N\srUNITED STATES

DEPARTMENT OF COMMERCE

TECHNOLOGY ADMINISTRATION

NATIONAL INSTITUTE OF STANDARDS

AND TECHNOLOGY

NATL INST, OF STAND & TECH R.I.C.

AlllDM aD73ET

SWlST)«f|t(» 1) f.T REFERENCE

NISI

PUBLICATIONS

NIST Technical Note 1354

The NIST 60-Millimeter Diameter

Cylindrical Cavity Resonator:

Performance Evaluation for

Permittivity Measurements

Eric J. Vanzura

Richard G. Geyer

IVIichael D. Janezic

The NIST 60-Millimeter Diameter

Cylindrical Cavity Resonator:

Performance Evaluation for

Permittivity Measurements

Eric J. Vanzura

Richard G. Geyer

Michael D. Janezic

Electromagnetic Fields Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

August 1993

A:

*

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, Director
National Institute of Standards and Technology Technical Note

Natl. Inst. Stand. Technol., Tech. Note 1354, 236 pages (August 1993)

CODEN:NTNOEF

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1993

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

List of Figures vii

List of Tables xii

Abstract 1

1 Introduction 2

1.1 Definition of Permittivity 2

1.2 Operation of Mode-Filtered Cylindrical Resonators 3

1.3 Methods of Measurement 5

2 Theory of Microwave Resonators 7

2.1 Introduction 7

2.1.1 Fields in a Right Circular Cylindrical Cavity 13

2.1.2 TE Modes 15

2.2 TEoip Mode-Filter Cylindrical Cavity 18

2.2.1 NIST 60-mm X-Band, Mode-Filtered Cavity Resonator . 21

2.2.2 Evaluation of Real Effective Permittivity 23

2.2.3 Evaluation of Dielectric Loss and Quality Factor 25

2.2.4 Port Coupling Coefficients for Determining Unloaded Cavity Q-Factor 30

2.2.5 Cavity Conductor Quality Factor 33

2.2.6 Separation of Cavity Endplate Losses from Side-Wall Losses

in Helical Cavity 38

iii
iv CONTENTS

3 Cavity Resonator Design 40

3.1 Helically Wound Circular Waveguide 41

3.2 Coupling Flange 41

3.3 Apertured Coupling Endplate 42

3.4 Tuner Flange 42

3.5 Tuning Endplate Assembly 43

3.5.1 Sliding Base 43

3.5.2 Sliding Base Extension 43

3.5.3 Tuner Endplate Assembly 44

4 Resonator Evaluation 45

4.1 Dimensional Characterization 46

4.1.1 Effective Electrical Length and Diameter 46

4.1.2 Sensing Micrometer Accuracy 48

4.1.3 Tuner Assembly Attachment Repeatability 50

4.1.4 Iris Perturbation 51

4.1.5 Resistive Loss Perturbation 56

4.1.6 Summary of Diameter and Length Uncertainties 56

4.2 Evaluation of Q 57

4.2.1 Port Losses - Coupling Factor {k) 58

4.2.2 Q Corrections: Fixed-Length Method 59

4.2.3 Q corrections: Fixed-Frequency Method 60

4.2.4 Stability of Q and Coupling Factors 61

4.2.5 Repeatabihty of Q and Coupling Factors 61

4.3 Systematic Uncertainty 65

4.3.1 Simplifying Assumptions 65

4.3.2 Resonance Model 66

4.3.3 Interfering Non-T^oi Modes 71

4.3.4 Imperfect Geometry and Wall Resistivity 71

4.3.5 Resonance Fitting - 5ii/522 Versus S21/S12 Measurements 73
CONTENTS V

5 Software 74

5.1 Instrument Control- MEAS_RES 74

5.1.1 Measuresweep Subprogram 75

5.1.2 Measurelist Subprogram 75

5.1.3 Measuretrigger Subprogram 75

5.1.4 Measurelistrig Subprogram 75

5.1.5 Measuredumdata Subprogram 76

5.1.6 Measurediskdata Subprogram 76

5.1.7 Fit_sparms Subprogram - Resonance Fitting 76

5.1.8 Levenberg-Marquart CSUB 77

5.2 Data Analysis - CAVITYPROG 78

5.2.1 Calc_c Subprogram - Speed of Light 79

5.2.2 Var_length Subprogram 80

5.2.3 Var_f req Subprogram 81

5.2.4 Cavity_dimens Subprogram 82

5.2.5 Calc_inodes Subprogram 84

5.2.6 Calc_freqs Subprogram 85

5.2.7 Calc_eps_re Subprogram - Permittivity Calculation ... 85

5.2.8 Calc_tand Subprogram 86

5.3 Other Software 87

6 Laboratory Environment and Equipment 88

6.1 Laboratory Environment 88

6.2 Laboratory Equipment 88

6.2.1 Network Analyzer 88

6.2.2 Computers 89

6.2.3 Dimensional Measurement 89

6.3 ANA Waveguide Calibration 90

6.3.1 X-band TRL Cahbration Kit 90

6.3.2 Calibration Procedure 92

6.3.3 Waveguide Port Extensions 93
vi CONTENTS

7 Sample Specifications and Measurement Procedures 94

7.1 Sample Specifications 94

7.2 Optimum Sample Thickness 95

7.3 Grinding and Lapping 95

7.4 Sample Cleaning 96

7.5 Measurement Procedures 96

7.5.1 Port Coupling Factors and Unloaded Qq 96

7.5.2 Fixed-Frequency Method of Permittivity Measurement . 97

7.5.3 Fixed-Length Method of Permittivity Measurement ... 99

8 Permittivity Measurements 102

8.1 Check Standard Measurements 102

8.2 Candidate Reference Materials 104

8.2.1 Cross-linked Polystyrene 104

8.2.2 Fused Silica 104

8.2.3 Alumina 105

9 Uncertainty Analysis 110

9.1 Permittivity Uncertainty (Ae'^) Ill

9.2 Loss Tangent Uncertainty 113

10 Future Work and Acknov^ledgements 118

References Cited 120

A Partial Filling Factor for Loss Determination 124

B Radial Air Gap Correction 126

C 60-mm Cavity Drawings 131

D MEAS_RES Program Listing 156

E CAVITYPROG Program Listing 198
List of Figures

1 . 1 The fixed-frequency/variable-length and fixed-length/variable-frequency

permittivity measurement methods rely on the empty-cavity dimensions and Q as a reference 6

2.1 Magnetic field hnes of the resonant mode TEoip in an isolated

dielectric resonator 8

2.2 Electric field distribution in equatorial plane for TEoip mode.

Breadth of dashed lines is proportional to the electric field. ... 8

2.3 Cylindrical coordinate system for waveguides 9

2.4 Rectangular coordinate system for waveguides 10

2.5 Circular cylindrical hollow cavity of length L and radius a. . . . 10

2.6 Normalized cavity Q versus diameter-to-length ratio for a right

circular cyhnder for TMmnp modes, bg is the skin depth of the

cavity wall material and A is the free-space wavelength 18

2.7 Normalized cavity Q versus diameter-to-length ratio for a right

circular cylinder for TEonp modes. 6s is the skin depth of the

cavity wall material and A is the free-space wavelength. ..... 19

2.8 Currents flowing in (a) cylinder wall and (b) cavity end plates for

TEoip and TMup modes 20

2.9 TEoip niode filter consisting of helix waveguide 21

2.10 NIST mode-filtered X-band cavity 22

2.11 Schematic of fixed-frequency cavity resonator measurement technique 23

2.12 Cavity transmission curve defining loaded Q-factor, Q — /o/ A/1/2. 28

2.13 Equivalent resonator circuit for refiection Q measurement [18]. . 31

2.14 Input reflection coeflficient versus to 32

Vll
viii LIST OF FIGURES

4.1 Cavity length versus diameter calculated from subsets of the 7-

14 GHz resonance spectrum, T£'oi(io)-T'£'01(37). Subsets consist

of five adjacent modes ( T£oi(io-i4), ^.^o^h-is), • • •, 7^^01(33-37) ) 47

4.2 Difference between micrometer-measured length and length calculated from X-band resonance spectra 48

4.3 Calculated diameter versus change in length as measured by the

sensing micrometer. A diameter uncertainty of ±0.002 mm is found from the range of results 49

4.4 Length versus diameter calculated from five-frequency subsets for seven different cavity lengths. Displacement of curves from one

another indicates sensing micrometer accuracy 50

4.5 Length versus diameter results showing cavity dimension repeatability for three tuner-base detach/reattachments. Cavity dimensions calculated from subsets of five adjacent TE'oip modes. Also

note the repeatability of the crossing pattern that occurs for the

middle subsets 52

4.6 Measured resonant frequency shifts when couphng irises are plugged

relative to measured resonant frequencies when both ports are un- plugged 54

4.7 Cavity length versus diameter calculated from an uncorrected res- onant frequency spectrum, and from the same frequency spectrum

corrected by the measured iris perturbation shown in Fig. 4.6. Cavity dimensions are calculated from subsets of five adjacent

resonant frequencies. Length and diameter uncertainty estimiates

for each calculation are also shown 55

4.8 Measured Q^ versus frequency for three cavity lengths. Data

show increasing Q and mode interference above 11 GHz 59

4.9 Values of Qq « 84 700 calculated from scattering parameters. The

cavity was opened and closed every three measurements for the

first nine measurements and every four measurements thereafter.

A new calibration was performed after measurements 9, 25, 41,

57 and 73. Horizontal hatched lines show 95% and 99% certainty

bounds 62

4.10 Calculated values of /q. The cavity was opened and closed every

three measurements for the first nine measurements and every

four measurements thereafter. A new calibration was performed

after measurements 9, 25, 41, 57 and 73. Horizontal hatched lines

show 95%. and 99% certainty bounds 63
LIST OF FIGURES ix

4.11 Calculated values of Ki. The cavity was opened and closed every

three measurements for the first nine measurements and every

four measurements thereafter. A new calibration was performed

after measurements 9, 25, 41, 57 and 73. Horizontal hatched lines

show 95% and 99% certainty bounds 63

4.12 Calculated values of K2- The cavity was opened and closed every

three measurements for the first nine measurements and every

four measurements thereafter. A new calibration was performed

after measurements 9, 25, 41, 57 and 73. Horizontal hatched lines

show 95% and 99% certainty bounds 64

4.13 Lumped-parameter equivalent resonant-circuit showing coupling

port impedances and reference plane a-a 67

4.14 Measured scattering parameters (dots) and locus predicted from

nonhnear least-squares fit to (4.9) 67

4.15 Magnitude of residuals to circuit model fit 68

4.16 Phase of residuals to circuit model fit 68

4.17 Magnitude of residuals to circuit model fit when an e^^ term is included in the circuit model to compensate for network analyzer

phase drift 69

4.18 Phase of residuals to circuit model fit when an e^^ term is included

in the circuit model to compensate for network analyzer phase drift. 69

8.1 Uncorrected permittivity e'^ of three glass samples measured in

the NIST resonator. Both fixed-frequency, and fixed-length re- sults compare favorably with previous results 103

8.2 Uncorrected loss tangent tan6 measurements of three NIST glass

samples compare favorably with previous results 103

8.3 Permittivity e'^ of cross-linked polystyrene 104

8.4 Loss tangent tan6 of cross-linked polystyrene 106

8.5 Permittivity e'^j of two fused silica samples (1 and 2) and one fused

quartz sample (3) 106

8.6 Loss tangent tan6 of two fused silica samples (1 and 2) and one

fused quartz sample (3). Note effect of interference by a TE02

mode between 11 and 13 GHz 107

8.7 Permittivity e'^ and estimated uncertainty of fused sihca sample

# 1 107
X LIST OF FIGURES

8.8 Loss tangent tan(5 and estimated uncertainty of fused silica sample

# 1 108

8.9 Measured permittivity e'^ of several alumina samples from the

same batch 108

8.10 Measured loss tangent tan6 of several alumina samples from the

same batch 109

9.1 Permittivity uncertainty (Ae'^) versus frequency due to the uncertainties in various measured parameters for a 10-mm thick sample

with e'^ = 10 - iO.OOl 112

9.2 Loss tangent uncertainty A(tan(5) versus frequency due to uncertainties in various measured parameters for a 10-mm thick sample

with e'^ = 10 - jO.OOl 117

B.l Unperturbed cylindrical cavity resonator with no air gap present

between sample under test and resonator side wall 127

B.2 Perturbed cylindrical cavity with uniform air gap between sample

and resonator wall 128

C.l Cutaway assembly drawing of NIST cavity resonator 132

C.2 Detail of tuner assembly and flange 133

C.3 Cutaway side view of tuner-base and extension 134

C.4 Orthogonal views of coupling flange 135

C.5 Side view of coupling flange 136

C.6 Cutaway side view of coupling flange 137

C.7 Top view of coupling flange 138

C.8 Bottom view of coupling flange 139

C.9 Orthogonal views of tuner flange 140

C. 10 Side view of couphng flange 141

C. 11 Cutaway side view of coupling flange 142

C.12 Top view of tuner flange 143

C.13 Bottom view of tuner flange 144

C. 14 Orthogonal views of tuner base 145

C.15 Top view of tuner base 146
LIST OF FIGURES xi

C.16 Bottom view of tuner base 147

C.17 Side view of tuner base 148

C.18 Coupling endplate 149

C.19 Temperature-probe bolt and coupling-endplate pressure ring. . . 150

C.20 Tuner-endplate 151

C.21 Tuner assembly backstop, torque nut, micrometer-drive holder

and sliding bushing 152

C.22 Tuner guide shaft, spring backstop and extension 153

C.23 Measurement-micrometer reference yoke 154
List of Tables

2.1 Zeroes t'^^ of the first derivative of the Bessel function of first kind and order m 16

2.2 Typical mode attenuation in helical waveguide at 35 GHz (from

Cook [2]) 21

2.3 Skin depth and relative loss of various metals 35

4.1 Repeatabihty of tuner assembly connection 51

4.2 Overall cavity dimension uncertainties 57

4.3 Stability of Qq and coupling factors. Precision is given as the

standard error of 10 measurements 61

4.4 Parameter estimates for data subsets. Qi values are calculated

from Qq.i^i and K2 70

4.5 Mode efficiencies (Qnieasured/^theoretical) ^ ^^^ ^^^ ^^^^ ^^^" quencies. Square brackets: older endplate, unbracketed values: newer endplate 72

6.1 Definitions and class assignments of an X-band TRL calibration

kit 91

9.1 Estimated parameter uncertainties for the NIST resonator. ... 110

Xll
The NIST GO-millimeter Diameter Cylindrical Cavity Resonator:

Performance Evaluation for Permittivity Measurements

Eric J. Vanzura

Richard G. Geyer

Michael D. Janezic

Antenna and Materials Metrology Group

Electromagnetic Fields Division

Abstract

Uncertainty estimates are developed for dielectric permittivity calculations made using the NIST 60-mm diameter cylindrical res- onator. A mode-filtering hehcal waveguide makes up the cavity's cylindrical wall, which permits the generation of high-purity

TEoip resonant modes for high accuracy permittivity measurements. The cavity's length can be varied from 408 to 433 mm.

Fixed-length and fixed-frequency measurements in the X-band fre- quency range are evaluated with particular emphasis on 10 GHz.

Resonator theory and design, measurement tolerances, and soft- ware are included.

Keywords: cylindrical cavity, dielectric permittivity, measurement

software, resonator theory and design, TEqi helical waveguide,

uncertainty estimates, x-band 10 GHz microwave.
Chapter 1

Introduction

1.1 Definition of Permittivity

Permittivity is a physical quantity that describes the relation between electric

field E in volts per meter and electric displacement D in coulombs per square

meter. In a vacuum, this relation is given by

D = eoE, (1.1)

where cq = 8.85418782 x 10"-^^ farads per meter is the permittivity of free

space. In matter, electric displacement is affected by the material's electric

susceptibility, xl-, which is a measure of the material's polarizability. In an

isotropic, hnear media the polarization vector P is parallel and proportional to

the electric field,

P-eoxlE, (1.2)

and the electric displacement vector then becomes

D = eoE + P

= eoi_l + x:)E = e*E , (1.3)

where

e* = eo(l + x:-) , = eoe^ (1.4)

is the permittivity of the material, and e^ = (1 + Xe) is the material's relative

permittivity or dielectric constant. In general, electric susceptibiHty has real
1.2. OPERATION OF MODE-FILTERED CYLINDRICAL RESONATORS 3

(i.e., dispersive) and imaginary (i.e., absorptive) parts which makes relative

permittivity a complex quantity. We define relative permittivity as 4 = 1 + x:

= ^'r-J^r- (1-5)

For low-loss materials e'^ is often loosely called the relative permittivity, with

the distinction between e^ and e'^ implicitly understood. The term e'^ is called

the dielectric loss factor.

If the induced conduction current is directly proportional to the electric field,

the curl of the magnetic field H is V X ;^ = {juje* + a)E , = jujeoe'^[l-j'-^-j^-)E , = jujeot'fiil - jt8in6)E . (1.6)

This relation for loss tangent tan6 now explicitly separates dielectric and con- ductive losses. In most discussions dielectric loss of nonconductive materials at microwave and millimeter wave frequencies is described by a frequency dependent conductivity of the material at the measurement frequency. In this case

loss tangent e'^ = a/utQ, and loss tangent becomes

tan(5 = -f = . (1.7)

^R ^^o^R

1.2 Operation of Mode-Filtered Cylindrical

Resonators

The cylindrical cavity resonator described in this Technical Note was built by

the NIST Electromagnetic Properties of Materials program to accurately measure the complex permittivity of dielectric materials. The application of cavity

resonator methods to measure the complex permittivity of dielectric samples is not new [1]. However, resonator methods are inherently accurate and to this

day prove to be the most accurate measurement method available. NIST chose

to build a cavity resonator to accurately characterize dielectric materials that in

turn can be used as check standards to test the validity of other measurement

methods. Special test services are performed regularly for those outside NIST,
4 CHAPTER 1. INTRODUCTION

and, since its introduction, the NIST resonator has characterized a large number

of materials both for internal and external use. Cavity resonators can measure permittivity only at fixed-frequencies and over a narrow range of frequency. This is the case especially for the shorter, solidwalled cavities because higher-order modes occur very easily. Measurement of

permittivity over a broader range of frequencies is partially solved by the use

of helical waveguide cavities [2, 3]. The mode filtering capability of helical

waveguides allows construction of cavity resonators that can support several

highly pure modes in a limited frequency range.

The NIST resonator has a 60-mm diameter helical wire-wound cylindrical wall,

with a variable length of approximately 408 to 433 mm. The cavity's helical

winding allows only circumferential currents to flow, which permits propagation

of only the TEon family of circular transmission line modes, where n describes

the number of radial variations. Mode-filtering capability permits the use of a

long cavity that would otherwise be highly overmoded. This allows permittivity

measurements at several TEqi resonant frequencies throughout X-band (8.2 to

12.4 GHz) rather than at only one or two specific frequencies.

The well defined geometry of the NIST cylindrical resonator allows standing waves to occur at certain frequencies. Power enters through a coupling iris lo- cated in one of the cavity's endplates and excites the TEqi waveguide mode.

The present coupling endplate connects to X-band waveguide feeds. The cylindrical TEqi waveguide mode has an easily calculable wavelength that depends on waveguide diameter and the permittivity of the medium inside the waveguide (7.1). Power propagates in this mode from the coupling endplate to the

other endplate and reflects back. At certain frequencies the cavity length will be

an integer number p of half-wavelengths, resulting in constructive interference.

The integer p is called the axial mode number. At these frequencies a standing

wave is set up, and a resonance condition occurs. For the NIST cavity, the first TEqi resonant mode occurs near 6.1 GHz, and the first TE02 resonant mode

occurs near 11.2 GHz.

The NIST cavity resonator is ideally suited for accurate permittivity measure- ments because the resonator's quality factor Q is very high (60 000 to 80 000).

Q is sometimes defined as the number of radians a wave propagates before the

wave's power decays to e~\ or approximately 37%, of its initial power. For a high-Q resonator this means that the excitation wave reflects off the cavity endplates and travels the length of the cavity several times before the wave's power

is significantly reduced. For example, if p = 25 and the Q of the T'£'oi(25) mode

is 80 000, then an excitation wave travels the length of the cavity approximately

Q/(257r) « 1000 times before decaying to e~^ of its initial power. If this high-Q

cavity contains a sample, the excitation wave also passes through the material
1.3. METHODS OF MEASUREMENT 5

more than 1000 times. In effect this increases the effective thickness of the sample and makes cavity resonator measurements sensitive to small changes in e'^ and tan(5.

1.3 Methods of Measurement

To determine the permittivity of a material, two measurement methods can be used, as shown graphically in Fig. 1.1. One method involves reducing the

sample-loaded cavity's length so that it resonates at the same frequency as the

empty cavity. The change in length is a measure of permittivity, and the change

in Q is a measure of the sample's dielectric loss (Sec. 2.2.-3). The second method

holds the cavity length fixed. The change in resonant frequency determines e'^, and the change in Q determines dielectric loss. The theory from which e'^ is calculated assumes that the cavity and sample have

zero loss (infinite Q), so field orthogonality can be assumed. In practice, this

assumption causes little problem because of the cavity's low surface impedance

and because dielectric samples are usually low loss. The cavity resonator is so

sensitive, however, that it is possible to determine and make corrections to fre- quency measurements that improve e'^ measurement accuracy. These corrections

are discussed in Chapter 4. In a sense, the concept of having two methods to

measure the real part of permittivity is artificial. All that is needed to calculate

e'^ are cavity length and diameter, sample thickness and axial mode number.

The empty cavity needs to be measured only because we have to determine the

cavity's dimensions, where cavity length and diameter are calculated from the

frequencies and axial mode numbers p of the TEoip resonance spectrum.

The fixed-frequency method is more accurate than the fixed-length method for

calculating dielectric loss. Cavity Q is frequency dependent. Skin depth (resistive losses) of the cavity walls decrease and coupling port losses increase as frequency increases. Presently, the measured empty-cavity Q is used as the ref- erence that determines the cavity's conductor and port losses. When we measure

the Q of the empty cavity, we determine the cavity loss at that resonant fre- quency. When a sample is inserted the resonant frequency is lowered. If we use the Q of the empty cavity resonating at a different frequency as our reference, our loss tangent calculation will be in error. This is especially true when the

sample is very low loss (tan6 < 0.0003), because small differences in cavity Q

become important. When sample loss is greater than tan5 > 0.0008 the differences between empty-cavity losses and the cavity losses of the sample-loaded

cavity are small enough that the fixed-length method can be used with nearly

the same accuracy as the fixed-frequency method.
CHAPTER 1. INTRODUCTION

I- AF -H

F IXED-LENGTH

AL or AF

t an6

e'

F I XED-FREQUENCY

Figure 1.1: The fixed-frequency/variable-length and fixed-length/variablefrequency permittivity measurement methods rely on the empty-cavity dimensions and Q as a reference.
Chapter 2

Theory of Microwave

Resonators

2.1 Introduction

Resonators create, filter, and select frequencies in oscillators, amplifiers, and

tuners. Recent advances in the miniaturization of microwave circuits have

spawned the development of low-loss, temperature-stable dielectric resonators.

These resonators replace waveguide filters in many communication systems where

microstrip and stripline resonators cannot be used due to their intrinsic high loss. Dielectric resonators are also very important fixtures for measuring the electrical

properties of low-loss solids in the microwave region. An important resonator

circuit at microwave frequencies is the closed cylindrical cavity operated in a

resonant TE^ip mode. The magnetic field for this mode is sketched in Fig. 2.1. For a distant observer this mode appears as a magnetic dipole, and for this rea- son some authors call it a "magnetic dipole mode." The electric field lines are simply circles concentric with the axis of the cyhnder (Fig. 2.2).

For a TE mode in a waveguide, the electric field is everywhere transverse to

the propagation direction or to the axial z-direction of the guide. Coordinate

systems for circular and rectangular waveguides are sketched in Figs. 2.3 and 2.4,

respectively. For a TEmn mode in a rectangular guide, m simply represents the

number of antinodes occurring in the electric field E in the x-direction and n

represents the number of antinodes in E in the y-direction. For a TEmnp mode

in a finite circular guide, m represents the number of antinodes occurring in E

in the 0-direction, n the number of antinodes in the radial r-direction, and p the

number in the z-direction. The same notation is used for transverse magnetic

or TMjnn modes, with the m and n indices representing the antinodes occurring
CHAPTER 2. THEORY OF MICROWAVE RESONATORS

Figure 2.1: Magnetic field lines of the resonant mode TEoip in an isolated dielectric resonator.

Figure 2.2: Electric field distribution in equatorial plane for TE'oip rnode.

Breadth of dashed lines is proportional to the electric field.
2.1. INTRODUCTION

Y

*-X

Figure 2.3: Cylindrical coordinate system for waveguides.

in the magnetic field H for a given coordinate direction. Generally, the choice

of cavity to be used in dielectric metrology is influenced by the shape and size

of the sample to be measured. Insertion and removal of the sample from the

cavity should be convenient and not significantly alter the Q [4] of the air-filled

cavity or increase the radiation from the cavity. Hollow circular cylindrical

waveguide resonators, which are terminated by two short-circuit endplates, are the most commonly used for dielectric measurements of low-loss solids. The

reasons for this are the relative ease of fabrication and the very high Q-factors

and accompanying narrow bandwidths that can be obtained with this fixture in

the microwave region [1]. If the cavity is made from helically wound waveguide,

it also acts as a mode filter in that all other waveguide modes except the TEqip

mode [3] are greatly attenuated.

In general, microwave energy is coupled to the cavity through transmission line

probes, as illustrated in Fig. 2.5. As in resonant transmission fines where res- onance occurs at many frequencies, the hollow cylindrical waveguide has many

resonance frequencies and associated field distributions or modes. The field of

the mode with the lowest or dominant resonant frequency is termed the dominant mode of that fixture. Only enough energy is provided by the input excitation to match cavity losses. Once the internal fields at a resonant frequency

are determined, an equivalent lumped parameter circuit of the fixture may be

ascertained. In addition, the internal power dissipation, stored energies, and

energy fiow out of the cavity can be determined.
10 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

^ X

Figure 2.4: Rectangular coordinate system for waveguides.

Transmission

Line ProbeFigure 2.5: Circular cylindrical hollow cavity of length L and radius a.
2.1. INTRODUCTION 11

As expected, the different microwave modes that can be set up in a hollow

right circular cyhnder depend on both the dimension of the cylinder and the

microwave frequency. A hollow right circular cylindrical cavity is supportive of

both TE- and TM-mode field structures. For steady-state exp{jLut) field time

dependence, the equations for the axial components of the electric and magnetic

field interior to a waveguide of arbitrary cross-section and filled with material of

complex permittivity e* and complex permeability /i* satisfy the homogeneous

Helmholtz equation

^1 hJ^^ [h] 1=^' (2.i;

where z is the direction of propagation and E., H^ are taken for TM or TE

modes, respectively. The differential operator V'^ is the Laplacian, which can be expressed in any curvilinear coordinate system by evaluation of the metric.

