Technical Report ECOM- 0138- 21- T Reports Control Symbol OSD- 1 366 February 1971 IMPEDANCE CHARACTERIZATION OF A WAVEGUIDE MICROWAVE CIRCUIT C.E.L. Technical Report No. 208 C ontract No. DAAB07- 68- C- 01 38 DA Project No. 1H021101 A042.01.02 Prepared by Robert L. Eisenhart COOLEY ELECTRONICS LABORATORY Department of Electrical Engineering The University of Michigan Ann Arbor, Michigan for U.S. Army Electronics Command, Fort-Monmouth, N.J. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior approval of CG, U. S. Army Electronics Command, Fort Monmouth, N. J. Attn: AMSEL-WL-S. THE UNIVERSITY OF MICH!GAN ENGINEERING LIBRARY

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ABSTRACT The induced e. m.f. method has been extended and applied to derive the driving point impedance of a common waveguide structure used for mounting small microwave devices. The resulting mathematical relationship has been conceptually interpreted as an equivalent coupling circuit, terminated by a set of impedances which are associated with the many modes within the waveguide. Properties of this circuit and its terminations are discussed in detail. In addition the multilateral nature of the circuit allows consideration of the mount in the waveguide as an obstacle to any incident propagating mode. The driving point impedance of this mount was also considered from the experimental viewpoint. An investigation was carried out to check and support the results of the theoretical analysis. A novel measurement technique was employed, based upon the use of subminiature coaxial line to gain electrical access to a terminal pair located inside the waveguide. An extensive model of the measurement circuit was developed, which enhanced the accuracy of the data interpretation, and provided excellent agreement between these values and the theory. Measurements of the mount as an obstacle to the H10 mode ~~111~~10 iii

It is anticipated that this formulation will permit accurate design of many components which previously required empirical methods based on limited experimental data.

FOREWORD This report was prepared by the Cooley Electronics Laboratory of the University of Michigan under United States Army Electronics Command Contract No. DAAB07-68-C-0138, "Countermeasures Research. " The research under this contract consists in part of an investigation to develop operational solid-state components in microwave circuits. The material reported herein represents a summary of a theoretical and experimental study which was made to determine the impedance characteristics of a commonly used waveguide mounting structure.

TABLE OF CONTENTS Page ABSTRACT iii FOREWORD v LIST OF ILLUSTRATIONS viii LIST OF SYMBOLS xi LIST OF APPENDICES xx CHAPTER I: INTRODUCTION 1 1.1 Statement of the Problem 1 1. 2 Topics of Investigation 4 1. 3 Review of the Literature 6 1. 4 Reporqt Organization 8 CHAPTER II: THEORETICAL ANALYSIS OF THE MOUNT 10 2.1 Introduction 10 2.2 General Analysis Procedure 10 2.3 Post Mount Analysis 13 2.3.1 Dyadic Green's Function 13 2.3.2 Expansion of J(r) 16 2.3.3 Determination of EG) 18 2.3.4 Expansion of EA 18 2. 3. 5 Spatial Harmonic Equations 20 2.3. 6 Determination of Impedance Components 22 2.3.7 Mode Impedances 25 2.4 ZR Low Frequency Limit 30 CHAPTER III: PROPERTIES OF THE EQUIVALENT C IRC UIT 35 3.1 Introduction 35 3.2 Convergence Properties 35 3.3 Assumptions and Error 37 3.4 Terminated Waveguide 41 3. 5 Multiport Characteristics 44 3.6 Impedance Characteristics 46 vi

TABLE OF CONTENTS (Cont.) Page CHAPTER IV: EXPERIMENTAL DEVELOPMENT 55 4. 1 Introduction 55 4.2 Equipment Development 55 4.2. 1 Mount Design and Construction 57 4.2.2 Waveguide Terminating Considerations 59 4. 3 Measurement Circuit Modeling 60 4. 3. 1 Statistical Comparison Technique 61 4. 3.2 Coaxial-Radial Line Transformation 64 4. 3. 3 Effects of Circuit Modeling 66 4. 4 Measurement Procedure 66 CHAPTER V: COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS 69 5. 1 Introduction 69 5.2 Driving Point Impedance Comparison 69 5. 3 Waveguide Obstacle Reactance Comparison 74 5. 3. 1 Post Inductance 76 5. 3. 2 -Tuned Post 81 CHAPTER VI: REVIEW, CONCLUSIONS AND RELATED FUTURE STUDY 84 6. 1 Introduction 84 6.2 Review 84 6. 3 Conclusions 86 6. 4 Suggested Areas of Related Future Study 87 6. 4.1 Theoretical Study 87 6. 4.2 Experimental Study 88 6. 4. 3 Application of the Circuit 88 APPENDICES 89 RE FERENCES 124 DISTRIBUTION LIST 127 vii

LIST OF ILLUSTRATIONS Figure Page 1.1 Typical mount configuration with device 2 mounted at the bottom of the waveguide. 2. 1 General mount configuration with description of parameters. 14 2.2 Circuit relationships. 23 (a) Generalized circuit for gap driving point impedance ZR. (b) Coupling network for a typical parallel set. 2.3 Mode pair impedance plot. 27 2. 4 Parallel effect of waveguide arms. 29 2. 5 Equivalent circuit of post mount. 31 2. 6 Cross-sectional view of (a) Waveguide mount for TEM mode (b) Standard stripline 32 3.1 Truncation error for Z. 37 n 3.2 Impedance comparison plot. C-Band waveguide a = 4.76 cm, b = 2.215 cm, s' = 0.500, h' = 0.0, w' = 0.115, g' = 0.069. M and N represent the number of terms retained in summing the respective series. 40 3.3 Post obstacle circuit for incident H10 mode. 45 3. 4 Driving point impedance for gap position (h') variation with s' = 0.333, w' = 0.115, g' = 0.069, a = 4.76 cm, b = 2.215 cm. 50 (a) Resistive component. (b) Reactive component. viii

LIST OF ILLUSTRATIONS (Cont.) Figure Page 3. 5 Driving point impedance for waveguide height (b) variation with h = 0. 076 cm, g = 0. 152 cm, w' = 0. 115, s' = 0. 500, a = 4.76 cm. (Note: h' and g' vary for each curve since normalized to b. ) 51 (a) Resistive component. (b) Reactive component. 3. 6 Driving point impedance for post position (s') variation with h' = 0.250, w' = 0.115, g' = 0.069, a = 4.76 cm, b = 2.215 cm. 52 (a) Resistive component. (b) Reactive component. 3. 7 Normalized obstacle reactance for gap size g variation in C-Band waveguide. a = 4.76 cm, b = 2.215 cm, s' = 0.500, w' = 0. 115. 53 4. 1 General mount configuration. 56 4.2 Measurement mount. 58 4. 3 Measurement circuit equivalent model. 62 4. 4 Coaxial - radial line transformation. 65 (a) Physical configuration. (b) Equivalent circuit. 4. 5 Measurement circuit modeling comparison for the driving point impedance. 67 (a) Resistive component. (b) Reactive component. 5. 1 Driving point impedance comparison - theoretical and experimental s' = 0. 500, h' = 0.035. 70 5. 2 Driving point impedance comparison - theoretical and experimental s' = 0. 500, h' = 0.500. 71 ix

LIST OF ILLUSTRATIONS (Cont.) Figure Page 5.3 Driving point impedance comparison - theoretical and experimental s' = 0. 250, h' = 0. 500. 72 5. 4 Driving point impedance comparison - theoretical and experimental s' = 0. 333, h' = 0.250. 73 5. 5 Mount obstacle equivalent circuit. 75 5. 6 Post cross-section comparison. 77 5.7 Normalized flat post reactance w' = 0. 058. 78 5.8 Normalized flat post reactance w' = 0. 115. 79 5.9 Normalized flat post reactance w' = 0.230. 80 5. 10a Normalized obstacle reactance for gap size g variation, s' = 0.500, w' = 0.115. (Theory) 82 5. 10b Normalized obstacle reactance for gap size g variation, s' = 0.500, w' = 0.115. (Experiment) 83 A. 1 Coordinate description for the rectangular waveguide. 91 C. 1 Gap impedance representation for use in the computer program. 103 D. 1. Line length equivalence for a compensated discontinuity. 115 (a) Discontinuity model. (b) Equivalent length of Z line. D. 2 Compensated step discontinuity in coaxial line. 117 E. 1 Standard circuit unit for data interpretation program. 120 x

LIST OF SYMBOLS Defined by or Symbol Meaning first used in A Region at z = 0 containing the post Fig. 2.1 A+, A Arbitrary coefficients Eq. B. 2 AY Current y-distribution normalized expansion coefficient Eq. 2. 6b Af Current x-distribution normalized expansion coefficient Eq. 2. 6c a Width of the waveguide Fig. 2. 1 B Region at z = 0 not containing the post Fig. 2.1 B, B Arbitrary coefficients Eq. B. 2 BQy Current y-distribution normalized expansion coefficient Eq. 2. 6b B Current x-distribution normalized expansion coefficient Eq. 2. 6c b Height of waveguide Fig. 2. 1 C1, C2 Effective discontinuity capacitances Fig. E. 1 CD Discontinuity capacitance Fig. D. la Cfl, Cf2 Fringing capacitances in the coaxial radial line transformation Sec. 4. 3.2 Ck Coaxial line capacitance per unit length Eq. D. 3 C Gap series capacitance Fig. C. 1 xi

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in C Gap parallel capacitance Fig. C. 1 p C Parallel plate capacitance in the radial line Sec. 4. 3.2 d Diameter of a circular post Sec. 2.3.2 E Electric field vector Eq. 2. 1 EA Gap electric field vector Sec. 2.2 EA nth harmonic of EA Eq. 2.14 n E Transverse magnetic mode mn designation Sec. 2.3.1 En nth harmonic of E Eq. 2.14 F(y) General distributed quantity Eq. 2. 13 f Frequency in GHz Sec. 4.3 G(rl r') Dyadic Green's function Sec. 2.2 G(rl r') y y portion of G(rl r') Sec. 2.3.1 GE(r r') Emn mode portion of G(rl r') Sec. 2. 3. 1 GH(rlr') Hmn mode portion of G(rlr') Sec. 2.3.1 GT(rI r') Green's function for terminated waveguide Sec. 3.4 G (z I z') Factor of G(rl r') which is a Z function of z Eq. B. 12 g Gap size Fig. 2.1 xii

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in g' Normalized gap size = g/b Fig. 2. 1 go(z I z) Free space one-dimensional Green's function Eq. B. 11 gT(Z I z') Green's function for terminated onedimensional line Eq. B. 1 H Transverse electric mode designation Sec. 2. 3.1 mn h Gap position (center from bottom) Fig. 2. 1 h' Normalized gap position = h/b Fig. 2. 1 I Total gap current Eq. 2. 16 In nth harmonic gap current Eq. 2.15 6(r - r') Unitdyadat r = r' Eq. 2.4 J Current density vector Eq. 2. 1 J~n nth harmonic of J Eq. 2.14 J Current density scale factor Eq. 2. 6a 0 j Complex coefficient = v Eq. 2. 1 K Scale factor for ~ Eq. 3.1 m K' Scale factor for Eq. 3.2 n k Wave number (phase constant) Eq. 2. 1 k Waveguide wave number (phase g constant) Eq. A.20 xiii

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in 2 2 2 k Parameter = k + k Eq. A. 9a mn x y k m - eigenvalue parameter Eq. 2. 5 x k n - eigenvalue parameter Eq. 2. 5 y L Gap series inductance Fig. C. 1 L1 Effective discontinuity inductance Fig. E. 1 L Transition inductance for coaxialc radial line model Sec. 4. 3.2 LEN Measurement circuit effective line Fig. E. 1 length L, Coaxial line inductance per unit length Eq. D. 3 L Discontinuity compensating 0 inductance Fig. D. la L Transition inductance for coaxialradial line model Sec. 4.3.2 Compensating length Eq. D. 4 1,' 2 Distances to terminations 1 and 2 from post mount plane z = 0 Eq. 3.6 M Number of terms in, Sec. 3.2 m M Vector function Eq. A. 5 m x-distribution eigenvalue index Sec. 2. 3. 1 m Vector function Eq. A. 9a xiv

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in N Number of terms in E Sec. 3.2 n N Vector function Eq. A. 5 n y-distribution eigenvalue index Sec. 2. 3. 1 n Vector function Eq. A. 9b A n Surface normal unit vector Eq. A. 7 P Arbitrary vector coefficient Eq. A. 10 Pn nth harmonic power flow at the gap Eq. 2. 15 PR Total power at the gap Eq. 2.16 R Q Arbitrary vector coefficient Eq. A. 10 R Gap series resistance Fig. C. 1 RR Resistive part of ZR Fig. 3. 2 r General field point Sec. 2.2 r' General source point Sec. 2.2 S General vector Eq. A. 4 S EA - field y-distribution normalized expansion coefficient Eq. 2. 10b s Post position (center from side) Fig. 2. 1 s' Normalized post position = s/a Fig. 2.1 T Simplifying factor Eq. A. 12 mn t Time/or compensating length Sec. 2.2/ (use is clear from context) Fig. D. 2 xv

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in u(x) Current x-distribution function Eq. 2. 6a u(y) Current y-distribution function Eq. 2. 6a V Voltage across the gap Sec. 2. 3. 4 ta(x) EA - field x-distribution function Eq. 2. 10a (y) EE - field y-distribution function Eq. 2. 10a w Post width (flat strip) Fig. 2. 1 w' Normalized post width = w/a Fig. 2. 1 XL Post inductive reactance for H10 mode Eq. 3. 10a XIBS Obstacle reactance in shunt across the waveguide normalized to the H10 mode Fig. 5.6 XR Reactive part of ZR Fig. 3. 2 x Rectangular coordinate unit vector Fig. 2. 1 YG Transformed gap admittance Eq. 3. 10b Y R Gap driving point admittance Eq. 2. 17b YR YR Approximate value of YR Eq. 3.3 Y' (n40) Reactive effect of n > 0 modes in R the H10 obstacle circuit Eq. 3. 10c A Rectangular coordinate unit vector Fig. 2.1 Z(cv) General frequency dependent impedance function Sec. 1.1 xvi

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in Z Characteristic impedance of onedimensional line Eq. B. 3 Z Waveguide mode (m, n) characteristic impedance Eq. 2. 24 Z Stripline characteristic impedance Eq. 2.26 CS ZE Impedance component due to E modes Eq. 2.21c ZG Terminal impedance of device in the mount Sec. 3.5 ZIN General input impedance at a IN terminal Eq. 2. 27 ZH Impedance component due to H modes Eq. 2.21b Z jmn Terminating impedance on waveguide arm j normalized to Z Eq. 3. 9 c mn ZQ1 Terminating impedance at l on one-dimensional line Eq. B. 3 Z Mode pair impedance Eq. 2.21a mn Zn nth harmonic impedance component Sec. 2. 3. 5 Z' Approximate value of Z Eq. 3.4 Z)BS Obstacle impedance in shunt across the waveguide, normalized to the H1i mode Sec. 5.3 ZR Gap driving point impedance Sec. 2.3 xvii

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in ZTmn Terminated mode pair impedance Eq. 3.7 Tmn A z Rectangular coordinate unit vector Fig. 2. 1 a Arbitrary scalar coefficient Eq. A. 17 Arbitrary scalar coefficient Eq. A. 17 F Attenuation constant for oneg dimensional transmission line Appendix B r Waveguide attenuation constant Eq. 2. 5 m n'Y General phase constant Eq. A. 4 A Err-or in Y- Eq. 3.3 A' Error in Z' Eq. 3. 4 n ~~6 _Mode coefficient Eq. 2.5 Waveguide phase parameter Eq. A. 8 77 Free-space impedance = 120ir ohms Eq. 2.9 o Post width parameter Eq. 2.8 0 Truncation parameter = argument of finalterm of Sec. 3.2 m K Gap coupling factor Eq. 2.22b gn K Post coupling factor Eq. 2. 22a pm x Free-space wavelength Eq. 2.25 xviii

LIST OF SYMBOLS (Cont.) Defined by or Symbol Meaning first used in Xg 9Waveguide wavelength Eq. 2.25 /O~ Permeability of free-space Eq. 2.1 P1, P2 Complex voltage reflection coefficient at terminals 1, 2 on onedimensional line Appendix B Plmn7 P2mn Complex reflection coefficients for terminations 1 and 2 Eq. 3. 6 T Green's function termination parameter Eq. 3.6 T. Single waveguide termination parameter Eq. 3.8 0p Gap size parameter Eq. 2.12 411' 42 Scalar wave equation solutions Eq. A. 6 co Angular frequency Eq. 2. 1 xix

LIST OF APPENDICES Page APPENDIX A: DETERMINATION OF THE DYADIC GREEN'S FUNCTION FOR RECTANGULAR WAVEGUIDE 89 APPENDIX B: DETERMINATION OF THE GREEN'S FUNCTION FOR TERMINATED' WAVEGUIDE 97 APPENDIX C: COMPUTER PROGRAM FOR'THEORETICAL IMPEDANCE CALCULATIONS 102 APPENDIX D: DETERMINATION OF APPROXIMATE AND LIMITING VALUES FOR SMALL COAXIAL LINE DIS CONTINUITIES 113 APPENDIX E:' COMPUTER PROGRAM FOR EXPERIMENTAL DATA INTERPRETATIONS 119 xx

CHAPTER I INTRODUCTION 1. 1 Statement of the Problem This report is concerned with the impedance characterization of the microwave structure shown in Fig. 1. 1, commonly used for mounting small microwave devices in shunt across a waveguide. The general term "impedance characterization" implies complete knowledge of the driving point and transfer impedances associated with and between each and every entry or terminal port of the mount. This information is best displayed by development of an equivalent circuit, representing the effects of the electromagnetic fields within the region of the mount. Once such a circuit is established, standard circuit analysis techniques can be applied when using the mount. In the theoretical analysis of parametric amplifiers and frequency converters, general impedance functions Z(wco) are assumed to be known and are utilized accordingly as parameters when determining such quantities as gain, bandwidth, stability and noise figure. When designing low frequency (< 100 MHz) circuits, the determination of the various impedance functions is normally straight forward and presents no particular problem. However, this is not the case when the frequencies of interest fall within the - 1

-2ILLL,> mi.crowave d evice Fig. 1. 1 Typical mount configuration with device mounted at the bottom of the waveguide.

