2L- q -5 DISCONTINUITY CHARACTERIZATION IN SHIELDED MICROSTRIP: A THEORETICAL AND EXPERIMENTAL STUDY by Lawrence Patrick Dunleavy A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering) in The University of Michigan 1988 RL-985 = RL-985

ABSTRACT DISCONTINUITY CHARACTERIZATION IN SHIELDED MICROSTRIP: A THEORETICAL AND EXPERIMENTAL STUDY by Lawrence Patrick Dunleavy Chairperson: Pisti B. Katehi The need for more accurate modeling of microstrip discontinuity structures has become apparent with the advent of Monolithic Microwave Integrated Circuits (MMICs) as well as the push to higher millimeter-wave frequencies. The development of accurate microstrip discontinuity models, based on full-wave analyses, is key to improving circuit simulations and reducing lengthy design cycle costs. In most applications, radiation and electromagnetic interference are avoided bv enclosing microstrip circuitry in a shielding cavity (or housing). The effects of the shielding can be significant, even when the top of the cavity is several substrate heights above the circuitry. Shielding effects are not adequately accounted for in the discontinuity models used in most microwave CAD software. A new integral equation method is described for the full-wave analysis of shielded microstrip discontinuities. The integral equation is derived by an application of the reciprocity theorem and then solved by the method of moments. Two types of circuit excitation are considered: a coaxial excitation method developed here, and the widely used gap generator method. Several numerical experiments are presented that lead to very useful relationships governing the convergence and stability of the method of moments analysis of this problem. Most significantly, an optimum sampling range is defined that provides the most accuracy in the matrix solution.

Numerical and measured results are presented for microstrip effective dielectric constants, and the network parameters of open-end and series gap discontinuities, and two coupled line filters. For the effective dielectric constant, as the size of the shielding cavity is reduced, the difference between the numerical and CAD package results becomes significant. Conversely, for an open-end discontinuity, choosing a smaller shielding cavity extends the frequency range over which the equivalent capacitance is relatively constant. The numerical results are in excellent agreement with measured data obtained using a variation of the thru-short-delay de-embedding technique. A case in point is that of a four resonator coupled line filter. The numerical results accurately predict the filter performance in every way, whereas discrepancies are observed in the CAD model predictions. It is demonstrated that these discrepancies are due to shielding effects.

To my parents and to the memory of my brother Michael ii

ACKNOWLEDGEMENTS I have many to thank for helping me along the Ph.D. journey. First I thank my advisor, her constant optimism and good nature greatly eased the difficult times. I also extend my sincere gratitude to the other members of my dissertation committee for their time and their consideration. There are several other professors whose inspiration and advise have left their mark on my career. Among these are Dr. T.B.A. Senior, Dr. Chen-To Tai (my favorite tennis rival), Dr. Val Liepa, and Dr. Emmett Leith. I also thank Dr. W.A. Fordon and Dr. Jon Soper of Michigan Technological University. I am indebted to my friends and colleagues who contributed in various ways to this work especially Mile. Emilie van Deventer, Mr. Tim Peters, Mr. Marcel Tutt, Mr. Mike Dunn, Ms. Jeanette Vecchio, and Mr. Jim Morgan. The insightful conversations held with Dr. Tom Willis, Mr. Norm VandenBerg, and Mr. Tom Livernois helped form some of the ideas presented. I also thank Mr. Jim Schellenberg, Mr. Ed Watkins and Mr. Doug Dunn of Hughes Aircraft Company for their inputs to the experimental part of this work. Finally, I thank my family. Though they openly feared I would be a professional student for life, my parents, brothers, sisters, and other relatives always gave love, support, and encouragement. For this, I also extend my appreciation to my friends The Buschellmans, The van Deventers, The Heeremas, and The Linquists. iii

This work was sponsored in part through a Howard Hughes Doctoral Fellowship. Additional support was provided by The National Science Foundation (Contract. No. ECS-8602530) and the Army Research Office (Contract No. DAAL03 -87-K-0088). iv

TABLE OF CONTENTS DEDICATION................................ ii ACKNOWLEDGEMENTS......................... iii LIST OF FIGURES............................. ix LIST OF TABLES.............................. xv LIST OF APPENDICES.......................... xvi LIST OF SYMBOLS.................................... xvii CHAPTER I. INTRODUCTION.......................... 1 1.1 Motivation 1.2 Thesis Objectives 1.3 Accomplishments 1.4 Brief Review of Theoretical Approaches 1.4.1 Quasi-Static Techniques 1.4.2 Planar Waveguide Models 1.4.3 Rigorous Full-Wave Solutions 1.5 Description of Theoretical Methods 1.6 Description of Experimental Study II. THEORETICAL METHODOLOGY.............. 13 2.1 Assumptions 2.2 Method of Moments Formulation v

2.2.1 Application of Reciprocity Theorem 2.2.2 Expansion of Current with Sinusoidal Basis Functions 2.2.3 Transformation of Integral Equation into a Matrix Equation 2.3 Derivation of the Green's Function 2.3.1 Geometry and Electromagnetic Theory 2.3.2 Solution to Boundary Value Problem for the Green's Function 2.4 Impedance Matrix Formulation 2.4.1 Evaluation of the Electric Field Due to the Test Currents 2.4.2 Evaluation of the Impedance Matrix Elements 2.5 Excitation Vector Formulation 2.5.1 Coaxial Feed Modeling by an Equivalent Magnetic Current 2.5.2 Evaluation of the Magnetic Field at the Aperture 2.6 Current Computation for Two-Port Structures 2.6.1 Application of Reciprocity Theorem for Dual Excitation 2.6.2 Expansion of Current and Modified Matrix Equation 2.6.3 Modifications to Impedance Matrix 2.6.4 Modifications to Excitation Vector 2.7 Determination of Network Parameters 2.7.1 Network Parameters for One-Port Discontinuities 2.7.2 Network Parameters for Two-Port Structures 2.8 Summary of Theoretical Methodology m. COMPUTATIONAL CONSIDERATIONS..........55 3.1 Formulation for Computer Solution 3.1.1 Formulation to Compute Impedance Matrix [Z] 3.1.2 Formulation to Compute Excitation Vector [IV] 3.1.3 Defining the Strip Geometry 3.2 Algorithm for Current Computation 3.2.1 Input Data File vi

3.2.2 Computation of LN(n) and MN(n) Vectors 3.2.3 Computation of the Impedance Matrix and Excitation Vectors 3.2.4 Computation of the Current Distribution 3.3 Algorithms for Computing Network Parameters 3.3.1 Algorithm to Compute One-port Network Parameters 3.3.2 Algorithm to Compute Two-port Network Parameters 3.4 Convergence Considerations Zqp and Vq 3.4.1 Convergence of Impedance Matrix Elements Zqp 3.4.2 Convergence of Excitation Vector Elements Vq 3.5 Convergence of Network Parameters 3.5.1 Numerical Experiment A: Effect of K-value 3.5.2 Numerical Experiment B: Leff, seff Convergence on n and m 3..5.3 Numerical Experiment C: Optimum Sampling Range 3.6 Summary of Computational Considerations IV. EXPERIMENTAL METHODOLOGY............. 96 4.1 Discussion of Experimental Approach 4.1.1 ANA Error Correction 4.1.2 Difficulties with Microstrip Measurements 4.1.3 Resonator Techniques 4.1.4 The De-embedding Approach 4.2 Comparison of De-embedding Methods 4.2.1 Fixture Equivalent Circuit Modeling 4.2.2 Time Domain De-embedding 4.2.3 Full Matrix De-embedding 4.3 Implementation of TSD De-embedding 4.3.1 Software and Hardware Considerations 4.3.2 Connection Approaches For TSD De-embedding 4.3.3 Measurement of Effective Dielectric Constant 4.3.3 Measurement of Open-end Discontinuity 4.3.4 Measurement of Series Gap Discontinuities vii

4.4 A Perturbation Analysis of Connection Errors in TSD Deembedding 4.4.1 Basic Approach to Perturbation Analysis 4.4.2 Perturbation Analysis and Results 4.4.3 Connection Errors in seff and Open-end Measurement 4.4.5 Connection Errors in Mieasurement of Series Gap Discontinuities 4.5 Summary of Experimental Methodology V. NUMERICAL AND EXPERIMENTAL RESULTS.....140 5.1 Cutoff Frequency for Higher Order Modes 5.2 Effective Dielectric Constant Results 5.3 Results for Open-end Discontinuity 5.4 Results for Series Gap Discontinuities 5.5 Results for Coupled Line Filters 5.6 Summary of Numerical and Experimental Res-nlts VI. CONCLUSIONS AND RECOMMENDATIONS.......167 6.1 Conclusions from Theoretical Work 6.2 Conclusions from Computational Work 6.3 Conclusions from Experimental Study 6.4 Conclusions Based on the Results 6.5 Recommendations APPENDICES.............................................. 174 BIBLIOGRAPHY...........................................224 viii

LIST OF FIGURES Figure 1.1 Typical millimeter-wave integrated circuit structure. Accurate modeling of discontinuities is key to cost effective designs............... 2 1.2 Three basic classes of microstrip........................ 4 1.3 Basic geometry for the shielded microstrip cavity problem....... 5 1.4 Flow chart illustrating theoretical approach for characterizing microstrip discontinuities................................. 11 2.1 Microstrip structures for which thin-strip approximation is valid... 14 2.2 Representation of coaxial feed by a circular aperture with magnetic frill current Mt.................................... 16 2.3 Total fields Etot, Htot inside cavity are produced by magnetic current source Mo at aperture and electric current distribution J, on the conducting strip..................................... 17 2.4 Test current Jq on conducting strip and associated fields Eq, Hq inside the cavity............................................. 18 2.5 Strip geometry for use in basis function expansion of current for the case of an open-ended line............................... 19 2.6 Geometry used in derivation of the Green's function......... 23 2.7 Geometry used to set up surface integral for excitation vector........ 32 2.8 In the case of dual excitation, the total fields Etot, Htot inside the cavity are produced by magnetic currents MA1, M<,, and the electric current J,. 38 2.9 Strip geometry for basis function expansion with dual excitation. The case shown. corresponds to a thru-line.................... 41 2.10 Representation for microstrip open end discontinuity......... 50 2.11 Equivalent network representation for generalized 2-port discontinuity. 52 ix

2.12 The even and odd mode excitations correspond to placing electric and magnetic walls in the center of the two-port structure........... 53 3.1 Determination of the computational parameters for a series gap is somewhat more complicated than for an open-end or a thru-line........ 60 3.2 Flow chart for program SHDISC to compute current distribution.. 62 3.3 Flow chart for subroutine that computes vectors LN(n), and MN(n). 63 3.4 A plot 3-dimensional plot of the impedance matrix for a typical open-end shows that it is diagonally dominant and well behaved........... 66 3.5 The amplitude distribution for the excitation vector is highest for Q=l, which corresponds to the position of the feed................ 67 3.6 Excitation vector for two port coaxial excitation............ 68 3.7 A comparison of computation times for two different methods of matrix equation solution shows the advantage of using alternatives to matrix inversion.................................... 69 3.8 This 3-dimensional plot of the magnitude of the elements of the inverted impedance matrix shows that each row and column has a sinusoidal shape. 70 3.9 The imaginary part of current distribution for an open-end discontinuity displays a sinusoidal behavior......................... 71 3.10 The current for gap generator excitation is discontinuous around the position of the source, but is otherwise well behaved............ 72 3.11 Current distributions for a typical series gap discontinuity...... 73 3.12 Flow chart for computation of one-port network parameters..... 76 3.13 Flow chart for computation of two-port network parameters..........77 3.14 3-dimensional plot illustrating computation of LN(n) over m and n. 79 3.15 The formation of an impedance matrix for one row (Q = 32) versus the summation index n.............................. 80 3.16 3-dimensional plot illustrating computation of MN(n) over m and n. 81 3.17 The formation of an excitation vector versus the summation index n. 82 3.18 Convergence of Leff versus sampling for several different K-values.. 84 3.19 Convergence of sef versus sampling for several different K-values.. 85 x

3.20 Reciprocal matrix condition number versus sampling........ 87 3.21 The convergence of Leff on n was found to depend on I., but is satisfied in all cases after 500 terms have been added................ 88 3.22 The convergence of Eeff on n shows similar behavior as that for Leff, and is satisfied in all cases after 500 terms have been added.......... 89 3.23 The convergence of Lef! on m is also satisfied after 500 terms..... 89 3.24 A plot of the reciprocal condition number versus NSTOP shows that the condition for erratic current depends on both NSTOP and 1...... 90 3.25 The effect on the reciprocal condition number of varying only 1 indicates the existence of an optimum sampling range................ 92 3.26 The optimum sampling range is seen to correspond directly with the flat convergence region for the L/!f computation............... 94 4.1 Microstrip test fixture approach for de-embedded measurements... 99 4.2 Fixture characterization by the TSD technique (Note: the "short" can be any highly reflecting standard)..................... 105 4.3 Procedure used in this work for measurement and de-embedding... 109 4.4 7mm coaxial/microstrip test fixture (partially disassembled)..... 110 4.5 The TSD Connection approach used for the present work, relies on repeatable coax/microstrip connections................... 111 4.6 TSD Connection approach relying on repeatable microstrip/microstrip connections.................................. 112 4.7 Sketch of TSD standards used for measurements of e//f, and open-end and series gap discontinuity circuit. Note: all dimensions of Figure are in mils (1 mil=.001"), h =.025", e, = 9.7.................. 114 4.8 Typical effective dielectric constant measurement resulting from TSD fixture characterization. (Shielding dimensions: b = c =.25")..... 115 4.9 Angle of open-end reflection coefficient resulting from a typical fixture characterization procedure......................... 115 4.10 Sketch of series gap discontinuity test circuit. Note: all dimensions in mils, h = 25, e, = 9.7. (Shielding dimensions: b = c =.25")...... 117 4.11 Sketch of two resonator coupled line filter. Note: all dimensions in mils, h = 25, = 9.7. (Shielding dimensions: b =.4",c =.25")...... 118 x1

4.12 Sketch of TSD standards for two resonator filter measurement. Note: all dimensions in mils, h =.025", e, = 9.7................... 119 4.13 Sketch of four resonator coupled line filter. Note: all dimensions in mils, h =.025", r = 9.7. (Shielding dimensions: b =.4",c =.25" )......120 4.14 Sketch of TSD standards for four resonator filter measurement. Note: all dimensions in mils, h =.025", E, = 9.7.................... 121 4.15 Flow chart illustrating approach for perturbation analysis of connection errors....................................... 124 4.16 Variation of S11 and S12 for 10 connections made to a typical thru (or delay line) standard. Error vectors may be defined as the vector perturbation of each of the measurements from the average................. 126 4.17 Error vectors for thru or delay line standards. These were also used to perturb the S-parameters of the two and four resonator filter structures. 130 4.18 Error vectors for measurement of open-end reflection standard.... 131 4.19 Combined error vectors for series gap measurements.......... 133 4.20 This plot of the final de-embedded result for a 5 mil series gap discontinuity illustrates the information obtained through the perturbation analysis (f=10GHz, 20 connection permutations)................... 137 5.1 The cutoff frequency for higher order modes in shielded microstrip may be approximated by analyzing an infinite dielectric-loaded waveguide... 143 5.2 Variation of cutoff frequencies with shielding for three commonly used substrates enclosed in a square waveguide (b = c). Also shown is empty guide case (, = 1, h = 0)............................ 144 5.3 Below the cutoff frequency f,, the microstrip current on an open-ended line forms a uniform standing wave pattern (f = 16GHz,, = 9.7, W/h = 1.57, h =.025", b = c =.275").......................... 145 5.4 As the frequency is increased above f,, more and more distortion is observed in the open-end current distribution (f = 22GHz,e, = 9.7, W/h = 1.57, h =.025", b = c =.275")..........................146 5.5 Effective dielectric constant comparison for an alumina substrate compared to measurements and CAD package results (e, = 9.7, h=.025", b = c =.25.................................... 148 5.6 The effects of shielding on ~eff are apparent as the size of the shielding cavity is reduced (see Table 5.1 for geometry.).............. 149 xii

5.7 Shielding effects are also significant for the quartz substrate shown here (see Table 5.1 for geometry)................................... 150 5.8 The numerical and CAD package results display excellent agreement for the case of a thin GaAs substrate (er = 12.7, h =.004", b = c =.07", fc = 81GHz)............................................ 151 5.9 Effective length extension of a microstrip open-end discontinuity, as compared to results from other full-wave analyses (e, = 9.6, W/h = 1.57, b =.305", c =.2", h =.025")......................... 152 5.10 A comparison of the normalized open-end capacitance for three different cavity sizes; shows that shielding effects are significant at high frequencies (see Table 5.1 for cavity geometries).................... 154 5.11 Nomalized open-end capacitance for three different cavity sizes for a quartz substrate. This data also shows an increase in the capacitance as the cutoff frequency is approached (see Table 5.1 for cavity geometries).. 154 5.12 Numerical and measured results show good agreement for the angle of S11 of an open circuit (e, = 9.7, W = h =.025", b = c =.25")...... 155 5.13 Magnitude of S21 for series gap circuit A (G = 15 mil)....... 157 5.14 Magnitude of S21 for series gap circuit B (G = 9 mil)......... 157 5.15 Magnitude of S21 for series gap circuit C (G = 5 ml)........... 158 5.16 Angle of S21 for series gap circuit A (G = 15 mil).......... 159 5.17 Angle of Sll for series gap circuit A (G = 15 mil).......... 159 5.18 Results for transmission coefficient S21 of two resonator filter (e, = 9.7, W = h =.025"; b =.4", c =.25").................... 163 5.19 Results for transmission coefficient S21 of four resonator filter (e, = 9.7, W =.012", h =.025"; b =.4", c =.25")................ 164 5.20 Results for lowering the shielding cover on the amplitude response of four resonator filter (e, = 9.7, W =.012", h =.025"; b =.4")........... 165 D.1 The current source is raised above the substrate/air interface to apply boundary conditions............................. 187 F.1 Strip geometry used in evaluation of surface integrals......... 201 H.1 K-connector (2.9mm) coaxial/microstrip test fixture......... 211 xiii

H.2 Coax/microstrip connection technique used with K-connector (2.9mm) launchers............................................ 213 H.3 K-connector multi-line test fixture used for testing microstrip/microstrip interconnects................................. 214 H.4 7mm Coax/microstrip connection repeatability measurements... 216 H.5 Standard deviation data for experiments with 7mm fixture...... 217 H.6 K-connector coax/microstrip repeatability measurements normalized to average........................19 H.7 Microstrip fabrication/mounting repeatability measurements normalized to average.................................. 220 H.8 Microstrip/microstrip interconnect repeatability measurements normalized to average................................ 221 H.9 Standard deviation data for experiments with K-connector fixture.. 222 xiv

LIST OF TABLES Table 2.1 COORDINATE TRANSFORMATION VARIABLES......... 36 3.1 RESULTS FOR OPTIMUM SAMPLING RANGE EXPERIMENT. 93 4.1 DIFFICULTIES WITH MICROSTRIP MEASUREMENTS..... 98 4.2 FULL MATRIX DE-EMBEDDING METHODS........... 104 4.3 CONNECTION PERMUTATION TABLE EXAMPLE....... 127 4.4 PERTURBATION ANALYSIS RESULTS FOR eff and OPEN-END MEASUREMENTS............................. 135 4.5 PERTURBATION ANALYSIS RESULTS FOR SERIES GAP MEASUREMENTS................................ 138 5.1 CAVITY NOTATION USED TO DENOTE DIFFERENT GEOMETRY AND SUBSTRATE PARAMETERS.................... 149 5.2 COMPARISON OF Leff/h COMPUTATION FOR THE TWO TYPES OF EXCITATION METHODS....................... 153 H.1 SUMMARY OF REPEATABILITY EXPERIMENTS........ 212 XV

LIST OF APPENDICES Appendix A. REVIEW OF METHOD OF MOMENTS................. 175 B. DERIVATION OF INTEGRAL EQUATION FOR ELECTRIC FIELD 177 C. EIGENFUNCTION SOLUTION FOR GREEN'S FUNCTION..... 180 D. BOUNDARY CONDITIONS AT SUBSTRATE/AIR INTERFACE... 186 E. EVALUATION OF MODIFIED DYADIC GREENS FUNCTION... 196 F. INTEGRATION OVER SUBSECTIONAL SURFACES......... 199 G. EVALUATION OF MAGNETIC FIELD COMPONENTS....... 206 H. MICROSTRIP CONNECTION REPEATABILITY STUDY....... 210 xvi

LIST OF SYMBOLS a = Dimension of cavity along longitudinal (x) direction of microstrip A' = Magnetic vector potential A() = Complex coefficients for series representation for G(/) b = Dimension of cavity along transverse (y) direction of microstrip B() = Complex coefficients for series representation for G(C) c = Dimension of cavity along direction perpendicular to substrate (z-direction) c1' = Distance from the center of the feed to the top of the cavity (c" = c - h,) Cnq = Variable multiplier function of n and q used to simplify expressions for magnetic field Hq at coaxial aperture cop = Normalized equivalent capacitance for open end discontinuity C(o) = Complex coefficients associated with the y-component of the magnetic field (i) (i) = Complex coefficients associated with the z-component of the magnetic field H5 dImn = First denominator function of complex Green's function coefficients xvii

d2mn = Second denominator function of complex Green's function coefficients i = Electric field in the i'h region of the cavity Eq = Electric field associated with the test current Jq Eqz = x-component of the electric field due to the test source Jq evaluated at the substrate/air interface Etot = Total electric field inside shielding cavity f = Frequency of operation (Hz) f = Cut-off frequency for non-evanescent waveguide modes (GHz) fn = Numerator function used in expression for rSS G = Dyadic Green's function of ith region G() = xx- component of dyadic Green's in region i G( = xz- component of dyadic Green's in region i h = Thickness of microstrip substrate h = z-coordinate of center of coaxial feed H' = Magnetic field in ith region of cavity Hq = Magnetic field associated with the test current Jq Hqo = Common variable multiplier function of q used to simplify expressions for magnetic field Hq at coaxial aperture. Hqt = Projection of magnetic field onto the plane of the feed aperture xviii

[I] = Unknown current vector comprised of complex coefficients for current [Id] = Unknown current vector for general dual excitation [I,] = Unknown current vector for even excitation [Io] = Unknown current vector for odd excitation Ip = Complex coefficients current expansion into series of basis functions ( components of I). Iqmn = Surface integral encountered in the evaluation of Eq. J = Electric current distribution in cavity Jq = Test current source (weighting function in method of moments) J, = Surface current distribution on microstrip conductors ki= Complex wave number in region i ks = Eigenvalues for the x-variation of the Green's function (= 2) ky = Eigenvalues for the y-variation of the Green's function (= mi) k() = Eigenvalues for the z-variation of the Green's function in region i K = Wave number used in the sinusoidal subsectional expansion of the current (equal to the real part of the wave number in the dielectric region) Is = Length of each subsection L1 = Length of input uniform line section section for series gap or filter L,ff = Effective length extension for open end discontinuity xix

Lin = Length of the microstrip line in inches (for an open end or thru line). L' = Normalized length of the microstip line (=-Ln/Ad) Lmn = mnth term of series used to compute LN(n) LN(n) = Storage vector used in computation of the impedance matrix Mmn = mnth term of series used to compute MN(n) JM/t= Integrand for numerical integration performed to compute Mmn MN(n) = Storage vector used in computation of the impedance matrix Ms = Magnetic frill current existing over aperture of coaxial feed Msl = Magnetic frill current existing over aperture of coaxial feed on left, for case of dual excitation MAr = Magnetic frill current existing over aperture of coaxial feed on right, for case of dual excitation 1,0 = Component of magnetic frill current in the < direction MSTOP = Index for truncation of m-series used to compute LN(n), and MN(n) Ms = Equivalent magnetic surface current distribution at coaxial aperture NSTOP = Index for truncation of n-series used to compute Zqp and Vq NV, = Number of subsections that strip is divided NV = Number of samples per dielectric wavelength (= 1/1,) r = Position vector anywhere inside cavity XX

r' = Position vector of assumed infinitesimal current source r= Radius of inner conductor of coaxial feed rb= Radius of outer conductor of coaxial feed RC = Reciprocal matrix condition number RIn = Argument of first Sinc function of expression for Tqmn (= [(kr + K)l ]) R2,= Argument of second Sinc function of expression for Iqmn (= [I(k, - K)] ) s = Estimated standard deviation (used for experimental data analysis) Sij Scattering parameters (i,j = 1,2) Sf= Surface of coaxial aperture S(') Portion of coaxial aperture surface lying in region i Sif = Surface of coaxial aperture for feed on left hand side (x = 0 side) of cavity Sq = Surface of the qth strip subsection Sf Surface of coaxial aperture for feed on right hand side (x = a side) of cavity Sstip = Surface of conducting strip SWR = Standing wave ratio on microstrip line tan Sd = Loss tangent of substrate material (= C ) V = Volume of shielding cavity Vo = Complex voltage present in the coaxial line at the feeding point [V] = Excitation vector used in method of moments solution,xxi

[V1] = Excitation vector associated with left hand feed [Vr] = Excitation vector associated with right hand feed [Vd] = Combined excitation vector for dual excitation [Ve] = Even excitation vector [Vo] = Odd excitation vector Vq = qth element of the excitation vector Vq= qt) element of the excitation vector corresponding to left hand feed for case of dual excitation Vqr = qth element of the excitation vector corresponding to right hand feed for case of dual excitation W = Width of microstrip line Xp = Position vector giving x-coordinate of the pth subsection Yo = y-coordinate of the center of the strip with respect to the origin Y = y-coordinate of center of coaxial feed yij =Normalized admittance parameters (i,j = 1,2) z(x)= Input impedance at position x looking toward the discontinuity zij = Normalized impedance parameters (i,j = 1,2) [Z]= Impedance matrix used in method of moments solution Zqp = The q - p element of the impedance matrix xxii

ag = Loss factor of microstrip transmission line a = Subsectional sinusoidal basis functions used for series expansion of current in the longitudinal (x) direction 3g = Phase constant of microstrip transmission line 7g = Complex propagation constant of microstrip transmission line r = Modified dyadic Green's function in region i r(x)= Input voltage reflection coefficient at position x looking toward the discontinuity r = x -component of modified dyadic Green's function in either region evaluated at substrate/air interface (z = h) o= Free space permittivity (= 8.S54 x 10-12 F/cm) i = Complex permittivity of material in ith region of cavity = Complex relative dielectric constant of substrate material (= - ) Er = Real part of er (q = Variable multiplier used in expression for Iq,, 8,o= Angle of reflection coefficient for open-end discontinuity Ao = Wavelength in air (e, = 1) Ad = Wavelength in dielectric (r, =,r1) Ag = Microstrip wavelength (Er = ef/) pi = Permeability of material in ih region of cavity xxiii

/uo = Free space permeability (= 47r x 10-7 H/m) p = Volume charge density in cavity a= Conductivity of substrate material ~, = Variable multiplier for complex Green's function coefficients 4' = Function describing variation of microstrip current in the transverse(y) direction w = Radian frequency (= 27rf) xxiv

CHAPTER I INTRODUCTION 1.1 Motivation Millimeter-wave integrated circuits are important for a variety of scientific and military applications, and a wide range of solid state circuitry has been developed in both hybrid and monolithic form. However, the inability to accurately predict the electrical characteristics of various circuit components is a serious barrier to the widespread. and cost effective application of these technologies. In fact, even at microwave frequencies, computer simulations of Monolithic Microwave Integrated Circuit (MMIC) components are often inadequate leading to lengthy design cycles with many costly circuit design iterations. The development of more accurate microstrip discontinuity models, based on full-wave analyses, is key to improving microwave and millimeter-wave circuit simulations and reducing lengthy design cycle costs. Typical microwave and millimeter-wave IC's contain various active and passive elements interconnected by microstrip transmission lines as illustrated in Figure 1.1. In the vicinity of transmission line junctions and other discontinuities, parasitic effects occur that can significantly modify circuit operation. These discontinuity effects can be modeled by the use of lumped equivalent circuits or by 1

2 Figure 1.1: Typical millimeter-wave integrated circuit structure. Accurate modeling of discontinuities is key to cost effective designs. generalized matrix representations; however, the models are only as accurate as the analysis technique used to derive them. Several techniques have been applied, and approximate models exist for most common discontinuities. However, for MMICs and for both hybrid and Monolithic Millimeter-wave Integrated Circuits (M3ICs), additional theoretical and experimental research is needed to establish the accuracy of these models and to develop improved models where needed. One area where this is true is for shielded microstrip discontinuities. As shown in Figure 1.2, there are three basic classes of microstrip. In open microstrip the top of the substrate is left open to the air. In this case, surface waves and radiation from circuit elements is unavoidable. In covered microstrip, a conducting cover is present, but no side walls are present, and in shielded microstrip the circuitry is enclosed in a rectangular waveguide.