All other field components can be expressed in terms of E^ and H,. In general,

any field in a hollow waveguide, however complicated, may be represented by a combination of TE and TM modes. The wavenumbers

e = k] - 7^ = uh*^l* , (2.2)

are the characteristic eigenvalues of (2.1). In other words, to each value of

/c^ there will correspond a function E, or //^ (for TM or TE modes) which

is a characteristic function from which may be derived the other components

of the fields. In general e* and ^* in (2.2) can be complex dyadics, but for

this discussion the material is assumed isotropic and each dyadic becomes a

single complex value multiplied by the unit dyadic. As defined in (1.7), the

conductivity of the medium is contained in the imaginary part of the dielectric

constant, e* = e' — je". For a plane-wave field H.{z) = i^^e"^' in which the

electric field is entirely transverse [TE waves),

where the value of A is the wavelength of a plane TE wave in the medium that

fills the hollow pipe and kc is the cutoff wavenumber of the TE mode. In order

to have propagation down the pipe, 7^ must be negative. The quantity k^ is always real because, for a circular waveguide of radius a,

(^c)_ = ^ , (2.4)

where t^„ is the nth root of the first derivative of the Bessel function J^^{kca) —

for a TEmn mode. When 7^ is positive or 9? (cj^e*/i*) < kl there is no propagation
12 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

of energy through the waveguide. This condition is termed cutoff. The cutoff or

critical frequency is given by

.? = 5r{^}.?. (2.5)

Substituting (2.4) into (2.5) gives the cutoff frequency for TEmn modes in a

circular waveguide:

,2 _{l\'l 1 \(t'„„''

27r/ U*//

or

(/c)rE„. = I ;^ ) «1 —MTM) . (2.6)

c c 2-na J'StU'a')

(Ac)r^„„ = JT^ = f^-^ , (2.7)

yJcJTEmn run where c is the velocity of light in the medium.

Similarly, for a TMmn mode in circular waveguide,

'^mn ikc)mn = -^ , (2.8)

where tmn is the nth root of Jm{kca) = 0. As before, the cutoff wavelength for TMmn modes is given by:

Mtm^. = r-^ (2-9)

If Xg is the wavelength in the guide and kg the guide wavenumber, then

l^jkg=J—, (2.10)

^9

or, from (2.3),

from which we derive the axial wavelength in the waveguide.

(2.11)

A, = , ^ ^ , (2.12)

/MAT
2.1. INTRODUCTION 13

where A is the wavelength of a uniform plane wave in the medium:

A = ^°

tOMO

V £0^0

where Aq is the free-space wavelength and we assume that 3f?{e*/i*}

Equation (2.12) then becomes

KAr Ar v/^-(t)' ^^>'.-ity'

(2.13)

'~~' t jJi .

(2.14)

Equation (2.14) shows that when the free-space wavelength is much less than

the cutoff wavelength, that waves propagate in the guide with a guide wavelength normalized by the square root of the product of relative permittivity and

relative permeability of the material filling the guide. Of course, in the empty

cavity situation, this means that the guide wavelength is essentially that of air. Similarly, when wavelengths are much greater than the cutoff wavelength for a

given mode, the guide wavelength becomes pure imaginary or, stated physically,

nonpropagating evanescent modes are set up.

2.1.1 Fields in a Right Circular Cylindrical Cavity

For the problem at hand, we are dealing with a cylindrical structure having

two metallic end plates. In other words, it is bounded in axial extent and the

associated internal fields may have variations in both the transverse and axial

directions. In the case of TE waves, the wave equation (2.1) for H^ expressed

in cylindrical coordinates becomes

r dr

where

and from (1.7)

d

. or

1 (92 d^ , r^ 0(p^ oz^

2 , 7.2 K -^r + k\

;2 2 * * 2*11 "\ k = uj II € = cj /i (e — je ) = a;V*eo4(l -J tan (5) .

(2.15)

(2.16)

(2.17)
14 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

The wavenumber is complex for lossy dielectric materials; that is,

K, — ryreal J '^imaginary 5

where it is easily shown that

'^real —

and

imaginary

J

(2.18)

(2.19)

;2.20)

In (2.19) and (2.20) ^ = ^'j^^q and e = e'^eo, where eo « 8.854 x lO'^^ Y/m

and /io = 47r X 10"^ H/m. For lossless dielectrics, cr = and e'^ = with

kreal = ^JT^ ^nd /C;imaginary 0. The method of separation of variables [5] yields a solution to (2.15) of the form

//,(r,0,2) = /?(r)$((^)Z(2). (2.21)

Substitution of (2.21) into (2.15) and then division by i/^(r, 0, 2) yields

1 d d 1 r R

Rdr dr J

1 d^^ Id^Z ,, (2.22)

The third term in (2.22) is explicitly independent of r and 0. It is also necessarily

independent of z if (2.22) is to sum to zero for all (r, 0, z). Therefore,

Id'^Z

Z dz'^ -/?^ (2.23)

where /? is a constant. Substitution of (2.23) into (2.22) and multiplication by

r^ results in

1 d

Rdr . dr

1 d'^(^ /,, ^.\ o (2.24)

The second term in (2.24) is a function of </> only, whereas the rest of the equation

is a function of r only. By the same argument the azimuthal component of H^

obeys the relation, = —m (2.25)

where m is a constant. Substitution of (2.25) into (2.24) and multiplication

throughout by R{r) yields

d d J r—R

dr I dr J + [{e-P^)r'-m']R = 0, (2.26)
2.1. INTRODUCTION 15

where P is the axial waveguide propagation constant and k is the medium

wavenumber. Now k'^—P^ represents the square of the transverse radial wavenumber kc, where

k, = ^Jk^ - /?2 , (2.27)

so (2.26) can be written

d

dr

dR

. dr

+ [{k^rf -m^]R = . (2.28)

The original Helmholtz equation (2.1) is now separated into three equations,

each of which determines only one of the functions R{r), $(0) or Z{z). Equations (2.23) and (2.25), are harmonic equations, whose solutions are harmonic

functions, or linear combinations of sines and cosines. Equation (2.28) is a Bessel

equation of the mth order with independent solutions Jmikcf) and Nm{kcr),

where J„, and A^^ represent Bessel and Newmann functions, respectively, of

order m. Because Nm{kcr) is not finite at r = 0, the solution for R{r) is

R{r) = Jm{kcr) . (2.29)

The choice of the constants /? and m, as well as the solutions for (2.23) and (2.25),

depends on the physical geometry of the fixture, conditions at the boundaries,

and the type of field to be supported by the fixture. For nonevanescent modes

in a right circular cylindrical cavity, kc can take on only characteristic or certain

discrete values that correspond to different modes of propagation. In the case

of the cylindrical cavity, the constant m must be an integer if the solution for

Ez is to be single-valued in (periodic). The radical that defines the transverse

radial wavenumber kc calls for some comment. The branch of the square root is usually chosen such that kc ^^ k SiS, |/?^| ^0 and /Cc ^^ |/5| as |A:^| —>• 0.

2.1.2 TE Modes

For TE mode structure {E^ = 0) we need solve for only H^. All other field

components interior to the cavity are derived from H^ by Maxwell's equations.

The complete solution for //^(r, 0, 2) is

Hz{r, 0, 2) = Jm {kcr) [A cos mcp + B sin m4>] sin Pz , (2.30)

which satisfies the boundary conditions at the cavity's cylindrical walls r = a and at the cavity endplate z — 0. This includes the case where the cavity is filled

with dielectric materials such that the transverse wavenumber kc is complex. A

and B are constants in (2.30) which determine the phase of the azimuthal field
16 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

Table 2.1: Zeroes t'^^ of the first derivative of the Bessel function of first kind

and order m.

m

n

1 2 3

3.8317 7.0156 10.1735

1 1.8412 5.3314 8.5363

2 3.0542 6.7061 9.9695

3 4.2012 8.0152 11.3459

orientation relative to the coupling port(s). We are currently considering only an empty cylindrical cavity with k = cuy/jloeo and arbitrary azimuthal phase. In

order to enforce the boundary condition on £'<^, derivable from H^ from Maxwell's

equations, we must have

E^{a,(P,z) = , (2.31)

or

J'^ (M) = , (2.32)

where J^ represents the first derivative of the mth-order Bessel function. If the zeroes of J^ are denoted by i'^„, where n = 1, 2, 3, . . . represents the zero- crossing number, then kc must be chosen to take only certain discrete values;

that is, f

kc = ^^ . (2.33)

Table 2.1 gives some representative values for t'^^. The boundary condition on the azimuthal electric field at 2 = L results in certain

allowable values for the longitudinal propagation wavenumber:

sin /?L = , (2.34;

or

(2.35)

where p = 1, 2,3, . . .. The final result for the axial magnetic field of the TE

mode within the cavity is:

-"z \-^ ^mnp) '-'n t'r

[A cos m0 + B sin m0] sin

pixz

17 (2.36)

where m, n, p are integers describing the TE mode. An identical analysis can be performed to derive the axial electric field for the TM case.
2.1. INTRODUCTION 17

The characteristic equation used to find the resonant frequencies for TE modes

in the empty cavity is given by Harrington [5] pp. 213-216 as,

/„(T£TM,) = ^[(%)%(f)|, (2.37)

where Cair is the speed of Hght under testing conditions in air. When / =

fo{TEmnp), we have a T^E-mode solution to Maxwell's equations. From (2.2),

the resonant frequency for the cyhndrical cavity filled with material of complex

permittivity e* and real permeability fi' is

/o(TE^.p)= .

^

, (M +(t) I

• (2.38)

27r\//.'6'(tan2 6 + l)^ L^ « ^ ^^M

As expected, when the length or diameter of a cylindrical cavity increases, the

resonant frequency decreases for any given mode.

The figure of merit for assessing the performance or quality of a cavity resonator

is the quality factor Q which is a measure or energy stored in the fields inside

the resonator compared to the energy loss or dissipation per cycle. The Q-factor

is defined by

maximum energy stored during a cycle

average energy dissipated per cycle

27rW _ luqW

PT ~ ~P~'

(2.39)

where W is stored energy, P is power dissipation, o^o is resonant radian frequency,

and T is period = —. The higher the axial TE-mode number for any given

azimuthal and radial mode number, the greater the cavity quality factor, for

any given cavity diameter or length. The quality factor Q will be discussed in

more detail in connection with the evaluation of dielectric loss measurements of

materials. Some examples of normalized cavity Q values are shown in Fig. 2.6

for some representative TMmnp mo.des and in Fig. 2.7 for some TEq^p modes as a function of cavity diameter-to-length ratio. In Fig. 2.7, the optimal (highest)

Q value is obtained for TEmnp modes when the diameter of the cavity equals its length. Our 60-mm diameter resonator's length is roughly seven times longer

than its diameter. Resonance Q-value is compromised somewhat by this length,

but the cavity's heUcal windings yield better than 30-dB mode purity. The

length of the resonator allows for the measurement of several TEoip modes in

the X-band frequency range.
18 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

Qxds

0.9

0.8

0.7

0.6

0.4

0.3

0.2

0.1

1 1 1 1

TMo23_________

V. -VW\vO,

- \\ \^:>< ''=^^^o^^

^^TwCb^

\ TM,,2

TMo,,

^-^

1 J

"tm—^

' 1

1 0.5 1.0 1.5 2.0 2.5 3.0

2a/L

Figure 2.6: Normalized cavity Q versus diameter-to-length ratio for a right

circular cylinder for TMmnp modes. 6^ is the skin depth of the cavity wall

material and A is the free-space wavelength.

2.2 T^oip Mode-Filter Cylindrical Cavity

One very useful cavity resonator for microwave dielectric property measurements

on low-loss materials is constructed to filter all modes resulting from current

other than that flowing circumferentially about the cavity wall. This yields a very high-Q cavity with a very pure TE'-mode structure for precision electrical

property measurements.

Consider, for example, the wall and endplate currents flowing in the cylindrical

cavity for TEoip and TMnp modes as shown in Fig. 2.8. For the TMup mode

there are both azimuthal and radial currents, whereas in the case of the TEoip

mode there are only azimuthal currents flowing in the wall and end plates. As

Cook [2] notes, the presence of unwanted modes produces larger data scatter of

loss tangent and permittivity with change in specimen length than when there

is only one effective mode propagating, such as the TEoip.

If the currents associated with the TMnp mode can be interrupted while at the

same time leaving those associated with the TEoip mode unchanged, then signif-
2.2. TEoip MODE-FILTER CYLINDRICAL CAVITY 19

QxdS 0.9 -

0.4 0-5 1.0 1.5 2.0 2.5 3.0

2a/L

Figure 2.7: Normalized cavity Q versus diameter-to-length ratio for a right

circular cylinder for TEq^p modes, bg is the skin depth of the cavity wall material

and A is the free-space wavelength.

icant attenuation of the TMup mode will take place. This can be accomplished

by constructing a cavity wall in which the conductivity is discontinuous along

the length of the cylinder but continuous around its circumference. This type of

cavity becomes a mode filter. One approach to the construction of a mode-filter

cavity is to construct a cylinder of annular copper rings electrically insulated

from each other [2] to form a continuous cylinder. In this case, the currents

fiowing from ring to ring will be greatly impeded while those flowing around the

rings are unimpeded.

Another approach is to make a helical waveguide by winding fine enameled wire

into a precision cylindrical former (Fig. 2.9). This type of waveguide has been

discussed in the literature by Morgan and Young [6]; Unger [7]; Young [8]; Cook
20 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

TEOlp

(a)

Olp

(tjj TMiip

Figure 2.8: Currents flowing in (a) cylinder wall and (b) cavity end plates for

TEqip and TMnp modes.

and Jones [9]; Cook [2]; Waldron [10] [11]; Waldron and Bowe [12]; Shimba [13];

Kazantsev, Kaznacheev and Meriakri [14]; Mikoshiba [15]; Piefke [16]; Waldron,

Bowe, Wackrill and Wescott [17]; and Noda, Yamaguchi and Suzuki [18]. In a helical waveguide the modes whose wall currents follow the conducting helix

possess attenuation constants which are essentially the same as for copper pipe.

All other modes, however, have a high transmission loss. In other words, all modes other than the TEqij, have very large attenuation constants in a helical

waveguide. The exact calculation of the attenuation constants for modes other

than TEoip depends on the helix pitch angle and the electrical properties of the

thick lossy dielectric jacket surrounding the helix; the fields of those modes penetrate into the jacket and are attenuated, whereas that of the TEoip mode does

not and is therefore minimally attenuated. The helical waveguide, then, is the

equivalent of an anisotropically conducting cylinder. Whether we use helically

wound wire as opposed to annular copper rings is a question of convenience in

manufacture.

Waldron [10], and Waldron and Bowe [12] have analyzed the azimuthal electric

field dependence as a function of the radius of the helical winding, the complex

permittivity of the jacket material, and the pitch angle of the helix. To a first- order approximation the TEoip mode is independent of the pitch angle. Waldron

and Bowe [12] also treat the effect of pitch for the TEoip mode to the second order

and find that the effect of a finite pitch does indeed cause a slight attenuation of

the TEoip mode. Cook [2] compares the attenuation of the TEoip mode to other

modes in a 50-mm diameter helix at 35 GHz; these results are given in Table 2.2. Note that the modes other than TEoip have attenuation constants larger by

several orders of magnitude than that for TEoip. Pieflvc [16] demonstrates that
2.2. TEoiP MODE-FILTER CYLINDRICAL CAVITY 21

INTERNAL THICK LOSSY

RADIUS a DIELECTRIC JACKET

Figure 2.9: TE^ip mode filter consisting of helix waveguide.

Table 2.2: Typical mode attenuation in helical waveguide at 35 GHz (from

Cook [2]).

Mode Attenuation (dB/km)

TE,, 2-3

TEn 4 000-5 000

TEn 6 000-7 000

TE21 10 000-12 000

attenuation for the TE^p mode exceeds by a factor of 3 x 10'^ the attenuation

of the same mode in a homogeneous circular copper cavity having the same

dimensions. An electromagnetic field analysis of propagation interior to a helix

of arbitrary radius and pitch that is loaded with a lossy dielectric sample is useful in the actual design of the mode-filter structure.

2.2.1 NIST 60-mm X-Band, Mode-Filtered Cavity Resonator

The 60-mm diameter X-band cavity constructed by NIST is mode filtered and

is similar to those developed in the United Kingdom and Germany [2, 3]. A

cutaway drawing of the fixture is shown in Fig. 2.10. The cylindrical walls of

the cavity consist of helically wound wires (0.15 mm diameter) which allow only

azimuthal current flow that, in turn, yields very pure TEoip mode structure and

empty cavity Q values of approximately 8 x lO'* at 10 GHz. Non-T£'oip modes

such as TEi-2p and T£'o2p, if excited, are attenuated by at least 30 dB, so high

precision dielectric measurements can be made. Details of the cavity resonator

design are given in Chapter 3, and drawings are given in Appendix C.
22 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

COUPL ING

APERTURES

UPPER CAVITY

FLANGE

REPLACABLE

FIXED

END PLATE

DIELECTRIC

SAMPLE

MOVEABLE

END PLATE

LOWER CAVITY

FLANGE

DETACHABLE

TUNER

ASSEMBLY

PVC BUSHINGS

TEMPERA TURE

SENSING PROBE

ACRYLIC PLASTIC

WATER JACKET

WATER

HELICAL WAVEGUIDE

TUNER POSI TION

SENSING

MICROMETER

MOTOR-DRIVEN

M I CROME TER

WATER INTAKE

FOR TUNER ASSEMBLY

PREC IS ION

L I NEAR BEARINGS

SPRING-LOADED

CARBIDE RODS

RESONA TOR

SUPPORT RODS

Figure 2.10: NIST mode-filtered X-band cavity.
2.2. TEoip MODE-FILTER CYLINDRICAL CAVITY 23

Transmission

Line Probe

(a) Without Sample (b) With Sample

Figure 2.11: Schematic of fixed-frequency cavity resonator measurement technique.

The relative real effective permittivity (e'^) and loss tangent (tan(5 = ^''/e')

calculations can be made by noting the change in a TEoip mode when the

resonator contains the dielectric sample. Either a fixed-length or fixed-frequency

technique can be used. The fixed-length technique uses the shift in resonant

frequency and change in resonant bandwidth when the sample is inserted into

the cavity to calculate e'^ and tan 6. With the fixed-frequency technique, the

cavity length is shortened so the cavity resonates at the empty-cavity frequency

(Fig. 2.11). In this case, the changes in cavity length and resonance bandwidth

are used to calculate e'^ and tan 6. In order to measure the change in length, the

change in the lower movable endplate's position relative to the cavity body must

be measured. This measurement is made using another micrometer attached to

the bottom cavity flange. We now address these calculations in more detail.

2.2.2 Evaluation of Real Effective Permittivity

The nonzero electromagnetic field components for the TE^ip mode interior to a

right circular cylindrical cavity are derived from (2.36) with m = and n = 1.
24 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

The axial magnetic field is

H. = H,J, (^) sin (?^) . (2.40)

The two other nonzero field components are derivable from H^ using Maxwell's

equations expanded in cylindrical coordinates:

1 d^H, pHoTra

Hr = ^^i.(¥)cos(^), (2.41)

/c2 dzdr t'o^L

The steady-state time factor exp(ja;t) is understood in (2.40)-(2.42). Application of boundary conditions at resonance for the tangential electric {E^) and

tangential magnetic {Hr) fields on the surface of a homogeneous, linear, isotropic

dielectric disk sample and for the tangential electric field on the cavity end plates

yields the following transcendental equation in terms of the axial wavenumber

/?i in the sample under test:

tan(/?i6) tan [/?o (Lr - ^)]

Pi Po

where

(2.43)

Pi = v/^Vi^e^/^oeo - k^c , (2-44)

with kc — ^01 /a and where the axial propagation wavenumber Pq in the air-filled

portion of the cavity resonator is

Po = ^[^) -kl. (2.45)

Here we allow for change in the speed of light in air due to ambient temperature and humidity. From (2.42) we may determine the radial variation of

the azimuthal electric field internal to the cavity, as shown in Fig. 2.2. For the

fixed-frequency technique, Lr is simply L, whereas for the fixed-length technique

Lr = Lq. The real effective permittivity is implicitly expressed in the axial propagation constant in the sample. The Newton-Raphson iterative method may be

used (Sec. 5.2.7) to solve for Pi in (2.43) from which the relative real effective

permittivity is given by

en PI + kl

Pi + kl

v2

Equation (2.43) is solved for either cavity length, keeping the resonant frequency

constant, or resonant frequency, keeping the cavity length fixed.
2.2. TEqip MODE-FILTER CYLINDRICAL CAVITY 25

2.2.3 Evaluation of Dielectric Loss and Quality Factor

Evaluation of the loss tangent tan 6 in terms of measured changes in the Q-factor

of the cavity is more complicated. For any given mode the resonator bandwidth

is proportional to dissipated power and therefore inversely proportional to Qfactor. Hence, high Q-factor resonators have narrow bandwidths. In this section,

we discuss resonators with respect to the Q factor and its measurement for

dielectric loss evaluation.

Useful insight into some properties of cavity resonators may be gained by ex- amining the differential equation for a simple damped resonant circuit with a

forcing input function f{t) and and output function v{t): _ + 2a-+.lv = m. (2.47)

Equation (2.47) may be obtained by suitable manipulation of Maxwell's equations. When (J = 0, the homogeneous solution to (2.47) is simply

v{t) = CismuJot + C2CosujQt . (2.48)

The case where cr > always corresponds to a resonator with losses. If we

Laplace transform (2.47) and solve for the transfer function of our resonator, we

obtain

T{s) = V{s)/F{s) = J^ ,

, (2.49)

S'^ + 2(75 + CJq or T{s) ^

1 1 (2-50)

2iUL Is + a + jujl s + a — jujlwhere uj^ is the loaded natural resonant frequency (either with or without sample) given by

a;^ = ^a;2_^2_ (2.51)

The presence of loss (a > 0), however small, in the cavity results in a change

in the resonant frequency. This is commonly called frequency pulling due to

loss. Under zero-loss conditions (a = 0) we get uJi — ljq. This indicates that

uncompensated losses in the resonator can introduce a positive bias in e'j^ results.

The natural response of a simple resonator described by (2.47) is

v{t) = Ve-'''smuJLt . (2.52)

The stored energy W is proportional to the average value of v'^{t) over a period,

which for small a is given by

W = iy2 -2cTt (2.53)

2

^

^
26 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

The average power P in the system isdW P= — = 2aW, (2.54)

at

so

From (2.39) we may now write the Q-factor in terms of the resonant unloaded

natural frequency and loss,

Q = ^ . (2.56)

From (2.51), the loaded natural resonant frequency now becomes

CJl = LJQ (2.57)

1

1 4Q2J

Thus, the frequency pulling due to loss can be determined by the Q-factor.

Substitution of (2.56) into (2.47) yields

d'^v Lu^dv ^ ^, , , ,

from which we see that in the case of a perfect resonator [Q —> oo), the first derivative term vanishes. For a practical resonator, however, Q is finite, and the

second term of (2.58) must be kept. Kajfez [19] has also treated this resonance

frequency correction for the case of a one-port cavity.

If the transfer function T{s) is written in the frequency domain rather than as a Laplace transform, we set s = juj and a = u)q/2Q from (2.56). Equation (2.49)

then becomes

T{to) = V{ju;)/F{juj) = ^^- . (2.59)

The relevance of this discussion on the resonator transfer function in terms of

the Q-factor of any resonator is now apparent. The denominator in (2.59) may

be written as

2 2 , .?^^0 JW^O

UJn — UJ +

— l+jQ {---)] . (2.60)

Q Q ^UJq U)

The frequency dependence in (2.60) can be factored in the following way,

LU uJq fUJ — LlIq\ (U)q

LUq UJ ^ UJq -^ ^ UJ + 1 ) . (2.61;
2.2. TEqip MODE-FILTER CYLINDRICAL CAVITY 27

For a high-Q resonator the measurement frequency is close to the resonance

frequency, u) « cjq, so ujq/uj + 1 « 2. Thus,

^ ~ 2 = 2(5a; , 2.62

where 6u) is known as the frequency tuning parameter of the resonator. Substitution of (2.62) into (2.59) then gives the following relation for the cavity

transfer function

-]Q/ [ujloq)

1 + jQ2{buj)

where the magnitude of the transfer function versus frequency is a bell-shaped

Lorentzian resonance line shape whose form is Q-dependent.

The half-power bandwidth of any resonator is defined as the frequency spread

Aw, where Acj is the difference in half-power frequencies uJi and u)2-, and with

' \T{uj)\ = ^\T{uj,)\ (2.64)

for either uj = uji oi uj — uj2- Equation (2.64) may be written as

Ql {ujujq) 1 Q

(2.65)

^1 + 4Q2 [Sojf v^^o

Equation (2.65) is only valid when ^Q'^{6ujY = 1 (assuming uj w ujq) or when

*c. = ±^. (2.66)

The half-power frequencies must be symmetric about the resonant frequency for

pure modal structure; that is,

^2 = ^0+^. (2.67)

The bandwidth by definition is Aw = \<jJi - W2I = 7^ = 2(7 . (2.68)

Hence, the quality factor is given by

Q = ^ = l~T-^ (2.69) Aw A/1/2
28 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

Frequency

Figure 2.12: Cavity transmission curve defining loaded Q-factor, Q = /o/A/1/2

This is shown in Fig. 2.12. Knowledge of the Q-factor at any resonant frequency

allows us to rapidly determine the resonator bandwidth and loaded natural

resonant frequency ujlWhen a resonant cavity is used as a dielectric measurement device in a mi- crowave system, several different Q-factors can be defined. The first Q-factor

accounts for internal cavity losses. It is defined as the unloaded Q-factor QoSecond, the external Q-factor Qe accounts for external losses due to the coupling

ports. These occur because the cavity resonator must be connected to external

microwave circuitry in order to be usable. Finally, the loaded Q-factor Qi is the

measured Q-factor that includes both internal and external losses.

Consider the unloaded Q-factor that is due to losses in the cavity itself. For the

unloaded Q,

LOqW

Qo =

p, (2.70)

where P, is the internal power dissipation. For cavity resonators, power dissipation Pc in the cavity conducting end plates and wall and power dissipation Ps

in any dielectric sample in the cavity contribute to internal power dissipation.

Hence,

1 Pc + Ps I I — . . = — ^ = \ 2 71)

Qo ujoW Qc Qs' ^ ^ where the cavity conductor quality factor is given by Qc, and the dielectric

quality factor by QsFor linear, homogeneous dielectrics the dielectric quality factor Qs is

Qs =

iJoW uJoeoe'j,fn\E\'dV

Ps aJIIlEl^dV

(2.72;
2.2. TEoiP MODE-FILTER CYLINDRICAL CAVITY 29

a tan 6

The loss tangent is simply the reciprocal of Qs- Clearly the lowest value of Qfactor in (2.71) dominates Qo, the unloaded Q. It is therefore to our advantage

to have a high-Q resonator for good measurement precision of low-loss dielectric

materials.

In practice, a cavity must deliver power to an external load through the coupling

ports. The power loss due to the presence of an external load in a cavity dielectric

measurement system results in the external quality factor Qe, which is defined

by

Qe = ^ . (2.73)

The stored energy in the numerator is still the energy stored inside the cavity,

but the power loss in the denominator is an external drain on the internal energy.

The loaded Q is the total Q for the system including power losses both internal

and external to the cavity resonator. The loaded Q is

Q. = ^, (2.74)

where Pt is the total power loss. That is,

PT = Pr + PE , (2.75)

so 111 , , -— = —- + —. (2.76)

Ql Qe Qo

The loss tangent is then

Qs Ql Qc Qe

or '^"^ = i-^' <^-'«'

with a sample inserted into the cavity. For the empty cavity the total loaded Q

losses are just — = — + —. (2.79)

Ql Qe Qc

The loaded Q-factor Ql and the unloaded Q-factor Qo are related by the coupling coefficient k = Pe/Pi- This relationship is given by Ginzton [20] and

Sucher and Fox [21] as Ql = -:^. (2.80)
30 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

For a two-port resonator like the NIST X-band cavity,

K = ACi -(- /C2 , (2.81)

where ki and K2 are the coupling coefficients for ports one and two. In terms of

the unloaded cavity Q with a sample, we have

^0 = i^n"". A

' (2.82)

1 + Qc tan

or, in terms of the loaded cavity Q with a sample under test.