-3microwave region. In particular, when using a structure or microwave circuit such as shown in Fig. 1. 1, determination of impedance seen by the device becomes very difficult due to the complex nature of the electromagnetic fields which are involved. The question is then asked, why use such a complex structure? The answer comes from experimental work which has shown this configuration to work well for many applications because of the strong coupling between the "post" current and propagated energy within the waveguide. The motivation for this topic came from work done by myself on parametric up-converters designed around such a waveguide circuit. It became apparent that lack of an adequate circuit description was sufficient to completely remove any chance of predictable success with the circuit. Although initial interest in this structure was for use in parametric circuits (especially wide-band design), it was obvious a complete description would have great significance in the design of circuits for a variety of applications. In particular this method of mounting solid-state microwave source elements is very common. The mode of operation of a transferred-electron oscillator (Gunn diode) has been shown to be strongly dependent on the circuit characteristics (Ref. 1) Also, studies of the TRAPATT mode of avalanche diode operation indicate that the impedances at harmonics

-4of the oscillation frequency affect the power output (Ref. 2). Frequency multipliers represent another area where the operation is influenced by impedances at frequencies other than the input and output (Ref. 3). The utility and application of the work described in this report may be expressed through the following quotation, taken from a design paper by Getsinger (Ref. 4). "In order to reduce the approximations involved in design work it is necessary to be able to describe the circuit under consideration in mathematical terms. In order to [ conceptually] relate the circuit to the real world, it is necessary to be able to interpret the circuit in terms of a physical configuration with reasonably accurate correspondence between the predicted behavior of the circuit and the measured behavior of the physical structure. This is particularly apparent in the microwave region where the electromagnetic fields are distributed throughout the entire structure constituting the circuit, rather than, as at low frequencies, being confined to individual circuit elements... Thus, the microwave engineer often tends to think more in terms of physical structures than in terms of conventional circuit elements. A major portion of his work is in selecting appropriate physical structures and finding dimensions which will cause the structure to yield the desired performance as a microwave component... Since microwave networks are made up mostly of distributed elements, impedance may be defined only at terminal surfaces. However it is possible to describe microwave network impedance variation with frequency at some terminal surface in terms of lumped-element mathematics..." 1. 2 Topics of Investigation The impedance characterization sought is determined by a theoretical approach, based on development of a formulation derived from a solution to Maxwell's equations. This theoretical

-5analysis is supported by an extensive experimental effort. Both the theoretical and experimental analyses are dependent upon new techniques or procedures which are developed. The theoretical analysis is discussed in detail. A general procedure is first developed, then applied to the specific problem outlined. All of the mount configuration parameters are left as variables, so that their significance in the resulting characterization can be determined. Of major importance in this study is the development of a thorough understanding of how these various parameters affect that characterization, allowing the possibility of some success in impedance synthesis, which is the key to successful mount design. To enhance this understanding many graphs are presented, representing various sets of parameter values, accompanied by detailed discussion of the dominant characteristics. The initial objective was to determine the impedance seen at the terminals of the mounted device. Once completed, it was possible to interpret the resulting formulation as an equivalent circuit relating the device terminals to a set of impedances representing all the possible modes in the waveguide. The terminals associated with the propagating modes can be considered as input ports for each respective mode, thus allowing description of the mount as a load to an incident mode. This capability results in the complete characterization desired.

-6The experimental analysis was carried out to check and support the results of the theoretical work. Each aspect of this analysis is discussed from the design of the necessary equipment to the final results of data interpretation. Among the several items included are measurement circuit description, coaxial to radial line transformation, and multimode matching considerations. 1. 3 Review of the Literature Prior to World War II, microwave technology was not the subject of extensive research, primarily because the state of the art in frequency sources, amplifiers and other general components was not sufficiently advanced to include a wide availability of devices at such high frequencies, (> 1.0 GHz). However with the advent of radar and its strong resolution dependence upon frequency, coupled to the wartime urgency, a large scale research and development requirement appeared on the scene. Through the concentrated effort of many scientists, engineers and associate coworkers the fundamentals of microwave technology as it is known today were established and documented (Ref. 5). During and following this period, very few solid-state devices were available for the microwave region. In June 1958, a paper published in the IRE Proceedings did much to introduce solid-state diodes to the microwave world (Ref. 6). This paper discussed the various properties of the diodes and uses to which their characteristics could be put.

-7Soon afterwards many technical papers appeared (Refs. 7- 12). In general, these represent theoretical analyses of different types of amplifier and frequency converter circuits (includes harmonic generation) to determine characteristics such as gain, bandwidth, noise figure, stability and frequency conversion efficiency. All are quite restricted by use of many assumptions, one being the use of filters to control energy flow in the circuit. For coaxial and stripline applications this is reasonably accurate; however, with waveguide no comprehensive circuit description has been outlined and such filters are hard to realize, presenting a major obstacle to the application of this theoretical knowledge to practical waveguide circuits. Getsinger, in 1966 and 1967 suggested a relatively simple equivalent circuit for the waveguide mount (Refs. 13, 14). A more complete description was discussed by Yamashita and Baird (Ref. 15) using the variational approach in solving for the impedance, considering the post as a radiating antenna element. Then Hanson and Rowe (Ref. 16) following a similar procedure to that of Yamashita and Baird introduced a coaxial line as a tuning stub for the circuit. An equivalent circuit was developed and used to help analyze the operation of the oscillator being investigated. These analyses have treated the impedance characterization of the mount but all have imposed unacceptable restrictions on the range of mount parameters and frequency. Usually the post

-8is considered to be located in the center of the waveguide, and the device is positioned at the bottom of the guide. In addition the waveguide height is often reduced for matching purposes, and the frequency range is restricted to that of the dominant mode. All of these restrictions are removed in the following analysis, which is therefore of considerable generality. No previous work was found discussing the measurement of the device terminal impedance. This is no doubt due to the inaccessability of these terminals, using standard measurement techniques. 1. 4 Report Organization Chapter II presents the theoretical analysis both as a general procedure and as an application of this procedure towards the resolution of the impedance characterization, resulting in the determination of an equivalent circuit for the mount. This chapter is supported by Appendix A, which contains the detailed development of the waveguide dyadic Green's function. This function is necessary to describe the electric field within the guide relative to an arbitrary current element. An interesting low frequency limiting property of the circuit is also discussed.

-9Chapter III points out various general properties of the equivalent circuit developed in Chapter II and considers the effect of having terminations other than a match on the waveguide ports. The necessary mathematical modifications to include the terminations are contained in Appendix B. The chapter is concluded with a comprehensive discussion of the mount impedance as it is affected by changes in the various mount parameters, using graphs to illustrate key points. Appendix C is a short discussion of the computer program used in determining the graphs. Chapter IV deals with the development of the equipment, techniques and procedures necessary to perform the measurement of the mount impedance. Appendices D and E support this chapter by respectively expanding the discussion of the measurement circuit and the data interpretation program. Chapter V presents the results of the experiment and compares these results to the theoretically predicted values. Chapter VI summarizes the report with a review consolidating the major points of the thesis to establish a perspective for the conclusions and discussion of possible areas of related future work.

CHAPTER II THEORETICAL ANALYSIS OF THE MOUNT 2. 1 Introduction The objectives of this chapter are to 1) develop a mathematical formulation for the terminal impedance seen by a device mounted as shown in Fig. 1.1, and 2) to successfully interpret that formulation into an equivalent circuit representing the distributed circuit effects as lumped elements. It is anticipated that such an equivalent circuit will enhance understanding of the mount by providing a means to conceptually associate various circuit elements with mount characteristics. 2. 2 General Analysis Procedure The procedure utilized here is based upon an extension of the induced e. m. f. method of Carter (Ref. 17). It is presented here in a form applicable to determination of the driving point impedance of an antenna located in a region having a defined boundary, through solution of the field equation - 2 V x V x E - k E =-j, J (2. 1) where E and J are the electric field and current vectors and an ejt time dependence has been assumed. For convenience, the -10

-11procedure is described in the following series of steps: Step 1: Determine the dyadic Green's function G(r I r') for the region. This is expressed in the form of a series of orthogonal functions defined by the region. Step 2: Express J(r) in a general set of orthogonal functions similar to those used in the expansion of G(rI r'). Step 3: Using the equation: E(r) = -j w f G(rl r') ~ J(r) dv' (2.2) determine an expression for the electric field intensity valid anywhere within the region. Step 4: Develop an expression for the electric field EA at the antenna feed. The Lorentz Reciprocity Theorem (Ref. 18) in a condensed form representing the case with perfectly conducting boundaries, establishes a direct relationship between the antenna feed EA field expression and the general E( r) described by step 3. Consider a source current element J(1) at point 1 inside the defined region which generates a field value E(2) at point 2. Conversely if a current element J(2) aligned with E(2) generated a value E(1) at

-12point 1, then f E(1). J(1) dv = E(2) J (2) dv. (2. 3) V V If desirable, the E - field elements can be considered the source functions and the current densities as induced, without changing the relationships in (2. 3). Step 5: Consider then the field EA from step 4 as the source function at point 1 and E(r) a general field value at point 2. Relating appropriate expansion terms through (2. 3) results in an infinite set of equations, each representing one of the spatial harmonic components defined by the orthogonal function expansion. These equations are individually interpreted to represent the equality between the power incident at the antenna feed point, and that radiated by the antenna for each harmonic being considered. Using the fact that the sum of each side of the infinite set of equations described represents the total power applied to the antenna terminals, an expression is found for the input impedance (antenna driving point impedance) as the sum of impedance terms representing all of the possible spatial harmonic components. An equivalent circuit is then obtained, providing interpretation of the impedance.

-132. 3 Post Mount Analysis The general procedure described in the preceding section is here applied to analysis of the post mount shown in Fig. 2. 1. Attention is directed initially to a mount in a waveguide which is infinite in the axial + z directions, and the results are modified to take account of terminations in Section 3. 4. The antenna driving point impedance ZR determined by this analysis represents the impedance seen at the terminals of a device located in the post mount. 2. 3. 1 Dyadic Green's Function. The dyadic Green's function G(r I r') required here is the solution of the equation V x V x G(rl r') - k2 G(rl r') = I 6(r - r') (2.4) for the waveguide boundary conditions (Ref. 19). I 6 (r - r') represents the unit dyad at r = r'. Each one of the nine components of G(r I r') may be interpreted as represent-A A A ing coupling between one component of J(x, y, z) and one of E(x, y, z). However the orientation of the post parallel to the yaxis limits J to only a y-component and also necessitates considering only the y-component of the resulting E field. Therefore only AA the yy-component of G(r r') is required here.

- 1 4A / a width of waveguide b height of waveguide h gap position (center from bottom) s post position (center from side) w post width (flat strip) g gap size s' = s/a normalized post position w' = w/a normalized post width h' = h/b normalized gap position g' = g/b normalized gap size Fig. 2.1 General mount configuration with description of parameters

-15i.e., G(rl r') y G(rl r') = yy [G (rl r') + GE(rl r') where GH(rl r') represents coupling from J(r') to Hmn waveguide modes and GE(rl r') represents coupling from J(r') to E waveguide modes. mn This Green's function derived in Appendix A, is dependent upon two independent eigenvalues which are related to the dimensions of the waveguide. Since a complete solution requires the inclusion of all eigenvalues, a double sum results as shown in (2. 5). 00 co (2-60) (k2-k 2) e mn G(rl r') = total coupling = 2 m=1 n=O ab k r mn sin k x sin k x' cos k y cos k y' (2.5) x x y y where k =-ff k n_ k= 2_ x a y b' It, n=O 2 2 22 r (k +k -k), 6 mn x y o (O, nO0 with all dimensions specified in Fig. 2. 1.