3 This last category, shielded microstrip, is the most common for practical applications. In fact, usually the microstrip circuit is completely enclosed in a shielding cavity (or housing) as shown in Figure 1.3. This prevents radiation and electromagnetic interference and it also supresses surface wave modes. The effect of the shielding on discontinuity behavior can be significant, and requires accurate modeling, at high frequencies. There are two main conditions where this is true. The first occurs when the frequency approaches or is above the cutoff frequency above which higher order modes can propagate within the shielded structure. The second occurs when the metal enclosure is physically close to the circuitry. A full-wave analysis is required to accurately model shielding effects. These effects are not adequately accounted for in the discontinuity models used in most available CAD packages. The theoretical part of this thesis addresses this area. There is also a great need for experimental data. Published experimental data on microstrip discontinuites is very limited, especially for high microwave (above X-band) and millimeter-wave frequencies. Such measurements are not trivial, but are essential to verifying the accuracy or at least the reasonableness of theoretical results. This was the motivation for the experimental part of this work. 1.2 Thesis Objectives The overall goal of this research is to use full-wave analysis techniques combined with experimental data to study some relatively basic shielded microstrip discontinuities. This research is intended to aid in the development of an accurate data base for all kinds of microstrip discontinuities. Such a data base has several uses: 1) existing CAD models can be checked against it to determine their regions of validity, 2) it; can be used to develop new and more accurate, C.A.D. models, and 3) it can be used to improve experimental methodology. For example, it can

4 h~\~\~,\\~,\\\\\\~,\I a) Open microstrip - I& 0\~,~~~~~,,\,~,~\\\ b) Covered microstrip I I 14 I.N\X c) Shielded microstrip Figure 1.2: Three basic classes of microstrip.

5 microstrip shielding cavity (or housing) Y coaxial / - coaxial input output / z=h " ' a X x=a dielectric substrate Figure 1.3: Basic geometry for the shielded microstrip cavity problem. be used to help characterize microstrip standards for measurement calibration and verification. The specific objectives of the present research may be summarized as follows: * Develop a new theoretical method for computational analysis of shielded microstrip discontinuities * Explore the use of a practical excitation mechanism * Investigate high frequency microstrip measurement techniques * Conduct an experimental study in close correlation with the theoretical work The primary objective is to develop an accurate and computer efficient method for full-wave analysis of discontinuities in shielded microstrip. This method is to be demonstrated by obtaining numerical results for practical discontinuity structures

6 for which the thin-strip approximation ' applies. Another objective is to investigate ways to use practical circuit excitation mechanisms in the theoretical solution. To date, all full-wave analyses of microstrip discontinuities use either a gap generator excitation method [5,11], or use a cavity resonance technique (without actually exciting the circuit)[8,10] to determine discontinuity parameters. Both of these techniques are purely mathematical tools. The former has no physical basis relative to an actual circuit. The latter is also abstract, since in any practical circuit some form of excitation is present. To derive a more realistically based formulation, the use of a coaxial excitation mechanism is to be explored and compared to the gap generator method. A final objective is to conduct an experimental study to aid in the verification of the theory. As part of this study, a number of techniques are to be assessed qualitatively as to their suitability for high frequency (X-band and higher) measurements. Limitations on measurement accuracy are also to be addressed. 1.3 Accomplishments In this thesis, the research objectives outlined above are accomplished as follows. A new method is developed for the analysis of discontinuities in shielded microstrip. In the analysis one of two different types of excitation can be used: the first is a coaxial excitation method, and the second is the gap generator approach discussed above. It is shown that with gap generator excitation, the current is disrupted over the region surrounding it; while with coaxial excitation, the current is undisturbed and uniform along the length of the strip. However, for the evaluation of discontinuity parasitics, the final results appear to be unaffected by the method of excitation. The validity of the thin-strip approximation for the The thin-strip approximation assumes that the width of the conducting strips is small compared to a wavelength.

7 structures considered in this thesis is also examined. To demonstrate the method, numerical results are presented for open-end and series gap discontinuities, and a four resonator coupled line filter. These results are compared to other full-wave analyses, to data from Super Compact and Touchstone 2, and to measurements. The measurements were performed by the author using a variation of the TSD de-embedding technique [44]. The contribution of this thesis is in three areas: theoretical, computational, and experimental. The theoretical contribution is in the derivation of the method of moments solution for the problem of Figure 1.3. This derivation is based on modeling the coaxial feed with a magnetic frill current (Chapter 2). To the author's knowledge, this is the first time that the frill current approach has been applied to the shielded microstrip problem. From a computational point of view, extensive numerical convergence experiments were performed that lead to some surprisingly simple relationships governing the convergence and stability behavior of the method of moments solution for this problem (Chapter 3). Finally, as part of the experimental study, a novel perturbation analysis was applied to study the effect of connection repeatability errors on microstrip de-embedding accuracy (Chapter 4). The experimental results demonstrate the accuracy and usefulness of the theory developed here and also suggest some areas where improvements can be made (Chapter 5). 1.4 Brief Review of Theoretical Approaches As mentioned above, several approximate techniques exist for microstrip discontinuity analysis. The published literature on this subject is voluminous. However, a few comments are in order before discussing full-wave solutions. 2 Super Compact and Touchstone are microwave CAD software packages available from Compact Software and EESOF respectively.

I 1.4.1 Quasi-Static Techniques Quasi-static techniques are well established and described in standard texts [1]-[3]. With these techniques, equivalent circuits are derived in terms of static capacitances and low frequency inductances. Convenient analytical formulas for discontinuity parasitics are possible, yet their accuracy is questionable for frequencies above a few GHz. Also, most existing models based on these techniques ignore shielding effects. 1.4.2 Planar Waveguide Models Planar waveguide models provide a frequency dependent solution. In this approach, an equivalent planar waveguide geometry is proposed for the microstrip problem. The transformed problem is then solved using an appropriate analytical technique, such as mode matching [4]. Models derived from this technique are generally considered accurate to higher frequencies than those derived from quasi-static models. However, the method does not provide an adequate model for shielded microstrip. As discussed in Chapter 5, the higher order modes in shielded microstrip are essentially rectangular waveguide modes. The mode behavior of a planar waveguide is quite different than for rectangular waveguide. Its applicability even for the dominant miicrostrip mode is questionable, since the dominant mode in shielded microstrip is the result of an infinite summation of evanescent waveguide modes. This is not to say that reasonable predictions are not possible with this method, only that its application for the shielded microstrip problem is not rigorous.

9 1.4.3 Rigorous Full-Wave Solutions As described above, for many applications, the limitations of the above two techniques cannot be tolerated, and a more accurate solution is required. A full-wave solution3 that meets this requirement was developed by Katehi to treat discontinuites in open microstrip [5,6]. This technique has so far been applied to solve for various discontinuities in open microstrip. The new analytical methodology presented here is an extension of the approach of Katehi to shielded microstrip configurations. While rigorous solutions to shielded discontinuities have been advanced by others, their is a need for further research in this area. The most extensive work has been performed by Jansen et al. [7]-[9]. Although reasonable results have been demonstrated for several microstrip structures, there has been little accompanying experimental verification, and only limited comparisons to other rigorous numerical solutions. 'The primary reasons for this have most likely been the difficulties of performing the measurements, as discussed herein, and the existence of only a few other rigorous solutions ( [10]-[12]). Further, the only published results from the other rigorous solutions that the author is aware of is for the open-end discontinuity. 1.5 Description of Theoretical Methods The theoretical method developed here is addressed to the shielded microstrip geometry shown in Figure 1.3. The shielding box forms a rectangular cavity, which -for most practical uses- is cutoff for the highest frequency of operation. That is, the cavity dimensions are usually such that the coupling of microstrip 3 A full-wave solution, refers to the application of a rigorous electromagnetic analysis to the microstrip geometry, making as few assumptions as possible.

10 modes into higher order waveguide modes is avoided. However, as far as the the current distribution is concerned, the solution presented here is accurate whether the cavity is cutoff or not. Above this cutoff frequency the onset of higher order modes distort the current distribution and the definition of circuit behavior in the usual way, by a transmission line model, becomes ambiguous. The theoretical method is based on Galerkin's implementation of the method of moments. A flow chart illustrating the method is shown in Figure 1.4. First, the required matrix equation is derived. In this derivation, the coaxial feed is represented by an equivalent magnetic current source. The reciprocity theorem is then applied to establish an integral equation that relates this magnetic current source and the electric current on the conducting strips to the electromagnetic fields inside the cavity. By expanding the electric current into a series of sinusoidal subsectional basis functions, the integral equation is transformed into a matrix equation. In a similar way, the reciprocity theorem is applied for the case of gap generator, and an identical form results for the elements of the impedance matrix. Next, the Green's function is derived and used to evaluate the electric and magnetic fields within the cavity. The Green's function is derived by applying boundary conditions to the problem of Figure 1.3 with the conducting strips replaced by an infinitesimal current source on the substrate/air interface. This is a common approach in solving boundary conditions of this type [16]. Finally, the matrix equation is solved to compute the current distribution. Based on the current, either an equivalent circuit or scattering parameters are derived to characterize the discontinuity being considered. 1.6 Description of Experimental Study To obtain verification data for the theoretical method of this thesis, an ex

11 RECIPROCITY THEOREM METHOD OF MOMENTS MATRIX EQUATION [Z][ i] =!V] CURRENT DISTRIBUTION ON CONDUCTING STRIPS I I SCATTERING PARAMETERS EQUIVALENT CIRCUIT MODEL Figure 1.4: Flow chart illustrating theoretical approach for characterizing microstrip discontinuities.

12 perimental study was conducted in cooperation with Hughes Aircraft Company 4 As discussed in Chapter 4, the lack of experimental data on microstrip discontinuities is mainly due to the difficulties with removing test fixture parasitics from the measurements (called de-embedding), and the non-repeatability of microstrip connections. To address these issues, a study of de-embedding techniques was conducted, from which it has been concluded that the most suitable technique for the measurements of this thesis is the thru-short-delay (TSD) method. Also, a connection repeatability study was carried out [14,15], and the results were used to decide on how to best implement TSD de-embedding, and they were used to approximate the associated uncertainty in de-embedding accuracy. S-parameter measurements were then obtained for an open-end discontinuity, three series gap discontinuities with different gap widths, and two coupled line band pass filter structures. 4 Hughes Aircraft Company, Microwave Products Division, Torrance, California.

CHAPTER II THEORETICAL METHODOLOGY 2.1 Assumptions In this solution, a few simplifying assumptions are made to reduce unnecessary complexity, and excessive computer time. Throughout the analysis, it is assumed that the width of the conducting strips is small compared to the microstrip wavelength Ag. In this case, unidirectional currents may be assumed with negligible loss in accuracy. While substrate losses are accounted for, it is assumed that the strip conductors and the walls of the shielding box are lossless, and that the strip thickness is negligible. For the computation of two-port network parameters (Section 2.7.2), the strip geometry is assumed to be physically symmetric with respect to the center of the cavity (in both the x and y directions of Figure 1.3). Also, to simplify notation, the assumed time dependence is ejdt, and it is suppressed throughout the dissertation. The above assumptions are valid for the high frequency analysis of the microstrip structures of Figure 2.1, provided good conductors are used in the metalized areas. The first three structures of Figure 2.1, the open-end, the series gap and the coupled lines are discontinuity structures. The last one, the thru-line, 13

14 N \\\\\\\'T - J w OPEN END T w SERIES GAP W << g -AL w COUPLED UNES T w THRU-LINE Figure 2.1: Microstrip structures for which thin-strip approximation is valid. is not. The thru-line is included since it is a useful test case, and it is used to determine the microstrip propagation constant 7g. 2.2 Method of Moments Formulation The method of moments is a well established numerical technique for solving electromagnetic problems [17],[18]. A review of the basic approach is given in Appendix A. This section makes use of the method of moments to set up a matrix equation that provides for a computer solution to shielded microstrip discontinuity problems. For the most part, the theory presented applies to the use of a coaxial excitation mechanism, based on the frill current model. A few comments are made, however, to indicate how the theory differs for the use of a gap generator excitation model. In the matrix equation, the only significant difference is the excitation vector used.

15 2.2.1 Application of Reciprocity Theorem Reciprocity Theorem for Coaxial Excitation Consider the geometry of Figure 1.3. In most cases the coaxial feed, or "launcher", is designed to allow only transverse electromagnetic (TEM) propagation, and the feed's center conductor is small compared to a wavelength. In these cases, the radial electric field will be dominant on the aperture and we can replace the feed by an equivalent magnetic surface current, sometimes called a "frill" current, whose only component is in the X direction (i.e.M, = MiVI) 1. This method of modeling the feed with a magnetic current source will be discussed further in Section 2.5. The magnetic current source is coupled with the current distribution J, on the conducting strip to produce the total electric field Et and the total magnetic field Htt inside the cavity as indicated in Figure 2.3. We now propose an independent test current source Jq existing only on a small subsection of the conducting strip as shown in Figure 2.4. Using the reciprocity theorem, the two sets of current sources are related according to I (J. Eq-Hq.M |dvo = | Edv (2.1) where the volume V is the interior of the cavity. Since Jq is x-directed and zero everywhere except over one subsection of the conducting strip, the right hand side of ( 2.1) reduces to J Jq Etotdv = Jj Jq\ Et(z = h)ds = 0 (2.2) 1 refers to the cylindrical coordinate referenced to the center of the feeding aperture (see Figure 2.2.)

16 magnetic frill coaxial current M feed annular aperture Figure 2.2: Representation of coaxial feed by a circular aperture with magnetic frill current iMV. where Sq is the surface of an arbitrary subsection and Et^(z = h) is the xcomponent of the total electric field which must vanish on the surface of the conductors (z = h) since they are assumed to be perfectly conducting. Reducing the remaining volume integrals in ( 2.1) to surface integrals results in s E(Z = h) Jd3S = / Hq( =0) * Mds (2.3) where Stip is the surface of the conducting strip and Sf is the surface of the coaxial aperture. Note that this equation is not explicitly in the form of the operator equation of (A.1). This is because in using the reciprocity theorem formulation we have inherently placed it in the inner product form of (A.3). Reciprocity Theorem for Gap Generator Excitation

17 STRIP CONDUCTOR Figure 2.3: Total fields Etot, Htot inside cavity are produced by magnetic current source M, at aperture and electric current distribution J, on the conducting strip.

18 Y Figure 2.4: Test current Jq on conducting strip and associated fields Eq, Hq inside the cavity. The formulation for the case of gap generator excitation can also be derived from reciprocity theorem. In this case, it is assumed that Es is non-zero at one point (xg) on the strip, between two subsections. Setting M, = 0 in 2.1, and reducing the resulting volume integrals to surface integrals yields J. q(Z = h). J3d = j. Jqj E~tt(z = h)ds (2.4).trip q Since E, = 0 everywhere except xg, the right hand side of 2.4 vanishes everywhere except over the the subsectional surface containing x2. This surface integral we arbitrarily set equal to unity. That is J It|t(z =-1 for=h)ds (2.5) w IS is 0 else where g is the index corresponding to the position of the gap generator.

19 y b I II X X, X,...X...~ X,X,X 1 2 3 P Ns-2 Ns-1 Ns - (p-l) 1 a P z Figure 2.5: Strip geometry for use in basis function expansion of current for the case of an open-ended line.

20 2.2.2 Expansion of Current with Sinusoidal Basis Functions In order to solve the integral equation (2.3), the current distribution J, is expanded into a series of orthogonal functions as follows. Consider the strip geometry shown in Figure 2.5, let N, i. =, (y) E Ipcp (x) (2.6) p= 1 where Ip are unknown current coefficients. The function 4, (y) describes the variation of the current in the transverse direction and is given by 2 - Yo-VW/2<y Yo +W/2 -(y) -[ (2.7) 0 else This variation was chosen to agree with that derived by Maxwell for the charge density distribution on an isolated conducting strip [20], and it has been used successfully by others to describe the transverse variation of microstrip currents [5,21,22]. The basis functions ap(x) comprise an orthonormal set and are described by six(K13) () in(K ) < < (2-8) sin(Klz) -1 e 0 else for p # 1, and sinf(K()l O < < a(x) sin((.s) 0 (9) 0 else for p = 1, where K = wv/ioSlio is the real part of the wave number in the dielectric region W is the width of the microstrip line

21 YO is the y-coordinate of the center of the strip with respect to the origin in Figure 1.3 xp is the x-coordinate of the pth subsection (= (p - 1)-l) I1 is the subsection length (I. = xp+l - xp) 2.2.3 Transformation of Integral Equation into a Matrix Equation The integral equation (2.3) can now be transformed into a matrix equation by substituting the expansion of (2.6) for the current J,. This results in the following: [Jj Eq(z = h). i (y) ap (x) xds] Ip = Hq. Mds (2.10) where Sp is the surface area of the pth subsection, and N, is the number of sections that the strip is divided into for computation. We may now express (2.10) as [Z] [I] =[V]. (2.11) In the above, [Z] is called the impedance matrix, and has the form Zll Z12... ZIN. Z21 Z22 Z2N. [Z]= 1 22N(2.12) ZN,1 ZN.2 * — ZIV. [V] is called the excitation vector and may be expressed as [V]=[ V V2... VNs. (2.13) [I] is the current vector comprised of the unknown current coefficients as follows: [I =[ II. 2 (2.14)

22 The individual elements of the impedance matrix are given by Zqp=J jEq(z =h) (y) cap( ds. (2.15) The elements of the excitation vector (coaxial excitation) are given by V7 Hq Mds. (2.16) We can now solve for the current vector by matrix inversion and multiplication according to [I] =[ZI1[V]. (2.17) For gap generator excitation, the impedance matrix elements are also given by (2.15); however, the elements of the excitation vector are given by the right hand side of (2.5). 2.3 Derivation of the Green's Function To compute the elements of the impedance matrix, we must derive the Green's function associated with the electric and magnetic fields Eq, Hq. We will first define the problem geometry and outline the electromagnetic theory to be used. Then, the boundary value problem will be solved for the Green's function. 2.3.1 Geometry and Electromagnetic Theory The geometry used in the Green's function derivation is shown in Figure 2.6. The cavity is divided into two regions: Region 1 consists of the volume contained within the substrate (z < h), while region 2 is the volume above the substrate surface (z > h). Notice, as discussed in Section 1.5, the conducting strips have been replaced by an infinitesimal current source J.

23 Y z=c -y=b x=a a) Cutaway view Z — — C z=c - z=h - (2) I (1).;~ x I x=a b) Cross section in x-z plane Figure 2.6: Geometry used in derivation of the Green's function.

24 The Green's function will be defined as the electric field due to an infinitesimal current source located on the substrate surface of Figure 2.6. After deriving the Green's function, the fields associated with the test source Jq will be evaluated by integrating over the surface of the qth subsection. The test source Jq and the associated fields within the cavity (Eq,Hq) are related through Maxwell's equations, which may be put in the following form: V xE1 = Jo^H' (2.18) V x AH = jw=iEi+J (2.19) V.J = -jwp (2.20) V ( ') = P (2.21) V (1iH) 0. (2.22) In the above, i = 1,2 indicates that these equations hold in each of the regions respectively. Also, to simplify the notation of this subsection, the subscript q is suppressed with the understanding that all the field quantities discussed here are associated with the test source Jq (i.e. E' = E; etc.). Further, since it is assumed that both regions are non-magnetic and that region 2 is air, we have l = - 2=/-0 (2.23) eeo for i = 1 ee for (2.24) Eo for i= 2 where C; = rl- j — = erl (1 - j tan Sd) * (2.25) In the above, tan 6d = is referred to as the dielectric loss tangent of the dielectric region, 8d is called the dissipation angle.

25 2.3.2 Solution to Boundary Value Problem for the Green's Function We now introduce the vector potentials Ai such that 1 -t = -- x A'. (2.26) A'0 In view of (2.26), the electric field may be written as (B.13) E'= -j (1 + VV) A (2.27) where A' satisfies the inhomogeneous wave equation v2 Ai + kAi = -uoJ. (2.28) The integral form of the electric field is derived in Appendix B, and is given by (B.21) = -jwcoJ [(1 + ) (G)T] Jdv' (2.29) where k? = W2loei, and G is a dyadic Green's function [23] satisfying the following equation V2 G' + ki = -6 (r - r'). (2.30) In (2.30), I is the unit dyadic given by ~x + y + zz. Because of the air/dielectric interface, a two component vector potential is necessary to satisfy the boundary conditions [24]. Accordingly, let A' = A:x + Ai. (2.31) From (B.17) A' is related to G by the following volume integral: Ai'= - J idv'. (2.32) G may be expressed in most general form as follows: Gxx + + y G xz G,= | y + GGy + G + yz (2.33) + Gzzx + GI2yz + Gzzz

26 Assuming an infinitesimal x-directed current source given by J = 6 (r-r')x (2.34) in (2.32) allows for reducing G to G G'=^ + G xzz. (2.35) The dyadic components of (2.35) are found by applying appropriate boundary conditions at the walls: x = 0, and a; y = 0, and b; and z = 0, and c. As detailed in Appendix C, these components may be expressed as 00 00 G(1)= - A cos kzx sin kyy sin k(O)z (2.36) m=1 n=O 00 00 G(I - B (1) sin k,:x sin ky cosk?)z (2.37) m=1 n=O 00 00 G(2)= E A (2) cos kx sin ky sin k2)(z - c) (2.38) m=1 n=O 00 00 G(2)= B(2) sin k=x sin ky cos k2)(z -c) (2.39) m-1 n=O where ks = n7r/a (2.40) ky = m7r/b (2.41) k(l = /k- k2 (2.42) k(2) - k k- k2 (2.43) kl = wV Toi (2.44) ko = wo;//OO. (2.45) The coefficients A), A,(2) B$(), and B(2) are found by applying boundary conditions at the air/dielectric interface. The details of this analysis can be found in Appendix D. The results are: A() = -pn cos k,:x' sin k,,y' tan k2)(h - c)(2.46) abdlnn cos k?') h

27 A(2) -pn cos k:cx' sin ky' tan )(2.47) abdln cos kI2)(h - c) B(-) -cn,(l - C)k, cos kx' sin ky' tan k?)h tan k(2)(h - c) abdlmnd2mn cos kc) h B(2) = -n,( - e)k, cos k:x' sin ky' tan k(c)h tan k(2)(h - c) abdmnd2mn cos k(2)(h - c) where 2 for n = 0 pn = (2.50) 4 forn 0 dim = k(2) tan k(')h - k) tan k(2)(h - c) (2.51) d2mn =- k)e tan k)(h - c) - kl) tan kc')h. (2.52) Having derived the Green's function, we are now ready to proceed to the formulation for the elements of the impedance matrix and excitation vector. 2.4 Impedance Matrix Formulation The elements of the impedance matrix are given by (2.15) ZqP = J / E(z = h). (y') a (x' ds' (2.53) which reduces to Zqp = Eqx(z = h)4 (y') ap (x') ds' (2.54) To evaluate the impedance matrix elements we need only Eq.(z = h); that is, the x-component of the electric field due to the test currents J, at the air/dielectric interface.