Thus, from

tan (5= — 2.84

(1 + «;)QlQc

we can determine the loss tangent if we know the coupling coefficient k, the

loaded quality factor Qi and the cavity conductor quality factor Qc2.2.4 Port Coupling Coefficients for Determining

Unloaded Cavity Q-Factor

Frequency domain techniques for cavity measurements can be a reflection method

or transmission method. An automatic network analyzer (ANA) displays the

magnitude and phase of the reflection coefficient Sn or transmission coefficient

521. The reflection coefficient can be displayed on the network analyzer in polar

coordinates. A Smith chart overlay for the polar reflection-coefficient display

enables the impedance to be read directly. Swept-frequency measurements can

also be performed and displayed with the network analyzer. Ginzton [20] considers various equivalent circuits for cavities with multiple resonant modes and

shows that a resonant circuit with an external load has a coupling coefficient

given by

« = §^-l = |^, (2.85)

where Rl and Re are the internal cavity and external resistive losses, respectively. For K < 1, the cavity is undercoupled to any external component. For

/^ > 1, the cavity and external component are overcoupled, and for k = 1, the

cavity is critically coupled. In order to determine the total empty cavity con- ductor loss Qc, we need only measure Ql (without sample) and Qe- Qc is then

given by

We — WL
2.2. TEmp MODE-FILTER CYLINDRICAL CAVITY 31

M

C

R

Figure 2.13: Equivalent resonator circuit for reflection Q measurement [18].

Consider now a reflection type of Q measurement with the equivalent circuit for a simple one-port cavity shown in Fig. 2.13. The input impedance is given by

Z^ = juLi +

{ujMf /R

1+^'Qofe-e)

(2.87)

The reactive component ujL\ due to the coupling port is usually ignored or incorporated into a phase term for data-fitting purposes as described in Sec. 4.3.2.

The factor Ri = (a;M)^/i? is defined as the induced input resistance. Using (2.62) we may rewrite the expression for input impedance as

Z,=

Ri Rr

l+jQo2^^WQ l+jQo2

(2.88)

The corresponding input reflection coefficient is

^i — Zq

r, Zi + Zq

(2.89)

where Zq is the characteristic impedance. When the cavity is detuned, we may

allow Aw ^^ oo so that Z^ -^ 0, and the input reflection coefficient becomes

r,:i — i Z3 1 (2.90)

At frequencies close to the resonant frequency wq, the locus of the input reflection

coefficient describes a circle (See Fig. 2.14). The complex number F^ — Yd is given by

2Z^/Zq

Tn = l+jQo2^ +

I Acj Zo

(2.9i;

Now the ratio of the input impedance to the characteristic impedance Zq is
32 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

Im (D

Re (D

Figure 2.14: Input reflection coefficient versus cu.

defined to be the coupling coefficient; that is,

(2.92)

Furthermore, the loaded coupling coefficient is defined in (2.80) and (2.85) so that (2.91) may be rewritten

r,-r

2k

D

1 + K + J(1 +

2

k^)Ql 2Aa;

wo

2

2Aa;

{i + ^)[^ + jQL2^]

' (2.93)

At resonance Auj = and the circle F^ — Fd intersects the real axis. From (2.93)

the diameter of this circle is

d = IF, - FD Imax 2k

1 + K

(2.94)

Consequently, if we determine the diameter of the circle on the polar display of

the network analyzer, we can compute the coupling coefficient from (2.94) to be

K, =

d

2- d

(2.95)
2.2. TEqip MODE-FILTER CYLINDRICAL CAVITY 33

For two ports,

Ql = ,^^\ , (2.96)

1 + Ki + AC2 and Ki and K2 can be determined from separate measurements, as noted above.

Then, since we know from transmission measurements that

Ql = ^, (2.97)

we can determine Qq. Kajfez and Hwan [22] give similar expressions for a one- port cavity with shghtly different couphng elements. This analysis has been

extended by Estin and Janezic [23, 24] to the case of a two-port cavity.

2.2.5 Cavity Conductor Quality Factor

The cavity conductor quality factor Qc is defined as

1 Pc

Qc ^0^ (2.98)

where Pc is the power loss caused by the finite conductivity of the cavity walls

and W is the total stored energy in the cavity. At resonance the total energy

interior to the cavity is constant. The average values of the magnetic energy

Wm{t) and electric energy We{t) are also equal. In other words, the energy al- ternates between being stored in the electric field and the magnetic field. Lastly,

the total stored energy in the cavity is twice the average value; that is,

W{t) = 2W^ = 2Wm . (2.99)

Equation (2.99) implies that the input admittance at the cavity ports is real

under resonance conditions. To find the total energy storage, the volume integral

of either the electric or magnetic field energy over the volume of the resonator

must be performed. The average stored electric energy in the empty cavity is

f' f We = ^^ \E\^dV . (2.100)

4 "'cavity

For the TEoip mode structure, the electric field Ecj, is given by (2.42), so

r in

W, = ^Hy{l+ f^ ^J„V„,). (2.101)
34 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

Equation (2.101) demonstrates that increased energy storage results as the

length L of the cavity is increased, particularly for the higher axial mode numbers. This fact was taken into account in the design of the NIST 60-mm diameter

X-band mode-filtered resonator.

In order to determine the cavity Q, the losses caused by the finite conductivity

of the cavity walls must now be evaluated. For small losses the surface cur- rents are essentially those associated with the loss-free field solutions in (2.40)

through (2.42). The surface current is given by

Js = nx H , (2.102)

where n is a unit normal to the surface and directed into the cavity. The wave

impedance Zm of the metal wall of a uniform cavity is given by

Zm = \l ^^^

,

, (2.103)

or

a

1 -7

•^ ^—'- +

2a 8 a

(2.104)

For metals a is usually greater than 10^ S/m and e' is of the order of 10~^^ F/m.

For uj equal to 10^° rad/s, uje' /a is only 10~^; for uj equal to 10^^, uje' /a is only

10~^. Hence,

^(1+j) (2.105)

la

is a very good approximation to metal wave impedance at microwave frequencies

and

Rm = ^ {Zm) = J^ = T^ , (2.106)

* za OgCJ

where

/ 2

(2.107)

UflU

is the plane-wave skin depth of the metallic cavity wall (depth at which fields

have decreased by a factor of e~^ from their surface values). The skin depth and

relative loss of various metals at 10 GHz are shown in Table 2.3. The total power loss in the walls of the cavity is given by

Pc = \l ., „ 3? [Zm) Js J:ds , (2.108)

z -^cavity walls

or, if the real part of the wave impedance is uniform over the wall surface,

^'^ = t/cavity walls

l^*-l^^^- ^'-'''^
2.2. TEq^p MODE-FILTER CYLINDRICAL CAVITY 35

From (2.41) and (2.42) we may write the equivalent surface currents at the top,

side, and bottom of our cavity. For TEoip mode structure they are

Js, top endplate —— ^0 -^^

I cos (pir) u^

t'oiL

Js,side — -^O-^O (^oi) sin

p7iZ

~17

Js, bottom endplate

(2.110)

(2.111)

(2.112)

where u^ is the unit vector in the azimuthal direction. The magnitudes of the

surface currents at the top and bottom end plates are the same. If the real part

of the wave impedance is the same for both the top and bottom endplates, the

total top and bottom power dissipation Pt^b is

Pf-ht,b — RtuHq

p-na

-toiL.

n2

'^r4'°(ir)F*- (2-113)

Since

we have

r^{4¥)r*=f^»('-''

Pt,b — ^m,epLlQ7Ta

2 2 f P^ra

IqiL

•^0 l^oi)

(2.114)

(2.115)

where Rm,ep is the surface resistance of the endplates.

Table 2.3: Skin depth and relative loss of various metals.

Metal Conductivity, a

(xlO^ S/m)

Skin depth, Sg, at 10 GHz (xlO-^m)

Relative loss

per meter

Ag 6.17 6.42 0.97

Cu 5.80 6.60 1.00

Au 4.10 7.85 1.19

Cr 3.84 8.11 1.23

Al 3.72 8.26 1.25

70-30 brass 1.57 12.7 1.92

P 0.9 17.0 2.5

solder 0.71 18.5 2.8
36 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

Similarly, the total side wall power loss can be determined as

^side — i^m^side^Q'^O.'JQ \^Ql) q ' (2.116)

so the total cavity conductor power dissipation is

r J / \^1

Pc — Pt,b + Pside = j/q^^'^O (^01 ) I P-m,side~ + Rrn.epO' I T^ T ) I

The cavity quality factor Qc for TEoip mode structure is then

(%)^(f)^

(2.117)

Qc =

a Va/"*" L \ L J

where

V = yi^o/i

(2.118)

(2.119)

Clearly, if the surface resistance of the endplates is the same as that of the

cylindrical wall {Rm,side — Rm,ep = Rm) of the cavity, then (2.118) reduces to

Q

V (%)' + (f)^

c

2Rr 'A'-fY^iifr

(2.120)

In general, for any TEmnp mode structure in an empty right circular cylindrical

cavity, we may in similar fashion determine

Qc [TEmnp) — i-(i?7)1k„f+(T)^

3/2

27r(5. (Cnf + 2f(T) +(l-T)fe)'

(2.121)

Equation (2.121) is valid when

^'^TTijSide ^^^m.ep ^m i (2.122)

for any transverse electric mode and reduces to (2.120) for m = 0, n = 1. In

practice, experimentally determined values of Q (whether the cavity is empty

or filled with a sample) are usually less than the theoretical value by an unpredictable amount. This discrepancy between measurement and theory usually is due to the effects of surface finish in the cavity interior. Surface roughness leads

to an effective penetration depth for electromagnetic energy which is larger than
2.2. TEqip MODE-FILTER CYLINDRICAL CAVITY 37

that given by the plane-wave 6s in (2.107). The effective penetration depth 6s^m

is that which is determined by the measured quality factor Qc,m of the (empty)

air-filled cavity. That is, from (2.120) and (2.121)

c Cair

X m.

a J

2 / X 2 {^Jn.\ ^ (£![)

's,m 27rQc,mf ii'-fY + ii^y

(2.123)

As we see from (2.123), the measured Qc,m is usually less than the theoretical

Qc for any given resonant frequency / or axial mode number p. Thus, Sg^rn is generally greater than the theoretical 65. In the case of the mode-filtered cavity

where the side wall is constructed with helically wound waveguide, the real part

of the side wall wave impedance Rm,side may not exactly equal the real part

of the end plate wave impedance Rm,ep- Through the use of (2.123), the loss

tangent evaluation can be corrected by considering the difference in cavity wall

losses with and without sample insertion (see Cook [2]).

A somewhat simpler approach for calculating the cavity quality factor Qc that

avoids the calculation of wall currents is the use of an incremental frequency

rule [25, 26]. The incremental frequency rule, valid only for rotationally symmetric modes {TEonp), replaces the detailed surface integrations outlined above

by a simple computation of increment in inductance per unit length when all metal walls are receded by a distance Sg. The steps are as follows:

1. Compute the resonant frequency /o of an ideal cavity with perfectly con- ducting walls.

2. Move (perfectly conducting) walls into cavity by one skin depth Sg. Compute change in resonant frequency for this reduced cavity (r.c), A/o, r.c3. Compute the cavity quality factor from Qc = /o/A/o, r.cFor the helically wound cavity, where 6s^side may differ slightly from 5^ ep, we

have

A/o = --7- = -^-6s,side - -^-r'^5s,ep (2.124)

do da oL

dfo dfo ^ dfo

^/o = -

For the TEqip mode,

Jo — ^^s-('iy^

TM
38 CHAPTER 2. THEORY OF MICROWAVE RESONATORS

so

Qc =

,' \2

(^) +(f)'

-|3/2

Rm,side I ^fll ) _i O Rm,ep ( pTT V

(2.126)

or, lOr rirn,side ^m,ep J^rm

Qc = (^)^(fy

3/2

2/2. n^) +!(?)

(2.127)

which is the same as (2.120). An alternate form of (2.126) is

Qc

27r t (f/ \2 , D (p7rD\\c

(2.128)

SiCp

where D is the cavity diameter and 6s is the skin depth at /q. Qc is the un- loaded quality factor of the air-filled cavity and that it decreases with increasing

frequency since Rm is proportional to s/Jq. Larger Qc values are obtained by

using higher values of p. The actual loaded Q for the empty cavity is simply

related to the theoretical Q by the factor {1 + ki + k,2), as discussed previously.

If ^s^side = 6s,ep = ^s, thcu (2.128) may be written

Qc^ Ao \t'o.f+{^y

I 3/2

2'k6s (+1 \2 I £> (pT^D\ (2.129)

2.2.6 Separation of Cavity Endplate Losses from

Side-Wall Losses in Helical Cavity

In some applications it is useful to separate the endplate cavity losses from the

side-wall losses. This is particularly true if one or both of the end plates are

physically changed. One reason to change the end plate(s) would be to obtain a

surface resistance measurement of a conductor. One approach to resolving this

problem would be to measure the cavity quality factor Qc with two rotationally

symmetric modes at resonant frequencies which differ by 10 MHz, say, where the

tuning factors are not asymmetrically distributed about the associated, resonant

frequencies and at the same resonant length. Those modes must be excited and
2.2. TEqip MODE-FILTER CYLINDRICAL CAVITY 39

be measurable. From (2.121) we have

Qc (TEonp) —

Ao it'onf+if)'

-|3/2

2'7r S (4-1 ^2 . 2a (

p-ira \ c (2.130)

and

Qc {TEoiq)

(t'oif+if)'

-]3/2

2^^.,..e(4)' + T(T) ^s,ep

(2.131)

where n ^ I, p ^ q. With these distinct modes there is only an azimuthal

electric field, so it is not necessary to consider any capacitive loading at the end

plate gaps. If these modes are 10 MHz apart, errors in the skin depth losses at X-band should be less than 3%. Solving (2.130) and (2.131) for Sg^g^de and

subtracting, we obtain the following relation for the skin depth loss of the end

plates for p

yi^ q:

where

XoL'

's,ep AB C-D

iir^a^ip -q) VQc {TEonp) Qc {TE,Olq, (2.132)

A

B

C

D

= 1 +

pi: a ITOn it,.fA^)\

= 1 +

qira

IT01

(f„.f +

''"L

(2.133)

(2.134)

(2.135)

(2.136)
Chapter 3

Cavity Resonator Design

This chapter describes the design of the NIST helically wire-wound cylindrical cavity resonator. The resonator is designed to give the best dimensional

stability and repeatability possible. Detailed machine drawings are given in Appendix C. The cylindrically shaped cavity is nominally 60.0 mm in diameter.

The resonator length varies from 408.5 mm to 433.3 mm. The inner wall of the

cylindrical waveguide consists of two insulated copper wires helically wound,

side-by-side. Flanges connect to the top and bottom of the cylindrical waveguide. The top flange of the resonator holds a replaceable coupling endplate.

The bottom flange connects to a retractable tuner assembly. This tuner assembly contains an optically flat endplate on three hardened steel rods that each run along a pair of instrument-grade linear ball bearings. This lower endplate

moves up and down inside the cavity by a motor-driven micrometer. A capacitive sensing micrometer detect resonator length by pressing against a yoke that

ties the three travelling rods together.

The entire tuner assembly disconnects and lowers away from the cavity flange.

In this way, disk-shaped samples can be placed onto the tuner endplate, then

raised up into the cavity. Surrounding the cylindrical waveguide is a water

jacket for temperature control. The water jacket is sealed at the top and bottom

flanges with O-rings. The base of the sliding tuner assembly is also temperature

controlled. Water circulates from a water bath to the bottom flange then into

the water jacket surrounding the helical windings and back out to the water

bath. Temperature is nominally controlled to within il:0.1°C.

40
3.1. HELICALLY WOUND CIRCULAR WAVEGUIDE 41

3.1 Helically Wound Circular Waveguide

The cylindrical waveguide section is 430.8 mm long. The cylindrical wall of the

cavity resonator is made from two insulated, helically wound copper wires. The

windings consist of two 0.165-mm diameter (0.0065 in) wires wound side- by-side.

The wires are embedded in an epoxy and surrounded by a 2-mm thick fiberglass

sheath. This sheath is then epoxied into a steel pipe. To prepare the waveguide

section for the resonator, the waveguide's steel outer wall was made concentric

with the inner wall by slipping a cylindrical mandrel into the waveguide and

turning the outer wall of the waveguide in a lathe. Shrink tubing was then

placed over the waveguide to act as a seal from the water jacket. Finally, the

ends of the waveguide were cut to give a good cylindrical geometry.

The helical waveguide allows currents to flow in the circumferential direction and

prevents currents from flowing in the axial direction. As such, the waveguide

acts as a mode-filter that allows TEqi modes to propagate, while attenuating

other modes. There are over 1170 possible TE and TM modes below 18 GHz

for a cylindrical cavity measuring 60 mm in diameter and 430 mm in length.

The mode-filtering characteristics of this resonator allows us to readily identify

the TEqi modes and alleviates problems caused by mode-degeneracy. As we

will see in Chapter 4, TE02 and TE12 modes are weakly excited and can cause

interference.

3.2 Coupling Flange

The resonator's top flange slides onto and butts up against the end of the helix waveguide. Figures C.4 through C.8 display this piece. The part that flts against the waveguide has a 60-mm diameter hole. The flange is machined to

fit very concentric and perpendicular to the longitudinal axis of the cylindrical

waveguide. Opposite the recessed side that butts against the helical waveguide

is another recessed surface which accommodates the coupling endplate. Proper

machining of these two surfaces assures good cyhndrical resonator geometry. A

threaded hole is placed in the fiange to accommodate a temperature probe. The

temperature probe shdes vertically into the water cavity jacket, through the

couphng flange. The knurled temperature-probe bolt shown in Fig. C.19 presses a small O-ring against the temperature probe and the coupling flange to seal off the water jacket. Three other threaded holes placed in the outer radius of the

flange hold steel rods that tie to both the top and bottom flanges, which hold

the entire cavity together.
42 CHAPTERS. CAVITY RESONATOR DESIGN

3.3 Apertured Coupling Endplate

The apertured coupling endplate which slides into the coupling flange is shown

in Fig. C.18. The couphng endplate is tightened to the couphng flange with a

pressure ring, shown in Fig. C.19. The inner side of the coupling endplate is polished to optical flatness. The outer side of the coupling plate contains precise

waveguide flange-bolt holes and two short (10.795 mm) precision waveguide

sections which continue from the flange-bolt holes down to the coupling slits. The two rectangular slits couple energy into and out of the cavity. The coupling

slits are located to optimally couple with the TEqi mode. This is done by

locating the slits at the point where the Bessei function Jg is a maximum for

the first zero.

Presently, the cavity's coupling endplate is equipped with X-band flanges and

is designed for nominal minus 30 dB resonance transmission at 10 GHz. The

slits measure 5.080 by 2.286 mm (0.2 by 0.09 in). At lower frequencies, the

apertures couple less energy into the cavity because of their reduced size relative

to the waveguide wavelength. Conversely, energy couples more strongly into the

resonator at higher frequencies. Weak coupling into the cavity is most ideal in

order to maximize cavity Q, and to minimize Q-value and frequency corrections.

Couphng could be made weaker by reducing the aperture sizes, but this would

make the transmission response at lower frequencies too weak to be measured

by a network analyzer.

3.4 Tuner Flange

The tuner flange shown in Figs. C.9, CIO, C.ll, C.12 and C.13 connects the

helical waveguide and water jacket to the tuner assembly. The helix waveguide

fits into a recess in the flange similar to the coupling flange. Two O-rings in the

flange press against the waveguide and the acrylic plastic tube to seal the water

jacket. Three threaded holes attach to tie rods that pull the coupling flange and

tuner flange against the helix waveguide. The entire cavity is supported from

the tuner flange by three 12.7-mm (|-in) diameter rods that rest in recessed

holes. A vertical hole near the edge accommodates the measurement micrometer,

which extends down to press against a measurement yoke connected to the

moveable tuner endplate. The micrometer is fastened to the tuner flange by

a lateral screw. There are three U-shaped recesses at the edge of the flange.

These are for the tuner assembly latches. A protruding ring in the tuner flange

lines up with the tuner assembly, and assures good alignment. Outside of this

ring is a recession in the tuner flange to catch dust and other contaminants.
3.5. TUNING ENDPLATE ASSEMBLY 43

This improves length repeatability when the tuner base is lowered and then

reattached. Finally, a sample centering ring is machined into the tuner flange.

This ring has a chamfered corner that helps to center a poorly aligned sample

when it is raised up into the cavity, which prevents damage to the tuner.

3.5 Tuning Endplate Assembly

The tuner assembly is designed to position the endplate inside the cavity with

utmost precision. This is accomplished by using a motor-driven micrometer

to move the endplate assembly up and down combined with a 0.1-/im precise

measurement probe connected to the cavity's tuner flange. The measurement

probe pushes against a yoke extending out from the endplate's guide shafts. The

resonance frequencies can be tuned by varying the effective length of the cavity.

3.5.1 Sliding Base

The base which holds all the parts of the tuner assembly is shown in Figs. C.3,

C.14, C.15, C.16 and C.17. This base has inserted PVC bushings (Fig. C.21)

which allow the base to easily slide up and down on the cavity-stand rods.

Three instrument grade linear ball bearings are press flt into this tuner base.

The tuner endplate guide shaft rods run along these bearings. The three latches

which connect to the tuner flange swing up from the edge of the base. Each latch

has a torque nut (Fig. C.21) that allows exact pressure to be placed on the latch.

The tuner base does not make contact everywhere around the flange. Instead,

raised surfaces on either side of each latch help ensure length repeatabihty by

preventing contamination. The micrometer which drives the tuner-endplate is fastened to the tuner base with a replaceable cup shown in Fig. C.21. This cup

is machined to fit the particular drive micrometer, and if a different micrometer

is used, one can design a diflFerent cup. The micrometer cup fits into a hole with

a recessed lip in the center of the tuner base. Finally, for temperature control,

the tuner base has a channel machined into it which allows water to circulate.

3.5.2 Sliding Base Extension

An extension to the shding base shown in Figs. C.3 and C.22 contains three more

press fit linear ball bearings, which help the alignment of the tuner endplate

guide shafts. The extension butts against the sliding base and is attached by

knurled nuts that screw onto the sliding PVC bushings.
44 CHAPTERS. CAVITY RESONATOR DESIGN

3.5.3 Tuner Endplate Assembly

The tuner endplate assembly consists of an optically polished endplate, three

hardened-steel guide shafts and a micrometer reference yoke. These and other

parts for the tuner base assembly are shown in Figs. C.20, C.21, C.22 and C.23.

A flat hardened-steel plug, press-fit into the center of the endplate rests against

the drive micrometer. The guide shafts are attached to the tuner endplate with

set screws. The guide shafts are stabilized linear bearings in the tuner base and

base extension, and by the micrometer reference yoke. This yoke has fingers

which extend radially outwards. The measurement micrometer attached to the

tuner flange presses against one of these fingers to give a differential length

measurement. The three guide shafts extend further through the tuner base's

linear ball bearings, and then through the linear ball bearings in the tunerbase extension. Between the tuner base and the tuner-base extension, the guide

shafts are spring loaded. The combination of yoke, linear ball bearings and

spring loading helps assure alignment stabihty of the tuner endplate.
Chapter 4

Resonator Evaluation

We have quantified the cavity resonator's dimensional and electrical performance

to ensure accurate calculations of permittivity and loss factor in dielectric samples. The uncertainty in resonator parameters such as cavity diameter and

length, resonance frequency, and Q have been determined in order to calculate

overall measurement uncertainty.

Knowledge of the behavior and performance of the resonator allowed us to develop several possible corrections that enhance measurement accuracy. These

corrections, which adjust for resonator losses and geometric deviations are included in this chapter. Even without corrections, the NIST resonator's permittivity measurement yield excellent agreement with previously published results, as demonstrated by the 1723 glass results given in Sec. 8.1 [27].

Resonator characterization also allows us to determine when the resonator's

performance has degraded. In this respect control charts have been developed

to help track the resonator's behavior.

The early portions of this chapter describe the results of cavity dimension ex- periments. In particular precision in the diameter and length is shown, and

the tuner-endplate assembly's travel accuracy and attach repeatability are evaluated. We then discuss the effects on cavity dimension calculation caused by

coupling aperture perturbations and resistive wall losses. Later portions deal

with the evaluation of resonance quahty factor Q. Methods for determining

Q and their associated uncertainties are discussed. Loss mechanisms are then

separated into parts in order to deduce sample loss.

45
46 CHAPTER 4. RESONATOR EVALUATION

4.1 Dimensional Characterization

This section presents results from experiments that determined the accuracy for

resonator diameter and length. In a geometric sense, the resonator diameter is fixed, and the length changes as the tuning endplate moves up and down inside

the cavity. In an electrical sense this is only partly true. The electrical skin

depth penetration varies with frequency. Presently, no work has been performed

to study the effective penetration depths into the helical wall and endplates.

Cavity dimensions are determined from resonance spectra, and therefore should

be considered the "electrical dimensions" of the resonator.

The change in position of the tuning endplate is measured with a sensing micrometer. The accuracy of this sensing micrometer is verified. The assembly

that holds the tuning endplate can detach from the resonator. We investigated

both the accuracy and repeatabihty of the length measurement assembly and

tuner assembly, and the effective electrical diameter and length of the resonator

chamber under varying conditions.

4.1.1 Effective Electrical Length and Diameter

The resonator's effective electrical diameter and length are found through linear

regression from the distribution of the resonant-mode spectrum as described in

Sec. 5.2.4. For example, if we know the resonant frequencies and axial mode

numbers p of at least three TEqx^ modes, we can estimate resonator length and

diameter. If we use more than three modes in calculating length and diameter,

their precision estimates decrease. The frequencies of the resonant modes can

be very accurately determined to ±500 Hz, which allows accurate determination

of the resonant cavity's dimensions. However, as we will show, various physical

phenomena can pull resonant frequencies up or down, which results in different

estimates of the cavity dimensions, depending on which modes are used in the

linear regression.

Figure 4.1 shows typical length versus diameter results which demonstrate the

frequency dependence of cavity dimensions calculation. Cavity length and diameter were calculated from subsets of the 7-14 GHz TE^ip resonance spectrum.

Each subset consisted of five adjacent resonant frequencies, and began with the

lower frequencies and moved upward (T£'oi(io-i4), T£'oi(i4_i8), • -, T£'oi(33_37)).

Cavity length and diameter calculated for each subset are slightly different, and

there is a systematic trend in which calculated diameter decreases and length

increases as the higher frequency modes are used to calculate cavity dimensions.

Also, the calculated cavity dimensions crosses back onto itself before the increas-
4.1. DIMENSIONAL CHARACTERIZATION 47

LENGTH VS DIAMETER CRLCULRTED FROM SUBSETS OF TE_, SPECTRUM

y In 340 1 1 1 1 1 1 28-3?. ?9-33, 30-31

11.,.-^"' ^ 3I-35.32-3G

335 '/ ^^\ — 3

33-37 27-^^^^^^^= = '^-'^

nOI—

330 — \\

X

CO

CO

325

1?-2I==>\

_(=! \< = =M-I8, 15-19,16-20 I3-I7-A

bO

0)

320 ~ I2-I6 = = >\

ll-15 = = >\

—

315 ~ I0-M = = >\ —

310 1 1 1 1 1 1 26 2? 29 30 32 33 Diameter minus 60x10^ (A'ni)

Figure 4.1: Cavity length versus diameter calculated from subsets of the 7-14

GHz resonance spectrum, T£'oi(io)-T'£"01(37). Subsets consist of five adjacent

modes ( TEq^io-h), 2^ -£^01(14-18), • • •, 2^ -£'01(33-37) )•

ing length/decreasing diameter trend continues. This is most likely to be caused

by mechanisms that are not smoothly varying with frequency, mode interference

for example. The systematic behavior demonstrated in Fig. 4.1 indicates that one or more frequency-dependent mechanisms are affecting our cavity dimension

calculation. We will discuss in Sec. 4.1.4 and Sec. 4.1.5 possible causes for this

behavior, and Sec. 4.1.4 demonstrates how coupling-iris perturbations reduce

cavity dimension uncertainties.