-162. 3. 2 Expansion of J(r). Consider first the current density J (r). Rather than assuming a specific distribution it is more useful to expand the current in a general orthogonal set, with trigonometric functions to correspond to G(r I r'). This can be done by taking intervals 0 -- 2a, 0 -- 2b for the x and y directions respectively. Then J(r) = y J u(y) u(x) 6(z-o) (2.6a) o 2-6 u(y) = L b (AY cos tb + BY sin b (2. 6b) u(x) o(Ax cos + Bf sin ) (2. 6c) f =1 a f a f a'1l z = 0 6 (z-o) = 0, otherwise with A and B as normalized expansion coefficients. While it is desirable to leave the y distribution in this general form, the x distribution may be specified more precisely. In particular, following the suggestion of Yamashita and Baird (Ref. 15) the circular post is here represented by an equivalent flat post or strip in the plane z = 0. An equivalent width (Ref. 20)

-17w = 2d, where d is the diameter of the post, was initially chosen but was subsequently reduced to w = 1. 8 d because of the proximity to the waveguide walls. This effect is discussed in Section 5. 3. 1. The current distribution is assumed to be constant across the width of the strip, although this distribution will actually be the sum of many components related to the modes present in the surrounding region. However this assumption should yield quite good accuracy since the width will normally be small (usually w' < 0. 25); this point is discussed further in Section 3. 3. Setting w 1, s <x<s+ u(x) = (2. 7) O, otherwise results in A = w cos f ( f) (2. 8a) f a B = w sin (T 0f) (2. 8b) where f2w f 2a

- 182. 3. 3 Determination of E(r). Substituting the expanded forms for G(rl r') and J(r) into (2. 2), and performing the integration, yields the equation 2 2 mn j 7 i j CO co (2-6) (k2 k ) e E(r) n abk n n=O m=l mn sin k x cos k y. (2. 9) x y The orthogonal properties of the integration remove dependence upon BY and AX 2. 3. 4 Expansion of EA. The assumption is made that a voltage V exists across the feed or gap g, thus specifying a conV stant spatial E - field = - g. For large gaps a set of spatially varying fields should be summed for an exact representation; however the gap considered here is sufficiently small that the approximation is good. This assumption is discussed in more detail in Section 3. 3. Considering an expansion for the field at z = 0 in the region A of Fig. 2.1, A V EA Y V (y) v(x) 6 (z-o) (2. 1Oa) EA =eg where

-190 (2-6 ) =Lb c~ S Cos (2. 10b) ~IP(Y) = ~ b Sp b p=o (x) = 1 (2. 10c) Sp = normalized expansion coefficient V = voltage across the gap. The y-distribution function s(y) is expanded only in cosine terms because the rectangular waveguide will not support y-directed sinusoidally varying (with y) E - fields. The field in region B,' shown in Fig. 2. 1, will not be considered further because the current J(r') does not exist in this region - consequently the E - field there makes no contribution to the power relationship which is to be developed. Describing the y-dependence then as 1, h 2< < h + (in the gap) (y) = (2.11) 0, otherwise (along the strip) which satisfies the zero condition along the strip, results in S= g cos bp7Ih (S P (2. 12) where 0p- = 2 for the expansion coefficient in (2. lob). p 2b'

2. 3. 5 Spatial Harmonic Equations. It is necessary to recognize that E(r), EA, and J(r) are all of the general form 00 o0 F(y) = F = F' cos (2. 13) n=O n=O When a component Ej of E(r') is considered as a source function at a point 1, the induced current at a point 2 must be of the same spatial harmonic, i. e., component J.. This follows from the relationship between E and J indicated in (2. 2). Therefore it follows that (2. 3) holds for each spatial harmonic of the quantities involved, or EA (1) Jn(l) dv = SEn(2) n(2) dv v n v for n = 0, 1, 2,.. o. (2. 14) The motivation for this separation of the general relationship (2. 3) into a set of equations will become more apparent as the analysis proceeds. Using (2.14), substitute in E(r), EA, and J(r) and integrate each side over the plane z = 0. The left hand side, representing the antenna gap field expansion becomes

-21a b /2-6 h sin0\ j j EA J dydx =- V J w Ay COs-. ( ) o o n nI V I =-Pn for n = 1, 2,. co (2. 15) n n or n In (2_ o_) AY J w cos k h ( 0) I~nb b n Y \ 0n / where P represents the power flow from the gap for the spatial n harmonic specified by the value of n, and Jn utilizes the description in (2. 7) for u(x) to correspond to to(x) in EA n The total radiated power PR will then be equal to the sum of all of these power terms and can be represented in terms of the gap voltage and total current I as 00 00 P =VI= P= V (2.16) R n n n=O n=O By defining Z Gap driving point impedance = I R T and P V2 th V n V Z - Impedance related to n harmonic = -- = n I 2 P n I n n we have

-22V2 = V = (2. 17a) R n=O n n=O n hence OR n Z YR - (2. 17b) R n=0 n This impedance relationship is represented by the parallel circuit of Fig. 2.2a. 2. 3. 6 Determination of Impedance Components. The values for the individual Z components are found by evaluating the right hand side of (2. 14) by integrating to get 22 2 2 2 2 2 71 JJ w (2- ) (k k2 Ay sin k s (sin 0 = n F O a k b m=l mn rn for n = 0, 1, 2,.. oo (2. 18) this time utilizing (2. 6c) for u(x) to correspond to E(r). Since P Zn 2 n it follows that

-23Z r Z, Z2 2 0 00 00 I =- In n=O (a) Inn PI- I -- ~* I - (b)..: - Fig. 2. 2 Circuit relationships (a) Generalized circuit for gap driving point impedance ZR (b) Coupling network for a typical parallel set.

-242 2 2 j 71 b(k k ) o sin k s sin 2 _ _ _ _ _ _ _ _ _ _ _ _ _ x _ _ _ n ~.sin h no 2 2 2 2 ( a k(2-6o) cos k ___ m1 y (h +) for n = O,1, 2,...o. (2. 19) The required impedance ZR is found by summation, in accord with (2. 17). It is very worthwhile to note at this time the lack of dependence of the impedance function Z on the assumed current distrin bution in the y-direction u(y). This agrees with the notion that both the current distribution and the impedance are independently determined by the physical configuration. However, as a consequence of the above development, the y-directed current expansion coefficients AY and hence the total current distribution can now be determined. n Using V I= n Z n and substituting from (2. 15) and (2. 19) results in

-25sin 0 V a k cos k hsin Y _n n 2 2 cc sin k s in 0 i J o wm=1 (k2- k k2) _ x n)2] for n = 0, 1, 2,...c o (2. 20) which can be substituted in (2. 6b). Equation (2. 20) has both real and imaginary parts representing respectively the current contributions along the post due to propagating and evanescent modes. 2. 3. 7 Mode Pair Impedances. It is clear from (2. 17) that the total impedance ZR is made up of the parallel connection of an infinite number of sets each one of which contains an infinite sum of terms, thus representing all of the possible modes in the waveguide. This formulation readily permits relating each impedance component to a specific mode pair impedance, defined as:'2 2 (k -k mn ak 2 2 k2 ZH ZE(2 21a) (2-s0) (k +k -k)2 o x y j 72 b k kx2 ZH a(2-6 o) k 2 + k 2 _ (2. 2kb) 2kx y (k +k -k) k

-26representing a series combination of the H and E mode conmn mn tributions for a (m, n) set. For frequencies below the cutoff frequency the impedance ZH contributed by the H mode is inductive and ZE for the Emn mode capacitive; consequently the combination has a resonant frequency, as shown in Fig. 2. 3. This resonance produces the zero in the reactive region which, as seen nvr from (2. 21), occurs when k = k. Since k = the zero is dey y b' pendent only on n and the guide height b. It should also be noted that (2. 21) is a function only of the waveguide dimensions a, b and the eigenvalues chosen. The equations for ZH and ZE are derived separately by using GH(rlr') or GE(rl r') respectively. They are not normally considered separately however, unless it is desirable to perhaps determine power levels of distinct modes. Z is more convenient mn as a composite effect in the circuit. The remaining terms in (2. 19) are then interpreted as coupling factors which determine the coupling of the strip and gap to a particular mode pair impedance, and which are a function of width and position of both strip and gap. In other words, all of the mode pair impedances Z exist for a given waveguide, but their coupmn ling to the mount is determined by the mount configuration. Defining Post coupling factor Km = sin k s (.. m (2. 22a) pm xs(si2Om)

-27Zmn= ZE + ZH +- Zmn'b ZE / \ ZH / / 0 | frequency // REACTIVE RESISTIVE REGION REGION (cutoff freq) Fig~ 2o 3 Mode pair impedance plot.

-28sin 0) Gap coupling factor K g cos k h (2. 22b) gn y On (2. 19) may be expressed as ZnZ ( n ( ) for n = 0, 1, 2,... c (2.23) m=1 gn which is shown schematically in Fig. 2. 2b. The symmetry of the post mount about z = 0 along the zaxis suggests separating Z into two parallel components, each mn representing the impedance associated with either the positive or negative z-direction. As shown in Fig. 2. 4 Z = Characteristic impedance of the waveguide = 2Z cmn mn (2. 24) In particular for the dominant H10 mode 10 Z 2 qb k 2 (' b (2.25) c10 a (k2 k 2 a \ x which agrees exactly with Schelkunoff's power-voltage definition of characteristic impedance (Ref. 21) for waveguide. Since the mode pair impedance can be associated directly with the impedances which the waveguide presents at the plane z = 0 for the + z-directions, it is most convenient to interpret Z as ~~~~~~~- ~mn

-2 9 Z m mn cron cmn Z direction Z direction Fig. 2. 4 Parallel effect of waveguide arms.

-30being a termination on the mount. With this in mind the equivalent circuit of the post mount is defined as that portion of the total circuit consisting only of the parallel sets of ideal coupling transformers which provide connection between the gap-post and all of the individual ports of the modes, exclusive of the terminations. In other words the equivalent circuit is strictly a coupling circuit defined at the plane z = 0 and therefore not containing any energy storage or dissipative capability. This equivalent circuit is shown in Fig. 2. 5. 2.4 ZR Low Frequency Limit R If the gap position is chosen to be adjacent to the bottom of the the waveguide, it is possible to consider the mount as a short length of transmission line with its axis in the y-direction, terminated by a short circuit at y = b (i. e., the top of the waveguide) (Ref. 22). The cross-section of this line would be represented by two infinite ground planes with a thin post as center conductor, as seen in Fig. 2. 6a. Standard strip-line, having a characteristic impedance of Zcs = 60 ln (w) (2.26) for a > 2.0, is shown in Fig. 2. 6b (Ref. 23). Fortunately, for such a restriction on w, the characteristic impedance is relatively

r -0 0--o 00 r —I-1: -IgO I Kg o Kpl: I, Kp: I: Km: I ZG'- r- - L- l- O t, o-'ZI 2 L IZ 01 Z I I L -o - T-~I J KPZ Kp~ K K:~ 12 I 1z22 P2 P2 2 o-r 7L 1 0-r I, o-r I -I'ft d ~~~~~IZ2,1 C/3 o- -T i — Kp3: I Kp3: I KP3 I O-~~~~~~~~~~~~~~~~~~~~~~~~~c "r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i~~~~~~~~~ I- I IZ309 I LZ_ _ L'-J L:_ I O.- -T - LT - Fig. 2. 5 Equivalent circuit of post mount0 0 Fig. 2. 5 Equivalent circuit of post mount

-32_/,, // / /,/ a I 1 T (a) / / /, /_./_/,. / a_ /-./ -/y/l// - (b) Fig. 2. 6 Cross-sectional view of (a) Waveguide mount for TEM mode (b) Standard stripline

-33independent (error < 3%) of the angular orientation of the center conductor with respect to the sides, so that (2. 26) also represents the characteristic impedance of Fig. 2. 6a. The impedance at the gap is then equivalent to the input impedance of this shorted transmission line, or ZIN = j Z tan(kl) (2.27) which becomes IN i ~ ln aw) (2.28) iN = bn) for 1 = b and kb << 1.0 in the low frequency case. Looking next at ZR we find that for k very small, Z for n > 1 becomes very large relative to Z (i. e., n = 0) such that -- 0 the parallel effects of Zn, n > 1, can be neglected in Fig. 2. 2a. Therefore 2 2 jb sin2 k s in 0 Z Z j_ _k x (SinO (2.29) R 0 a k (2.29) m=l x Placing the center conductor at s = a/2 to correspond to Fig. 2. 6a gives a further reduction of 2 0o sin 0 R X (-#;~) m=l, 3 m (2.30) odd

-34Using the trigonometric identity 2 l sin y = 2 (1 - cos 2y) (2.31) and the necessary series expansion relationship (Ref. 18, p. 580) results in R n (, (2. 32) showing very good agreement with (2. 28). This demonstrates that the reactive energy stored in this physical structure is, as expected, independent of the method of analysis.

CHAPTER III PROPERTIES OF THE EQUIVALENT CIRCUIT 3. 1 Introduction The objective of this chapter is to develop familiarity with the equivalent circuit through discussion of the various properties. Initially we review the more general properties such as convergence, error and reciprocity; then we conclude by considering the effect of variation in the circuit parameter values on the detailed impedance behavior. 3. 2 Convergence Properties Since the developed circuit (Fig. 2. 5) is made up of a doubleinfinite number of terms, convergence of the various impedance functions must be insured before practical application is possible. First consider the convergence of the set Zn for arbitrary n. For large values of m the terms in the series decrease as.' m7WW 2 sin - 2 Lim Zn K(~) (in 7) (3.1) m 0 2 / which is determined by the value of the argument 2. The significance of width w must be noted at this time. As long as w / 0 the series eventually converges as 1/m3, but an increasing -35

-36number of terms must be included as w — O. Consider M to be the number of terms included in the summation. Then 0 M= M2 M 2 is the argument of the final term. Fig. 3. 1 shows the error due to truncation of the summation as a function of 0 M. This curve was obtained by calculating a set of partial sums for Z using progressively greater values of M. When the difference between two succeding Z values was less than 0. 1%, the larger value was considered as the limit or sum of the series, and all previous partial sums normalized to this value. Percentage error was determined relative to the final Zn for all partial sums and plotted as shown. If the first zero of (sin 0 m/0 m) is chosen as the truncation point, (i.e., 0M = r) then M = w' represents the number of modes to consider for error 1%. As w - 0 the configuration approaches that of an infinitely thin post which is characterized by infinite inductance, i. e., the divergent series resulting from a 1/m term sin 0 variation since lim = 1. 0 — 0 m m Secondly, the parallel combination of an infinite number of sets is considered. From (2. 17) the total admittance is the result of summing the admittances of the individual sets whose terms decrease for large values of n as Lim YR 2 K' (n) (sin 2 2 )(3. 2) n -co n~rrg2

-37PARAMETER eM 0.0 1.0 2.0 3.01 4.0 ERROR I I* /I -20 Fig. 3. 1 Truncation error for Z n

-38This is obviously of the same form as (3. 1), with different parameters involved. The dependence on the gap size is the controlling factor in the number of model sets N to include in the circuit, and similarly divergence is approached as g --- 0. The divergent admittance would represent a short-circuited gap which is appropriate for g = 0. As with (3. 1) consider the truncation point as the first zero of the sin 0n/0n function. Therefore N = 2 represents the number of sets to include. It should be noted that whenever dealing with a physically realizable mount, i. e., finite post and gap dimensions, the impedance functions are well behaved although slowly converging. The slow rate of convergence is not a problem however, since most precision analyses are carried out using a computer, as was done here. 3.3 Assumptions and Error In (3.2) it is possible to see the consequences of the assumed y-distribution of the E - field in the gap. The rate of convergence for YR is controlled directly by the expansion coefficient Sp expressed in (2. 12), and therefore will reflect any error in the distribution as error in the result. Then the approximate value Y' is related to the true value YR by L = YR(1 + I) (3. 3) n n

-39where A is due to convergence error from the (sin 0n/0n) description for the field distribution. However from (3. 1) it is seen that the approximate x-distribution assumed for the current on the post will produce similar errors resulting in Z' n Then z' = Z (1 + A') (3. 4) n n where A' is due to convergence error from the (sin 0 /0 m) description for- the current distribution. Reconsidering gives " - Z' (1+ y 1_ (1 + a) (3 5) -R n Z1 ( (1 +') Z (1 + A') YR n nn n or R YR for A A a', since both errors are due to comparable assumptions in parameter distribution. In fact, the error A can also be attributed to premature truncation of the j summation. But quantitan tively, if the ~, summation is likewise cut short, the change in the m error a' should follow A so that (3. 5) is still good. This is demonstrated in Fig. 3. 2. Use of the "first zero" criteria previously mentioned dictated M = 20, N = 30 for the limits of the respective

-40100 ohms 0. -100 M=20, N=30 M=10, N 15 RR -200- MZ 8,N=12 XR - 2 6 10 14 18 22 frequency GHz Fig. 3.2 Impedance comparison plot. C-Band waveguide a = 4.76 cm, b = 2.215 cm, s' = 0.500, h' = 0.0, w' = 0.115, g' = 0.069. M and N represent the number of terms retained in summing the respective series.

-41summations. This was first reduced to M = 10, N = 15 and then M = 8, N = 12 with the resulting impedances compared, indicating excellent agreement. The truncation criteria can then be re1 1 duced to M = I, N = without loss of accuracy. This error compensation effect is due to the stationary nature of the impedance formulation with respect to the current and fields (Ref. 18, pp. 260-261). 3. 4 Terminated Waveguide The derivation presented so far has only considered infinite guide length or matched conditions. This was sufficient to establish the circuit representing the post mount as shown in Fig. 2. 5. A more practical case would include the possibility of terminating the waveguide arms in something other than a match, e.g., sliding short, filter element,etc. To do this a new Green's function is required, which takes the terminating boundary conditions into account. To satisfy this requirement, GT(rl r') i. e., the Green's function for terminated waveguide, was derived. This derivation is presented in Appendix B. Fortunately GT(rl r') is directly related to the previous G(rl r') by GT( r') = G(rI r') T where

-42mn1 mn 2 -2mn1 2 1 + Plmne + P2 mne + PmnP2 mne -2r (f +k2 ) PlmnP2mne (3. 6) representing the scattered energy effect of the terminations, and Plmn' P2mn = complex reflection coefficients for terminations 1 and 2. 1i' Z2 = distances to terminations 1 and 2 from post mount plane. This factor T being independent of the x, y coordinates, is carried through all of the mathematics (equations 2. 9 - 2. 21) to act directly in determining a terminated mode pair impedance, Z as Tmn Tmn Zmn T (3. 7) This can be separated to represent the two arms of the waveguide with Z T1 - representing arm #1 cmn and Z T2 - representing arm #2 cmn2

-43where -2 r mn j 1 + p. e T.: (3.8) \j - 2 r.' mn j jmn An interesting and expected form of Tj is Z. + tanh r Q. T. =jmn mn (3.9) J 1 + Z. tanh r Q. jmn mn j with Z. = terminating impedance on arm #j normalized jmn to Z cmn which is the impedance translation transmission line formula. Note that Z is imaginary for non-propagating mode pairs. c mn The denominator of T accounts mathematically for the possible resonance between the two terminations; i. e., -2 rm(f +12) - Plmn P2mn e n only when Ip11 = Ip21 = 1 and the proper phase exists with a propagating mode.

