measurement Results 71

7.5.7 Measurements of Magnetic Materials 72

7.5.8 Effects of Air Gaps Between the Specimen and

the Waveguide for Magnetic Materials 72

7.6 Uncertainty Determination of Combined Permittivity

and Permeability Measurements in Waveguide 75

7.6.1 Independent Sources of Uncertainty for Magnetic Measurements ... 75

7.6.2 Measurement Uncertainty for a Specimen in a Transmission Line ... 76

7.7 Uncertainty in the Gap Correction 76

7.7.1 Waveguide Air-Gap Uncertainty for Dielectrics 78

7.7.2 Coaxial Air-Gap Correction for Dielectrics 78

8 Short-Circuited Line Methods 78

8.1 Theory 78

8.2 Measurements 80

IV
9 Permeameters for Ferrites and Metals 81

9.1 Overview of Permeameters 81

9.2 Permeability of Metals 82

9.3 Ferrites and Resistive Materials 83

10 Other Transmission-Line Methods 84

10.1 Two-Port for Thin Materials and Thin Films 84

10.1.1 Overview 84

10.1.2 Scattering Parameters 85

10.2 Short-Circuited Open-ended Probes . 86

10.2.1 Overview of Short-Circuited Probes 86

10.2.2 Mode-Match Derivation for the Reflection Coefficient

for the Short-Circuited Probe 87

11 Measurement Methods for Liquids 89

11.1 Liquid Measurements Using Resonant Methods 89

11.2 Open-Circuited Holders 89

11.2.1 Overview 89

11.2.2 Model 1: TEM Model with Correction to Inner Conductor Length . . 92

11.2.3 Model 2: Full-Mode Model Theoretical Formulation 93

11.2.4 Uncertainty Analysis for Shielded Open-Circuited Holder 97

11.3 Open-ended Coaxial Probe 99

11.3.1 Theory of the Open-ended Probe 99

11.3.2 Uncertainty Analysis for Coaxial Open-ended Probe 101

12 Dielectric Measurements on Biological Materials 102

12.1 Methods for Biological Measurements 102

12.2 Conductivity of High-Loss Materials Used as Phantoms 103

13 Capacitive Techniques 106

13.1 Overview of Capacitive Techniques 106

13.2 Capacitance Uncertainty 107

13.3 Electrode Polarization and Permittivity Measurements 108

13.3.1 Overview 108

13.3.2 Four-Probe Technique 109

14 Discussion 110

15 Acknowledgments 110

16 References 111
A Review of the Literature on Dielectric Measurements of Lossy Materials 136

B Gap Correction in Dielectric Materials 141

B.l CoELxial Capacitor Model for Dielectric Materials 141

B.2 Rectangular Waveguide Model 144

C Gap Correction for Magnetic Materials 145

C.l Coaxial Capacitor Model for Magnetic Materials 145

C.2 Waveguide Model for Magnetic Materials 147

C.3 Magnetic Corrections for Gaps in the X-Z and Y-Z Planes 148

C.4 Magnetic Corrections for Gaps Across the X-Y Plane 148

C.4.1 Mitigation of the Effects of Air Gaps 148

D Permittivity and Permeability Mixture Equations 148

VI
Measuring the Permittivity and Permeability

of Lossy Materials: Solids, Liquids, Metals,

Building Materials, and Negative-Index

Materials

James Baker-Jarvis, Michael D. Janezic, Bill Riddle

Robert T. Johnk, Pavel Kabos, Christopher L. Holloway

Richard G. Geyer, Chriss A. Grosvenor

National Institute of Standards and Technology,

Electromagnetics Division, MS 818.01, Boulder, CO

e-mail:jjarvis@boulder.nist.gov

Abstract

The goal of this report is to provide a comprehensive guide for researchers in the area of

measurements of lossy dielectric and magnetic materials with the intent of assembhng the

relevant information needed to perform and interpret dielectric measurements on lossy materials in one place. The report should aid in the selection of the most relevant methods for

particular applications. We emphasize the metrology aspects of the measurement of lossy

dielectrics and the associated uncertainty analysis. We present measurement methods, algorithms, and procedures for measuring both commonly used dielectric and magnetic materials,

and in addition we present the fundamentals for measuring negative-index materials.

Key Words: CaUbration; coaxial line; loss factor; metamaterials; microwave measurements;

NIM; permeabihty; permittivity; uncertainty; waveguide.
1. Introduction

Many materials used in electromagnetic applications are lossy. The accurate measurement of

lossy dielectric materials is challenging since many resonant techniques lose sensitivity when

applied to such materials and transmission-line methods are strongly influenced by metal

losses. Our goal is to assemble the relevant information needed to perform and interpret

dielectric measurements on lossy materials in a single report. Therefore we include sections

on the underlying electromagnetics, circuit theory, related physics, measurement algorithms,

and uncertainty analysis. We also have included a section on dielectric measurement data we

have collected on a wide selection of lossy materials, including building materials, hquids, and

substrates. This report should aid in the selection of the most relevant methods for particular

applications. With emphasis on metrology, we will introduce the relevant electromagnetic

quantities, overview a suite of measurements and methods, develop the relevant equations

from first principles, and finally include the uncertainties in the measurement processes.

There is a continual demand for accurate measurements of the dielectric and magnetic

properties of lossy solids and liquids [1]. T5rpical applications range from dielectric measure- ments of biological tissues for cancer research, building materials, negative index materials,

electromagnetic shielding, to propagation of wireless signals. In this report, the term loss

will refer to materials with loss tangents greater than approximately 0.05.

Measurements without well-characterized uncertainties are of dubious value. Variations

in the repeatability of the measurement are not sufficient to characterize the total uncertainty

in the measmement. Therefore, in this report, we pay particular attention to uncertainty

analysis.

Over the years, there has been an abundance of methods developed for measuring electromagnetic permeability and permittivity. These techniques include free-space methods,

open-ended coaxial-probe techniques, cavity resonators, dielectric-resonator techniques, and

transmission-Une techniques. Each method has its niche. For example, techniques based

on cavities are accurate, but not broadband, and are usually limited to low-loss materials.

Nondestructive techniques, although not the most accurate, are very attractive since they
maintain material integrity.

Dielectric properties depend on frequency, homogeneity, anisotropy, temperature, surface

roughness, and in the case of ferroelectrics and ferromagnetics, applied bias field [2, 16]. Field

orientation is important for measurements of anisotropic materials. Measurement fixtures

where the electromagnetic fields are tangential to the air-material interfaces, such as in TEqi

cavities, generally yield more accurate results than fixtures where the fields are normal to the

interface. Unfortunately, for many applications it is not always possible or even preferable to measure in-plane field orientations. For example, circuit boards and printed-wiring boards

operate with the electric field primarily normal to the plane of the panel and therefore this

component of the permittivity is of paramount interest. However, measurements with the

electric field perpendicular to the specimen face may suffer from the effects of air-gap depolarization. Air gaps around the electrodes in transmission-line measurements produce a large

systematic uncertainty since they introduce a series capacitance that results in a severe bias

in the calculated permittivity. In cases such as in coaxial matched-load termination measurements, the eflFects of air gaps can be reduced by metalization of the specimen surfaces

or application of conductive pastes to the sections of the specimen in contact with the electrodes; otherwise they may be corrected by numerical techniques [3,4]. The characterization

of anisotropic materials generally requires two techniques, one for the normal component of

permittivity and one for the in-plane component. However, the loss measurement is usually

not strongly anisotropic and one accurate measurement usually suffices.

In recent years, research into negative index materials (NIM) has increased. These lossy

materials support negative permittivity and permeability over narrow-bands of frequencies.

In this report we will address a number of metrology issues related to this field of study.

We begin with an overview of relevant electrical parameters in Section 2 and with an

overview of constitutive relationships and the local, applied, evanescent, and macroscopic

fields in Section 3. We then discuss calibration issues, sample holders, and sample preparation

in Section 4. Next, we proceed with an overview of measurement methods in Section 5 and

transmission-line methods in Sections 6 through 8. Some of this was given in Reference [1], but we have rewritten this for lossy materials. We then overview the measurement of the
permeability and resistivity of ferrites and metals in Section 9. In Section 10 we discuss

two-port and short-circuited coaxial probes. Liquid measurement methods are overviewed

in section 11. Finally, we overview the measurement of phantom and biological tissues in

Section 12.

2. Electrical Properties of Lossy Materials

2.1 Electromagnetic Concepts for Lossy Materials

In this section, we overview the most basic concepts needed to study and interpret dielectric

response in lossy materials. We use the following notation for the permittivity, e* = eo(e^ —

je'l), where eo = 8.85 x 10"^^ (F/m) is (approximately) the permittivity of vacuum and

ian5d = 4V4 is the loss tangent in the material. A similar relationship holds for the

permeabihty, /i* = HoilJ'r ~ JfJ'r)^ where ^o = 47r x 10~^ (H/m) is the (exact) permeabihty of

vacuum and the magnetic loss tangent is tan 6m = fJ'r/fJ'rIn the time domain, the most general causal linear relationship between the displacement

and electric fields is

J

rex

' f{r)E{r,t-T)dT, (1)

u

where / is the impulse-response function. The permittivity is defined in terms of the Fomrier

transform of the impulse-response function e{u) = eo{l + J^[f]{u)). An analogous equation

can be written between the induction and magnetic fields.

Electric and magnetic fields are attenuated as they travel through lossy materials. The

plane-wave attenuation coefficient in an infinite medium is denoted by the quantity a and

the phase by (3. The propagation coefficient is'y = a + jP = jk = juj-y/eJI. For waves in a

guided structure 7 = jx/k"^ — k^, where kc = uJc/c = 27r/Xc is the cutoff wave number. The

attenuation is given by

a = —^y/e[.fi'^\ ( (-| - 1 +tanSdtanSmy + (tan 5^ + tan 5^)2 j + --| _ 1 +tanSdtan5m,

(2)
where kr = ujy/e'fi'. In a dielectric medium where kc —* 0, we have

a

:V2

y^yyr+w^-i. (3)

In low-loss dielectric media, taUdS <C 1, o; reduces in the limit to a —^ ujy/d^{tain6d)/2c.

The phase coefficient /? is given by

/3 = ±-r^y/e!~i4.\ ( (rf ~ 1 +tan5d tan 5^)2 + (tan 5^ + tan 5^)2 j _ (_| _ i + tan^dtanJ^).

(4)

Note that at cutoff, /? is a function only of the loss factors. In dielectric media this reduces

to

p = i-^y^^y/l + tan2 5d

~1.

(5)

The imaginary part of the propagation coefficient defines the phase of an electromagnetic

wave and is related to the refractive index n = ±y%/4. In normal dielectrics the positive

square root is taken in eq. (4). In NIM, which have effective negative e^ and /i^, we take

negative square root in eq. (4). This is the origin of the term NIM. This keeps the real part

of the wave impedance positive. The wave impedance for a transverse electric and magnetic

mode (TEM) is vZ/^/e, transverse electric mode (TE) is juj/i/'j = fiu{p + ja) / {f3'^ + a"^) , and

for a transverse magnetic mode (TM) is 'j/juje. As a consequence of conservation of energy,

a must positive. Below cutoff the propagation coeflScient becomes 7 = Jk^ — A;^, since a

factor of j is factored out from under the square root.

The wavelength in a material is influenced by the permittivity; for example, for a TEM

mode, A^ « C^ac/V^'rl^'rfThe skin depth, the distance a plane wave travels where it decays to 1/e of the initial amplitude, is related to the attenuation coefficient by Ss = 1/a. The wave amplitude decays due to losses as |E| oc exp{—az). The power in a plane wave, E{z,t) =

Eq exp {—Oiz) exp {jut — j(3z), attenuates as P oc exp {—2az). In metals, where the conductivity is large, the skin depth reduces to

Ss = I

^ (6)
where adc is the dc conductivity and / is the frequency. We see that the frequency, conductivity, and permeabihty of the material determine the skin depth in metals. The surface

resistance for metal is

OsCTdc V CTdc When the conductors on a substrate are very thin, the fields can penetrate through the

conductors into the substrate. This increases the resistance because the field is in both the

metal and the dielectric. As a consequence of the skin depth, the internal inductance decreases with increasing frequency, whereas the surface resistance Rs increases with frequency

in proportion to y^.

Some materials exhibit ionic conductivity, so that when a static (dc) electric field is

applied, a current is induced. This behavior is modeled by the dc conductivity cidc, which

produces a low-frequency loss (oc 1/uj) in addition to polarization loss (e"). In some materials,

such as semiconductors and disordered solids, the conductivity is complex and is firequency

dependent. This will be discussed in Section 2.3.

The effective permittivity for linear, isotropic materials, that includes dc conductivity

is defined from the Fourier transform of Maxwell's equation: juT) -|- J = jcoeE + cr^cE =

jueljf'E = V X H, so that

6:,, = e:6o-i(e:'eo + ^). (8)

The group velocity in a transmission fine of length L is

duj ^ .dcf) c ,„.

where n is the index of refraction, </> is the phase, and the group delay is — |^. For anomalous

dispersion such as in NIM the index of refraction becomes negative, predicting a group

velocity faster than c. This result is traced to the approximations made in deriving eq. (9)

and a detailed anlaysis shows this is no longer a problem. The phase velocity is defined from

the definition of a constant phase 9 = ut — /3z so that the time derivative yields

dz u){k) ^ = ^P =

-J- ^(^)

^.

^= -Z7U- (10)
Waves where the group and phase velocity are in the same direction are called forward

waves. If they are opposite in sign then they are called backward waves as shown in Figure 1. Backward waves occur in regions of strong dispersion such as in NIM materials near resonant

frequencies where the effective permittivity and permeability can become negative. A typical

backward wave may have anomalous dispersion of the form uJ^ = 1/k'^CC^ with phase velocity

u/k = IjlP-yJLC, and group velocity duj/dk = —l/k'^\/CC. NIM are formed by resonant

structures embedded in a matrix, which form regions where the effective permittivity and

permeability become negative at specific frequencies. The derivative of the refractive index

in eq. (9) is needed in calculating the behavior of backward waves. The effective permittivity

and permeability of resonant systems and backward waves can be characterized by harmonic

oscillators. Since the polarization satisfies the driven harmonic oscillator proble and ^{u) =

XE = {e{uj) — 1)E, we have

e(a;) = co (c, + f^ .^ )

(11)

fiiuj) = Mo (C2 + f\ -p )

, (12)

where the constants are determined by fits to the resonance curve. In the special case where

Cr = /ir = —1, Pendry and Maslovski et al. [90] have conjectured that nearly perfect-lens

properties are possible. Using the effective permittivity and permeability in the form of eqs.

(11) and (12), the the dispersion relations, phase, and group velocities can be obtained.

2.2 Overview of Relevant Circuit Theory

The phase velocity in a transmission line with distributed capacitance C and inductance C

is given by Vp = u/(3. At cutoff the phase velocity becomes infinite (whereas the physical

speed, Vg < c). For a uniform lossless line, Vp = 1/y/CC. For a lossy fine

Vp ~ —. D2—

(!"/

y/CC 1 — g;;;2£2

Therefore, because of the relationship between e <-^ C and p^ "r^ C, the propagation velocity

varies as a function of the permittivity, permeability, and frequency. In lossy low-temperature
Forward Wave Backward Wave

(D CO

Figure 1. Forward and backward wave dispersion curves.

co-fired ceramics (LTCC) and printed wiring board (PWB) applications signals with high

harmonic content, such as pulsed waveforms, will exhibit dispersion as they propagate along

the hne. The propagation delay per unit length found from eq. (13) is td ~ y/CC{l —

The characteristic impedance of a transmission line is given by [5, 6]

Zc =

'R + juC

(14) G + JLOC

where G is the conductance, u = 27r/, and C and R are the distributed inductance and

resistance that may be frequency dependent. For a TEM transmission line this would reduce

the characteristic impedance to Jfi/e. The capacitance C for low-loss dielectrics depends

very weakly on frequency. The reason £ is a function of frequency is that it can be expressed

as £ = Cint + Cexti where Cint is the internal self inductance of the conductors and Cext is

the external inductance between the conductors. R and Cint depend on the skin depth.

8
In an ideal lossless transmission line at typical wireless frequencies, we can approximate

eq. (14) as

Zc{ideal) ~

Y

7^ "^ Y

(1^)

In eq. (15) the characteristic impedance of the lossless transmission line is real and depends

only on C and C. Zc can be complex [7,8]. An approximation for the characteristic impedance

at high frequencies can be obtained by a Taylor-series expansion in R. Assuming G — 0,

. R R^

+ (16)

The propagation coefficient 7 is also important in transmission-line performance because it

describes the attenuation and phase response of the line

R

^ = a + j(3=^{R + jujC){G + juC) ^ ^r^ + J^VCC

2Zo

1- R'

%UJ'^C?

(17)

where C, G, £, R are measured distributed circuit parameters per unit length. The last

expression in eq. (17) is a Taylor-series expansion of 7 with G = in powers of R. In eq.

(17), a and /? are the attenuation and phase coefficients, where a includes loss due to the

conductors, dielectric, and radiation: a = ac + ad + cXr- Here, ac can be approximated by [6] R{u)

Oir — (18)

^Zc{ideal)

Equation (18) is very approximate, and in microstrip, stripline, coplanar and other waveguides,

there are significant corrections [9]. The dielectric attenuation is

eoe:T^r = -yJeoc'^^ofJ'rtandd. (19)

Note that ad increases linearly with frequency, whereas ac increases as the square root of

frequency. The bulk dc resistance is given by

1 L.

Rdc =

Cdc ^e

(20)

where Lg and Ag. are the length and effective area of the conductor where the current passes.

Using eq. (6), we see that resistance increases with frequency, approximately as R ^ Rdc +

RfyJJ, where Rf is the increment of the alternating current (ac) resistance per square root
Table 1. Conductivities of metals. The skin depths are for 10 GHz.

Material Conductivity, cTdc/ lO'^{S/m) Skin depth 6 x 10^ (m)

6.60

8.26

6.42

7.85

17.0

12.70

19.0

Cu 5.80

Al 3.72

Ag 6.17

Au 4.10

In 0.87

70-30 brass 1.57

Typical solder 0.71

Table 2. Resistivities of metals [10, 11 .

Material Cu Ag Au Mo Pb Al Sn Pd Pt

Bulk resistivity {fxQ,- cm) 1.7 1.6 2.3 5.2 20.6 2.65 11 10.8 10.6

Thick-film resistivity {fiQ- cm) 4 5 5 12 -

- - - Maximum temperature ("C) 950 950 950 1500 327 660 230 1550 1768

10
of frequency from dc. Bulk electrical conductance and resistivity of commonly used metals

are given in tables 1 and 2 [5]. The conductance G of the transmission line depends on the loss in the dielectric substrate. At low gigahertz frequencies or below, these losses are smaller than conductor losses.

However, the dielectric losses increase roughly in proportion to to or faster, as given in eq.

(19), whereas conductor losses increase as y/uj. Therefore, at high frequencies, above approximately 10 GHz to 30 GHz, the dielectric loss can exceed metal losses. This is a reason why low-loss substrates are used at high frequencies.

The dielectric and magnetic loss tangents can in some cases be expressed in terms of

circuit parameters and frequency /: tan Sd — G/{27rfC). For magnetic materials the effective

loss tangent is tan J^n = R/{27rfjC).

Lossy dielectric and magnetic materials are dispersive. As a consequence, du/dt =

E • dB/dt -f- H • dB/dt ^ l/2{ed\'E\'^/dt + fid\'H\'^ /dt) . The energy stored in a dispersive

material is [12]

Ueff = Re(^^{u;o)\ < E(r,t) • E(r,t) > -t-i?e (^^^(a;o)^ < H(r,t) • H(r, t) > . (21)

The time average <> is over the period of the carrier frequency ljq. Equation (21) reduces

to the normal stored-energy expression when the material parameters are independent of

frequency; also, in this limit Ue/f = u. The conservation of internal equation We//, for the

Poynting vector S = E x H in linear, dispersive media is given by

dt

-h V • S = -J • E - 2ujQlm[e{uJo) < E • E > -f/i(a;o) < H • H >]. (22)

2.3 DC and AC Conductivity

The alternating current (ac) conductivity cr*^^ = Udc + j^{^{^) ~ ^o) has been used in the

literature to model either the ac effects of the free charge and partially bound free charge

in hopping and tunneling conduction, or as another way of expressing the complex permittivity. Since some charge is only partially bound, the distinction between conductivity and

permittivity can, at times, get blurred. This is particularly an issue in quantum-mechanical

11
analysis where energy states are discrete. In this section we consider the ac complex conductivity (7*^^ — a' — ja" of the free, bound, and partially bound charge. Most models of ac

conductivity are based on charged particles in potential wells, yielding a percolation threshold, where energy fluctuations determine whether the particle can surmount the barrier and

thereby contribute to the conductivity.

Semiconductors are materials where ac conductivity is commonly used. GaUium arsenide

and gallium nitride are highly insulating and therefore have less loss. Figures 2 and 3 show

» measurement results on the permittivity of high-resistivity galUum arsenide as a function

of frequency. These measurements were made by a mode-filtered TEqi X-band cavity. As

the frequency of measurement increases to the gigahertz range, the free-charge loss in many

semiconductors decreases. At high frequencies, lossy semiconductors and metals have a complex free-charge ac conductivity, explained by the Drude model. This can cause the

effective permittivity to be negative [12]. To understand this, consider Maxwell's equation -- + J = V X H. (23)

Usually the constitutive relations for the fi:ee-current density and displacement are J = crE

and D = eE, where a is the conductivity due to the motion of free charge and not the boundcharge polarization, either through ionic movement, hopping, or tunneUng phenomena. As

frequency increases the conductivity can be complex, where the real part is approximately

the dc conductivity and the imaginary part relates to the phase of the charge movement.