28 2.4.1 Evaluation of the Electric Field Due to the Test Currents Since Jq is a surface current distribution, the volume integral in (2.29) is reduced to the following surface integral: E -q J (1 + VV) (G )] Jqdx'dy' (2.55) where Sq is the surface of the qth subsection. For best accuracy in applying the method of moments, the test currents Jq are expressed in terms of functions which are identical to the basis functions (Galerkin's method) J, = b(y')aq(x') (2.56) where 4(y') is given by (2.7) with y replaced by y', and aq,(') is given by (2.8) and (2.9) with p replaced by q and x replaced by a'. We now substitute from (2.56) for Jq in (2.55) to yield E,' = -jW/ [( + k ) (G) (y') (x')xdx'dy' (2.57) Let us define a modified dyadic Green's function r by ( = -jo [1 + -VV. (). (2.58) Then, EB can be expressed as = I| rJ. (Y')q(X')d'dy'. (2.59) The dyadic transpose of (2.35) yields (G)T = G=xx + Gzx. (2.60) When this expression is substituted in (2.58) and the divergence and gradient operations are performed we can express r as ( see Appendix E, (E.3) ) = 5 zi ~ +,x+ (2.61)

29 where '~ G}~+ w-+-o (09; o(26) r Gs +: z (2.62) Xk x k, TX a: az rs ~ = k2'Oy Gx + a ) (2.63) a (a + G i) (2.64) r =;+ - ox + az Substituting this expression into (2.59) gives E\ = / r't o' a),(x'dx'dy' x J I r= (y') q(x')dx'dy' A + J S Y.(Y')aq(x')dx'dy' y + j ' rF )a(y),(x')dx'dy'. (2.65) Recall that we only need the x-component of this field which is given by Eis= J JSr (y) aq(x')dx'dy'. (2.66) Furthermore, at z = h, boundary conditions require that E()(z = h) and E2)(z = h) be identical. From the above equation it is obvious that this implies r)(z = h) = r2(z = h) = P(z = h). (2.67) This equality is verified in Appendix E, and the result of (E.7) may be put in the form rr(z = h) = jwlo Z bd' mnd [cos k,x sin kyy cos kx' sin kyyl (2.68) m=I n=O lmnd2mn where fmn = pn tan k')h tan k(2)(h - c) [k(2)e (1 - k tan k2)(h - c) -k) ( l- ) tan k1. (2.69) 0 Q

30 If we place (2.68) into (2.66) we obtain E()(z - h) = E(2)( = h)= E(z = h) = j o E E abdimnd2mn cos k sin kVy zqmn (2.70) m=l n=0 Omn where Iqmn = s / cos k:x' sin kyy'(y')Caq(zx)dx'dy'. (2.71) This surface integration is evaluated in closed form in Appendix F; the result is (qIKl2 cos krxq Sinc 1 *1 Iqn = -4 sin1 Sinc [(k + K)l]J Sinc [j2(k - K)l]. 4 sin KtI, L2 sin kYo Jo(ky, ). (2.72) where sin for t 0 Sinc(t) = t (2.73) 1 for t 0. 2 for q= 1 -q = (2.74) 4 else K = wv/#ooiQQ, (2.75) 1 Rn, = ~(k+k,)t, (2.76) R2n = -(k - k)l:. (2.77) The position vector Xz gives the 2-coordinate of the qth strip subsection. As will be outlined in Section 3.1.3, the functional dependence of this vector varies with the type of structure being analyzed. We are now ready to evaluate the impedance elements. 2.4.2 Evaluation of the Impedance Matrix Elements From (2.70) and (2.54) we have p = fmn n (.78) m=l n=O abd- mn

31 where =pm cos kx sn k y i(y)op(x)dxdy. (2.79) Since (2.79) is the same integral as (2.71), we can obtain the expression for Ipmn by substituting p for q in (2.72). From (2.78), the entire expression for Zqp may be written as follows: Zqp - jwuoK 214 0 6ab sin2 Kl ~ c E ^n cos k;q cos kxp[ SincRn SincR2n]2 16absi K m- 1 n —n=o [sin(kYo)Jo (' )]2 tan k(i)h tan k(2)(h - c) [k(2) tan k)h - k() tan k(2)(h - c)] [k2)e; (1 -) tan k2)(h - c) - kl) (I - k) tan k)h] (2.80) (2.80) [k 2)r tan k(h - c) - ck) tan k()h] With the theory for the computation of the impedance matrix complete, we turn to the theory for the excitation vector 2.5 Excitation Vector Formulation In this section, a surface integral will be set up that provides for evaluating the elements of the excitation vector according to (2.16). This equation may be re-written as Vq = H(x = 0) * Moqpdpdb (2.81) where p, and q are cylindrical coordinates referenced to the center of the feeding aperture, as shown in Figure 2.7. An expression for MO will be presented first. Then, the magnetic field components parallel to the plane of the aperture (i.e. the y - z plane) will be derived based on the Green's function of Section 2.3. The actual integration of (2.81) is performed numerically, and this is described in Section 3.1.

32 z z i.........-'.... "".. '"'lt...... —' ^^..::...: I Figure 2.7: Geometry used to set up surface integral for excitation vector.

33 2.5.1 Coaxial Feed Modeling by an Equivalent Magnetic Current If the radius of the coaxial feed's inner conductor is assumed to be much smaller than the wavelength (kra << 1), and the coaxial feed line is designed to allow only transverse magnetic (TEM) propagation, we can represent the aperture by an equivalent magnetic frill current given by [25,18] M = - (2.82) where Vo is the complex voltage present in the coaxial line at the feeding point rb is the radius of the coaxial feed's outer conductor ra is the radius of the coaxial feed's inner conductor p, f are cylindrical coordinates referenced to the feed's center Substituting from (2.82) into (2.81) yields (with ds = pdpdq) seen in Figure 2.7. The integration can be broken up as follows: = -' n'id) [ H" )dpd + p H])dpd (2.84) In I q 2) ] where

34 S5') is the portion of the feed surface lying below the substrate (z" = p sin <^ -t) S(2) is the portion of the feed surface lying above the substrate (z" = p sin k ~ -t) The evaluation of the magnetic field components H(') and H(2) is described next. Once these have been evaluated, the integration of (2.84) is carried out numerically as discussed in Section 3.1. 2.5.2 Evaluation of the Magnetic Field at the Aperture To evaluate the magnetic field component Hq,, we will first determine the y and z components, and then perform a coordinate transformation to the cylindrical coordinates p and q. Determination of y and z components of Hq.- The magnetic field Hq anywhere inside the cavity is given by (2.26) H=V x Aq. Ipo The y and z components are given by Ha' _ (Oz (2.85) A -o a z Ox H' - 1 OA' Hq = -- (2.86) 9z IAo ay where, from (2.32) and (2.56), As = A J J (Y')tq(x(')G ds' (2.87) Ar, = po )(Y')(') ds'. (2.88) Combining the last four equations, we have H =s (G 4-G (y')q (x')ds' (2.89) - aay

35 These components are evaluated in Appendix G (for i = 1,2). Setting x = 0 in the resulting expressions yields: H()(x= ) = HoE c,c(') sin ky coskl)z (2.91) m=l n=O H(l)(x = 0) Hqo E E cnl) cos kYy sin k0c)z (2.92) m=l n=O H(2)(- = o) Ho E E nq c) sin kyy cos k(2)(z - c) (2.93) m=l n=O H(-)(x ) = Hqo > j cnqc(2) cos kyy sin k(2)(z - c) (2.94) m=1 n=O where -qK2 O 4ab sin KI, cnq = cos kzxq Sinc [(k + K)l., Sinc [- (k,- K)l] and -C) k(mn d k(l)ki(2) ) tan k(2)(h - c) ymn ~ d j 1 - [(kl))2 + k2(1 - _)] tan k()h} (2.95) c() - nky tan k2)(h - c),, W. Cn =sin kVYo )Jo(k-) (2.96) dzIm cos k('h 2k (2) - c {k)k2)tan ck()h V 2mn - (k(2))2 - k(1 - e:) tan k2)(h - c)} (2.97) dim, cos k(2)(h - c) 2 the y and z components of the magnetic field anywhere inside the cavity. To find the C component we will perform the necessary coordinate transformation in two steps: 1. move the origin from the corner of the cavity (Figure 1.3) to the center of the coaxial feeding aperture.

36 Table 2.1: COORDINATE TRANSFORMATION VARIABLES VARIABLE RELATIONS UNIT VECTOR RELATIONS x" =x x" = y" = - Y = p cos y" == cos op - sin qb " = - h, = pcos q z = sin 'p + cos,$,.,, 2. perform a cartesian to cylindrical coordinate transformation. Referring to Figure 2.7, let us denote a new coordinate system by (x", y", z") whose origin is at the feed's center (x, y, z) = (0, Yc he). The relationship between the new and old coordinates are outlined in Table 2.1. Using these relations, we will make the following substitutions in (2.91)-(2.94) to move the origin to the feed's center, and transform to cylindrical coordinates: Y —y"+Yc = pcos++Y, z - z + h, = psin + Yc Now, let Ht represent the projection of Hg onto the plane of the aperture such that Hst H= y + HI = H1Q0 + H p (2.99) = q Y qz q qpP where Hp' and H, are the p and k components respectively. Using the relations of Table 2.1, we readily obtain H = - sin OH', + cos OH,. (2.100) If we substitute from (2.91)-(2.94) into the above, and transform to cylindrical coordinates we obtain H()(x = 0) =

37 00c 00 Hqo - sin I f E C nqC n sin ky(pcos + Y) cos kl)(p sin k + h,) m=1 n=O + cos n E c(n CZ)n cos (p cos X + Y,) sin k()(p sin < + hc) (2.101) m=1 n=O H (2,) (X0)H 0 ): 00 00 Hqo - sin Ib i E CnC(2)n sin ky(p cos b + Yc) cos k(2) (p sin b - c") m=l n=O + os CnqC)C (,, cos (p c + Yc) sin k(2)(p sin c") (2.102) m=l n=O where c = c-hc. (2.103) 2.6 Current Computation for Two-Port Structures In the preceeding sections, the theory has been advanced for computing the impedance matrix and excitation vector associated with a one-port discontinuity, such as an open-ended transmission line. This section will present the modifications necessary to extend the theory for treatment of two-port structures. Our approach for computing the network parameters (scattering parameters etc.) of two-ports, requires simultaneous excitation of the strip conductors from both sides of the cavity. We will refer to this as "dual excitation". 2.6.1 Application of Reciprocity Theorem for Dual Excitation In section 2.2.1 an integral equation (2.3) was derived by applying the reciprocity theorem to the one-port network of Figure 2.5. In an analogous fashion, we will now apply the reciprocity theorem to the two-port network of Figure 2.8. In Figure 2.8, both magnetic current sources M01 and MO. are coupled with the electric current source J, on the conducting strips to produce the total fields

38 y STRIP CONDUCTOR Figure 2.8: In the case of dual excitation, the total fields Etot, Hfot inside the cavity are produced by magnetic currents M+1, MO, and the electric current J,.

39 Eot, Htot inside the cavity. As before, we consider an independent test source Jq, and associated fields Eq and Hq as shown in Figure 2.4. Applying reciprocity theorem between these two sets of sources yields j - (J*Eq-Rq - qV(M -H8 q Mf) dv = J J J EtBtdv = 0 (2.104) where the volume V is the interior of the cavity and Mi = MA1-, (2.105) M/,r = M~,4. (2.106) The right hand side of (2.104) vanishes as described by (2.2). Reducing the remaining volume integrals of (2.104) to the appropriate surface integrals gives J Eq(z = h). Jdds Hq(x = 0) Mads etrip Sf + J Hq(x= a) Mds. (2.107) In the above, Sif = Surface of coaxial aperture on left hand side (x = 0 side) of cavity Srf = Surface of coaxial aperture on right hand side (x = a side) of cavity. By comparison with (2.3), (2.107) can be seen as a natural extension to the theory for the case of single excitation. 2.6.2 Expansion of Current and Modified Matrix Equation The current J, is again expanded according to (2.6) N. Js = (y) s Ipp (x) p=l

40 The only difference is that we now must consider the basis function for the xdependence on the last subsection (i.e. closest to the right-most feed) as a special case. This is necessary since at each end of the cavity only a half sinusoidal basis function is required as illustrated in Figure 2.9. Hence, for the right-most subsection we let S~inI-afl X <x<a <, (x) = s i (K) N, x < (2.108) 0 else where the quantities K and I. are as defined in Section 2.2.2, and N. represents the index for the right-most subsection. The rest of the basis function expansion is the same as given by (2.7)-(2.9). Substitution of (2.6) into (2.107) yields E | E,(z = h). +(y)ap(x)xdIds = /I B,(x = 0)- MYds + H,(x = a) - M r4.109) which can be expressed as [Z] [I] = [V] + [V,] = [Vd] (2.110) where (Id] is the vector containing the current coefficients for the case of dual excitation [VI] is the excitation vector of the feed on the left (N. x 1) [V,] is the excitation vector of the feed on the right (N, x 1) (Vd] is the combined excitation vector for dual excitation. The elements of the excitation vectors [VI], and [V,] are given by Vq, = Hq(x}0) Msids (2.111) Vq = Hq(zx = a). AMsds (2.112)

41 Y b,, ^, % % %% % % ^, %, % 0 x x... X... X X X 2 3 4 p Na-2 IN-1 NH. I x s (p-l) 1 a P Figure 2.9: Strip geometry for basis function expansion with dual excitation. The case shown corresponds to a thru-line.

42 We can solve for the current vector by matrix inversion and multiplication according to [I] = [ [Vd] (2.113) 2.6.3 Modifications to Impedance Matrix The elements of the impedance matrix for the case of dual excitation are given by the same integral equation as for the one port case, namely (2.15). The difference is in the integration over the last subsection (p = Ns) where ap(x) is now given by (2.108). The integration for Eq. given by (2.70) is also modified for the last subsection (q = Ns) in a similar way. It can readily be shown that the surface integration over the last subsection (i.e. closest to the feed on the right) is equivalent to the integration over the first subsection (i.e. closest to the feed on the left). Hence, the elements of the impedance matrix are again given by (2.80) the only change being that Cq is as redefined below rather than by (2.74) { 2 for q= 1 or q = N (2.114) (, - (2.114) 4 else and Cp for the dual excitation case is given from (2.114) with q replaced by p. 2.6.4 Modifications to Excitation Vector We now consider the integrations of (2.111), and (2.112). By analogy with (2.82) we may express the two magnetic currents as follows: M, = - Vi (2.115) Ml,, = + - (2.116) In () p

43 where the positive sign in the second current source indicates that it is taken to be in the opposite sense (Figure 2.8). In the above two equations, Vol is the complex voltage in the coaxial line at the left-hand feed, and V, is the complex voltage in the coaxial line at the right-hand feed. Substituting from (2.115) into (2.111) yields Vqi = -( ] ] Hq(x = O)dpdk. (2.117) Similarly, from (2.116) and (2.112), qr = v ) jc Hqo(x = a)dpd4. (2.118) Now, the integration required for Vqi is identical to that carried out in Section 2.5 for Vq (single excitation case). The computation of Vqr is only slightly modified as we need to shift the origin to (x', y', z') = (a, Y, he) instead of to (0, Ye, hc). After examining the x-dependence of Hq given in Appendix G, it becomes obvious that we need only multiply the result for Vql by cos n7r to get the result for V,,. That is Vqr = -- cosn7r Vq. (2.119) Vor0 Let Vqd represent the elements of [ Vd] given by Vqd=Vql + Vr. Then, Vqd = (1 -V cov s nr) Vq (2.120) where Vq represents the excitation vector of Section 2.5 with VO set equal to unity. As discussed in Section 2.7, two-port scattering parameters are found by applying even and odd mode excitations to the circuit. The next step here is to find the excitation vectors for these two cases. For the even mode excitation, we let Vol = Vor = 1. In this case, from (2.120) we have Ve = V,(1 - cos nr)

44 0 for n even (2.121) 2Vq for n odd. For the odd mode excitation, we let Vol = -Vo, = 1. Now, using (2.120) V 0= Vq(l+cosnr) 2Vq for n even (2.122) 0 for n odd where Vqe represents the elements of the even excitation vector [Ve] Vqo represents the elements of the odd excitation vector [Vo] Using these two excitations, we can compute both the even and odd mode current distributions using the following matrix equations: [Ie] = [Z]- [Ve] (2.123) [Io] = [Z]-l [Vo] (2.124) where [Is] represents the current vector for even excitation [Io] represents the current vector for odd excitation. 2.7 Determination of Network Parameters The preceeding sections presented the theoretical methods for computing the current distributions for one- and two-port shielded microstrip discontinuities. The next step is to use these currents to determine the associated network parameters that can be used to represent them.

45 The relevant network parameters include the parameters of uniform microstrip line sections, and the parameters associated with discontinuity structures. The parameters for the uniform line sections are the complex propagation constant (-y), and the characteristic impedance (Zo). For the discontinuity structures, the relevant parameters are one or more of the following: * input impedance * reflection coefficient * scattering parameters * impedance parameters * admittance parameters * equivalent circuit parameters. Since the aim of this thesis is to concentrate on discontinuity effects, the characteristic impedance is not considered. For a microstrip line Zo cannot be strictly defined, and there has been considerable controversy over the most appropriate definition to use [27]-[31]. To avoid potential ambiguities caused by comparing results which may have been normalized to a different Zo, the author has chosen to work with normalized network parameters where possible. In comparing scattering parameters, the normalizing impedance is whatever impedance corresponds to the microstrip line width in use (i.e. it does not need to be calculated to compare scattering parameters). This is true for both the measurements and the numerical results of the present research, although it may not be true for the results presented from CAD packages.

46 2.7.1 Network Parameters for One-Port Discontinuities The simplest one-port discontinuity is an open-ended microstrip line. We will use the open-end as an example to illustrate the methodology for determining oneport network parameters. These include the propagation constant of the line, the reflection coefficient, and the input impedance at various points on the line. Calculation of the Propagation Constant In general, the complex propagation constant -g is given by -C= Ca + j /3 (2.125) For the computation of the current distribution, the theory for which is described in the preceeding sections, In the development of the preceeding sections, which describes the theory for computation of the current distribution, the dielectric material was assumed to be lossy in general. However, for the discontinuity structures treated in this thesis, only the lossless case is considered for the computation of network parameters. The network parameter theory for the lossy case is a relatively straightforward extension, hence the theory of this section can easily be generalized to handle the lossy case. In the lossless case 79 = jo,, where the phase constant 0g is given by 2r 2ir = (2.126) Ag and A, is the microstrip wavelength. This can be determined by calculating the distance between adjacent current maximums. Another parameter that is often used to describe microstrip propagation characteristics is the effective dielectric

47 constant eff. This may be found from Ag according to =(o) 2 (2.127) where Ao is the free space wavelength. Determination of the Input Reflection Coefficient and Impedance To determine the input reflection coefficient we first need to establish a reference plane at some point on the line. Two convenient points are just to the right of the coaxial input (x = 0) and at the point where the discontinuity begins. Choosing the reference plane at x = 0, the voltage reflection coefficient (looking towards the discontinuity) anywhere on the transmission line is given by [5] r(x) = SWR - 1 e_ - 2,(^ -) (2.128) SWR+ 1 where Xm,, is the position of a current maximum. SWR denotes the standing wave ratio, which is given by the ratio of the maximum current amplitude Ima,,l to the minimum current amplitude IminI. That is, SWR = limi. (2.129) Imin I The e-ic term in (2.128) arises since, on a lossless line, a current maximum corresponds to a short circuit (i.e. r(xm,,) = e-"). From (2.128), the reflection coefficient at the end of the line(x = Lin) is given by r(L = SWR- 1 iej2g, a (2.130) r(L) = SWR + 1 where dma = I - Xma is the distance from the end of the line to a maximum.

48 The normalized input impedance at any point on the line may be found from the reflection coefficient according to z(x) +(x) (2.131) Equivalent Circuit for an Open-End Discontinuity An open-end discontinuity in microstrip can be represented as either an effective length extension Lcff or by an equivalent capacitance cop as shown in Figure 2.10. Effective length extension for open-end discontinuity.- The effective length extension represents the length of ideal open circuited transmission line which, if as a continuation of the strip, would present the same reflection coefficient at x = Li, as the open-end discontinuity (Figure 2.10). This length is deduced from the fact that the current on an ideal open circuit would go to zero at a distance -J- from the last current maximum. Hence, the effective length extension is given by LeAf = - d,, (2.132) - -4 This representation gives an intuitive feel for the magnitude of the end effect. On the other hand, the equivalent capacitance representation is better for circuit design purposes. Equivalent capacitance for open-end discontinuity.- For an open-end in a lossless microstrip environment, the standing wave ratio is infinite (SWR -- oo); hence, from (2.130) rop = rop= ej e (2.133) where 8op = 2gdmax- r. (2.134) The associated normalized equivalent capacitance (Figure 2.10) can be expressed

49 as sin 2gd,, _ sin 2gL,ff = w(1 - cos 23gdmax) w (1 + cos 239Lef) ' (2.135) An algorithm for calculating one-port network parameters from the microstrip current, computed numerically, is discussed in the following chapter. 2.7.2 Network Parameters for Two-Port Structures The network parameters for two-port structures determined by analyzing the currents from the even and odd mode excitations discussed in Section 2.6, and illustrated in Section 3.2.4. General Representation by Equivalent T-network In general, a passive symmetric two-port structure can be represented by the T-network equivalent circuit of Figure 2.11. The T-network parameters are given in terms of the normalized impedance parameters according to VI z\1 Z12 I, [. (2.136) L V2 L 21 22 I2j Since we have assumed that the two-port structure is passive, reciprocal, and symmetric Z11 = Z22 and z12 = z21. These impedance parameters are found from the input impedances of the even and odd current distributions. The even mode excitation (Vg = Vg2 = Vo) corresponds to placing an electric wall in the center of the circuit as shown in Figure 2.12a. The odd mode excitation (Vg1 = -VK2 = VO) corresponds to placing a magnetic wall in the center of the circuit as shown in Figure 2.12. The normalized input impedances ZIN and ZN, for these two cases are found by analyzing the two separate current distributions as described in Section 4.2 for one-port analysis.

50 V op 1 0 i —& 0 0 I C r OR ri ~ p op I 0 o 4T —41 -r *r op op Figure 2.10: Representation for microstrip open end discontinuity.

51 Once these impedances have been determined, the impedance parameters are found according to Zll Z + Z- (2.137) 212 N. (2.138) Derivation of Scattering and Admittance Parameters The scattering parameters may be derived from the normalized z-parameters by using the following relations Sl = S22Z= 11 - 2 (2.139) D 12 = S21 = (2.140) where D = z21 + 2z1- z2 (2.141) Furthermore, it may also be desirable to compute the normalized y-parameters for the network. These may be derived from the scattering parameters using the following relations: 1- S2 - sh22 1 = Y22 = 11 S2(2.142) (1 + Sl)2 S12 2Sll512 Y12 = Y21 = (2143) / = 21 = (1 + S11)2 - (2.143) 2.8 Summary of Theoretical Methodology In this chapter, the theoretical approach used to compute network parameters for shielded microstrip discontinuities has been described. A combination of reciprocity theorem and the method of moments is used to derive a matrix equation

52 -------- ---- - - po 2-PORT p II,DISCONTINUITY I! -I -. ----. --- —-- —, 11 so Z1l ' 12.22'Z12 12 4. —E Figure 2.11: Equivalent network representation for generalized 2-port discontinuity.

53 ELECTRIC WALL Zo I zo A) Z1 - Z12[ l[ Z22-Z12 T 91 I, — ~ — 'J -o,-Z12 oI z e I e ZlN ZIN a) Case for even excitation. MAGNETIC WALL V gi gi 0 I ' I 0 IN I IN I b) Case for odd excitation. Figure 2.12: The even and odd mode excitations correspond to placing electric and magnetic walls in the center of the two-port structure.

54 that consists of an impedance matrix, an unknown current vector and an excitation vector. Two types of circuit excitation mechanisms are considered: 1) a coaxial excitation method, and 2) a gap generator method. The matrix equation is solved to compute the current distribution on the conducting strips. Based on this current distribution, the network parameters for one- and two-port discontinuities are calculated.