By looking at the range of length and diameter values calculated from the entire

7 to 14 GHz resonance spectrum, we conclude from Fig. 4.1 that the uncertainty

in our length determination is less than or equal to AL = ±0.012 mm, and that

diameter can be determined within AD = ±0.003 mm. These numbers also

correspond to approximately 1.5 standard deviations estimated by least-squares

regression of all X-band TEoip frequencies. Chapter 9 will demonstrate that

these estimated length and diameter uncertainties are significant contributors

to the total permittivity uncertainty. The length and diameter uncertainties
48 CHAPTER 4. RESONATOR EVALUATION

BB

o

CO

O.IU

0.112 O.U

0.108

0.106

^3 0.104

i-i

0.102

0.1

0.098 s

o

0.096

0.094 i 0.092

3,

0.09

0.088 -a

s

1-5

0.086

0.084

0.082

Normalized Lenglb ts Sensing Micrometer

I 4 I 8 I 12 I 16 I 20 I 24 I 21 i 17 I 13 I 9 I 5 I 1

I 4 I 8 2 6 10 14 18 22 23 19 15 11 7 3 2 6 Sensing Micrometer Reading (mm)

Figure 4.2: Difference between micrometer-measured length and length calculated from X-band resonance spectra.

can be reduced if we limit our measurements to a narrower frequency range. For

example, the range of cavity dimensions calculated from subsets of the T£'oi(2o)

to T£'oi(29) modes (approximately 9 to 11 GHz) is about one third smaller than

the range of values calculated for 7 to 14 GHz subsets. This fact allows us to

calculate cavity length and diameter from the T£'oi(20) to T£'oi(29) modes that

gives AL = ±0.004 mm and AD = ±0.001 mm.

4.1.2 Sensing Micrometer Accuracy

When the fixed-frequency method is employed to measure permittivity, the tuner

endplate must be adjusted to shorten the cavity when the sample is in place.

The change in cavity length determines e'^, and the sensing-micrometer measurement accuracy is therefore critical. To assess the accuracy of the sensing

micrometer, we measured the TEqi X-band resonance spectra at different cavity

lengths in 1 mm increments over the tuner endplate's 24-mm range of travel.

Cavity dimensions were calculated from the X-band resonance spectra at these

different cavity lengths. Figure 4.2 shows the difference between the length measured by the micrometer and the resonator length calculated from the resonant

frequencies. Over the entire 24-mm endplate travel, the sensing micrometer

reading and the calculated cavity length agree to within ±0.011 mm.

Figure 4.3 shows how the calculated cavity diameter changes by approximately
4.1. DIMENSIONAL CHARACTERIZATION 49

Calculated Diameter versus Sensing Micrometer

I 4 I 8 I 12 i 16 I 20 I 24 I 21 I 17 I 13 I 9 I 5 I 1

I 4 I 8 2 6 10 14 18 22 23 19 15 11 7 3 2 6 Sensing Micrometer Reading (mm)

Figure 4.3: Calculated diameter versus change in length as measured by the

sensing micrometer. A diameter uncertainty of ±0.002 mm is found from the

range of results.

±0.002 mm as the cavity length changed by 24 mm. Because length changes

when the fixed- frequency method is used, this result determines the diameter

uncertainty that should be used in calculating permittivity uncertainty. Because

frequency changes when the fixed-length method is used, the same diameter uncertainty of ±0.002 mm was found from the range of diameter results calculated

from TEqip subsets shown in Fig. 4.1. A similar experiment which was performed two years earlier indicates that som.e deterioration of the length-measurement system has taken place. In this ex- periment cavity dimensions were calculated for seven different cavity lengths

from five-frequency subsets of the 7-14 GHz resonance spectra, as explained for

Fig. 4.1. The sensing micrometer was used to change the cavity length, and variations in calculated cavity length give an indication of the length-measurement

accuracy. Figure 4.4 shows how cavity dimensions calculated from different parts

of the resonance spectrum vary as the cavity length is changed. The difference

between dimension results calculated from identical subsets shows a length variation less than ±0.004 mm and a diameter variation of less than ±0.001 mm.

The resonator has been opened and closed several hundred times during those

two years, and the sensing-micrometer end created a very small pit in the aluminum yoke that is part of the tuner-endplate assembly. If the micrometer had

not been settled into the pit during the 0-mm measurements, the anomalous 0-
50 CHAPTER 4. RESONATOR EVALUATION

CRVITY DIMENSIONS CRLCULRTED FOR SEVEN LENGTHS

o

4-=

o

hjO

340

334

2 328-

32?-

315 -

310

1 1 1

MODE NUMBERS ^"\?~~~^^^^^^^^

Normalization

+ 433.0tntti

Factor: \\\^sS \ \\ \

\l ~

S 4 31.5mtn \\\ ti ^ \ * 430. Bmm \\ \ Vv\ "

H 428.5mtii X\ N \ ^

X 421.0nim \\

4IG.5mm \\

I 4 12.0mm

1 1 1

LOWER = = > ^\

MODE NUMBERS

1 1 1 26 2? 2B 29 30 31 Diameter minus QO.OxlO^(iim)

32 33

Figure 4.4: Length versus diameter calculated from five-frequency subsets for seven different cavity lengths. Displacement of curves from one another indicates

sensing micrometer accuracy.

mm result shown in Fig. 4.2 could occur. A temporary solution to this problem

has been to rotate the endplate so that another one of the yoke's three fingers,

shown in Fig. C.23, presses against the sensing micrometer. A permanent solution to this problem is to press a hardened steel plug into the aluminum yoke

to serve as the reference surface for the micrometer. Meanwhile the sensing micrometer accuracy is given to be ±0.011 mm. We shall see in Chapter 9 that

this contributor to length uncertainty is a major contributor to permittivity

uncertainty.

4.1.3 Tuner Assembly Attachment Repeatability

To determine whether the small pit in the tuner-endplate yoke affects the tuner

assembly attachment repeatability, an experiment was performed in which the

tuner assembly was detached and then reattached to the resonator 23 times.

Each time the tuner assembly was detached, the tuner endplate was lowered.
4.1. DIMENSIONAL CHARACTERIZATION 51

Table 4.1: Repeatability of tuner assembly connection.

Mean micrometer reading (mm)

0.000 00 -0.00100 -0.00010

0.000 00 -0.000 90 -0.000 80 0.00122

0.00115 0.000 20 0.000 72 0.000 25

0.00107 0.00117 0.00160 0.000 70

0.000 90 0.00135 0.000 40 -0.000 95

0.00017 0.00143 -0.000 22 -0.00125

Average ± standarc error: 0.000 31 ± 0.000 87

then raised until the T£'oi(25) mode resonated as close to 10.63 GHz as possible. The micrometer reading was then recorded. Results are given in Table 4.1.

The standard error of the 23 micrometer readings is 0.000 9 mm, and the 95%

certainty interval is ±0.001 8 mm. This demonstrates excellent tuner assembly

attachment repeatability in which the cavity length can be repeated to within

2 /im. This means that the pit in the tuner-endplate yoke does not affect length

repeatability when the length is held fixed. Only when the length varies do

we see the discrepancy in length measurements. These results are in agreement with an experiment performed two years previously in which the cavity's

tuner base was reattached three times. Cavity dimensions were calculated from

the five-mode subsets described for Fig. 4.1 and results are shown in Fig. 4.5. For cavity dimensions calculated from identical subsets, length is repeatable to

within ±0.002 mm and diameter is repeatable to within ±0.0003 mm.

4.1.4 Iris Perturbation

The coupling irises cause two perturbing effects on cavity dimension calculations.

First, power loss through the coupling irises shifts the resonance frequencies

downward. Second, field distortion near the coupling irises also causes reso- nance frequencies to shift downward. Both of these perturbations are frequency

dependent and increase in magnitude as frequency increases. We quantified both

of these perturbations under typical resonator operating conditions.

Power loss through irises

The radiated power lost from the cavity through the coupling irises is analogous

to an added resistance in the coupling loop of a resonant circuit, the result
52 CHAPTER 4. RESONATOR EVALUATION

REPERTRBILITY OF CAVITY TUNER BASE CONNECTION

340

335

^ 330

1—

I

X

CO

CO

325

•5 320

bjO

1-4 315-

310

33-3;' 2?-3i

l?-16">

I0-M =

26 2? 28 23 30 31 Diameter minus 60x10^ (A^m)

32 33

Figure 4.5: Length versus diameter results showing cavity dimension repeatability for three tuner-base detach/reattachments. Cavity dimensions calculated

from subsets of five adjacent TEqi^ modes. Also note the repeatability of the

crossing pattern that occurs for the middle subsets.

being that the resistance lowers the resonant frequency and Q. Gallop and

Radcliffe [28] show that the insertion loss / and unloaded Qq can be used to

estimate the frequency shift due to power lost through the coupling apertures.

The frequency shift due to coupling aperture loss is given by

/o

v7

4Qo

(4.1)

At 10.63 GHz, the insertion loss is minus 25.4 dB and the unloaded Qq is approximately 84 000. This results in a frequency shift of approximately minus

1.7 kHz. The effect of this perturbation is much less than the field-distortion

perturbation described next. We have made no attempt to make this correction

to the measured resonance frequencies. Nonetheless, the generalized method for

correction is to measure the insertion loss at several frequencies in the resonator's

frequency range, calculate the theoretical frequency shift for each resonance fre- quency, and plot a chart of frequency shift versus resonance frequency. Then,
4.1. DIMENSIONAL CHARACTERIZATION 53

for any given resonance frequency, we may refer to the chart to determine the

appropriate frequency shift to be applied.

Field distortion

The electromagnetic field bulges into the coupling irises and distorts the ideal

cylindrical field pattern inside the resonator. The resulting field distortion in- creases the effective volume of the cavity, and therefore lowers the resonant fre- quencies. Because the distortion occurs at the coupling endplate, we expect the

cavity length calculation to be affected. Perturbation theory given by Slater [29]

and Waldron [10] can be used to estimate the frequency shift due to field distortion at the coupling irises. In general, perturbation theory can be applied

when the change in volume is much less than the total volume {AV <C V). The

frequency shift due to field distortion near the coupling irises can be written as Af ^ I^y i^H' - eE') dV

f Iy{iiH^ + eE^)dV ' ^

• ^

where in this case /i = /xq, e = cq. E and H are the electric and magnetic fields

inside the cavity. The integral in the numerator is over the volume added to

the cavity by the coupling irises, and the integral in the denominator is over the

entire cavity volume. The volume AV added to the cavity by the coupling irises

is difficult to estimate because the depth of field penetration in the coupling iris

is frequency dependent. Use of (4.2) to estimate the frequency shift will be prone

to errors in most cases. Therefore we used an experimental approach to estimate

the magnitude of the frequency shift due to the coupling irises. We plugged one

iris and measured the frequency shift of the TE^ip modes at the other port.

Results for both coupling irises are given in Fig. 4.6. The frequency shift due to

field distortion into the coupling irises increases as frequency increases because

field penetration into the irises increases as wavelength decreases.

Iris perturbation corrections

The tendency for calculated length to increase and diameter to decrease as frequency increases as seen in Figs. 4.1, 4.4, and 4.5 can be explained by field

pattern distortion near the coupling irises and to a lesser degree, by radiated

power loss through the coupling irises. The coupling irises are located in a cavity

endplate, and the field distortion in the vicinity of the coupling endplate will

make the calculated resonator length larger than its actual physical length. The

magnitude of the perturbation increases as frequency increases, so we expect to

see an increase in calculated cavity length at higher frequencies.
54 CHAPTER 4. RESONATOR EVALUATION

RESONANCE FREQUENCY SHIFT AFTER PLUGGING COUPLING IRIS

550 1 1 1 1 1 1 1 1

1

500 - /

150 -

/

/W -

400 - -

£ 350 0: PORT + PORT 2 r""^^^^ -

:; a-—^ L 300 , ^r"^ -

X J^ /x /

S 250

UJ^

^

200 ^^

Or-

,c.^^

^

\

150 ^ —

100 ^.*-^

,*-»-'^ fl/a: PORT 1 PLUGGED

B/b: PORT 2 PLUGGED

-

50 - —

1 1

i 1 1 1 1 1 1 9.5 10 10.5 11 11.5 RESONANCE FREQUENCY (GHz)

12.5 13

Figure 4.6: Measured resonant frequency shifts when coupling irises are plugged

relative to measured resonant frequencies when both ports are unplugged.

Correction for the iris perturbation can be made by adding the frequency shifts

given in Fig. 4.6 to the measured resonant frequencies. Figure 4.7 demonstrates

the result of this correction. For a given length, the cavity's length and diameter

are calculated from subsets of five adjacent modes from the 8-13 GHz TEoip res- onance spectrum. Estimated length and diameter uncertainties that come from

the linear regression are also shown. The uncorrected cavity dimensions vary

with frequency by more than the precision estimates, indicative of a systematic

error. When the iris-perturbation frequency correction is applied cavity length

decreases and diameter increases slightly, as expected. Most importantly, when

the correction is applied, the systematic relation between length and diameter

is reduced, and the scatter of cavity dimension results is in general agreement

with precision estimates. This finding is important for demonstrating that the

frequency dependence of the cavity dimension calculation can be largely eliminated by applying the frequency shift corrections from Fig. 4.6. Without the

iris-perturbation frequency correction, cavity dimensions would have to be re- calculated at each frequency, and the data reduction process would be slowed

down. With the frequency correction. Fig. 4.7 shows that the range of calculated
4.1. DIMENSIONAL CHARACTERIZATION 55

0.37

0.365

0.36

E 0.355 E

o 0.35 oCO^

1 0.345

enc 0.34 D(D(0

0.335

13O

(0 0.33 o

0.325

0.32

0.315

ouncorrected dimensions

-D- corrected dimensions

.

I .... I ^ 5^ 8 gj s oo

odod d

o

calculated diameter - 60.0 (mm)

Figure 4.7: Cavity length versus diameter calculated from an uncorrected reso- nant frequency spectrum, and from the same frequency spectrum corrected by

the measured iris perturbation shown in Fig. 4.6. Cavity dimensions are calculated from subsets of five adjacent resonant frequencies. Length and diameter

uncertainty estimates for each calculation are also shown.

cavity length values improves from approximately ±0.009 mm to ±0.005 mm,

and the calculated cavity diameter range slightly improves from approximately

±0.0013 mm to ±0.0010 mm.
56 CHAPTER 4. RESONATOR EVALUATION

4.1.5 Resistive Loss Perturbation

Slater p. 70 [29], as well as many other autliors[10][30][31] apply perturbation

theory to find the first-order frequency correction due to radiation skin-depth

penetration into the cavity walls (a form of resistive loss). Skin-depth penetration makes the effective cavity length and diameter larger by approximately one

skin depth at all cavity walls [25, 26]. This effect is analogous to resistive damping R in an LRC circuit that lowers the resonant frequency by approximately

one-half of the half-power bandwidth,

Therefore, the proper correction is to add this amount to the measured resonant

frequency. By using corrected resonant frequencies, we calculate the physical

dimensions of the cavity. Without the correction we calculate the effective electrical dimensions of the cavity.

Use of the cavity's physical dimensions is important so that the same simple

correction can be made when the dielectric sample is measured. While the

physical dimensions remain constant, the electrical dimensions of the cavity are frequency dependent, and change because of coupling iris perturbations and

wall losses. Additionally, when power is absorbed by the sample the resonant

frequency is further reduced and the half-power bandwidth is broadened. By

correcting the empty-cavity resonant frequencies to calculate the cavity's physical dimensions, we can apply the same frequency-shift correction given in (4.3)

to the sample-loaded cavity to calculate e'^. If this correction were not applied,

the frequency shift caused by the sample in the cavity would be slightly too

large because the sample's power absorption had further pulled the frequency

down. This increased frequency shift biases permittivity results making them

somewhat high, on the order of 0.5%, depending on the sample permittivity

and thickness. This correction becomes especially necessary when measuring

medium-loss, low-permittivity samples, because the magnitude of the correction

becomes more significant relative to the frequency shift caused by the sample.

4.1.6 Summary of Diameter and Length Uncertainties

When the fixed-length method is used to measure permittivity, length uncertainty is primarily affected by frequency-dependent perturbations caused by the

coupling irises. Figure 4.7 shows the ±0.009 mm length uncertainty can be

improved to ±0.005 mm when the iris perturbation correction is applied.
4.2. EVALUATION OF Q 57

Table 4.2: Overall cavity dimension uncertainties.

Parameter

uncertainty

Measurement

method

With

corrections

Without

corrections

A Length (mm)

A Length (mm)

A Diameter (mm)

Fixed length

Fixed frequency

Either

0.0054

0.0054

0.0010

0.0092

0.0112

0.0013

If the fixed-frequency method is applied, length uncertainty depends on micrometer accuracy, which is approximately ±0.011 mm over the micrometer's entire

24-mm travel range. The sensing micrometer's accuracy was previously shown

to be approximately ±0.005 mm. The resulting change indicates that some wear

occurred in the length measurement assembly.

The attachment repeatability of the tuner assembly should also be figured into

the overall length uncertainty when either method is used. The attachment

repeatability is less than ±0.002 mm. This repeatability is taken to be independent of the frequency- and length-dependent accuracy estimates given above,

and the overall length uncertainty is calculated as the root sum squares of dimensional calculation accuracy and tuner assembly repeatability, as shown in

Table 4.2.

4.2 Evaluation of Q

Resonance quaUty factor is proportional to the energy stored in the resonating

system W divided by the average power loss of the total system

Q =

UJqW

(4.4)

Power is consumed by several mechanisms including conductor losses Pc (endplates and helical wall), dielectric sample losses Ps and coupling losses Pe- Conductive and dielectric losses are considered internal that define the unloaded Qq

of the resonator. Losses caused by the iris coupling ports are considered external

losses. Each one of these loss mechanisms has a characteristic Q. The sum of

the reciprocals of all Q values "loaded" Ql value:

1 PC + PS + E, PEr

CUqW
58 CHAPTER 4. RESONATOR EVALUATION

1 1 „ 1 7^ + 7^ + E

Qs Qc iQ

1 ^ 1

Ei

Qo i Qei

(4.5)

where the sum is over all coupling ports. Loaded Qi is the quantity measured.

To calculate dielectric loss of a sample, we take the measured "loaded" Qi

and remove conductor and aperture losses to find Qs- The aperture coupling

losses are defined by Qei (see (2.71), p. 28, and Sec. 2.2.5, p. 33). A common

approach for finding Qs is to subtract the empty-cavity Qi from the value of

Qi of the cavity with sample. For our cavity, this approach is usually sufficient

if we measure samples with loss tangents greater than 0.001. When measuring

very low-loss samples, however, slight differences in cavity wall and port losses

between the empty cavity and the cavity with sample can lead to errors that

are of the same order as the sample's loss tangent. For example, Fig. 4.8 shows

how Qi increases as frequency increases up to approximately 11.7 GHz where

the TEq2i mode interferes with the TE^ip modes we wish to observe. When a sample is placed in the cavity, the resonant frequency will be lower unless the

cavity is shortened. Figure 4.8 shows that if the resonant frequency lowers, Qi

will also lower even if the sample were completely lossless. The scatter in Qi

measured at or near the same frequency for different cavity lengths is smaller

than the change in Qi when frequency shifts by a typical 200-300 MHz. Because

Qi remains more constant with changing length than with changing frequency,

we can understand that errors in loss tangent calculation are minimized if the

cavity with sample is shortened so that it resonates at the same frequency as the empty cavity.

4.2.1 Port Losses - Coupling Factor {k)

If the variable-frequency measurement method is used the largest correction to

the measured Qi is for power lost through the coupling ports. The Qe\ and Qe2

of the coupling ports can be determined from the coupling factors (/^i,/s:2), so that the internal Qo can be determined from (4.5). We simultaneously determine

Ki and K-i from measured scattering parameter data obtained from a network

analyzer. The reflected scattering parameters, Sn and 522, are measured. The

locus of resonance S-parameter data is a circle in the complex reflection plane,

and the radius of this "Q-circle" depends on the coupling factors. If we measure

the ^ii and ^22 resonance responses, ki and K2 can be solved simultaneously
4.2. EVALUATION OF Q 59

VERSUS FREQUENCY FOR THREE CHVITY LENGTHS

1 r

««« L0=A 12.33 mm

XXX L0=115.62 mm

ttt LPI=132.08 mm

?0

B0

50

• )(

"t

,

30

X

3 10 II 1? RESONRNT FREQUENCY (GHz)

I r

X

X

J L

13 M 15

Figure 4.8: Measured Ql versus frequency for three cavity lengths. Data show

increasing Q and mode interference above 11 GHz.

from the equation [23]

txj —

1 - ri - r2

2 = 1 or 2 , (4.6)

where r^ are the radii of the Q-circles in the complex reflection plane. From (2.96),

the coupling factors and Q are related by

(1 + Ki + K2) , J_ _ 1

1 1 1 = ^T- + T^ +

Qo Qei QE2

(4.7)

4.2.2 Q Corrections: Fixed-Length Method

If we employ the fixed-length method, the resonant frequency decreases when the

sample is inserted. When frequency decreases, coupling port losses decrease and

internal conductor losses increase slightly. To compensate for the coupling port
60 CHAPTER 4. RESONATOR EVALUATION

losses, the coupling Qe must be determined at both the empty-cavity and the

cavity-with-sample resonant frequencies, then the internal (unloaded) Qq can be

separated from the measured (loaded) Qi, as shown in (4.5). This correction for

changes in coupling port losses improves loss factor accuracy by approximately

15%. When frequency changes, internal conductor losses change slightly, and a

correction to Qq can be made to obtain the best accuracy possible. To correct for

changes in internal conductor loss, the skin depths at both resonant frequencies

must be known, and the change in Qq has to be calculated from the volume and

surface integrals that comprise the numerator and denominator of (4.4).

Power absorbed by the resonator's conductive walls is proportional to skin depth

6s given in (2.107) as

6s =

^

• (4.8)

Equation (4.8) shows that a one-percent decrease in frequency leads to a halfpercent increase in skin depth. This leads to a half-percent increase in conductor

loss and, from (4.5), a half-percent decrease in Qq. For example, with the fixedlength method, frequency decreases in our resonator by approximately 300 MHz

when a half-wavelength thick sample is inserted. At 10 GHz this would lead

to a 1.5% increase in conductor loss when the sample is in place. To correct

for this change, the empty-cavity Qo should be decreased by 1.5% in order to

approximate the conductor loss in the sample-loaded cavity.

4.2.3 Q corrections: Fixed-Frequency Method

Fewer corrections to Q are needed if the fixed- frequency method is used to measure a dielectric material. No corrections to the coupling Qe are needed

because the iris coupling factors do not change. At a fixed frequency, skin depth

does not change and conductor loss in the resonator's air-filled section remains

constant. A resonator containing a dielectric sample must be shortened in order

to resonate at the empty-cavity frequency. The shortened length of the cavity

leads to less conductor loss. This decrease in resonator wall loss is determined

from surface integral equations given in Chapter 2. The calculation is time

consuming and results in only a minor correction to Qq on the order of 0.5-

2%. Moreover, the uncertainties in the endplate and helical wall skin depths

leads to significant uncertainty in the correction. For this reason corrections to

Q because of reduced wall losses becomes undesirable when several samples or measurement frequencies are involved.
4.2. EVALUATION OF Q 61

Table 4.3: Stability of Qq and coupling factors. Precision is given as the standard

error of 10 measurements.

Measurement

Numbers

Mean

Qo ± precision

Mean Coupling

Factor {k) ± precision

Micrometer

Reading (mm)

1-10 port 1 port 2

84 633 ± 33

84 650 ±44

0.028 457 ±0.000 012

0.028 110 ±0.000010

0.01808

11-20 port 1 port 2

84 676 ± 42

84 648 ± 63

0.028 4601 ±0.000 019

0.028 111 ±0.000013

0.01810

4.2.4 Stability of Q and Coupling Factors

To perform a precise measurement of a sample's loss factor, the resonator's Q

and coupling factors must remain stable throughout the time required to make

a measurement, usually less than one hour. In this experiment the stability of

the coupling factors and resonance Q was observed for a 4-h period to confirm

that these parameters were not drifting by more than 4 parts in 10 000.

The network analyzer was calibrated with a TRL two-port calibration. The

tuner endplate was adjusted until the T£'oi(25) mode resonated at 10.630 GHz.

Waveguide port extensions were manually adjusted until the resonance peak

passed through the real axis in the reflection plane. The resonance response was

then measured once every 10 min for a total of 10 measurements. Without detaching the tuner assembly from the cavity, the tuner endplate was then lowered

and raised back up until the cavity resonated at the same frequency. The sensingmicrometer reading was recorded, and port extensions were again adjusted and

recorded. The resonance S-parameters were then measured another 10 times at

10-min intervals without adjustments. Coupling factors and unloaded Qq were

calculated from the zero-order equations given in (4.6) and (4.7). Results from

this measurement are given in Table 4.3 and indicate an excellent stability of

both coupling factors and the unloaded Qq.

4.2.5 Repeatability of Q and Coupling Factors

In the same experiment that tested the tuner assembly attachment repeatability described in Sec. 4.1.3, Qq, ki and k,2 were also calculated from Sn and ^22

scattering parameter data. The network analyzer was calibrated with one-port

response calibrations on both port 1 and port 2 in which the detuned short of the

coupling apertures was the calibration standard. The resonator tuner assembly
62 CHAPTER 4. RESONATOR EVALUATION

o

o

C\J oo

CD

o

o

o

oCM

.<P<Pr

00 Q)Cb

cP ^

oOq 00 ^ {fi O

°o°o ^

qr ^ %

CP o ^aSfe

(?

20 40 60

Measurement Number

80

Figure 4.9: Values of Qq ^ 84 700 calculated from scattering parameters. The

cavity was opened and closed every three measurements for the first nine measurements and every four measurements thereafter. A new calibration was performed after measurements 9, 25, 41, 57 and 73. Horizontal hatched lines show

95% and 99% certainty bounds.

was re-attached and the r£'oi(25) mode was tuned to 10.630 GHz. A set of four

consecutive measurements was taken to test stability, then the resonator tuner

assembly was detached and reattached. The open/close measurement was re- peated four times for each cahbration, and after the fourth set of measurements,

the network analyzer was recalibrated. An exception to this procedure was

made in the first nine measurements, during which three consecutive measure- ments were made before the tuner assembly was opened, and the tuner assembly

was detached and reattached three times, rather than four, before recahbration.

There were a total of 89 measurements, 23 opening and closings of the tuner

assembly, and 6 different calibrations. Results from this experiment are given in

Figs. 4.9-4.12.
4.2. EVALUATION OF Q 63

CD

<O

N

HI

CD

CD

O

<b^

OCP o oo ^

(fi6b

^0

-0--

o (^ 'ODD'O) oqo 0' -(fi-

(SS>,(P

20 40 60

Measurement Number

80

Figure 4.10: Calculated values of /q. The cavity was opened and closed every

three measurements for the first nine measurements and every four measure- ments thereafter. A new calibration was performed after measurements 9, 25,

41, 57 and 73. Horizontal hatched lines show 95% and 99% certainty bounds.

in

in

15

(POf)OP

^

-J O Or\ O

20 40 60

Measurement Number

80

Figure 4.11: Calculated values of ki. The cavity was opened and closed every

three measurements for the first nine measurements and every four measure- ments thereafter. A new cahbration was performed after measurements 9, 25,

41, 57 and 73. Horizontal hatched Unes show 95% and 99% certainty bounds.
64 CHAPTER 4. RESONATOR EVALUATION

IT) <O

in

in<o

in

CD

CP

xj---

00 CJP

o

GD°0 ^

(P

?^

on 0°

oo aj^^

o

OD

20 40 60

Measurement Number

80

Figure 4.12: Calculated values of K2. The cavity was opened and closed every

three measurements for the first nine measurements and every four measure- ments thereafter. A new calibration was performed after measurements 9, 25,

41, 57 and 73. Horizontal hatched lines show 95% and 99% certainty bounds.
4.3. SYSTEMATIC UNCERTAINTY 65

Figures 4.9, 4.10, 4.11 and 4.12 show the estimated parameters for Qq, /q, Ki

and «2, respectively. The mean value of Qq is very stable within the scope of a single calibration and changes slightly from cahbration to calibration. The

standard error for Qq over all measurements is ±62.2. The standard error for /o

repeatability is ±395 Hz. The discrepancy of Ki in the first nine measurements

from all the other ^i results cannot be explained at this time. To the best of our knowledge, no physical changes were made during the measurements. Most

importantly, however, Qq was not significantly altered by the change in ni.