Combining adc with the displacement field produces an eflfective real part of the permittivity

that can be negative over a region of frequencies

^'^^'" = <-£-^- (<+£) (^^)

where a' « ade- There are a number of distinct models for a^^^. The Drude model for the complex conductivity of metals results in a^^^ = foNe^/{m{'^Q+ juj)) = a' —ja", where 70 is the collision

frequency, foN is the electron density, m is the ion mass, and e is the electronic charge [12].

Note that the dc conductivity is ddc = foNe^/m'^Q. The net dielectric response is a sum of

12
10

12.97

12.96

12.95

12.94

12.93

12.921-

12.91

Start

Middle

Terminus

-*—*- ''' I ''» I * ' ' 8 9 10

Frequency (GHz)

11 12

Figure 2. e^ of gallium arsenide measured by an X-band cavity [13]. Start, middle, and

terminus refer to different specimens taken from the same boule.

the dipolar contribution and that due to the ions where e' = e'^ — foNe^/m{'yQ + a;^), and

e" = Ne'^jo/muj{'yQ+uj'^). Therefore, for metals, the real part of the permittivity is negative

for frequencies near the plasma frequency, Up = JfoNe'^/eom. The plasma frequency in

metals is usually well above 100 GHz; in gases it can be lower since the density of charged

particles is lower.

For disordered solids, where hopping and tunneling conduction takes place with a relaxation time Tg, the ac conductivity can be expressed as [14, 15]

(^toti^) = (^oJ^Te/ hi (1 + jUTe)

ujTe arctan {coTe) = (To a;reln(l + (a;Te)^)

• (25) i In^ (1 + a;2r2) + arctan^ (uTe) ^ ^ In^ (1 + u^r^) + 2 arctan^ {ujTe)_

For complex conductivity the combination of permittivity and conductivity in eg// satisfies the Kramers-Kronig causality condition. In Section 5, we discuss the Kramers-Kronig

relation with dc conductivity included.

13
COO

X

C

0.80

0.75 F0.70 r

0.65 r

0.60 '-

0.55 r

0.50 E0.45

8 9 10 11

Frequency (GHz)

Figure 3. Loss tangent of gallium arsenide measured by an X-band cavity [13].

3. Applied, Macroscopic, Evanescent, and Local Fields

In this section, we overview a number of pertinent concepts used in dielectric analysis. This

includes the electromagnetic fields in materials, field behavior, NIM materials, and types of

material property averaging in disordered solids.

3.1 Microscopic, Local, Evanescent, and Macroscopic Fields

If an electromagnetic field is applied to a semi-infinite media, the fields in the material include the effects of both the applied field and the particle back-reaction fields. The charges

and spins in a medium interact with the local fields and not necessarily with the total applied field. For example, when an applied electromagnetic field interacts with a dielectric

material, the macroscopic and local fields in the material are modified by surface-charge

dipole-depolarization fields that oppose the apphed field. When considering time-dependent

high-frequency fields, this interaction is more complex. In addition, the internal energy affects the resulting electromagnetic behavior. For example, depolarization, demagnetization,

thermal expansion, exchange, and anisotropy interactions can influence the dipole orientations and therefore the fields. Usually these effects of internal energy are modeled by an

14
effective field that modifies the appHed field. In constitutive modeling in Maxwell's equations,

we must express the material properties in terms of the macroscopic field, not the applied

or local fields and therefore we need to make clear distinctions between the fields [16].

Fields may be propagating, of form e^^'^*~^^\ damped propagating e^('^*~^^)~'*^, or evanescent e-^'^*""^. Evanescence occurs at frequencies below cutoff, that is in a waveguide, where

transverse resonance occurs [17]. Below cutoff, 7 = Jk^ — k^, where kc is the cutoff wave

number calculated from the transverse geometry and k = Uy/ejl. Evanescent electromagnetic fields also occur at apertures and in the near fields of antennas. Near (evanescent) fields

consist of stored energy and there is no net energy transport over a cycle unless there are

losses in the medium. In the case of an antenna the evanescent fields are called the near field.

Evanescent fields can be detected only if they are perturbed and converted into propagating

waves or transformed by dielectric loss. Electromagnetic waves may convert from evanescent

to propagating. For example, in coupling to dielectric resonators, the evanescent waves fi^om

the coupling loops produce propagating waves in the dielectric resonator.

The relationships between the applied, macroscopic, local, and the microscopic fields

are important for constitutive modehng. The applied field originates from external charges,

whereas the macroscopic fields are averaged quantities in the medium. The macroscopic

fields within the material region in Maxwell's equations are imphcitly defined through the

constitutive relationships and boundary conditions. The macroscopic field that satisfies

Maxwell's equations with appropriate boundary conditions and constitutive relationships

is generally not the same as the applied field. The macroscopic field is approximately the

applied field minus the field due to surface depolarization. In a homogeneous, semi-infinite

slab, illmninated by an applied field, the macroscopic field is generally of smaller magnitude

than the applied field. The local field is the averaged electromagnetic field at a particle site

due to both the apphed field and the fields from all of the other sources, such as dipoles [18,

19]. The microscopic field represents the atomic-level electromagnetic field where particles

interact with the field from discrete charges. At the first level of homogenization, particles

interact with the local electromagnetic field. The spatial and temporal resolution contained in

the macroscopic variables are directly related to the spatial and temporal detail incorporated

15
in the material parameters obtained when the charge density is homogenized and expanded

in a Taylor series. If Maxwell's equations are solved precisely at the microscale level with

exact constitutive relationships, then the macroscopic electromagnetic field is the same as

the microscopic field.

Theoretical analysis of the effective local electromagnetic field is important in dielectric

modehng of single-molecule measurements and thin films, since at this level we need to know

some, but not all, the details of the molecular structure. The electromagnetic fields at this

level are local, but not atomic scale. Since electrical measurements can now be performed to

very small spatial resolutions and the elements of electrical circuits approach the molecular

level, we require good models of the macroscopic and local fields at all spatial scales. This is

particularly important since we know that the Lorentz theory of the local field is not always

adequate for predicting polarizabihties [20,21]. Also, when solving Maxwell's equations at

the molecular level, definitions of the macroscopic field and constitutive relationships are

important. The formation of the local field is a very complex process whereby the apphed

electric field polarizes dipoles in a specific molecule and the applied magnetic field causes

precession of spins. Then the molecule's dipole field modifies the dipole orientations of other

molecules in close proximity, which then reacts back to produce a correction to the molecule's

field in the given region. This process gets more complicated for time-dependent behavior.

We define the local electromagnetic field as the effective, averaged field at a specific point in a

material, which is a function of both the applied and multipole fields in the media. The local

field is related to the average macroscopic and microscopic electromagnetic fields in that it

is a sum of the macroscopic field and effects of the neax-field. In ferroelectric materials, the

local electric field can become very large and hence there is a need for comprehensive field

models.

Mandel and Mazur developed a static theory for the local field in terms of the polarization

response of a many-body system using a T-matrix formalism [22] . Gubernatis extended the

T-matrix formalism [23]. Keller's [24] review article on the local field uses an electromagnetic

propagator approach. Kubo's hnear-response theory has also been used for electromagnetic

correlation studies [18,25].

16
In the literature of dielectric materials, a number of specific fields have been introduced

to analyze polarization phenomena. The electric field acting on a nonpolar dielectric is commonly called the internal field, whereas the field acting on a permanent dipole moment

is called the directing field. The difference between the internal field and directing fields is the

averaged reaction field. The reaction field is the result of a dipole polarizing its environment.

Electromagnetic fields in materials can either be freely propagating, propagating with

attenuation, or evanescent. At a macroscopic level the effects of the material are modeled by

an effective permittivity and permeability. The formation of the permittivity or permeability

from the basic charges and atoms can be modeled using a complicated statistical-mechanical

averaging procedure that contains microstructural effects. However, in general the constitutive modeling is less ambitious and places the microstructural effects into the definition of

the impulse-response function in the time domain or the permittivity and permeability in

the frequency domain. In nonhomogeneous random composite materials the modeling of the

permittivity depends on whether the wavelength in the material is long or short relative to

material inclusions. In long wavelength approximations, the material can be modeled with

an effective permittivity and permeability. If there is a periodicity in composite materials,

then the electromagnetic propagation will manifest this in the scattering response and there

may be band gaps. If the applied field has a wavelength short in relation to particle size then

the material parameters axe spatially dependent and resonances can occur in the inclusions.

If resonances occur in the medium, then the effective real parts of the permittivity and

permeability can exhibit anomalous behavior; for example, they may obtain negative values

(see eqs. (11) and (12)). If only the effective permittivity or permeability have negative

values, then evanescent fields are produced, analogous to fields below cutoff in a waveguide

[26].

Evanescent and near fields in dielectric measurements are very important. These fields

do not propagate and therefore can be used to measure materials of very small spatial

dimensions; for example, in evanescent microwave probing [27].

The various types of wave propagation in heterogeneous materials are summarized in

Figure 4. Region 1 in Figure 4 corresponds to the quasistatic region. This imphes low

17
Floqoet-Blocli modes

Resonance associated

Willi periodicity Efleclive media

Resonances associated

Willi individoalscatterers

Elfeclive media

Classical mixing llieory

Freqoency

(or length scale)

Figure 4. Length scales in eflFective medium.

frequencies or, more importantly, frequencies where the wavelength is much larger then the

periodicity of the scatterers that compose the composite media. These scatterers may possess induced or permanent dipole moments, as is the case for atoms or molecules of classical

materials, or these scatterers could be generic in shape and placed in a host matrix to obtain

a man-made composite material having some desired property. Using asymptotic techniques

we can show that the electromagnetic field, in the low-frequency limit, behaves as if the

composite material is an equivalent effective medium with homogenous material properties.

The effective materials properties are derived from a quasistatic solution of the periodic

structure. The basic result is that the effective permittivity is obtained by taking a ratio of

some averaged displacement field to an averaged electric field. The effective permeability is

obtained by taking a ratio of some averaged induction field to an averaged magnetic field.

Region 2 in Figure 4 corresponds to a region where the scatterers are designed in such a manner such that the scatterers themselves can resonate. This occurs in NIM materials.

In Region 3 in Figure 4, we see that as the wavelength approaches the dimensions of the

inclusions, the fields no longer longer respond as in an effective medium. At these frequencies, a more complicated field behavior exists and more elaborate techniques to analyze the

electromagnetic field interaction with the composite periodic structures must be used; that

18
is, usually a full-wave approach. The classical approach that is used to analyze periodic

structures is the Floquet-Bloch mode approach, where the fields are expanded in an infinite

sum of plane waves.

3.2 Averaging

If we consider electromagnetic wave propagation from macroscopic to molecular to submolecular scales, the effective response at each level is related to different degrees of homogenization. The wavelengths of appHed fields can be comparable to particle size either for

molecules, at very high frequencies, or in macroscopic composites, when the inclusion size

is of the order of a wavelength. In microelectrodynamics, there have been many types of

ensemble and volumetric averaging methods used to define the macroscopic fields obtained

from the microscopic fields [12,22,28,29]. For example, in the most commonly used theory of

microelectromagnetics, materials are averaged at a molecular level to produce effective molecular dipole moments. The microscopic electromagnetic theories developed by and Jackson,

Mazur, Robinson [12,28,29] average multipoles at a molecular level and replace the molecular

multipoles, with averaged point multipoles usually located at the center-of-mass position.

This approach works well down to near the molecular level, but breaks down at the molecular to submolecular level. The next level is the averaging of molecular moments to produce

effective dipole moments on the supramolecular scale [26].

In the different theories, the homogenized fields are formed in different ways. The averaging is always volumetric and not over time. In a number of approaches, the volumetric

averaging is accomplished by convolving the unit step function with the fields. Jackson uses

a truncated averaging test function to proceed from microscale to the macroscale fields [12].

Robinson and Mazur use ensemble averaging [28, 29] and statistical mechanics. Ensemble

averaging assumes there is a probability that the system is in a specific state. In the volumetric averaging approach, the averaging function is never explicitly determined, but the

function is assumed to be such that the averaged quantities vary in a manner smooth enough

to allow a Taylor-series expansion to be performed. In the approach of Mazur, Robinson,

and Jackson [12, 28, 29] the charge density is expanded in a Taylor series and the multipole

19
moments are identified. These moments are calculated about each molecular center of mass

and are treated as point multipoles. However, this type of molecular averaging hmits the

scales of the theory to above the molecular level and limits the modeling of induced-dipole

molecular moments [16]. Usually the averaging approach uses a test function ft where

E=|dr'e(r-r')/t(r'). (26)

However, the distribution function ft is seldom exphcitly needed or determined in the analysis. In general, this distribution function must depend on the material properties since it is

the constitutive relations that determine ft. The averaging function must also be a function

of the level of homogenization used.

3.3 Constitutive Relations

In materials. Maxwell's equations are not complete until we specify the constitutive relationships between the macroscopic polarization, magnetization, and current density as functions

of the macroscopic electric and magnetic fields. The relationship of the polarization, magnetization, and current density to the applied and macroscopic electric, magnetic fields can be expressed as {P,M, J} <-> {Ea,Ha} <r^ {E,H} where subscript a denotes apphed. The

double-headed arrow in this relation indicates that the relationship could be nonlocal in time

and space and the constitutive relation may be a linear or nonlinear function of the driving

fields [30,31], and contain the contributions from both the electrical and the mechanical

properties such as stress-strain, as well as thermal contributions such as temperature. When

used in Maxwell's equations, the displacement field D, the induction field B, and current

density J must be expressed in terms of the macroscopic electromagnetic fields.

The fields and material-related quantities in Maxwell's equations must satisfy underlying

symmetries. For example, the dielectric polarization and electric fields are odd under parity

and even under time-reversal transformations. The magnetization and induction fields are

even under parity transformation and odd under time reversal. These symmetry relationships

place constraints on the nature of the allowed constitutive relationships and requires the

constitutive relations to manifest related symmetries [28,32-38]. The evolution equations

20
for the constitutive relationships need to be causal and in linear approximations must satisfy

time-invariance properties.

In any complex lossy system, there is conversion of energy from one form to another;

for example, electromagnetic to thermal energy through photon-phonon interactions. The

coupling of electromagnetic fields to phonons, that is, lattice vibrations, is through the

polariton quasiparticle. Magnetic coupling is mediated through magnons and spin waves. These effects are manifest in the constitutive relations and the resultant permittivity and

permeability.

Fields E and B have been well established as the fundamental electromagnetic fields,

which originate firom charge and spin. However, when free charge is present, there are both

free and bound currents, and we feel it is more instructive to deal with E and H as fields

that drive D and B. This approach separates the free-charge current density (J) fi:om the

bound-charge current density {dP/dt).

The macroscopic displacement and induction fields D and B are related to the macroscopic electric field E and magnetic fields: H, M, and P by

D = eoE-hP-V- Q+... = eoE + P (27)

and

B = //oH + //oM. (28)

In addition,

J = J(E,H), (29)

where J is a. function of the electric and magnetic fields, cq and /io are the permittivity

and permeability of vacuum, and Q is the macroscopic quadrupole moment density. P is

the dipole-moment density, whereas P is the effective macroscopic polarization that also

includes the macroscopic quadrupole-moment density [12,28,29,31,39]. The polarization

and magnetization for time-domain linear response are convolutions.

21
3.4 Local Electromagnetic Fields in Materials

In the literature, the effective local field is commonly modeled by the Lorentz field, which

is defined as the field in a cavity that is carved out of a material around a specific site,

but excludes the field of the observation dipole. A well-known example of the relationship

between the apphed, macroscopic, and local fields is given by an analysis of the Lorentz

spherical cavity in a static electric field. In this example, the apphed field, depolarization

field, and macroscopic field are related by

E = E, - -i-P. (30)

For a Lorentz sphere the local field is well known to be a sum of apphed, depolarization,

Lorentz, and atomic fields [24, 40]

Ez = Eq + EdcpoZ + ^Lorentz -f" ^atom- (31)

For cubic lattices in a sphere, the apphed field is related to the macroscopic field and polarization by

Ez = E + -i-P. (32)

In the case of a sphere, the macroscopic field equals the apphed field. Onsager [18] generalized

the Lorentz theory by distinguishing between the internal field that acts on induced dipoles

and the directing field that acts on permanent dipoles. Some of the essential problems

encountered in microscopic constitutive theory center around the local field. Note that

recent research indicates that the Lorentz local field does not always lead to the correct

polarizabihties in some materials [20]. Near interfaces we expect the Lorentz local field to

break down.

The field that polarizes a molecule is the local field p « oEi. In order to use this

expression in Maxwell's equations, the local field needs to be expressed in terms of the

macroscopic field E; = /3E, where /3 is some function. Calculation of this relationship is not

always simple. Since the local field is related to the macroscopic field, the polarizabihties,

permittivity, and permeabihty absorb parts of the local field; for example, p ^ a/?E. The

local field is composed of the macroscopic field and a material-related field, as in eq. (32).

22
Part of the local field is contributed by effects of additional parameters such as thermal

expansion and quantum effects. These additional degrees of freedom are contained in the

internal energy.

3.5 Effective Electrical Properties £ind Mixture Equations

Composite media consist of two or more materials combined in various ways. Usually, in

order to define an effective permittivity and permeability, the inclusions must be smaller

than the wavelength. Examples of mixing are periodic arrays of spherical particles, layered

media, or mixtures materials. The calculation of the effective material electrical properties

depends on the frequency of the application and the periodic or random state of the mixture.

Over the distance of a wavelength, the material may contain many inclusions or particles.

Various mixture equations for composites have been derived over the years for specific

symmetries and material properties. Equations for the effective permittivity or permeability of composites axe summarized in Appendix D. Lewin's mixing theory [41] is valid for

wavelengths on the order of the inclusion size. Metamaterial behavior can occur when the

wavelength approaches the inclusion size. At frequencies corresponding to one-half wavelength across the inclusion, resonances occur that can produce negative permittivity and

permeabihty.

3.6 Structures that Exhibit Effective

Negative Permittivity and Permeability

Evanescent or near fields originate from electromagnetic fields below cutoff near antennas or

in waveguides. The evanescent field behavior can be modeled exactly by solving the boundary

value problem. Structure-related evanescent field behavior can effectively be obtained from

a structure filled with a bulk negative permittivity since for negative permittivity, ujy/ejl

is pure imaginary and therefore the fields are damped. As a trivial example, consider a

waveguide below cutoff. This is an approximation since it uses an effective material property

to characterize elements in an electromagnetic boundary-value problem. However, this type

of analysis works on the scale of the waveguide. NIM electromagnetic material properties

23
are effective in that they are not separated from effects due the guiding structures.

The electromagnetic foundations of NIM materials are very simple. Consider a plane

wave where the Poynting vector is in direction k. If we take time and spatial transforms of

Maxwell's equations we get

k X E = a;/iH (33)

and

k X H = -weE. (34)

Therefore, for a material with positive permittivity and permeability, E and H form a righthanded coordinate system and the phase and group velocities are in the same direction,

consistent with the propagation direction of the Poynting vector. Now, if the permittivity

and permeability are effectively negative, then we have a left-handed coordinate system and

the group velocity is in the direction of k and phase velocity is in the opposite direction

(due to the negative square root for the index of refraction). However, the permittivity and

permeability cannot be negative for all frequencies since causality, modeled by the KramersKronig relations, would be violated. Anomalous behavior can occur over limited, dispersive,

bands as long as energy-momentum is conserved and causality is maintained.

Whereas the imaginary part of the permittivity must always be positive for energy conservation (e-^'^* sign convention), the real part of the effective permittivity can take on negative

values through various resoances. We must emphasize that in NIM the permittivity and

permeability are effective quantities and not intrinsic properties, which are negative at the

frequencies and regions near structure resonances. The most common way is when the complex current density in Maxwell's equations is combined with the permittivity to form an

effective permittivity e — a" — ja' . This approach has been studied in plasma motion in the

ionosphere. It also occurs in semiconductors in the milhmeter range, and in superconductors.

The Drude model in metals is a Lorentzian-based model of this type of effective negative

permittivity (see Figure 5) The permittivity for a plasma has a high-frequency behavior that

can be negative. Another way of producing an effective negative permittivity is with an array

of inclusions embedded in a media. The inclusions resonate at a specific frequency related to

their size [26]. Lorentzian-based models of a resonator can produce negative permittivity. In

24
0.02 0.04 0.06 0.08 O.IO k„a

Figure 5. A material where the permittivity and permeabihty are both simultaneously

negative (NIM) [26].

lumped jC — C circuit parameter models, the effective capacitance and inductance can also

become negative.

In periodic structures, the eigenvalue spectrum is well known to have propagating and

cutoff regions. This is analogous to cavity resonators, where there are propagating and

evanescent modes. Propagating cavity modes are analogous to the propagating modes in a

periodic structure whereas the cutoflF modes are analogous to the evanescent cavity modes

and the stop bands in models of semiconductors. When both effective permittivity and

permeability are negative simultaneously then propagation is again possible in —. The

contribution to the permittivity from microscopic dipoles should not be confused with the

effective permittivity that is a bulk effect and not an intrinsic characteristic of the material.