CHAPTER III COMPUTATIONAL CONSIDERATIONS This chapter is divided into two main parts. The first part describes the software design for implementing the theory developed in Chapter 2. First, the formulation used to compute the current distribution is re-arranged to facilitate computer solution. Next, the computer algorithm is discussed, and the computation of the current distribution is described. Algorithms for computing one- and two-port network parameters are also included. The second part of the chapter focuses on numerical convergence considerations. First, the convergence of the elements of the impedance matrix and excitation vectors is considered. Then a series of numerical experiments are described which are designed to test the stability and convergence of the final results. 3.1 Formulation for Computer Solution To compute the current distribution given by (2.17) we must first compute the elements of the impedance matrix Zqp and the elements of the excitation vector V1. The formulations for these elements, given in Chapter 2 are put in a form more convenient for programming below. 3.1.1 Formulation to Compute Impedance Matrix [Z] 55

56 The elements of the impedance matrix are given by (2.80). This equation may be re-written in the following form jwuoK 214 NSTOP Zq, = '6ab K sin2 KIcs esc [Sinc [l(k + K)l] Sinc [(k, - K)l]] [LN(n)] (3.1) where the vector LN(n) is given by the series MSTOP LN(n)= c Lm (3.2) m=l with the series elements Lmn given by qc,[sin(kYo)Jo (k..W)]2 tan k'1)h tan k(2)(h - c) Lmn a [ tl a - ) n - k2(h - c)] [k(2)e; (1- ) tan k(2)(h - c) - k() (1- ) tan k(')h] [k(2): tan k2)(h - c) - k tan k ()h] ( All of the other parameters are as defined in Chapter 2. 3.1.2 Formulation to Compute Excitation Vector [V] Excitation vector for coaxial (frill current) excitation Formulation.- The elements of the excitation vector for the coaxial excitation mechanism (2.84) may be written as TVo/ V K2 NSTOP Vq = E- X * cos k,=x q In(^) 4absinK!,,=o Sine [X(k, + K)l] Sinc [l(k,- K)!] [MN(n)] (3.4) L^ kJ LK1 Sc k., - JK)IX where the vector MN(n) is expressed in terms of the series given by MSTOP MN(n) = E Mm,. (3.5) m=l

57 The series elements Mmn are given by the following integral Mmn = fs M ndpd4 J= lM ldpdo +J]M(2) dp d (3.6) In the above, Sfl) is the portion of the coaxial aperture surface lying above the substrate (region 2) and S(2) is the portion of the coaxial aperture surface lying within the substrate (region 1). The integrands AM4 of (3.6) are given by M() - cos c)n os k,(p cos b + Y,) sin k) (p sin 0 + hc) -sin <f c( sin ky(p cos b + Yc) cos k(l)(p sin ' + h,) (3.7) for p and > in region 1, and M(2) = cos cos ky(pcos + Y) sin k(2)(psink - c") -sin & c(,2 sin ky(p cos < + Y,) cos k(2)(p sin - c"). (3.8) for p and < in region 2. Note that the coefficients cYLI, CZ)n, m2), and n2), which are given by (2.95)-(2.98), are independent of the integration variables p and q. Numerical Integration.- The elements Mmn of the series MN(n) needed to compute the excitation vector for the coaxial (frill current) method are calculated numerically using a 16 point Product Gauss formula approximation [33]. Let us define a pair of dummy variables s and u and a function F(s, u) such that JOmea2w rPm-a=b SmMa fUy a=l J J M dpdo>= J ] F(s,u)ds (3.9) Ji"s=0 iPmi r =-l umsw, -l where the correspondence between (u, s) and (p, k) is given by the following relations U 2p- (Pmo + Pmin) 2p -(b (3- 10) Pmax - Pmin rb - ra P = [u(rb-r) + (rb + r)] (3.11) =2 = (m +m1) - -1 (3.12) = Pmax - min = r(s+ 1). (3.13)

58 The numerical integration can be carried out by generating a set of 16 pairs of points (uj,sj) and adding up their contributions according to J/i< —=2r tPma =rb 16 in M mdpd n=E Sj(uj, sj) =r, j=1 (3.14) where ( uj, sj ) is the jth pair of integration points Bj is the weighting factor associated with the jth pair of integration points, and.F(sj, uj) is the transformed integrand found by performing a coordinate transformation on Mmn - Alternatively, once we have chosen the 16 points (sj, uj), we can find the corresponding values of p and f by (3.11) and (3.13) and obtain the same result. That is Im a Ps=2r pmas=rb. 16 / I Mdpd- = Z BjMmn(pj, -j) Jmin ~ J Ain =ar jl (3.15) where 1 p = [j(, - r,) + (rb + r,)] Oj = r(j + 1) and J M (p(, 'j) error condition for - h: < pj sin Oj < -t for -t < pj sin qj < c" = c - h(3.16) else. Gap generator excitation

59 For the excitation vector, all of the elements of the impedance matrix are set to zero except one, which is given a value of unity. That is, [V] = [ o0... 1 *.. o o ]. (3.17) Hence, the elements of the excitation vector for this case are given by v 1 for xq = X3 Vfo=rxg= (3.18) 0 else where x = Xg is the position of the gap generator. 3.1.3 Defining the Strip Geometry The formulations derived above for the computation of the current is general. That is, the same formulation can be used for all of the discontinuities considered in this thesis. However, the strip geometry differs between the cases. The strip geometry is specified by the position vector Xq (or xp) and the number of sections N,. The following discussion illustrates how these parameters are determined for various structures. For an open-ended line (Figure 2.5) and a thru-line (Figure 2.9) the position vector is given by 2p=(p-P l) 4 (3.19) The only difference between the two cases is the number of subsections N,. The strip geometry for a series gap discontinuity is shown in Figure 3.1. Let N. be the number of subsections required to compute the current on the left hand strip, with Ns being the total number of subsections. The position vector may be written as f(P-1)1 p < N.. P = ( P N(3.20) { pl+G p > Na.

60 y, b ---- Li ---- G ---- Li ---- 0 X,..,X X,...,X 1 Na Na+1 No I I X Figure 3.1: Determination of the computational parameters for a series gap is somewhat more complicated than for an openend or a thru-line.

61 where G is the length of the gap space. The strip geometry definition for coupled line structures is only slightly more complicated, than those discussed above. Further details have been omitted. 3.2 Algorithm for Current Computation The method of moments solution for the current distribution was implemented in a Fortran program named SHDISC. The various computation steps are outlined in the flow chart of Figure 3.2 and are summarized below. 3.2.1 Input Data File A data file is set up first to input the analysis frequencies and the geometrical parameters of the problem to be solved. Also, in this data file several flags may be set or cleared to direct the program flow, and file names are assigned to output files that will contain the results of the computations. One important flag indicates whether or not the LN(n) and MN(n) vectors are to be calculated or read in from a storage file. These vectors are independent of the subsection length I, and the strip geometry in the x-direction described by xp. Hence, once they have been calculated they need not be re-calculated unless the cavity geometry or the frequency has changed. For example, the same LN(n), MN(n) vectors may be used to calculate the current distributions for an open-end discontinuity, a thru-line, and series gap discontinuities with different gap spacings. This is important because approximately 50% to 90% of the computation time (depending on the size of the impedance matrix) is spent on evaluating the vectors LN(n) and MAN(n) vectors. There is also a flag to choose between gap generator and coaxial excitation.

62 I m, READ OD,., I Il/ K L!)II% Figure 3.2: Flow chart for program SHDISCto compute current distribution.

63 COMPUTE VARIABLES COMMON TO BOTH Lrn ANO Mmn COMPUTATIONS NO Figure 3.3: Flow chart for subroutine that computes vectors LN(n), and MN(n).

64 3.2.2 Computation of LN(n) and MN(n) Vectors After the input data file is read and the global parameters and arrays have been calculated, the program flow is directed to the subroutine "COMPLNMN", if the LN(n) and MN(n) vectors have not been previously computed. A flow chart for this subroutine is given in Figure 3.3. The first step is to set up the integration points pj and Oj to be used in the numerical integration for computing Mmn as discussed in Section 3.1.2. After this, the computation of LN(n) and MN(n) for each n is carried out by adding up the Lmn's and Mmmn's over the summation index m. The summations over m are terminated in one of two ways. The first is to test error functions which describe the fractional change in LN(n) and MN(n) with the addition of the last Lmn and Mmn respectively. The other way that the summations over m are terminated is if the specified maximum index value MSTOP is reached before the error goals have been satisfied. The truncation of the computations over the n-index is determined in a similar way. This time, however, the error functions describe the fractional change in the summation of LN(n) and MN(n) over n. As in the summation over m, the computation is also terminated if the maximum n-index NSTOP is reached before the error goal has been reached. For the convergence experiments which are described shortly, the error goals are set to very low values so that the computations on m and n are carried out to MSTOP and NSTOP respectively. 3.2.3 Computation of the Impedance Matrix and Excitation Vectors After the LN(n) and MN(n) vectors have either been calculated or read in from a storage file, the strip geometry is defined. This is done based on the

65 type of discontinuity to be analyzed and the geometry specified in the input file, as discussed in Section 3.1.3. Next, the elements of the impedance matrix and excitation vector are computed. To gain insight into the nature of these matrices, we will examine plots of a typical impedance matrix and typical one- and two-port excitation vectors. Typical impedance matrix.- Figure 3.4 shows the amplitude distribution of a typical impedance matrix. It is seen that the amplitude of the diagonal elements is the greatest and the amplitude tapers off uniformly as one moves away from the diagoxal. Another observation is that the matrix is symmetric such that Zqp = Zpq for any p and q, and the amplitude for any row or column displays the same distribution with respect to the diagonal element. Typical excitation vectors.- Figure 3.5 shows the amplitude distribution for the excitation vector for a typical one-port (open-end) structure. It is seen that the amplitude is highest over the first subsection, which is closest to the feed. The amplitude then tapers off rapidly over the subsequent subsections. The even and odd excitation vectors for a typical two-port (thru-line) structure is shown in Figure 3.6. The amplitude distribution is symmetric around the center of the cavity for the even case, and asymmetric for the odd case. Next to each feed (i.e. near x = 0 and x = a) the amplitude distribution has the same shape as for the one-port case, which is expected from (2.120). 3.2.4 Computation of the Current Distribution In the matrix equation (2.17), which describes the solution for the current distribution, matrix inversion and multiplication is implied. While this is correct mathematically, it is not efficient numerically. There are several approaches for solving systems of equations without matrix inversion. Most of these are readily

66 /ZQP/.A Q P Figure 3.4: A plot 3-dimensional plot of the impedance matrix for a typical open-end shows that it is diagonally dominant and well behaved.

67 > w O. I 83E+00 0. I 46E+00 0. 110E O+00 0.730E-01 0.365E-0 1 -0.647E-05 -.0.000 0.315 0.630 0.945 1.260 X(WAVELENGTHS) 1.575 Figure 3.5: The amplitude distribution for the excitation vector is highest for Q=1, which corresponds to the position of the feed.

68 0. 57E+00 0.1 25E+00 bJ 0.940E-01 0 z 0.627E-01 0.313E-01 -0.460E-05 -- 0.000 0.320 - 0.640 0.960 1.280 1.600 X(WAVELENGTHS) a. The amplitude distribution for the even case is symmetric 0.1 57E+O0 0.941E-01 O 0.315E-01 > -.'-\ ---y -0.31 1E-01 -0.937E-01 -0.156E+00 0.000 0.320 0.640 0.960 1.280 1.600 X(WAVELENGTHS) b. The amplitude distribution for the odd case is asymmetric Figure 3.6: Excitation vector for two port coaxial excitation.

69 8 6 u 4 / INVERSION - UDU METHOD 2 0' 0 100 200 300 400 Ns Figure 3.7: A comparison of computation times for two different methods of matrix equation solution shows the advantage of using alternatives to matrix inversion. available as subroutines in standardized libraries (e.g. Numerical Analysis and Applications Software -NAAS). For this work, a subroutine was employed that uses UDU factorization and back substitution. This method (also called LDL in some texts) takes advantage of the symmetry of the impedance matrix to speed computations. Figure 3.7 shows a comparison of computing times observed for a typical problem using matrix inversion versus the UDU factorization method. These computations were performed on an Apollo DN3000 work station. The speed advantage of the latter method becomes increasingly significant after a matrix size of about 150 x 150. Matrix sizes for problems studied in this thesis typically vary between about 50 x 50 and 250 x 250. Typical open-end current distribution.- When the impedance matrix of Figure 3.4 is inverted, the amplitude distribution is as shown in Figure 3.8. Multiplying by the excitation vector of Figure 3.5 yields the current distribution of

70 P=64 Figure 3.8: This 3-dimensional plot of the magnitude of the elements of the inverted impedance matrix shows that each row and column has a sinusoidal shape.

71 0.560E-02 0.336E-02/ \ 0.1 1 E-02 / \ I 1 / \ I - -0.113E-02 - \ -0.338E-02/ \ / -0.563E-02 - -I- I-I '-I 0.000 0.320 0.640 0.960 1.280 1.600 X(WAVELENGTHS) Figure 3.9: The imaginary part of current distribution for an openend discontinuity displays a sinusoidal behavior. Figure 3.9. It can be seen that the sinusoidal shape of the current has the same general shape exhibited by the first column of the impedance matrix. This is not surprising given the shape of the excitation vector. The multiplication can be thought of as a weighted summation of the first few rows of the inverted impedance matrix. A typical current for gap generator excitation is shown in Figure 3.10. For this computation the gap generator was located at a short distance ( - 3Ad )from the wall. The current is seen to be discontinuous around the region of the gap generator, however, is otherwise well behaved. The effect of this discontinuity can be minimized by locating the gap generator source at the beginning at the beginning of the first subsection (q = 1). Typical two-port current distributions.- The current computation for two-port structures is similar. In this case, even and odd currents are computed using excitation vectors shaped like those of Figure 3.6. The current distributions for a

72 0. 1 3E-01 - 0.778E-02 / \ - 0.260E-02 \ / - -0.258E-02 - -0.776E-02 - \ / -0.129E-01 - 0.000 0.320 0.640 0.960 1.280 1.600 X(WAVELENGTHS) Figure 3.10: The current for gap generator excitation is discontinuous around the position of the source, but is otherwise well behaved. series gap are shown in Figure 3.11. Due to the symmetry of the structure, the even current is symmetric, and the odd current is asymmetric around the middle of the cavity.

73 0. 714.E-02 -- 0.418E-02 - ~-, 0.121 E-02 - \ / a. - -0. 175E-02 \ -0.472E-02 -0.768E-02 I --- - l 0.000 0.350 0.660 0.990 1.320 1.650 X(WAVELENGTHS) a. Even current 0.596E-02 0.357E-02 0 1 9E-02\ -0.1 19E-02 \ -0.357E-02 - \ / \ -0.59e~-02 o.oao 0.330 0.660 0.9.3 1.020 X(WAVELENGTHS) b. Odd current Figure 3.11: Current distributions for a typical series gap discontinuity

74 3.3 Algorithms for Computing Network Parameters 3.3.1 Algorithm to Compute One-port Network Parameters A Fortran program was written to compute one-port network parameters as discussed above 1. A flow chart for the program ONENET is shown in Figure 3.12. The program first reads in the current distribution from a storage file created with SHDISC. The next step is to perform a cubic spline fit to the current. This is necessary to accurately determine the positions of the minimum and maximum current values. The subroutine used to perform this spline fit is a modification of a program appearing in [34]. In a cubic spline fit, each interval between two points is represented by a different cubic equation of the form I(x) = a,(x - xp)3 + bp(x - Xp)2 + Cp(x - xp) + dp (3.21) Once the spline fit coefficients ap, bp, cp, and dp have been found, determining the positions of the current minima and maxima is straightforward. First, the two points surrounding an extremum are found by searching for a sign change in the slope of the current ( I'(sp) = c, ) evaluated at successive points. Next, the actual position of the extremum is found to within the accuracy of the curve fit. This is done by finding the root of I'(x) = 0 which lies wthin the interval (xp < z < p+). The microstrip wavelength (AX) is then computed as the average distance between each pair of current minima and maxima. With the knowledge of the wavelength, the effective dielectric constant Eeff, and the phase constant /P can be calculated according to (2.127) and (2.126). 1 In this thesis the only one-port structure considered is the open-end discontinuity.

75 Finally, the equivalent circuit parameters for the open-end are computed. To do this, the reflection coefficient Top and impedance zop are calculated. Then, the effective length extension and equivalent capacitance are computed from (2.132) and (2.135) respectively. 3.3.2 Algorithm to Compute Two-port Network Parameters The program for computing two-port network parameters TWONET is very similar to that for one-port parameters. A flow chart is given in Figure 3.13. The first step is to read in the even and odd current distributions. These are analyzed separately. The analysis is performed in the subroutine SPLIMP following the same steps as in the one port case to compute Ag, fig, seff, r(Li), and z(Li) for each of the distributions. Next, the impedance, scattering, and admittance parameters are calculated using the relations of Section 2.7. This whole process is repeated for each of the analysis frequencies. 3.4 Convergence Considerations Zqp and Vq In this section, the convergence of the elements of the impedance matrix Zqp and the excitation vector Vq are discussed. The series involved in the computation of these elements are functions of two summation indices m and n. In theory, the summations are infinite; however, for computation we must truncate them at some point where the error due to this truncation is negligible. 3.4.1 Convergence of Impedance Matrix Elements Zqp We will now consider the convergence behavior of the matrix with respect to the summation indices m and n.

76 READ IN CURRENT DATA FROM STORAGE FILE FITTO CURRENT CUBSP FND POSmONS OF CURRENT SUBROUTINE MINIMA AND MAXIMA MINMAX g,ef and 13 i 3COMPUTE I.. I STORE NETWORK Lf and Cp PAR'S IN RLE STOP FREUE NCY ITO Figure 3.12: Flow chart for computation of one-port network param eters

77 fSTART READ IN CURRENT DATA FROM STORAGE RLE [START FREQUENCY LOOP] — ANALYSIS OF CURRENT FOR SUBROUNE - EVEN EXCITATION -— hp ANALYSIS OF CURRENT FOR.| SUBROUINE ODD EXCITATION SPUMP I COMPUTE NORMALZED * IMPEDANCE PARAMETERS * SCATTERING PARAMETERS * ADMITTANCE PARAMETERS L. --------- STOP FRECY LOOP ] Figure 3.13: Flow chart for computation of two-port network parameters

78 Convergence of Zqp with mn.- As can be seen from (3.1), the convergence of the impedance matrix with respect to m is described by the convergence of LN(n) with m. Recall that LN(n) is given in terms of the series of (3.2) MSTOP LN(n)= E Lmn. (3.22) m=l Figure 3.14 shows the typical variation of LN(n) with m and n. Most of the contributions from LN(n) to the impedance matrix are concentrated in the first several n values. The convergence over m is good. Further analysis, for this case shows that the change in LN(n) appears to be negligible after about m = 500. Convergence of Zqp with n.This computation of Zqp over n is illustrated for a typical impedance matrix in Figure 3.15. This figure shows the convergence behavior for one row (the 32nd) of the 64 x 64 element impedance matrix of Figure 3.4. This behavior is representative of that for any row. After only a few terms the diagonal element (p = q = 32) rises above the others, and after adding 100 terms the amplitude distribution is well formed. 3.4.2 Convergence of Excitation Vector Elements Vq We now turn our attention to the convergence of the excitation vector. In the following discussion, we consider only the one-port excitation vector since the two-port excitation vectors are derived from the one-port case (Section 2.6.4). Convergence of Vq with m.As can be seen from examining (3.4), the convergence of Vq with respect to m is described by the convergence of MN(n) with respect to m. MN(n) is given by (3.5) MSTOP MN(n) = E Mmn. (3.23) m=l

79 mdependenc '\ /LN(N)/ m=200 n-dependence n=100 Figure 3.14: 3-dimensional plot illustrating computation of LN(n) over m and n.

80 /ZQP/ P=1 P=32 N=O N N=100 Figure 3.15: The formation of an impedance matrix for one row (Q = 32) versus the summation index n.

81 /MN(N)/. dependence n-dependence Figure 3.16: 3-dimensional plot illustrating computation of MN(n) over m and n. A plot of MN(n) with m and n, shown in Figure 3.16, shows that the convergence of MN(n) on m is not as good as that for LN(n). The series exhibits a damped oscillation around the convergent value, which may take more than 1000 terms to determine. However, good results for network parameters are obtained by stopping at m = 500. This is discussed further in the section 3.5. Convergence of Va with n.- Figure 3.17 illustrates the computation of the elements of Vq as a function of the summation index n. As in the case of the impedance matrix, the amplitude distribution for the excitation vector has been well defined after adding the first 100 terms. 3.5 Convergence of Network Parameters

82 /Vq/, \ Q=64 Q N \N=100 Figure 3.17: The formation of an excitation vector versus the summation index n.

83 The previous section described how the elements of the impedance matrix and excitation vectors converge on the summation indices m and n. This convergence is important to examine; yet the question remains: how are the final results affected by various convergence related parameters? To answer this question, a series of numerical experiments were carried out. These experiments investigate the convergence behavior of the network parameters for an open-end discontinuity with respect to the number of samples per wavelength N. (= 1/1,), and the truncation points NSTOP, MSTOP for the summations over n and m respectively. From an efficiency point of view, we would like to minimize both the sampling rate and the truncation points. To examine these issues, of numerical experiments were performed at different frequencies and for different geometries, and a summary is given here. In this summary, the numerical experiments have been grouped into three separate categories which are named Experiment A, Experiment B, and Experiment C. Each of these explores a different aspect of the convergence behavior 2. 3.5.1 Numerical Experiment A: Effect of K-value Objective In the first experiment, the objective was to investigate how the value for K (K-value) used in the basis functions (2.8) and (2.9) affects the convergence on N, of the Leff and e.ff computations. Procedures and results 2 Unless otherwise noted, the parameters used for the plots shown in this section are the following: A, = 9.7, W = h =.025", a = 3.5", b = c =.25", f = 18GHz.

84 0.45 0.40 -; K=PI/10 - K=2PI 0.30, s K=4*P1 -- K-6'P 0.25 K1'*P1 0.20 0.15.... 0 25 50 75 100 125 150 Nx (sampleswavelength) Figure 3.18: Convergence of Lf/f versus sampling for several different K-values. Using the programs discussed previously, data was generated to plot Leff and ~erf versus Ns for several different K-values. Figure 3.18 shows the convergence behavior of Lf f for a typical case. It is seen that a relatively flat convergence region exists for all the K-values between about 40 and 100 samples per wavelength (Ad). Outside this region the solution behaves differently for different K-values. At first glance, it appears that the best convergence is achieved for higher K-values (e.g. K = 8r); however, quite the opposite conclusion results from examining the eIff computation. As can be seen from Figure 3.19, the best convergence for Eeff is obtained for low K-values. Based on these observations, it was theorized that it may be possible to improve the Lef computation by choosing a larger K-value at the end of the line, while keeping the K-value over the rest of the line at a low value. This was investigated by modifying the program SHDISC to use K = 27r for all the subsections except

85 7 -6-l /- 4 K-PI/10 - K-6'P1 K..10-Pt 2 a constant value of = 2 over all the subsections yields the best convergence 0 50 100 150 Nx (samples/wavelength) Figure 3.19: Convergence of Eeff versus sampling for several different K-values. the last Ad/8 portion of the strip. Over the last Ad/8 portion, the K-value was varied to examine its effect on the Leff and seff computations. It was found that a constant value of K = 2r over all the subsections yields the best convergence behavior for Le11. As expected, changing the K-value just at the end of the strip had very little effect on the e/ff result. Some comments will now be made on the Leff convergence behavior for low and high sampling rates. Referring back to Figure 3.18, it was found that a minimum sampling limit was observed that varies with the K-value. This limit may be expressed by the following Observation 11.1 For the sinusoidal basis functions of (2.8) and (2.9), there exists a lower sampling limit which depends on K and lI as follows KI 2 < (3.24) or N, > K. (3.25) T

86 If the sampling rate is lower than this limit, obvious errors in the current distribution and the network parameters (e.g. negative Leff) result. This limit has a physical basis in that if it is violated, more than one quarter of a sinusoid is represented by each half basis function. This means that the basis function tries to peak before it reaches the end of the section. Before examining the convergence behavior for high sampling rates, it will be useful to define the matrix condition number. The matrix condition number gives a measure of the relative sensitivity of the errors in the matrix solution to a change in the matrix elements. It may be defined as [35] Cond(Z) = lZi\ * iZ-11l. (3.26) Matrices with low condition numbers are said to be well conditioned; matrices with high condition numbers are said to be ill conditioned. In general the condition number, and therefore the accuracy of the matrix solution, degrades as the matrix size increases. This degradation in the condition number is responsible for the non-convergent behavior exhibited in Figure 3.18 for high sampling rates. Figure 3.20 shows how the reciprocal condition number (RC = l/Cond(Z)) is degraded as Nx increases. As Ns increases, the matrix order (N,) also increases for a fixed physical length of line. Hence, the gradual degradation in the condition number observed out to about N. = 130 was expected. Not expected was the large jump in RC which occurs at about 140 samples per wavelength. This jump indicates that the matrix suddenly becomes ill conditioned and, the associated current distribution is completely erratic. This erratic current condition was found to be independent of K, however, as discussed in the next section it is directly related to the length of the cavity a, and the number of terms added in the summation over n (NSTOP).