4.3 Systematic Uncertainty

The excellent results from the Q stability and repeatability experiments of

Sees. 4.2.4 and 4.2.5 indicate that the loss tangent of a sample should be measurable to high accuracy. Unfortunately, systematic errors reduce the accuracy

of our loss tangent measurements.

In general, systematic errors are difficult to detect because the instrument appears to work properly. If detected, systematic errors are usually very difficult

to quantify and correct. This section describes some systematic errors inherent

in the NIST 60-mm cavity resonator and offers some discussion on how to avoid

permittivity and loss tangent measurement errors. These errors arise from:

1. interfering resonant modes,

2. simplifying assumptions used in the cavity theory,

3. the resonance model chosen to fit measured scattering parameter data,

4. imperfect resonator geometry and

5. limitations of the network analyzer.

There may be other systematic errors, and their detection must rely on future

comparisons with other instruments.

4.3.1 Simplifying Assumptions

Permittivity (e'^) Model

The model to calculate e'^ assumes that the cavity and sample are lossless. As

discussed in Sec. 4.1, loss lowers resonant frequency, which causes the cavity's
66 CHAPTER 4. RESONATOR EVALUATION

electrical dimensions to be larger than the physical dimensions. The corrections

that can be made for this systematic error due to limitations in the e'^ model

have already been discussed.

Loss Tangent (tan(5) Model

The loss tangent model we use assumes we know the Q' of a resonator that

contains a lossless sample with the same permittivity as the real sample. Typically the empty-cavity Q is used instead of the Q' of the cavity with a "lossless"

sample, which leads to errors in tan6 calculation because the empty-cavity Q is slightly different than Q'. For example, the fixed-length method leads to errors

in Q because of the frequency-dependent behavior of the coupling ports and

skin-depth. In the case of the fixed-frequency method, Q differs slightly from Q'

because the length of the cavity with sample is shorter than the empty cavity.

4.3.2 Resonance Model

To find resonance parameters such as /o, Qo, ^^i and K2 a nonlinear least-squares

fitting routine [32] is used to fit scattering parameter data to a resonance equation. The resonance equation is derived from the two-port, lumped-element

equivalent circuit [33] shown in Fig. 4.13. This represents a more accurate and

detailed representation of the NIST cavity than that given in Fig. 2.13. The

input impedance at the port 1 coupling port is given by

^=#^ = n+x, + -, ^^^^ , (4.9)

where the normalized parameters

uJO^^ ' l+r2+ja;A'2

r, = 1^ and X, = |^ , f = l,2 (4.10)

characterize the self-impedances of the coupling ports; the coupling factors

^r = -^rjt^ ^ = ^'2 (4.11)

are functions of the mutual inductances Mi. The resonant frequency, ujq and the

decoupled "unloaded" quahty factor C^o are defined by
4.3. SYSTEMATIC UNCERTAINTY 67

PORT 2

Figure 4.13: Lumped-parameter equivalent resonant-circuit showing coupling

port impedances and reference plane a-a.

in

q

5

in ,^ o

</> 00

0)

as

<D 1-

cn

Q- in

-0.99 -0.98 -0.97 -0.96 -0.95

Magnitude

Figure 4.14: Measured scattering parameters (dots) and locus predicted from

nonlinear least-squares fit to (4.9).
68 CHAPTER 4. RESONATOR EVALUATION

Magnitude Residual for Port 1 Impedances

c;

INJ

oo

o

o

CMO

O

.-. .•. . ...•-!••

•

• • •

. W*w V** v..

- •• • /

• •

•

• •/ •

•• •

•- ••

./

•

•

10.6298 10.6299 10.6300 10.6301 10.6302

Frequency (GHz)

Figure 4.15: Magnitude of residuals to circuit model fit.

Phase Residual for Port 1 Impedances

CO

O)

10.6298 10.6299 10.6300 10.6301 10.6302

Frequency (GHz)

Figure 4.16: Phase of residuals to circuit model fit.
4.3. SYSTEMATIC UNCERTAINTY 69

Magnitude Residual for Port 1 Impedances

c:

fNI

o

o

o

d

C\J o

o

o

d

10.6298 10.6299 10.6300 10.6301 10.6302

Frequency (GHz)

Figure 4.17: Magnitude of residuals to circuit model fit when an e^^ term is included in the circuit model to compensate for network analyzer phase drift.

Phase Residual for Port 1 Impedances

CO

0)^

Q T10.6298 10.6299 10.6300 10.6301

Frequency (GHz)

10.6302

Figure 4.18: Phase of residuals to circuit model fit when an e-'^ term is included

in the circuit model to compensate for network analyzer phase drift.
70 CHAPTER 4. RESONATOR EVALUATION

Table 4.4: Parameter estimates for data subsets. Ql values are calculated from

Qo.Ki and k,2.

Data used Ki K2 Qo Ql /o (GHz)

All points 0.028 47 0.02815 84 232 79 719 10.630 001462

2/o ± 20 points 0.028 57 0.028 24 83 988 79 473 10.630 001971

z/o ± 3 dB points 0.028 51 0.02818 84199 79682 10.630 001508

z/o ± 100 points 0.028 51 0.02819 84 049 79 539 10.630 001676

First half of data 0.02817 0.02782 83 856 79 410 10.630 001685

Second half of data 0.028 70 0.028 31 85 042 80 456 10.630 001882

Figure 4.14 shows typical results when we fit the magnitude and phase of the resonance equation to measured S-parameters. Close examination of Fig. 4.14

reveals a systematic pattern in the residuals to the fit. A residual is defined as the distance between a measured datum and its predicted value. Figures 4.15

and 4.16 more clearly show the magnitude and phase of the residuals. If our model perfectly described the measured data, the scatter in the residuals would

be random. This systematic pattern points to a discrepancy between the lumpedparameter circuit model and the measured resonance data. Isolating the cause

of this discrepancy is difficult because the precision is so good. Either the circuit

model or the S-parameter data could be exhibiting their limitations. We suspect

that phase drift in the network analyzer's calibration is the cause of this residual

error. This is so because when (4.9) is multiphed by a variable e^^ term, the

residuals to the fit become random, as shown Figs. 4.17 and 4.18.

The potential error due to the systematic pattern in the residuals has been

estimated [34] . Nonlinear regression was performed on an S-parameter data set to estimate the parameters in (4.9). The full data set consisted of 401 evenly

spaced frequencies, with /o at the center frequency. We compared parameter

estimates from five subsets of the data to the estimates obtained using all 401

frequencies. The first three data sets were centered about /q. The last two data

sets consisted of the first and second halves of the data. Results are shown in

Table 4.4. Results show that the systematic pattern has little effect on the final

parameter estimates (« 0.3%) when the regression uses data evenly distributed

about /o. However, when the frequencies are not evenly distributed about /o,

Q-value and coupling factor estimates can differ by 2% or more.
4.3. SYSTEMATIC UNCERTAINTY 71

4.3.3 Interfering Non-T^oi Modes

The resonator's helical waveguide wall acts as an excellent mode suppressor,

but other modes such as the TE02 and TE12 still exist. Although these modes

are excited at much lower power than the dominant TEqi modes (nominally

minus 30 dB), their existence still can cause mode interference. If a non-TE'oi

mode causes interference, no corrections can be made. Instead, we must try

to resonate the cavity at either a different frequency or a different length in

order to adequately separate the TEqi mode from the undesired mode. Mode

interference changes the resonant frequency and Q of a TEqi resonance, which

introduces significant permittivity measurement errors. An interfering mode

makes the TEqi resonance response shghtly asymmetric. This asymmetry is revealed when resonance scattering-parameter data are fit to characteristic res- onance equations. Even shght distortions to the resonance shape can be seen

in a plot of the residuals to the fit [34]. The computer program MEAS_RES described in Chapter 5 calculates resonance parameters from S-parameter data.

To aid in detecting mode interference, MEAS_RES gives residual plots and prints

out the average residual variance (x). Presently, the threshold S-parameter

residual variance at which mode interference becomes suspect is on the order of

X = 2.3 X 10-^

In the rare instances when two modes resonate at exactly the same frequency,

mode interference may not be detected by distortion of the resonance response.

When this situation occurs, the iris coupling coefficients aci and ^2 change from

their expected values.

4.3.4 Imperfect Geometry and Wall Resistivity

As given in (4.4), resonance quality factor is defined as the ratio of the energy

contained in a resonating system divided by its energy dissipation rate. The

energy contained in the cavity depends on its cylindrical geometry. If the res- onator's geometry is poor, the cavity Q will be reduced. Also, if the conductivity

of the resonator walls is different than what we expect, resistive wall losses will

make the cavity Q change from the expected value. To get some idea of the

cavity's "mode-efficiency," we can compare the theoretical Qo for an ideal cylindrical copper cavity to the Qq of our resonator. Disparities between the two Q

values can be attributed to a combination of imperfect geometry and resonator

conductivity different from copper. Table 4.5 gives the percent discrepancy from

an ideal copper cavity with a bulk conductivity of 5.80 x 10'' S/m. Four different

frequencies were tested, and two different endplates were used. Values in square

brackets were found using an older endplate, and the unbracketed values were
72 CHAPTER 4. RESONATOR EVALUATION

Table 4.5: Mode efficiencies (Qmeasured/^theoretical) ^ ^^^ ^*-*^ ^^^^ frequencies. Square brackets: older endplate, unbracketed values: newer endplate.

Axial

mode no. 10.05

Frequency (GHz)

10.34 10.63 10.92

20 78.46

21 80.49 78.27

22 79.97 78.70

23 [78.13] 79.42

[76.77]

24 78.84

[75.68]

76.86

25 [75.50]

26 [74.93]

found using a newer endplate.

The measured Q-value is higher by approximately 2000 (86000 vs. 84 000) with

the new endplate, which indicates that the surface conductivity of the older

endplate is approximately 15% less than that of the newer endplate [23]. This

comes as no surprise because the older endplate has become scratched from use and the cavity Q has steadily degraded over the course of three years. The fact

that the cavity resonator does not yield Q values expected for a copper cavity is no surprise. The helical windings that compose the cylindrical resonator wall are made of enamel-insulated copper wires embedded in an adhesive binder. Power

loss in the binder and insulation will increase the effective surface resistivity of

the cavity wall, thereby lowering the Q from the ideal copper-cavity value. Also,

the helical walls are not perfectly smooth, which would allow less energy storage

in the resonator.

Whether the cavity Q is lower than that of a copper cavity because of geometrical imperfections or increased surface resistivity does not matter. It is important only that the cavity Q remain constant. We have seen a steady de- crease in cavity Q since the resonator was built and now have shown that a new

endplate increases the Q to levels found previously. The resonator still has the

old endplate installed and is functioning properly. However, the scratches on the

endplate can lead to systematic errors. For example, the scratches create an air

gap between the endplate and the sample, which will very slightly increase the

calculated permittivity of thinner samples. When the resonator is refurbished,

as discussed in Chapter 10, the tuner-endplate will be replaced.
4.3. SYSTEMATIC UNCERTAINTY 73

4.3.5 Resonance Fitting - 811/822 Versus 821/812 Measurements

The MEAS_RES computer program obtains data from a network analyzer and applies a Levenburg-Marquart least-squares fitting routine to estimate /o, Qo, /^i, >^2 etc. These resonance parameters can be estimated from either 5ii, ^22, 521 or

5i2 data. The transmitted signal of a resonance response is well out of the

network analyzer's noise floor, and Q estimates calculated from 52i-only data

generally yield precision estimates of approximately ±50. Q precision estimates

from combined 5ii,522 fits are on the order of ±300 because the resonance

response appears as a small dip in a high reflection.

Although precision estimates from least-squares fits to S21 data are better than

combined ^n , 5*22 precision estimates, Sn , 522 measurements deliver better overall repeatability because one-port calibrations require less cable flexing. Oneport cahbrations result in better phase and amplitude stability than two port

calibrations. Additionally, one-port calibrations take about half the time to

perform as two-port calibrations. These considerations make combined Su , 522

measurements preferable to through 52i or 5i2 measurements.
Chapter 5

Software

This chapter describes our computer programs and routines as applied to the

resonator method of permittivity measurement. The programs are modular and

certain routines are used in several places. All programs are written in BASIC.

Notation denoting the actual variable used in computation are used throughout

this section. The two primary programs, MEAS_RES and CAVITYPROG are

listed in Appendices D and E.

The MEAS_RES program utihzes some subprograms many times in different

places. Zoom_on_peak adjusts the network analyzer center frequency and span.

Fit_sparms is a nonlinear Levenberg-Marquart fitting routine that finds reso- nance parameters. The two subprograms which actually send commands and

receive data from the network analyzer are named Set_nwa and Read_nwa. These

subprograms interpret mnemonics sent to them by higher level routines, then

send the properly formatted commands to the network analyzer.

5.1 Instrument Control - MEAS_RES

The program MEAS_RES controls the network analyzer and automatically ac- quires resonance parameters of one or several resonances. Any of the four Sparameters measured by the network analyzer may be used to calculate reso- nance parameters. The operator controls the program's flow by pressing softkeys.

74
5.1. INSTRUMENT CONTROL - MEAS-RES 75

5.1.1 Measuresweep Subprogram

If the resonance frequencies are unknown, the computer slowly steps through

a frequency range and searches for resonances. Once a resonance is found, the

computer zooms in on the resonance, acquires the S-parameter data, and fits the

data to the appropriate Lorentzian resonance equation (reflected or transmitted resonance response).

5.1.2 Measurelist Subprogram

This subroutine steps through a list of frequencies and finds the resonance parameters of any resonances found. Resonance frequencies are typed into data

statements before the program is run. The data statements are located at the

line labelled Resonance_data. The computer reads the frequency hst from the

data statements when the subroutine is invoked. Once the resonance parameters

are found, the computer sets the network analyzer to the next frequency and

repeats the process.

5.1.3 Measuretrigger Subprogram

If the operator wants full control of both the network analyzer and computer,

this method is useful. The computer waits until the operator presses the trigger softkey, obtains the S-parameters of interest and calculates the resonance

parameters. This is useful when the fixed-frequency, variable-length method is employed. With this method, the operator has to move the tuner-endplate to

achieve the desired frequency.

5.1.4 Measurelistrig Subprogram

This routine sets the network analyzer to a predefined frequency given in a datastatement list and then waits until you are ready to measure the resonance. When you have made the necessary adjustments to the resonator, press the

trigger softkey to measure the resonance. When the resonance parameters have

been found the computer sets the network analyzer to the next frequency in

the frequency list. This capability is useful when the network analyzer is to be

calibrated at each resonance, or when the operator wants to adjust the resonance

frequency of the cavity.
76 CHAPTERS. SOFTWARE

5.1.5 Measuredumdata Subprogram

This subroutine is useful for program development and to test the effects of noise

inserted into ideal data. The driver contained in this subroutine can be modified

to use different resonance-fitting models. S-parameters and frequency list data

are loaded from disk.

5.1.6 Measurediskdata Subprogram

This routine loads previously measured S-parameter data from disk and calculates the resonance parameters. Sometimes this is useful if the network analyzer

is calibrated in frequency list mode. Several frequencies can be measured at

once, the data stored on disk and then the measurement can be repeated. By

changing the Start_index and Stop_index, the operator can select the proper

resonance from the S-parameter data set.

5.1.7 Fit_sparms Subprogram - Resonance Fitting

In both the variable-length and the variable- frequency sections of CAVITYPROG,

and throughout MEAS_RES the Fit_sparms subprogram calls a CSUB that calculates resonance parameters and their uncertainties from measured scattering

parameter resonance data. Fit_sparms is the CSUB driver that sets up the Sparameters, iterates the CSUB to parameter-convergence, and prints and plots

the results.

Presently there are three Lorentzian resonance equations to which the scattering

parameters can be fitted. Two are transmission equations, and the third is a

reflection equation. Each describes a circle in the complex reflection plane.

The two transmission-response equations describe a circle near the origin in

the complex impedance plane. The third is a reflection-response equation that

describes a circle near | F |= 1 on the impedance plane. The variable Sss selects

which equation to fit to the data.

When Sss = 1, a simple resonance transmission expression given in many text- books [35] is applied:

This is a Lorentzian line-shape equation with an arbitrary phase and magnitude

term. The loaded Qi depends only on the distribution of data around the circle.

The scattering parameters trace out the circle faster with respect to frequency as
5.1. INSTRUMENT CONTROL - MEASURES 77

resonance Q increases. Unfortunately, (5.1) does not decouple Qq from Q^. To

do so, the S-parameters' magnitudes must be accurately measured. Shulten [36]

gives both reflected and transmitted resonance line-shape equations that express

Qo separately as follows:

Sr{f) = ^f—-—-jA-Aexp{je) and (5.2)

1 + 2jQl^ -Jo /o

In the Fit_sparms subprogram, the reflected S-parameter expression (5.2) cor- responds to Sss = 2 and the transmission equation (5.3) is used when Sss = 3. To accurately determine Qq, the amplitude factor A must be very close to 1. The

arbitrary phase term compensates for phase delays due to residual calibration

error and drift. Equations (5.2) and (5.3) express Qq separately from Ql, which

allows us to calculate the coupling port Q-values (Qe)- For an empty cavity,

1 _ 1 1 1 Ql Qo Qei Qe2

5.1.8 Levenberg-Marquart CSUB

To find the resonance parameters from measured S-parameter data, we implement a Levenberg-Marquart nonlinear least-squares fit technique [37]. The user

supplies the constitutive equation (5.1), (5.2) or (5.3) and its derivatives with

respect to the unknown parameters. The derivatives are used to provide a set of linear equations which are used to increment the approximated values of the

unknown parameters in the correct direction. The CSUB performs one iteration

that computes the derivatives and increments the estimated values of the un- known parameters. Fit_sparms iterates the CSUB until convergenceJs reached.

Once the parameter values are determined, Fit_sparms makes a final iteration

to compute the covariance matrix. The square root of the diagonal elements

of the covariance matrix provide the estimated standard errors in the fitted

parameters.

This implementation of the Levenberg-Marquart method uses source code written in FORTRAN. Since BASIC and FORTRAN are inherently incompatible

because of integer, string and array operations, the creation of the CSUB re- quires some restructuring of I/O operations and array pointers [38], [39].
78 CHAPTER 5. SOFTWARE

5.2 Data Analysis - CAVITYPROG

This section describes the CAVITYPROG computer program. The program

contains all the utilities needed to fit resonance data, calculate permittivity and

loss tangent and perform error analysis. Either the variable-length or variablefrequency technique may be applied. Structured programming methods have

been used.

CAVITYPROG is designed to be interactive and facilitates user interaction

through the use of menus and softkeys. Various parameters such as cavity and

sample dimensions, and resonance frequencies can be changed through the use

of softkeys. Descriptions of the softkey functions are located at the bottom of

the computer display.

Because of the complexity of incorporating error-correction routines into CAVITYPROG, be aware of the need to enter corrected data into the calculations.

Specifically, the unloaded Qo niust be used to properly calculate loss tangent, and

resonant frequency must be corrected to reflect the downward frequency shift

due to finite cavity losses. These corrections are discussed more thoroughly in

Sec. 4.2. CAVITYPROG uses softkeys to direct program flow. The various utilities are accessed by the operator pressing softkeys to move up and down through the

program's branches. For example, when the program is run the computer flrst displays the main menu and a set of softkeys. The main menu displays:

1. Laboratory temperature, air pressure and relative humidity

2. Speed of light in the laboratory

3. Sample description and thickness

4. Cavity diameter and length

The operator may use the main-menu softkeys to call specific utilities. Each one

of these utilities contains its own menu and softkeys which allow the user change

parameters or call other utilities. The utilities directly accessible from the main

menu include:

1. Calculation of permittivity by the fixed-frequency technique

2. Calculation of permittivity by the fixed-length technique

3. Calculation of speed of light from environmental parameters
5.2. DATA ANALYSIS - CAVITYPROG 79

4. Calculation of resonator diameter and length from the empty resonator's

TEqi resonance spectrum

5. Calculation of all TE and TM modes for a given cavity

6. Calculation of the TEqi frequency spectrum for either an empty or a

dielectric-loaded cavity

7. Calculation of resonance frequency and Q from S-parameters loaded from

disk

Sections 7.5.2 and 7.5.3 give typical step-by-step procedures for performing fixedfrequency and fixed-length permittivity measurements.

When CAVITYPROG is first run, verify that all parameters have their proper

values. For instance, the speed of light in the cavity should be recalculated

from the current environmental conditions before proceeding with any other

calculation. Also, check the cavity's length and diameter, and if necessary,

recalculate them.

Permittivity calculations rely on the user to provide a first guess of e'^ from

which a convergence routine finds the proper value. To aid the user in making

this guess two utilities are provided. One utility calculates the frequencies of

all TE and TM modes for the cavity. The other calculates the TEqi modes for

either an empty or dielectric-loaded cavity. These routines are also useful during

the data-acquisition process in helping to locate desired modes.

5.2.1 Calc_c Subprogram - Speed of Light

The Calc_c subprogram calculates the speed of light and its uncertainty from

the resonator temperature, barometric pressure, and relative humidity. The

speed of light in the laboratory should be the first calculation performed when

running CAVITYPROG. Nearly all other calculations depend on the speed of

light. Calc_c displays a menu showing the values of temperature, pressure and

relative humidity. Their values and uncertainties may be changed using softkeys.

The speed of light and its uncertainty is recalculated and displayed each time one of the values is changed.
80 CHAPTER 5. SOFTWARE

Calculation of speed of light

We implement the following equations to calculate the speed of light, given by

Liebe [40]:

n

where

n^n' - jn" = 1 + Nt , (5.6)

Nt = (n - 1) • 10^ ppm , = No + N'{f)-jN"if), (5.7)

A^o = iVi + A^2 , = 2.588p^ + (41.63^ + 2.39)e^. (5.8)

The dispersive term in (5.7), N'{f) — jN"{f), is neglected. In the frequency

range 8-27 GHz at our typical laboratory conditions, the dry-air refractivity

A^o — 269.03, whereas the dispersive term affects the total refractivity Nt in the

fifth digit. A''! and A^2 are the nondispersive refractivity contributions of dry

air and water vapor. The terms for the partial pressures for dry air and water

vapor, p and e, are calculated from the barometric pressure P, relative humidity

[/, and temperature T in degrees Celsius as follows:

p = P-e, (5.9)

^ = 300/(7 + 273.15), (5.10)

t/ = (e/e,)100 < 100 , (5.11)

e = 2.4076 • lO^UO^ exp(-22.64^) . (5.12)

The saturated water vapor pressure e^ is found by setting U = 100 in (5.12).

5.2.2 Var_length Subprogram

The Var-length subprogram calculates complex permittivity using the variablelength, fixed-frequency technique. The variable-length technique is generally

considered to be more accurate than the variable-frequency technique. Frequency dependent perturbations such as coupling factors and skin depth remain

constant if the frequency is fixed.

When running the variable-length section of CAVITYPROG, the computer displays a menu containing the values of all the necessary parameters. The user
5.2. DATA ANALYSIS - CAVITYPROG 81

may change the sample description, thickness and thickness uncertainty. The

resonance frequency is required to be a single frequency for both the empty

and dielectric-loaded conditions. The resonance frequency is very sensitive to

changes in cavity length, and a high accuracy micrometer must be used to make

this constraint experimentally achievable. The micrometer readings and un- loaded Q-values along with their uncertainties may be entered for the empty and

loaded resonances. If desired, the user can call a fitting routine (Fit_sparms)

which calculates the loaded and unloaded Q-values, center frequency and their

uncertainties from measured resonance S-parameter data. A description of this

routine is found in Sees. 5.1.7 and 5.1.8.

When all the necessary parameters have their proper values, the user then presses

the "Calc Epsiloh" softkey to invoke the subroutine Calc_eps_len. The equations to calculate complex permittivity are nearly identical for both the variablefrequency and the variable-length techniques, so both techniques use the same

subprograms. A flag Do_eps_freq informs the subprograms which technique

to use. The subroutine Calc_eps_len sets Do_eps_freq equal to zero, then

calls the subprograms Calc_eps_re, Calc_tand and Calc_eps_im which calculate permittivity and loss tangent. After e'j^ and tan6 are calculated, pressing

the "Calc Errors" softkey calls the subroutine Calc_errs_len which again sets Do_eps_freq equal to zero and calls the subprogram Calc_errors. The computer then displays an error table showing the error contributions due to the

uncertainty of each variable while holding all other variables fixed.

5.2.3 Var_freq Subprogram

The variable-frequency section of CAVITYPROG closely resembles the variablelength section except that the micrometer readings are forced to a single value

and the resonance frequencies for the empty and loaded cavity are allowed to be

different. The computer displays a menu identical to the variable-length menu

containing all the parameters and their uncertainties needed for permittivity

calculation. Use the softkeys to set the empty and dielectric-loaded cavity dimensions and Q-values (or bandwidths), sample thickness, and initial guess for

e'ff. Like the variable-length technique, the empty or dielectric-loaded cavity

Q-value, resonance frequency and their uncertainties can be calculated by using

the "Fit S parms" softkey. When you have set the necessary variables to their

proper values, the subroutine Calc_eps_freq is run by pressing the "Calc Epsilon" softkey. Calc_eps_freq sets the flag Do_eps_freq equal to one and calls

the subprograms Calc_eps_re, Calc_tand and Calc_eps_im. After calculating permittivity, uncertainties can be calculated by pressing the "Calc Errors"

softkey.
82 CHAPTER 5. SOFTWARE

To enhance accuracy, correct the resonance frequency and resonator Ql- Energy

absorption by the cavity and sample pull the resonant frequency downward and

will affect permittivity results. To compensate, the resonant frequency of the

cavity with sample should be increased by one-half the 3-dB resonance bandwidth, and the cavity dimensions should be decreased to remove the perturbation due to skin-depth penetration. These corrections are especially important

for lossy samples. To properly calculate loss tangent, the frequency-dependent

coupling factors should be removed from Ql. By using Qq rather than Ql, you

can remove the effect of frequency-dependent coupling factors. For utmost losstangent accuracy, you should also compensate the empty-cavity Qq to reflect the

change in frequency-dependent skin-depth losses.

5.2.4 Cavity_dimens Subprogram

The Cavity_dimens subprogram allows the operator to enter the cavity's diameter and length from the keyboard or to calculate cylindrical cavity dimensions. The Cavity_dimens subprogram is a driver for the Calc_dimens subprogram that calculates cavity dimensions from the measured TEqi^ frequency

spectrum. Because the cavity dimensions affect many other calculations, this

subprogram can be called only from the main menu. To calculate the cavity

dimensions from the TEqi spectrum, the operator first must find at least three

TEqi frequencies for a given cavity length. The operator enters the axial mode

number and frequency for each of these modes, then presses the "Find Dimens"

softkey. The cavity diameter and length uncertainties are calculated from the

variance/covariance matrix returned from the regression operation. If the operator wants to calculate the resonator's dimensions with skin-depth included, the

measured resonance frequencies are entered directly. If the cavity dimensions

without skin-depth are desired, the operator must enter the resonant frequencies

plus one-half the unloaded resonance Qo bandwidth. For example:

-'compensated -^measured

1

1 +

2Qo

;5.i3)

This does not include frequency pulling due to coupling port perturbations or coupling port losses. The need for and method of frequency compensation for

these effects is under investigation and is described in Sec. 4.1.