-443. 5 Multiport Characteristics Since it was possible to isolate as terminal effects all of the various mode pair impedances, the equivalent circuit of the post mount was defined as a multiport coupling network. The plane z = 0 defines the position of all ports with respect to the waveguide; noting, as shown in the preceding section that mode terminations must be considered as the shunt combination of the two waveguide arms. For a propagating mode the plane z = 0 is accessible as an input port to the circuit so that the post mount may be considered as an obstacle in the waveguide. Normally only the H10 mode will be propagating so that all other mode ports will be terminated in Zcmn Termination of the opposite waveguide arm to the H10 mode, and knowledge of the characteristics ZG of the particular device as placed in the gap, will permit accurate obstacle description. The circuit in Fig. 3. 3 results from considering waveguide arm No. 1 as the input port to the post mount equivalent circuit for the H10 mode, with combined components specified as follows: 0 K0 jX m2 m (1- w) (3.10a) t K2 / /Z (3. 1ob)

-45-.- arm 1 arm #2 I I I I XL ZIN Z10 T Fig 3~3 o obstacle cr cuit o i Fig. 3o 3 Post obstacle circuit for incident H10 mode.

- 462 00 Yj (n#0) = (Kpl) 1 (K 2 (3. 10c) Z z [m=l mn Kgn/ Since the formulation for the modified driving point admittance Y'(nf0) is the same as that discussed in Section 3.3 for YR, the 1 1 less restrictive summation criteria M - N = may be used because of the compensating action. However this does not hold for the single summation of the inductive reactance XL. Any error in the summation will directly affect XL. Therefore it is necessary to use M = w2 for this special case. In fact it is also necessary to consider the error A' due to the assumption for the current distribution. While this has not been analyzed theoretically, the experimental work discussed in Section 5. 3. 1 indicates accurate results are obtained by use of the correction factor (1 - w'). Since wT < 0.25, this factor never becomes very large, simply adjusting for slight tendency due to the assumed current distribution to predict high values. The special case where the gap is shorted out, ZG = 0, allows description of a post-in-waveguide. This holds for any incident mode by considering the proper input port of Fig. 2.5. 3. 6 Impedance Characteristics Using (2. 17b), (2.21a), (2. 22), (3. 6), and (3. 7) a concise form for the driving point impedance is found as

-47Z:1 (3. 11) R n=0O 1 K Tmn K gn which is represented by the circuit in Fig. 2. 5 with terminated impedance ZTmn. With this relationship established, it is desirable to take a closer look to determine what are the dominant and lesser characteristics, and how are they controlled by parameter values. A broad frequency range has been considered because of high interest in determining impedance characteristics for harmonics of pump and mixing frequencies in the design of parametric amplifiers and frequency converters (Refs. 24, 25). The dominant mode frequency range for the (C-Band) waveguide considered here is 4 - 6 GHz. Matched conditions are presumed in the following discussion so that Z = Z. It is clear from (2. 21a) that once the Tmn mn waveguide dimensions a and b are chosen (as has been done in specifying C-Band waveguide), the characteristic impedances for all modes are established. In particular this means that the scaling and placement of the zero and pole for each mode impedance pair Zmn (Ref. Fig. 2.3) has been fixed. Next the coupling to the gap must be considered. In general if arbitrary values are picked for s' and h', both K and K will be non-zero and all modes pm gn will have a non-zero contribution to ZR. However special cases do

-48exist where either one or both of the coupling factors may become zero for various specific values of m, n. The best recognized example of this is the lack of coupling to all m = even modes when the post is centrally located (s' = 0. 500) in the guide since sin m7Ts' = 0 for these conditions; i. e., K = 0. Other post popm sitions can likewise decouple modes for specific values of m. More significant however to the general characteristic is the placement of the gap with h'; this parameter controls the coupling of the Z n sets to the overall circuit. Equation (2. 19) indicates a zero point for each Z which becomes the dominant characteristic seen by n the gap because of the parallel nature of the sets. Fig. 3. 2 shows this effect quite well with zeros present at f = 6. 77, 13. 54, and 20.31 GHz, corresponding to the zeros of Z1, Z2 and Z3 respectively for the general case. It is however possible to choose h' such that cos nfh' = 0. When this is done,the associated set Z is decoupled so the zero is not present. This is possible for the following conditions, h' decoupled sets for 0.500 n = 1, 3, 5....o 0.250 n = 2, 6, 10... 0 0.166 n = 3, 9, 15....o 0.125, 0.375 n = 4, 12, 20... co

- 49which represent the most significant cases. The strong effect of this change in h' on the impedance characteristic is shown in Fig. 3. 4, drawn for the first three h' values in the list above. Note the lack or presence of zeros at 6. 77, 13. 54, and 20. 31 GHz for each curve and how this feature predominantly sets the pattern. In Fig. 3. 5 the dependence upon waveguide height is shown. Here again the dominant characteristic is the placement of the zeros which shift to increasing frequency as b decreases. This shift produces an increase in the real part of the impedance in the H10 region because of reduced shunting by the higher order n > 0 modes. Fig. 3. 6 demonstrates the relatively slight effect produced by shifting the post sideways in the waveguide. The gap position chosen for this graph h' = 0. 250 decoupled the Z2 set which resulted in a more slowly varying impedance through the mid-frequency range. Fig. 3. 7 indicates what can be done with the mount circuit as an obstacle to the H10 mode. Plotted is a family of curves representing a "tuned post" in the waveguide. The gap size is varied from zero to slightly larger than 1/4 the guide height. The gap impedance Z is determined simply from the parallel-plate capaG citance of the end of the post, which was centered to decouple the H20 mode. This extended the dominant mode region to 7.46 GHz

-50300 200 1 1 RR " I I I00- i - \Ii 2 6 10 14 1 8 22 frequency GHz, 0.166 6 Gap Position h= 0.250 - L0.500 - - I~ I I [:~ / / ~,,~~i ~ -.-.' -oI I'100 I// i./.~!.t -200 2 6 10 14 18 22 frequency GHz Fig. 3. 4 Driving point impedance for gap position (h') variation with s' = 0.333, w' = 0.115, g' = 0.069, a = 4.76 cm, b = 2. 215 cm. (a) Resistive component. (b) Reactive component.

-51300 (a) II 100-: 2 6 10 14 18 22 frequency GHz Waveguide r2.215 cm Heiht b 1.400 cm 0.870 cm -. fO I. I 0 A. iI I /IoI (b) -200 2 6 10 14 18 22 frequency GHz Fig. 3. 5 Driving point impedance for waveguide height (b) variation with h = 0.076 cm, g = 0.152 cm, w' = 0.115, s' = 0.500, a = 4.76 cm. (Note: h' and g' vary for each curve since normalized to b). (a) Resistive component. (b) Reactive component.

-52300-, l ii /(a) if RR ~ ~ 2 6 l0 14 18 22 frequency GHz r 0.250 Post Position S 0.333 0.500 100 (b) 0XR - 100 — -200 2. 6 10 14 18 2 frequency GHz Fig. 3. 6 Driving point impedance for post position (s') variation with h' = 0.250, w' = 0.115, g' = 0.069, a = 4.76 cm, b = 2.215 cm. (a) Resistive component. (b) Reactive component.

-53I' a obstacle configuration g 1.0.8.6.4 S.2 80 g=.035.2I 40 —-0 g6.0 70 4.0 5.0 6.0 7.0 freq (GHz) I 6.77 Fig. 3. 7 Normalized obstacle reactance for gap size g variation in C-Band waveguide. a = 4.76 cm, b = 2.215 cm, s' = 0.500, w' = 0.115.

-54the cutoff frequency for the Hll and Ell modes, thus permitting observation of the characteristic at 6. 77 GHz, where the reactance is independent of the gap size. Actually the reactance is independent of any impedance ZG which happens to be present at 6. 77 GHz because the admittance function Yh(n0O) is infinite at this frequency due to the zero of Z1. This interesting feature would not be present if the gap were centered halfway up the post. Therefore a great variety of passive waveguide elements may be obtained through variation of the mount parameters. The curves for Figs. 3. 1 - 3.2, 3. 4 - 3. 7 were all determined using a special computer program developed for that purpose. This program is presented in Appendix C.

CHAPTER IV EXPERIMENTAL DEVELOPMENT 4. 1 Introduction The purpose of this chapter is to discuss the work which went into the development of the equipment, techniques and procedures necessary to arrive at an accurate means of measuring the driving point impedance ZR, of the waveguide mount shown in Fig. 4. 1. This measurement information is desired to both aid and support the theoretical analysis discussed in the previous chapters. 4.2 Equipment Development Measurement of this terminal impedance would not normally be considered possible because of the inaccessibility of the terminals, which probably accounts for the lack of published material dealing with the problem. However, with the advent of subminiature coaxial cable and connectors it is now possible to isolate the terminals electrically without affecting the surrounding field conditions, by running the measurement circuit cable inside the post. Initially the equipment was designed for X-Band waveguide over the frequency range 6 - 22 GHz. This range proved insufficient to fully test the validity of the theoretical results in that the maximum number of propagating modes pairs was limited to five by the 22 GHz equipment limitation. Therefore it was necessary -55

-56-F cg / ZR Fig. 4.1 General mount configuration

-57to reduce the lower limit by going to larger waveguide. The next lower standard waveguide band was considered (C-Band), increasing the frequency range to 3 - 22 GHz. For this range the C-Band waveguide would support up to nineteen propagating mode pairs (Hmn and E mn), adequate for our purposes, thus leading to adoption of this range for the study. 4. 2. 1 Mount Design and Construction. The design and construction of the mount were considered as a single problem, because the two are so interrelated: in effect, designing to utilize the available construction capabilities while simultaneously skirting around construction difficulty. Most of this difficulty is due to the flexibility desired with the position parameters h and s, which is necessary to permit complete analysis. It is neither practical nor necessary to provide for variation with d and g as these values would normally be established by other criteria such as the size of the device to be mounted. Typical values are used here for d and g. Since excellent electrical contact is required at all junctions between movable parts, it is not feasible to have the parameters s and h continuously adjustable; rather these parameters occupy a set number of discrete values. In this manner it is possible to ensure adequate electrical and mechanical integrity within the system. The mount described in Fig. 4.2 proved to work very well, demonstrating excellent symmetry in the data for values of h

- 58a = 1.874" (4.760 cm) b= 0.872" (2.215 cm) d = 0.120" (0.305 cm) g = 0.060" (0.153 cm) h'= h/b = variable S'= S/a = 0.250, 0.333 or 0.500 _ _ _ _ _ a _b g4 0.065" (1.65 mm) TEFLON 50 ohm COAXIAL LI NE Fig. 4.2 Measurement mount

-59symmetrical about the center (b/2) of the guide. This data symmetry was sought as an indicator of proper construction, since differing fabrication methods were required for the top and bottom of the post. The "measurement probe" was made by epoxy-soldering the subminiature coaxial line in a 0. 120 inch sleeve and carefully attaching the center conductor to the center of the matching 0.120 inch rod. This is then held in place by a clamp around the coaxial portion, allowing vertical movement of the probe. Three sets of holes had to be drilled for the three positions across the guide. The holes were plugged when not in use. The probe has a standard subminiature connector on the outside end on which was attached an adapter to precision 7 mm coaxial line. 4.2.2 Waveguide Terminating Considerations. Inherent in the objective to analyze this waveguide circuit over a wide frequency range is the necessity to describe accurately the terminating conditions for each of the propagating modes. From practical considerations only a short-circuit or a match could be provided for experimental purposes. The short-circuit was not used primarily because choosing its position would introduce another circuit parameter, adding undesirable complexity. Therefore a matched termination was required for both waveguide arms.

-60To satisfy this requirement, a standard H10 waveguide termination for C-Band was modified to improve its matching characteristics for higher order modes. This was done by supplementing the original load with four more tapers, providing a multiple load effect on the end wall similar to the configuration commonly used in anechoic chambers. Although equipment was not available to test the individual terminating characteristics for each mode (above H10), it was possible to demonstrate the adequacy of the system by shifting the termination with respect to the mount while making ZR impedance measurements. If reflections for any of the propagating modes were present, they would be seen as a change in the impedance at the mount. No changes were noted. 4. 3 Measurement Circuit Modeling To measure the desired impedance ZR it was necessary to provide connectors and adapters to get from the subminiature cable up to the standard 7 mm size available on precision test equipment. These elements introduced irregularities into the measurement line which, if ignored, would produce errors in the data interpretation. Additional difficulty was created by the necessary transmission mode transformation from coaxial line to radial line at the gap. In order that a high degree of confidence could be put in the data and its interpretation, it was necessary to model these effects as an equivalent measurement circuit. A statistical comparison technique,

- 61described below, was used to arrive at values defining the assumed lumped discontinuities and line lengths involved. The resulting equivalent circuit is shown in Fig. 4. 3. The circuit loss was empirically determined to be attenuation = 0.15 + 0.005 fl 45 dB where f is the frequency expressed in GHz. This effect was considered as a perturbation on the measured standing-wave-ratio rather than as additional elements in the equivalent circuit, because the overall effect was small and could be more conveniently handled this way. Modeling this loss in the measurement circuit would require the use of a complex characteristic impedance for the transmission line, complicating the analysis unnecessarily. 4. 3. 1 Statistical Comparision Technique. The analysis of the measurement circuit was based upon the concept that the source of error or inconsistency in a set of data could be separated statistically into two groups. First is the inherent inaccuracy in the measurement equipment, which has hopefully been minimized by proper measurement technique. The second is the interpretation error due to improper assumptions about the circuit. If a simple transmission line assumption is made for the circuit, this is in effect a "model" which, if different from the true circuit, will introduce interpretation error. Ais more complex models are

A B 1 0.921 0.692 1 0.692 0.615 0.920 1.199 1 ZIN 2.0 2.0 4.0 - 7.0 slotted line j 7mm connector 7 mm to 1.65 mm adapter..... I B C 1 6.0 0.934 8.0 1 4.874 1 360 187 17'i. -1 15.0 1400 ZR T + + T T +, 1 1.65 mm connector 1.65 m coax-radial line load coax_ transformation note: a) capacitor values x 10-15 F b) inductor va lues x 10-12 H c) line lengths in cm Fig. 4.3 Measurement circuit equivalent model.