87 16C2 10'3 -10 io-B. 101.........,... ~. 0 50 100 150 200 Nx (samples/wavelength) Figure 3.20: Reciprocal matrix condition number versus sampling. 3.5.2 Numerical Experiment B: Lff, Ceff Convergence on n and m Objective The objective of this next experiment was to determine the appropriate truncation points NSTOP and MSTOP for the double summations involved with computing the impedance matrix and excitation vector elements. Procedures and Results First, several program runs were executed for different values of NSTOP, with MSTOP fixed at 1000. Data was generated to plot Leff and e/ff versus n for several I, values. Figure 3.21 shows that for all the 1, values NSTOP = 500, gives good convergence. The same can be said for the convergence of eff with n

88 0.32 \ -.01 z.02 0.30 \ - -.03 / - ---.04 V.-o ---<.05 0.28 0.26 0 200 400 600 800 NSTOP Figure 3.21: The convergence of Leff on n was found to depend on l., but is satisfied in all cases after 500 terms have been added. (Figure 3.22). To investigate the convergence behavior with respect to the summation index m, NSTOP was fixed at 500, and the program was run for different values of MSTOP. Figure 3.23 shows that Leaf converges well on m also after about 500 terms. Unlike the convergence on n behavior, the convergence on m does not depend on l1. The convergence behavior of Ceff (not shown) on m, was found to be similar to that for LeffThe dependence of the n convergence on 1I observed in Figures 3.21 and 3.22 warrants further consideration. It is seen that for large 1 values the solution converges faster on n. This variation of the convergence behavior with different I, values is also reflected in the reciprocal condition number as seen in Figure 3.24. It is also seen that the condition for erratic current, discussed in Section 3.5.1, is a function of both NSTOP and I,. Further experimentation with different geometries and frequencies, lead to the formulation of the following observation:

89 7.24 7.22 7.20 0 7.18 7.16 7.14 7.12 0 I I. x.----- it,..-.,..01.02.03.04.05 U -. -. - 200 400 600 200 400 600 800 NSTOP Figure 3.22: The convergence of el on n shows similar behavior as that for Lef, and is satisfied in all cases after 500 terms have been added. 0n 30 0.325 0.300 0.275 -0j. 0.250 -0.225 - I. x = -.01 --.02 -.03 *-c.04 =.05 0. 2........................ 0 200 400 600 800 1000 1200 MSTOP Figure 3.23: The convergence of LI/ on m is also satisfied after 500 terms.

90 10'2 10-3 -4. 10 x lo's.01 10.5 i.02 NSTOP 10.6.0 10both NSTOP and.04 10-8 Os-.05 10-9 10'1 10'1 IF 0 200 400 600 800 NSTOP Figure 3.24: A plot of the reciprocal condition number versus NSTOP shows that the condition for erratic current depends on both NSTOP and I,. Observation 111.2 The condition for erratic current is given by a simple relation between the cavity length a, the truncation point NSTOP, and the subsection length l1 which may be expressed as NSTOP * 1 < a (3.27) or NSTOP NT < a (3.28) This will be referred to as the erratic current condition. This condition places an upper limit on N. and, correspondingly, a lower limit on the I1 value that can be used to generate useful current results. The erratic current condition was tested against several cases where erratic currents were observed, and it appears to give an exact prediction.

91 3.5.3 Numerical Experiment C: Optimum Sampling Range Objectives The objective in this last experiment was to examine the effect of varying I. on the reciprocal matrix condition number, while keeping the matrix size constant. It was hoped that an optimum value, or range of values, for l- could be found for which the matrix condition number is maximized. Procedures and Results To keep the matrix size constant, the number of sections N, was fixed at 100, and the length of the open-ended line (L' = Li,/Ad) was allowed to vary such that L' = 100 * Is for all cases. Data was then generated to plot RC versus Is as shown in Figure 3.25. As suspected, an optimum sampling range was found, outside of which the matrix is ill conditioned. The experiment was repeated at three different frequencies and for several different shielding geometries. In all cases, an optimum sampling range was found, but it was different for each of them. To examine this sampling range further the following postulate was advanced, based on the erratic current condition of Experiment B. Postulate 1.l Postulate:.- Let N,, and Nr2 correspond to the minimum and maximum desired sampling rates. These rates are defined as those values between which the reciprocal condition number is greater than 10-4 for a fixed matrix size

92 * 10 - Nxl=30 Nx2-135 10'. - 0 25 50 75 100 125 150 175 200 225 NX Figure 3.25: The effect on the reciprocal condition number of varying only Is indicates the existence of an optimum sampling range of 100 x 100. 3 Let 11, and 12 be the subsection lengths corresponding to Ns1 and N:2 respectively. That is 1s1 = 1/Nzi (3.29) 1,2 = 1/NI 2. (3.30) We now postulate the existence of two constants r1 and r2 such that 1 = NSTOP (3.31) 412 = NSTOP (3.32) The definition of the sampling range is illustrated in Figure 3.25. This postulate 3 Based on observation, a value of RC = 10-4 appeared to be a good value to use as a lower limit.

93 Table 3.1: RESULTS FOR OPTIMUM SAMPLING RANGE EXPERIMENT f (GHz) a 1t2 1i 1:avg r2 r | rav 2 1.58.0035.0125.008 1.1 4.0 2.5 8 6.33.017.07.043 1.3 5.5 3.4 8 1.0.0029.0125.008 1.5 6.3 3.9 18 3.5.0074.033.020 1.1 4.7 2.9 18 1.0.0023.0125.0074 1.2 6.3 3.7 min. - - - 1.1 4.0 2.5 max. - - - - 1.5 6.3 3.9 avg. - - - - 1.2 5.4 3.3 was applied to the observations from several test runs and the results for the sampling range parameters are summarized in Table 3.1. The data of Table 3.1 shows that, although Postulate 3.1 is only approximately true, an optimum sampling range can be specified in terms of the multipliers rl and r2. This range is formulated in the observation below. In selecting the multiplier r2, the maximum observed value from Table 3.1 is used, and in selecting the multiplier rl the minimum observed value is used. This is done to define a range for which condition number for the cases of Table 3.1 is always greater than 10-4. Observation 111.3 An optimum sampling range (or criteria) may be defined by the following choice of subsection length I. 1.5a 1 < 4a NSTOP - NSTOP' (' )

94 0.40-. 0.38 ~'36 ' JAOPTM SAMPLING REGION - 0.32- __ 0.28 0.24 NSTOP.5 a NSTOP/4a 0.20 -.. '....,...... 0 25 50 75 100 125 150 Nx Figure 3.26: The optimum sampling range is seen to correspond directly with the flat convergence region for the Leff computation. A good average value to use is 3a 3-34 = NSTOP (3*3) The above observation is very significant. Based on the knowledge of only two parameters, a and NSTOP, an optimum sampling range (or range of subsection lengths) can be determined. The erratic current condition is automatically avoided by sampling within this range, and the best accuracy in the matrix solution should be guaranteed. To support this last claim, consider the plot of Figure 3.26. It is seen that the optimum sampling region specified by 3.33 coincides directly with the flat convergence region for the Leff calculation! This consistency between the optimum sampling region and the flat convergence region for the Le!! calculation was observed in all the cases of Table 3.1.

95 3.6 Summary of Computational Considerations In this chapter, the software design for the computation of the current distribution, and the network parameters for discontinuities has been described. The programming language used is Fortran. The convergence of the solution with respect to the relevant parameters has been explored extensively by performing a series of numerical convergence experiments. The results from these experiments lead to some simple, but very useful relationships governing the convergence and stability of the solution. A summary of the main findings is given in Chapter 6.

CHAPTER IV EXPERIMENTAL METHODOLOGY As part of this research an experimental study of microstrip discontinuities was performed. In conducting this study, the author spent fifteen months at Hughes Aircraft Company1. Hence, parts of the study were performed at Hughes; the rest was performed at the University of Michigan. The chapter gives a description of the study and the procedures used to obtain measured data. First, a general discussion of the experimental approach is given. This consists of the use of Automatic Network Analyzer (ANA) techniques in conjunction with a method for de-embedding (or removing) the effects of the test fixture from the measurements. Next, a comparison of various de-embedding methods is presented. It is concluded that the method most suitable for the measurements of this thesis is the thru-short-delay (TSD) method. The implementation of this method for use with the present research is then explained. Finally, the procedures used to obtain measurements of effective dielectric constant, open-end and series gap discontinuities, and coupled line filter structures are explained. A complete error analysis for these measurements is beyond the scope of this thesis, however, an attempt is made to give a reasonable estimate of measurement uncertainties. To this end a perturbation analysis approach is developed and applied to approximate the effect of connection repeatability errors on de-embedding accuracy. 1 Hughes Aircraft Company, Nlicrowave Products Division, Torrance, CA 96

97 4.1 Discussion of Experimental Approach 4.1.1 ANA Error Correction A basic ANA provides for two-port, error corrected S-parameter measurements in a coaxial or waveguide environment. Error correction is achieved by using a set of standards whose electrical characteristics can be determined to a high degree of certainty. For example, the standards used in a typical coaxial calibration are a 50 ohm load, a short circuit, an open circuit (whose fringing capacitance has been determined apriori), and a thru connection. Measurements on these standards are used to construct an error model for the ANA system which accounts for various system imperfections such as finite coupler directivities, connector mismatches, unflat frequency responses, and source and load impedance mismatches [36]. 4.1.2 Difficulties with Microstrip Measurements Measured data on microstrip discontinuities is very limited, particularly at higher frequencies (above lOGHz). This is due to the many difficulties involved with performing accurate microstrip measurements. The key difficulties associated with these measurements are summarized in Table 4.1. These difficulties are not unique to microstrip and apply to measurements in other planar transmission media as well. The main difficulty is that in order to measure a microstrip circuit, it is generally mounted in a test fixture with either coaxial-to-microstrip or waveguide-tomicrostrip transitions (also called launchers). Figure 4.1 shows the basic configuration for test fixture measurements. The transitions invariably introduce unwanted

98 Table 4.1: DIFFICULTIES WITH MICROSTRIP MEASUREMENTS * Separation of microstrip test fixture parasitics from measurements * Inadequate microstrip calibration standards * Non-repeatability of microstrip connections * Effect of variations in substrate material properties * Effect of variations of metallization dimensions and substrate thickness * Effect of substrate mounting techniques parasitics and a reference plane shift to the measurements. These effects must be accurately accounted for and removed from the measurements, or incorporated into the ANA system error model. One alternative to the use of a test fixture is to employ coplanar wafer probing [37]. However, to measure microstrip structures with coplanar probes requires the use of coplanar-to-microstrip transitions. Since the issues with removing the effect of these transitions are the same as for the other transitions mentioned above, the term "test fixture" as used below will be assumed to include the case of coplanar probing. Another main difficulty with the measurements is the inadequacy of microstrip calibration standards. Conventional calibration standards are much more difficult to realize in microstrip than in waveguide and coax. Perfect short circuits are complicated by the non-uniform nature of the fringing fields in microstrip, thin-film resistors do not provide the same quality of 50 ohm terminations as in conventional media, and the open-end capacitance is not known to a high enough degree of accuracy for it to be used directly as a calibration piece 2. Hence, conventional ANA 2 A rigorous numerical solution, such as that developed here may provide

99 [ III I —I ANA SYSTEM i II COAX (OR WAVEGUIDE) TEST PORTS MICROSTRIP TEST PORTS 41 AL, I I INPUT HALF O L. OUTPUT HALF OF TEST FIXTURE, TEST FIXTURE ISAI r- -,s91 I O.U.T. I L — I I.UT, Figure 4.1: Microstrip test fixture approach for de-embedded measurements. calibration (Section 4.1.1) in microstrip is not easily performed. This difficulty is overcome through the use of de-embedding techniques as discussed below. The third factor complicating these measurements is the non-repeatability of microstrip connections. Microstrip connections are much harder to make, and much less repeatable than connections in coax and waveguide. This is a key limiting factor to the accuracy of microstrip measurements at higher frequencies. To address this issue, a microstrip connection repeatability study was carried out [14,15]. Details of this study are included in Appendix H, and the results are summarized in Section 4.3 below. The remaining factors of Table 4.1 can also be important. These factors include the effect of variations in substrate material properties, variations in metalization and substrate thickness dimensions, and substrate mounting techniques. The efthe required accuracy to alleviate this problem.

100 fects of these factors on measurements is considered further in Section 4.3, and some consideration is also given in the repeatability study (Appendix H). Their effect can be minimized by paying careful attention during material selection, and during the fabrication and mounting of test circuits. Before discussing de-embedding methods, a few comments will be made on resonator techniques, since they have been widely used for microstrip measurements. 4.1.3 Resonator Techniques One way of minimizing transition effects is to incorporate the transition into a resonant circuit which is lightly coupled to a source and detection system [l]. The majority of existing experimental microstrip data have been obtained using such resonator techniques [38]-[41]. The main advantage of this approach is that unwanted transition effects, such as poor VSWR or contact repeatability have minimal effect on measurement accuracy. Also, useful measured data can be obtained without the use of an ANA. However, the coupling necessary for adequate sensitivity is sufficient to provide some reactive loading of the resonant system which is not easily accounted for. In addition, the measurement procedure is tedious and is not practical for measurements over a broad range of frequencies. 4.1.4 The De-embedding Approach The preferred approach to removing test fixture effects is called de-embedding. De-embedding refers to the process by which test fixture effects are removed from the measurements. With the exception of time-domain de-embedding, which will be discussed in Section 4.2.2, the de-embedding process consists of two steps:

101 1. Test fixture (or error) characterization 2. Error extraction. A microstrip test fixture can be characterized either by an equivalent circuit model (this is generally valid only at low frequencies) or by performing measurements with a number of planar standards inserted within the fixture. The electrical parameters of the fixture, for example the S-parameters for each fixture half, are then used to mathematically move the effective calibration reference planes from the coax or waveguide test ports to the desired microstrip test ports (Figure 4.1). 4.2 Comparison of De-embedding Methods In this section, several de-embedding methods are compared in terms of their applicability for our measurement requirements. Specifically, the method used must: * be usable at high frequencies (10GHZ and higher) * allow the de-embedding of arbitrary transition effects * allow the connection of any standards used to be similar to those made to the discontinuity structures * use standards whose electrical characteristics are "known" to a reasonable degree of certainty. Also, the electrical characteristics of any standards used must be They must be well established theoretically, or determined from independent measurements. 4.2.1 Fixture Equivalent Circuit Modeling

102 One method of fixture characterization is to propose an equivalent circuit model for the fixture. The simplest model uses a section of ideal transmission line to model each half of the test fixture. In this case, launcher parasitics are completely neglected and de-embedding is performed by simple transmission line rotations around the Smith chart. Launcher parasitics can be at least partially taken into account by using more complicated equivalent circuit for the fixture whose parameters are fitted to a set of measurements on a reference circuit (e.g. a straight section of transmission line [42]). This approach while an improvement over simple transmission line rotations, lacks generality. It also tends to be unreliable for high frequency use, since fixture parasitics become more difficult to model as frequency increases. 4.2.2 Time Domain De-embedding Another method of de-embedding makes use of transformations back and forth between the frequency and time domains. This method, which follows from Hines and Stinehelfer [501, involves the use of a Fourier transform to obtain a time domain response from frequency-domain data. In concept, fixture effects can be eliminated by isolating (or gating) the time response of the desired circuit and then transforming back to the frequency domain to obtain the de-embedded frequency response. However, for adequate resolution, it is necessary to use long input and output lines and collect data over a broad range of frequencies. Even then, although useful for many applications,such as fixture development, this technique is not as accurate as full matrix de-embedding in the frequency domain [51]. 4.2.3 Full Matrix De-embedding

103 The remaining de-embedding methods to be discussed fall into the category of full matrix de-embedding. In full matrix de-embedding, each half of the test fixture is characterized by matrix parameters (e.g. scattering matrix [S], or transmission matrix [T]) as a "black box". This is done by measuring planar standards for which the network parameters are known, or can be determined to a reasonable degree of certainty. Examples of such standards are microstrip delay lines, offset microstrip open-ends, and varactor diodes Once the fixture has been characterized, inverse matrix operations can be used to deduce the electrical parameters of the unknown device or circuit. These matrix operations vary little between full matrix de-embedding methods; conversely, the fixture characterization process differs considerably depending on the type of standards used. Most methods use one of the following combinations of standards: 1. Delay line and reflection standards [44]-[47] 2. 1-port offset reflection standards [52] 3. Varactor diode standards [53]-[55] The relative merits of different fixture characterization approaches based on the use of these standards are compared in Table 4.2. Based on this comparison, the use of delay line and reflection standards, and in particular, a method based on the thru-short-delay (TSD) approach was adopted for this thesis. A discussion of this choice is given below. In the TSD method (Figure 4.2), two-port measurements made on a thru (zero length delay) line, a "short" circuit, and a delay line provide enough information to characterize the fixture. As byproducts of the procedure, calculations of the propagation constant 9, and the reflection coefficient r, are provided. Since the original paper [44], it has been pointed out that the "short" implied

Table 4.2: FULL MATRIX DE-EMBEDDING METHODS APPROACH TSD (TRL, LRL etc.) S.C. + 4 O.C.'S VARACTOR DIODE AUTHORS Franzen and Special ianco e. a., da Siva and McPhun Bauer and Penfield. Pedc Engen and Hor, Maury et. al. and Peterson o Thru, delay and reflection sds. o Offset reflection stds. o Reverse bias varacor diodes CONCEPT used as variable capacitane o Network pars of stds. theoredcally o Short used to establish sids. known or not ailcal relerence plane. o Stds. characterized by o Fixture network parameters o Fxture network parameters independent measurements computed rom 1- and 2-port meas. computed from 1-port meas. o Fixture network parameters computed from 1-port meas. o Stds. easy to realize in mcrostrip o. ADVANTAGES o Open circuit stds. easy o C ecions ds. same as realize o Multiple known loads possible oConnections to stpdisconanum s | with a single connection connections to mstrip discontinuies Connections to stds. same as o Connections to stds. same as connections to mstrip discontinuities o Connections to stds. similar to o Provides calculation of propagation connections to active devices constant 1 g o Provides calculation of propagation o Provides calculation of reflection constant Y coefficen r o Perfct hrt circuit difficult o Connections to diodes dissimilar DRAWBACKS o Connection repeatability to reize to mstrip discontinuities. portanfor a racy o Connection repeatability o Characterization of stds. important or accuracy tends to be complicated o-4 4,^

105 UNKNOWNS: S1A. S22A.S21A (1S2A) - 3 S1IS 228 S21- 3 7 COMPLEX \ \ UNKNOWNS J I "SHORT" TH-U I 321 M I I \ 11M 1 2M S Ss S11 2 2MM Figure 4.2: Fixture characterization by the TSD technique (Note: the "short" can be any highly reflecting standard). in TSD, need not be perfect. In fact, any highly reflecting standard may be used in its place [45,46]. The only requirement is that the same reflection coefficient r, must be presented to both microstrip test ports. This is important since, as discussed above, a perfect short is difficult to achieve in microstrip. This variation of the TSD method, that uses an arbitrary reflection standard, has been called the thru-reflect-line (TRL) method. Another variation, called the line-reflect-line method (LRL) [47], allows the use of a non-zero length thru line. Since both of these variations are derived from TSD, the overall approach will be referred to as TSD in this thesis. The understanding will be that the "short" is non-critical. The main advantage of the TSD approach over the alternatives of Table 4.2 is that the TSD standards are easiest to realize in microstrip. da Silva and McPhun, suggest the use of 4 microstrip open circuits, and a short circuit. As far as this author can determine a perfect short is required to accurately establish the reference plane. To use varactor diode standards a rather complicated modeling procedure

106 must be carried out to characterize the diodes prior to use. In addition, the fixture hardware may be designed so that the connections made to the TSD standards are made in the same way as the connections to the microstrip discontinuity test circuits. This is not the case with varactor diode standards as the introduction of the diode mounting structure adds parasitics during fixture characterization that, in general, are not present when measuring discontinuity circuits. One drawback to the TSD method is that good microstrip connection repeatability, and microstrip circuit fabrication repeatability are important for accuracy. This is also true (probably to a greater degree since more standards are required) for da Silva and McPhun's approach. In this area, the varactor diode approach is attractive since several different capacitances can be realized by varying the diode's reverse bias voltage. However, this advantage may be partially offset by the nonrepeatability of returning to the exact same bias voltage during calibration that was used during the characterization of the diodes. One-tier vs. Two-tier De-embedding.- As discussed by Lane [43], full matrix de-embedding can be performed using either a one-tier or a two-tier approach. The choice between these two options is controversial, and it depends on the application and fixture hardware. Referring to Figure 4.1, one-tier de-embedding involves making a direct calibration at the microstrip test ports. In this approach, fixture effects are included in the error model used to represent the ANA system imperfections. This approach has been used in coplanar waveguide [37], and also in microstrip [56]. One-tier de-embedding is more straight forward and in some ways easier to implement than two-tier de-embedding. Also, with the speed of current ANAs, the one-tier approach can display de-embedded measurements in essentially real time. In contrast, two-tier de-embedding involves calibrating first at the coaxial or

107 waveguide terminals. Measurements are then made on various microstrip standards to determine the S-parameters of each half of the test fixture. These are stored and later used to mathematically transform the measurements to the microstrip test ports. One advantage of two-tier de-embedding for the present application is that it provides for better monitoring of the de-embedding process. Measurements can be made with several different connections made to each of the standards used for fixture characterization. These measurements can be stored in files and then compared in order to screen out bad connections. The remaining connection trials can then be averaged to reduce connection repeatability errors. For this reason the TSD method was implemented as a two-tier procedure, as described next. 4.3 Implementation of TSD De-embedding 4.3.1 Software and Hardware Considerations Computer programs for performing TSD fixture characterization and de-embedding were provided by Hughes. As part of this work, modifications were made to these program to customize them for the present application. The mathematics used are described in detail elsewhere [48,49]. A flow chart for the measurement procedure used is shownin Figure 4.3. The programs used to carry out this measurement procedure are set up to process several measurement frequencies simultaneously. First, ANA system calibration is performed with coaxial standards. The S-parameters of each of the TSD standards are then measured and stored in data files. To reduce connection repeatability errors, repeated measurements are performed with two to five connections made

108 to each of the standards. These measurements are then used to obtain an average set of S-parameters for each fixture measurement. The averaged S-parameters of the standards and the physical length of the delay line are then input to the fixture characterization program. This program performs calculations to provide measured values for Eeff,r,(= ( op), and the S-parameters [Sa], [Sb] for each half of the test fixture. Finally, measurements are made with the D.U.T. (i.e. the desired discontinuity circuit) connected within the fixture. Again, repeated measurements are made to reduce connection errors. The averaged D.U.T. S-parameters are then processed along with the fixture characterization data to obtain the de-embedded measurement. If necessary, the reference planes established by the de-embedding (in the middle of the thru line) are moved by performing simple transmission line rotations. The phase constant used for these rotations is based on the measured effective dielectric constant. The instrumentation used for the measurements of this thesis was almost exclusively HP8510 ANAs which were available to the author at both the University and at Hughes. The only exception is that an HP8409 ANA with a high frequency waveguide extension was used to perform connection repeatability measurements in the 26.5-40GHz frequency range. The test fixture that was used for the discontinuity measurements was also provided by Hughes. A picture of the test fixture is shown in Figure 4.4. The fixture employs a pair of 7mm coaxial "Eisenhart" launchers [57] and is usable to 18GHz. The shielding is provided by placing U-shaped covers on top of the microstrip carriers. This forms a cavity similar to Figure 1.3. Another fixture, usable to 40GHz was also developed by the author for use with this thesis. This fixture was used in the connection repeatability study (Appendix H), but not for discontinuity measurements. The reason for this was logis

109 PERFORM SYSTEM CALIBRATION W/COAXIAL STDS. MEASURE S-PAR'S OF FIXTURE W/STDS. PERFORM FIXTURE CHARACTERIZATION INPUTS: OUTPUTS: - S-PAR'S OF - eeff TSD STDS -rS - PHYSICAL LENGTH OF - FIXTURE S-PAR'S DELAY LINE I [SA],S I] MEASURE FIXTURE W/ D.U.T. DE-EMBED FIXTURE EFFECTS INPUTS: I OUTPUTS: - S-PAR'S OF D.U.T. - DE-EMBEDDED I S-PAR'S OF D.U.T. - FIXTURE CHARACTERIZATION DATA ii I_ Figure 4.3: Procedure used in this work for measurement and deembedding.