Calculation of Cavity Dimensions

The resonant frequency for a TEmnp mode is given in a linearizable form by

fL, = B{l) + B{2)v\, (5.14)
5.2. DATA ANALYSIS - CAVITYPROG 83

where

^a) = (^)''^(2) = ( 2L

(5.15)

The speed of light in the resonator is c; the length and diameter of the resonator

are L and D, and t'^^ = I'q^ = 3.831 705 970 2. By measuring at least three TEqi modes, we can use the mode numbers and

frequencies to perform a least- squares estimate to find the unknowns B{1) and

B{2). Let

Y=\ : \ ,x =

1 Pi ]

1 pI

I

J B

B{1)

B{2) J

• (5.16)

-1

The vector B is found by

The estimated variance-covariance matrix is given by

(5.17)

(5.18)

where s^ is given by the residual sum of squares divided by the number of degrees

of freedom

Y^Y - B'^X'^Y , , s^ ^ . (5.19)

A^

The diagonal elements of V(B) yield the square of the standard uncertainties of

B. The noncovariant uncertainties in the fit parameters are:

AB(1) = sj^^) and AB(2) = \l^^) (5.20)

95% uncertainty bounds for the fit parameters are given by the program by

multiplying the standard uncertainties by the 95%-area value of a t-distribution

with {N — 2) degrees of freedom. An analysis of this problem is given in Draper

and Smith [41].

These uncertainties are appropriate for specifying the possible ranges for an individual parameter without regard to the value of the other parameter. Be aware

that the values B{\) and B{2) are covariant. In a two-dimensional (5(1) versus B{2)) space, the uncertainties given by the covariance matrix diagonals describe a rectangular, noncovariant region of possible values of B{\) and B{T). The

variance/covariance matrix defines the elliptically shaped (63%) standard uncertainty contour interval in a B{\) versus 5(2) plane. The rectangle described
84 CHAPTERS. SOFTWARE

by the noncovariant uncertainties bounds most of the covariant, elliptical con- tour interval. The areas inside the rectangular area and outside the elliptical area are regions where the probability of simultaneous values of B{1) and B{2)

are highly unlikely because of the covariant relationship between the two fit parameters. For example, if B{1) increases it is most likely that 5(2) will decrease.

This means that the 95% noncovariant rectangle will be a good approximation

of the covariant elliptical confidence interval for most regions. Areas outside of

the elliptic locus but within the prescribed rectangle are accounted for, but have

low probability of occuring.

5.2.5 Calcjnodes Subprogram

The Calcjfiodes subprogram can calculate all TE and TM modes which can

exist in a cylindrical cavity up to the T£'8,20,40 rnode and the TMs,20,39 niode.

This subprogram is often useful in helping to locate cavity modes during an

experiment, and can help verify cavity performance. In X-band the non-T£'oi

modes of greatest concern are TEu and TE02. Before pressing the "Calc Modes"

softkey, the operator must set the cavity dimensions to the desired values. After

the operator presses the Calc Modes softkey, the computer displays the cavity

dimensions and the start and stop frequencies for the calculation. The operator sets the start and stop frequencies, then proceeds with the calculations by

pressing the "OK Begin" softkey.

Calculation of TE and TM modes

TM resonance frequencies are calculated from the equation:

'ttir . / / f TTin \ , / P" fr{TM„^,) =

^-^\l[^) +[j^) . (5.21)

The TE resonance frequencies are calculated from the equation:

27r V V a / V L fr{TEmnp)^i^^[^} +lV) ' (5-22)

where

tmn is the n-th root of the equation Jm{x) — 0,

t'^„ is the n-th root of the equation J^(a:) = 0,
5.2. DATA ANALYSIS - CAVITYPROG 85

m, n, and p are integer mode numbers,

a is the cavity radius in m,

L is the cavity length in m and

Cair is the speed of hght in the laboratory in m/s.

tmn and t'^^ are in the form of look-up tables in data statements found in the

subprogram Modefreqs2. These equations are from the book by Jackson [42].

5.2.6 Calc_freqs Subprogram

This subprogram calculates the TEqi resonance frequencies for either an empty

or dielectric-loaded cylindrical cavity.

The calculations for the empty cavity finds all TEmn resonance frequencies from

the given start frequency up to axial mode number 50, or up to the maximum

frequency of interest. On the other hand, the calculations for a dielectric-loaded

cavity find only one TEqi frequency at a time. This is because an iterative

technique is needed to match boundary conditions. The operator sets the theoretical parameters for the cavity dimensions, sample thickness, and permittivity

and then enters into the computer an approximate value for the resonance fre- quency. The computer then iterates on the frequency using Newton's method

until resonance boundary conditions are satisfied.

5.2.7 Calc_eps_re Subprogram - Permittivity

Calculation

The subprogram Calc_eps_re is used by both the variable-frequency and the

variable-length sections of CAVITYPROG to calculate e'^. It is also used by the

subprogram Calc_errors. The same code is used for the variable-frequency and

variable-length technique because only the sample thickness, dielectric-loaded resonance frequency and cavity dimensions are required to calculate e'^. The initial guess for e'^^ must be near the actual value for the calculation to

converge to the proper value. This is usually a simple matter because we usually

know the approximate permittivity. Sometimes it may be necessary to measure

a very thin sample, which assures us that we are on the first root no matter

how high the permittivity may be. A way to check whether the correct value of

permittivity has been found is to calculate e'^ for several adjacent TEqi,-, modes.

If the results agree with one another, then the proper permittivity has been

found.
86 CHAPTERS. SOFTWARE

Iterative Method for e'^

To find e'^, the proper value of the axial wavenumber inside the resonator sample

(/?i) must be obtained from the transcendental equation (see Sec. 2.2.2)

./^x tan/?i6 tauBnY

fiPi) = = —f- + —f^ , (5.23)

where b is the sample length (thickness); Y = L — b is the length of the air-filled

portion of the resonator, and Po is the axial wavenumber in the air-filled portion

of the resonator given in by (2.45)

/?o = \ld^\itaxr - ^c

UJ ^2

—) -kl. (5.24)

with kc = tQ^/ci.

Every term in (5.23) is known except for Pi. The equation has many roots and

cannot be solved directly. However, if we know on which root the solution lies,

Newton's iterative method may be applied to find the proper solution. For this case, Newton's method takes the form

The derivative of (5.23) is

S; ,, - tan(/?i6)

/'(A) =

'"^'^'"'^ ,,

. (5.26)

Pi

We begin with a guess for e'j^ and compute the starting value for Pi . We repeatedly apply (5.25) to converge to the proper value of Pi. Convergence is deemed

appropriate when

f(RA

< 10"' . (5.27)

f'iPi)

5.2.8 Calc.tand Subprogram

The calculation of loss tangent is performed using equations given in Cook [2]. The Calc_tand subprogram is used by both the variable- frequency technique

and the variable-length technique. Dielectric loss tangent is given by

tan. ^

pP^-^)H-(1/^«)[2(^.-^)-.1 . (^ - 1) , (5.28) p{2b-s) \Qd QJ
5.3. OTHER SOFTWARE 87

where Qd is the Q for a cavity filled with sample, and Q' is the theoretical Q

for a cavity filled with a "lossless" sample of the same permittivity as the lossy

sample. Lr is the total resonator length and b is the sample thickness, with

sin^ PoiLr - b) ^ - L. ^

(5.29)

sin /?i6

sin 2/?o {Lr - b)

Pi

, and (5.30)

sin2/?i6

^ = -^

(5.31)

The wavenumber Pq is given by (2.45) and pi by (2.44). As stated in Cook,

the theoretical Q is determined by measurement of the air-filled cavity so that

the effective skin depth d is determined from

^

''^''^^''^

(5.32)

Qo[k^, + {2a/Lo)Pl

where Lq is the length, and Qo is the unloaded Q of the air-filled cavity. From this

calculation of the effective skin-depth, the theoretical value of Q' is determined

from

Q = 7 ^ -^-^ , (5.33)

{^(^)}{^'{^ + B] + 2a{pPl + Pi)}^

with

A = p{2b-s) , (5.34)

B = 2{Lr-b)-q. (5.35)

5.3 Other Soft^vare

A computer program named GRAPH-DATA was developed by the NIST Electromagnetic Fields Division. Most of the figures and graphs produced for this

report were made by GRAPH_DATA. Also, many of the subprograms found

in GRAPH_DATA are incorporated into MEAS_RES and CAVITYPROG to

aid in disk I/O and graphing. Some of these subprograms are Enterf ilename

,

Enter_id, Select_disk, Save_file, Enter_file, Menu_scroll, Plot.all,

Pack.data, Setup_hp7475a and Setup_laserjet.
Chapter 6

Laboratory Environment and

Equipment

6.1 Laboratory Environment

Atmospheric temperature, pressure and relative humidity affect the speed of

light in the laboratory, and to calculate the permittivity of a material relative

to a vacuum, we must use the proper value for the speed of light in various

equations. Our laboratory maintains 40% relative humidity to within ±2%.

Temperature is maintained at a nominal 23°C, but variations can be more than

±3°C. Temperature variations can change the cavity resonator's dimensions,

and Chapter 9 shows that uncertainty in speed of light can contribute a sizable

uncertainty to permittivity calculations.

6.2 Laboratory Equipment

6.2.1 Network Analyzer

The automatic network analyzer (ANA) is used to obtain scattering parameter data for the cavity resonator. These scattering parameters are used to find resonance parameters such as resonant frequency, resonance bandwidth and coupling factor. Certain cavity measurements do not require ANA calibration. For

example, the resonant frequency is found at the frequency where the resonance

response is a maximum. Also, the half-power bandwidth can be accurately measured with an uncalibrated ANA because the response of the network analyzer is very linear. Measured Q can be accurately calculated from the half-power band88
6.2. LABORATORY EQUIPMENT 89

width obtained from uncalibrated ANA data because the ANA has a very linear

response. To measure unloaded Q and coupling factors, the network analyzer

must be calibrated.

6.2.2 Computers

The electromagnetic properties of materials (EPM) program currently has an HP 9000 series 300 computer that runs the instrument control and data analysis

programs described in Chapter 5. This computer runs the HP BASIC (RMB)

operating system. Additionally, we have an 80486-based computer that runs TransEra HTBasic that emulates the RMB environment. This computer and

other DOS based computers are used for running DOS based programs (word

processing, typesetting, spreadsheets, graphics).

Mention of specific products does not constitute a product endorsement by

NIST. Product makes and models are mentioned to give the reader an understanding of the equipment currently in use by the EPM program.

6.2.3 Dimensional Measurement

Endplate Sensing Micrometer

The sensing micrometer that measures the cavity's change in length can be

detached and connected to a flat granite surface in order to measure sample

dimensions.

High-Precision Micrometer

The endplate sensing micrometer compares well with a hand-held micrometer

with a 2.5 /im (0.0001 in) precision and gives excellent agreement with the Fowler

measurement micrometer.

Gage Blocks

To check the accuracy of our micrometers, we have a set of A+ rated gage blocks.
90 CHAPTER 6. LABORATORY ENVIRONMENT AND EQUIPMENT

6.3 ANA Waveguide Calibration

We calibrate the network analyzer to obtain the most accurate measurements of resonance S-parameters. Calibration removes the effects of mismatches, losses

and phase delays which occur before the measurement reference plane. The res- onator, as discussed in Chapter 3, is excited by coupling energy through small

apertures that are connected to a waveguide feed system. We wish to define

the measurement reference planes at the resonator's coupling ports. The precise

rectangular X-band waveguide connects to waveguide-to-coaxial cable adapters,

which in turn connect to the network analyzer. In uncalibrated mode the refer- ence plane is undefined. Moreover, in uncalibrated mode the phase and magnitude at our waveguide measurement ports varies with frequency in a nonlinear

way. Even for narrowband responses, such as for the empty cavity, nonlinear

phase dispersion may affect the Q-value measurement. Measurements of wider

resonance-width responses, such as our glass measurements (see Sec. 8.1), can show significant phase and amplitude distortion due to the nonlinear response

characteristics of the cavity's feed system.

6.3.1 X-band TRL Calibration Kit

To calibrate the network analyzer, the Through-Reflect- Line (TRL) method

is employed. This calibration technique is easy to implement and has minimal

hardware requirements. The calibration kit requires only three standards which,

in this instance, are very easy to obtain. The first is a "through" connection,

in which the two waveguide flanges of both measurement ports are connected

together. The second is a "reflect," which is a simple waveguide short that is connected to both ports, one at a time. The third standard, the "line," is a 1-cm

long electroformed section of X-band waveguide. Table 6.1 gives the definitions

of the standards and class assignments of our X-band TRL calibration kit.

Optimum Line Length

The difference in phase delay between the "through" and the "line" must be

between (20 and 160°)±n xl80° over the desired frequency range. The exact

length of the waveguide line is arbitrary, as long as this criteria is met. In fact,

the network analyzer needs an approximate value for the line delay only so it can determine the above value of n. The optimum phase delay for the TRL

method is 90° The optimal line length yields as close to a 90° phase shift over our frequency range of interest. Phase delay of a rectangular waveguide section
6.3. ANA WAVEGUIDE CALIBRATION 91

Table 6.1: Definitions and class assignments of an X-band TRL calibration kit.

Standard Offset Freq. (GHz) Type Name

No. Class Length

(mm)

Delay

(ps)

Zo Min Max

9 Load 6.5554 16.0 W/G XLOAD

13 Delay 6.5554 16.0 W/G ADAP

14 Thru 6.5554 16.0 W/G THRU

15 Line 9.977 33.38 6.5554 16.0 W/G LINEl

16 Line 25.387 84.681 6.5554 16.0 W/G LINE2

18 Reflect 6.5554 16.0 W/G SHORT

is given by

(6.1)

where the line length is ^ and the guide wavelength is

\/l - (A/A,)^

(6.2;

For the TE^q mode, the cutoff wavelength Ac is simply two times the width of

the guide.

Because of the nonlinear phase shift in rectangular waveguide, we cannot use a simple center-frequency argument to determine the optimum line length. A

straightforward way to find the optimum length goes as follows:

1. Calculate the line length ii for a 20° delay at the lowest frequency of

interest.

2. Calculate the line length (/?2) for a 160° delay at the highest frequency of

interest.

3. If ^1 < ^2, choose a hne length anywhere between di and ^2-

4. If ^1 > ^2, either reduce the frequency range of interest, or split it into two

separate frequency ranges. Each frequency range then needs a separate

line with its own optimum length.
92 CHAPTER 6. LABORATORY ENVIRONMENT AND EQUIPMENT

Minimum Frequency

In specifying the minimum frequency of each standard, the network analyzer

needs to be given the exact waveguide cutoff frequency under the conditions of

operation. The waveguide is air-filled, and the cutoff frequency depends on the

speed of light. To perform a calibration of the utmost accuracy the calibration

kit definitions must reflect this fact. The cutoff frequency for the TEiq mode is

/c = 5^ , (6.3)

where A is the waveguide width and the speed of light c is determined from the

equations given in Sec. 5.2.1.

Offset Impedance

The line impedance of rectangular waveguide varies nonlinearly with frequency.

Unlike coaxial systems a single impedance cannot be specified. To calibrate the

network analyzer properly, we must set the network analyzer system impedance

equal to the line impedance of the calibration standards. (This is true only if one

wishes the apparent line impedance of the ports to be the same as the defined

line impedance of the standards.) If we define the system and line impedances

equal to 1 Q, calibrated impedance data will be normalized to 1 Q. Calculation

of the true impedance of a device then involves simply multiplying the calibrated

impedance data by the waveguide line impedance at that frequency.

6.3.2 Calibration Procedure

Network analyzer calibration mathematically removes the effects of losses and

phase delays prior to the measurement ports. This permits us to define dB,

0° phase reference planes at the measurement ports. S-parameter data are then

measured relative to these reference planes. We define our reference planes at

the cavity's coupling ports. During resonator measurements, both waveguide Hbends curve downward to feed energy to the resonator. During the calibration

process, the configuration of the waveguide H-bends is necessarily different. One

of the H-bends must be turned 180° This allows the two measurement ports to be

connected together. To avoid misalignment and repeatability problems when the

H-bend is turned over, alignment holes are placed in all flanges. This includes

the coupling endplate and coaxial-to-waveguide adapters as well as the H-bends.

1. Set the network analyzer system impedance to 1 Q.
6.3. ANA WAVEGUIDE CALIBRATION 93

2. Load the X-band TRL calibration kit definitions into the network analyzer.

3. Disconnect the waveguide feed system from the resonator, and turn one H-bend 180°

4. Support the waveguides so that the measurement planes are aligned. Bend

the cables as little as possible. Do not twist.

5. Connect the measurement planes together.

6. Enter the TRL calibration menu by pressing the appropriate keys.

7. Press "thru." Wait for the network analyzer to finish.

8. Disconnect the through connection and attach the line. Press "line."

9. If more than one line is needed for the ANA's frequency range, connect

the lines and measure as is appropriate.

10. Disconnect the standard fine.

11. Connect the short to one measurement port. Measure.

12. Connect the short to the other measurement port. Measure.

13. Save the calibration in one of the network analyzers calibration registers.

6.3.3 Waveguide Port Extensions

When the network analyzer is calibrated we often wish to use port extensions

to translate the calibration reference plane to the cavity's coupling apertures. If port extensions are applied the following steps should be taken.

1. Set the type of port extensions to "waveguide" and set the waveguide

cutoff frequency to that calculated for a vacuum.

2. Set the relative velocity factor to the number obtained when the speed

of light in the laboratory is divided by the speed of hght in a vacuum.

In Boulder, Colorado, this number is approximately 0.9997 as calculated

from equations by Liebe [40].

3. When a resonance is being measured by the network analyzer, set the

display to show phase and adjust the port extensions until the phase of

the peak response (the resonant frequency) passes through zero.
Chapter 7

Sample Specifications and

Measurement Procedures

7.1 Sample Specifications

This section gives specifications for resonator samples, and describes how to

calculate the proper sample thickness. Because resonance conditions inside a resonator are sensitive to a sample's geometry, accurate permittivity and loss

tangent measurements require stringent sample specifications. A sample must

have very flat and parallel faces. Tolerances on the sample diameter are less

stringent. Because the electric field near the cavity wall is small, the air gap

between the sample and the wall causes minimal perturbation for nonmagnetic

materials. The sample diameter is allowed to be comfortably smaller (0.2 mm)

than the lowest cavity diameter which permits easy movement of the sample up

and down in the cavity.

1. Thickness: ±5 /jra (±0.0002 in), maximum.

2. Flatness: ±0.01 //m, nominal, depending on lapping capability.

3. Parallelism: ±2.5 /^m, (±0.0001 in).

4. Diameter: 59.7 ±0.03 mm, (2.350 ± 0.001 in).

5. Edge breaks: Sharp to 0.5 mm, (0.02 in).

94
7.2. OPTIMUM SAMPLE THICKNESS 95

7.2 Optimum Sample Thickness

Error analyses presented in this report show a minimum measurement uncertainty when the sample thickness is an integer multiple of one-half wavelength.

The axial wavenumber of a TEqi mode in a dielectric-loaded cylindrical waveguide is given in (2.44) as

/?2 = a;2eop,oe>'« - % . (7.1)

The permittivity and permeabihty of free space are cq and fiQ. The relative

permittivity and permeability of the sample material are e'^ and ^'^. The cavity

radius is a. The term Iq^ = 3.831 705 970 2 is the first nonzero root of the

derivative of the zero-order Bessel function Jq{x) = 0. Using (7.1), we calculate the axial guide wavelength inside the material Xg^ =

Pi/2tt at a desired frequency. Any integral multiple of Ape/2 yields minimum

uncertainty due to sample thickness uncertainty.

7.3 Grinding and Lapping

A typical procedure for grinding a resonator sample is given below.

1. Cut out rough sample with a diamond saw. Proceed slowly to avoid cracking or stressing the sample.

2. Grind diameter to specification in a rotating grinder.

3. Coarse surface grind a face to within 0.05 mm (0.002 in) of final specification.

4. Fine surface grind to 0.01 mm (0.0005 in) of specification.

5. Polish one side to flatness specification.

6. Flip sample over and mount on optically flat tool.

7. Polish second side to final specifications.

Flatness should be checked with a laser interferometer.

Parallelism is checked with a dial indicator. If thickness differences over the surface are small, an autocolhmator may be used. On optically

clear parts, the overall flatness and parallelism can be checked by

observation of interference fringes caused by front and rear surface

reflections.
96 CHAPTER 7. SAMPLE SPECS. & MEASUREMENT PROCEDURES

7.4 Sample Cleaning

A sample should be free of contaminants before it is measured in a cavity. Grinding can leave embedded particles and lubricants on the sample surface. Handling

the sample can leave oils. Samples should be handled with lint-free gloves and

cleaned with 99% pure isopropyl alcohol before being placed in the resonator.

Also, contaminants can be removed by ultrasonic cleaning. This is not a rec- ommended practice because of possible porosity or hydrophilic nature of the

material to be measured. To prevent the sample from being damaged by the

ultrasonic cleaner, gauze is wrapped around the sample. After cleaning, the

sample is rinsed off with distilled water and allowed to dry. A desiccator may

be used if the sample is porous or hydrophilic. After cleaning and drying the

sample, observe its condition under a microscope.

7.5 Measurement Procedures

7.5.1 Port Coupling Factors and Unloaded Qq

The port coupling factors K,i and unloaded resonance Qo are found from this

experiment. Port coupling factors are calculated from reflection S-parameters

(5ii and S'22)- Transmitted power can be calculated from port coupling factors

and compared to measured values.

1. Perform two-port TRL cahbration.

2. Detune cavity by completely lowering bottom endplate from cavity.

3. Look at phase of Sn and 522- Adjust port extensions until phases equal

zero at resonant frequency.

4. Run software which reads S-parameters.

5. Least-squares fit S-parameters 5ii and 522 to circle to find the center and

radius of circle.

(a) x-offset plus radius should equal one. The difference of this quantity

from one is a measure of the resistive loss of the coupling port. If the

number is very near one, network analyzer uncertainties will predominate. The x-offset plus radius is near 0.997 ± 0.005, which indicates

that the correction for r^ is indeed small (no greater than the order

of 0.001).
7.5. MEASUREMENT PROCEDURES 97

(b) y offset sliows how close zero phase adjustment is. The small y offset

is a combination of uncorrected reactance of the coupling port, phase

delay and ANA uncertainties.

(c) coupling factor ac is a function of the Q-circle radii given by

«^ = -. ~ , (7.2)

1 - ri - r2

as discussed in Sec. 4.2.1

6. Calculate theoretical transmission coefficient from ki and K2.

I T-,/ x |2 I c |2 4/^1/^2

I T{lj) I =1 521 I -

(1 + Ki + K2f

7. Transform transmission S-parameters, S21 and 5i2,

8. Perform linear least-squares fit to transformed parameters.

(a) Slope is a function of loaded Ql

9. Calculate unloaded Qq from loaded Ql and coupling factors Ki, K2.

10. Calculate transmission coefficient from (loaded or unloaded) Q and coupling factors. Compare to measured values.

7.5.2 Fixed-Frequency Method of Permittivity

Measurement

To measure dielectric loss, the fixed-frequency method is more accurate than

the fixed-length method because the effects of frequency-dependent variables are minimized. When the frequency is fixed, conductor skin-depth and iris coupling coefficients remain constant. The following procedure outlines how to use the program MEAS_RES to measure several resonances on an uncalibrated

network analyzer. This allows for permittivity calculation across the entire Xband frequency range.

1. Turn on water-bath circulator and network analyzer at least an hour before

starting the experiment.

2. Open and close the empty cavity at the tuner-base. Tighten the three

tuner-base nuts with a fixed-torque wrench. This assures the tuner base

is properly affixed. Lower the tuner-endplate about 0.1 mm and raise it again. This will assure that the sensing micrometer is properly seated.

Zero the sensing micrometer.
98 CHAPTER 7. SAMPLE SPECS. & MEASUREMENT PROCEDURES

3. Load MEAS_RES into the computer. RUN. Press "Sweep nwa." A menu will

appear that gives the start, stop and step frequencies for the sweep along

with other information about data routing. Change the sweep parameters

as desired and press "OK/DONE."

4. Enter a description of the experiment. For example, give information that

it is the empty cavity, give date and time and why you are making the

measurement. As many lines may be entered as desired. When finished,

ENTER a null string to begin the sweep.

5. Control the network analyzer by computer to find resonances in the specified frequency range, and print the computed values of resonant frequency

and Ql6. Open the resonator when the program finishes. Clean the sample. With

an edge of the sample on an edge of the tuner-endplate, slide the sample

laterally onto the endplate. This minimizes the amount of air between the

sample and the endplate. Make sure the sample is centered. Clear samples

allow one to see an interference pattern to emerge.

7. Close the tuner-base and tighten the nuts with the torque wrench. Lower

the endplate at least 0.1 mm and raise it up until the measurement micrometer is once again zeroed.

8. Run MEASJIES again, but this time press the "Measure on trigger" softkey.

9. Adjust the center frequency of the network analyzer to the first resonant

frequency on the empty-cavity frequency list. Move the endplate up so that the TEoin mode once again resonates at the empty cavity frequency.

Begin with a large frequency span and scale. As you raise the endplate

to obtain the resonant frequency, reduce the frequency span and scale until the resonance display is very fine (for low-loss materials the frequency

span should be about 200 kHz and the scale should be 0.2 dB). With

this configuration it is easier to make the upper and lower frequency re- sponses symmetric about the center frequency than to try to center the

peak frequency. If you suspect that there may be more than one axial

half-wavelength inside the sample (the sample is thick or has high permittivity), you may have to move the micrometer up so that more than one

resonant mode passes the network analyzer center frequency. If you know

the approximate permittivity of the sample, you can use the Calc_freqs

section of CAVITYPROG to estimate the cavity length in order to have the

TEqiu mode re-resonate. An alternative is to use the fixed-length method

to calculate permittivity. With the fixed-length method you can guess at

the axial mode number until the proper one is found.
7.^. MEASUREMENT PROCEDURES 99

10. Press the "Zoom Peak S21" softkey once the resonant frequency is cen- tered at the desired frequency. Mark down the micrometer reading. The

computer will set up the proper span, obtain the data and calculate the resonance parameters.

11. Set up the next center frequency while the analyze calculates resonance

parameters. The desired resonance can be either higher or lower than the

desired center frequency. If the resonance is at a higher frequency, make

sure to lower the endplate at least 0.1 mm past the necessary point, and

then raise the endplate up so that the resonance is at the desired frequency.

This helps to eliminate any drive-micrometer backlash.

12. Repeat the above steps for all desired frequencies. Be aware that the

micrometer travel is less than 1 in, and you may not have enough travel

to obtain the needed frequency.

13. Obtain both empty and with-sample resonance spectra. Load and run the

CAVITYPROG program.

14. Calculate speed of light from barometric pressure, relative humidity and

temperature.

15. Calculate the empty-cavity dimensions from the empty-cavity X-band reso- nance spectrum. Discount interfered modes. If x^ > 3x 10~^ the resonance

is most likely being interfered by a nearby non-T£'oi mode.

16. Select the fixed-frequency softkey.

17. Calculate e'^ and tan(5 from the resonant frequency, the micrometer reading

and the empty and with-sample resonance Qs.

18. Compute the measurement uncertainties for e'^ and tsmS by pressing the

"Calc Uncert" softkey. A table of uncertainties due to the uncertainties in

individual parameters will be shown. These uncertainties can be printed

by pressing the "Print Uncert" softkey.

7.5.3 Fixed-Length Method of Permittivity

Measurement

If measurements are taken with the network analyzer uncalibrated, the procedure for the fixed-length method is much simpler than the fixed-frequency

method. The two methods yield comparable measurement accuracy for e'^, but
100 CHAPTER 7. SAMPLE SPECS. & MEASUREMENT PROCEDURES

to obtain accurate loss measurements with the fixed-length method, the network analyzer must be calibrated in order to determine Qq. The following is a

fixed-length measurement procedure that uses an uncalibrated network analyzer.