The polarization is usually modeled by a damped harmonic oscillator equation that in

the frequency domain yields eq. (11)- The simple harmonic oscillator equation for the

polarization P for single-pole relaxation is

d'^P dP 1_

+ 7- +

Co E(i),

df^ ' ' dt T T

inertial damping restoring driving

(35)

25
where 7 is a damping coefficient, r is a relaxation time, and cq is a constant. In the Debye

model the inertial term is absent. The eflFective permittivity has the form in eq. (11), which

can become negative.

Since no intrinsic magnetic charge exists, an approach to realize negative effective magnetic permeability has to be different from that for the negative effective permittivity. Here

again, we can consider intrinsic magnetic material properties or proceed with construction

of the electrical circuit configuration that would give an effective permeabihty containing a

negative real part in a certain frequency range. Thin metalHc ferromagnetic materials have a

large saturation magnetization that yield reasonable values of permeabilities in the gigahertz

frequency range.

The constitutive relation for the magnetization of materials is given by the LandauLifshitz equation of motion ^^^'^^ « - /io|7|M(r,0xH(r,t) - ^^M(r, t) x (M(r, t) x H(r, t)), (36)

precession '

;—

' damping

where H is the magnetic field a is a damping constant and 7 is the gyromagnetic ratio. This

equation models the intrinsic response of spins to applied magnetic fields. In NIM materials,

instead of obtaining the magnetization and permeability from eq. (36), effective properties

are identified for a resonant system modeled by eq. (12) [26]. The field averaging used in

NIM analysis is based on the magnetic field and material properties components along the

axes of a unit cell. In his analysis of NIM, Pendry averages the magnetic field in a cube [42],

For each magnetic field component, he gets

<H>i=- f B. dr. (37) a Jn

The induction field is averaged as

< B >i= 4 / B • dS, (38)

where i takes on x, y, and z. Following these averaging definitions, the effective relative

permeability is then defined as: <B>,

flQ < ti>i

26
In order to obtain a negative permeability in NIM applications, the circuit has to be

resonant, which requires the introduction of capacitance into the inductive system. Pendry

[42-44] introduced the capacitance through gaps in coupled-ring resonators. The details

of the calculation of effective permeability are discussed in Reference [42]. In principle,

any microwave resonant device passive and/or active can be used as a source of effective

permeability in the periodic structure designed for NIM apphcations [45].

We should note that the composite materials developed by Pendry have a scalar permeability. However, in general, the permeability is tensorial.

4. Instrumentation, Specimen Holders, and Specimen

Preparation

4.1 Network Analyzers

Automatic network analyzers (ANAs) have become the preferred data acquisition system for

many researchers. When making material measurements we need to understand the errors

associated with the scattering parameters measured by the network analyzers. Network

analyzer systems have various error sources. These include [46]:

• Imperfect matching at coimectors

• Imperfect cahbration standards

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

analog to digital converter

• Imperfect tracking in dual-channel systems

Generally, the manufacturer furnishes specifications for its measuring system. The choice of

network analyzer is crucial for accurate phase data over the frequency band of interest.

27
4.2 ANA Calibration

4.2.1 Coaxial Line Calibration

Various components of the ANA introduce phase and magnitude uncertainties. CaUbration

of the ANA removes the systematic uncertainties through measurements of a set of standards;

for example, shielded open-circuited termination, short-circuited termination, or load. Information on the difference between a specific measurement of these standards and the expected

values that are stored in the ANA system generates a set of error-correction coefficients. The

calibration coeflacients are determined by solving a set of simultaneous equations generated

from the linear fractional transformation. After calibration, when the system is operated

with the errors correction, the measurements are updated by the cahbration information.

The 7 mm line calibration kit contains the following standards:

• Open-circuited termination

• Short-circuited termination

• Low-and high-frequency loads

• Sliding load

4.2.2 The Waveguide Calibration Kit

For calibration of waveguides we can construct a calibration kit for the ANA. The transmissionthrough-hne (TRL) calibration consists of measuring a through^ a reflect, and a section of

line. The length of line to be used as the through is calculated as follows: the phase delay

{(f)) in waveguide is related to line length {£) and guided wavelength (A^) by

. = ^, (40)

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

A, = , ^ (41)

A procedure for calculating line length is to calculate £ 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,

28
and then choose a Hne length between these extreme values; typically, A/4 at a geometric

center frequency {/center = Vfminfmax)-

When using waveguide for measurements we can insert sections of waveguide, each approximately two wavelengths in length, between the coax-to-waveguide adapter and the

specimen holder. The function of the waveguide sections is to damp out any evanescent

modes generated at the coaxial-line to waveguide adapters. The X-band calibration kit

contains the following items:

• Coax to waveguide adapters

• Short-circuited termination

• A section of waveguide line to be used as a specimen holder

• A section of calibration line

• Two lengths of X-band waveguide approximately two wavelengths long acting as mode

filters

• load

• Calibration coefficients provided on an appropriate computer disk

4.2.3 On-Wafer Calibration, Measurement,

and Measurement Verification

Work at NIST led to the development of accurate multiline through-refiect-line calibrations

for on-wafer measurement. The method uses multiple transmission-line measurements to

improve both the accuracy and bandwidth of the calibration over that of conventional TRL

techniques [7]. The multiline method is implemented in the NIST MultiCal software suite. The software

also determines the complex frequency-dependent characteristic impedance of the transmission lines used in the calibration and allows the user to set the calibration reference

impedance to the characteristic impedance of the hne, tjrpically 50 D,, or to any other real

value. The calibration comparison method assesses the accuracy of a working caUbration

29
by comparing it to an accurate reference calibration, usually the multiline TRL calibration.

This method has led to an understanding of the systematic uncertainties present in many

common on-wafer calibrations. The calibration comparison method has also proved to be a

valuable tool for the development of working calibrations with an accuracy comparable to

the multiline TRL calibration [7,47].

4.3 Specimen-Holder Specifications

4.3.1 Specimen Holder

The specimen holder for transmission-reflection (TR) measm-ements should consist of highprecision waveguide or coaxial line. There should be a length of waveguide between the

waveguide-to-coaxial line adapter and specimen holder to damp out evanescent waves generated at these discontinuities. The length of the specimen holder should be measured with

a high degree of precision. Nicks and other abrasions in the specimen holder will, of coiuse,

degrade performance by increasing the surface resistance. When a 7 mm coaxial headless air

line is used, APC-7 connectors are usually preferred. The specimen holder should be treated

with extreme care and stored in a protected area.

»(

Figure 6. Cross section of a coaxial line and specimen.

30
bA~^-^—:-—~^^^ J:^. -

.

Figure 7. Cross section of a waveguide with specimen.

The real part of the surface impedance determines the loss in the hne and is given by

Z,=

27r +

b{^dcS)oc a{adc5)i

(42)

where b is the inner radius of the outer conductor, a is the outer radius of the inner conductor,

adc is the conductivity, 6 — l/y/irf/j,, andudc is the skin depth [48].

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

of approximately ±2 //m, varies slightly with frequency from 50.25 1^ at 0.1 GHz to 49.95 Q,

at 20 GHz [48].

4.3.2 Specimen Preparation

Specimens intended for measurements must be carefully, prepared. Scratches, nicks and

cracks may alter the measured dielectric properties. Minimize any unnecessary wear and

tear on the specimen by placing it in a secure area between measurement sessions. The

specimen length measurement is critical and should be performed carefully with a precision

micrometer at laboratory temperatures. Dimensions for 7 mm line and X-band waveguide

specimens are given in Figures 8 and 9. The following list summarizes the preparation procedure:

31
6.95 mm 1 3.04 mm

Figure 8. Machining details for a 7 mm coaxial-line specimen. Uncertainties in dimensions

approximately ±0.02 mm.

• Carefully select a piece of material free of unusual inhomogeneities or cracks.

• Have the specimen machined to fit as tightly as possible into the specimen holder. The

machining process should not leave metallic residue on the specimen. Note that gaps

near the center conductor of a coaxial line are more serious than gaps near the outer

conductor (by a factor of 2.3). Specimens that fit very tightly on the outer conductor

can be inserted more easily by prior cooling.

• Measure the specimen length with a high degree of precision at a temperature very

close to that realized in the laboratory. The resulting strain, AL/L, from increased

temperature can be calculated from the linear thermal expansion coefficient ar by

using the relation AL/L = arAT.

• Keep the specimen very clean and store in a secure area with required humidity. If the

specimen requires cleaning, an ultrasonic cleaner will suffice and be careful to avoid

the effects of absorbed water. Alcohol usually works better than distilled water.

• Keep the gap between specimen and guide walls to a minimum. We have found that

clearances of 2-7 /xm (0.0001 to 0.0003 in.) are acceptable for low-permittivity materials. However, even with these tolerances the measurement of high-permittivity

32
22.80 mm ± 0.05 mm

10.09 mm ±0.05 mm

Figure 9. Dimensions of a waveguide specimen. Uncertainties in dimensions approximately

±0.02 mm.

materials is limited. For better results the specimen faces should have copper or another conductor deposited on them, or the specimen should be soldered into the line.

5. Overview of Measurement Methods for Solids

Each dielectric measurement method has its frequency range of applicability. Each method

also utilizes a specific electric-field orientation. Since there is such variability in measure- ment fixtures, frequencies of interest, field orientation, and temperature dependence, we will

give a broad-brush overview here of many of the important features of the most important

measurement techniques. These features are summarized in Tables 3, 4, and 6. The full characterization of anisotropic materials generally requires two measurement

techniques, one for the component of permittivity normal to the specimen face, and one for

the in-plane permittivity. The loss tangent measurement is not usually as strongly affected

as e^ by anisotropy, and a single measurement of loss tangent usually suffices. For example,

resonant transmission-line methods can be used for e^ and the loss could be obtained by more

accurate in-plane techniques such as a TEoi resonator (for modes see Figure 5). However,

there axe materials where the loss is significantly anisotropic, and therefore two independent

methods for loss must be applied.

Measurement of magnetic materials requires a strong applied magnetic field. Magnetic

33
Table 3. Dielectric measurement categories for medium-high-loss materials compared with

typical uncertainties for each method [49].

Technique Field Advantages A6;/4, A tan Sd

Coaxial line, waveguide TEM,TEio Broadband ±1 to 10 ±0.005

Slot in waveguide TEio Broadband ±1 to 10 ±0.005

Capacitor Normal E-field Low frequency ±1 ±10-4

Cavity TEoi Very accurate ±0.2% ±5 X 10-5

Cavity TMo„ 4. ±0.5 ±5 X 10-4

Dielectric resonator TEoi Very accurate ±0.2% ±1 X 10-5

Coaxial Probe TEM, TMoi Nondestructive ±2 to 10 ±0.02

Fabry-Perot TEM High frequency ±2 ±0.0005

Table 4. Magnetic measurement methods compared with typical uncertainties.

Technique Field Advantages A/i^K, % A tan 5d

Coaxial line, waveguide TEM,TEio Broadband ±2 ±0.01

or waveguide

Cavity TEoii Very accurate ±0.5 ±5 X 10-4

Cavity TMiio /^;^ ±0.5 ±5 X 10-4

Dielectric resonator TEoii Very accurate ±0.5 ±1 X 10-5

Whispering-gallery Hybrid Very accurate ±1 ±5 X 10-6

Courtney TEoi Very accurate ±1 ±5 X 10-5

materials can be measured in coaxial lines and waveguides, split-post magnetic resonators,

TMiio [50] or TEqh cavities, whispering-gallery modes, or other dielectric resonators (see

Table 4).

Over the years certain methods have been identified as particularly good for various

classes of measurements, and these have been incorporated as standards of the ASTM (American Society for Testing and Materials) [51] and the European Committee for Electrotechnical

Standardization (CENELEC). However, the list is rather dated and doe not include some

of the more highly-accurate, recently developed methods. The ASTM standard techniques

applicable to thin materials are summarized in Table 6. Measurement fixtures where the electromagnetic fields are tangential to the air-material

interfaces, such as in TEoi cavities and dielectric resonators, generally yield more accurate

results than fixtures where the fields are normal to the interface. This is because the tan34
Table 5. Cavity Fields

Cylindrical TEqi Cavity E<^ Hp, B.Z Coaxial TEM Line and Cavity Ep H<^ Cylindrical TMqi Cavity Ep, Ez H^

Rectangular Waveguide TEio Ey Hi, B.Z

Table 6. ASTM standard techniques for dielectric measurements.

ASTM No. Applicability Method Frequency

D150 discs capacitor 1 Hz to 10 MHz

D1531 thin sheeting bridge network 1 kHz to 1 MHz

D1673-94 polymers capacitor 10 kHz to 100 MHz

D2149 discs capacitor 50 Hz to 10 MHz

D2520 small rods rectangular resonator 1 GHz to 50 GHz

D3380 clad substrates stripline 8 GHz to 12 GHz

D5568 cylindrical specimens co£Lxial line 1 MHz to 20 GHz

gential electric field vanishes on a conductor, so any air gap will be in a low-field region. In

contrast, if the field is normal to the electrode, the air gap is in a region of strong electric field.

In fact the electric field is more intense in the air gap region (Ea, Ca) than in the specimen

region {Eg, Cs), since the continuity of the normal component of the displacement field yields:

Ea = CsEs/ca- If the electric field is tangential to a gap between a material and a metafile

wall of a measurement fixture, then the measurement of e^ is minimaUy infiuenced by the

gap and no gap correction is usually required. Unfortunately, for many applications, in-plane

field measurements are not the preferred one. For example, circuit boards and PWBs using

microstrip or stripline, operate with the electric field primarily normal to the plane of the

sheet and, therefore, this component of the permittivity is of primary interest. However,

measurements with the electric field perpendicular to the specimen face suffer from air-gap

depolarization. In such cases, the effects of the air gap must be either accepted or mitigated

by metalization of specimen surfaces or application of conductive pastes, or corrected for by

numerical techniques [3,4]. The gap correction is due to the capacitance of air in the gap

and effectively depends on frequency only through the dependence of e^ in the gap-correction

35
«'r

/7\

ffigh-loss material V.,^^^^^

Vv

«v/ v^___

C'r

—11. Low-loss material « r

Frequency

Figure 10. Typical frequency dependence in low loss and Debye materials.

formula. TMon resonators are particularly good for measuring the permittivity along the

axis of a rod specimen (see ASTM D2520).

The real and imaginary parts of the permittivity, excluding the contribution of the dc

conductivity, are related by the Kramers-Kronig dispersion relations [52]. This relationship

is a requirement of causahty that the fields are turned on at some time and are not static

fields. Kramer-Kronig relations are an integral relationship between the real and imaginary

parts of the permittivity. A consequence of these relations is that frequency-dependent

changes in the real and imaginary parts of the permittivity are correlated (see, for example,

the correlation in Figure 10). One example of a Kramers-Kronig relation is

e>o) = ^^

+ -Pr[<M-l]-^^^- (43)

cqujq it Jo uJq— w^

Although eq. (43) is exact, it requires very broad frequency information and is relatively comphcated to use. A few simple consequences of Kramers-Kronig are the following. The static

value of the permittivity, for a material with no dc loss, is e^ = 1 + {2/'k)P J^{e"{u)lij)du.

36
Frequency

Figure 11. Typical permittivity variation for polar materials.

If there is no dc conductivity, e"(0) = 0. Lynch [53] developed a simplified version of this

relationship that relates changes in permittivity between frequency points to the expected

loss tangent. His analysis allows a consistency check on measurements in the radio-frequency

to low microwave bands for measurements where the loss tangent changes slowly with fre- quency and the loss contains no appreciable contribution from the dc conductivity. If we

have two measurements of the permittivity at different frequencies, then

« m tan 6 log^o ( T (44)

In this equation, 1.0 < m < 2.3, and Lynch determined m = 1.5 is usually optimal.

In the radio-frequency and microwave bands, e^ is almost always a monotonically decreasing function of increasing frequency (see Figures 11 and 10). The permittivity of low-loss

materials decreases slower (dispersion) with frequency than high-loss materials. In the radiofrequency and low microwave bands, any increase in measured e^ as the frequency increases

is almost always due to a fixture geometrical resonance or other fixture-dependent artifacts.

On the other hand, e" does not necessarily decrease as frequency increases. It may increase as

frequency increases or form a Debye loss peak. The loss tangent in the radio-frequency and

microwaves bands of most ceramics, fused silica, some plastics and glasses, increases nearly

37
6x10

I ' I I I I I I I I I I I I

,

• 14 GHz Split-Post ResonSor

' • 2.0GHzS[^it-PostResonaor

» 3.3GHzSpiit-Po5tResonator • 5 GHz Split-Post Resonator » lOGHzSjiit-CsdinderResonaor

» 10 GHzParallel-Plie Resonator

• 10 GHz Cylindrical Canty

• 35 GHz Split-Qrtinder Resonator

I I I I I I I I I

I I I I I I I I I I I I I I ' I 10 15 20 25 30 35 40 45 50

Frequency (GHz)

Figure 12. The loss tangent of fused silica for various fixtures operating at different frequencies. Notice a nearly linear dependence on frequency

linearly as frequency increases. In such cases, we can extrapolate loss-tajigent measurement

data from one frequency range to another (see Figure 12). Materials of very low-loss have

loss tangents and permittivity in the microwave spectrum that are nearly frequency independent (see, for example. Figure 13)). This is a direct consequence of Kramers-Kronig relations

since the changes in the loss and the real part of the permittivity are strongly correlated

over a band of frequency. In materials with a finite dc conductivity, e" increases at low

frequencies as 1/uj as frequency decreases, where / = 27rct; is frequency. Conducting materials are difficult to measure at low frequencies (/ < 1 MHz). This is because the boundary

layer between the specimen and electrode produces an electrode-polarization double-layer

capacitance that biases the measurement result.

There are many mathematical models used for extracting permittivity and permeabihty.

38
a

4.0

3.9

3.8

3.6

; • 1.4GHz^lit-PostResonior

• » 2.0 GHz Split-Post Resonator • 3.3 GHz SpUt-Post Resonator • 5GEzSplit-PostReson2tor • 10 GHz Si^it-CylmdsT Resonator » 10 GHzParallel-Plale Resonator • lOGHzCylinchcalCawty • 35GHzSplit-CyimderResonior •«. • •4.. » •

» •

'

*

* »

10 15 20 25 30 35 40 45 50

Frequency (GHz)

Figure 13. The permittivity of fused silica for various fixtures operating at different frequencies.

Methods based on Maxwell's equations use single- or full-mode models that relate the resonant frequency or transmission and reflection coefficients to functions of the permittivity

(see Table 7). These types of models are accurate, but require complicated solutions to

Maxwell's equations and can be computationally intensive.

At lower frequencies, dielectric and magnetic material properties can be extracted from

a transmission-fine model using the impedance Zc and propagation coefl&cient 7 (see Figure

14). For cases where both dielectric and magnetic properties are required we must use both

the impedance and propagation coeflacient simultaneously [4, 54]

.

For dielectrics, either the impedance or propagation coefficient may be used in isolation

for determining the permittivity. In both cases, a model of the transmission-line structure is

required that relates the material properties to the reflection and transmission coeflacients.

39
Table 7. Models used for the permittivity extraction.

Model Advantage Disadvantage

Lumped element simple limited to low frequency

Distributed element simple medium-low frequency

Single-mode based on Maxwell's eqs cannot handle discontinuities

Full-mode based on Maxwell's eqs can handle discontinuities

Portl

Figure 14. Waveguide transmission line with specimen

The models may be based on lumped circuits or distributed elements. For distributedelement models, we usually use an approximation to Maxwell's equations in terms of current

and voltage waves on the transmission line.

Surface roughness of the conductor increases the capacitance, resistance, and inductance.

The capacitance is increased because the effective thickness of the rough material between

two conductors is less than the measured thickness [55] . The roughness of the surface of the

conductors can also affect the resistance due to the increased surface area.

6. Transmission-Line Techniques

Transmission-line methods can be broadly grouped into categories where (a) the cross section

of specimen and holder is constant and (b) fixtures where the cross section of the specimen

holder or specimen contains discontinuities. The first category is easier to treat since the

electromagnetic modes decouple. Usually a dominant mode analysis is sufficient for these

types of problems. The second category requires a full-modal solution due to mode generation at interfaces. Fixtures without discontinuities include coaxial and waveguide two-port

40
systems. Examples of fixtm^es with discontinuities include shielded coaxial line, open-ended

waveguide, and open-ended coaxial probes.

In many cases, transmission-line techniques require specimen preparation such as machining or removal of copper cladding. In the case of printed-wiring boards the copper cladding

can be etched to form the specimen into a microstrip for measurements. Bulk specimens

must be machined to fit tightly into a waveguide or coaxial line.

Transmission-Une measurements use various terminations that produce different resonant

behavior in the transmission hne. Maximal electric field is required for good permittivity

measurements. This can be achieved by an open-circuited or other capacitive termination.

Magnetic permeability measurements should be carried out in regions of high magnetic field.

This can be achieved by a short-circuited or other inductive termination.