110 Figure 4.4: 7mm coaxial/microstrip test fixture (partially disassembled). tical and not technical. All of the discontinuity measurements were performed at the University, and at the time these experiments were planned (substrates ordered etc.), facilities were not available to make measurements above 18GHz. Hence, the 7mm fixture was used for all the discontinuity measurements. 4.3.2 Connection Approaches For TSD De-embedding There are three basic connection alternatives for TSD characterization of a coaxial fixture. Each of these must rely on at least one of the following assumptions: 1. repeatability of connections made from the coaxial-to-microstrip transition (launcher) to the microstrip line (i.e. coax/microstrip connection repeatability) 2. repeatability of microstrip/microstrip interconnects

111 w12 C11. C21 H- L2j \ H H _ __LA L13LALa) THRU b! "SHORTR -i /2 L /2 - f/2- H C33 1 C i4 C24 I m' [LA ^**UT La i.U.T I I d) O.U.T. MEASUREMENT e) DELAY REPEATASIUTY ASSUMPTIONS: Cl1 -C12" C13 Cl4 ANO C21- C22 - C23 C24 Figure 4.5: The TSD Connection approach used for the present work, relies on repeatable coax/microstrip connections. 3. uniformity of electrical characteristics between different transitions (launcherto-launcher uniformity). In the first connection approach (Figure 4.5), continuous substrates are used to realize the three standards, which are connected between the same pair of launchers (L,, Lb). In the second method (Figure 4.6), the TSD standards, and the D.U.T., are connected between the same two launcher/microstrip assemblies using microstrip-to-microstrip interconnects (e.g. ribbon bonds). In the third approach, each of the standards would be constructed complete with their own intact launchers. While this has practical advantages, such as improving the durability of the standards, launcher-to-launcher uniformity is not generally a good assumption for microwave de-embedding (Appendix H). In all of the above approaches, it is important to have good uniformity between various microstrip line sections as mounted in the fixture hardware. If the fixture must be removed from the coaxial measurement ports to insert either the stan

112 [^ C'2La1 C^^12 a C'22 LA.-X LS k LS L - _ II-I_ - a) THRU be "SHORT1/2 -1 -i 1/2 --- /2 H/2g C- ' ''!. I -,- D23 I ' — C C'21 — L'A aS E ^ Z e LA O L6T ____ ______ LS I I d) O.U.T. MEASUREMENT c) OELAY REPEATABILITY ASSUMPTIONS: C'11 C'12- C'13- C'14 ANO C'21 - C'22 - C'23 C24 Figure 4.6: TSD Connection approach relying on repeatable microstrip/microstrip connections. dards or the D.U.T., de-embedding accuracy is also subject to uncertainties due to repeated coax/coax connections. However, the study results suggest that these uncertainties are negligible compared to other errors. The results of the repeatability study (Appendix H) clearly favor the connection approach of Figure 4.5 relying on repeatable coax/microstrip connections, and this was the approach adopted for the present work. 4.3.3 Measurement of Effective Dielectric Constant As mentioned previously, one of the byproducts of the TSD method is the calculation of the propagation constant -7(= a29 +jt3). For the alumina substrates used, the loss factor a was found to be too small to measure by this method, or conversely the measurement sensitivity is not great enough. Using Super Compact this loss (combined dielectric and conductor losses) is estimated to be.05dB/cm

113 at lOGHz. For the delay length used here this translates to a total loss of.01dB which is less than the error due to connection repeatability. The phase constant 13 is measurable, and may be used to calculate the effective dielectric constant Eeff by the following relation: ()= W (4.1) where c is the velocity of light and w is the radian frequency. The value of se/f calculated from the TSD procedure relies mainly on the delay line length A/l, and the difference in measured transmission phase between the thru and delay line standards. Figure 4.8 shows a typical Eeff measurement resulting from a single TSD fixture characterization procedure. A sketch of the standards used is given in Figure 4.7. The "raw" measurement shown is the actual measurement, and the "fitted" result was obtained by performing a least squares polynomial curve fit to the measurements. As supported by the theoretical results of Chapter 5, the actual e.ff should follow a smooth curve. Hence, deviations from the fitted curve indicate imperfections in the measurement. In this work, the fitted curve is generally used for any post de-embedding transmission line rotations. 4.3.3 Measurement of Open-end Discontinuity A measurement of the reflection coefficient and related capacitance of a microstrip open-end discontinuity was obtained based on using an open-ended microstrip line as the reflection standard in place of the "short" of Figure 4.2. A measurement of the reflection coefficient r,(= /rOp/e93") results from the fixture characterization procedure. Figure 4.9 shows a representative plot of the measured (de-embedded) reflection coefficient of the open-end standard of Figure 4.7. The phase angle shows an increasingly negative phase shift with frequency, as expected.

114 K A 244 STD. DIM. A THRU 730 DELAY 824 a) Thru and delay line standard 730 J b) Open-end standard Figure 4.7: Sketch of TSD standards used for measurements of Ceff, and open-end and series gap discontinuity circuit. Note: all dimensions of Figure are in mils (1 mil=.001"), h =.025", e, =9.7.

115 7.4 -7.2 -7.0 - 6.8 - 0 w -N- MEAS. (RAW) -a- MEAS. (FITTED) 6.6 - 6.4 I * 0 2 4 I I I,.. I I I I a I I I 6 8 10 12 14 16 18 20 FREQUENCY(GHZ) Figure 4.8: Typical effective dielectric constant measurement resulting from TSD fixture characterization. (Shielding dimensions: b = c =.25"). nO I -5 C3 8 8 -(3 -10 - -15 --20 --"C I... -sv) 0 2 4 6I 0 2 4 6 I ' I ' I ' I ' I - 8 10 12 14 16 18 FREQUENCY(GHZ) 20 Figure 4.9: Angle of open-end reflection coefficient resulting from a typical fixture characterization procedure.

116 The reflection coefficient angle Oop is related to the normalized capacitance by the relation given by (2.133) - sin 0op p (l+cos0p) (4.2) A calculation of c0p based on measurements is shown compared to numerical results in Chapter 5. 4.3.4 Measurement of Series Gap Discontinuities Measurements were performed on three series gap test circuits of different gap widths 15mil, 9mil, and 5mil. A sketch of the test circuit layout is shown in Figure 4.10. The TSD standards of Figure 4.7 were used for fixture characterization. Because the length of the input/output lines L1 and L2, to the left and right of the series gap respectively, are shorter than half the thru line of Figure 4.7, the reference planes established by the de-embedding are not at the desired positions of a-a and b-b (Figure 4.10). A transmission line rotation is used, in each case, to move the reference planes to the desired positions. The phase constant used for this rotation is calculated from the fitted effective dielectric constant discussed in Section 4.3.2. The length of rotation is given by the difference in physical length between L1, L2, and half the thru line length. The measured results are discussed in Chapter 5. 4.3.5 Measurement of Coupled Line Filters The last measurements to be described were made on two different coupled line band pass filters. The first filter is the two resonator coupled line filter depicted in the sketch of Figure 4.11. The TSD standards used for this measurement are

117 H 730 i ] TEST CKT. A 15 B 9 C 5 a b Figure 4.10: Sketch of series gap discontinuity test circuit. Note: all dimensions in mils, h = 25, r, = 9.7. (Shielding dimensions: b = c =.25").

118 S - 5 1 S2 25 L * 100.2 Figure 4.11: Sketch of two resonator coupled line filter. Note: all dimensions in mils. h = 25, r- = 9.7. (Shielding dimensions: b =.4", c =.25" ). shown in Figure 4.12. As in the case of the series gap measurement, a transmission line rotation was required after de-embedding to adjust the reference planes to the desired positions a-a and b-b of Figure 4.11. The other filter measured is the four resonator filter of Figure 4.13. In this filter, the line widths of the filter structure are 11.9 mils (.302mm) which corresponds to about a 65 ohm impedance level. Since the ANA provides for measurements in a 50 ohm environment, a quarter wave transformer was designed to transform from this 65 ohm impedance level to approximately 50 ohms at the coax/microstrip connection points. By including the same transformer on the input and output portions of the TSD standards (Figure 4.14), its effects can be effectively removed through the de-embedding process. The result is that microstrip calibration reference planes are established at the

119. A 393 STD. DIM. A TR U 730 DELAY 824 a. Thru and delay line standards. 393 --- l I b. Open end standard. Figure 4.12: Sketch of TSD standards for two resonator filter measurement. Note: all dimensions in mils, h =.025", cr = 9.7.

120 a i 212 - - 106 243.9 -4153 1 1 b I 1232 I I I I L 72.6 -v -- - --------- -------- -— S ------------- I ' I II I Ie I AtI 393 t!. i I I I I I a LL-2 I L1 I 4 -b W * 23.1 1 W2 11. W3 16.9 SI ~ 3.4 S2 ' 25. S3 27.2 LI * 90.9 L2 a 113 L3 a 93.9 Figure 4.13: Sketch of four resonator coupled line filter. Note: all dimensions in mils, h =.025",, = 9.7. (Shielding dimensions: b =.4", c =.25" ). planes a-a and b-b of Figure 4.13. The de-embedded scattering parameters are referenced to approximately 65 ohms. Note that no transmission line rotations are required to adjust the reference planes for this measurement since the input and output line sections L1 and L2 are the same length as half the thru line. The measured results for both of these filters is discussed in the next chapter.

121 A 1 STD. DIM. A T 730 DELAY 824 Ii3E a. Thru and delay line standards. 730 - 72.6 I I W1 = 23.1 -— 1 --- ------— 2 --- —----------------------- 393 W = 1.9 i W = 16.9 b. Open end standard. Figure 4.14: Sketch of TSD standards for four resonator filter measurement. Note: all dimensions in mils, h =.025", = 9.7.

122 4.4 A Perturbation Analysis of Connection Errors in TSD De-embedding As part of this thesis, an approach was developed to analyze the uncertainties in TSD de-embedding results arising from connection repeatability errors. The analysis consists of perturbing the S-parameters of the TSD standards and the D.U.T. with a set of error vectors that are representative of the variations of each S-parameter (S11, S12 etc. ) measurement with repeated connections. Software was written to allow processing the perturbed S-parameter data in the same way as the measurement data is processed (Figure 4.3). This perturbation analysis, allows for an approximation of how connection errors -which are inevitable- propagate through the TSD mathematics and limit the precision of the final results. The precision of a measurement process is not the same as accuracy. Accuracy refers to how close the result of an experiment comes to the true value. Since the true value is usually unknown, as it is for the present measurements, true measurement accuracy is often impossible to evaluate. Precision on the other hand is a measure of how reproducible a result is. Measurement reproducibility is something for which a reasonable estimate can usually be made, and this is the purpose of the following discussion. An examination of the measurement procedure (Figure 4.3) shows that any variations in the final results are due to uncertainties in the fixture measurements of the standards and the D.U.T.. The precision of these fixture measurements is affected by two factors. First is the non-repeatability of the fixture connections, which for the present measurements includes both coax/coax and coax/microstrip connections. The second factor affecting the precision is random errors due to ANA instrumentation or the non-repeatability of the system calibration procedure, which is believed to be negligible compared to other error sources. For convenience, the following definition is advanced:

123 Definition IV.1 The combined effect of the random errors in the ANA instrumentation and the errors due to connection non-repeatability will be referred to as connection errors. The repeatability experiments presented in Appendix H explore the errors in microstrip fixture measurements caused by non-repeatable microstrip connections. In these measurements, the resulting spread of the data in each case includes both random ANA errors, and those due to coax/coax connection non-repeatability in addition to the repeatability issue being studied. Hence, the results presented actually represent the total connection errors associated with each repeatability issue. These connection errors are present during the measurement of each of the three TSD standards (Figure 4.2) and also during the measurement of the D.U.T., which in the present application is a discontinuity structure. 4.4.1 Basic Approach to Perturbation Analysis The perturbation approach developed to analyze the effect of these connection errors is outlined in Figure 4.15. First, an analysis frequency is chosen. The analysis is performed at a single frequency to prevent the data processing from becoming too cumbersome. In this work a frequency of 10GHz was chosen since it is about in the middle of the frequency range used for the discontinuity measurements. After the analysis frequency has been chosen, the next step is to derive a representative set of error vectors for the type of connection used. Figure H.4 shows one way of looking at the coax/microstrip connection repeatability data. There, the average set of S-parameters are used to normalize the S-parameters from each of the connection trials by way of vector division. Another way to look at this data is to perform a vector subtraction between the S-parameters from each

124 Figure 4.15: Flow chart illustrating approach for perturbation analysis of connection errors.

125 of the connection trials and the average S-parameters. In this case the result is a set of error vectors that represents the vector perturbation of each of the S-parameter measurements from the average. This idea is illustrated in Figure 4.16. This figure shows a typical set of Sn and S12 measurements resulting from 10 connections on a thru line3. The error vector AS, shown is given by A511 Sm - Sii1avg (4.3) where Slim is the measurement for a single connection trial and S11avg is the average of 10 connections. The error vectors A512, AS21, and AS22 are defined analogously. Thus, for each connection trial we may define four error vectors. Error vectors so derived, are used in the analysis to perturb the S-parameters for each of the TSD standards and the D.U.T.. To do this, a nominal (or average) set of S-parameters are obtained from measured data. The different error vectors are then added to the nominal S-parameters in an order determined by setting up a permutation table similar to that shown in Table 4.3. To understand how this permutation table is used, consider the following example. Assume that each fixture connection can be made in one of ten possible ways and let these connections be numbered 1 through 10. Associated with each connection is a set of four error vectors may be derived as discussed above. For this example, assume that the same set of error vectors can be used for all fixture measurements. The final de-embedded result will depend on which of the 10 possible connections was made to each of the standards during fixture characterization and to the D.U.T. before measurement and de-embedding. For 4 fixture measurements, and 10 possible connections for each, there are 104 permutations of different con3 Note: In Figure 4.16 and the other polar representations that follow, the Smith chart lines drawn only have meaning when the scale is 1.0. When magnified scales are indicated (e.g. SCALE=0.1), only the relative magnitudes and phases of the points plotted are important.

126 S11 * SCALE * 0.1 S12 o SCALE * 1.0 Figure 4.16: Variation of Sil and S12 for 10 connections made to a typical thru (or delay line) standard. Error vectors may be defined as the vector perturbation of each of the measurements from the average.

127 Table 4.3: CONNECTION PERMUTATION TABLE EXAMPLE CONNECTION THRU "SHORT" DELAY D.U.T. COMB. NO. (OPEN) 1 1 2 5 5 2 6 5 1 9 3 1 6 10 10 4 3 3 7 4 5 7 5 6 4 nection combinations. Table 4.3 only shows 5 of these permutations. Each of the connection combinations is assigned a number, and each row of the table describes the corresponding connections for each of the fixture measurements. The order of the connections is chosen randomly by using a a pseudo-random number generator to set up each column of the table. In this way a large number of repeated deembedding procedures can be synthesized with a relatively small set of measured data. Once the perturbed sets of S-parameter data on the standards and the D.U.T. have been obtained, the fixture characterization and de-embedding operations are the same as those of Figure 4.3. The only difference is that instead of processing measured data for different frequencies, the software is used to process perturbed S-parameters for different connection combinations. 4.4.2 Perturbation Analysis and Results

128 The above gives a brief description of the basic approach used to analyze connection errors. A critical step in the analysis is the selection of a representative set of error vectors for the connection errors of a given microstrip fixture measurement. To be statistically rigorous, a different set of error vectors is required for each different fixture measurement that is made, since the error vectors may differ with the device or circuit being measured. In addition, these error vectors should be derived from a large number of connections in each case. Obtaining this extensive of a data set would be a formidable task. One problem is that the measurements themselves, and the associated data processing is very time consuming. Second, there is a limit on the number of connections that can be made between a particular set of connectors and a particular microstrip test line. As the number of connections is increased, the wear on the fixture hardware (and the experimenter) gradually degrades the performance of the connection and from a statistical point of view the population average Cp will not be a constant. Because of these difficulties, some simplifying assumptions are made to allow an approximate analysis to be carried out. In doing this, the author has attempted to make the best use of the available connection repeatability data to derive error vectors for each of the fixture measurements. These error vectors are described below. The main assumption made is that the error vectors used represent a random sample of the possible connection errors. Within the limits of this assumption, the resulting analysis gives a reasonable approximation to the related uncertainties. In the analysis which follows an analysis is carried out to estimate the precision of the measurements made of 6eqf, and the open-end and series gap discontinuities. An approximation of the precision of the coupled line filter structures is not included because the repeatability data currently available is not sufficient to derive

129 error vectors for these measurements. Error Vectors used for Delay Lines, and Filters Based on the repeatability measurements for a microstrip thru line presented in Figure H.4, a set of error vectors were derived in the manner described above. These are shown plotted in Figure 4.17. The difference observed in this Figure between the S12 and S21 error vectors (Figure 4.17b) may concern some readers, since for a passive two port structure we would expect the S12 and S21 measurement to be identical. However, The difference is very small and not considered significant. They are due to a residual systematic error in the ANA that is not removed by the calibration. In the deembedding algorithm (Figure 4.3) the two measurements are averaged and set equal. However, to avoid amiguities in the perturbation analysis to follow only the S12 data is processed. Also, in the perturbation analysis it will be assumed that the error vectors of Figure 4.17 are representative of the connection errors in the measurement of both the thru and the delay line standards of Figure 4.7. The only difference between these standards and the thru line that was measured for the connection repeatability study is the length of the line. Otherwise, the type of connection used for each is the same. The vector subtraction performed in deriving the error vectors essentially normalizes them to the average (though in a different way than a vector division does). Because of this it is reasonable to assume that the error vectors do not vary greatly between different fixture measurements, provided the magnitudes of the average S-parameters are similar. This is true for both the thru

130 S $l * SCALE - 0.05 S22o SCALE * 0.05 a. Error vectors for 511 and S22 measurement. Sl * SCALE ~ 0.02 S12 a SCALE * 0.02 b. Error vectors for S12 and S21 measurement. Figure 4.17: Error vectors for thru or delay line standards. These were also used to perturb the S-parameters of the two and four resonator filter structures.

131 S1 * SCALE * 0.05 Figure 4.18: Error vectors for measurement of open-end reflection standard. and delay line standards. Error Vectors For Open-End Standard In contrast, the reflection coefficient measurement for an open-end is not similar to that for a thru line and a different set of error vectors is needed. Hence, a new set of error vectors were derived from measurements of 11 repeated connections made to the open-end standard and these are shown in Figure 4.18. Since the magnitude of S11 and S2 are about the same, it is assumed that the same error vectors can be used to perturb both of these measurements. Error Vectors For Series Gap Measurements. For the series gap measurements, repeatability data was obtained for four con

132 nections on two of the gap circuits, and five connections on the third. Error vectors were derived separately for each of the gaps and compared. It was seen that the magnitude of the error vectors did not differ significantly between the three sets. These error vector sets were then combined to form a larger set of error vectors representing 13 possible connections. These are shown in Figure 4.19. 4.4.3 Connection Errors in Eeff and Open-end Measurement Using the method described above, a perturbation analysis was carried out to analyze the effects of connection errors on the Eef f rop, and cop measurements which are calculated as part of the TSD procedure as described previously. To do this, nominal S-parameter measurements were taken from measurements on the standards of Figure 4.7 at f = 10 GHz. The error vectors discussed above were used to perturb these nominal parameters according to two different permutation tables, one with 20 connection permutations and the other with 100 permutations. These permutation tables are similar to Table 4.3, except that a different set of error vectors are used for the open-end standard. Also, the connection combinations for the SI, and S22 measurements were allowed to vary independently so that the permutation tables had 5 columns instead of 4. The statistical data are calculated for each parameter based on the observed results for the different connection combinations. The average is calculated by summing up the results for each of. the different combination numbers and dividing by the number of permutations. The estimated standard deviation s for the data is given by S = Std.Deviation = N (4.4) N -i

133 S1 * SCALE * 0.05 22 o SCALE * 0.05 a. Error vectors for S11 and 522 measurement. S2 * SCALE * 0.02 S12 o SCALE * 0.02 b. Error vectors for S12 and S21 measurement. Figure 4.19: Combined error vectors for series gap measurements.

134 where N = number of trials xi= observed result for the parameter of interest for connection combination i The results of the perturbation analysis are summarized in Table 4.4. It is seen that the results for 100 connection permutations are not significantly different than those for 20 permutations. The range of observed values increase slightly for each of the parameters. This is reasonable since more worst case connection combinations are possible with a greater number of permutations. On the other hand, the standard deviation values are seen to change by a much smaller amount. It appears that 100 is a sufficient number of permutations from which to base the statistical observations. The results of the analysis indicate that the uncertainty in fef f and the open-end parameters due to connection errors can be appreciable. The standard deviations for these parameters is about.5% (of the average) for eeff, and about 8% for the open-end parameters. These standard deviation values were used to derive error bars for the measurements of f.f and the open-end parameters presented in the next chapter.

135 Table 4.4: PERTURBATION ANALYSIS RESULTS FOR e1ff and OPEN-END MEASUREMENTS PARAMETER PERMU- MIN MAX AVG RANGE STD. TATIONS VALUE VALUE VALUE DEV. Eeff 20 6.851 6.978 6.922.127.0302 100 6.850 6.998 6.922.147.0312 I op 20 -11.8 -9.2 -10.3 2.6.799 (DEG) 100 -12.0 -8.7 -10.2 3.3.837 cop 20 1.285 1.650 1.440.365.110 (pF-Ohm) 100 1.210 1.670 1.420.465.115 4.4.5 Connection Errors in Measurement of Series Gap Discontinuities Next, the perturbation analysis was carried through the de-embedding of the three series gap discontinuity circuits measured for this thesis. For this part, nominal S-parameters were taken from each of the series gap measurements at f = 10GHz. The error vectors of Figure 4.19 were then used to perturb this data, and both 20 and 100 connection permutations were synthesized. Figure 4.20 illustrates the information that can be gained from this perturbation analysis. Shown is the spread in final de-embedded 511 and S12 data for gap circuit C (G = 5mil) caused by connection errors. The perturbation analysis results (100 permutations) for all three gap circuits are summarized in Table 4.5. From these results it is seen that the SI5 data shows relatively constant behavior with respect to the statistical parameters, while the change in the S12 data is significant. This is because in each case the amplitude of the S1 measurement is relatively large

136 compared to the corresponding error vectors (Figure 4.19). One important observation is that for large gap widths the uncertainty in the phase of S12 can be appreciable. This is because the magnitude of the nominal S12 value begins to approach the magnitude of the connection error. At first glance it appears that the uncertainty in the magnitude of the S12 measurement increases as the gap width is reduced. This is not the case since if the range and standard deviation values are divided by the average to calculate these parameters on a percentage basis, the opposite is true. Hence, the measurement uncertainty caused by connection errors increases as the magnitude of the S-parameter being measured decreases. The standard deviation data from this analysis is used in constructing error bars for the series gap measurements presented in Chapter 5.

137 S11 * SCALE a 1.0 S12 o SCALE ' 0.3 Figure 4.20: This plot of the final de-embedded result for a 5 mil series gap discontinuity illustrates the information obtained through the perturbation analysis (f=10GHz, 20 connection permutations).

138 Table 4.5: PERTURBATION ANALYSIS RESULTS FOR SERIES GAP MEASUREMENTS TEST PARAMETER MIN MAX AVG RANGE STD. CKT. VALUE VALUE VALUE DEV. Si|i.964 1.02.991.056.014 A LS,1 -8.2 -2.5 -4.7 5.7 1.33 (G=15mil) IS121.07.088.081.019.004 LS12 76.9 95.1 83.5 18.2 3.2 |IS|.963 1.00.982.041.011 B LS,1 -14.2 -7.7 -11.1 6.5 1.29 (G=9mil) 1S121.117.141.132.023.005 LS12 77.1 88.8 81.3 11.7 2.0 IS 1.940.983.962.043.011 C LSl -20.1 -14.2 -16.7 5.9 1.28 (G=5mil) 5121.220.254.240.034.007 LS12 72.7 78.9 75.2 6.2 1.1.,

139 4.5 Summary of Experimental Methodology The experimental methods used in this thesis are summarized below. * The TSD method is used to de-embed the test fixture effects from the measurements. An open-end is used in place of the "short" circuit as the reflection standard * The measurements of ~eff and rop are obtained as by products of the TSD procedure * A perturbation analysis approach is used to help approximate measurement uncertainties

CHAPTER V NUMERICAL AND EXPERIMENTAL RESULTS In this chapter numerical and experimental results obtained through the present research are presented for the network parameters of shielded microstrip discontinuities. Included here are results for the effective dielectric constant, open-end and series gap discontinuities, and coupled line filters. Where possible comparisons are made to available data from other theoretical solutions. One case for an open-end is compared to to other full-wave solutions. However, since the emphasis in this study is to compare with measured data, extensive comparisons are made to available CAD models since this data is easier to generate for an arbitrary test case. Also, it is useful to include data from these CAD models in the study since they are widely applied to design shielded microstrip circuits. The CAD models of Super Compact and Touchstone are based on a combination of different theoretical techniques, most often embodied in simplified closed form solutions, curve fit expressions or look-up tables 1. These models do not provide a means to account for the effects of the shielding box of Figure 1.3. In the case of Touchstone, no shielding effects are included, and for Super Compact only a cover height is provided for, and this does not apply to the open-end or series In the manuals for these programs, references are listed for each discontinuity model, the reader is referred to these manuals for further information about the theoretical basis for the CAD models. 140

141 gap discontinuities. It is generally believed that as long as the dimensions of the shielding are large relative to the substrate thickness that the shielding will have a negligible effect. To simulate a complicated circuit containing many discontinuities, the discontinuities are assumed to be independent of one another and their respective models are used to generate a matrix representation for each discontinuity. The overall circuit performance is predicted by mathematically cascading the matrices together. In contrast, the full-wave solution presented here accurately treats the entire geometry of the shielded inicrostrip circuit as a boundary value problem. Any interactions between, for example, the fringing fields on an open-ended line and an adjacent conducting strip are automatically included in the analysis. Because of this, the method is expected to provide better accuracy than CAD model predictions. Still, as will be seen shortly, the CAD models give quite reasonable results in many cases. However, in other cases, particularly where shielding effects become significant, the accuracy of the CAD models is questionable. One case where shielding effects are noticeable is when the frequency approaches the cutoff frequency for the onset of higher order modes. As will be discussed next, as the size of the shielding box increases the cutoff frequency for the onset of higher order modes decreases. 5.1 Cutoff Frequency for Higher Order Modes Higher order modes occur in open microstrip in the form of surface waves and radiation modes. Surface wave modes may be minimized by keeping the substrate electrically thin at the operating frequency. However, the first surface wave mode has a cutoff frequency of zero. This is not the case for shielded microstrip, where the nature of higher order modes are quite different. The shielding forms a waveguide structure that elimi

142 nates radiation and surface waves. Instead, the higher order modes take the form of waveguide modes. As a consequence, below the waveguide cutoff frequency, only the dominant microstrip mode can exist. For the present problem of Figure 1.3, the cutoff frequency for higher order modes may be approximated by analyzing the infinite dielectric-loaded waveguide of Figure 5.1. The closeness of the solution for the propagating modes of the dielectric-loaded waveguide to the solution for shielded microstrip has been observed in the past, both analytically [58], and numerically [59]. The solution for the propagation characteristics of the dielectric-loaded waveguide takes the form of trancendental equations [26,60] that must be solved either numerically or graphically. The analysis is carried out with LSE and LSM modes which are TE and TM respectively relative to the normal to the air-dielectric interface (i). The cutoff frequency for the first propagating mode within the structure depends on the geometry. The following definition will help clarify what is meant by cutoff frequency as used in this thesis. Definition V.1 For the purposes of this thesis the cutoff frequency fc will be defined as the first frequency where non-evanescent waveguide modes can exist inside the cavity. It will correspond to either an LSM or an LSE mode depending on which has the lowest cutoff frequency. A numerical solution to the dielectric-loaded waveguide problem was formulated by a student at the University [61]. This program has been used to analyze the cutoff frequencies for several of the shielding geometries considered in this thesis. The plot of Figure 5.2 shows the variation of the cutoff frequencies with shielding sized for a square waveguide three different substrates enclosed. It is seen that for the alumina (e, = 9.7) substrate, the highest achievable cutoff frequency is limited to about 50GHz, while for a given shielding size, much higher cutoff frequencies are possible with the use of the thinner quartz (C, = 3.8), or GaAs (e, = 12.7)

143 Figure 5.1: The cutoff frequency for higher order modes in shielded microstrip may be approximated by analyzing an infinite dielectric-loaded waveguide. substrates. These cutoff frequencies have been found to give a good prediction of the onset of higher order effects observed in the current distributions computed with the new method. As an example, Figure 5.3 shows the current distribution on an openended line operating below the cutoff frequency. For the indicated geometry, fc is about 17.9 GHz. As the frequency is raised above the cutoff frequency, the current becomes more and more distorted as shown in Figure 5.4. This distortion may be explained as follows. Above cutoff, the microstrip current excites a waveguide mode which travels down the cavity until it reaches the wall at x = a, it is then reflected back and forth inside the cavity and interacts with the microstrip current. This waveguide behavior can take place, even though the cavity is not at resonance, because the microstrip current has an external energy source via the coaxial excitation.