-63assumed, effectively providing better approximation to the true circuit, the error due to the modeling decreases. In theory then, if a perfectly true model is assumed for the circuit, the total error will be minimized and the remaining error is due solely to the equipment (e.g., frequency drift, mechanical tolerances, etc.). Initially a short-circuit was placed at plane A of Fig. 4. 3. A set of data was taken to determine electrically the location of this short with respect to a given reference plane, assuming a simple airline case. By using a wide frequency range (8 - 18 GHz for 40 points) a certain amount of data scatter was present which could be defined by the mean value and average deviation from the mean for the short position. Approximate values for the discontinuity capacitances of the connector support bead were determined by physical considerations and used as initial values for modeling the circuit up to the short-circuit. The data was then interpreted through the new model and the scatter in the resulting short-circuit position compared. This trial and error procedure was followed, using a computer, until a stable minimum in the deviation was established. As expected only a small improvement of 3. 0% was found by including this first capacitor. Next the adapter replaced the short-circuit at plane A and the short-circuit moved to plane B. A new data set was taken, and the whole procedure repeated considering only the element values between planes A anid B as variables. This time a very substantial improvement, i. e., a 66. 0% decrease in the average

-64deviation, was noted. Finally the short-circuit was moved to plane C, which is actually a cylindrical surface across the gap at the surface of the post. A 35. 0% improvement in the deviation resulted by using the values shown for the elements. The method used for determining the initial approximate and limiting values for the line elements is discussed in Appendix D. The minimizing procedure was insensitive to changes in the values of the coaxial-radial line transformation elements because of their proximity to the short-circuit at plane C, with the result that these values could not be verified by measurement. Derivation of this transformation is discussed in the following section. 4. 3.2 Coaxial-Radial Line Transformation. Determination of the transformation circuit was accomplished by following closely the concepts discussed by Getsinger (Ref. 13). A direct analogy is not possible however, because of the difference in configurations, i. e., the coaxial line here feeds the center of the radial line, not the outside. Fig. 4. 4 shows the correspondence between the lumped element circuit and the actual circuit. Cfl and Cf2 are fringing capacitances, with Cr the parallel plate capacitance in the radial line region. Lc is due to the region in the corner formed by extending the outer boundary of the coaxial line. Lr represents the volume in the radial line. This relatively simple equivalent circuit provides a good approximation as long as the dimensions involved are small with respect to wavelength.

-65K 0.06 0" -_ Cr 0.020. f2 ______________ __________ _ \ 0.120" Lr (a) Lc Lr Cf -- Cf2 - _- Cr (b) Fig. 4. 4 Coaxial - radial line transformation (a) Physical configuration. (b) Equivalent circuit.

- 664.3.3 Effects of Circuit Modeling. To determine the usefulness of the measurement circuit model, comparisons were made of the various circuit effects on the data interpretation. Three situations were considered. First, a simple transmission line was assumed between the load and the measurement equipment; second, the coaxial-radial line transformation was introduced; and third, the complete circuit model was used. Fig. 4. 5 indicates the differences in interpretation for the last two cases compared with the theoretical case for a typical data set. Lines are used here to represent the trends in the data interpretation. Data points are individually shown when the measurements are discussed in detail in the next chapter. The simple transmission line case is not shown because it bore so little resemblance to the other cases. Its only characteristic in common with the others was the placement of zeros at 6. 77 and 20. 3 GHz. Very worthwhile improvement is noted by including the total circuit model, justifying the effort involved. 4. 4 Measurement Procedure All of the impedance measurements were made using standard slotted-line techniques. The equipment was continually recalibrated to ensure precision and repeatability. Once the coaxial line adapter was placed on the probe and characterized, it was not removed. The data necessary to determine impedance, i. e., frequency, standing-wave-ratio (SWR), and standing-wave- minimum

-6720~..... 2 6 10 14 18 22 frequency GHz Line Transf ormati on Measurement Circuit Model only LTotal Model ~~I f\t~;I \(b) 0. I I I - 2 6 10 14 18 22 frequency GHz Fig. 4. 5 Measurement circuit modeling comparison for the driving point impedance. (a) Resistive component (b) Reactive component ~.

- 68position, was processed through a computer program to be interpreted as ZR versus frequency for each of the many configurations under test. This program is discussed in Appendix E.

CHAPTER V COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS 5. 1 Introduction This chapter brings together the results of the theoretical and experimental analyses, using the driving point impedance ZR and the H10 obstacle reactance as points of comparison. 5.2 Driving Point Impedance Comparison The driving point impedance of the mount shown in Fig. 4.2 is a function of the six dimensional parameters a, b, s', h', d, and g. Only the two here referred to as the position parameters s' and h' are varied in these measurements. Relatively standard values were chosen for the others to conform to a typical mount, i. e., a, b represent C-Band waveguide with d = 0. 120" and g = 0. 060", proper for mounting a "pill-type" device package. In addition all measurements were made under matched conditions on both waveguide arms. Theoretically-determined values are compared to those resulting from the measurements in Fig. 5. 1 - 5. 4. Fig. 5. 1 represents the impedance for the most typical mounting configuration, with the post centered and the gap at the bottom. Fig. 5.2 then shows the result of moving the gap halfway up the post. Since the post is centered, only coupling to the - 69

-70300, I 2000 ZR 0o -100' -200 2 6 10 14 18 22 frequency GHz Theoretical Result Resistive ~ o * 0 Experimental Result Reactive X x X x x Fig. 5. 1 Driving point impedance comparison - theoretical experimental s' = 0.500, h' = 0.035.

300 0 0 200 * 0 0e x 0 0 100 ~ ZR 00 lXReactive x x xx x x w X -100 x = x -2002 6 10 14 18 22 frequency GHz Theoretical Result Resistive 0, ~ ~ Experimental Result Reactive x x x x x Fig. 5.2 Driving point impedance comparison - theoretical and experimental s' = 0. 500, h' = 0.500.

-72300200d- - 0 o 0 00 X 0 0 X 010 -2002 6 10 14 18 22 frequency GHz Theoretical Result Resistive o o, o o Experimental Result Reactive x x x x x Fig. 5. 3 Driving point impedance comparison - theoretical and experimental s' = 0.250, h' = 0.500.

-73300 2000 -~~~~100 —~0 0-00 2 6 10 14 18 22 -200, 2 6 10 14 18 22 frequency GHz Theoretical Result Experimental Result Reactive xx x x x Fig. 5. 4 Driving point impedance comparison - theoretical and experimental s' = 0. 333, h' = 0.250.

-74m = odd modes exists. One of these modes, the H30 mode, having a pole at 9. 45 GHz, causes the "resistive component" to have a zero at that frequency by decoupling energy to the H10 propagating mode. Leaving the gap centered in the post and shifting the post to s' = 0. 250 results in Fig. 5. 3. Note the additional "resistive component" zero at 6. 3 GHz compared with Fig. 5. 2. This is due to the H20 pole (cutoff frequency) which now must be considered for the off center post. Fig. 5. 4 was included to show a characteristic markedly different from the others, produced by choosing different position parameters, i.e., s' = 0.333, h' = 0.250. In all figures the data clearly shows the zeros and damped poles predicted by the theory, providing the verification desired. 5. 3 Waveguide Obstacle Reactance Comparison The measurements in this section consider the waveguide mount as an obstacle in the waveguide to an incident H10 mode, with a matched termination on the opposite arm. This condition is represented by Fig. 5. 5, defining ZOBS as the normalized obstacle impedance in shunt across the waveguide. Z'BS includes the effects of the impedance ZG placed in the gap plus all higher order modes, and is valid for all frequencies above H10 cutoff. Propagating higher order modes are seen as resistive elements in ZOBS

-75ZI.? ZOS I 1.0 ohm I Z=O Fig. 5. 5 Mount obstacle equivalent circuit.

- 765. 3. 1 Post Inductance. If the gap is shorted out (i. e., ZG = 0), all coupling to n > 0 modes is removed, leaving only m > 1, n = 0 modes for consideration. These modes all exhibit inductive properties below cutoff so that in the dominant mode region the post will, as expected, appear inductive. This configuration allows isolated study of the post cross-section and lateral current distribution effects, leading to a general description of a postin-waveguide. Initially the relationship between a circular post (d = diameter) and a flat strip post (w = width) was considered. Although this relationship depends slightly on frequency, post size and location, good results are obtained by setting w = 1. 8 d for the equivalent width of a circular post. Fig. 5.6 compares the measured values for a circular post with d = 0. 120 inches and a flat post with w = 0. 216 inches. Next, the inductive reactances of flat posts, varying both size and position, were measured and compared against the predicted values. In general the predicted values were a little high, depending directly on the post width. This is due to the constant current assumption across the post width, which becomes more erroneous as the width increases. Therefore a small correction factor (1 - w'), was introduced to compensate for this error in the theory, resulting in the graphs shown in Figs. 5.7 - 5. 9.

-774.0- X-.120 inch dia. x x 0 -.216 inch wide 2.0, 2 5 0 S'=.500 0 x x2 | 4.0 5.0 6.0 freq (GHz) Fig. 5.6 Post cross-section comparison.

-788.0 6.0o W.058 Data 0 4.0 Theory Marcuvitz S'=.250 X OBS 1.0-.8 333.6. 0- \S v 5 00.4 4.0 5.0 6.0 freq (GHz) Fig. 5.7 Normalized flat post reactance w' = 0.058.

-794.0 - W.115 Data 0 Theory Marcuvitz / 2.0XOBS s' =.50 1.0Q8'.6t- S =.333 0/ S =.500 4.0 5.0 6.0 freq (GHz) Fig. 5. 8 Normalized flat post reactance w' = 0. 115.

-80W'-.230 2.0Data 0 Theory Marcuvitz -.8 S=.250.6 XOBS 0.4 S5=.333 S =.500 4.0 5.0 6.0 freq (GHz) Fig. 5.9 Normalized flat post reactance w' = 0.230.

-81These graphs also include predicted values by Marcuvitz (Ref. 26) for the centered post position showing good correlation to the theory for Figs. 5. 7, 5.8 while deviating approximately 10% for the wide case of Fig. 5. 9. (Marcuvitz does not predict for off-centered flat posts. ) The noticeable upswing of the off-centered curves of all three widths for the high frequency end is due to the action of the H20 mode impedance approaching infinity at 6. 3 GHz. 5. 3.2 Tuned Post. The final example considered is that produced by leaving the gap open, resulting in ZG being the end capacitance of a circular post. This configuration is commonly called the tuned post and is described in Fig. 5. 10 for different gap sizes. The gap dimensions are varied from zero to slightly larger than 1/4 the guide height. By centering the post, the H20 mode is decoupled so that the dominant mode region is extended to 7. 46 GHz, the cutoff frequency for the H11 and E11 modes, thus allowing the observation and verification of the characteristic at 6. 77 GHz where the reactance is independent of the gap size.

-82a obstacle configuration b 1.0.8e (a).6 g'= 0.0.4.g=.035 g'.070 4.0 5.0 6.0 7.0 freq (GHz) 6.77 Fig. 5. 10a Normalized obstacle reactance for gap size g variation, s' = 0.500, w' = 0o115. (Theory)

-831.0 (b) 0.6 o 4 000 0 0 9.035,121 1 1 1 i Io ~.2 g'9~0140 0 — g-.070I 0 I 4.0 5.0 6.0 7.0 freq (GHz) 6.77 Fig. 5. 10b Normalized obstacle reactance for gap size g variation, s' = 0.500, w' = 0.115. (Experiment)

CHAPTER VI REVIEW, CONCLUSIONS AND RELATED FUTURE STUDY 60 1 Introduction All of the work is reviewed briefly, highlighting the prominent concepts and results introduced. Conclusions are drawn, followed by a short discussion of related future areas of study. 60 2 Review The objective of this report was to characterize a common waveguide mount,'resulting in a description which was convenient for use in circuit design. This goal has been satisfied. In the process, many additional thoughts have been introduced which either present an original idea or add support to a concept previously recognized. It is felt worthwhile to provide a short summary of these points, including the section number where each is discussed in more detail. 1) General Theoretical Analysis. By assuming a general expansion of the current density in the same orthogonal functions as the Green's function, it is possible to develop some knowledge about a radiating element without knowing the actual current distribution~ Section 2.2 - 2o 3o -84

-852) Current Density y-dependence. As a consequence of the mathematics the y-dependence was easily developed, giving the current distribution on the post as a function of the mount parameters and the mode indices. It is most interesting how the phase information for current components associated with the H and E modes mn mn above and below cutoff is contained in the formulation. Section 2. 3. 6. 3) Mode Related Impedances. The development of the mode impedances ZH and ZE offer a strong case for adopting these formulations as the basis for waveguide characteristic impedances for all modes. Section 2. 3. 7. 4) Series Resonance of Mode Pairs. The resonant effect resulting from the series combination of ZE and ZH for the same m, n set is also interesting, particularly the fact that the resonant frequency is independent of m. Section 2. 3. 7. 5) Mount Equivalent Circuit. This circuit simply provides a means of defining the coupling between the impedance present in the gap and the mode impedances present in the waveguide arms. It is a linear, passive, reciprocal, doubly infinite network whose elements are a function only of the mount parameters. Section 2. 3. 7.

-866) Impedance Measurement Technique. This simple concept of running a subminiature coaxial line inside the post to gain access to the gap was the heart of the thesis. All other attempts to measure ZR failed. If the experimental effort had not produced such reliable and selfconsistent results, the necessary insight to develop the theory would never have been obtained. Now that the technique has proven itself by measuring predictable impedance variations accurately, its real value will be in applications where the configuration cannot be handled theoretically, thus providing unique information. Section 4. 2. 1. 7) Measurement Circuit Modeling. Although the statistical method used was slow, tedious and expensive on the computer, it appeared to be the only way to isolate the extremely small effects due to the discontinuities in the line. The work was well justified by the resulting improvement in the data interpretation. Section 4. 3. 6. 3 Conclusions The various graphs in Figs. 5.1 - 5.4 and 5.6 - 5.10 show a high degree of correlation between the theoretical plots and the measured data. Although only a limited number of situations were presented, they were of sufficient diversity to fully test the theory

-87and the experimental procedure. The theory is further supported by the agreement with Schelkunoff (Ref. 21) and Marcuvitz (Ref. 26) for those special cases, as well as the low frequency formulation discussed in Section 2. 4. On this basis it is reasonable to conclude that the theory presented is valid and that the measurement technique developed was highly successful. 6. 4 Suggested Areas of Related Future Study The following suggestions are submitted as areas in which investigations could be carried out using the information developed in this thesis as a basis upon which to expand. 6. 4. 1 Theoretical Study. 1) Application of the theoretical analysis to other physical configurations, e.g., gap driving point impedance for a gap in a circular post between two parallel ground planes. This configuration represents the feed point of a radial transmission line. 2) Removal of the current density x-distribution assumption in order to enhance accuracy and permit consideration of larger posts. 3) Removal of the gap voltage distribution assumptions in order to enhance accuracy and permit consideration of larger posts and larger gaps. 4) Application of the analysis to develop an equivalent circuit for a mount containing 2 independent gaps in the post.