1. Open the network analyzer at the tuner base, and clean the endplate with

a lint-free cloth and pure isopropyl alcohol. Close the endplate and tighten

the nuts with a fixed-torque wrench.

2. Zero the endplate by lowering then raising it. Lower the endplate at least

0.1 mm and raise it back up to take out any drive micrometer backlash

and to seat the sensing micrometer. Zero the sensing micrometer.

3. Run the MEAS_RES program, and press "Sweep nwa." Setup the start, stop

and step frequencies and the minimum signal level and signal-to-noise as desired and press "DONE/OK" to begin the measurement sweep. The

computer will control the network analyzer and find all resonances in the

specified frequency span. If the flag Printer_on is true, results will be

printed out at the printer specified by Print_addr.

4. Open the resonator and place the sample onto the endplate. Place the

edge of the sample onto the edge of the endplate and slide sideways onto

the endplate. This minimizes any air gaps between the sample and the

endplate that may occur.

5. Close the tuner assembly and tighten the nuts with the torque wrench.

Lower the endplate at least 0.1 mm and raise it up again so the sensing

micrometer reads zero. The resonator length is now the same as the emptycavity length.

6. Run MEAS_RES again to measure the resonance spectrum and print out

measurement parameters.

7. Run the CAVITYPROG program. Calculate the speed of light from barometric pressure, relative humidity and temperature. Calculate cavity dimensions from the empty-cavity TEqi resonance spectrum. From the main

menu press "Fixed length," and the fixed-length method menu will appear on the computer screen.

8. Calculate permittivity for each resonant mode by entering the emptycavity resonant frequency and Ql, the sample-loaded resonant frequency

and Ql and all associated uncertainties. The uncertainty in Ql for the

uncalibrated fixed-length technique is 8%. The cavity length and diameter uncertainties are approximately 0.011 mm and 0.002 mm, respectively,
7.5. MEASUREMENT PROCEDURES 101

as described in Sec. 4.1. After entering in all necessary parameters press

the "Calc Epsilon" softkey, then press the "Calc Uncert" softkey. Uncertainties are not recomputed until you press this softkey. An uncertainty

table will appear on the screen, and these results can be printed out by

pressing the "Print Uncert" softkey. When you return to the fixed-length

menu the uncertainties given for e'^ and land are the root-sum-square of

all uncertainties given in the uncertainty table.

9. Repeat the above step for each mode. If you have not matched the sampleloaded mode with the correct empty-cavity mode, permittivity (e'^) results

will still be correct, but loss tangent will vary with frequency.
Chapter 8

Permittivity Measurements

This chapter presents results of permittivity e'j^ and loss tangent tanS measure- ments made in the 60-mm diameter resonator. The first section compares results

of measurements made on a reference glass material that was previously characterized by the National Bureau of Standards (now NIST). The second section

gives results of measurements made on candidate reference materials. NIST is evaluating these materials for external distribution.

8.1 Check Standard Measurements

Since 1974 the National Bureau of Standards and then NIST have issued an

alumino-sihcate glass as a dielectric reference material. This glass was subjected

to international intercomparision [43]. Three resonator samples made from the

existing glass stock have been measured, and our results show excellent agreement with accepted values. For all three samples, the fixed-frequency method

was used. Figure 8.1 shows permittivity e'j^ and Fig. 8.2 shows loss tangent tan5

results from several measurements. The error bar shown in each plot shows the

internationally agreed upon value at 9.2 GHz and its associated uncertainty.

No error corrections have been applied to these results. Coupling factors and

resistive losses remain more-or-less constant when the fixed-frequency method is used. Frequency-shift corrections that affect e'^ tend to cancel out, and loaded

Q's (Ql) were used to calculate loss tangent. (Boron and 0H~ impurities are responsible for the decreasing permittivity and increasing loss tangent as fre- quency increases.)

102
8. 1 . CHECK STANDARD MEASUREMENTS 103

PERMITTIVITY MERSUREMENTS OF NIST GLRSS

G . 1 B

=

G . 155 -

.0:^ MJ

G . 1 35

I I 1 1 1

""» SAMPLE GL 1 - FIXED FREQUENCY

" + SHMPLE GL2 - FIXED FREQUENCY

X SRMPLE GL3 - FIXED FREQUENCY

RGREED NB5 VHLUE AND UNCERTRINTY

*

+ X

+ X M

+ * X X ** M ~

" X * * ^

' * *

*

1 1

1 1 1 1 10 1 1 12 1 3 FREQUENCY ( GHz

)

Figure 8.1: Uncorrected permittivity e'^ of three glass samples measured in

the NIST resonator. Both fixed-frequency, and fixed-length results compare

favorably with previous results.

I-zu

(Jz

cr

Lfl o

_l

LOSS TRNGENT MERSUREMENTS OF NIST GLASS

0. 005 3 1 1 1 1 1

0. 0052 _ -K SRMPLE GLl - FIXED FREQUENCY

+

* —

+ SRMPLE GL2 - FIXED FREQUENCY il

-f

0. 005 1 - " SRMPLE GL3 - FIXED FREQUENCY +

« *

" .

'

+ 0.005 -

+ + *

-

0. 0049 - -

0. 004S - -1

X «

« -

0. 004 7 -

• « +

+ -

0. 004B - + RGREED NBS VRLUE RND UNCERTRINTY -

0. 0045 -

+ -

. 004 4 1 1 1 1 1 9 10 11 12 FREQUENCY C GHz )

13 14

Figure 8.2: Uncorrected loss tangent tan6 measurements of three NIST glass

samples compare favorably with previous results.
104 CHAPTER 8. PERMITTIVITY MEASUREMENTS

2.533

2 . 532

2.53 1

2 . 53

FIXED FREQUENCY MEHSUREMENT

I :

I 1 I I 1 I

1 ^ I I

1

1—r—i r- -1 1 1—

1

1 1 r-

*<**« Measured values

Estimated k>ncertdinty

10 11 12 FREQUENCY (GHz

)

_, L

13 M

Figure 8.3: Permittivity e'^ of cross-linked polystyrene.

8.2 Candidate Reference Materials

We selected three low-loss materials for evaluation and qualification as standard

reference materials [4]. These materials and their nominal permittivities are: cross-linked polystyrene (2.53), fused silica (3.83) and alumina (10.0). Some

criteria for selection include isotropy, homogeneity, purity and stability with

frequency and temperature. After further evaluation, NIST plans to issue one

or several of these materials to industry and governmental agencies.

8.2.1 Cross-linked Polystyrene

Two 19 X 60.9 X 60.9 cm (0.75 x 24 x 24 in) blocks of cross-linked polystyrene

are under evaluation by NIST. Figures 8.3 and 8.4 present e'^ and tan5 data

from a fixed-frequency measurement.

8.2.2 Fused Silica

Three samples of recently manufactured fused silica were measured by NIST.

Samples 1 and 2, from different manufacturers, are made of the highest-purity

synthetic fused silica available. Sample 3 is a commercial-grade fused quartz.
8.2. CANDIDATE REFERENCE MATERIALS 105

Figure 8.5 gives e'^ results for these three samples. Samples 1 and 2 yield nearly

identical results, while sample 3 has a noticeably lower permittivity. Figure. 8.6

gives loss tangent results from these same three samples. All three samples have

a noticeably different loss, and, surprisingly, the fused quartz sample has lower

loss than the fused silica. We normally expect impurities to cause greater loss. The discontinuity in the results near 11.7 GHz are due to mode interference by

TEo2p modes.

These results indicate that the permittivity and loss of high-purity, high homogeneity fused silica are consistent from manufacturer to manufacturer. However

there are repeatable differences in permittivity and loss between the two fused

silica samples. The fused-quartz sample has a noticeably lower permittivity

and loss than the fused-silica samples. This could be because the fused-quartz

sample is less dense and has more seed crystals and inclusions, or because of

impurities. Also, the fixed-frequency and fixed-length results for sample 2 are nearly identical.

Uncertainty estimates for sample number 1 are given in Figs. 8.7 and 8.8.

8.2.3 Alumina

Eleven samples of 99.9% pure AI2O3 are under evaluation. Samples 8-11 were made separately from the first seven samples, and permittivity results for samples 1-7 are given in Figs. 8.9 and 8.10. A demonstrable difference between

samples from the same batch can be seen. This is not due to measurement

variations because results for Sample 6 are repeatable. Although further investigation is needed, the higher permittivity results at the lower frequencies are most likely due to our having used an effective cavity length that was too long

for those frequencies (Sec. 4.1). Loss tangent for this high-purity alumina is the

lowest of any material we have measured. Also, the fixed-frequency method gives

stable and repeatable loss tangent results for most of the frequency range, while

the fixed-length method varies much more. This is due to the limitations in

the loss tangent model which assumes an equivalence between the empty cavity

and a cavity containing a hypothetical lossless sample of the same permittivity

as the real sample (Sec. 4.2). The negative loss tangent values near 11.5 GHz

are due to the interfering TEq2\ mode. Uncertainty estimates for samples with

e'^ « 10 are given in Chapter 9, so no uncertainty results are given here.
106 CHAPTER 8. PERMITTIVITY MEASUREMENTS

FIXED FREQUENCY MER5UREMENT

1 30O 1 1 1 1 I

1250 -

-

U)

rg 1200 - _

H

zu

(Jz

a.

IS) o

_l

1 150

« «

« « « «

M

M

«

1 100 _

-

« «

• M tf

-

1050 - -

1000 1 I I 1 1 10 11 12 FPEOUENCY C GHz

1 3

Figure 8.4: Loss tangent tan6 of cross-linked polystyrene.

PERMITTIVITY OF THREE FUSED SILICH SHIiPLES

3 . B4

3 . &35

3 . 83

3 . 825

\U 3.82

3 . B 15

3.81

3 . 805

1

1

I 1 1 1

I I 1

POINTS NOT CONNECTED - FIXED LENGTH METHOD

POINTS CONNECTED - FIXED FREQUENCY METHOD

3 3 3

3 3 3 3 3 ^333^

9 11 13 FREQUENCY ( GHz15

Figure 8.5: Permittivity e'j^ of two fused silica samples (1 and 2) and one fused

quartz sample (3).
8.2. CANDIDATE REFERENCE MATERIALS 107

LOSS TRNGENT OF THREE FUSED SILICR SAMPLES

350

3QQ

li.ilUTS NOT CONNCCTED - FIXED LENGTH METHOD

PUINTS CONNECTED - FIXED FREQUENCY METHOD

9 10 11 \2 13 FREQUENCY ( GHz

)

M 15

Figure 8.6: Loss tangent tan(5 of two fused silica samples (1 and 2) and one fused

quartz sample (3). Note effect of interference by a TEq2 mode between 11 and

13 GHz.

PERMITTIVITY or HIGH PURITY FUSED SILICR

3 . S'lS "-.

3.835

3.825

3 . B 1 5

UPPER UNCERTRINTY

LOWER UNCERTRINTY

9 10 11 12 FREQUENCY ( GHz

)

13 14

Figure 8.7: Permittivity e'j^ and estimated uncertainty of fused silica sample #

1.
108 CHAPTER 8. PERMITTIVITY MEASUREMENTS

LOSS TRNGENT OF HIGH PURITY FUSED 5ILICR

30O

<

Zu

o

z

CE

O

200

9 10 11 \2

FREQUENCY ( GHz

)

13 M

Figure 8.8: Loss tangent tan6 and estimated uncertainty of fused silica sample

#1-

10. 04

10. 03

10. 02

10.01

9 . 99

9 . 9E

PERMITTIVITY tlEHSUREMENTS OF RLUMIIJH bHMPLES

1

\

:— II

J •» 4 a i

i i 1

9 1 1 FREQUENCY ( GHz

)

13

Figure 8.9: Measured permittivity e'j^ of several alumina samples from the same

batch.
8.2. CANDIDATE REFERENCE MATERIALS 109

L.0S5 TRNGENT ME RSUREMENTS OE RLUMINR

-

in

ino -20

-40 -

-60

9 1 1 FREQUENCY (GHz)

1 3

Figure 8.10: Measured loss tangent tan6 of several alumina samples from the

same batch.
Chapter 9

Uncertainty Analysis

This chapter describes how uncertainty estimates are obtained from the computer program CAVITYPROG. Typical uncertainty estimates are given for a 10-

mm thick sample with a relative permittivity value of 10 — j 0.001. CAVITYPROG

generates uncertainty tables for e'^ and tan6 due to individual parameter uncertainties. The uncertainties in cavity diameter and length, resonant frequency,

micrometer reading, and resonance bandwidth were found in Chapter 4 and are summarized in Table 9.1. The e'^ and tan6 results for the samples presented in Chapter 8 show estimated

total uncertainty. This chapter presents estimated e'^ and tanS uncertainties due

to uncertainties in individual parameters. Because there can be any number of

combinations of sample permittivity, loss, thickness, cavity length and resonant

frequency, for the sake of brevity, we present results for a hypothetical sample

with a relative permittivity of 10 — jO.OOl. Uncertainty estimates are given for

several frequencies in the X-band range. The uncertainty estimates are similar

Table 9.1: Estimated parameter uncertainties for the NIST resonator.

Cavity diameter a ±2 iim

Cavity length Lr ±9//m

Micrometer reading ALr ±1 fim

Sample thickness b ±3/xm

Resonance frequency ^sample ±1000 Hz

Speed of hght '^air ±1000 m/s

Empty cavity Q Qo ±3%

With-sample Q Qs ±3%

110
9.1. PERMITTIVITY UNCERTAINTY (Ae'ji) 111

for other permittivities, and must be evaluated on a case by case basis. For

example, the measurement uncertainties for a thin sample with low dielectric

constant can be much larger than one might expect (on the order of 20%),

whereas a thicker sample or one with higher permittivity would have much

smaller estimated measurement uncertainty.

9.1 Permittivity Uncertainty (Ae^^^)

To find uncertainties in e'^^, CAVITYPROG calculates the change in e'^j when one parameter is changed by its given uncertainty, while all other parameters are held

fixed. This is repeated for each parameter to find the estimated uncertainty

Ae'^ due to individual parameters. CAVITYPROG then finds worst-case total uncertainty by summing the individual uncertainties. This variational method is equivalent to taking the total differential of e'^ with respect to all parameters.

Figure 9.1 shows the estimated uncertainty Ae'^ due to the individual measured

parameter uncertainties given in Table 9.1. Cavity diameter and length uncertainties are the greatest sources of uncertainty

for e'^ permittivity calculation. As shown previously in Fig. 4.3, the calculated

cavity diameter changed less than 0.002 mm when the tuner endplate varied the

cavity length by 24 mm. This result is corroborated at all frequencies in X-band

as shown in Fig. 4.7 of Sec. 4.1.1 where the cavity dimensions were calculated

from different subsets of the 7-14 GHz mode spectrum.

When we measure samples using the fixed-frequency method, the cavity length can be shortened by its maximum 24 mm range. From dimensional evaluation

experiments described in Chapter 4, the uncertainty in diameter is less than

±0.002 mm and the uncertainty in length is no greater than ±0.009 mm. The

overall uncertainty in cavity length is taken as the root-sum-square of the cavity

length and the micrometer reading. These uncertainties are primarily due to

endplate-travel accuracy (Sec. 4.1.2) and the agreement of calculated cavity

dimensions from subsets of the 7-14 GHz resonance spectra (Sec. 4.1.1).

Sample thickness uncertainty is another significant error source that tends to

have a maximum contribution at the frequencies at which the other error sources

are minimized, as shown in Fig. 9.1. If the sample is thin < 2mm, this estimated

error can become very significant especially if the sample surfaces are rough.

The speed of light in Boulder, Colorado is approximately Cair = 2.99709 x 10^

m/s at 10 GHz, which corresponds to a relative velocity factor of 0.99972. At

standard temperature and pressure (23°C, 1.013 x 10^ Pa (1013 mbar)) and 50%

relative humidity the speed of light is approximately 2.996 95 x 10^ m/s with
112 CHAPTER 9. UNCERTAINTY ANALYSIS

0.016

>^

i 0.014 +-• CD^ 0.012

.^ 0.01

>

I 0.008

CD

CL

CD

>

0.006

i5 0.004

CD

CL

0.002

8 9 10 11 12

Frequency (GHz)

^^ frequency ^^ speed of light -- micrometer

-^

length

diameter

sample thickness

13 14

Figure 9.1: Permittivity uncertainty (Ae'^) versus frequency due to the uncertainties in various measured parameters for a 10-mm thick sample with

e'^ = 10-jO.OOl.
9.2. LOSS TANGENT UNCERTAINTY 113

relative velocity factor 0.99967. In the calculations for e'^, erroneously setting

the speed of light in the air-filled portion of the cavity to the speed of light in a vacuum gives a significant upward bias to e'^ results. If we use the proper value

for the speed of light is calculations, the residual uncertainty in Cair is approximately ±300 m/s. To accommodate for miscellaneous atmospheric disturbances

we have specified the uncertainty in the speed of light to be approximately three

times greater (« ±1000 m/s). Figure 9.1 shows that this uncertainty in the

speed of hght can indeed contribute an appreciable error to e'^. Resonant frequency can be very accurately determined to within a few hundred

hertz. However, if the resonant frequency is set and observed for one hour the

resonant frequency drifts no more than 1 kHz due to changing cavity length

caused by settling of the endplate drive micrometer or because of slight changes

in cavity temperature. The reading of the sensing micrometer changes less than

0.001 mm to reflect this change in length. Because this problem occurs when the

resonator is left alone for a period of time that is approximately the same as the

time the computer takes to measure the X-band resonance spectrum, we use a

1 kHz uncertainty for the empty and sample-filled resonant frequencies. Fig. 9.1

shows the uncertainty in e'^ due to a resonant frequency uncertainty of ±1 kHz.

There are cases in which the Q of a resonance can be low (~ 2000) because of a sample's extreme thickness or lossiness. In this case, resonant frequency and Q

can be difficult to define because of asymmetries that may occur in the resonance

response. When this is the case, one must use special precautions in defining the

uncertainties in resonance frequency and Q because systematic errors having to

do with the resonator's frequency dependent characteristics may be occuring.

9.2 Loss Tangent Uncertainty

Loss tangent uncertainty is calculated from the total differential of the loss tangent equation given by Cook [2]. We have verified that this method is identical

to the variational method used in calculating Ae'^ in which the uncertainties are added to individual parameters, one at a time, to find the change in tan6. The

resulting individual estimates for A tan6 are added together to get a worst-case

uncertainty estimate. The total differential method can be used to estimate

Atan5 because we have an explicit equation for tan^. The equation is given by

, , pi2b-s) + il/('^)[2(L^-b)-q] (

1 1
114 CHAPTER 9. UNCERTAINTY ANALYSIS

where

sm2Po{Lr-b)

Q = n

. (9-3)

Po

sin 2/^1

6

The total differential is found from

f (9 tan 6 d tan 6 dp d tan b dq

O tan — oLr \ —z-: 1 ;: 777 h

(9.4)

dLr dp dLr dq dLr

. (9 tan b d tan b dp d tan b dq d tan b ds

db dp db dq db ds db

r. , j d tan b d tan b dp dP^ d tan b ds dp^

"^ ^^1 de'ji ^ dp Wed?^^ ds Wed?^

^^ I 5 tan (5 1 +^^sample 1 ^ ' \ (9-5)

L <^^sample )

1 d tan b I +^Qempty \

^

{

( ^Vempty ) id tan b dp dPo d tan b dq dfSo dp dpo dcair dq dPo dCar

^ , d tan b dp dPo d tan b dp dp^ d tan b dq dPo d tan 6 ds dp^ +da i — TTTT^ \ ;; t^tt-t: \ ;:; tttt-^; \- dp dPo da dp dP^ da dq dPo da ds dp,, da

^ , , d tan b dp dPo d tan b dp dp. d tan b dq dPo d tan b ds dp.

+df < — -T^^^^^—^—^^

+ —^—^^

+

dp dPo df dp dp, df dq dPo df ds dp, df

Uncertainty is found by replacing the partials dLr, db, 5e'^, ^Qsample' ^^emptv

dcair, da and df with their respective estimated uncertainties AL^, A6, Ae'^,

"^^samole' ^^empty ^^air, Aa and A/. The partial derivatives are given by

d tan b _ ^ l^empty ~ ^sample/

dLr ^'rPQemptyQsdiTnple^'^^ ~ ^^

5 tan 5 _ "^ vQempty ~ ^sampleJ (^-^'' ~ ^

~~ ^i db <^'rPQempty Qsample (^^ - s)

dt&nb _ (Qempty -<5sample)(2^- 2Z/r + 9)

^4 (4)' PQemptyQsample(2^ - «)

(9.6)

(9.7)

(9.8)
9.2. LOSS TANGENT UNCERTAINTY 115

(9 tan 5 _ 26(e'^p — 1) — e'/^ps + 2Lr — q

<5^sample " ^^^sample^^^ - ^)

5 tan 6 26(e'^p ~ 1) — ^'rPS + 2Lr — q -^Qempty ^fl^^^^gmpty^^^ - s)

The partial derivatives of p, ^ and s are given by

dp 2{b - Lr) sm{Po{b - Lr)) cos{f3o{b - Lr))

OPo

dp

We

dp

db

dp

dLr

dq_

dPo

dq

db

dq

dLr

sm^bP,)

2bcos{bP,)sm^{Po{b - Lr))

sin^ibP,)

'

2Posm{Po{b-Lr))cos{Po{b-Lr))

sin2(6/?J

2P,cos{bp,)sm^iPo{b - Lr))

sm^{bP,)

'

2posm{Po{b - Lr))cos{Po{b - Lr))

sm^ {bp,)

sm{2Po{b - Lr)) 2{b - L,) cos{2Po{b - Lr))

Pi

-2cos{2Po{b - Lr)) ,

2cos(2/?o(6- Lr)) and

ds _ 2bcos{2bP,) 26/?, We

Po

Pe ds— = 2 cos 2bp,

db

PI

'

The the wavenumbers by the guide wavelength defined as

P = ^(2./)V-(^) ,

/^o =!^

\

27r/ t'r

0e =^

\

27r/

Co 'R

+1

'-01

(9.9)

(9.10)

;9.11)

f9.12)

^9.13)

^9.14)

^9.15)

^9.16)

[9.17)

;9.18)

(9.19)

;9.20)

;9.2i)

(9.22)
116 CHAPTER 9. UNCERTAINTY ANALYSIS

where e" -C e' so that e* ^ e' — jc". The uncertainties in the wavenumbers are then found from the total differentials

A/?o = -^ ^Carr + ^"Att + -^T^f ^nd (9.23)

A/J, = |^Ac„„ + f^Aa + ^A/

(9.24)

ae}^ da aJ

where the partial derivatives are

dp, iir'fe]R '^ '' ^(^)%«-(%y

2 5A (^Ol)^ 1 da a^ //9,rA2 (^)^'.-(^)'

<9/?e (27r/)^ 1

and

CO

<9/?o 47r2/

(^r^-fr)"

5/ c;

5/?o (^'oi)

2air

V Cair / V a /

5a tt'^

a^o (27r/)

v(i)M¥y

OCair ^air y(^r-(%y

(9.25)

(9.26)

(9.27)

(9.28)

(9.29)

(9.30)

Loss tangent uncertainty due to the uncertainties in the measured parameters

is shown in Fig. 9.2. We can observe a highly frequency-dependent behavior

in loss tangent uncertainty due to cavity dimensions uncertainties. Typically

we measure samples with much lower loss than our hypothetical 10-mm thick

sample with permittivity 10 — jO.OOl, and loss tangent uncertainty is usually

dominated by uncertainties in the measurement of resonance Q.
9.2. LOSS TANGENT UNCERTAINTY 117

180

160

COb

X 140

>>

*-• c 120

CO

+-• 1-

CDO

100

cD

+-• 80

c

(D

O)C 60

03

h-

(0 40 (/) o

_l

20

8

frequency

speed of light

micrometer

length

diameter

sample thickness

empty Q

with-sample Q

9 10 11 12 13 14

Frequency (GHz)

Figure 9.2: Loss tangent uncertainty A(tan6) versus frequency due to uncertainties in various measured parameters for a 10-mm thick sample with.

e'^:= 10- jO.OOl.
Chapter 10

Future Work

The change in length of the cavity is monitored by the sensing micrometer. As

described in Sec. 4.1.2, the sensing micrometer's tip gradually wears a small

pit into the extended finger of the tuner-endplate yoke. The yoke is made of

aluminum and is therefore softer than the sensing micrometer's tip. To improve

the repeatability of cavity length measurement, we will press fit a hardened steel

plug into the yoke for the sensing micrometer to press against.

The uncertainty estimates given in Chapter 9 assume that systematic errors have been accounted for. To determine whether our accuracy estimates are

truly correct, we must compare our results with the results of other instruments

of similar accuracy. The glass results given in Sec. 8.1 compare favorably with

results from an international intercomparison completed in 1974. NIST plans

to engage in future international comparisons not only to qualify the NIST

resonator for calibration service, but also to qualify our stock of cross-linked

polystyrene, fused silica, and alumina as a Standard Reference Material.

118
Acknowledgements

The authors express their sincere appreciation to Hans J. Liebe of the U.S. Department of Commerce Institute for Telecommunication Services for his valuable

contributions in the design and evaluation of the NIST cavity resonator. Thanks

also go to Ulrich Stumper of the German Physikalisch Technische Bendesanstalt

for his consultations on the initial design of the NIST resonator. We also express our appreciation to William A. Kissick. His leadership of the NIST Electromagnetic Properties of Materials Program and of the Broadband Microwave Metrology Group has contributed a great deal to the progress and successes of many

individuals. Thanks also go to Howard E. Bussey, whose many contributions to

the field of electromagnetic properties of materials measurements continue to be

standards by which new techniques are judged.

The authors thank Lydell L. Frasch of Boeing Defense and Space Group, Charles

Wilker and Ron Riegert of E.I. du Pont de Nemours & Company, and Mark D.A.

Rosen and Mark Lizza of Grumman Corporate Research Center for their careful

review of this Technical Note. Critical technical review of NIST publications is an important component of NIST manuscript review, and their contribution is greatly appreciated.

119
References Cited

[1] F. Horner, T. Taylor, R. Dunsmuir, J. Lamb, and W. Jackson, "Resonance

methods of dielectric measurement at centimetre wavelengths," J. lEE,

vol. 93, Pt. Ill, pp. 53-68, 1946.

[2] R.J. Cook, "Microwave cavity methods," in High frequency dielectric measurement, (J. Chamberlain and G.W. Chantry, ed.), pp. 12-27, Guildford, U.K., IPC Science and Technology Press, 1973, Mar. 1972.

[3] E. Ni and U. Stumper, "Permittivity measurements using a frequency-tuned

microwave TEqi cavity resonator," lEE Proceedings, vol. 132, Pt. H,

pp. 27-32, Feb. 1985.

[4] R. G. Geyer, "Dielectric Characterization and Reference Materials," NISTTN 1338, Natl. Inst. Stand. Technol., 1990.

[5] R.F. Harrington, Tim.e- Harmonic Electromagnetic Fields, New York:

McGraw-Hill Publishing Co., 1961.

[6] S. Morgan and J. Young, "Helix waveguide," Bell System Tech. J., vol. 35,

pp. 1347-1384, 1956.

[7] H.G. Unger, "Helix waveguide theory and application." Bell System.

Tech. J., vol. 37, pp. 1599-1647, 1958.

[8] D. Young, "Measured TEqi attenuation in helix waveguide with controlled

straightness deviations," Bell System Tech. J., vol. 44, pp. 273-282,

1965.

[9] R.J. Cook and R.G. Jones, "Precise Dielectric Measurement Techniques

for the Frequency Range 10 GHz to 150 GHz," pp. 528 531, Proc. 8th

European Microwave Conf., 1978.