144 400 - 300 - -b- E — = 9.7, h.025' a3 r 200 0 E = 3.8, h=.010' E -- ~ =12.7, h =.004' 100- - 1E =l.Oh=O."0 (EMPTY GUIDE) - 0.0 0.1 0.2 0.3 0.4 b (INCHES) Figure 5.2: Variation of cutoff frequencies with shielding for three commonly used substrates enclosed in a square waveguide (b = c). Also shown is empty guide case ( = l, h = 0). 5.2 Effective Dielectric Constant Results Effective dielectric constant results are presented in this section for 50 ohm lines on three common substrates: alumina, quartz and gallium arsenide (GaAs). As discussed in Section 2.7, the microstrip effective dielectric constant eeff is computed from the current distribution. The current can be associated either with a thru line or an open-ended line. Although a calculation of the effective dielectric constant was not among the primary objectives of this work, its determination is an integral part of the solution for discontinuity effects, and the comparisons which follow also lend insight into the relationship between substrate geometry and shielding effects. Effective Dielectric Constant Results for an Alumina Substrate Figure 5.5 shows feff for a 25 mil thick alumina substrate. The numerical re

145 - m - 0.874E-03 - 0.523E-03 0. 1 73E-03 - -0. 1 78E-03 -0..526E-03 -0.879E-03 - 0.000 1.275 2.550 3.825 5.100 X (WAVELENGTHS) 6. 375 Figure 5.3: Below the cutoff frequency f,, the microstrip current on an open-ended line forms a uniform standing wave pattern (f = 16GHz,, = 9.7,W/h = 1.57, h =.025",b = c =.275).

146 0 - 0.496E-03 - 0. 03E-03 - - -0.937E-04 -0.290E-03 -0.487E-03 0.000 1.750 3.500 5.250 7.000 8 X(WAVELENGTHS) Figure 5.4: As the frequency is increased above f, more and more distortion is observed in the open-end current distribution (f = 22GHz,4e = 9.7, W/h = 1.57, h =.025", = c =.275")..750

147 suits are compared to measurements, and to CAD package results. Note that Super Compact allows only the cover height to be varied while the calculation provided by Touchstone neglects shielding effects. For the shielding geometry used here, it is seen that the difference between the numerical and CAD package results are within experimental error. However, interestingly enough, better agreement between the CAD results and the numerical results is observed at higher frequencies. This may be due to the fact that the side walls, which are not included in the Super Compact analysis, are electrically closer to the strip at low frequencies. The measured data is obtained as a byproduct of the TSD fixture characterization procedure (Section 4.3). The data shown represents the average of ten separate procedures. These procedures were conducted over a period of about four years at both Hughes (not all by the author) and at the University. At least three or four different sets of TSD standards were used over this period, however, the mechanical dimensions and substrate parameters are designed to be identical. The error bars shown in Figure 5.5 represent the standard deviation (~s) of the different measurements. This data is shown here in lieu of the result from a single measurement, since it gives a more representative view of the involved measurement uncertainty. In this case the error bars shown represent the combined effect of connection errors, variations in.,, differences in substrate mounting, and errors in specifying the physical difference between the length of the thru and delay line standards. The major error source is believed to be the variations in v, which can be significant for alumina substrates [62,63]. To see how eff varies with shielding, consider the plot of Figure 5.6. This plot compares numerical and Super Compact results for three different shielding geometries. The notation used to describe different shielding and substrate geometries is explained in Table 5.1. The case for cavity CA is the same as that of

148 7.6 7.4 7.2 7.0 -5 — SUPER COMPACT 6s -- TOUCHSTONE: -* — THS RESEARCH 6.6 1 6.4 t MEASUREMENT 6.2 6.0 -0 2 4 6 8 10 12 14 16 18 20 22 FREQUENCY(GHZ) Figure 5.5: Effective dielectric constant comparison for an alumina substrate compared to measurements and CAD package results (,r = 9.7, h =.025", b = c =.25"). Figure 5.5. For the other two cases, where the shielding is closer to the microstrip, the agreement is not as good. Effective Dielectric Constant for a Quartz Substrate The effect of shielding on cef for a quartz substrate is displayed in Figure 5.7. In this case the Super Compact analysis is seen to give good results for both of the two larger shielding geometries. However, the numerical results again show a reduced value as the size of the shielding is reduced further. The reduction of the effective dielectric constant, relative to Super Compact, can be explained as follows. For a larger shielding geometry, the field distribution on the microstrip more closely resembles the open microstrip case, with most of the electric field concentrated in the substrate. In this case, most of the electric field lines originate on the microstrip conductors and terminate on the ground plane

149 Table 5.1: CAVITY NOTATION USED TO DENOTE DIFFERENT GEOMETRY AND SUBSTRATE PARAMETERS CAVITY W (in) in)in) c (in) fb (GHz) CA 9.7.025.025.250.250 21.8 CC 9.7.025.025.100.100 37.5 CF 9.7.025.025.075.075 41.7 QCB 3.82.0157.010.122.080 45.8 QCE 3.82.0157.010.100.100 73.0 QCG 3.82.0157.010.050.05 102.5 8.5 8.0 7.5, 7.0 6.5 6.0 5.5 - CT -0 --0 — A 0 CC: SUPER COMPACT CF: SUPER COMPACT CA: SUPER COMPACT CC: THS RESEARCH CF: THIS RESEARCH CA: THIS RESEARCH 0 8 16 24 32 40 48 56 64 FREQUENCY (GHZ) Figure 5.6: The effects of shielding on cef are apparent as the size of the shielding cavity is reduced (see Table 5.1 for geometry.)

150 3.3 -3.2 - - * -— c QCB: SUPER COMPACT 3.1 - * - QCE SUPER COMPACT 3.0 -- - QCG SUPER COMPACT * QC8:THIS RESEARCH * QCE THIS RESEARCH - g QCG: THIS RESEARCH 28 27..., ~ ~, -. ~,..,..,.. 0 20 40 60 80 100 120 FREQUENCY(GHZ) Figure 5.7: Shielding effects are also significant for the quartz substrate shown here (see Table 5.1 for geometry). below. As the cavity size is reduced, the ground planes of the top and side-walls are brought closer to the microstrip lines. The electric field distribution is now less concentrated in the substrate, as more field lines can terminate on the top and side walls. As a result, a proportionally larger percentage of the energy propagating down the line does so in the air region, and the dielectric constant is reduced. Effective Dielectric Constant for a GaAs Substrate Figure 5.8 shows a comparison of the effective dielectric constant for a 4 mil thick GaAs substrate. This is a typical substrate geometry used for MMIC purposes. The agreement between the numerical and CAD model predictions this case is excellent. The differences observed in the case of the other two substrates was not seen for the GaAs substrate of Figure 5.8. because of how thin the substrate is relative to the size of the cavity. Hence, all of the effective dielectric constant results presented above demon

151 9.1 8.9 8.7 _ -... --- THIS RESEARCH 8.5 --- TOUCHSTONE 8.5 --- SUPERCOMPACT 8.3 8.1~~',~~,~~,~~,~ 0 20 40 60 80 100 FREQUENCY(GHZ) Figure 5.8: The numerical and CAD package results display excellent agreement for the case of a thin GaAs substrate (6r = 12.7, h =.004", b = c =.07", f, = 81GHz). strate that the CAD package predictions are valid when two conditions are met: 1) the shielding is large with respect to the substrate height, and 2) the frequency is below the cutoff frequency. When the dimensions of the shielding becomes cornparable to the substrate height, the CAD results are no longer accurate. This suggests the need for an improved CAD formulation valid for small as well as large shielding geometries. The present method could be used as the basis for deriving such a formulation. 5.3 Results for Open-end Discontinuity As discussed in Section 2.7, an open-end discontinuity can be represented by an effective length extension Leff, by a shunt capacitance cop, or by the associated reflection coefficient Fop (= Sll). Each of these three representations will be used in this section.

152 Q 3 -- - JANSEN & KOSTER 0.30 - THIS RESEARCH -J U r -TOH 0.25 0.20 0.15-,,' — *' 0 4 8 12 16 20 24 28 FREQUENCY (GHZ) Figure 5.9: Effective length extension of a microstrip open-end discontinuity, as compared to results from other full-wave analyses (c, = 9.6, W/h = 1.57, b =.305",c =.2", h =.025"). The plot of Figure 5.9 compares Lff results to those of Jansen et al. [64] and Itoh [10]. The results from this research are almost identical to those obtained by Jansen et al. for frequencies above 8 GHz, but show a reduced value for lower frequencies. Jansen et al. speculate that the large difference in the results obtained by Itoh are due to an inadequate choice of basis functions. The case of Figure 5.9 was chosen to compare the coaxial and gap generator excitation methods used in the method of moments solution 2. Table 5.2 shows that the results computed for this case by the two methods are equivalent. This equivalence is also observed for the computations of two-port scattering parameters for the structures considered herein. Hence, as far as computing network parameters is concerned either method gives good results. Since the coaxial method is more 2 The inner and outer radii of the coax feed was taken to be.007" and.016" respectively.

153 Table 5.2: COMPARISON OF Leff/h COMPUTATION FOR THE TWO TYPES OF EXCITATION METHODS / (GHz) 4 8 12 14 16 18 20 GAP GENERATOR.298.305.309.321.324.344.353 COAXIAL EXCITATION.299.304.309.322.327.344.352 realistically based, this conclusion lends validity to the use of the gap generator method. The results shown in Figure 5.10 illustrate the effect of the shielding on the open-end discontinuity. The normalized open-end capacitance cp is plotted for three different cavity sizes. The results show that reducing the cavity size raises fc (as expected), and it lowers the value of cp. For comparison, data obtained from Super Compact and Touchstone and measurements (see Section 4.3) are included. The errors bars on the measurements represent the estimated standard deviation (~s) of the connection errors for this measurement from Table 4.4. Similar shielding effects are observed for an open-end on a a quartz substrate as shown in Figure 5.11. In this case it is seen that the Super Compact result gives a good value for low frequencies, and where the frequency is well below the cutoff frequency for a given shielding size. Or stated another way, the shielding effects are less severe for a smaller shielding cavity! This conclusion defies common sense, but is strongly supported by the numerical results. Note that Jansen's results (Figure 5.9) show a similar rise in the open-end effect as the cutoff frequency is approached.

154.-O. E 0 0. a 0 0v 3.5 3.0 2.5 2.0 1.5 - 1.0 -0. - fc = 20.8GHZ I1 SUPER COMPACT TOUCHSTONE CA: THIS RESEARCH CC: THIS RESEARCH CE: THIS RESEARCH CA: MEASUREMENT / f = 41.7GHZ fc 37.5GHZ I. - - I - I I I I. --.. - - %d.%d 0 10 20 30 40 50 60 70 FREQUENCY (GHZ) Figure 5.10: A comparison of the normalized open-end capacitance for three different cavity sizes shows that shielding effects are significant at high frequencies (see Table 5.1 for cavity geometries). 0-1 M 9 ar. 0 0 0.7 -0.6 -0.5 -0.4 - --- SUPER COMPACT - *- CE THIS RESEARCH — Q- CG THIS RESEARCH - *- QCB: THIS RESEARCH 0.3 I -................ 0 20 40 60 80 FREQUENCY (GHZ) 100 120 Figure 5.11: Nomalized open-end capacitance for three different cavity sizes for a quartz substrate. This data also shows an increase in the capacitance as the cutoff frequency is approached (see Table 5.1 for cavity geometries).

155 0 --5 -10 -- TOUCHST ONE - 15 ---- SUPER COMPACT THIS RESEARCH -20 J MEASUREMENT -25 -30- ] i. i i 0 4 8 12 16 20 24 FREQUENCY(GHZ) Figure 5.12: Numerical and measured results show good agreement for the angle of S11 of an open circuit (I = 9.7, W = h =.025", b = c =.25"). As a last example of the open-end effect, Figure 5.12 shows results for the angle of S1, of an open-end compared to measurements (see Chapter 4). The measurement is seen to favor the numerical results, although the differences observed are not overly significant for this shielding geometry. The error bars the approximate standard deviation of the connection errors associated with this measurement (Table 4.4). 5.4 Results for Series Gap Discontinuities Numerical and experimental results have been obtained for series gap discontinuities with three different gap spacings (G) 15 mil (i.e..015"), 9 mil, and 5 mil. The test circuits and the shielding dimensions used for the measurements are those of Figure 5.12.

156 Numerical results for the magnitude of S21 for these gaps are shown plotted in Figures 5.13- 5.15. For comparison, results obtained using Super Compact, and Touchstone are also shown plotted along with measured data. The error bars associated with the standard deviation of connection errors (Table 4.5), are on the order of ~.5dB and are too small to show on the plots. With one exception, the measured data best follows the results of this research. In contrast, the Touchstone analysis had the least agreement with the measurements. This may in part be due to the fact that Touchstone does not include any shielding effects either the side walls or the shielding cover into account. However, the Super Compact model for the series gap does not appear to include the effect of the cover. The one exception where the numerical result appears to be slightly off from the measurement is in the plot of the magnitude of S21 of the 5 mil gap. Based on numerical investigation it appears that by using a smaller subsection length in the method of moments computations, the value for S21 can be improved (i.e. it approaches the measurement). However, as discussed in section 3.5, the subsection length cannot be decreased arbitrarily as other implications must be considered. The best approach may be to minimize the size of the matrix by using a small subsection length around the region of the gap and a larger subsection length over the uniform line sections. Results for the angle of S21 and SI1 for the 15 mil series gap are shown in Figures 5.16 and 5.17. The error bars in these charts represent the estimated standard deviation from the perturbation analysis (Table 4.4). Although the measurements tend to favor the numerical results, the differences are not too significant. The phase of the S-parameters for the other two series gaps behave in a similar way as

157 0 'I% 0.0 " -7.5 --15.0 --22.5:30.0 --37.5 I -a — g-U — N TOUCH-SrONE SUPER COMPACT THIS RESEARCH MEASUREMENT a,.I I.I va.r I......I- I..I. — r %.F. %# 0 2 4 6 8 10 12 14 16 18 20 FREQUENCY (GHZ) 22 Figure 5.13: Magnitude Of S21 for series gap circuit A (C = 15 mil). 0.0 -7.5 000 N C,) -15.0 -22.5 -30.0 -37.5 -45.0 -a —%.r N TOUCHSTCNE SUPER COMPACT THIS RESEARCH MEASUREMENT FREQUENCY (GHZ) Figure 5.14: Magnitude of S21 for series gap circuit B (G = 9 mil).

158 0.0 -7.5.. -15.0 '0 i- -- TOUCHSTCNE -22.5 -. — SUPER COMPACT I — THIS RESEARCH X -30.0 N MEASUREMENT -37.5 -45.0 ' i I I I | ' i I I 0 2 4 6 8 10 12 14 16 18 20 22 FREQUENCY (GHZ) Figure 5.15: Magnitude of S21 for series gap circuit C (G = 5 mil). that for the 15 mil gap and have been omitted from this treatment. The measurements made on the series gap discontinuities are seen to further verify that the theory developed here gives good results. For the large shielding dimensions used for the measurements (b, c > h) the CAD models are seen to give reasonable results. The behavior of series gaps for small shielding dimensions was not studied, instead emphasis was placed on obtaining results for coupled line filters since their behavior is more complicated and therefore more interesting. 5.5 Results for Coupled Line Filters The last results to be discussed were obtained for the two and four resonator filters discussed in Section 4.3. For brevity only the amplitude and phase of the transmission coefficient S21 will be discussed. Note that for the shielding geometry of both filters, the cutoff frequency f,. is approximately 13.9GHz. Above this, the

159 w I-fo I,) en v 95 -85 -75 -65 a I 16 -a TOUCHSTONE SUPER COMPACT THIS RESEARCH MEASUREMENT 55.-...... I. n 0 I T I ' I I I I I * I '* 2 4 6 8 10 12 14 16 18 20 FREQUENCY (GHZ). 22 Figure 5.16: Angle of S21 for series gap circuit A (G = 15 mil). 0 0 W Q Cn V v -0 — lb TOUCLSTONE SUPER COMPACT THIS RESEARCH MEASUREMENT 0 2 4 6 8 10 12 14 16 18 20 22 FREQUENCY (GHZ) Figure 5.17: Angle of SI, for series gap circuit A (G = 15 mil).

160 filter measurements are distorted due to waveguide moding within the test fixture3. The measured and numerical results of this research are compared to CAD model predictions. The CAD package analysis for coupled line filters is performed by cascading two different types of discontinuity elements together: coupled microstrip lines, and open-end discontinuities. As mentioned previously, neither of the packages studied here account for shielding in the open-end discontinuity, however, Super Compact does include the effect of the cover height in the model for coupled lines. Two Resonator Filter Figure 5.18 shows a comparison of the measured and predicted response of the two resonator filter. From this plot, it is seen that the analysis from both the numerical and the CAD packages give a very good prediction of the response of the filter in the pass band. Outside the pass band the amplitude response is seen to more closely follow the numerical results. Close examination of the phase plot (Figure 5.1Sb) shows that the the numerical results are shifted up in frequency by a small amount as compared to the measurements. It is believed that this discrepancy, though small, is related to the thin-strip approximation used for the current distribution. In the theoretical solution, the current is assumed to be uni-directional and to have a symmetric variation in the transverse direction as described by (2.7). In the coupling region 3 Because energy can propagate in a waveguide mode, significant coupling occurs between the two coaxial feeds of the fixture, and the resulting measurement uncertainty is large. Hence, for all the filter results presented, the measurements are only good up to fC = 13.9 GHz.

161 of the filter the close proximity of the adjacent strip conductors will cause the current to become non-symmetric and may require a more general definition of the current as the strip becomes wide. However, note that the thin-strip approximation used here already gives a very good result. If in the magnitude plot of Figure 5.18, the response for the numerical results of this research are shifted down slightly, the agreement with the measurement will practically be exact. Four Resonator Filter The results for the four resonator filter are shown in Figure 5.19. In this case, numerical results for S21, demonstrate excellent agreement with measurements up to the cutoff frequency. Note that for this filter the strip widths are about half as wide as those in the two resonator filter. Hence, the error due to the thin strip approximation is reduced. As in the case of the two resonator filter, the CAD models fail to predict the filter response in the rejection band, whereas the numerical results follow the measurements closely. For the four resonator filter, this is true for both the phase as well as the magnitude of S21. In the phase response, the CAD models display a large error compared to measurements between about 6 and 8.5GHz, while the numerical results track the measured phase very well. Below about 5.5GHz, the measured phase is seen to be different from the predictions of both the CAD models and the numerical results. This is most likely due to a phase error in the measurements. In the TSD technique, the delay line for the measurements should ideally be - at the measurement frequency. When the electrical length becomes either too short or

162 too close to a multiple of l phase ambiguities can result. A good rule of thumb is for the delay line to be between - and 4. At 5.5GHz the delay line used for the measurements is slightly less than A; hence, this is most likely the source of the phase error. We will now examine what happens as the top cover is brought closer to the circuitry. The results of Figure 5.19 show that even for large shielding dimensions the CAD models do not adequately predict the filter response in the rejection band. Figure 5.20a shows Super Compact predictions for the four resonator filter with two different cover heights. These predictions indicate that lowering the cover height should significantly narrow the pass band, and reduce the amplitude in the rejection band. A significantly different prediction is observed in the numerical results for this case presented in Figure 5.20b. A narrowing of the pass band response is also observed in the numerical predictions, but not by nearly as much as in the Super Compact prediction. More importantly, the amplitude in the rejection band is seen to increase instead of decrease! To prove that the numerical prediction is indeed the correct one, an additional measurement was made of the filter for the low cover height case. As can be seen from Figure 5.20b the measured data falls practically on top of the numerical predictions for both cover heights. 5.6 Summary of Numerical and Experimental Results In this chapter results were presented for the effective dielectric constant of uniform microstrip lines, and the network parameters for open-end and series gap 4 Multiple lines are needed for broadband measurements.

163 0 r 0 - -10 --20 --30 --40 --50 --60 - -aN TOUCHSTONE SUPER COMPACT THIS RESEARCH MEASUREMENT f c 13.9GHZ.7n _. 2 4 I 8 10 12 14 1 1 2 4 6 8 10 12 14 16 18 20 FREQUENCY (GHZ) a. Amplitude of S21 200 0 O CIto V v 100 - 0 --100 - N NN, 0 A.tn TOUCHSTONE SUPER COMPACT THIS RESEARCH MEASUREMENT N -200 I —~ -V==Nr= 2 4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20 FREQUENCY (GHZ) b. Phase of S21 Figure 5.18: Results for transmission coefficient S21 of two resonator filter (e = 9.7, W = h =.025"; b =.4", c =.25").

164 -20 - -40 - -60 - -80 - N NO N -0 —&. -a -a — N TaUCHSTONE SUPER COMPACT THIS RESEARCH MEASUREMENT C,, -100 I.I.I.I.I.I.I 2 4 6 8 1 0 12 14 FREQUENCY(GHZ) I I I I 16 18 20 a. Amplitude of S21 AM 0 1U c0 %loo V= 200 -100 -102 I N TOUCHSTONE SUPER COMPACT THIS RESEARCH MEASURBMENT -200,..I. -.. - 2 4 6 8 1 0 12 14 16 18 20 FREQUENCY (GHZ) b. Phase of S21 Figure 5.19: Results for transmission coefficient S-21 of four resonator filter (e, = 9.7,I'V =.012", h =.025"; b =.4", c =.25")

165 ~i | SUPER COMPACT -20 RESULTS c- - - c =.25' 0 - 40.', ".. --- c=.075' cn -60 -80 ~ /...... -100 ', 2 4 6 8 10 12 14 16 18 20 FREQUENCY(GHZ) a. Super Compact predictions ~ NUMERICAL DATA -20 C =.075'.. *, - c-.25S 60 ^ MEASURED DATA f 13.9GHZ -80 C e C.075" N c.25" -100 -,,,, 2 4 6 8 10 12 14 16 18 20 FREQUENCY(GHZ) b. Numerical results of this research compared to measurements Figure 5.20: Results for lowering the shielding cover on the amplitude response of four resonator filter (e, = 9.7, W =.012", h.025"; b =.4").

166 discontinuity and for two coupled line filters. The higher order modes in shielded microstrip are described to be essentially waveguide modes. In fact, the cutoff frequency fC for a partially filled rectangular waveguide gives a good prediction of the onset of higher order mode behavior in the computed microstrip current distribution. A wealth of numerical and experimental data is presented for the above mentioned structures, and comparisons are made to other full-wave analysis and to commercially available CAD packages. Conclusions based on these comparisons are summarized in the next chapter.