-885) Development of complex reflection coefficient descriptions for standard waveguide obstacles excited by higher order modes. 6. 4. 2 Experimental Study. Use the measurement technique on other types of mounting structures, including coaxial line, strip line,etc.; especially on configurations which cannot be handled theoretically. 6. 4. 3 Applications of the Circuit. The circuit developed to describe the mount can be used to improve the design of waveguide oscillators, amplifiers, frequency multipliers and converters, phase shifters, mixers, attenuators and various filter elements. Electronically tunable elements can also be designed using voltagetunable devices mounted in this manner. An example of such an element is the common switching element employing a PIN diode as a variable Z. G

APPENDIX A DETERMINATION OF THE DYADIC GREEN'S FUNCTION FOR RECTANGULAR WAVEGUIDE The following derivation was discussed in an advanced Electromagnetic Field Theory course given by Professor Tai, but the general method is not well known. For completeness a detailed description is given here. We are interested in solving the vector wave equation 2 - V x V x E(r) - k E(r) = -j w co J(r') (A. 1) to develop a relationship for the electric field E(r) as a function of the current density J (r'). The method to be used is called the Green's Function method which is based on the proposition that a function G(r I r') can be found which will satisfy the equation V x V x G(rl r') - k G(rl r') = I 6(r - r') (A. 2) similar to (A. 1), subject to the boundary condtions of the region of interest. Assuming such a G(r r') exists, (A. 1) and (A. 2) can be combined to develop an integral equation giving the necessary relationship between E (r) and J(r'). This result is E(r) = -j " /o f G(rl r') ~ J(r') dv' (A.3) v -89

-90inviting interpretation of G(rl r') as a coupling factor between E(r) and J(r'). To solve (A. 2) let us investigate the solutions of the homogeneous equation - 2 - V x V x S-yS - S = O. (A. 4) S in general can be represented by A 1A M = V x (z1) N = x V x (z) (A. 5) where 41 and V'2 are two independent solutions satisfying the scalar wave equation 2 2 V2 4, + = O. (A. 6) The region under consideration is the inside of a rectangular waveguide, described in Fig. A. 1. It is considered infinite in extent in the + z - directions requiring Sommerfeld s radiation condition (Ref. 27) for the boundary conditions at large I z. The waveguide is assumed to be perfectly conducting, consequently A - n x E = 0 is the internal boundary condition on the guide wall surface. Since G(rl r') is associated directly with E, this boundary condition must hold for G(r r') in (A. 2), therefore requiring

~aplnoastA mtInS.utpoae aqe;ioj uoldldaosap aO4uipLooo I'V'2i x D 0 -16

-92in x - = 0 (A. 7) N on the walls. Equation (A. 6) can then be solved by separation of variables resulting in the functions v1 = cos k x cos k y e j z (A. 8a) x y and 2 = sin k x sin k y e-i z (A. 8b) 2 x y with m r nff k k = x a y b 2 2 2 2 % =y - k - k y which when substituted in (A. 5) satisfy these boundary conditions. The general solution is represented by = m e-j z (A. 9a) mn -- A A m = -x k cos k x sin k y + yk sin k x cos k y y x y x x y and

-93N =n e-j z (A. 9b) mn n = [-x j k cos k x sin ky y j 9 k sin k x cos k Y x x y y x y A 2 + z k sin k x sink y mn x y where 2 2 2 k k +k mn x y We now assume an expansion of the delta function I(r - r') in terms of these general functions allowing arbitrary vector coefficients P, Q. -O n f d [I mn mn Q m=0 n=O -oo (A. 10) To determine P, take the dot product of Mm,n,(s') with both sides of (A. 10) and integrate over the enclosed volume, yielding J M,,(') I (r - r') dv = M', (') vol m n m'n vol = f d Tm P e + ) d z (A. 11) -o0 -00 where ab 2 T,, = k,,(1+ ) (A.12) m'n' 4 m'n (l~go) (A. 12)

- 94and M' indicates evaluation at the primed coordinates. Then knowing that [ ()] z= e )Z d z (A.13) -00 results in M' (') = 2ii P(-') T, m'n' rn n or M'm(-c) (2-6) p (0 mn2. (A. 14) mn k2 ab mn Following the same procedure with Nm,n,(') we find that N'T (-0) (2 - 6 ). Q () = mn. (A. 15) k 2T a b mn so that laOF-F3= C C M C+ (Cd rN' I(r-r') = 3 [2r T Mn mn( ) mn( m)Nn( m=O n=O -oo mn (A. 1 6) Next assume an expansion of G(rlr') in the same functions including necessary arbitrary scalar coefficients (a, 3) to represent a general solution. (A. 17) -__ _ oo oo ood (i)M' _ _ (A.,17 G(rr')= L f 2wr T aMmn(Mmn( )+3Nmn( )Nmn(m=O n=-oo mn

-95Now by substituting (A. 17) into (A. 2) and carrying out the indicated operations we find that 2 2 2 (k + - k ) G( rl r') I 6(r - r'), (A. 18) which specifies (a, 3) as ( = 2 2 2 (A. 19) mn giving 0_ o o o (-0)+N ( N' G(rl r') = mn mn mn mn m=0 n=O -oo mn ( - k ) (- + k ) g g (A. 20) 2 2 2 2 where k = k - k = (waveguide wave number) g mn This is integrated by contour integration; by closing our contour in the lower half plane for the case z > z' we include only the residue from the pole at ~ = k. For z < z' the contour ing cludes the pole at ~ = -k. This convention is the result of assuming ej t time dependence. We have then as a final result O 00 (2-6 ) -r- iz-zZ G(rlmr') = ab k2 rm' +nik ) n'( +k e m=O n=O ab g mn mn (A. 21)

-96top sign for z > z' where we use bottom sign for z < z' and rmn = j k. The functional portion of (A. 21) is determined by substituting for m, n from (A. 9) resulting in m m' + n(+k ) n'(+k (k2 2 2 2 ycs Y Y[kx + 2 j sinkxxsinkxx'cosk ycosk y' (A. 22a) or k 2 A Ay n (k2k y2)] sink xsink x'cosk ycoskyy' (A.22b) A k for the y y component. Equation (A. 22a) has two terms representing respectively the functional part of GH(rl r') and GE(rl r'), while (A. 22b) is the combined form used in (2. 5).

APPENDIX B DETERMINATION OF THE GREEN'S FUNCTION FOR TERMINATED WAVEGUIDE During the process of solving for G(r I r') in Appendix A, the technique of separation of variables was used to solve the homogeneous equation (A. 6) in three dimensions. This concept can also be applied in interpreting G(r I r') to determine dependence upon the spatial variables. In particular the y-dependence corresponds directly to that of a one dimensional transmission line situation with matched terminations. By considering the waveguide phase constant (kg = -j rg) as the phase constant of a hypothetical transmission line, it is possible to investigate the terminated properties of this one-dimensional line and apply the solution to the waveguide; thus resulting in a modified function of GT(r I r') which shall be called the Green's function for terminated waveguide. Consider the one dimensional wave equation d gT(zIz') 2 d +z g gT(zIz) = - 6(z - z') (B. 1) d z which has the solution -97

- 98A j k z + -j k z A e g + B e z > z' (B. 2a) gT(z I z) = - jk9 z -jk z Ae + B e z < z (B. 2b) where A, B, A, B are coefficients to be determined by application of boundary conditions. Here gT(z Iz') is associated with the voltage on the line so that the appropriate termination or boundary condition is (Ref. 19), ZI dgT(Zlz I z) gT(11 z') =j.k dz(B. 3) g(21IzZ k dz c g z =1 with Z1 = effective terminating impedance at f1 and Z = characteristic impedance of the line. A more convenient form for describing the termination is obtained using the concept of complex voltage reflection coefficient p where

-99k1 c Pi 71l + Z (B. 4) Applying (B. 3) to (B. 2a) results in A+ B+ -j 2k f A= B e g 1 (B. 5) Similarly for (B. 2b) we have j 2k 2 B =A jP2 e g (B. 6) where pi, P2 are'reflection coefficients at planes 1 and 2 respectively and'1' 2' are the coordinate values at planes 1 and 2. At z = z' the voltage must be continuous so that g(zz) =I gT(z lz') (B. 7) resulting in a third relationship +j 2k [z [i iV- j2kg(l -z')] j2k (2 -z)] e g g 2 (B. 8) The fourth and final condition requires a step discontinuity in the voltage derivative at z = z' giving

-100B kZe +A g 1[ 2g2 ] - g 2 =k g (B. 9) Solving simultaneously (B. 5), (B. 6), (B. 8), and (B. 9), and substituting in (B. 2) results in j2k -j 2k (1 - _ )> 9 1- g (1..... (B. 10a) gT(z z') = | kj2kg [ ( -2kge1-e2)) J Z<Z (B. lOb) For p1 = p2 = 0 (matched conditions) this reduces to -j k Iz-z' I -r Iz-z e g g g'(z z')= j2k - 2 rg (B. 11) g which is the one-dimensional free space Green's function. The multimode case must consider a summation of effects; therefore

r -r g mn and -r Iz-z'I mn go(zl(zlzG(zlz) = Z e G. (z 2') (B. 12) o z 2 IF mn mn representing the portion of G(rlr') which is a function of z. By direct association between (B. 12), (B. 11) and (B. 10) we see that the modification necessary to consider terminations for the waveguide is the term in brackets in (B. 10). However, this generalized form is unnecessary since the present application establishes both the source and observation points to be in the z = 0 plane of the waveguide, requiring the substitution z = z' = 0. In addition it is more convenient to define!1 and f2 as distances (magnitude only) from z = 0 to the termination planes. Therefore mn 1 mn 2 + Plmne 1 + P2mne ( ) = ) T. (B. 13) w rite GT~rl r') - G~r[ r') T. (B. 14)

APPENDIX C COMPUTER PROGRAM FOR THEORETICAL IMPEDANCE CALCULATION The driving point impedance ZR is calculated using (3. 11) and the summation limit criteria developed in Section 3. 3. Only the matched condition is considered so that Z = Z. Multiple Tmn mn loops are used in the program allowing variation of all configuration parameters plus frequency. Plots of impedance versus frequency are obtained along with individual value listings for each complete parameter set. The obstacle impedance elements (3. 10) can also be calculated but only relative to the H10 mode. A complex circuit is used to represent ZG to allow great flexibility in program use. This is seen in Fig. C. 1. Simpler circuits are represented by setting the appropriate elements in Fig. C. 1 equal to zero or infinity. The waveguide input impedance ZIN (see Fig. 3. 3), normalized to Z is also determined for the terminating condition of a match, short, or open on the opposite waveguide arm. All of the information concerning parameter values and iterations is contained in the necessary "data deck"which must follow the program. The description for this "data deck' format is given below. This is followed by a listing of the full program. -102

-103ZG => TCP c Fig. C. 1 Gap impedance representation for use in the computer program.

-104Data Deck Format Card Format Description 1 2513 Fifteen numbers, all > 1 representing the number of iterations for each variable; plus the branching code 5 includes obstacle program ZQT = 1 skips obstacle program Order of variable iterations: M, N, H', G', S', W', A, B, C, D, CP, L, R, CO, ZQT 2 10F8. 4'Values of H', G'/2 < H' < 0. 500 3 10F8. 4 Values of G', 0.0 < G' < 0.250 4 10F8. 4 Values of S', 0.0 < S' < 0. 500 5 10F8. 4 Values of W', 0.0 < W' < 0.250 6 10F8. 4 Values of A, in centimeters (4.76/C, 2.286/X) 7 10F8. 4 Values of B, in centimeters (2. 215/C, 1.016/X) 8 10F8. 4 Values of C, usually = 0. 0, allows for ex-15 ternal fringing fields at the gap, in 10 farads 9 10F8. 4 RHO, values of D, RHO = -1, 0, +1 D in centimeters 10 10F8. 4 Values of CP, in 10 farads 11 10F8.4 Values of L, in 10 henrys

-10512 10OF8. 4 Values of R, in ohms -12 13 10F8. 4 Values of CO, in 10 farads 14 2F10. 5, FSTART, FINT, FMAX, TITLE; where 14, 14A4 t FSTART + FINT = 1 FREQUENCY value, FINT = frequency increment, FMAX = number of increments, and TITLE = describes frequency range

FOFTPAN IV r C' 1PILF' " A I N C~ - TIIS PR.,R A'/.. I[ OF'SI(;G!FD TO- IGIVE, AP IMPED'ANC- PLOTS (W I THOUT C TF.RIINATI.''NS) AS A FUNCTIr'Nr OF THE MA'JY MOU-tNT PARAMETERS. IT C WMILL GIVE INDIVIDUAL 14.DE PLOTS ALS.. IF MODIFIED SLIGHTLY.. C SFC!,,F.;L9LY IT GIVES NORr.A.L IZE') INPUT IMPEDANCE VALUES AND PLOTS C TO1 A H( 10) } MODE INCIDENT FROM ONE SIDF WITH A SHORTDPEN OR C M.ATCH (_it THE OTHER., VITH COMPLF X LOAD IN THE GAP. OCO COM.M nC..N R. (MN10,15, 101 ),XMN( 10, 15, 10o1) 0002 D IMF N.S I ON FRrO(110),.D 15 00), H ( 1I0),S (5 ), W ( 5), RN(15, 10l1 ),t i XN( 1 5,r 1 01) t,GN( 15, 101),NN( 15,I 101) RLOADI 110,) XLOAD( 110)IA(5), 2 IBIS',FTITLE 4),CAP(lO)4,, LOAD( 110)llO, 3 B L 0 A D ( I10 I),BC A P I I 0v),R P L T (110 0) X P L T ( I 1 ),CF 5 1),( 5 ),D I S ( 2 o), 3~~~~~~~~~~~~~~~~~~~~~~~~~~6 4 CP(5), (5),P (5),C 5),RSN( 22),XSN(2),RTN(22), XTN( 22) 0003 P FA1. L'-0004..................IT.... AXMAX SAXWMAXMAAX AM-AXCMAX7TGMAXCPM —AX- A 1 LMAXRMAX,CGMAXO 0005 MK 99 0006 I FORVAT (2513)1 0007 2 FORMAT (10F8.4) 0008 3 FORMAT (2F10.5,t14,14A4)_.00, 09 4 FC1R.MAT, (1H1 0XIFSTAPT =',FIO. 5, 7X,'FINT =',FIO.5, 7X, I'FMAX =',I4,10XI14A4) 0010 5. FORMAT (H,4X,'MMAX =1',I395X,'NMAX = 13, 55,'AXX,'tIHMAX =',13,5X,1'CGAX =',I3,5X,'SMAXX =', 13,5XWMAX =',I3135XAMAX =',I3,SX, 2'BMAX =',1I3,5X,'CMAX =', 13/5X,'DMAX =',I3,5X,'CPMAX=',1I3,5X, 3'LMAX I',I3,5X,'RMAX =',I3, 5X,'COMAX=', I3,5X,'FLAG =1',13) 0011 6..FOP]MAT -lHO"-,IOX,'GAP HE IGHT VALUES ARE =',10(TF.4,',,)) 0012 7 FORM.AT (IHO,IOX,'POSITION VALUES ARE -=',10(F8.4,' )) 0013 8 FORMA T (IHOLOX,'POST WIDTH VALUES ARE =',10(F8.4,-',' ) 0014 9 FORMAT (lH0DIOXOvG. WIDTH VALUES ARE =',tl0(F84,1,')) 0015 10 FORM'~AT (ItHO,1OX,'G. HEIGHT VALUES ARE',10(FR84,'' ) ) 0016 11 FORMAT (lHl) 0C 1 7 - 12 FOR'MAT (1OX'i - -' M 2, OX,'N ='1 2,IO X'GU I OF W I DTH CM =', I F, 4 1.)X,'GUIDF HEIGHT (CM) ='TF8.4//f)

0 C 1 8 1 3 FJAT( VA i H, t', r' FP F0G F t O CY IN HZ') 14 FORMA0-T ( IH,'9 I lOX,'FPEe (-HZ) IIOXI R N (O IS )',1I0 X t I (GHvS ) / 0 C, 2 C, 1 5 FFOPMFk A T (lI XF1I. I,?XI,.3, 12 Xvf- 8.I,1? XI F %. 3,I I 2X, F9. 3) 00 2 1 1 6 F C. Fk FM 4AV ( IP " ), 1O X,' F P I NJ GE CA P V AAL FJS ARF =', IFI(ER. 4'I J C 22 17 FIU`- A T I X 2' = t 11 X,' I =' I I?,lO 1 0 X' I GUI DE W I Di (, C) =' I F 3 4, l JX H G j FH I (HIT (CM) ='I F. 49LX' 1X =W' F 84/X,' ='4 2 FkP- 4//) 0023 13 FOIAIT Ux, =' 12, lX, C' XpN = I, I?, 1OX,'GUIDE WIDTH (CM) =' I Ff3.4, iJX,'G (U0IDF HFIGCHT (CM) =,FF.4, IC0X,' X 4 =',F 9.4/ 3X, IS=' F 8,,r 1 I X 9I =l,,8'.4rlO,Xrl =ljF8.4j1.3XX,,CAP =I-pF84//) 0024 19 F (RM, vAT U X, =I I'1 2,1IC X,"' I N 2; IC0,'GUX G [ I 9 F W IDTH (CM) M, 1 FB81. 4,t 1 O! X,'G). ~;iJIf)E~ H~E IGf;HT ( CM~~ ) =',t F8.4r,, l C)Xr' W =',I F8.4/ 3X,'S =' 2 F F.4,1OX,'H 0X,F P.'+,4L I X, F 8.4 I'0X,'C AP =',F.4,t I I X v'D iT S' 3 FF.4,1 XOI C P =',F8.4/ 3XT'L ='IF8.41 OX,'P =',F3.4, IOX9 4 ir =', F 34ff) 0025 20 F FlPAT (1HOICIXC A SIZE VALUES ARE I,10(F3. 4,'t') 0026 2 1 FrnPIM AT ( 1HOv 1C Xv'FHFPl FO I STANCEI TO Lfl0AD='"t 10 FRI P4,''1 0027 22 FO RM AT (1,1OH 0,. 0 X'PAC K AGE CAP AC I TAN\CE'10(F.F 4,'')) _ OC28 23 - FnRMAT (lHCIOX9'L-FAD INDUCTANCE =',10(Fl3.4'I,') ) 002129 24 FfIRMAT ( lHOIOX9, SISTANCE VALUES =', 10(F8.4,',')) I 00 30 25 F (IRM AT ( 1c40 (0. Xv,'SEP IF S CAPAC IFAI TAN CEE =i)( F-. 4,,')) 0C;'3 1 26 F CM- I MT( 1140,t 1 OX,' FPFO (GH7 )'I 10X,'IRSN MS (HMS 1), X, XSN (OHMS)', 1 1OX,)t.P'BTN (UHMS),S 1OXXTN (O1H1MS)'//) 0.032 40(I R EA[) 5 I)V YA X, N "AAX v,HM-AAXvG-,GMAAX,SkX,4AXAMAX, HMGAXCMAXf)MAXi 1 CPt AXL'.1AX,P MAX,CCV A,70 A _7_ _ _ _ T 0033 REA'D ( 9,r 2 )(H(K),K=L, HMAX) 0034 REA,-) (5,2) G(KK), 1=i, GMAX) 0(-35 P F AD (5,2 )(S(K),K=i=,SMAX) 0036 READ ( 5,2 )(W(K),K=1,WMA Y __ _ 0037 P An 5 t2 )(AUK)t,K=l,AMAX) 003 3 READAf 5 B ( K) 1,W=l, M RMAX) 0C39 R[An (5,? )(CAP(K),K=1vC MAX) 0C 401 dF AD ( 5,? ) P K I, ( L. I S T ( K ), K= 1, IMA X) 0041 RFA" (5,2) (CP(K),K=1,CPMAX) nC 42 P (F 5,' ) ( K ), K=1 L-! —iAX)