[10] R. Waldron, Theory of Guided Electromagnetic Waves, New York: Van

Nostrand Reinhold Co., 1970.

[11] R. Waldron, "A helical coordinate system and its applications in electromagnetic theory," Quart. J. Mech. Appl. Math, vol. 11, pp. 438-461,

1958.

120
BIBLIOGRAPHY 121

25

26

19 D.

20 E.

21 M

22 D.

23 A.

241 A.

R. Walclron and D. Bowe, "Normal modes of helix waveguide.," Marconi

Rev., vol. 28, pp. 29-64, 1965.

M. Shimba, "Measurement of spurious mode losses in helix waveguide,"

Rev. Electron. Commun. Lab., Tokyo, vol. 11, pp. 111-120, 1962.

Y. Kazantsev, I. Kaznacheev, and V. Meriakri, "Investigation of helix

waveguide," Radio Eng. Electron, (translation of Radiotekh. Elektron.),

vol. 4, pp. 88-94, 1959.

K. Mikosliiba, "TE'oi loss characteristics in multimode waveguide with ran- dom imperfections," ./. Inst. Electron. Com.m. Eng. of Lapan, vol. 4,

pp. 2-3, 1964.

G. Piefke, "The influence of helix wire diameter on the modes in a helix

waveguide," lEE paper No. 2858, vol. E, pp. 110-118, Sep. 1959.

R. Waldron, D. Bowe, P. Wackrill, and B. Wescott, "Helix waveguide theory," in Proc. lEE Conf. Transm,ission Aspects of Com.rn.unication Networks, pp. 210-215, 1964.

K. Noda, K. Yamaguchi, and N. Suzuki, "Circular electric wave transmission through helix waveguide," Rev. Electron. Com.m.un. Lab., Tokyo,

vol. 10, pp. 49-70, 1962.

Kajfez, "Correction for measured resonant frequency of unloaded cavity,"

Electron. Lett., vol. 20, no. 2, pp. 81-82, 1984.

L. Ginzton, Microwave Measurements, International Series in Pure and

Applied Physics, New York: McGraw-Hill, 1957.

. Sucher and J. Fox, Handbook of Microwave Measurem.ents, Vol. H,

Brooklyn, N.Y.: Polytechnic Press, 1963.

Kajfez and E. Hwan, "Q-factor measurement with a network analyzer,"

IEEE Trans. Microwave Theory Tech., vol. 32, no. 7, pp. 666-670, 1984.

J. Estin, "Resonator characterization and measurement," Cyberlink Corporation, Boulder, CO, Final Report to Sponsor, July 1990.

J. Estin and M.D. Janezic, "Improvements in dielectric measurements

with a resonant cavity," in IEEE IMTC-S Conf. Proc, Atlanta, GA,

May 1991.

D. Kajfez, "Incremental frequency rule for computing the Q-factor of a

shielded TEornp dielectric resonator," IEEE Trans. Microwave Theory

Tech., vol. 32, pp. 941-943, Aug. 1984.

H.A. Wheeler, "Formulas for the skin effect," Proc. IRE, vol. 30, pp. 412-

424, Sep. 1942.
122 BIBLIOGRAPHY

[27] H.E. Bussey, J.E. Gray, E.G. Bamberger, E. Rushton, G. Russell. B.W. Pet-

„ley, and D. Morris, "International comparison of dielectric measurements," IEEE Trans. Instrum. Meas. TechnoL, vol. IM-13, pp. 305-311,

Dec. 1964.

[28] J.C. Gallop and W.J. Radcliffe, "Dimensional measurement by microwave

resonances," J. Phys. E: Sci. Instrum., vol. 14, pp. 461-463, 1981.

[29] J.C. Slater, Microwave Electronics, New York: D. Van Nostrand Company,

Inc., 1950, p. 75.

[30] H. Bussey, "Standards and measurements of microwave surface impedance,

skin depth, conductivity and Q," IRE Trans. Instrum.., vol. 1-9, pp. 171-

175, Sep. 1960.

[31] H. Bethe and J. Schwinger, "Perturbation theory for cavities," Tech.

Rep. Dl 117, Natl. Defense Res. Committee, Washington, DC. Mar.

1943.

[32] P. Boggs, R. Byrd, J. Donaldson, and R. Schnabel, "Algorithm 676 — ODRPACK: software for weighted orthogonal distance regression," ACM

Trans. Mathematical Software, vol. 15, no. 4, pp. 348-364, 1989.

[33] C. Montgomery, R. Dicke, and E. Purcell, Principles of Microwave Circuits.

Vol. 8 of MIT Radiation Laboratory Series, McGraw-Hill Book Co., Inc.,

1948.

[34] E. Vanzura and J. Rogers, "Resonant circuit model evaluation using re- flected S-parameters," in IEEE IMTC-S Conf. Proc, Atlanta, GA, May

1991.

[35] H. Nussenzvieg, Causality and Dispersion Relations, Vol. 95 of MathemMics

in Science and Engineering, New York: Academic Press, Inc., 1972.

[36] Von G. Schulten, "Resonatoren fur millimeterwellen und ihre verwendung

zur beobachtung von gasresonanzen," Frequenz, vol. 20, no. 1, pp. 10-22,

1966.

[37] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes:

The Art of Scientific Computing, Cambridge: Cambridge University

Press, 1986.

[38] E.J. Vanzura, "Creating CSUBs written in FORTRAN that run in BASIC,"

in INTEREX Proceedings of the 1988 Conference of HP Technical Computer Users, INTEREX International Association of Hewlett-Packard

Computer Users, Paper 20, 18 pages, Aug. 1988.

[39] E.J. Vanzura, "Creating CSUBs in BASIC," HP Design and Automation,

Oct. and Nov. 1988, Austin: Wilson Pubhcations, Inc.
BIBLIOGRAPHY 123

[40

[41

[42

[43

[44

[45

H. Liebe, "An updated model for millimeter wave propagation in moist

air," Radio Science, vol. 20, pp. 1069-1089, Sep.-Oct. 1985.

N.R. Draper and H. Smith, Applied Regression Analysis, J. Wiley and Sons,

Inc., second ed., 1981.

J.D. .Jackson, Classical Electrodynamics, New York: J. Wiley and Sons,

Inc., second ed., 1975.

H. E. Bussey, D. Morris, and E. Zal'tsman, "International comparison of

complex permittivity measurement at 9 GHz," IEEE Trans. Instrum.

Meas. TechnoL, vol. IM-23, pp. 235-239, Sep. 1974.

P.C.R. Grivet Acad. Sci. (Pans), vol. 218, p. 71, 1944.

H. Bussey and J. Gray, "Measurement and standardization of dielectric

samples," IRE Trans. Instrum., vol. I-ll, pp. 162-165, Dec. 1962.
Appendix A

Partial Filling Factor for Loss

Determination

Previously we showed that the unloaded quality factor of the cavity with sample

inserted is 111 .* X ^0, sample ^c Qs

where tan (5 = l/Qs- The empty unloaded cavity quality factor is

1 1

^0, empty ^c

(A.2)

In practice, a loaded cavity quality factor is always measured, which includes

coupling port losses Qe

where Qo is either with or without sample insertion. The loaded quality factor

is related to the unloaded quality factor by (2.80).

The analysis in Sees. 2.2.5 and 2.2.6 derived the theoretical Qc and the measurement approach for evaluating total power loss in the cavity walls, as well as a technique for separating sidewall from end plate losses. However, the loss in

the sidewall of the cavity is different when the sample is inserted as opposed to

when the cavity is empty (air filled). Therefore, it is necessary to determine the

energy stored and power dissipated both (1) in the sample of arbitrary thickness

having relative complex permittivity e'^j — J^'r and (2) in the air-filled portion of

the cavity when the sample is inserted. This allows us to account for the fact

that the cavity is only partially filled with dielectric.

124
125

Reference [2] shows that the theoretical quality factor Qq of a cylindrical cavity

of radius a that is partially filled with a sample of thickness h having a relative

complex permittivity e^ = e'^ — je'^, is given by

^

{F{2h - G) + [2(L - 6) - U]/e'^] auj'i,'^e'j, ^°

{SsC^ {kl [(26 - G)F + 2(L -h)-U] + 2a [F(5l + /?2]) ^ ^^

+aLj^li'jie'jiF{2b - G) tan (5} ,

where c is the speed of light in air, ui is angular frequency for any TEqip resonant

mode, 6s is the effective penetration depth given by (2.123) of the cavity's wall

and end plates, and

sm [/3ib\

G = ?^. (A.6)

^

^ sin[2/3o(L-6)]

_ ^^_^^

Equation (A. 4) is valid for a dielectrically lossy sample and takes into account

the effect of finite wall and end plate loss intrinsic to the cavity. The measured

unloaded quality factor (5o,m with the dielectric sample is generally somewhat

lower than the theoretical Qo in (A. 4). From (2.123) and (A. 4) Cook [2] has

further shown that the loss tangent may be precisely calculated from

tan (5

where

1 J F{2b-G) + [2{L-b)-U]/e'j,

I F{2b-G) J LQo,m Q\

(A.8)

^ ^

{F{2b - G) + [2(L -b)-U] /e'^] au^jiy^ ^ 6s,mc' [k'c [F{2b -G) + 2{L -b)-U]+2a {F(3l + dl)]

' ^ ' '

Equation (A.8) is similar to previous analysis except it now contains a filling

factor because the specimen only partly fills the cavity.
Appendix B

Radial Air Gap Correction

One possible source of error in dielectric property measurements at microwave

frequencies is that due to the air gap between the sample under test and the wall

of the waveguide or resonator fixture. The solid dielectric under test is usuallymachined into the shape of a parallel-sided disk for a right-circular cylindrical

resonator, and clearance must be left between the sides of the disk and the wall

of the resonator so that the disk may move freely in the resonator. Exactly how

much clearance is permissible before the error in dielectric characterization of

the sample becomes significant is a problem that must be examined.

Consider two situations for our cavity resonator in which the walls are modelled

as perfect electrical conductors. The first case (Fig. B.l) illustrates a dielectric

sample in a cylindrical resonator with no air gap. The second case (Fig. B.2)

shows the actual situation in which an air gap exists. The presence of this gap

results in an increased length of Li -I- AL at resonance, where Li = L — 6 for the

ideal case. This, in turn, leads to a measured permittivity of the sample under

test that is too small.

One approach to correct for the presence of an air gap is to consider the change

in resonant frequency due to the gap with perturbation theory, where the ideal

situation shown in Fig. B.l is perturbed. Since we know the measured length

Li + AL at measured resonant frequency fm and the radii of both the sample

under test and dielectric resonator, we can estimate the resonant frequency for

the resonator with no air gap but of length Li+AL for use in the transcendental

equation for permittivity calculation. Here a method described by Grivet [44]

and Bussey [45] is outlined.

We noted that at resonance the total energy interior to the cavity is constant,

as a function of time and that the average values of the energy stored in the

126
127

z

Volume Vo

I

/^o.^o

^^..^ -<

^ \ \ \ \ \ \ \T\T\T\Tr\T\vCT\ \ \ \ \ \ \ ^\\\\\\\\\^\\\^ / /v\\\\\\\\\ ^^\\\\\^\^\^\\ // ^ v\\\\\\\\\

Volume Vu,:

Figure B.l: Unperturbed cylindrical cavity resonator with no air gap present

l)etween sample under test and resonator side wall.

electric or magnetic field are equal. Hence, for TEo]„ mode structure,

cavity

e'\E,E;\dV

cavity

fi.\HH*\dV , (B.l)

where the * denotes complex conjugate and the integrals are taken over the

entire volume of the cavity. For a magnetically impermeable sample under test,

//, may be taken outside of the above integral and since, by Faraday's law.

V X E = -j2'KffiH , 'B.2:

'B.l) can be written

e'\E\^dV

cavity 47r2/2//, '^ '' ''cavity

[ V X ^] ^ dV 'B.31

If we now make the assumption that the electric field in the perturbed cavity is approximately that of the unperturbed, we may write

Ev, , (B.4]
128 APPENDIX B. RADIAL AIR GAP CORRECTION

Figure B.2: Perturbed cylindrical cavity with uniform air gap between sample

and resonator wall.

since we know that Eu satisfies the correct boundary conditions at the conducting walls of the cavity and is continuous at the boundary of the sample under

test. This approximation is valid only if the air gap is not too large relative to Xg

and if the air gap is uniform, so azimuthal symmetry is maintained. From (B.4),

V X ^„ = -j2TTfu^Hu , where /„ denotes the unperturbed resonant frequency. Hence,

;b.5)

J n u, air I

-' \F dV

Vu.s-AV

+ ni,ME...\'dv]

J u

Vo f^o\H.^u, air I 'd\'-IIIVu,s-AV

lio\HuJ^dV

+ JJ/^^l^o\Hu,s\'dv} :b.6)

where fm represents the measured resonant frequency, the subscripts v and

s denote unperturbed and sample, and where the integration over AV is an

integration over the volume of the air gap which is AV = -Kah - irr^b = nb (o^ - r^) , :b.
129

where a and r are the respective radii of the resonator and the sample under

test. Equation (B.6) may be simphfied as follows:

= li{jll/\"^.^u\'dV + jjj^^^,\H^,\'dv] . (B.8)

We now note that

from (B.l), so

/^ [III,, ^»l^», airP''!^ + //4 _ <\EuA'dV

= /«{///,/o|£„,a,rP'i^ + //4/'.I^..Pdv} (B.IO)

We now have a correction formula for the unperturbed resonant frequency /^ in

terms of the measured resonant frequency fm in the presence of an air gap of

volume AV; that is,

A = /„|i !!!^y(t'.-i,)\E^,?dV^

1

I S!h,MEu.^u\''dV + !ISv^^J-'AE^,?dv\

Of course, when there is no air gap ISV = and /„ = /„,. For the modefiltered TEoip cavity of concern here, we have only an azimuthal component of

the electric field that is of concern in the above integrations.

If an air gap is present between the sample under test and wall of the waveguide

of the cavity resonator, the cavity will be at resonance at an increased length

(compared to the situation where no gap exists). This leads to a measured

dielectric constant that is too small. The procedure from the analysis above,

which is valid if the air gap is not too large relative to the guide wavelength and

if the air gap is uniform so that azimuthal symmetry is maintained, is as follows:
130 APPENDIX B. RADIAL AIR GAP CORRECTION

1. Measure the resonant frequency fm and measured length of the cavity at resonance.

2. Compute the unperturbed resonant frequency /^ for a resonator with no

air gap but with the length measured in step 1 above.

3. Use the unperturbed frequency derived in step 2 for e'^ determination

in (2.46).

An alternate form of (B.ll) for computing the correction to the measured relative

dielectric constant in the presence of a radial air gap for TEqip mode is

.. .> ,, i:io''lMk,nfsm'{P,z)r,dr,d<P

Jo Jo [^il^^c^i)] sm iPiz) ridr^dcp

which reduces to

with ^ = JliKr) - ^Mk,r)Mk,r) + .J^,{Kr) . (B.14)
Appendix C

60-mm Cavity Drawings

To be consistent with shop toohng, dimensions for resonator parts were specified

in inches. Apologies are offered for our deviation from Standard International

units. In the following drawings, dimensions are in inches unless otherwise specified.

131
132 APPENDIX C. 60-MM CAVITY DRAWINGS

COUPLING /

APERTURES

UPPER CAVITY

FLANGE

REPLACABLE

FIXED

END PLATE

DIELECTRIC

SAMPLE

MOVEABLE

END PLATE

LOWER CAVITY

FLANGE

DETACHABLE

TUNER

ASSEMBLY

PVC BUSHINGS

TEMPERA TURE

SENSING PROBE

Acrylic Plastic Water Jacket WA TER

HELICAL WAVEGUIDE

TUNER POS IT ION

SENSING

MICROMETER

MOTOR-DRIVEN

MICROMETER

WATER INTAKE

FOR TUNER ASSEMBLY

PRECISION

LINEAR BEARINGS

SPRING-LOADED

CARBIDE RODS

RESONATOR

SUPPORT RODS

Figure C.l: Cutaway assembly drawing of NIST cavity resonator.
133

25.4 111111

Figure C.2: Detail of tuner assembly and flange.
134 APPENDIX C. 60-MM CAVITY DRAWINGS

Figure C.3: Cutaway side view of tuner-base and extension.
135

TOP VIEW

H

I

I n

^1-rri^'

b =tt

S I DE VI EW CUTAWAY S I DE V I EW

BOTTOM VI EW

Figure C.4: Orthogonal views of coupling flange.
136 APPENDIX C. 60-MM CAVITY DRAWINGS

. 500 ^0.005

. 306 *0 005

. 400 ^0.005

. 500 ^0-005

. 600 *o 005

. 050 ^0.002

-^

I N I /*

FOR 16 R I NGS

. 100 ^0.005

Figure C.5: Side view of coujjliiig flange
137

FLAT TO 3 /ill! (0.0001 in.)

3/8 - XX TAP

. 250 ^0.005

. 100 ^0005

. 306 iooos

. 400 ^0005

FLATT0 3 A/m (0.0001 in.)

45 DEGREE BEVEL

FOR TEMPERATURE

PROBE 0-R I NO

Figure C.6: Cutaway side view of coupling flange.
138 APPENDIX C. 60-MM CAVITY DRAWINGS

1 . 7

2.200 R

0.13 R :g: 3 PLACES

CLEARANCE EOR TIE ROD

120.0° '0 3 PLACE

2-56 X 1/4" BOTTOM TAP

0.040 R '0 00, 6 PLACES

R to. CIO

R ^0010 i

188 R «o.oo5 3/8"-XX TAP

. 250 R '0 005 CLEARANCE FOR KNURLED

0-R I NG BOLT

0° '0 2

6 PLACES

Figure C.7: Top view of coupling flange.
139

1 . 180 R ^0.001

1.414 R !S:SSg

FIT TO 0. D . SHRINK TUBING

1 . 464 R ^0.002 F I T 1/16" 0-R I NGS

1 . 750 R *o.oo2

1 . 800 R ^0.002

FIT TO ACRYLIC PLASTIC

WATER JACKET

1 . 363 R *oooi

FIT TO O.D.

OF WAVEGU I DE

0.09 2 R !S:^io°

Figure C.8: Bottom view of coupling flange.
140 APPENDIX C. 60-MM CAVITY DRAWINGS

TOP VIEW

—rrm—

j

TTT

I--

Ti TTTT a-tfa- ^^^i____L, I^K

^

S I DE VI EW CUTAWAY SIDE VIEW

BOTTOM VI EW

Figure C.9: Orthogonal views of timer flange.
141

FIT FOR 1/16" 0-RING

. 306 'ooos

. 500 ±0.005

. 400 '0 002M

. 400 ±0.005

. 500 ±0.005

n r^

.J. -r-li,n f-

. 1 54 ±0.002

. 750 ±0.005'

0.02" X 20 CHAMFER

Figure C.IO: Side view of coupling flange.
142 APPENDIX C. 60-MM CAVITY DRAWINGS

FIT TO O.D. OF SHRINK TUBING

AND HEL I CAL WAVEGU I DE

FLAT TO 3 /iiii (O.OQOl in.) 00 1 I NCH

. 050 ±0.001

. 100 ±0.001

. 900 ^0 005

. 300 ^0.005

, 05 X 26 . 5 CHAMFER

TOP AND BOTTOM OF

CENTERING RING

Figure C.ll: Cutaway side view of coupling flange.
143

3. 25 R '00

2 . 100 R -000'

2 . 200 R '0005

THRU HOLE

0.125 R ±0.002

3 PLACES

. 250 '0 002 3 PLACES

. 200 R ±0.002

3 PLACES

2 . 401 R ±0.001

1 58 R ±0.002

THREADED 10-32

THROUGH

Fit to Acrylic Plastic Water Jacket

800 R '0.002

1 . 750 R '0.002 467 R '0.002

^ p +0 002 ^ l\ -0.000 363 R '0-001

Figure C.12: Top view of tuner flange.
144 APPENDIX C. 60-MM CAVITY DRAWINGS

1 . 180 R ^0-001

1.414 R l°o:°o°ol FIT TO O.D.

SHRINK TUBING

1 . 464 R ^0.002

FIT 1/16" 0-R I NGS

1 . 750 R *o.oo2

1 . 800 R ^0 002 FIT TO ACRYLIC PLASTIC

WATER JACKET

1.363 R *o.ooi

FIT TO O.D.

OF WAVEGUIDE

.092 R ^S:Cc

Figure C.13: Bottom view of tuner flange.
145

TOP VIEW

-i=r i^ "^^-^^

-t^-^

S I DE V I EW

BOTTOM VIEW

Figuro C.14: Orthogonal views of tuner 1)ase.
146 APPENDIX C. 60-MM CAVITY DRAWINGS

3 . 25 R »o

2.6000 RllS

1 . 00 «o c' 3 PLAC

f

. 1 56 iO.005 . 063 ^o.ooi 3 PLACES 3 PLACES

0.41 «<ioi 3 PLACES

8 R iO.ooi LACES

. 5500 R ±0.0005

, 2 PLACES

2 . 100 R 'ooo'

43 10.01 PLACES

Figure C.15: Top view of tuner base.

I
147

15.0° ^0.1

0. 126 D I A ±0.001

3 PLACES

F I LLET

. 047 R i0.oo2

6 PLACES

15.0° io.i

. 572 ±0.005

2 PLACES

2. 75 R ±0.01

. 8000 R '0 o°o5

. 700 R '0.001 FOR MICROMETER DRIVE CUP

1.31 '0°

6-32 BOTTOM TAP

3 PLACES

Figure C.16: Bottom view of tuner base.
148 APPENDIX C. 60-MM CAVITY DRAWINGS

. 25 ±0.01

. 35 *o.oi 3 PLACES

. 094 D I A ±0.001 3 PLACES

~l^kfF

FLAT TO 0.0001"

6 PLACES

. 030 ±0.001

. 50 ±0-01 J

I T ii4 T \— "nlit^i ~~

I I'

l - + -r 1-1

tr - ^

1 75 ±0.01 2. 125

±0.005 2.50

lO.Ol

Figure C.17: Side view of tuner base.
149

BOTH RADII USING 16 PLACES

SMALLEST MILL AVAILABLE

6-32 BOTTOM TAP

6 PLACES

1

1

t i , 090 'oooi ^

2 PLACES %

2 PLACE

0.400 •<>

. 250 R -o-oo^

. 6099 '0 000'

GNMENT PIN

S

40 '0 °o' 0.6396 '0 0001

ACES 4 PLACES

USE PRECISION

AL IGNMENT PI

SPEC I

ICAT I ONS

CUTAWAY S IDE V I EW

. 39 3 '0.005 TOP VIEW

vm 4j|_4^^

0.460 '0 005

iMi I <:i—rrrnri—q

tm V'i4J'f"l-,t-'t

. 040 '0 001 2 PLACES

. 200 »o.ooi 2 PLACES

CUTAWAY SIDE VIEW SIDE VIEW

Figure C.18: Coupling endplate.
150 APPENDIX C. 60-MM CAVITY DRAWINGS

UJo

C/)

CD

in

I

1^o

CD

o

o

Ql

z>

(/) 00

LlJ DC

CL

crO

o

CLo

o

UJo

o <

5 Li_ 6 ct: < CO Q Q

UJ O _j

o en ^ Z)

• z

o ^

X

oo

•-o oo<

oo

00o

UJ>

CLo

°-^

CNO

o

o

o

M / o / "*" o / "*"

/ o

/ - h

-J^'jTZ ffl

Dd

<

X

o

CNO

X

X

UJ>

UJQ

CO

O

m

m

o

CL

a:

Z)

<

a:

UJ

Q_

00 \

Q

UJQ

<

UJ

lO

O ce

z>

Figure C.19: Temperature-probe bolt and coupling-endplate pressure ring.
<

X

on

151

o

o

Hm

o

->

UJ

>

UJQ

00

CO

O o

CNO

O

-H o

o

in

o

oO

o

CN

>

o

o

CD

Figure C.20: Tuner-endplate.
152 APPENDIX C. 60-MM CAVITY DRAWINGS

0.875

0. 125 ±0.002/

. 740 ±0.002

i-^ 0.367 ±0.002

. 750 ±0.002

^

/

^'' 1 . 1000 ±0.0020

^1.1 880 ±0.0005

TYPICAL DESIGN FOR

MICROMETER DRIVE HOLDER

. 75 ±0.01

THREADED -

Dl A

. 875 ±0.005

1 . 25 ±0.05

Dl A

.50 ±0.05

. 500 ±0.005

Dl A

BUSHING FOR TUNER BASE

1 .00 DIA

0.50 DIA

THRU HOLE

8-32 ALLEN HEAD

THREADED HOLE

TOP VIEW

0.414

0.75 1^-

0.375

SIDE VIEW

BACKSTOP FOR TUNER ASSEMBLY

HEX HEAD

0.750 -

THREADED

0.500

0.250

1 .000

DIA

KNURLED

0.500 -

DIA

Ul

0.250

TUNER BASE TORQUE NUT

Figure C.21: Tuner assembly backstop, torque nut, micrometer-drive holder and

sliding bushing.
Lb6

/ . UU _o. 10

DIA

5 . 5000 ±0.0002

V-y//Y/////A

2 .08 to. 01

1 . 31 ±0.01

1

. 06 ±0.01 R

2 . 85 ±0.01 4

R

-< >

. 25 ±0.01

2 . 88 ±0.01

R

. 7992 ±0.0002

3 PLCS

EXTENSION FOR TUNER-ROD BEARINGS

0.500 DIA

0. 189 DIA

2-56 ALLEN HEAD

THROUGH HOLE

THREADED HOLE

TOP VIEW

0.083

0.250 0. 123

0. 125

. 188 ±0.001

SIDE VIEW

HARDENED STEEL GUIDE SHAET backstop for tuner spring

Figure C.22: Tuner guide shaft, spring backstop and extension.
154 APPENDIX C. 60-MM CAVITY DBAWINGS

120. 0° ^0-1 3 PLACES

F ILLET

6 PLACES

. 250 R ±0.002

. 375 R 0-002 1.065 *o.oo2

6 PLACES

. 500 *o.oo2

3 PLACES

. 094 R '0 001 3 PLACES

. 8000 R io.0005

TOP VIEW

. 370 to. 005

1

f

I—?i

^^

2 . 500 '0 002 3 PLACES

S I DE V I EW

Figure C.23: Measurement-micrometer reference yoke.
155

Other resonator parts not shown are:

1. 1 ea. 60-mm diameter helical waveguide.

faced off perpendicular to axis.

outside diameter turned to be concentric with inner diameter.

ends flat and perpendicular to axis to within 3 ^m (0.0001 in).

2. 1 ea. 4 in outside-diameter, 3.5 in inside-diameter acrylic plastic tubing

with tangential inlet and outlet ducts at opposite ends.

3. 1 ea. plug for tuner-base water channel.

4. 6 ea. 2-56 machine screws for coupling-endplate pressure ring.

5. 3 ea. 2-56 set-screws for tuner-endplate.

6. 3 ea. Allen-head | — 20 bolts to attach stand to tuner flange.

7. 1 ea. 10-32 knurled screw for micrometer attachment.

8. 2 ea. ]^-in inside flange o-rings.

9. 2 ea. ^-in outside flange o-rings.

10. 1 ea. temperature probe o-ring.

11. 3 ea. |-in diameter tie rods. Both ends threaded ^ — 20. For attaching

coupling and tuner flanges.

12. 3 ea. washers and 1 — 20 tie rods.

4

13. 1 ea. base for resonator stand. 18-in diameter, 1-in thick aluminum with

3-in diameter center-hole and 3-|-in diameter holes located 120° apart at a 2. 1-in radius.

14. 3 ea. stands to connect from base to tuner flange. 12-in long, | ± 0.01

in diameter stainless steel rods.
Appendix D

MEAS_RES Program Listing

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