6.1 Coaxial Line, Rectangular, and Cylindrical Waveguides

6.1.1 Overview of Waveguides Used in Transmission/Reflection

Dielectric Measurements

Due to their relative simpHcity, the off-resonance waveguide and coaxial hne transmission

or reflection and short-circuit Hne methods are widely used broadband measurement techniques [3,54,56]. In these methods, a precisely machined specimen is placed in a section of

waveguide or coaxial line, and the scattering parameters are measured, preferably by an automatic network analyzer. The relevant scattering equations relate the measured scattering

parameters to the permittivity and permeability of the material. Network analyzers have improved over the years to a point where broad frequency coverage and accurate measurement

of scattering parameters are possible.

Transmission-line measurements usually axe made in rectangular or circular waveguide

or coaxial lines. The three major problems encountered in transmission hne measurements

are air gaps between the sample and fixture electrodes, half-wavelength resonances, and

overmoding.

Coaxial lines are broadband in the TEM mode and therefore are attractive for permittivity and permeabihty measurements of lossy materials. Coaxial-hne measurements can be

41
made using short-circuited, open-circuited, or matched-load terminations. The short-circuit

hne (SCL) method was used by Roberts and von Hippel [57] over 50 years ago as an accurate

broadband measurement procedure. Shielded open-circuited line (SOCL) methods are used

for measurements of powders and liquids. Full-scattering matched-load methods, where the

transmission line is open on both ends, are usually used to measure solids.

Coaxial lines can also be used as permeameters for measurements of the magnetic properties of ferrites. In this approach, measurements are obtained by placing a specimen in

a coaxial line terminated in a short-circuit and measuring the change in inductance and

resistance using an impedance analyzer. This procedure is summarized in Reference [58].

The permeability of metals can be determined by winding around a metal toroid of the

material under test with a fine wire and measuring the change in inductance compared to

a measurement in free space. Open-circuited terminations are useful for dielectric one-port

measurements of hquids and powders.

Corrections for the effects of air gaps between the specimen holder and the sample can be

made by analytical formulas or mitigated by use of conducting pastes or solder that is applied

to the external surfaces of the specimen before insertion into the specimen holder [54, 59]

.

The field model for closed waveguide structures usually assumes a single mode of propagation in the specimen. Propagation of higher-order modes becomes possible in inhomogeneous specimens of higher permittivity. Air gaps play an important role in mode conversion. Higher-order modes require a coupling mechanism in order to begin propagating. In

waveguide and coaxial line the asymmetry of the specimen and machining precision promotes

higher-order mode propagation. Generally, the appearance of higher-order modes manifest

themselves as sudden dips in the reflection coefficient, which is a result of resonance of

the excited higher-order TM, or TE modes. The TEM mode will also resonate at integral

multiples of one half wavelength in the sample. The single-mode model, developed in the

next section, can handle the higher-order TEM resonances in dielectrics, but not the TE

or TM resonances. We can expect point-by-point numerical reduction techniques to break

down for single mode models near higher-order mode resonances. These modes propagate in

materials of high-permittivity or inhomogeneous specimens [56] . Optimized, multifrequency

42
Table 8. Rectangular waveguide dimensions and operating frequencies in air. EIAWR Band a (cm) b (cm) TEio Operating Prequency(GHz)

650 L 16.510 8.255 1.12- 1.70

430 W 10.922 5.461 1.70 - 2.60

284 S 7.710 3.403 2.60 - 3.95

187 c 4.754 2.214 3.95 - 5.85

90 X 2.286 1.016 8.20 - 12.4

42 K 1.067 0.432 18.0 - 26.5

22 Q 0.569 0.284 33.0 - 50.0

solutions fare better in this respect [60] since these frequencies can be given less weight. In

order to minimize the effects of higher-order modes, shorter specimens can be used to maintain the electrical length less than one-half the guided wavelength in the material. Also,

well-machined specimens produce fewer spurious modes.

Effects due to unwanted higher-order modes can be minimized by mode filters. This can be particularly helpful in cylindrical waveguides. One way to construct a mode filter is to

helically wind a fine wire about the inner surface of the waveguide specimen holder, thus

eliminating longitudinal currents. Another approach is to insert cuts in the waveguide walls

to minimize axial currents along the waveguide. These types of mode filters do not work for

TEM-mode structures.

6.1.2 sections of Waveguides Used as Specimen Holders

TE and TM waveguides can be only used over a band of frequencies. Therefore the selection

of a waveguide depends on the frequencies to be measured. The cutoff frequency is a lower

bound on frequencies that can be transmitted in the guide without the fundamental mode

becoming evanescent. At cutoff, the long transverse dimension of the rectangular waveguide

encompasses one half wavelength, which is a resonant condition. Whereas coaxial lines have

no cutoff, at low frequencies the effects of skin depth can affect the measurement of the TEM

mode.

A list of operational frequencies for various waveguide sizes are given in Table 8 and for

various coaxial line sizes in Table 9. The cutoff frequency for either TE or TM waves in

43
Table 9. Air-filled coaxial cable operating frequencies.

Coaxial cable outer diameter (mm) Upper operating frequency (GHz)

3.5

7.0

14.0

34.5

18.2

8.6

rectangular waveguide is given by

(45)

where a is the long dimension of the guide, b is the short dimension of the guide, and

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

1 c yjcjmn — (46)

2y/f7Fa 2a'

Rectangular waveguide operated in the TEio mode becomes overmoded at twice the cutofT

frequency. Operating frequencies and waveguide dimensions axe given in Tables 8 and 9.

For a circular waveguide of radius a the attenuation is given by [5]

Rs 1 OiTMmn =

a7y^l_(^)2'

R.o

OiTEr,

m

where p'^^ are the roots of J'^ip'run) = and r/ = ^Jflo/eo.

The cutoff frequencies for TMmn in circular waveguide are given by

(47)

(48)

\Tc)mn

Pr. y/eJl2'Ka

where p^n are the nth roots of Jm{Pmn) = 0. For TEmn waves we have

(/c),

Pr

(49)

(50)

y/eji2'Ka

where p'^^ are the nth roots of Jmip'mn) — 0- Coaxial line has the distinct advantage of

having no cutoff frequency. However, coaxial line becomes multimoded above the cutoff

44
frequencies of the TE and TM modes (for a review of cutoff frequencies in TEM structures

see Weil [61]). Higher-modes can be formed by bead inhomogeneities, discontinuities, and

mismatches that couple to the TEn and higher modes. The approximate upper frequency

limit is given in Table 9. The propagation of the TE and TM modes 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 [5,61],

JnikcO) Jn{kcb)

and for TE waves in coaxial line by

Kik^a) iV;(/c,6)

= 0, (51)

0, (52)

Ji,{k,a) J^{k,b)

where Jn and iV„ denote the Bessel functions of the first and second kind, and a and b are

the inner and outer radii, respectively [5] . The cutoff wavelengths are given approximately

by

2

Ac w -(6-a) , ^ = 1,2,3,---. (53)

For example, the TM mode cutoff frequency in 7 mm coaxial line for eq. (53) is approximately

34 GHz. For beadless air line it is possible to exceed the frequencies given in Table 9.

6.2 Slots in Waveguide

A useful method for both low- and high-loss substrates is the slot-in-a-waveguide technique

where the material is inserted as shown in Figure 15. The material is parallel to the TEio

electric field; therefore the effects of air gaps are minimal. The slots are in the center of the

wide wall of the waveguide. The sample is inserted into the slots and protrudes out both

sides. To minimize radiation from the slot, the slot should be kept as narrow as possible [62].

6.3 Microstrip, Striplines, and Coplanar Waveguide

In the previous section we considered closed transmission lines. However, in many appHcations the transmission lines are planar with open surfaces. For example, conductors may be

45
Figure 15. Measurement of substrates using a slot in a waveguide.

deposited directly on thin films or substrates to form transmission lines. Open structures can

radiate and have a comphcated field structure. Measuring the permittivity of lossy materials, circuit boards, thin films, and substrates nondestructively is frequently of interest [63].

These materials may be thin and may be clad in copper, making measurements with closed

transmission lines difficult. Generally, an effective permittivity is defined by relating the

measured capacitance with and without air filhng. Therefore Ce// depends on the fields in

the specimen and the fringing fields. Since the structures axe open, approximate permittivity

models are usually used. The effective permittivity eg// maps complicated field structure

into a single parameter. If the effective permittivity is used in isolation then it has httle

fundamental significance. However, if effective permittivity is coupled to more extensive fixture models, as Getsinger and Wheeler have done [64,65], then it can be used to obtain the

permittivity [66-68]. Loss is usually difficult to measure in transmission lines and the cases

of microstrip, coplanar waveguide, and stripline are no exceptions. Resonant microstrips or

striplines allow a more accurate calculation of the loss tangent than do nonresonant methods.

The permittivity is obtained from the shift in resonant frequency. In microstrip as depicted in Figure 16, the total conductor loss has contributions from both the strip conductor

and the ground plane. For narrow microstrip, the ground plane loss is a small part of the

total loss, whereas for wide microstrip, the ground-plane loss becomes more important, see

References [69] and [70]. In fact, for w/h » 1, the ground-plane loss nearly equals the strip

46
Figure 16. Cross section of a microstrip.

loss. The conductor loss can be obtained from [71]

Q!r =

Rs '°°

J —oo dx

2ZJ- ;-»

—' - ' (5^'

where Jsr is the current density on the strip given in reference [69] , and Rs is the Leontovich

surface resistance [72] and is given by eq. (7). This integral can be evaluated to yield

Oigr —

Rs ( w h f ^ f w\^

-— < arctan — m 1 + — w-kZq \ 2h w \ \2hJ ^ Thus, the total conductor loss is given by

(^Tlmicro) ^strip "r OLqt-

(55)

(56)

For a microstrip line, the attenuation constant associated with the strip conductor is expressed as

Oistrin i?,^ln(ff-l)

(57)

where w is the strip width, t is the strip thickness, and A is the hmit for the integrations,

as defined in Reference [69].

Reference [73] gives values of A for superconductors. Rgm is the modified Horton impedance boundary condition as discussed in [69] and given by

'cot [kct] + CSC {kct)

Rsm = ^A*c^ I"^

Kct (58)

where kc is the wavenumber in the conductor. Zo is the characteristic impedance given

by [74]

32/i

Zo =

47rVer + l \n{l +

w

h

w

fhr^' Me. + l) y \\w) \

16e, )

(59)

47
where h is the height of the substrate, 770 is the impedance of free space and

L .nro 0-5160 0.0788

r. = W0.4052 + \ . (60)

Copper

2h

.#;,

i i J

, t \Sample

'-Y H

*r.-E

Figure 17. Cross section of a covered stripUne [75].

As with the microstrip, the hne loss for a stripHne consists of components from both the

strip and its ground planes. Holloway has calculated the attenuation constant for stripline

[69,70], which is

Ol'T {stripline) — Oistrip > ^groundi V^^)

where

Rf. w

(^fitrin ^strip In(x-l). (62) 27r2Zo^"^^A

Once again, A is the stopping distance and is given in Reference [69,70]. Rsm is given by

eq. (58) and Zq is the impedance

Zo =

16e K{k') (63)

where K is the elliptical integral of the first kind, k = sech(7r7i;/4/i), k' = y/1 — A;^, w is

the width of strip, and h is the distance between the inner and outer conductors. The

ground-plane losses are

"="-"" = Zot^ (f)

' ^®^'

where Rs = l/crdcS is the surface impedance, ddc is the metal conductivity, S is the conductor

skin depth, and

.w^ 2h

h TT^W 7-00 •- -' (65)

48
H

^-E

-t^

J J Copper

-' y ^^^^ Sample

Figure 18. Coplanar slot cross section.

In coplanar slot waveguide (CPS), as shown in Figure 18, the characteristic impedance for

a thin stripline is given by [70]

Zo =

2/io K(k)

eo{er + l)K{Vl^^) (66)

The attenuation is [70]:

p u2 O^CPS

1 2a + A6-Q- A 1 2b- Ab-a- A

AZoK^k^b"^ - o?) [a ^^ A b + a-A'^b ^^ A b + a + A'

(67)

The variation of the attenuation with frequency for stripline, microstrip, CPS, and CPW

hues is given in reference [9]. The attenuation in coplanar waveguide (CPW) (see Figure

Figure 19. Coplanar waveguide fixture cross section.

(19)) was calculated by Holloway [73,76]

OiCPW ~ R^

F

i+yrr AF (68)

49
where

with

1 /2a b-Q\ 1 f2h h-a

a ^ V A 6 + a/ 6 ^ l A6 + a

(69)

7mo = ju^/qlox^:—, (70)

2 _ riI^K{k') .

(72)

(73)

(74)

Gdc is the conductor's conductivity, and Cc is the conductor's permittivity that is small in

comparison to a and can be neglected. Also,

W/- 2 '

Zs = T / ^° rn^/'/" f^

Zm = iW . ; CSC(M).

A;' = VT^^^. (75)

6.4 Ground-Penetrating Radar

Specialized antenna systems have been used to detect buried objects. The use of the reflection

coeflacient allows the dielectric measurement of buried material and possibly the reconstruction of the geometry of buried objects such as archaeological artifacts, pipes, conduit, or

land mines [77-83].

6.5 Free-Space Measurements

Free-space measurements are commonly performed using antennas, as shown in Figures

20 and 21. The Styrofoam pylons have a permittivity close to that of freespace, roughly

1.05. Multiple reflections from the room walls are commonly suppressed by use of absorbing

tiles. Free-space measurements, depending on the type of antenna, generally operate in the

band from 50 MHz to 30 GHz. In these methods the TEM mode transmission and reflection

coefficients are measured using algorithms similar to those used in closed transmission lines to

obtain the material properties [84-87]. The calibration method used is generally a variation

50
Reference Measurement

Sample Measurement

Figure 20. Free-space measurement setup of a planar sample.

of through-reflect-line (TRL). The short-circuited termination is a metal plate. Although

the wave front is spherical, Muscil and Zacek [87] suggest that the electromagnetic fields

can be assumed to be plane waves in the antenna far field. This assumption simplifies

the inversion algorithm. The measurement may involve reflection-only data using a short

circuit termination or reflection and transmission data using a specimen open on both sides.

For electrically thin specimens a short-circuit termination is not useful since the tangential

electric field component approaches on the metal. In the case of electrically thin materials, a

transmission measurement is preferred since both ^n and S21 can be used [4]. The technique

lends itself well to elevated temperature measurements. In Figure 22 we show time-domain

data collected on a sheet of Rexolite material.

7. CoELxial Line and Waveguide Measurement

Algorithms for Permittivity and Permeability

The goal of this section is to present various approaches for obtaining both the permeabiUty

and permittivity from transmission-line scattering data. We will also overview transmission51
Figure 21. Picture of horn system for free-space measurement of a planar sample.

line theory for NIM materials. In the transmission/reflection (TR) measurement, a specimen

is inserted into either a waveguide or a coaxial line, and the specimen is subjected to an

incident electromagnetic field [see Figure 14]. The scattering equations are determined by

analyzing the electric field at the specimen interfaces. In order to determine the material

properties from scattering data, we must solve the electromagnetic boundary-value problem

in waveguides and coaxial lines. In developing the scattering equations, only the fundamental

waveguide mode (TEio in waveguide and TEM in coaxial line) is assumed to propagate.

Various transmission-Une techniques are compared in Table 10.

7.1 Specimen Geometry and Modal Expansions

In the case of perfectly homogeneous and isotropic specimens and holders, 7 is independent

of the transverse coordinates. Therefore, the eigenfunctions for the transverse components

in the air and specimen regions are orthogonal. In such cases, we can match fields, mode

by mode, and the modal coefficients decouple. However, when the specimens and specimen

holder are not perfectly formed or are slightly inhomogeneous, both ^i and e depend weakly on

52
0.02 -I

0.01 -

a.E

•0.01 -

-0,02

Time domain signature of rexolite sample Time domain signature of refe re nee

Possible edge effects

P"liy "I 1§ lH|~fc|j*~~

8 12 Time (ns)

Figure 22. Time-domain data on a Rexolite slab.

the transverse coordinates of the guide and therefore the different transverse eigenfunctions

in the specimen are not completely orthogonal to the transverse eigenfunctions in the air

section. In such cases, the modes of imperfect specimens cannot be exactly separated and

matched mode by mode. The imperfections in the specimen generate evanescent waves at the

specimen-material interface. These modes may propagate in the specimen and are noticeable

when they resonate, but they decay exponentially outside of the specimen. However, minor

imperfections can be treated by an effective single-mode approximation we can correct for

air gaps.

The fields in Regions I, II, and III of Figure 14 are found by analyzing of the electric

field at the specimen interfaces. We assume that the incident electric field is the TEio

mode in rectangular waveguide and the TEM mode in coaxial line. If the specimen has

air gaps near the electrodes or is inhomogeneous, then some of the energy carried in the

wave will be converted into higher-order modes. In the following we assume that gaps or

other imperfections can exist in and around the specimen. We further assume that the

53
Table 10. Dielectric and magnetic transmission/reflection (TR) measurement techniques for

lossy materials.

Technique Apphcability Advantages Disadvantages

Full 2-port (e only) e* (1 MHz-30 GHz) Requires one specimen e^ only

Full 2-port, one specimen ^^m;: Requires one specimen Unstable at nXm/2

(1 MHz to 30 GHz) for low-loss materials Open-ended coaxial probe e; (500 MHz to 30 GHz) Dominaint electric field e* only

Capacitor e; (dc to 50 MHz) Dominant electric field Low firequency

Shielded open circuit e* (1 MHz to 30 GHz) Dominant electric field e* only Short circuit

(100 MHz to 20 GHz)

Domineint magnetic field fi* only

Permeameter n; (dc to 50 MHz) Dominant magnetic field Low frequency

Free space e;, m; (GHz) Nondestructive low firequency

imperfections are limited and are such that the Laplacian can be separated into transverse

and longitudinal components. If we assume that the vector component of the normalized

radial electric fields Ej, En, and, Em^ in the Regions I, II, and III, we can write for N

modes

TV Ei = exp(-7oia:) +5ii exp(7oia:) + J2^i exp(7^a;)],

1=2

incident wave evanescent

N

En = ^[A exp(-7^ix) + Ei exp(7^i2;)],

i=i

N

Em = S21 exp(-7oi(a: - L)) + J] [Fi exp{-joi{oc - L))].

' i=2"

(76)

(77)

(78)

transmitted wave evanescent

We assume that we are operating the waveguide at a frequency where only the fundamental mode is a propagating mode in the air section of the guide. The higher modes are

evanescent in the air section of the guide, but may be propagating in the material-filled

section. There may be additional modes produced by mode conversion for the other components of the electric field, but these are not necessary for specification of the boundary

conditions. In general, the amplitudes in eqs. (76) to (78) are functions of the transverse

coordinates. To find the coefficients, we must match tangential electric and magnetic fields

^TEM mode in a coaxial line or the TE\q mode in a waveguide (with a time dependence of exp(ja;t)

suppressed).

54
at the interfaces and integrate over the cross-sectional area. Various cutoff frequencies and

operating frequencies are given in Tables 8 and 13.

7.2 Completely Filled Waveguide

7.2.1 Materials with Positive Permittivity and Permeability

As a special case of the formalism developed in the previous section we consider a perfectly

homogeneous, isotropic specimen in a perfectly formed waveguide, as shown in Figure 14.

In this case no mode conversion occurs because the eigenfunctions in the air and specimen

regions are orthogonal with respect to cross-sectional coordinates. Therefore, the modes

may be decoupled and the evanescent modes are not of concern. In this case we need to be

concerned only with the fundamental mode in the guide. The normalized electric fields in

the specimen region x G (0, L) for a coaxial line with a matched load axe

Ei = exp(-7o2;) + 5ii exp(7oa;), (79)

En = C2 exp(-7ia:) -f C3 exp(7ix), (80)

Ein = S2iexp{-j,{x-L)). (81)

The tangential component can be calculated from Maxwell's equations, given an electric

field with only a radial component with the following boundary conditions:

Ei{x = 0) = En{x = 0), (82)

En{x = L) = Eni{x = L). (83)

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:

r(^ = o) = --^(^ = o), (84)

//o dx /i dx

'^ dE[i 1 dEjii

55
For a two-port device the expressions for the measured scattering parameters are obtained

by solving eqs. (76) through (78) subject to the boundary conditions.

When these equations are integrated over the cross-sectional surface area, the radial

dependence is the same for each region of the waveguide.