04 3 F A:) H,?) ( Q g)K R:, AX) 0044 P, E:, ) ( C (<K ):,CMK: CIMAX) 0C45 - t 3 F. S AP T _ AXv (FTITLE(K) pK __ _ 4 -0046 P I T fI(6,p4) FSTAPT,9FINIFMAX, (FTITLE(K),,PK=-II1) 0(C47 WP I'r E ( TF(,5 ) IMMAX,INMAXt, WIAX, IGMAX, S AAXWMAXAMAXX tA 8I MAX, CMAXI DMAXt I CPMAXIAIAX,PMAXC0MAX7iQT 3048 WP TTE(696) (H(K),jK =lHMAX) 004q W Rf WPITFI6,20)( G(K)tK=1v GMAX) 305') WRITE(6,7) (S(K),K=1 SMAX) 0051 WRITF(6,F3) (W(K),K=1,WMAX) 0052 WRITE(6,9) (A(K),K=1,tAMAX) 0053 WR __ ITE (6L 0)(R( K),K= IiBMAX) 0054 WR I T E 6,1 16) (CAP (K )K=1 CMAX) 0055 WRI _-E (6 t 2 )_RT RHO I (DISIT( K ) K= 1 t OMA X) - _ 0056 FITEt-(6,22)CP(K),K=lCPMAX) 0057 WRITF(6,23)( L(<),vK=l, LMAX) 0058R IPTE(6,?4)( T ( K ),i2=, I MAXR)MA 0059 WRWRIIE(6,25)( C(K),K= I CMAX) 0060 50 )O 60 KK=I1FMAX 0061 - FPEO(KK) F fSTART + FINT*KK 0062 60 C 0OINT I NUE 0063 DO 900 J =AIvAMAX 0064 D0 950 JR =lFMAX 0065 __O 601 IN =1,NMAX 0066 N = JN-1 0067 90 600 _ =1,MMAX 0068 65 IF (N) 70,70, 7 5 0069 70 DEL = 1.0 0070 GO TO 8 0071 75 DEL = 0.5 0072 80 DO 510 F =1,FMAX 0073 DENI =( M*14.986/A(JA) )**2 +(VN*14.986/3(Jf) )**2-FREQ(F)**2 0074 IF (ABS(DENI).LT..0000001) G0 TO 120 0075 L=376.700* (J9)-*0EL*(FfEQ(F)**?-(Pl 46,98 /e(J8) )**2)f(ALJA)* 1 FF0(F)*SPT(FA S ( 0EN1))) C076 IF (DENL),1?0, 100 _

0077 0.,.,,J:,F) = 7 Oi7~ Rl 4,,JIF) = 0.0 ____ C079 (, Tl, f5_, )....0813 10k) f,'MN(?4,JN,f:) =(.r__ 0081 XMN(M,JN, F) =7 0082 GIT TO 5On 0083 120 x Nrj( W-',JNJF-) = l9tn).O 0084.R..N ( M.... t f =I 1 00.- 0C085 500 CONT INUE _ __ _ _ __ 0086 510 CONTTIN UE 0087 600 CFNTINUE 0088 601 CONTINUE 0089 D- 800 JW =1,WMAX -0-090.............DO 75(C JS =1 SMAX 0091 603 DO 650 JN =1,NMAX 0092 N, = JN- l 0093 DO! 605 F =1IFMAX~ 0094 RN(JN,F) =0.o 0095 XN(JN,F) =0.0 0096 605 CONT I NUE0097 D(, 640 F =l1FMAX )G98 On, a3 I = 1,MMAX 0099 IF (F.GT. 1) GO TO 627 0100 CF (M)=( SINl( M*3. 1416*s( JS) ) *SIN( M*1.5708*W( JW) )/( M* 1.5708*W(JW ) )) **2 __ 0101 627 CONTINUE 0102 RN(JN,F) = RN (JN,F) +CF M)*RMN[(M,JNJ,F) 0103 XN(JNF) = X ( JNF ) +CF(M) *XMNY(MJNF') 0 104 630 COr NTINUE 0105 DEN2 = RN(JNF)**2 + XN(JN,F)**2 0106 GNJN,F) = RN(JN,F)/DFN2_ _ __ _ 0107 N( JN,F) =-XN(JNF) /DEN2 0108 640 CrNT INUE 0109 650 C.ON TI NUE 0110 DO 700 JH =1,HMAX

0 11 I UF 9A JG)=19,GMAX 0112 Dn ('53 F-=1FMAX__ 0 113 3 GI 17AflAP( F) = 3.0 0 114 653 C(NTIN ___ _ I I 1 5 j)I (CV J ~ =1 vC ~ -, X ~ _ _ _ _ _ _ _ _ _ _ _ 0116 WPIhI (6,11) 0 1 1 7 W I TF (6, 1.) F K,9 MK,9 A (JA)vB (J LI),W(JW) S (JS),H( JH) G(JG -P ),rCAP (J C) 118 IisWRITE (6,14) 0 1*19 Dr 655 F=1,FMAX 0120 BLOAD(F) 0.0 0121 6 5 C ONTINUE -_ _ _ _ _ _ _ _ _ _ _ _ _ _ 0122 DO 690 F=1,FMAX 0123 DO 680 JN=lNMAX 0124 N=JN- I_ 0125 IF (ON.EQ. 1) GO TO 658 0126 _ CFN (COS(N*3.1416*H(JH))*SIN( N*1.5708*G(JG))/(N*.1.5708*G(JG))) 1 **2 0127 GO TO 659 0128 658 CFN = 1.0 - 0129 659 IF(JC.GT. 1) GO TO 660 0130- GLOAD(F) = GLOAD(F) + CFN*GN(JNF) 0131 660 IF (JN.GT. 1) GO TO 670 0132 BCAP(F) = 6.2832 *FREQ(F)*CAP(JC)/(10.**6) 01-33, BLOAD(F) = BCAP(F) + CFN*BN(JNF) 0134 GO TO 6R0 0135 670 BLOAD(F) BLOAn(F) +CFN*BN(JN,,F) 0136 680 CONTINUE 013 OEN3 = GLOAD(F)**2 * BLOAD(F)**2 0138 RLOAD(F) = GIOAD(F)/DEN3 0139 YLOAD(F) = -BLOAD(F)/DEN3 0140 WPITE (6, 15) FRFQ(F),RLO0AD(FhvXLOAD(F) 0141 690 CONTINUE 0142 WRITF_(6,14) 0143 WRITE (6,11) 0144 WRI TE (6, 18) MK9MK, A (JA) B(J3B) W(JW) t S (JS ),H(JH)_, G(JG),PCAP(JC)

0145 CALL PLCT2 ( D, F FP Q F(AX), FR F[( 1 ), 300o. 0,-200.0) 0146 CALL PLOT.3 ('Rt',FP FQ, RtP. A FFil AX,4)] 0)147 CALL PLOT3 (I X',FQEQ,XLCAt,FMAAX,4) 0148 CALL PLOT 4 (1 6,I6H HfEAL & IT AG, TI ARY) 0149 6q5 C.NT ITNI T115n0 _F ( 7T. T. 3) G, ( TO 699 n.1 i1 Dn 150n.C- 1,cpMAX 12 n 149 Kl.=LMAX ___ _ __ ___ __ 1153 DO'148,9 KP=1!,t PMAX 01 54 -DC" 1470 KC = I, C f7MAX.01155 nor 1460 K-=,lOf)MA X 01l56 WRIE 61) ____ __IT...... 01 57 WR. TE (6t19) MKMK, 4( JA),BJ-B3), W(JW),S ( JS H(JH), GA(JG),CAP(JC), 1 DIST(IKD)),CPIKCDP),L(KL,RI(KR),C(KC) n158 WR ITE (6,26) 0159 DO 1300 F=1,FMAX 6n PHI =2* IST( K n )*SQPT(ABS(. 2096*(FR EO( F **2- ( 14 986A I JA) ) **2 ) ) 01 61_ RMN( 1, 1F)=376.'7, B J B)/(A(JA)*QRT A; 1sl.-( 14.99/A( JA *FREO( F)! 1 **f2) ) 0162 DFJ11 = 1.0 + 2.0*PH0*COSfPHI) + RHnt*RHn 0163 GL = (.n-PRHn,*RH) /(2.0,*MN(1,1,F)*DEN1O) 0164 BL = RHn.*STN(PHI)/(RMN(I,l,FJID) *PEN1 l __ 0165 OnMEGA = 6.2832*FREQ(F)/(10.**3) 0166 Dn__ 1 = (nMEGA**2CKC. C P(KCP)*R (KR) )**2 +(OMEGA*(C(KC)+CP KCP) ) 1. - OMFGA'**3*C(KC)*CP(KCP)*I. (KL)*1O..**'3)**2 01 67 PG = (M EGA*C KC ) )*2 *R (K R) / DEN 1 1 0168 XG = -(OMFGA**5*C(KC)**2*CP(KCPKC*(L(KL)*1.**3)**2 + MEGA*( C(KC 1 + CP(KCP)) + tMEEGA**3*(C(KC)**?*CP(KCP)*R(KR) C(KC)**2*L(KLt)* 2 f10.**3 -2, 0*C(KC)*CP(KCP)*L(KI )**1,***3))/DEN11 0169 DENI? = RG*RG + XC.*XG 0170 GG = RG/DENt2.171 BcG = -XG/DEN12 017?2 BCA(F) = 6.2832 *FREQ(F)*CAP(JC)/( 10.**6) 0173 GAPO = (SIN1..7.08,*G Jt;?/l.578*GIJG)) )*2 0174 BG1 = fG + BLOAD(F) - RN(l.,F)*GAPO + BCAP(F) n175 ___ nFN13 = GG*GG + BG1!*P.1

0176 PP1 = GG*GAPO/)EN13 0)1. 77 XPI = -R*G AP0)/POEN1i3 1178 PS1 = RoIICF(1) 0179 XSI = ((XPI + XN(1,F))/CF(l)) -XMN(1,1,F) 180n R F.= SNF) S 1(P MN ( 1,I F)2. 0~ 181 XSN(F) ).SI/(RMN(l,1lF)*2.fl) 0182 DENL4 RSI.*RSI + XS-1*XS1'1 83 GT = GL + S1/DEN4 P184 8T L- )(S1 /-DFN14 18 5 DEN1; (T*G'T + 8 T*RT 0186 RTN(F) = GT/(DFN15*2o*RMNN(l,1,F)) 0187 TN'((F) = -BT/1(DEN15*?.*PMN(1,, F)) 01883 WRITE (6915) FREQ(F),PSN(F),XSN (F),RTN(F),XTN(F) 0189 13T1 CONTINUE 0190 WRITE (6,26) 01~ r 9 1, WRITE (6,11) - 0192 WRITE (6,19) MKMKA(JA), B(JB), W(JW), S(JS),H(JH),rG(,JG), CAP(JC), 1 DIST(KO)),CP(KCP) L(Kt)R(KR) C (KC) 0193 CALL PLO'T? (D,FREQ(FMAX),FPFQ(1),?.5O0) _ __ 0194 CALL PL T3 - (''FR EQ0,PSN, FMAX 4) 0195 CALL Pt0T3 ('X~ FPFQ,XSNqFMAX,4) 0196 CALL PLOT3 ('S'FPEQT PTNFMAX,4) 01 97 CALL PL OT3 ( IY V' 0 F F R F 0, XTN IF ) AX 94 )')198 CALL PLOT4 (16,l6HREAL 9!MAGINARY) 9 i. 99 ~)1460 CONTINUE 0200) 1470) CONT I NUE 0201 14 8 C ONT INUE 02n2 1490 CONTINUE n2n3 1500) CONTINUE 0204 699 CONTINUE'3205 700I CONPT INUE n206 750 CONTINUE 0207 8=00 CONTINUF 0208 851) CONTINUE 020~9 O"~sC0 CC?~IONT! NUE 0?10 GO TO 40 0-?11 _[T)_

APPENDIX D DETERMINATION OF APPROXIMATE AND LIMITING VALUES FOR SMALL COAXIAL LINE DISCONTINUITIES The necessity to support the center conductor of a coaxial line generally results in the presence of small discontinuities in the line due to the supporting beads. The disruptive effect of such a support bead has been minimized considerably in the last few years with the development of precision 7 mm connectors; however the effect has not been totally removed. Additional discontinuities are introduced when the size or nature of the line is altered. Both of these types of discontinuities, when small, can be approximated by a lumped shunt capacitor. First order compensation is then possible by introducing a small inductance of the proper size adjacent to the discontinuity in series in the line. In the measurement circuit there are many such discontinuities present, all somewhat compensated to reduce their effect. All of the discontinuities result from step changes in the diameter of the inner or outer conductor. The uncompensated effect can readily be characterized by the equivalent capacitance of such a step using the chart provided in the Microwave Handbook (Ref. 28). A compensated discontinuity will have a somewhat smaller effective capacitance, depending on the design of the circuit; the handbook value thus represents an upper limit. -113

Consider the circuit of Fig. D. la where the shunt capacitance is small but fixed, we desire to choose the series inductance so that the input will see a match or Zc the characteristic impedance of the line. If IN RIN+ j IN (D. 1) then X 0.0 and R Z RIN c Z for the condition 2 Lo Z CD (D.2) This is not totally unexpected since a differential length of transmission line satisfies the same relationship, e.g., c 4 (D. 3) where L., Ca are inductance and capacitance per unit length. With this in mind, it is possible to interpret the compensated discontinuity as a length of transmission line X,

-115Lo _L A ZIN C ZC (a) ZCe ZIN 2> Z (b) Fig. D. 1 Line length equivalence for a compensated discontinuity. (a) Discontinuity model. (b) Equivalent length of Zc line.

-116L _ = 0 (D. 4) L~ as shown in Fig. D. lb. A partially compensated shunt capacitance would result in a reduced capacitance with a short length of line; while an over compensated capacitance would look like an inductor in series. Both of these conditions are seen in the equivalent circuit of Fig. 4. 3. For an example let us consider one of the step discontinuities in the 7 mm to 1. 65 mm adapter. The step considered results from changing the diameter of the inner conductor to match a 50 ohm airline to a 50 ohm Teflon line, as shown in Fig. D. 2. To compensate, a short length of the Teflon side is without Teflon, thus increasing the characteristic impedance in order to appear inductive relative to 50 ohms. The discontinuity capacitance will be somewhere between 8. 8 and 17. 9 x 10 farads (Ref. 28) depending on the distribution of the fringing fields between the air and the Teflon. This distribution will depend on the length t of the high impedance section between the two 50 ohm sections. This short section of 65. 2 ohm line can be considered as 50 ohm line with an excess inductance of 2. 47 n H/inch. The required compensating inductance for CD will be DL =(50) (8.8 x 10-15)= 22.0 pH (D.5)

outer conducter CD \ \ \\: ~~center - -o -\ conducter 0.120" 0276"T t o.010" 0,84 /A / ~~~/ /< ~~ ~ A,'Teflon (not to scale) Fig. D. 2 Compensated step discontinuity in coaxial line.