The constants in the field equations are again 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. The higher modes in eqs. (76) to (78) are

evanescent in the air-filled section of the guide. TM modes can be treated similarly. The

details of the boundary matching for the TEiq case are described in a previous report on

dielectric materials [54]. For a two-port device the expressions for the measured scattering

parameters are obtained by solving eqs. (76) through (78), subject to the boundary conditions. We assume we have isotropic materials so that 5i2 = 521. The exphcit expressions

for a specimen in a waveguide a distance Li from the port-1 reference plane to the specimen

front face and L2 from the specimen back face to the port-2 caUbration plane are related by

a phase rotation. The S-parameters are defined in terms of the reflection coeflacient F and

transmission coefficient Z by

r(l - Z2)

^22 = R2

S21 = R1R2

1 - F2Z2

F(l-Z^)'

1 - F2Z2

Zjl-V)

1 - F2Z2

where

Ri = exp(-7oLi)

R2 = exp(-7oL2)

(86)

(87)

(88)

(89)

(90)

are the respective reference plane transformations. We also have an expression for the

effective transmission coeflacient Z:

Z = exp(—7L). (91)

56
We define an effective reflection coefficient for TE and TEM modes using

^ = T-K^-T^- (92)

For TM modes in eq. (92), replace /i/7 by 7/e. For a coaxial line, the cutoff frequency

approaches 0, {Uc —> 0) and therefore F reduces to

r = ^'°'' ^X—

(93)

Cvac !±r_ I 1

^ ciab ye*

Additionally, S21 for the empty specimen holder yields additional information

52°i = i?ii?2exp(-7oLa). (94)

For nonmagnetic materials, eqs. (86) through (88) contain variables e^, e", L, and La, and

the reference plane transformations i?i, i?2- Equations for 512 and ^21 are theoretically equivalent for isotropic nongyromagnetic materials; however due to specimen inhomogeneities the

measurements will not be exactly the same. We have five complex equations, eqs. (86)

through (88), plus (94), plus the equation for the length of the air line La — Li + L2 + L, or

equivalently, 11 real equations for the 6 unknowns: e(,, e", /i^, //", Li, and L2. The equations

consist of an overdetermined system for the unknowns. This abundance of information will

be exploited in the next section. However, since we are dealing with isotropic materials and

^ii and ^22 are nearly redundant, and if we know the reference planes and sample length,

then we have four equations for the four materials parameters: e^, e", fi'^, /i".

7.2.2 Negative-Index Materials

For materials with negative real parts of the permittivity and permeabihty there are only

slight changes from the results in the previous subsection. When the propagation coefficient

7 = Q; + j/3 is calculated, normally the positive square root is taken for /? (see eq. (4)). The

real part of the impedance is always positive (note: there are very special cases where this is

not true, see Reference [12], pp. 266), since the real part of the integral of the time-harmonic

Poynting vector over the surface, where the energy flows through, is related to the dissipated

power in the media and the radiation resistance. 7 can be imaginary for propagating modes,

57
and real for evanescent modes or a combination of these. However, in NIM, if we consider the

wave impedance Z.^ = jtofi/'f = u{Pji +j afi) /

{a'^ + P"^) , since the real part of the impedance

is positive semi-definite and the real part of the permeability is negative, the negative square

root for P in the propagation constant must be used. This is the origin of the term negativeindex materials. However, a, which contains the material loss plus cutoff behavior, must

always be positive, a can contain both effects from stored energy in the evanescent modes

and losses due to dielectrics. Therefore, we require the physical solutions to have

Re{Z^) > 0, (95)

i?e(7) > 0. (96)

Evanescent waves travehng in the -\-z direction to oo must be damped. Evanescent waves

in a slab may travel in both directions due to reflections. At an interface between materials

the evanescent waves may be reflected and then travel in the —z direction. In addition,

there can be interference between attenuated waves traveling in opposite directions. The

energy flow is in the direction of the Poynting vector. The phase velocity in NIM is opposite

to the group velocity (backward waves). Another issue encountered when measuring NIM

materials, since the material is very inhomogeneous, the thicknesses of the slab and the unit

cell are not apparent. This presents a metrology problem in defining specimen thickness [88].

Pendry [89,90] has shown that for evanescent waves entering a NIM with e^ = //^ = —1, and of thickness d, the transmission coefficient is 52i = exp (ad). This can be obtained

from eq. (88) where a for a cutoff TE waveguide is a = Re{Jk1 — uj'^e^i). Therefore, for a

lossless slab, energy is stored at resonance. This stored energy could be used for enhanced

lens focusing. However, any loss in the material will retard this effect. This effect of stored

energy is commonly called amplification of evanescent waves, but it is probably more accurate

to say that there is an increase in stored energy due to resonance.

58
7.3 Methods for the Numerical Determination of Permittivity

7.3.1 Iterative Solutions

There are various ways of solving the scattering equations depending on the information

available to the experimenter. In cases where the specimen's length and reference plane

positions are known to high accuracy, taking various linear combinations of the scattering

equations and solving the equations in an iterative fashion yields a very stable solution

for specimens of arbitrary length. However higher-mode resonances may occur at specific

frequencies. A useful combination for dielectric materials is

2{[5i2 + 52i] + /?[5n + 522]} = ^

/_ ^2^2

-^ (97)

pa _ ^2

SnS22 - S2iSn = exp{-2jo{Lair - ^)} ^_p2^2 - (98)

In eq. (97), the 5-parameters to be used need to be transformed from the calibration

plane to the specimen face by use of eqs. (89) and (90). Here, Pi is a constant that varies

as a function of the specimen length, plus the uncertainty in scattering parameters and loss

characteristics of the material. The constant /?i is a weighting function for the 5-parameters.

For low-loss materials, the ^21 signal is strong and so we can set Pi = 0, whereas for materials

of high loss, 5ii dominates and a large value of Pi is appropriate. A general relation for Pi

is given by the ratio of the uncertainty in 5'2i divided by the uncertainty in 5ii.

For cases when the positions of the reference plane are uncertain, we find that eq. (98)

is robust. When we use eq. (98), no reference plane transformation need be performed since

it has been eliminated by use of the relation LaiT — Li + L2 + L. Equation (98) works well

for both low-loss and high-loss materials. That is, we have four real equations for the four

unknowns: e^, e", L, LairWhen both the permittivity and permeability are solved for in 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 due to TEM

mode resonance.

59
7.3.2 Explicit Method for Nonmagnetic Materials

In the last sections we detailed an iterative method for determination of permittivity. There

also are explicit methods for the case /ir = 1. Each method has its advantages and disadvantages. For the case of nonmagnetic materials there are two independent transmission-line

expressions for the permittivity (see Section 7.5, eqs. (114) and (122)). Boughriet [91] has

studied the second equation for permittivity. This equation is relatively stable for low-loss

materials

e* = —

1^r A2

1 , ,1

2.L^(;'

The alternative solution, which is relatively stable for high-loss materials is

c2

e* = — 2; -hi, 9 . /27rY &-{aTJ

(99)

(100)

where

7.4 Corrections to Data

-S (^°^)

Once a set of measurements is obtained we need to correct the data, taking into account

systematic uncertainties. These known uncertainty sources include air gaps around specimens, waveguide wall imperfections, short-circuit and waveguide losses. Air gaps corrections

are particularly important for coaxial specimens; in particular, the gap near the center conductor is important since the electric field is higher in this region. The outer air gap also

contributes, but to a lesser extent. In order to make corrections we need to make precise

measurements of both the specimen and the specimen holder; air gauging equipment is used

to measure the air line dimensions.

7.4.1 Influence of Gaps Between Specimen and Holder

Air gaps around specimens affect the 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 as in coaxial line, the gap

near the center conductor contributes more to the permittivity correction than a gap of the

60
same thickness near the outer conductor. These correction formulas are approximate, and

generally imder-correct for the effects of the air gap. Formulas for correction are given in

Appendix B.

7.4.2 Attenuation Due to Imperfect Conductivity of Specimen Holders

Since no waveguide at ambient temperature is perfectly conducting, all propagating modes

are attenuated to some degree. The finite conductivity of waveguide walls promotes power

loss in the guide. Waveguide and coaxial line losses can be probed by measuring the scattering

parameters of the empty waveguide.

The power flow propagating down the waveguide can be expressed in terms of the Poynting vector

P = l

f Re[E X H*] • d5 = % / \Ht\'dS, (102)

Z Js Z Js where Z^j is the wave impedance and subscript (f) denotes the tangential component [5]. The power loss in the guide is

PL = lf\lfds = ffjH,\'ds, (103)

where Rg = y'^f^ is the effective surface resistance, a is the conductivity, and c is a closed

integration path. The effective attenuation coefficient of the waveguide can be defined as ^^ a = (104)

2P"

For rectangular waveguide the integrals in eq. (103) can be performed to obtain [5]

OtTEr, 2Ra "^m'^HfiVsm

where a and 6 are the guide dimensions, fc is the cutoff frequency, m, n = 0,1,2,.

is the impedance of free space. For n = the attenuation is

(«)

-R.q TE„,o — 1 +

26 //c

&r/v^i

{fc/f? [ « V/

The attenuation coefficient in coaxial line is

"' = IV2^U+6

(105)

and 7]

(106)

(107)

61
7.4.3 Appearance of Higher Order Modes

The field model developed in this section assumes a single mode of propagation in the

specimen. At frequencies corresponding to one-half wavelength in low-loss specimens the

fundamental mode will resonant. In addition, due to cutoff conditions, higher modes of

propagation become possible in inhomogeneous specimens of high permittivity (usually for

relative permittivities greater than about 10). 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 section breaks down to varying degrees, particularly at

resonant frequencies of these modes. 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 transmission/reflection models to break down (including the one

developed in this section) for inhomogeneous specimens materials of high permittivity.

In order to minimize the effects of higher modes, shorter specimens can be used. Higherorder mode resonances have little effect if the specimen length is less than one-half guided

wavelength in the material. They may still propagate in the specimen at any frequency

above the cutoff of the mode.

7.4.4 Mode Suppression in Waveguides

The higher-order modes can be suppressed by mode filters. This would be particularly

helpful in cylindrical waveguide. One way to do this is to helically wind a fine wire about

the iimer surface of a waveguide specimen holder, thus eliminating longitudinal currents and

therefore TM and for example TEn modes. Another approach is to insert grooves in the

waveguide walls. These methods however do not work for coaxial line.

7.4.5 Uncertainty Sources and Analysis

Errors are the difference between the true vedue of a measurement and the measurement

result. The true value is a theoretical construct. The uncertainty is an estimate of the

region around the true value where a measurement will fall. In this section the uncertainty

62
incurred when using the equations expounded in this section will be estimated. The sources

of error in the TR measurement include

• Random uncertainties and error sources

(a) Errors in measuring the magnitude and phase of the scattering parameters

(b) Error in specimen length

(c) Error in reference plane positions

• Systematic uncertainties

(a) Gaps between the specimen and specimen holder and specimen holder dimensional

variations

(b) Line losses and connector mismatch

Correction for errors arising from gaps around the specimen are obtained from equations

available in the hterature [3, 92, 93]. We assume that all measurements of permittivity

have been corrected for the effects of air gaps around the specimen before the following

uncertainty analysis is applied. In order to evaluate the uncertainty introduced by the

measured scattering parameters, a differential uncertainty analysis is appUcable with the

uncertainty due to Sn and ^21 evaluated separately. We assume that the total uncertainty

can be written as M-1 e' ~ e'

Ae'l 1 i§^M-[S:^^'h{lt-^h&^'^J^ (-'

e'; e:

^ / f^ II \ ^ / f\ II \ ^ /o// \^

/ d^l ^,^ ,V /5<' ^r. \ (de'l ^ ,\ (de'l

where o; = 11 or 21, A^ is the uncertainty in the phase of the scattering parameter, /S.\Soc\ is

the uncertainty in the magnitude of the scattering parameter, Ac? is the uncertainty in the

air gap around the specimen, and AL is the uncertainty in the specimen length. The uncertainties used for the ^-parameters depend on the specific ANA used for the measurements.

The measurement bounds for 5-parameter data are obtained from specifications for a

network analyzer. The dominant uncertainty is in the phase of 5ii as | 5ii |—» 0. The

uncertainty in 521 is relatively constant until 15211 < -40 dB and then increases abruptly.

63
In Figs. 23 and 24 the total uncertainty in e* computed from ^21 and 5ii is plotted as a

function of normalized specimen length for low-loss and high-loss materials at 3 GHz with

various values of e*, and the guided wavelength in the material given by

27r Am —

In Figures 23 through 25 the uncertainty in permittivity determination is presented.

(110)

a4

0.016

0.012

0.00s

aoo«

T

3J3 35

Figure 23. The uncertainty in permittivity as a function of normalized length for a low-loss

specimen.

We see that the minimum uncertainty in the permittivity for low-loss dielectric materials,

using the iterative procedure developed here, occurs at multiples of one-half wavelength.

The reason for this can be determined by examining eqs. (86), (88) in the limit that Sn -^

0, 521 -^ 1 with F 7^ 0. These equations then reduce to

Z^-1^0. (Ill)

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

This increase occurs because A5'2i increases when the transmitted signal is less than -40

dB from the reference value. For the case of high loss the uncertainty in 5ii approaches

a constant value. This is so because for high-loss materials where the wavelength is much

64
Xm

Figure 24. The uncertainty in permittivity as a function of normalized length for a high-loss

specimen.

smaller than the specimen length; only weak signals penetrate through the specimen, 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.

7.4.6 Systematic Uncertainties in Permittivity Data Related to Air Gaps

Once a set of measurements has been made, we need to correct the data for known uncertainties. Known error sources include air gaps around specimens, 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.

Gap correction formulas that are relatively easy to implement [3, 92, 93] can be found in

the literature and in Appendix B. An electrical measurement of the air gap correction can

be made by using a additional resonator measurement of the same material in the frequency

band of the waveguide measurement. This measurement is then used to determine the

required gap in the correction formulas to match the data at that particular frequency.

Waveguide losses can be corrected for by measuring the scattering parameters of the empty

65
Oj09

0.07

OX3B

<ym -

QXH

A

%

3 Szt -II

'I -

\ y' X —' I y^ —

«, « (5A 2Ja>/^

' V-,. 1 \/^ *

\v A y^

^^^.x"""''^ «« ' OOA 2X»~

V|V^.-,.v^ <:?^"""**—•^ ^" -

.

, , ,f,

Figure 25. The uncertainty in imaginary part of the permittivity as a function of normahzed

length for a lossy specimen.

waveguide and calculating the appropriate attenuation coefficient of the guide. Also, the

calculated permittivity data can be smoothed.

7.5 Permeability and Permittivity Calculation

7.5.1 Nicolson-Ross-Weir Solutions (NRW)

Nicolson and Ross [94], and Weir [56] combined the equations for Sn and 521 and developed

a formula for the permittivity and permeability. Their procedure works well for frequencies

away from the TEM mode resonances where the specimen length is not a multiple of onehalf wavelength in the material. Near resonance, however, the method loses sensitivity for

low-loss materials.

We will now develop the equations for this method.

Zi = exp(-7L). (112)

We define a reflection coefficient for TEM and TE modes:

ri =

7 70 Z + Zq

(113)

66
For TM modes in eq. (113), replace /i/7 by 7/e. Therefore

K /l + Ti^'

In the NRW algorithm, the reflection coefficient is

(114)

Vi^X± VX2 - 1. (115)

The square root sign is determined from causality in eqs. (95) and (96). Also X is given

expUcitly in terms of the scattering parameters where

and

1^1 = 5-21 + 511, (117)

^2 = 521 - 5ii. (118)

Note that in the Nicolson-Ross solution, the S-parameters must be rotated to the plane of

the specimen faces in order for the correct group delay to be calculated. The correct root

is chosen in eq (115) by requiring |ri| < 1. The transmission coefficient Z\ for the NRW

procedure is given by

5ii + 521 - Ti

If we define

^=-i2^'°(i)i^' (^20)

then we can solve for the permeability using -- a-ro^Jbl' '"^'

where Aq is the free space wavelength and Ac is the cutoff" wavelength.

The permittivity is given by

In magnetic materials, eq. (120) has an infinite number of roots, because the logarithm of

a complex number is multivalued. In order to pick out the correct root we need to compare

67
the measured group delay to the calculated group delay (L/vg). The calculated group delay

is related to the change of the wave number k with respect to the angular frequency, by

analogy with the general group-velocity formula du/dk

'^calc. group -^ -ir '

A?

The measured group delay is

'^meas.group "Z T7 5 V J ZTT a/

where ^ is the phase of Zi. To determine the correct root, the calculated group delays are

found from eq. (123) for various values of n in the logarithm term of eq. (120), where

LnZi =\n\Zi\+ j{9 + 27rn), n = 0, ±1, ±2, • • •. The calculated and measured group delays

are compared to yield the correct value of n. For dispersionless materials the group delay is

constant with frequency.

Many researchers think of the NRW solution as completely explicit. However, due to

the phase ambiguity and causality constraints, it is not in the strict sense. For low-loss

specimens, the NRW solution is divergent at integral multiples of one-half wavelength in the

specimen. This occurs because the phase of Sn cannot be accurately measured for small

IS*!!]. Also in this limit both of the scattering equations reduce to the relation Zf —> l,which

is a relation for only the phase velocity and therefore solutions for e* and fi* are not separable.

This singular behavior can be minimized in cases where permeability is known a priori, as shown in previous work performed by Baker-Jarvis [54].

For magnetic materials there are other methods for solution of the S-parameter equations. In the next section we will describe various solution procedures. A comparison of the

Nicolson-Ross results for the permittivity of Teflon compared to the iterative permittivity

from eq. (97) is given in Figure 26.

7.5.2 Modified Nicolson-Ross: Reference-Plane Invariant Algorithm

In measurements, keeping the sample faces at the reference planes is difficult. Therefore an

algorithm that is independent of the reference planes would be very useful.

68
*^-=^-J

t I I I

, iy40\. 4^

7 mm

coaxial

sample

3.279 cm

long

±

6 9 12

Frequency (GHz)

16 18

Figure 26. The results of permittivity as a function of frequency using the Nicolson-Ross

algorithm versus eq. (97).

In order to obtain both the permittivity and the permeabihty from the S-parameter relations, we can have at least two independent measurements. Most commonly we use a

two-port measurement of 5ii and 521 on one specimen , or a one-port SCL measurement

of a specimen at two different positions in the line. Alternatively, we can use independent

measurements of two specimens of different lengths. In the full S-parameter solution developed, we solve equations that are invariant to reference planes for e and /i. This is an enhancement over the Nicolson-Ross equations. A set of equations for single-specimen magnetic measurements is eqs. (97) and (98). The LHS of eq. (98) is the determinant of the

scattering matrix.

7.5.3 Iterative Solution

Equations (97) and (98) can be solved either iteratively, or by a technique similar to the NRW

technique. In an iterative approach, Newton's numerical method for root determination

works quite well. To solve the system we can separate the system into four real equations.

The iterative solution works well if good initial guesses are available.

69
7.5.4 Explicit Solution

We can also possible to obtain an explicit solution to eqs. (97) and (98). Let x = {S21S12 —

<?i 1 522 )exp{270 (Lair " L)} and y = {{S21 + Si2)/2} exp{'yo{Lair - L)}: then we can show

that the roots for the transmission coefficient are Z = ±1 and

x + l

22/ \ " + '^'-1.

(125)

2y J

The correct roots are selected such that eqs. (95) and (96) are satisfied. The reflection

coefficient is

7 70 X — Z"^ Z — Zq ^ — ^

I " 7 70 For TM modes replace fx/y by 7/e in eq. (126).

The ambiguity in the plus-or-minus sign in eq. (126) can be resolved as for Z. The

permeability and permittivity are then

M: = -^^-^(lnZ + j27rn), (127)

J- — 1 2 7o-i^

< = -Ai^f - T^(lnZ + j27rn)2]//.:. (128)

u \r/ L2

The correct value of n is selected using the group delay comparison, as described in the

Nicolson-Ross-Weir technique. Usually we fit the measured group delay around a specffic

point by a least-squares polynomial. At low frequencies and low permittivity the correct roots

are more easily identffied since they are more widely spaced. However, high-permittivity

materials have many possible roots and discerning the correct n in the logarithm is sometimes

difficult.

7.5.5 NIM Parameter Extraction

For NIM, the correct sign on the index of refraction is required. In these measurements we

can use eq. (125) to obtain the index of refraction. As previously, the correct roots must

be selected by requiring that the real part of joo^/^ > and Re{'j) > 0. Also, the correct

branch of the logarithm function must be determined through the group delay, as discussed

previously for other methods.

70
7.5.6 Measurement Results

The measurement consists of inserting a well-machined specimen into a coaxial line or

waveguide and measuring the scattering parameters. For waveguide measurements we need

to have a section of waveguide of length about two free-space wavelengths between the

coax-to-waveguide adapter and the specimen holder. This acts as a mode filter for damping

evanescent modes. There are many roots to the equations for the permeability and permittivity, and caution must be exercised when selecting the correct root. At lower frequencies (less

than 1 GHz), the roots are usually more widely spaced and therefore root selection is simplified. Another approach to root selection is to measure two specimens of diflFering lengths,

where the results are compared to determine the correct root. In Figure (27), we show data

on 7 mm coaxial line-measurements on several polymer materials. For comparison purposes,

the corresponding measurements of permittivity and loss tangent on the same materials at 10

GHz using a highly-accurate dielectric resonator are Teflon: (2.05 ±0.02, 2 x 10"'* ±8 x 10~^),

Rexohte: (2.54±0.01,4 x lO'^iS x 10"^), Nylon: (3.04 ±0.02, 7 x 10"^ ±8 x 10"^), PMMA:

(2.61 ±0.02, 5 X 10-3 ±8 X

^q-S)^ polyvinylchloride(PVC): (2.70 ±0.02, 5 x 10-3±8 x 10"^).

w

2-

-t—* ^ A -*—*—*—A—*

*—A- A—A—A—A

^—A—*-

Nylon

PMMA

PVC

Rexlolite Teflon J I I I L. 5 10 15

Frequency (GHz)

Figure 27. Measurements of e^ on polymers.

20

71
to

c

CO

0.03

0.02-

0.01 -

0.00

^

—•- Nylon -•- PMMA

-*- PVC

\ -- RexlolitB

^ ^^*'~*->«—»~

-- Teflon

i—1—

L

1 r 1 1 1 1 1 1 L L 1 1 1 1 1 1 1 5 10 15 20

Frequency (GHz)

Figure 28. Measurements of tan 6 on polymers.

7.5.7 Measurements of Magnetic Materiads

Waveguide measurement data on magnetically loaded polymers are given in Figures 29

through 32 using the full iterative and explicit S-parameter techniques. The measurements

reported in this section have not been corrected for gaps around the specimen. The effect

of the air gaps is a bias in measured values of the material parameters that are below the

actual values. In the next section we will discuss ways of mitigating the effects of air gaps

on measurements.

7.5.8 Effects of Air Gaps Betvireen the Specimen and

the Waveguide for Magnetic Materials

Systematic uncertainties due to air gaps between the specimen holder and the specimen

may be either corrected with the formulas given in the Appendix C or mitigated using a

conducting paste can be applied to the external surfaces of the specimen that are in contact

with the specimen holder.

72
6.00 I r

5.75

rp 5.50

5.25

5.00

8.3 8.9

S—eOi^. .»*.»CT;5ln

J L

Nicolson-Ross

Iterative _

J L 9.5 10.1 Frequency (GHz)

10.7 11.3

Figure 29. e^ of a loaded polymer in an X-band waveguide with the full S-parameter iterative

technique.

9-6 10.2 Frequency (GHz)

10.8 11.4

Figure 30. e" of a loaded polymer in an X- band waveguide with the full S-parameter iterative

technique.

73
1.500

1.375 —

fl-f 1.250

1.125 —

1.000

r

1 1

. 1 1 - —

..—.. Nicolson-Ross —

—

' Iterative

.

1 I 1 1 , :

-

8.5 9.08 9.66 10.24 10.82 11.4

Frequency (GHz)

Figure 31. /^^ of a ferrite loaded polymer in an X-band waveguide with the fuU S-parameter

iterative technique.

1.00

0.89

0.78

0.67

0.56

/iR 0.45

0.34

0.23

0.12

0.01

-0.1

1 rNicolson-Ross

Iterative

....^..^^~''""'" "" '

8.4 9.6 10.2

Frequency (GHz)

10.8 11.4

Figure 32. //" of a ferrite loaded polymer in an X-band waveguide with the fuU S-parameter

iterative technique.

74
7.6 Uncertainty Determination of Combined Permittivity

and Permeability Measurements in Waveguide

7.6.1 Independent Sources of Uncertainty for Magnetic Measurements

In this section an uncertainty analysis is presented. The complete derivations of the uncertainty equations are given in [4] . The sources of error in the permeability and permittivity

TR measurement include

• Uncertainties in measuring the magnitude and phase of the scattering parameters

• Gaps between the specimen and specimen holder

• Specimen holder dimensional variations

• Uncertainty in specimen length

• Line losses and connector mismatch

Techniques for correcting for uncertainties arising from gaps around the specimen are given

in Appendix B [95-100]. We assume that all measurements of permittivity have been corrected for air gaps around the specimen before the uncertainty analysis is applied. In coaxial

hne, the permeability is less affected by air gaps than the permittivity since the magnetic

field is tangential to the air gap and therefore there is less stored magnetic energy in the

air-gap region when compared to a magnetic field normal to the air gap. In order to evaluate

the uncertainty introduced by the measured scattering parameters and specimen dimensions, a diflierential uncertainty analysis is assmned applicable with the uncertainty due

to Sii and S21 evaluated separately. We assume that the S-parameters are functions of

Sij{\Sii\, \S2i\,0u,62i,L,d). The root-sum-of-squares (RSS) uncertainties for the physically

measured parameters are

Ael

Ae"

(129)

(130)

75
where ck = 11 or 21, A^ is the uncertainty in the phase of the scattering parameter, Al^al

is the uncertainty in the magnitude of the scattering parameter. Ad is the uncertainty

in the air gap around the specimen, and AL is the uncertainty in the specimen length.

The A indicates one standard deviation in the independent variable. The derivatives with

respect to air gap, de'^jdd, have been presented previously [54]. The uncertainties used

for the S'-parameters depend on the specific ANA used for the measurements. This type

of uncertainty analysis assumes that changes in independent variables are sufficiently smaU

that a Taylor series expansion is valid. Of course there are many other uncertainty sources of

lesser magnitude such as repeatability of connections and torquing of flange bolts. Estimates

for these uncertainties could be added to the uncertainty budget.

7.6.2 Measurement Uncertainty for a Specimen in a Transmission Line

The measurement bounds for S-parameter data are obtained from specifications for a network

analyzer. The dominant uncertainty is in the phase of 5ii as | 5ii |—> 0. The uncertainty in

15211 is relatively constant until |S'2i| < —50 dB, when it increases abruptly.

In Figures 33 and 34, the total uncertainty in e* and /i*, computed from 521 and 5ii is

plotted as a function of normalized specimen length.

In Figures 33 and 34, the uncertainty due to the gap correction is not included, nor are

there uncertainties included for connector repeatability or flange bolt torquing. The maximum uncertainty for low-loss materials occurs at multiples of one-half wavelength. Generally,

we see a decrease in uncertainty as a function of increasing specimen length. Also, the uncertainties in the S-parameters have some frequency dependence, with higher frequencies

having larger uncertainties in phase.

7.7 Uncertainty in the Gap Correction

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

dominate the uncertainties of the measurement. In this section the uncertainty in the gap

76
AeR

0.11

0.09 -

0.07

0.05 -

0.03 -

0.01 -

«; = (5.0.05)

; = (2,0.05)

3.5

Figure 33. The relative uncertainty in e'j.{u) for a low-loss material as a function of normalized

length, with (i; = (2, 0.05), e* = (10, 0.05) and (5, 0.05).

0.11

0.09

0.07

0.05

0.03 -

0.01

«R = (10,0.05)

^^l = (2,0.05)

3.5

Figure 34. The relative uncertainty in /x^(a;) for a low-loss material as a function of normalized length, with /i* = (2,0.05), e* = (10,0.05) and (5,0.05).

77
correction will be worked.

7.7.1 Waveguide Air-Gap Uncertainty for Dielectrics

The uncertainty due to an air gap between specimen 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

dd '-""h-ih-d^e'J 'rnR^h-{h-d)eUr ^ ^

de" b

7.7.2 CoELxial Air-Gap Correction for Dielectrics

For coaxial line the relevant derivatives are given by

?^ = c'^^

^ + cg^

^^

(133)

5i?2 "" R2{L3 — e^fl-t'i) "" R2{L3 — e'^jiLxY

^fk = e' \

l_ e'2 ^ ^ Q34^

5i?3 "" R3{Lz — e'^j^Li) "" i?3(L3 — e^^Li)2'

^^cfi _ ,// / r -^

I -^1

1 /'1Qc;\

'^E~ ~ ^mR^mRl T p "^ 7-2p J' V-'^"^^/ (7X1.2 L2-rC2 J-i2^2

f)p" = -CR^'mRl-r^ + ^]- (136)

where

dR^ ^"^ '^'''L2R3 LIR3

L,=ln|^+ln|i, (137)

Ri R.3

L2 = ln|^, (138)

-fL2

1^3 = In 1^. (139)

8. Short-Circuited Line Methods

8.1 Theory

In this section we review the mathematical formalism for short-circuit measurements and

develop a measurement algorithm. We consider a measurement of the reflection coefficient

78
Li L AT,

* ^^^W I 1i1

III

x=o X=L

Figure 35. A transmission line with a short-circuit termination.

(5ii for a shorted two-port) as a function of frequency. We begin with a mathematical

analysis of the electromagnetic fields in the specimen (see Figure 35). The details of the

field model have been presented previously [54] and only the most essential details will be

presented here. In terms of hyperbolic functions, the reflection coefficient is _ tanh7L + ptanhjoAL - p{l + /?tanh7Ltanh7oAL) ~ tanh7L-h/? tanh7oAL -!-/?( H-/?tanh7Ltanh 7oAL)' ^

^

Although in the derivation of eq. (140), we assumed that the specimen plane coincides

with the measurement calibration plane, this is not in general the case; however, we can

transform the reference plane position by a simple procedure. To accomplish this, we write

the most general expression for the reflection coefficient as

ril(frans) = -^ir, (141)

where Tiitrans is the reflection coefficient at the calibration reference plane position,

Ri = exp(-7oI'i), (142)

and Li is the distance from the calibration plane to the specimen firont face. Equation (141)

transforms the reflection coeflacient from the calibration plane to the plane of the specimen's

front face. For many applications we wish to eliminate the distance Li from eq. (141). This

can be accomplished by measuring Tempty of the empty specimen holder,

Femptj/ = - exp(-27o[Li + AL + L]) = - exp{-2^QLair)

(143)

79
and consequently the ratio of the filled to empty holder reflection coeflacient is

l^^^L^:!!^ = - exp(27o[AL + L])r. (144)

' empty

If both the permeability and the permittivity are required, measurement data for two different short-circuit positions are needed. Note that standing waves can be formed in the

region between the specimen and short-circuit and between the calibration plane and specimen's front face. Therefore, certain frequencies, depending on specimen length and the other

lengths, will give better results for permittivity, whereas other frequencies will yield better

results for permeability.

The position of the short circuit for a coaxial Une is in a low electric field and high

magnetic field region, and a position A/4 from the short- circuit is a high electric field and

low magnetic field region. Therefore, as frequency permits, for permittivity measinements

the specimen should be moved away from the short-circuit termination. Permeabihty in

isolation can be obtained with the specimen at the short-circuit position. This depends on

the sample thickness and the wavelength. Of course when an ANA is used measurements

will be taken at many combinations of field strengths and therefore the uncertainty will

vary with frequency. A hmitation of the SCL method is that the electric field will be very

small over the entire waveguide section at low firequencies. Therefore dielectric materials are

difficult to measure at low frequencies in a SCL system.

8.2 Measurements

In the SCL technique the scattering parameter ^n is measured broadband, with the specimen

at a given position in the specimen holder. The distance from the specimen to the shortcircuit termination must be known to a high degree of accuracy. If both permeability and the

permittivity are required then the specimen must be moved in the line and the S-parameters

again measured.

Depending on the position of the short circuit, the specimen may be immersed in either

a region of high electric field or high magnetic field. A strong electric field is advantageous

for determination of permittivity, whereas a strong magnetic field is advantageous for permeability determination. Generally, the specimen end wiU be in a region of high magnetic

80
^R

10 11 12

FREQUENCY (GHz)

13

Figure 36. e^ without gap correction using SCL for a loaded polymer mixture.

field when the specimen is in closest physical contact with the short. We can take advantage

of the fluctuating electric and magnetic field distributions when measuring permittivity and

permeability. When taking broadband measurements on an ANA we can predict when the

specimen is immersed in the various field strengths. Then one can select the measurements

to be used for permittivity and permeability calculations [93].

Measurements were made on an ANA for various specimens. Using eq. (140) we obtained

the permittivity and permeabihty data shown in Figures 36 and 37.

9. Permeameters for Ferrites and Metals

9.1 Overview of Permeameters

In the past, permeameters have been used for low-frequency measurements on medium-high

permeability materials. For ferrites, the skin depth is large and is easier to model than in

metals where it is only a fraction of a millimeter. Therefore we use different approaches

for metals versus ferrites or other resistive materials. Rasmussen et al. [101], Hoer and

Rasmussen [102], Powell and Rasmussen [103], and Goldfarb and Bussey [58] have all described various ferrite permeameter designs. The conductivity of metals can be measured

81
H-k

10 11 12

FREQUENCY (GHz)

13

Figure 37. ^!^ without gap correction using SCL for a loaded polymer mixture.

with a four-probe system using voltmeters. An overview of this technique is given in [104].

Alexander et al. [105] have studied permeameters for ferromagnetic and ferrite thin films.

9.2 Permeability of Metals

For metals, the characterization of the electrical properties includes the surface resistance,

conductivity, and permeability. The high-frequency surface resistance can be determined by

end-plate substitution methods [106]. For metals, an accurate way to measure permeability

is to wind a toroidal specimen with wire and measure changes in inductance as shown in

Figure 38. These types of measurements are summarized in [104]. The main difficulty with

these measurements is that the skin depth is short and needs to be included in the model.

If a toroidal specimen is inserted into a region containing an azimuthal magnetic field, the

inductance is changed. If the inductance of the empty specimen holder is compared to the

inductance of the filled holder then we can extract the complex permeability of the material.

At very low frequencies, if the skin depth is much larger than the sample size, it need not

be modeled.

82
iHl^ «>)

(4

Figure 38. Toroid wiring over metal or ferrite core.

9.3 Ferrites and Resistive Materials

For magnetic materials such as ferrites at low frequencies, the skin depth usually is not important in the measurement. We consider a magnetic toroid in a short-circuited coaxial line.

For long wavelengths near the short circuit we have a strong, relatively uniform, magnetic

field. In such cases the following simple model can be used to determine the permeability.

Consider a toroid of inner diameter a and outer diameter h and height h at low frequencies (depending on properties, for frequencies < 100 MHz). The material contributes an

inductance of [58]

'~"m. C =—

H'h\n{b/a)

2tt and the inductance of the air space taken up by the toroid is

/loh ln(6/a)

Ca =

27r

(145)

(146)

The net change in the specimen inductance when the specimen is inserted into the holder is

i-^*~' — *-'m *~'a (147)

and therefore

/^; = i +

27rA£

liQh\n.{h/a)

(148)

83
p

probe

f*s, 5i>eciinenII

III

probe

Figure 39. Two open-ended coaxial probes with specimen inserted.

The magnetic loss may be obtained from consideration of the core loss Ai? or resistance

Ai?

^^'r =

IJ,ofh\n{b/a)

(149)

10. Other Transmission-Line Methods

10.1 Two-Port for Thin Materials and Thin Films

10.1.1 Overview

We can use two coupled coaxial probes to monitor the dielectric properties of thin materials.

The two-port measurement fixture is depicted in Figure 39. In this case there may be

transmission as well as reflection. An analj^ical solution for this problem has been derived

previously [107].

The scattering parameters can be obtained by matching tangential field components at

interfaces. The analytical solution follows the same Hne of reasoning used in the full-field

single coaxial probe problem. Experimental results indicate that this method is useful for

broadband material measurements on thin materials.

84
10.1.2 Scattering Parameters

The scattering parameters may be constructed if we match at the material-conductor interfaces, the Hankel transforms of Helmholtz equation for the electric and magnetic fields,

solve for coefficients, and then talce inverse transforms. In particular, two independent vector

relations can be obtained

Qi^ii + Q2521 = Pi (150)

and

Q25^11 + Ql^21 = P2. (151)

Here 5ii and 521 are the reflection and transmission vectors. The components of the matrices

are n A . ^27n(c) r CDnDm{exip{2^2d) + 1) ^^ Qlmn = Omn H Z / 7 JT: T; 7T d,Q, (152)

e* Jo 75(exp(272d) - 1)

r> _ n ^Mc) r CDnDmexp{j2d) ^^

,

e* 7o 72(exp(272d) - 1)

Pin = Ono

^ /

7 777—r\ T\—"^- vl54)

e* Jo 72(exp(272d) - 1)

The propagation constants are 7^(0) = JKi(c) ~ ('^/c)^^rc/^*c' where kn{c) are cutoff wavenumbers and subscript (c) denotes a coaxial line. Note that all of the TMon modes are evanescent.

Also 72 = j^k^ - C if 5R(/c2) > C. and 72 = yC-M if 5R(/c2) < C- Also Rm are radial

eigenfunctions given by eq. (184). Further,

e|2jMr^Ato^ (155)

e* Jo 72(exp(272(i)-l)

for 771, n = 1,2, ,N. The coefficients are calculated from the eigenfunctions Rn (see eq.

(161)) in the coaxial line of outer dimension b and inner dimension a

DniO = [' pMCp)Rn{p)dp (156)

J a

can be found analytically. For n = 0,

^o(^) = -rrr ['MCp)dp = -^i=i[jo(Ca) - MCb)] (157)

Jin b/a "'° yIn b/a ^

85
and otherwise

J a 2 C„ 1 C :[M(:b)Mkn{c)a) - Jo{Ca)Jo{knic)b)]. (158)

Straightforward matrix manipulation of eqs.(150) and (151), assuming that no singular matrices or inverses are encountered, yields the following solution for the forward problem:

521 = [Qi - Q2Qr'Q2]"'(^2 - Q2Qr'A), (159)

and

Sn-Qi\Pi-Q2S2i). (160)

The first components in 5ii and ^21 are the measured TEM mode reflection coeflficients

obtained by a network analyzer.

The fixture's electromagnetic behavior depends on whether the two flanges are connected

by conducting or nonconducting clamps or pins. If the two halves are electrically insulated,

then a radial waveguide TEM mode (£'2, H^) is present. If, on the other hand, the halves

are shorted, then this mode should be suppressed.

10.2 Short-Circuited Open-ended Probes

10.2.1 Overview of Short-Circuited Probes

Another method that has been used over the years is a variation of the coaxial probe,

where the sample is positioned at the end of the inner conductor as shown in Figure 40(a)

[97, 108, 109]. A full-mode model of configuration in Figure 40(b) has been published [110].

Since the sample rests on a metal surface, the method has limited loss resolution. The

method has the advantage that the sample has a strong electric field in the z direction at the

end of the inner conductor. The sample at the conductor end is in a small radial waveguide

with a TEM mode that has Ez and H^f, components. At the end of the coax the TEM mode

can propagate between the sample and short for [0 < p < a] and can resonant for high

firequencies (/ « c/2ay/e!^). Whereas for [a < p < b] there are only cutoff TMom modes.

86
Connector

Outer

Conductor

nner Conductor

(a)

Sample

(b)

Figure 40. Open-ended sample method, (a) depicts the sample only under the inner post

alone, and (b) the sample extends across the entire diameter of the coaxial line.

10.2.2 Mode-Match Derivation for the Reflection Coefficient

for the Short-Circuited Probe

The full mode-match solution for the reflection coefficient for case (b), where the sample fills

the coaxial Une at the short position, can be derived by matching the tangential electric and

magnetic fields at the coaxial-line-sample interface, and by requiring the tangential electric

field goes to zero on the short-circuit termination. The radial eigenfunctions in coaxial line

of inner radius a and outer radius b are

Rin{p) = co/p

for n:=o (TEM mode) = Cn[Mkinp)No{kina) - Ni{kinp) Jo{kina)] (161)

> ^

' n>o (TMo„ modes)

87
where Jo, Ji, Nq, Ni are the Bessel functions of the second kind. Also, ^ irkin 1

where the eigenvalues in the coaxial line are found from the condition that the z-electric field

component vanishes on the side wall. This condition requires for n = 0, kio = 0. For n j^

we have TMom modes with eigenvalues determined from

Jo{klnb)No{klr^a) - No{kinh)Jo{kina) = 0. (163)

The propagation coefficients are

\

a;2 7ln=.? CiP'ti-^ - kfn- (164)

The reflection coefficient of all modes, TEM plus evanescent, is

f = [7 - L4L3]"' •[L4Q + P]- (165)

The measured TEM reflection coefficient is Fq. Given Fq, the first component in eq. (165) is

solved by a nonlinear root finding algorithm for the permittivity. The unknown permittivity

is 62 and has thickness L and radius b. The inner conductor of the coaxial fine has radius a. The coefficients are given by

coshA;2n-^ /

and

P = Smn. (167)

The matrix elements are given by:

L4mn= 7^^COSh.k2nL T Ji{X2nr)Rlm{p)dr (166)

IUJCa Ja

f\R,oMX2nr)dr = . ^ f J,{\2nr)dT = ^

'^o(A2na)

^ ^^gg^

^" ^ln(6/a) "^a ^In (6/a) ^2n

'•«'_,,, s , 2 Crr. 1 A.

/ rRimJ\{\2nr)dr = -t^^77—777:2—^\2"['^(A2n&) Jo(fcima) - Jo(A2na) Jo(^im&)],

Ja TT Kim 'JQ\'^\mO) l^im ~ ^2n

(169)

88
7 _ J^^2 la rRimJi{X2nr)dr

^ ^

J^^2larRioJi{X2nr)dr

Ni = j\j!iX2.r)dr='y^. (172)

where A;2n = y A:! " '^in- ^2 = i^y/^2^2- The eigenvalues A2n are roots of

MX2nb) = 0. (173)

11. Measurement Methods for Liquids

There are both resonant and nonresonant methods for Hquids. Resonant methods use either cavities or dielectric resonators. Transmission-line methods include open-circuited or

matched-load waveguides. A good database for hquid dielectric measurements is given in

Reference [111].

11.1 Liquid Measurements Using Resonant Methods

A resonance can be supported in a low-loss dielectric cylinder within a metallic shield. It

will also resonant if a hole is drilled into the center and liquid is inserted as shown in Figure

41 [112,113]. The dielectric loss of the specimen influences the resonant frequency and Q.

The electric field is proportional to the Bessel function Ji(Ar), which goes to as r ^ 0.

Hence, by placing the liquid at the center of the dielectric resonator, the loss of the specimen

will not damp out the Q, and a resonant mode can propagate.