-118assuming all the fringe fields are in the air. Then from (D. 4), 22. 0 x 10 t 22.0 x -9 = 0. 009 inch 2.47 x 10 representing the minimum compensating length. If we had assumed all of the fringe fields were in the Teflon, then Lo = 44.8 pH and t = 0. 018 inch. In the adapter used, t = 0. 010 inch by measurement, leading us to believe that the discontinuity might be slightly under compensated, since it is reasonable to expect that most of the fringe field will in fact be in the Teflon. As it turns out, the effective capacitance was 4. 0 x 10 farads, determined by the statistical procedure outlined in Section 4. 3. 1, and is shown as the center capacitor in the 7 mm to 1. 65 mm adapter in Fig. 4. 3. This procedure was used to establish approximate values for all of the various discontinuities in the measurement circuit.

APPENDIX E COMPUTER PROGRAM FOR EXPERIMENTAL DATA INTERPRETATION A computer program was written to provide an accurate and rapid means of interpreting the data from the impedance measurements. By knowing the standing-wave-ratio, minimum position and frequency, in conjunction with the measurement circuit, it is possible to specify the impedance terminating the circuit which would generate the measured data values. For convenience, the measurement circuit is broken up into a number of smaller circuits, each of a standard form to simplify the mathematics. The impedance translation through the measurement circuit can then be accomplished by a repetitive operation through each of these standard circuits. Fig. E. 1 describes the standard circuit used. Where appropriate various element values are set equal to zero to properly represent a given section of line. The measurement circuit in Fig. 4. 3 is made up of eight such standard units. The results of the program are both listed and plotted as real and reactive parts versus frequency. -119

-120I|_ LEN I LI Zc = 50 ohms C, = == C2 LEN = length of line in centimeters CI, C2 = capacitances in 10-15 Farads El = inductance in 10o12 Henrys Fig. E. 1 Standard circuit unit for data interpretation program.

___ _._~__~ ~ C _rT1HIS PROGRAM T AKES MEASURFMFNT DATA AS INL SWR& __NLSR...... C FREQUENCY FROM SLCTTED LINE; FIRST CURRECTIN" THE SWRTHEN C TRANSLATING DOWN THROUGH LENGTHS OF LINE PLUS PI NETWORKS UNTIL C WE GET A RESULTANT IMPEDANCE. PROVISIONS FOR VA IOUS PLOTS C ARE MADE. WE GET OUT FREQ,WAVL,SWR(DB)SWRWRSWRM,RHOTHETA, C RLUAD, AND XLOAD. 0001 D IMENS ION TITLE ( 20),FREQ(99),RLOAD(99 XLOAD99), E153),RH 99 1 THETA(99 ),TL(9),L(9),C1(9 ),C2(9) tT.2(2 0) 0002 REAL MINLtL 0003 1 FORMAT (12,2X,19A4) 0004 2 FORMAT (11HO0,I,'NUMBER OF DATA PUINTS =',I2,15Xt19A4) 0005 3 FORMAT (LHO,2X,1OHFREQ (GHL),3X,15HWAVELENGTH (CM),4X,4HMlNL9tX, I 7HSWR( DB,5X,3HSWR, 5X 4HSWRM, 5 X, 3HRHO, 8X, 5HTHETA,5X 5HRLOAD, ___ 2 5Xt 5HXLOAD// ) 0006 4 FORMAT (5XFt7. 36XF7.4,i7X l F7.4 8X, Fo.2,4XF6.2,4XF6.2, 3Xt 1 F6.4, 4X, F7.2,4X, F8.3,3X,F 8.3/ 0007 5 FORMAT (F8.3,2X,F8.4,2XtF5.2) 0008 6 FORMAT ( IHl) 0009 7 FORMAT ( 1H, tlOX,' NUMBER OF LINE SECTIONS =', 12,5X,19A4) 0010 8 FORMAT (F7.3t3XF6. 1,4XF61,4X F6 1) 0011 9 FORMAT(lHO,60X,'FREQUENCY IN GHZ' ) 0012 READ (5,1 )LMAX,(TITLE(J) J=1 19) 0013 WRITE (6,6) 0014 DO 20.I=1,LMAX 0015 READ (5,8) TL(I),C(I),L(I ),C2(I) 0016 20 CONTINUE 0017 WRITE ( 67) LMAX, ( TITLE(J),=1, 19) 0018 45 READ (5,1) NN,(TI2 (J),J=1,19) 0019 46 WRITE(6,2) NN,(TI2 (J),J=1,19) 0020 47 WRITE (6,3) 0021 48 00 50 JJJ=1,NN __ 0022 RLOAD ( JJJ) =5000. 0023 XLOAD (JJJ) =5000. ___ ___ 0024 50 CONTI NUE 0025 DO 200 K=1,NN

0026 READ (5,5) FREQ(K),MINLDB 0027 WAVL = 29.875/FREQ(K) 0028 SWR = iO0O**( D /20o0) 0029 ER = 0.3 +.01*(FREQ(K)**1*45) 0030 FAG = 10.0**IALOGIO((SWR+l..:)/ISWR-1.j )-ER/20.) 0031 SWRM = (FAC+1.O)/(FAC-1.O) 0032 G = SWRM 0033 B = 0.0 __ _ __ 0034 P HI = 6.2 832*M I N -TL(l)) I WAVL 0035 DO 75 KK=1,LMAX 0036 IF (KK:Q. 1) } GO TO 62 0037 60 PHI: 6.2832*TL(KK)/WAVL 0038 62 DEN1 = (B*SIN(PHI) +COS(PHI I)**2 +(G*SIN(PHI )**2 0039 G = G/DEN1_ __ 0040 B1 = (B*COS(2.*PHI) +.5*S INZ*.PHI *( B*B+G*G-G 1I /DENI 0041 FTR =-(1.-( 395E-8)*FREQ(K )**2*Cl(KK)*L(KK +.30012566*FRE,( 1) *L(KK ) *BI 1) 0042 DEN2 = (.00012566b*FREQ(K )*L(KK)*G)l**2 +FTR**2 _ o 0043, G2 = G1/DEN2 0044 82 =(((3.95E-'8)*FREQIK )**2*C2(KKI*L(KK)*Bl-(.24E-11)*FREQ(K ) _ _ _.__....... _._._**3*.C1_KKL*L (KK*,C2. 16 F R F q K____ (KK(2_-. 2 B1)*FTR +.00012566*FREQK )*LKK)*Gl*** 2 ( 1 -(3.95E- 8 )*FR E Q(K 3**2*L(KK) *C2( KK) ) )/DEN2 0045 G = G2 0046 B = B2 0047 75' CONTINUE 0048 80 DEN3 = G2*G2 +.2*82 0049 RZ = G2/DEN3 0050 XL = -82/DEN3 0051 TEST = RZ*RZ + XZ*XZ -1. 0052 RHO(K = SQRT(TEST**2 +4.*XZ*XZ)/((RZ +1.)**2 +XZ*XZ) 0053 IF (ABSITEST) LT..001) GO 10 93 0054 THETA(K ___ = ( 180./3.14'16)*ATAN2(2.*XZ, TEST) 0055 GO TO 92 0056 90 IF (XZ) 91,91,93 0057 91 THETA(K ) = -90.0

0058 92 IF (THETA(K ))94995,t95 0059 93 THETA(K ) = 90.0 0060 94 THETA(K ) = THETA(K + 360. _0 0061 95 RLUAO(K ) = RZ*50. 0062 XLOAD(K( ) = XZ*50 _ 0063 100 WRITE (6,4) FREQ(K)9WAVLt4INLvOBSWRSWRMtRH0(K)jTHETA(K)v 1 RLOAD(K),XLOAD(K) 0064 200 CONTINUE 0065 W~ARITE (6 3 0 0 6 5 -- _ _ _c. _ WRITE (6,3)~ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __.__ __ __ 0066 WRITE (6,6) 0067 300 CONTINUE 0068 CALL PLOT2 (E,22*0,2*Q,30*0 0-20*3.) 0069 CALL PLOT3 VIR',FREQ,RLOADNN,4) 0070 CALL PLOT3 (lX' FREQXLOAD9NN,4) 0071 CALL PLOT4(lbI6HREAL & IMAGINARY) 0072 WRITE (6,9) 0 0 7 3 _ _ _ _ _ _W R I T E _(6, 6 ) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 0074 GO TO 20 0075 END

RE FERENCES 1. J. A. Copeland, "Theoretical study of a Gunn diode in a resonant circuit, " IEEE Trans. on Electron Devices, Vol. ED-14, No. 2, February 1967, pp. 55-58. 2. W. J. Evans, "Circuits for high-efficiency avalanche-diode oscillators, " IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-17, No. 12, December 1969, pp. 10601067. 3. D. H. Steinbrecker, "Efficiency limits for tuned harmonic multipliers with Punch-Through Varactors, " International Solid-State Circuits Conference Digest of Technical Papers, University of Pennsylvania, Philadelphia, February 15-17, 1967. 4. W. J. Getsinger and A. H. Kessler, "Computer-design of diode using microwave components, and a computerdimensioned, X-band parametric amplifiers," Microwave Journal, March 1969, pp. 119-123. 5. MIT Radiation Laboratory Series, McGraw-Hill, New York Vols. 1-28, 1946-1948. 6. A. Uhlir, Jr., "The potential of semiconductor diodes in high frequency communications, " Proc. IRE, Vol. 6, No. 6, June 1958, pp. 1099-1115. 7. D. Leenov, "Gain and noise figure of a variable-capacitance upconverter," B.S.T.J., July 1958, pp. 989-1008. 8. H. Heffner and G. Wade, "Gain bandwidth and noise characteristics of the variable-parameter amplifier, " J. Appl. Phys., Vol. 29, No. 9, September 1959, pp. 1321-1331. 9. B. J. Robinson, "Theory of variable-capacitance parametric amplifiers, " IEE Monograph, No. 480 E, November 1961. 10. K. Kurokawa and M. Uenohara, "Minimum noise figure of the variable-capacitance amplifier," B.S. T. J., May 1961, pp. 695-722. -124

-12511. D. Leenov and A. Ulhir, Jr., "Generation of harmonics and subharmonics at microwave frequencies with P-N junction diodes," Proc. IRE, Vol. 47, No. 10, October 1959, pp. 1724-1729. 12. D. B. Leeson and S. Weinreb, "Frequency multiplication with nonlinear capacitors - - A circuit analysis, " Proc. IRE, Vol 47, No. 12, December 1959, pp. 2076-2084. 13. W. J. Getsinger, "The packaged and mounted diode as a microwave circuit," IEEE Trans. on Microwave Theory and Techniques, Vol. 14, No. 2, February 1966, pp. 58-59. 14. W. J. Getsinger, "Mounted diode equivalent circuits," IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-15, No. 11, (Correspondence), November 1967, pp. 650-651. 15. E. Yamashita and J. R. Baird, "Theory of a tunnel diode oscillator in a microwave structure, " Proc. IEEE, Vol. 54, No. 4, April 1966, pp. 606-611. 16. D. C. Hanson and J. E. Rowe, "Microwave circuit considerations of bulk GaAs oscillators, " IEEE Trans. of Electron Devices, Vol. 14, No. 9, September 1967, pp. 469-476. 17. P. S. Carter, "Circuit relations in radiating systems and applications to antenna problems, " Proc. IRE, Vol. 20, June 1932, pp. 1004-1041. 18. R. E. Collin, Field Theory of Guided Waves, New York: McGraw-Hill, 1960, pp. 39-40. 19. C. T. Tai, Dyadic Green's Functions in Electromagnetic Theory, International Text Book Company (in press). 20. H. Jasik, Antenna Engineering Handbook, McGraw-Hill, 1961, Chapter 3, p. 7. 21. S. A. Schelkunoff,'"Impedance concepts in waveguides," Quart. Appl. Math., Vol. II, April 1944, p. 7. 22. M. K. McPhun, "Comparison of TEM with waveguide-below cutoff resonators," Electronic Letters, Vol. 5, No. 18, September 4, 1969, pp. 425-426.

-12623. S. P. Cohn, "Characteristic impedance of the- shielded-strip transmission line, " Trans. IRE, Vol. MTT-2, July 1954, pp. 52-57. 24. R. L. Eisenhart, "Derivation of parametric multifrequency coupling coefficients, " Proc. IEEE (Letters), Vol. 58, No. 3, March 1970, pp. 495-496. 25. P. J. Khan, "Advanced phase-shift amplifier study, " Tech. Report ECOM-0029-F, Contract No. DA 28-043 AMC-00029(E) Cooley Electronics Laboratory, The University of Michigan, Ann Arbor, May 1967. 26. N. Marcuvitz, Waveguide Handbook, Vol. 10, MIT Radiation Laboratory Series; 1951, McGraw-Hill, pp. 227-228. 27. A. Sommerfield, "Partial differential equations in physics," Lectures on Theoretical Physics, Vol. VI, Academic Press, New York, N.Y., 1967, p. 189. 28. Microwave Engineers Technical and Buyers Guide Edition, Horizon-House, Dedham, Massachusetts, 1969, p. 36.

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Security Classification DOCUMENT CONTROL DATA- R & D (St'ectrity classification of title, body of abstract and indexing annotation must be entered when the overall report is classified) 1. OR I G I N A TI N G ACTIVITY (Corporate author) 2a. REPORT SECURITY CLASSIFICATION Cooley Electronics Laboratory University of Michigan 2b. GROUP Ann Arbor, Michigan 48105 3. REPORT TITLE Impedance Characterization of a Waveguide Microwave Circuit 4. DESCRIPTIVE NOTES (Type of report and.inclusive dates) Cooley Electronics Laboratory Technical Report No. 208 5. AU THO R(S) (First name, middle initial, last name) Robert L. Eisenhart 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS February 1971 28 8a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) DAAB07-68-C-0138 01482-21-T TR208 b. PROJECT NO. 1H021101 A042. 01.02 - C. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) L d. it ECOM-0138-21-T 10. DISTRIBUTION STATEMENT This document is subject to special export controls and each transmittal to foreign governments or foreign nationals may be made only with prior saproval of CG, U. S. Army Electronics Command, Fort Monmouth, N. J. Attn: 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U.S. Army Electronics Command Fort Monmouth, New Jersey 07703 Attn: AMSEL-WL- S 13. ABSTRACT The induced e. m. f. method has been extended and applied to derive the driving point impedance of a common waveguide structure used for mounting small microwave devices. The resulting mathematical relationship has been conceptually interpreted as an equivalent coupling circuit, terminated by a set of impedances which are associated with the many modes within the waveguide. Properties of this circuit and its terminations are discussed in detail. In addition the multilateral nature of the circuit allows consideration of the mount in the waveguide as an obstacle to any incident propagating mode. The driving point impedance of this mount was also considered from the experimental viewpoint. An investigation was carried out to check and support the results of the theoretical analysis. A novel measurement technique was employed, based upon the use of subminiature coaxial line to gain electrical access to a terminal pair located inside the waveguide. An extensive model of the measuremen circuit was developed, which enhanced the accuracy of the data interpretation, and provided excellent agreement between these values and the theory. It is anticipated that this formulation will permit accurate design of many components which previously required empirical methods based on limited experimental data. DD1 NOV 651473 (PAGE 1) S/N 0101 -807-681 1 Security Classification A — A- 31(08

Securitv Classification 14. LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Waveguide device mounting structure Driving point impedance Equivalent circuit Multimode Mode impedance Obstacle impedance Impedance measurement I f:1 DfD f FOR e 473 ( BACK ) S/N 0101 -807-6921 Security Classification A- 31409

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