11.2 Open-Circuited Holders

11.2.1 Overview

The shielded open-circuited coaxial line specimen holder has been used for years for dielectric

measurements of liquids and powders in the microwave band [4, 114-117]. The fbd;ure is composed of an outer conductor that extends beyond the end of the inner conductor, as shown in Figure 42. The specimen holder algorithm is based on an accurate model of a

89
Fill Tube

Teflon Spacer

g

o

"i c

J

Teflon Spacer

oCoupling Loop

Figure 41. A dielectric resonator for liquids.

coaxial line terminated in a shielded open circuit. The advantage of an open-circuited holder

is the ease of specimen installation, the broad frequency capability, and the strong electric

field in the specimen region. The specimen holder has been found useful in both ambient

and high-temperature measurements. Von Hippel [118] used an open-circuited specimen

holder for hquid measurements. Bussey [114] extended the open-circuited holder technique to

higher frequencies using an admittance model. Scott and Smith [119] studied the instabilities

encountered in solving the relevant nonlinear open-circuit equations. Hill [120] studied in situ

measurements of soils using open-circuited transmission lines. Jesch [121] used the shielded

open-circuited holder for measurements on shale oil. Biological tissues have been measured

using the shielded open-circuited line, for example, see Stuchly and Stuchly [97] . The model

we use is based on a full-wave solution. The full-mode model is more accurate than the

commonly used capacitance expansion [114,122]. Here, we will develop the theory behind

the shielded open-circuited holder, and present measurement results and uncertainties.

There are various limitations of the measurement method. For low-permittivity materials

90
o

z^R

o

Outer Conductor

Port 1^ ^1 shield z = L

Outer Conductor

o

o

Figure 42. A shielded open circuit termination. The inner and outer radii of conductors are a and b.

i:.i.„..,.

Figure 43. Open-circuited holder for liquids.

{e'j. < 10) 14 mm coaxial line can be used up to approximately 1 GHz; 7 mm coaxial line can be

used up to approximately 10 GHz. For a given coaxial line there are cutoff conditions where

the high-order TMom niodes start to propagate. In addition, due to the bead properties, the

TEii mode propagate. Any deviation of the inner conductor from the vertical will cause an

asymmetry that may couple in to other higher modes. In theory, if the shield is infinitely

long and filled with specimen, the higher-order mode models can be used at frequencies

above cutoff of the shield section. However, the method, in practice, is limited to frequencies

where the higher modes are cut off in the specimen. Sufficiently lossy samples attenuate the

fields so that the measurement sees only a semi-infinite line. Less lossy samples will have

91
multiple reflections in the line. We can modify the algorithms so that the liquid is only in

the coaxial region. This enhances the measurement frequency range.

11.2.2 Model 1: TEM Model with Correction to Inner Conductor Length

Bussey derived a simple and useful model for the shielded open-circuited holder with a

frequency-dependent correction to the length of the inner conductor [114, 116]. For the

special case of a compensated bead where the connector air section length is Li , bead section

length is L2, and the inner conductor length is L3, we have

r = exp (-2(7:L. + 7.^.)) ^^ ' ^Tf^

(174)

^2 + tanh(73L3)

where

P2 =

\ ^ri3)

(175)

The propagation constants of the line sections and bead sections are

UJ

'^^ = ^ —\/<(^- (176)

'^vac Due to the open-circuit termination the inner conductor has an effective length longer than

the measured L3. In this TEM model, the higher modes at the discontinuity are modeled as

a correction in length L3 due to the fringing capacitance at the end of the iimer conductor.

Somlo [122] derived an approximate expression for a correction to the inner conductor length

AL, that accounts for the fringing capacitance:

AL={b- a)(0.6034 -f 0M64x^ + IS.igrE^-^^"^), (177)

where x = bJe^JX^ A is the free-space wavelength, and h and a are the radii of the outer

and inner conductors. AL is an increasing function of frequency. Equation (177) is vahd

for X < 0.3. We assume that the frequency of operation is chosen such that the TMoi

mode is below cut-off in the cylindrical waveguide section. This requires x < 0.383 or

fiGHz) < 0.115/6^5(3)^ [114]. The total length of the holder is L = Li + L2 + L3 + AL.

The effective bead thickness L2 near the connector needs to be determined to minimize

uncertainty.

92
11.2.3 Model 2: Full-Mode Model Theoretical Formulation

Consider a sample in the transmission line shown in Figure 43. The shielded open-circuited

holder consists of three sections. Region 1 contains the bead in the airline, which is at 50 CI,

Region 2 contains the sample, and Region 3 represents the shield region. The problem is to

accurately characterize the shielded open circuit. We will develop a compact expression for

the reflection coefficient in terms of the air line and the sample parameters.

We assume a time dependence of exp{jujt) in the coaxial hne. Due to azimuthal symmetry,

we need to consider only TEM and TMomn modes. We also assume that an incident TEM

wave travels from the input port towards the sample. At the sample discontinuity the TMomn

modes are reflected, and a TEM wave travels into the material. The problem is to obtain

an expression for the TEM reflection coefficient as a function of the specimen's complex

permittivity.

The radial component of the electric fields in Regions 1 through 3 are

oo Ep(i) = E^{p) exp (-710^) + Y^ ^nRnip) exp(7in2), (178)

n=0

oo oo ^P(2) = Yl AnRn{p) exp {-^2nZ) + J] BnRn{p) exp{j2nZ), (179)

n=0 n=0

oo ^P(3) = E QnGnip) exp {-j3nZ), (180)

n=l

where for n = 0,

and for z = 1, 2, if n > 0,

Jio^j^^, (181)

C^ac

Ifin J N

a;2

lab for 2 = 1,2 and for all of the modes in Region 3. If the argument is negative, then the wave

is evanescent and

lin \

a;2

"'in ~ ^riP'ri~2~

(l°"^j

^lah The radial eigenfunctions in Regions 1 and 2 for the TEM mode and evanescent TMorn modes

are, for m = 1,2, • •.

Rm{i) = Cm{i)[Ji{km{i)P)yo[km{i)a) - Fi(A:^(i)yo) Jo(^m(i)a)], (184)

93
and -Ro(i) = Co(i)/p. In eq. (184), Rm{i) are radial eigenfunctions, c^ac is the speed of light

in vacuum and

Co = , (185)

and ^

Ct7,

The eigenvalues are determined from

(186)

Mkm{i)a)Yo{kmii)b) - yo{km{i)a)Jo{km{i)b) = 0. (187)

Gn are normalized eigenfunctions in the cylindrical waveguide and Rn are the normalized

eigenfunctions in the coaxial line defined by

Gn{p) = SnMksnP), (188)

where Sm are normalization constants are given by

The eigenvalues are found when the tangential electric field on the side wall vanishes

Mksnb) = 0. (190)

Therefore k^n = Pn/b, where Pn is the nth root of Jo{x) = 0, for n = 0, 1, 2... . The azimuthal magnetic fields in Regions 1 through 3 are

CXD Hmi) = -Ro{p) exp (-7102:) - Y^ Tn -Rn{p) exp(7i„2), (191)

7lO ri=0 7ln

00 00 ^H^) = 51 -AnRnip) exp {-j2nZ) - Y^ -BnRn{p) exp(72n-^), (192)

n=0 72/1 n=0 72n

oo

^<^(3) = 5Z QnGn{p) exp (-73n-2)- (193)

n=0 73n

94
The tangential components of Ep and H^ must be continuous across the interfaces. If we

match the tangential components of the electric field from eqs. (178) and (179) at 2 = we

obtain the following equations:

Am + Bm = 6mQ + T^, (194)

where m = 0,1,2 ••• . If we match the magnetic field in eqs. (179) and (180) and the

tangential electric fields at 2 = L, we obtain

oo Qm = exp {j3mL) J2 l^ri CXp {-j2nL) + B^ CXp {'^2nL)] < GmRn >, (195)

n=0

or Q = MiA + M2B, (196)

where

M(^l)mn = exp {'^ZrnL) CXp {--f2nL) < GmRn >, (197)

M(2)r,jn = exp (73mi^) eXp (72nl') < GmRn > (198)

If we match eqs. (191) and (192) at 2 = we obtain

Am-Bm= ^^^^ [5mO " T^] , (199)

e27lm

where m = 0, 1, 2. If we match eqs. (192) and (193) at z = L we find

-^ ^

exp {-j3mL)-^Qm < GmRn >= ^n exp {-j2nL) - Bn CXp {j2nL)

,

(200)

e2 rn^o 73m

or M3Q = M4A - M55, (201)

where

M^S)mn = exp (-73ml') < GmRn > ^^^^^. (202)

73mer2

We can solve for the coefficients in eqs. (194) and (199) using

-'in — «2

SnoH + '-^^) + r„(i - '-^^)

^r2lln er27ln

95

(203)
and

Let

and

where

Bn = - (>nO[l ) + i n(H )

A = C-h MeT

B = D + MtF,

A/r

^

[ 1 ^rl72n \ . JVi(6)mn = o

-•• "rnn,

2 V e^27iny

2 \ e^27ln /

C^n —

^"=2

er27ln

r /. gr-l72n ^ 0n0{i- ) er27ln

We obtain the following equation for F:

where

Therefore,

[MgMs + M9M7]f + MgC + MgD = 0,

Ms = M3M1 - M4,

Mg = M3M2+M5.

p • f = f.

(204)

(205)

(206)

(207)

(208)

(209)

(210)

(211)

(212)

(213)

(214)

Applying measured results for the reflection coefficient of the TEM mode To, we use Cramer's

rule in eq. (214) to get a nonlinear equation for e^r of the form To = f{^2r)-

In the measurements, the reference plane must be transformed through the bead and air

sections. The transformed reflection coefficient to be used for the TEM mode in eq. (214) is

Ttrans = ^o exp {-2{jaLi + 76^2)), (215)

96
6-

S, 5

.

; • • 7nim Open-Circuited Holder

- NPL Results

|- Liquid Resonator

1

-

•

•

• • • : :

1 1 1 1

12J..L.1 1 1 3 4 5 Frequency (GHz)

Figure 44. The real part of the permittivity for 1-butanol.

where Li and L2 are the effective axial lengths of the air and bead sections of the connector.

The bead is usually nonuniform, so an effective length needs to be determined from a standard

measurement. A way to ehminate the bead rotation problem is to measure the sample holder

in air and divide that result into the result for the measurement with the sample in the holder.

The TEM mode admittance and reflection coefficient are related by

Yo-Y

ro =

^0 +

^' (216)

where Yq is the admittance of the air-filled line and Y is the admittance with the sample in

the line.

In Figures 44 and 45, permittivity and loss tangent data for butanol are plotted for

measurements with an open-circuited holder and the TEoi resonator.

11.2.4 Uncertainty Analysis for Shielded Open-Circuited Holder

Uncertainties include the reflection coefficient, conductor and bead dimensions. Since this

specimen holder is usually used for measurements of hquids, gaps between conductors and

specimens are neglected. These independent uncertainty sources are tabulated in Table 11.

97
tog

1

• 7mm Open-Circuited Holder NPL Results Liquid Resonator

*

t

J l__l L

' t .

J I—1—1—I—l_I—I I I L-l I 1—1 I I—I

' ' ' I L-l—1—I—I

Lj—I L_J I 12 ' ' I l_J E_] L. 3 4 5

Frequency (GHz)

Figure 45. The loss tangent for 1-butanol.

Table 11. Uncertainties for shielded open-circuited holder

Error Source Magnitude

Inner conductor length ±0.0002 m

Outer conductor diameter ±0.0002 m

Inner conductor diameter ±0.0002 m

Bead thickness ±0.0004 m

Max(|r|) ±0.01

Max(l^l) ±0.1°

98
In order to evaluate the uncertainty introduced by the measured scattering and length

parameters, we assume that a differential uncertainty analysis can be used. We assume that

the total RSS relative uncertainty with no cross-correlation can be written as

^<2 1

^r2 ^<2a|-pA^ , f^<2.n V , f9e'r2.rV , f9<2

VA l^^'^lj ^ [W.^'-) ^ l^^^J ^ laZf ^^3j , (217)

Ae';, 1

e"r2 -r2 \ (lff-i--0'*(S:«-)'*(t-)"-(g"-)' <»)

where A^n is the uncertainty in the phase of the reflection coefficient, A|r| is the uncertainty

in the magnitude of the reflection coeflicient, and AL, AL3 are the uncertainties the in

lengths of the inner conductor and bead. The bead holds the center conductor in place and

is usually made of polj^etrafluoroethylene.

The independent variables are the dimension of the conductors, the thickness of bead,

and the phase and magnitude of scattering parameters. The uncertainties used for the

scattering parameters depend on the specific ANA used for the measurements. If <Ts is the

standard deviation in N measurements of the independent variable r, then the standard error

is Ar = as/y/N- The partial derivatives in eqs. (217) and (218) and complete uncertainty

analysis is given in Baker-Jarvis et al. [116].

11.3 Open-ended CoELxial Probe

11.3.1 Theory of the Open-ended Probe

Over the years, the open-ended coaxial probe has been studied extensively from both theoretical and experimental aspects. Open-ended coaxial probes are commonly used as nondestructive testing tools and for liquid measurements over a frequency band of 200 MHz to

20 GHz. The method, although nondestructive, does have limitations. For example, the

fields at the probe end contain both Ez and Ep components. If there is an air gap between

the specimen and the probe, the discontinuity in the normal electric field causes a large

99
(1)

p

r

ground plane

Mro gro air gap (2)

/^^.^e" specimen

termination (4) e;^, /x*^

.2 =

.z = d

.z=L + d

Figmre 46. The open-ended coaxial probe over a specimen with an air gap between specimen

and probe. In Hquid measm^ements there is usually no gap.

error in the predicted permittivity. For this reason, the probe has been used primarily for

liquid and semisohd measurements, where good contact can be obtained. Equations for the

reflection coefficient can be obtained from solutions to the Helmholtz equation [98-100]. In

hquid measurements, the coaxial probe is submersed in the specimen and the reflection coefficient is measured. The permittivity is then obtained using a Newton-Raphson root finding

technique. The size of the probe is related to the frequency band of interest. A typical

probe with hftoff capabihty is shown in Figure 47. The higher the frequency, the smaller the

coaxial probe size. We use a full-mode model of the coaxial probe. In matrix form we can

write the reflection coefficient vector as Q • r = p, (219)

or

/ ^ ^mn^ n -* mj (220)

n=0

where

e;372 • e;s72 • e;s72 e.*c72(e272^((i + |iS) + e(i - Ji^)) + (C^ - 1) - 0(1 + 1£)))

+^—Smn, (221)

7n(c)

100
Figure 47. Dielectric coaxial probe for liquids.

Jo

OO ^ 6* Do(e^^^'^((l +

,73- ~rg e;... ) + e(i - 5!^)) + (d - i;^) + e(i + gf)))^^

£;o72(e2-^''((i + 1?5) + e(i - !?5)) + ((g'"'

s72 3 72- 72 1)- 0(1 + 1^)))

+—5^0, (222)

71

for m, n = 0, 1,2, ...A'^, where N is number of modes. We define propagation coefficients

7- = j^kf - C^, for 3fi(A;0 > C, ^ = 2, 3, 4, and 7^ = y/C^^ if ^^(/Ci) < C- For a shorted

termination at z = d + L, Q = exp (—273L). For a material termination at this position we

define O = exp (—273L)(1 — 62)/(l + 62), where 02 = (e*s74)/(e*d73). In the limit of semiinfinite material, L —> 00, and since e"^ > we obtain O —> 0. The reflection coefficient of the

TEM mode is of primary interest since the other modes are evanescent in the coaxial line.

The permittivity is calculated using a Newton-Raphson procedure on Fq. An uncertainty

analysis is given in reference [1].

11.3.2 Uncertainty Analysis for Coaxial Open-ended Probe

Uncertainties in open-ended coaxial measurements include calibration and measurement uncertainties [99]. We consider that the uncertainty in a measurement due to phase, magnitude

and Uft-oflF. These independent sources of error axe tabulated in Table 12. We consider that

calculated permittivity is a function of the following independent variables: e*3(|Fo|, ^o,*^)-

The estimated magnitude and phase uncertainties introduced by the measiuement standards

are combined with the network analyzer uncertainties. A worst-case differential imcertainty

101
Table 12. Sources of error for the open-ended coaxial holder

Error Source Magnitude

Outer conductor diameter ±0.0002 m

Inner conductor diameter ±0.0002 m

Permittivity of bead ±0.02

Max(|r|) ±0.01

Max(l^l) ±0.1"

analysis assuming no cross-correlations is

N(lfeH^^(SS^^(^^')^- ^'''^

A similar equation can be written for e^'^. Implicit differentiation can be used to find the

necessary derivatives. The details are given in Reference [99].

12. Dielectric Measurements on Biological Materials

12.1 Methods for Biological Measurements

The dielectric properties of human tissues have been studied extensively over the years.

Therefore the range of conductivities that occur in the human body are well known. In

the past, phantoms have been used for: (a) studies on the electromagnetic interaction with

human tissue, (b) health effects of microwaves, and (c) interaction of wireless transmitters

with human tissue [123]. The effects of electromagnetic waves on biological tissues is related

to the conductivity and dielectric properties of tissues. The application of electric fields

produces an overall increase in the molecular kinetic energy that is manifested in small

amounts of heating, bond breaking, and production of free-radical, unpaired electron spins.

Interactions of magnetic fields are less strong and are related primarily to paramagnetic

effects and magnetic elements in the tissue.

The body consists of muscle, fat, bone, organs, and cavities. Since the dielectric properties of human tissues are heterogeneous, materials can be made that simulate the electrical

response of a particular area of the human body. Most of the previously developed phantoms attempted to mimic both the conductivity and the permittivity of the human body.

102
Gabriel [124] has compiled extensive data on various human tissues and also developed a

database. This database reports measurements on various human organs such as head, brain,

muscle, fat, and bone.

12.2 Conductivity of High-Loss Materials Used as Phantoms

In this section we summarize the measurements on high-loss liquids that simulates the properties of biological tissues. For comparison purposes, typical ac conductivities and permittivities of the human body are shown in Figures 48 and 49. For the measurements on liquid

conductivity, we used a dc conductivity probe. In Figure 50, the conductivity of KCl in

500 ml deionized water is plotted versus concentration for minor variations in temperature.

The uncertainties in the liquid measurement are due to uncertainties in the measurement

device and uncertainties in temperature variations. Our lab has a temperature stability on

ifcl^C. We found for the conductivities: at cr = 0.5 the Type B expanded relative uncertainty

U = kuc = 0.02 (k-2). The dependence of the conductivity on the KCl concentration is

Table 13. Previous work on human phantom materials.

Frequency Materials Citation and Date

1 Hz to 1 GHz TEATFB Broadhurst et al. 1987 [125]

900 MHz graphite and resin Kobayashi et al. 1993 [126]

10 to 100 MHz glycine, NaCl Hagmann et al. 1992 [127]

500 MHz to 3 GHz sihcone rubber, carbon black Nikawa et al. 1996 [128]

200 MHz to 2.45 GHz gel, aluminum powder, NaCl Chou et al. 1984 [129]

750 MHz to 5 GHz polyacrylamide, NaCl Andreuccetti et al. 1988 [130]

5 to 40 MHz agar, sodium azide, PVC powder Kato et al. 1987 [131]

10 to 50 MHz gelatine and NaCl Marchal et al. 1989 [132]

300 MHz to 1 GHz liquid CENELEC Standard

nearly linear (see Figure 50) . Conductivities close to that of the human body components

can be reproduced by either carbon-black composites, glycine composites, or various salty

solutions. The liquids can be mixed easily to obtain the desired response. These liquids also

show good time stability in the conductivity and a predictable temperature response. The

103
1.2

1.0

^^_^ 0.8

^>: 0.6

<-j =3 -oO 4 O

0.2E*

0.0

blood bone cortical

-rir- lung inflated -©— muscle

grey matter -T- skin dry

heart

liver -»- stomach

lung deflated

10 10~

Frequency (Hz)

rrrrnff

10'

Figure 48. Conductivity measurements (S/m) on various body components [124] (no uncertainties assigned).

CO

blood bone cortical grey matter

heart

liver

ii lung inflated o muscle

-V— skin dry

lung deflated

10 10^

Frequency (Hz)

Figure 49. Permittivity measurements on various body components [124] (no uncertainties

assigned).

104
KCI Solution

1.2 KCI added to 500 ml deionized water

O = -0.002 + (.322 X(gmnns ofKCLl] (20.5 - 21 .7 C) O a = Q.011 + [.ae.-) X (g films of KCL)] (22.0 -22.3 C) = 0.034 +(354 X{granisofKCL)J (22.6-22.8 C)

I I I I I I I I I I I I I 1 I

KCI (grams)

Figure 50. Conductivity of potassium chloride solution. Also shown are regression fits, in

the straight lines, for the conductivity as a function of concentration.

1.2

1.0

M

c^ 0.8

-^>a 0.6 ^nrja

oO 0.4

0.2 -

0.0

Percent carbon black in silicone by weight

Figure 51. Conductivity as a function of carbon-black concentration for Cabot Black Pearls.

105
1.2

1.0 ^N

f, Vi 0.8 ^

>

o 0.6 ^

T5a

oU 0.4

Silicone + 5% Carbon Black by weight

Silicone + 6% Carbon Black by weight

Silicone + 7% Carbon Black: measurements of6 different samples

Silicone + 8% Carbon Black

Silicone + 10% Carbon Black

[* D D DDD DDDD DDD DDD DDDD D D D DD D 1 3

k k kkk kkiiA4 kkk kkk kkkk k A AAA An

A AAA AAAA AAA AAA AAAA A k AAA An

23456 23456 23456

10^ 10" 10^ 10^

Frequency (Hz)

Figure 52. Conductivity as a function of caxbon-black concentration for Cabot Black Pearls.

carbon-black conductivity exhibits a percolation threshold where the conductivity increases

abruptly, as shown in Figures 51 and 52. There is a need to develop new phantom materials

that are stable over time and exhibit the required electrical behavior over frequency bands

of interest.

13. Capacitive Techniques

13.1 Overview of Capacitive Techniques

Capacitance techniques (see Figures 53 and 54) are useful at frequencies extending from fractions of a hertz to megahertz frequencies. If the structures are deposited on chip so that they

have very small conductors, specimens can be measured up to gigahertz frequencies [133,134].

Capacitance models work well if the wavelength is much longer than the conductor separation. Of course, most transmission lines such as coplanar waveguide, microstrip, striphne,

and coaxial line can be used to make capacitive measurements.

A difficulty with capacitor measurements is the effects of the fringing-fields and, at low

frequencies, electrode polarization. The fringe field is usually partially eliminated by using

guards or making another measurement using a reference material or air. The fringing field

106
is minimized if the potentials of the guards and center electrode are at the same potential

and only the feed potential is at an elevated potential.

The capacitance for a parallel plate with no fringing fields near the edges is given by

The conductance at low fi:equency is given by

G = u

e"A

d

'

(224)

(225)

This model assume no fringing fields. A more accurate model would include the effects of

fringing fields. Use of guard electrodes as shown in Figure 53 minimizes the effects of the

fringe field.

Capacitor Nfeasuraneot

^^aidGap GuaidedHectrode GuandHectnode

\ T I

Samfii

Uiguarded Electrode

Figure 53. A specimen in a capacitor with electrode guards.

The permittivity can be obtained from measurements of C and G and is given by

^ C-jG/uj

13.2 Capacitance Uncertainty

In a capacitor measurement, the components of the permittivity are given by

^r ~ ^ l^airi

and

4' = ^-.

107

(226)

(227)

(228)
Table 14. Uncertainties for capacitive fixture

Error Source Magnitude Uncertainty

Plate separation ±0.0002 m

Area of plate ±0.0002 m

Max {C/Cavr) ±0.01

Max {G/Cair) ±0.01

The independent sources of uncertainty are tabulated in table 3. These include uncertainties

in C/Cairi G/Cair and fringing field effects, parameterized by a variable F. The total relative

uncertainty in the real part of permittivity is

de', diC/Cair) A4 =

^ (AC/a„)^ + (_^^ii^AFj , (229)

and

Ae:' = ^(AG/(.a.))^+(^^^M^A^)^. (230)

13.3 Electrode Polarization and Permittivity Measurements

13.3.1 Overview

Measurements on conducting liquids are complicated by ionic conductivity and by the effects

of electrode polarization [135]. Electrode polarization is caused by the build up and blockage

of conducting ions on the capacitor plates, producing a capacitive double-layer electric field. The double layers on the electrodes produce a very large capacitance in series with the

specimen under test. Since the electrode capacitance is not a property of the material under

test, it must be removed from the measurement. The presence of electrode polarization

results in the measured real part of the permittivity being many orders of magnitude greater

than the correct value.

The effects of electrode polarization can be modeled by the following relations [136]

R^Rs + Rp + Rsio^R'^C^, (232)

108
current probe

double layer

specimen

double layer

current probe

jzoltage probe

_i^oltage probe

Figure 54. Capacitor with specimen for minimizing electrode polarization.

where C, R are the measured capacitance and resistance, Cp^ Rp are the electrode doublelayer capacitance and resistance, and Cg, Rs are the specimen capacitance and resistance.

A way to partially eliminate electrode polarization is by measuring the capacitance at two

separations di, d2^ capacitances Ci and C2, and resistances Ri and R2. Since Cp is the same

for each measurement and Cg can be scaled as Cs2 = {di/d2)Csi, we can obtain the specimen

capacitance using

C-[s —

C2 — Ci^

dx _ til. d2 ^

R^ (233)

Another way of minimizing the effects of electrode polarization is to coat the capacitor plates

with platinum black [137]. This lessens the influence of electrode polarization by decreasing

the second term on the right hand side of eq. (231). Probably the best approach is to bypass

the electrode polarization problem altogether and use a four-probe capacitor system.

13.3.2 Four-Probe Technique

The four-probe capacitance technique (see Figure 54) overcomes electrode problems [138]

by measuring the voltage drop away from the plates and thereby avoiding the double layer.

The permittivity for a substance such as DNA contained in a solution, is calculated by

AZ' = -±(!2)2A(a;."),

Co cr (234)

109
AZ" = -l.i^)\ue% (235)

Go cr where Ae" is the difference in loss between fluid with material versus fluid alone and AZ'

and AZ" are the differences in real and imaginary parts of the impedance.

14. Discussion

In this technical note we have attempted to review all of the relevant elements of metrology

for measuring lossy materials. Some methods have been neglected, for example, many of

the nondestructive measurement methods. We have seen that only specific types of reso- nant methods work for lossy materials, and as a result some variations of transmission-Une

methods must be used.

15. Acknowledgments

We acknowledge discussions with Ed Kuester and Dave Hill, and Jerzy Krupka for designing

and writing software for the resonant liquid fixtures. We acknowledge the support of the

Departments of Justice, Homeland Security, and Dr. George Lieberman of the Office of Law

Enforcement Standards of NIST.

110
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A. Review of the Literature on Dielectric Measurements of Lossy Materials

In this section we briefly overview the existing literature on dielectric and magnetic material

measurements of a wide variety of lossy materials. General references include the databases

at the Center for Information and Numerical Data Synthesis Analysis (CINDAS) and Microelectronics Packaging Materials Database [139]. Both of these databases compile dielectric

measurement results on a large number of materials.

For the interested reader we cite various references to material measurements below. We

have included the most robust studies for which we have the most familiarity and concentrated on sources that include uncertainties.

• Measurements on concrete, rock, and asphalt are given in References [79, 140-160].

Figures 55 to 58 include measurements we performed on concrete and mortar with a

shielded-open-circuited holder.

• Measurements on plastics are given in References [9, 161-172].

• Measurements of organic materials such as wood depend strongly on the water content.

Measurements on various types of wood are given in References [173, 174]. Dielectric

137
18

Day 1

.J I L I 12I L. -1 \ I I 1 L_L I I I L. J I I I L. 3 4 5

Frequency (GHz)

Figure 55. The real part of the permittivity for (5000 PSI) concrete as a function of cure

time.

properties of wood, as a function of moisture content, axe given below.

• Many measurements on building material and in-door reflection data are given in

References [79, 175, 176, 259].

• Dielectric properties of grains and food products are given in [177-179]. Dielectric

properties of some foods are summarized in References [180-184].

• Dielectric properties of meat are given in References [185].

• The dielectric properties of various liquids are given in References

[3, 100, 117, 118, 137, 186-197].

• Ground-penetrating radar methods and measurements are summarized in References

[77-83,142,198].

• Nondestructive techniques are summarized in References

[199-207].

138
= .

CO £^>5 -5

a

>

u

CmO

a

-10-

15-

20

\ ^^^^^^^^^^^^^^=^=^

: -t-Dayl

-^-Day2

-4- Day 6

" -^ Day 15 -O- Day 28 * -B- Day 38 ^ 1 1 1 1 1 1 J r 1 1 1 1 ] 1 1 1 1 i 1 1 1 1 1 1 -25 12 3 4 5

Frequency (GHz)

Figure 56. The imaginary part of the permittivity for (5000 PSI) concrete as a function of cure time.

• Dielectric measurements on deionized water are given in References [208-215]. Water is commonly modeled by a Debye curve. For frequencies below 100 GHz we can use [208]:

e^ = Coo + , . 1-2- f (236)

ID

12 n _ where / is in GHz, jd = 20.27 + 146.5^ + 3149^, 9 = 1^ 300/(273.15 + T), e, =

77.66 - 103.3^, eoo = 0.0666^.

Measurements on salt water are given in References [216-219].

Measurements on ice are given in Reference [220] . The real part of the permittivity is

approximately e'^ = 3.15. The loss factor is shown to be

4 = 6.14 X 10-^F(1.118(^ -f 1))/g, (237)

where fc is the frequency in gigahertz and F = xe^ /{e^ — lY, 9 = 300/(273. 15+Tc) — 1,

where Tc is the Celsius temperature.

139
«- 1^ 1^ '^ '^'^'^'^-r^flDD 1

' 12 -I I I I I I I I I I I L. 3 4 5

Frequoicy (GHz)

Figure 57. The real part of the permittivity for mortar cement as a frinction of cm^e time.

• Dielectric measmrements of snow are smnmarized in Reference [221]. The permittivity

of snow depends on the density. Prom the data of Hallikainen at al. Reference [221]

we can fit the permittivity of snow at 10 GHz as a function of density:

e; = 1 + 1.2(p,/1000) + 1.2(/9/1000)J, (238)

where the density is in kg/m^. The loss factor at 18 GHz is modeled as

e'; = 15.5m^, (239)

where m^ is the percent liquid water content; in these formulas m^ ranges from to

16.

• Soils have been measured by many researchers as functions of organic, sand, clay and

water content [222-232] . Although soils vary by the amount of moisture, loom, sand,

and clay the variation of a typical soil as a function of the percent of volumetric moisture

content (P^) has been fit by Reference [232] at 78 MHz and at 20 °C as

e' = 0.00189P^ + 0.690P^ + 0.3. (240)

140
10

a

>

u

o

§>

Day 1 Day 2 Day 3 Day?

Day 16 -B- Day 29 Day 39

J I I L. 2 3

Frequency (GHz)

Figure 58. The imaginary part of the permittivity for concrete as a function of cure time.

• Dielectric properties of oil shale and char have been characterized by a number of

authors in References [233-238]. Dielectric measurements of coal are given in [239].

• Measurements on biological phantoms are summarized in References [124-132,240].

• Measurements of biological tissues, DNA, proteins, and reference liquids are summarized in References

[115,123,241-243].

• Dielectric properties of composites and building materials axe summarized in References

[176,244-258,260-270].

The dielectric properties of various building materials were measured by our group and

reported in Table 15

141
Kind of

wood

drv wood moist wood

e,. lanS conleni of

water £, lano

Beech 1.888 2.6E-02 9.5% 2.536 U5E-01

Ash 2.(M5 3^E-e2 10.2% 2.901 1.5E-01 Walnut I.S54 3.0E-02 10.]% 2.486 1,4E^1

Malioeanv 1.830 2.9E-02 10.9^t. 2„520 L8E-01

Svcamore 1.885 3.0E-02 10.3% 2.547 LSE-Ol

Birch 2.072 3.5E-02 10.3% 2.S60 1.8E-01

Pine 1.721 2.9E-e2 10.5% 2.211 1.3E-01 Alder 1.900 3.3E-02 1 1 .6% 2.510 1.5E-01 Oak 1.962 3.5E-e2 10.8% 2.584 U5E-01

Chern- 1.954 3.r.E-(B2 9 8% 2.697 1.5E-01

Figure 59. The permittivity of various types of wood at 3.6 GHz (Krupka personal communication 2004).

B. Gap Correction in Dielectric Materials

B.l Coaxial Capacitor Model for Dielectric Materials

Consider a capacitor consisting of layers of dielectric and layers of air in a coaxial line [59]

The dimensions are shown in Figure 60. We treat the system as capacitors in series, so

1 _ J_ J_ J_

Cm Ci C2 C3

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

F ^

and the potential difference between the conductors is given by

-6

(241)

(242)

V=- f E{r)dr.

Ja (243)

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

27reL

C = Inf (244)

142
Table 15. Dielectric measurements on various building materials between 700 MHz and 10

GHz.

Material Preq.(GHz) Temp.(C) < tan 5 Method Density(g/cc)

White marble 1.42 22 ±0.5 8.64 5.44 X 10-^ SPR 2.68

Stained ceramic tile 1.44 22 ±0.5 3.52 2.52 X 10-3 SPR 1.65

White marble 2.02 22 ±0.5 8.63 5.09 X 10-^ SPR 2.68

Stained ceramic tile 2.04 22 ±0.5 3.50 2.54 X 10-3 SPR 1.65

Green granite 2.03 22 ±0.5 7.75 1.44 X 10-2 SPR 3.02

White marble 3.21 22 ±0.5 8.61 5.99 X 10-4 SPR 2.68

Stained ceramic tile 3.31 22 ±0.5 3.56 2.82 X 10-3 SPR 1.65

Green granite 3.31 22 ±0.5 7.76 1.93 X 10-2 SPR 3.02

White marble 5.44 22 ± 0.5 8.51 1.13 X 10-3 SPR 2.68

Stained ceramic tile 5.50 22 ±0.5 3.24 3.78 X 10-3 SPR 1.65

Green granite 5.39 22 ±0.5 6.92 1.92 X 10-2 SPR 3.02

White marble 9.59 22 ± 0.5 8.56 1.05 X 10-3 SPR 2.68

Stained ceramic tile 9.81 22 ±0.5 3.24 3.77 X 10-3 SPR 1.65

Glazed tile 9.80 22 ±0.5 5.28 7.92 X 10-3 SPR

Unglazed tile 9.90 22 ±0.5 3.79 4.11 X 10-3 SPR

Green granite 9.41 22 ±0.5 6.90 1.46 X 10-2 SPR 3.02

Red brick 9.56 22 ±0.5 10.78 4.95 X 10-3 SPR

Dry pine wood 1.46 22 ±0.5 1.84 9.11 X 10-2 SPR

Linoleum tile 0.71 22 ±0.5 5.93 3.96 X 10-3 SPR

Asphalt shingle 0.71 22 ±0.5 3.64 2.01 X 10-2 SPR

Plywood 0.71 22 ±0.5 2.91 1.08 X 10-^ SPR

G)T)sum board 0.71 22 ±0.5 2.81 1.89 X 10-2 SPR

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

i?4 In ^ In ^ In R2

Ri R

pi

'- +

El

R2 hi^

+

R3

(245)

where e^^, e(.^ are the relative corrected and measured values of the real part of the permittivity and e'j.1 is the real part of the permittivity of the air gap. The corrected permittivities

are

, _

In (f)erg(l + tan2 Sjg) - In (g^)erm(l + tan2 bjm)

(hi (|!))(1 + tan2 big){\ + tan2 bim)

'Q(lf) g'-g +

^—^) firm (ln(g|))2e,^(l+tan2 5i^) (In (^))2e,g(l+tan2 5,3)(246)

143
Figure 60. A coaxial specimen in holder with air gaps near conductors with diameters

denoted by Di.

The corrected imaginary part is - In {^)erg{l + tan^ 5ig) tan J^^ + In (^^)erm(l + tan^ Sim) tan^i

e -^ = Crc tan 5,

ig

(In (g))(l + tan^ 5ig){l + tan^ 5im)

An approximate expression is given by

l"(it) ^r. l"(|f^) ^'-"^ (ln(^))2e^^(l+tan2 5i„) ^ (In (^))2e.p(l+tan2 5^^) (247)

re rnr L,

^3 — Crm-^l

where

tan (5c = tan(5^[l + e^^-^],

Li = hi— + hi—

L2 = hi

L3 = In

-R3

(248)

(249)

(250)

(251)

(252)

4

Equation (248) breaks down when e[^ > L3/L1. Correction data are shown in Figure 61.

144
1.56

Ec'/Em'

0.4045

Figure 61. The gap correction calculated for various values of e^, where i?2, Ri are the radii

of the inner conductor and specimen, respectively.

B.2 Rectangular Waveguide Model

In rectangular waveguide the electric field has two components. The first component E^ is

across the thin dimension and is sensitive to gaps at the sample-waveguide interface. The

Hz component has a discontinuity at the sample-air interface. This causes an increased

sensitivity to sample-length errors. For the case of a rectangular guide of narrow dimension

h with specimen thickness b — w and air gap width w{see Figure 62),

berg{l + tan^ 6ig) - wcrmi'i- + tan^ dim)

^rr. 62 e +

W^Crm

€ir. — Cj> tan 0. —herg{l + tan^ 5ig) tan 5im + werm{^ + tan^ dim) tan 5.

(253)

ig b^e (6 - W){1 + tan^ 6,g){l + tan^ ^..n )( (,_.)..I(t;un^,^^) + (,_.). ^aT^an^...) ) (254)

145
Figure 62. specimen in rectangular waveguide.

C. Gap Correction for Magnetic Materials

C.l Coaxial Capacitor Model for Magnetic Materials

For the calculation of the gap correction for the permeability a pure inductance model is

useful. We model the transmission line as a series of inductors for the E-field gap

(255)

where c, m, and air denote corrected value, measured value, and air space. Therefore the

corrected value is

The inductance is the flux penetrating the circuit divided by the current flowing in the circuit

/'

where

Ampere's law gives

-I (j)= I B-dS.

fB.-dI'=I,

146

(257)

(258)

(259)
50

-o

a>

SBo

10 15

Gap (|LLm)

20 25

Figure 63. Corrected permeability and permittivity as a function of inner conductor gap for a 7 mm specimen. The gap around the outer conductor is assumed to be zero. In this case the uncorrected (measured) value was /x^ = 25, e^ = 25.

which yields

Therefore

so

Bj. =

27rr

(t>= —fi'I]nb/a,

C = -—11 In h/a.

Therefore we can write for the corrected permeability

fi'^ji hi RjRi - [hi R2/R1 + In R4/R3]

f^cR lnR3/R2 _ „ lniR,/R,)

h'cR — H-mR]

(260)

(261)

(262)

(263)

(264)

Gap correction data are given in Figure 63. The corrections for permeability in coaxial line

are much less than for permittivity. This is due to the fact the azimuthal magnetic field is

continuous across the discontinuity, whereas the radial electric field is discontinuous across

the discontinuity.

147
Rectangular Waveguide

Figure 64. Rectangular waveguide.

C.2 Waveguide Model for Magnetic Materials

The TEio fields is rectangular waveguide are

lUJa TTX h„ = iiosin —

,

TT a

'y

ikuafi . TTX six = -no sm —

,

TT a

Hz = liocos —

.

a

(265)

(266)

(267)

(268)

By looking at the fields we see that the electric field is perpendicular to the X-Y plane

and therefore the permittivity is strongly influenced by the gap. The magnetic fields go

to zero at the Y-Z and X-Z planes, so the correction for permeability should not be large

unless there are significant gaps. For magnetic materials, the z component of the magnetic

field is discontinuous across the sample faces. Therefore we expect a strong dependence of

uncertainty in sample length on the uncertainty in permeabihty.

148
C.3 Magnetic Corrections for Gaps in the X-Z and Y-Z Planes

For magnetic materials in waveguide for the TEio mode, gaps in the X-Z plane are not severe

because the magnetic field circles around, and is tangential to small gaps (as shown in Figure

64). The corrections can be obtained using inductances in series. These are

l^'cR = f^'LR Q) • (270)

C.4 Magnetic Corrections for Gaps Across the X-Y Plane

These corrections can be obtained using inductances in parallel. As a result, uncertainties

in the sample length can be amplified because of the discontinuity in the permeability from

the sample to air. These corrections for permeability would be of the same form as for the

electric-field correction in the previous section.

C.4.1 Mitigation of the Effects of Air Gaps

We can minimize the effects of air gaps by placing a conducting material in the air gap. This

material may be a conducting paint or a conducting grease. The conducting material will

change the line impedance and line loss to a degree. However, for relatively small gaps, the

improvements in dielectric and magnetic property measurements far outweigh any changes

in the line impedance. The loss measurement will be influenced by this procedure. The

effects of gaps in the X-Z plane on the permittivity can be reduced if the gap is filled with

a of low-loss material with a high-permittivity. This works since the gap correction acts as

capacitances in series.

D. Permittivity and Permeability Mixture Equations

Many lossy materials can be fit with a modified Debye response

€ [uj) = e^ + ——-.—rjz^-z , 271)

149
eoo is the optical permittivity, Cg is the static permittivity, and r is the relaxation time.

If a == 0, then the response is Debye with dc conductivity and if o; 7^ it is a Cole-Cole

response. Robert [271] has shown that Cole-Cole fits the relaxation data of concrete data

very well.

We can readily estimate the permittivity of a mixture of a number of distinct materials.

The effective permittivity of a mixture Cg// of constituents with permittivity e^ and volume

fractions di can be approximated in various ways. For an arbitrary number of constituents

the following mixture equations apply [272]:

34 (4 + 2el)-

The Bruggeman equation [273] is useful for binary mixtures:

p' — f' f' — f'

(272)

''7f^='^ZT^^ (273)

or the Maxwell-Garnett mixtiue equation [273] can be used:

where €[ is the permittivity of the matrix and e'2 is the permittivity of the filler [246]

.

Over the years a number of phenomenological equations have been developed to model

liquids with permanent dipole moments. One of the most general of these is the HavrihakNegami equation [18]

In the special case where a = 0, /? = 1, the classical Debye distribution is obtained. The

Cole-Cole distribution is obtain when P =^ I. The Cole-Davidson model is obtained when

a = 0. In the cases of the Cole-Cole and Debye models, if e^ is plotted versus e" a semicircle

is realized. Cole-Cole plots can be very informative for many Hquids. Since we know that

to a very good approximation many liquids are described by this model, we can use the

distribution to extrapolate dc permittivity, thereby minimizing electrode-polarization eflFects.

150
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