CHAPTER VI CONCLUSIONS AND RECOMMENDATIONS The results from the research presented in this thesis lead to several conclusions regarding different aspects of shielded microstrip discontinuity characterization. These conclusions have been separated into appropriate categories as addressed herein. 6.1 Conclusions from Theoretical Work In the theoretical work (Chapter 2) a method of moments formulation for the shielded microstrip problem is derived based on a more realistic excitation model than used with previous techniques. This result follows directly from the reciprocity theorem, with the use of a frill current model for the coaxial feed. The reciprocity theorem is applied in a similar way for the case of gap generator excitation. The impedance matrix~formulation for gap generator excitation is seen to be identical to that for the coaxial excitation method developed here. The difference between the two methods is therefore only in the excitation vector used on the right hand side of the matrix equation. In solving for the impedance matrix elements, all of the required integrations are solved in closed form. Conversely, for the excitation vector of the coaxial excitation method, numerical integration is required. 167

168 6.2 Conclusions from Computational Work Computational considerations for implementing the theoretical solution are explored extensively (Chapter 3). A graphical presentation is given that illustrates the different steps in the computation of microstrip currents. The current for the coaxial excitation method is shown to be uniform along the strip, while a discontinuity in the current is evident in the current for the gap generator method. The effect of this discontinuity is minimized by positioning the gap generator at the beginning of the first strip subsection. The gap generator method is more computationally efficient than the coaxial excitation method since numerical integration is not required. Several numerical experiments are presented (Chapter 3) that explore the convergence and the stability of the solution. The parameters explored are the subsection length I,, the sampling rate N!V(= 1/l=), the value for K used in the basis functions, the summation truncation points NSTOP and MSTOP, and the cavity length a. A value of K = 2ir appears to be the best choice. The main conclusions drawn from these experiments may be summarized as follows: * A value of K =- 27r is a good value to use in the sinusoidal basis functions. * A minimum sampling limit exists such that the condition Kl < I must be satisfied in order to obtain useful current results. For a value of K = 27r, IC =.25 is the largest subsection length that can be used. * Good convergence on n and m is achieved after 500 terms have been added on each. * A maximum sampling limit exists in the form of an erratic current condition which may be defined by the simple formula NSTOP*Il < a. (NSTOP*Il > a must hold for useful results.)

169 * An optimum sampling range may be specified that automatically avoids the erratic current condition, and guarantees the best accuracy in the matrix solution. This range is given approximately by NSTOP < I NSToP< * The optimum sampling range was found to correspond directly with the flat convergence region for the Leff and ~eff computations. 6.3 Conclusions from Experimental Study From the experimental study (Chapter 4), a comparison of various measurement techniques lead to the choice of the TSD de-embedding method for the measurements of this thesis. Various microstrip connection repeatability issues are explored experimentally (Appendix I). The results indicate that, for the hardware tested, the best connection approach is to make the required fixture connections at the coax/microstrip connection points, rather than relying on the repeatability of microstrip/microstrip interconnects. Still, connection errors remain an important consideration for the measurements. To examine this, a perturbation analysis is developed that allows for an approximation to be made of the effects of connection errors on the precision of the final de-embedded results. It is seen that connection errors affect the phase of the S-parameter measurements most, and the amount of resulting measurement uncertainty increases as the magnitude of the parameter being measured decreases. 6.4 Conclusions Based on the Results Numerical and experimental results are presented for several microstrip structures (Chapter 5). Conclusions based on these results are described below. The effect of higher order modes on the current distribution is demonstrated. As the frequency is increased above the cutoff frequency, the current becomes

170 increasingly distorted due to higher order modes. On the other hand, as long as the cavity size is such that the frequency is below the cutoff frequency, the current is uniform and undistorted regardless of how thick the substrate is. This is in contrast to the case of open microstrip, where the first surface wave mode has a zero cutoff frequency and the onset of higher order modes depends strongly on the substrate thickness. The cavity resonance technique, mentioned in Chapter 1, does not allow for the current distribution to be studied. A comparison of effective dielectric constant results shows good agreement between the CAD package results, the numerical results from this research, and measurements for large shielding dimensions (b, c > h). The effects of shielding on the effective dielectric constant are examined. Only one of the packages studied takes shielding into account for the effective dielectric constant calculation, and then only cover effects are considered. A comparison of the CAD package results with numerical results show that good agreement is obtained when the shielding dimensions are large with respect to the substrate thickness, while for small shielding dimensions, the difference between the CAD package results and numerical results becomes significant. For the open-end discontinuity, good agreement with other full-wave solutions and with measurements is demonstrated. A comparison of open-end capacitance for different cavity sizes shows that, as the cutoff frequency is approached, the capacitance increases in each case. Choosing a small cavity with a high cut-off frequency extends the region where the capacitance is relatively constant. The openend models used in the two CAD packages studied here do not include shielding and cannot show this effect. A comparison of numerical and measured results for series gap discontinuities portrays good agreement for all three gap widths studied. The agreement with measurements is very good for the two larger gap widths, but for the smallest gap

171 width it is seen that one of the CAD models give a slightly better prediction of the behavior of the magnitude of S21 than the numerical results. The effects of shielding on the behavior of series gaps is not explored extensively. Good agreement is also demonstrated for the two and four resonator coupled line filters. For both filters, the numerical results of this research give a better prediction of the overall filter response than provided by the CAD models. This is especially true in the rejection bands of the filters. For the two resonator filter, a slight frequency shift is observed in the numerical results compared to the measurements. This is most likely due to the thin-strip approximation used for the current distribution, and is not a limitation of the method itself. In contrast, for the four resonator filter, which has thinner strip widths, the present method gave an excellent prediction of the filter performance in every way, whereas discrepancies are observed in the CAD model predictions. Reducing the cover height is seen to narrow the pass band response and raise the amplitude of the filter's rejection band response. The numerical results of this research give an excellent prediction of this effect, and this is proven by measured data. 6.5 Recommendations The data comparisons performed here have demonstrated the validity of the theoretical methods. However, there are a few areas where improvements can be made to the theory. Further numerical investigation for series gap discontinuities indicates that as the gap width becomes small, a smaller subsection length is required for accurate computation of the scattering parameters. However, the improvement in the resuit is limited because a smaller subsection length increases the matrix size and degrades the condition number (Section 3.5). One way to extend this limit would be to use a variable subsection length in the computations. That is, use a small

172 subsection length close to the discontinuity where the current is disturbed, and use a larger subsection length over the uniform sections of strip. In this way the size of the matrix would be maximized for a given problem. The thin strip approximation as used here provides an accurate solution for the discontinuity structures studied. In one case, that of the two resonator filter, it appears that a more general current distribution may improve the solution. The thin strip approximation uses a unidirectional current. The method presented here may be generalized in a straight forward way to allow for two component current distributions. This will also allow for more complicated structures to be considered, yet it will be more computation intensive. It may be possible to improve the filter analysis by changing the formulation to allow the transverse variation of the longitudinal component of the current to be non-symmetrical without actually adding a transverse current component. To be able to discern which approximations are most appropriate for a given problem detailed comparisons should be made between results obtained with a more general current distribution and those obtained with the present method. The experimental work has shown that good measured data on microstrip discontinuities can be obtained with the TSD technique. On the tnother hand it is shown that the measuremesunt uncertainties due to connection errors, and other repeatability issues can be appreciable. In applying the technique for millimeter-wave measurements close attention is required to minimize the associated uncertainties: The following suggests some guidelines for doing so: 1. Connection repeatability errors can be reduced by first selecting the best available fixture and connection technique, and then by averaging measurements from repeated connections made for each fixture measurement. 2. Variations in a, can be minimized by fabricating the standards. and the discontinuity circuits from the same substrate, or by using substrates from

173 the same "lot", or manufacturing run. 3. Variations in substrate and line width geometry, though not a major error source, should be monitored carefully during all steps of fabrication. This can be done by performing careful measurements of the fabrication masks, the substrates to be used, and the surface metalization geometry after etching. 4. In the area of substrate mounting, care must be taken to use the exact same technique in bonding the substrates onto carriers. 5. Finally, if the physical delay line length is used in the fixture characterization procedure, it should be carefully measured and input to the program for each set of standards that are used. Because of its susceptibility to connection repeatability errors, the TSD technique is not necessarily the best technique for high frequency measurements, and research into improved de-embedding methods should continue. In considering the trade-offs between alternative approaches that already exist, the sensitivity of each technique to the unavoidable measurement repeatability issues discussed here should be analyzed and compared. Ultimately, the best de-embedding method for a particular application depends on the required accuracy, the type of test fixture, and the nature of the device or circuit being tested.

APPENDICES 174

175 APPENDIX A REVIEW OF METHOD OF MOMENTS The general steps involved for in the computation of surface currents using the method of moments can be summarized as follows: 1. Formulate an integral equation for the electric or magnetic field in terms of the surface current density J, on the conductors. It is generally possible to put this equation in the form Lop (J.)i (A.1) where Lop is an integral operator, and g is a vector function of either the electric field E or magnetic field H associated with Jo. 2. Expand J, into a series of basis functions Jp so that N,. = E Ip J (A.2) pl1 where the Ip's are complex coefficients and N, is the number of sections the conductor is divided into. 3. Determine a suitable inner product and define a set of test (or weighting) functions Wq. The result may be expressed as E Ip (IVq Lop (J)) (Vq, (A.3) p=1

176 where the inner product is defined as (a,6) = Ja. l ds In Galerkin's method, the weighting functions are taken to be test currents Jg which are identical in form to the basis functions Jp. 4. Solve the inner product equation (A.3) and form a matrix equation of the form [Z] [I = [V] (A.4) where [Z] is termed the impedance matrix, and [V] is called the excitation vector. 5. Solve for the current coefficient vector by matrix inversion and multiplication according to [I]= [Z]- [V] (A.5)

177 APPENDIX B DERIVATION OF INTEGRAL EQUATION FOR ELECTRIC FIELD Starting with Maxwell's equations V xE = -jwuH V xH = jweE+J V = -jwp (B.1) (B.2) (B.3) (B.4) (B.5) = 0, We define A such that - 1 H=- xA. p (B.6) Substituting (B.6) into (B.1) yields x (E + jA) =. Since V x V7 = 0 for an arbitrary vector function <, we let E + jwA= - V-q. Making use of (B.6) and (B.S) in (B.2) yields x -x A= -j (jA + V ) + J V -V7 x A = -jwe(jwA + V+) + J p1 (B.7) (B.8) (B.9)

178 or - V2A + V(V A) = w2eA - jwlekf + uJ. (B.10) We use the Lorentz condition V(V A) = -jwYe'7q (B.11) in (B.10) to obtain 2 A + k2A = -j (B.12) where k2 = W2C1E. From (B.9) and (B.11) the electric field may be expressed as E = -jwA+ - V(V A) = -j(1 + 2VV'.)A. (B.13) We now define a dyadic Green's function G to be a solution of V2 G + k2G = -6( - r'). (B.14) To relate G to A, we will derive a vector-dyadic extension of Green's theorem. The vector Green's theorem is given by [23,19] j j jV Q [V x V x P-P * [V x V x Q]} dv = / * [P x (V x Q)- Q x (V x P)]d (B.15) For the shielded microstrip cavity problem of Figure 2.4, the volume V is the interior of the cavity, and S,,, and S, are the surface of the cavity walls, and the surface of a small volume enclosing the source region (J) respectively. The unit normal vector n is directed outward from V, and we will denote n' = -n' to be the inward directed normal vector. P and Q are arbitrary vector fields. After applying a few vector identities, (B.15) may be re-written as follows: P * [V x V x Q-[V x V x P Q} dv = / [(n' x P) (V x Q) + n' x ( x P) Q ds (B.16)

179 With the equation in this form, we can replace Q by a dyadic function T. The result is the vector-dyadic Green's theorem given by v{P * V[V x T]-[V x V x P]. T} dv = |f [(,t x P) (v x T) + n, x (v x p). T] ds. (B.17) We may now replace P with A and T with G and make use of vector-dyadic identities to yield If f ((vA G A. )dv I | {(n' x A) V x + (n' x Vx A) * G +n'. [A(V*Gn)] -n' [(V A)G]} ds. (B.18) If we require that the components of A and G satisfy the same boundary conditions on S, and S,, it can be shown that the entire surface integral on the right hand side of (B.18) vanishes. Substitution from (B.11) and (B.14) for V2A and V2G we obtain I j (V A -A.2G)dv = I v {( J-k2A) G -A [-I(r - ') -;2 * G]}dv = - J. Gdv + A(r) = 0. (B.19) Hence, A =,I J.Gd. (B.20) Finally, substituting from (B.20) into (B.13) produces the following integral equation for the electric field E= -jW(1 + 2v ) J J Gdv w= -J J J [(1 + hVV )(G)T] * Jdv (B.21) where (G)T represents the transpose of G.

180 APPENDIX C EIGENFUNCTION SOLUTION FOR GREEN'S FUNCTION The boundary conditions on the cavity walls are applied here in order to derive the functional form of the Green's function. First, the general solution to the homogeneous differential equations for the components of the Green's function is presented. Then, the boundary conditions on the walls are used to arrive at an eigenfunction expansion for each of the Green's function components. The particular solution for the Green's function is found by integrating the inhomogeneous differential equation across the source region. GENERAL SOLUTION TO HOMOGENEOUS D.E.'s FOR GREEN'S FUNCTION Consider the homogeneous forms of equations (2.28) and (2.30) V2'+ k,2A = 0 (C.1) V2l+k ' = 0 (C.2) where i = 1,2 denotes that these equations hold in each region respectively. The relationship between A' and Gi is given by (B.20) which reduces to A1 =., (C.3) A'o

181 since we consider an infinitesimal current source J for the Green's function derivation. The components of Ai and G are related as follows: A' = /oG' (C.4) As = - oG,. (C.5) With G' given by (2.35), it can readily be shown that (C.2) implies V2G_ + k2 G = 0 (C.6) VG, + k2, = 0. (C.7) We apply the method of separation of variables with Gs = X~(x)Y(y)Z:(z) (C.8) Ga = X(x)Y (y)ZZ(). (C.9) The well known general solution of each of the above differential equations may be put in the form A = Al cos ktt + A2 sin k't (C.10) where t = z, y, or z; = X, Y*, orZ\ (where s = z or z) and kf is complex in general. The eigenvalues are related by k? = k + k + k.' (C.ll) APPLICATION OF BOUNDARY CONDITIONS ON THE CAVITY WALLS In applying the vector-dyadic Green's theorem of (B.18), it was imposed that A' and iG satisfy the same boundary conditions. Hence, As and G, must satisfy the same boundary conditions on the waveguide walls and on the substrate/air

182 interface and, must have the same functional form in terms of spatial variation. The same holds true for A' and G\,. In order to establish what conditions A' (and correspondingly G ) must satisfy at the walls, we need first to establish more explicit relations between A' and i. From (B.13) 1 -- E = -jwA' + -- V(V. A'). (C.12) The vector potential A' may be expressed as (2.31) A' = A,: + A'z. (C.13) The use of (C.13) in (C.12) yields the following expressions for the electric field components: Er = -j A'+ 2+(V Ai)] =-jw A' +-a( ] (.14) k? x z (C.14) E - = a ( +OA+ ) (C.15) E = -jW [As+ (-a+ ) Z (C.16) We now will consider the boundary conditions at each of the cavity walls. Boundary Conditions at x = O0 a Since the cavity walls are assumed to be perfectly conducting, the tangential components of the electric field must vanish at the walls. We have E(x = 0,a) 0 (C.17) E'(x=: 0,a) = 0. (C.18)

183 In view of these two equations, (C.14) and (C.16) lead to a OA: OA E'(x a) = o, — + j)2 =O 0 T Ox z 8z I (C.19) is satisfied if the following condition is imposed: (A w aAe (2) l s t in which case (C.20) leads to (C.19) = 0. (C.20) (C.21) A'.(x = 0,a) = 0. (C.22) If (C.22) is placed into (C.21) it is seen that = 0. (C.23) ax rTO,0 The boundary conditions of (C.22) and (C.23) can be satisfied by choosing the following eigenfunction solutions for the x-dependence: X C = coskx (C.24) X -= sin k'x. 3 z (C.25) for i = 1,2, where for n = 0,1,2,... (C.26) Boundary Conditions at y = 0, b The tangential component of the electric field must vanish on the walls y = 0 and b; hence, E:(y =,b) = 0 ES(y=O,b) = 0. (C.27) (C.28)

184 From (C.14) and (C.15) E'(y 0, b) = -jw [A + L.2 ( +. az~ (C-29) I~Ob. =, 0 (C.29) ES(y = 0, b) = -jw A' + z( + Az)= = (C.30) [A+ ~ + Y=O =0 (Cc.30) It follows from the above two equations that the eigenfunction solution for the y-dependence is given by Y = sin ky (C.31) Y, = sin ky (C.32) (for i = 1, 2), where k() = k(2)k = r for m = 1,,3,.... (C.33) Note that it is easily shown that m = 0 leads to a trivial solution for the ydependence of both components. Boundary Conditions at z =0, c Similarly at the walls z = 0 and c we have E,(z =,c) = 0 (C.34) Ev(z=0,c) = 0. (C.35) Making use of (C.14) and (C.15) yields 1 a 0A, aA~] E(z = O,c) = -jw [A + n)(] + =Z),o 0 (C.36) F-jw aA 8AA aA ] E(z = O,c) = [ 2 ( + aZ ]0 (C.37) 1k, ( aO+ a )zo It can readily be shown that the eigenfunction solution for the z-dependence can be written as Z41) = sin k)z x z (C.38)

185 Z() = cosk )z (C.39) Z() = sin k2)(z-c) (C.40) Z2 = cos 2)(z-C) (C.41) where, from (C.11), (C.26), and (C.33), k'l) and k(2) are given explicitly by k = (/) - - ( )2 (C.42) k) = ko - ()2 ( )2. (C.43) kc and ko are the wave numbers in region 1 and 2 respectively, and are given by kl = w//Jel (C.44) ko = wV/0o. (C.45) REPRESENTATION OF GREEN'S FUNCTION BY EIGENFUNCTION SERIES We now combine the results obtained above, so that the Green's function may be written in series expansion form. Substituting from (C.24), (C.31),(C.38), and (C.40) into (C.8) and taking the summation over all the possible modes, results in the following for G(: 00 00 G( = A(), cos k, sin k,y sin k(1)z (C.46) mal n=O 00 00 mil n=O Similarly, if we substitute from (C.25),(C.32),(C.39), and (C.41) into (C.9) we obtain the following for G', 00 00 G( = B,() sin k, sin ky cos k)z (C.48) mal nsO 00 00 G(2) = B(2) sin k,x sin ky cos k2)(z - c). (C.49) ms1 n=O The complex coefficients An and Bn (i = 1,2) are determined in Appendix D by the application of boundary conditions at the substrate/air interface (z = h).

186 APPENDIX D BOUNDARY CONDITIONS AT SUBSTRATE/AIR INTERFACE The complex coefficients A,, and B' (for i = 1, 2) for the Green's function components given by (2.36)-(2.39) are found here by applying boundary conditions at the substrate/air interface (z = h). Figure D.1 shows a cross section of the cavity in the x-z plane. The application of boundary conditions at the interface is made difficult by the presence of the infinitesimal current source on the substrate surface. We will avoid this difficulty by first solving a similar problem with the current source raised a distance Ah above the substrate. After solving for the boundary conditions at z = h and z = h + Ah, the equations required to determine the coefficients Al, and Bm are obtained by letting Ah go to zero. FORMULATION From the consideration of the boundary conditions on the waveguide walls the components of the Green's function are given as G() = A cos kC sin ky sin k()z (D.1) mal n=O G() = B() sin kx sin ky cosk)z (D.2) m=l n=O GC = A cosx sin kc,y sin k(2)(z - c) (D.3) m-=1 n=O

187 z=c z=h (2) I x I x=a a) Actual position of current source 1 z z=c z=h' z=h (2) J h --------- l^ ----h (3) Ah - ------ I x M-W I x=a b) Current source raised above interface Figure D.1: The current source is raised above the substrate/air interface to apply boundary conditions.

188 G() - E B(2) sin kx sin kyy cos k(2)(z- c). (D.4) m=l n=O For region 3, the Green's function must satisfy the same differential equation (2.30) as in the other two regions. We will use the form of the general solution given by (3) = (3) ejks+(3) e-k-jk ), G(3 E cos kzx sin kvy [Am() +B(e ik n (D.5) G = A,,o^ e + B?^e- (D.5) m=l n=O 3,, = ( k +3) (3) -jk(3] (D.6) G(3 = E Z sin k=x sin k,y [C(3), '+ 1D( (D.6) m=l n=O where k = nr/a (D.7) ky = ml/b (D.8) k() = kT- - k (D.9) k() = k k (D.10) k3) = k(2) (D.11) k = w/o/T; (D.12) ko = W/o. (D.13) Recall, the electric field solution in terms of the vector potential components (C.14)-(C.16) E, = -jw A' + k2 z ( +A ]) (D.14) E = -k. ( a + a ) (D.15) k — yO + Oz) E = -jw A, + 8kZ( ]) (D.16) kl a O a az These equations hold in each region respectively (i.e. for i = 1,2, 3). The solution for the magnetic field can be written using (2.26) and (2.31) as follows: H' —V x A' =-([ x (A'^ + Ai)]. (D.17) 0 so0

189 Separating this into x, y, and z components gives Hz - z (D.18) H- --- Io Oy 1 (8 A 8A ) (D.19) H --. (D.20) ~o Oy ' APPLICATION OF BOUNDARY CONDITIONS Boundary Conditions at z = h At z = h, the following boundary conditions apply: E(1) = E3) (D.21) E() = E 3) (D.22) H = H3) (D.23) Hf(f) = H(3) (D.24) AoH(l) = - oH(3) H(1) = H(3 (D-25) clE(') = oE3). (D.26) We will make use of (D.22)-(D.25) to formulate four of the eight equations needed to solve for the complex coefficients in (D.1)-(D.6). We start with (D.25), then substitute from (D.20), and recall the correspondences of (C.3) to obtain dGc') l G(3) = \ -; - \h -i i (D.27) ay 0y When (D.l)and (D.5) are used the above, and orthogonality is applied, the following is obtained A(') sin k(')h = A(3 eL) + Be-)h mn z mn + mn (D.28)

190 where k(2) has been substituted for k(3) in accordance with (D.11). Next, from (D.23),(D.18), and (C.3) 0GW( ay OG(3) - y x '9y (D.29) From (D.2),(D.6) and the above B(') cos k)h = C(3)ej h + D(3) e-jk(,)h The combination of (D.24), (D.19) and (C.3) yields (D.30) aG(1 az aGOz J aG(3) Iz - aG ) (3z= ax (D.31) Making substitutions from 1),(D..2),(D.5) and (D.6) into this expression and using (D.30) leads to A(g) k) cos k -j)h = jk2) [A( ) ei)h - B( e-jk h] (D.32) Now consider (D.22). From (D.15) and (C.3) 1 a(8G1) 8G'( f;ry 9x az 8 aO9G(.aG(3) y(( Ox + Oz ),=l, ~ ax dz 9; (D.33) After appropriate substitutions and some algebra, the following equation is obtained A) (1 -, )k, sin k(1)h + B(l) k() sin k()h = -jk(2. C(3), - D(3)e-j(2)h 2 r mn?nn] (D.34) Equations (D.28), (D.30), (D.32), and (D.34) represent 4 of the 8 equations we need. Boundary Conditions at z = h'

191 We now proceed to the boundary conditions at z = h' (see Figure D.1) we have E(2) E(3) (D.35) E(2) = E(3) (D.36) E(2) - E(3). (D.37) (2) = H(3 (D.38) z x (H() - (3)) = J,. H = H(3) (D.39) -(H2)- H(3)) J,. (D.40) Of the above, we will use (D.35), (D.38), and (D.39) to derive three more equations for the complex coefficients. We start with (D.39) and use (D.18) and (C.3) to obtain G( jZ)= = Grz- \z=' (D.41) ay ay which yields after substituting from (D.4) and (D.6) B(2) cos k(2) (h' - c) = C(3) + D^-. (D.42) Next, consider the boundary condition of (D.38). This leads to 090 l = a'3) (D.43) ay ay Substitution from (D.3) and (D.5) yields A4,) sin k(2) (h'- c) = A),, ) h' + B(3) e-j''. (D.44) This equation, when combined with (D.3) and (D.5), shows that G(2(z = h') = G(3)(z = h'). (D.45) With the above equality, we can substitute from (D.14) into the boundary condition of (D.35), and make use of (C.3) to produce aG(2) + G(2) = 8G(3) aG(3) (D.46) ax( Ox az a 9x oz '

192 Suffering through the details again we obtain o(2) Cik)( ) (3).2hijk?,',(3)e-jk(')hi B sin k2)(h - c) = - [C eJ - ^ ]. (D.47) At this point we have 7 independent equations -(D.28), (D.30),(D.32), (D.34),(D.42),(D.44), and (D.47)- and we have 8 unknown complex coefficients. The other required equation is obtained by integrating the differential equation of (2.30) across the boundary at z = h'. Integration Across the Source Region at z = h' From (2.30) we have V2 Gi + kG' = -I5 ( - '). (D.48) Substitution from (2.35) for Gi yields (V2 + k?) (G^ x + G =xz) -6(r - r'fX (D.49) Hence, (V2 + k?)GL =-6(f - r') = -6( - ')6(y - y')6(z - z'). (D.50) We now integrate both sides of this equation over a line passing through the source point r', and then take the limit as the length of this line vanishes lim /h (V2 + k2)Gt dz = -6(z - x')6(y - y'). (D.51) This may be written as -o L902 +92 /\ 1j,_gZo j - (D.52)

193 If we make use of (D.3) and (D.5), we can show that the first integral vanishes as follows: h'+cr. ' 1h4a lim G'Gdz = lim [I Gdz + G ]dz =lim [i. - ((2)) COS kx sin kGd, (A(i)d2z B(e is)] K' + li [ EE (-2) i A cos k=x sin ky cos km) )] -h:+a.m=I n ---"O ks av..,O k) mn z=h' 0 (D.53) (since each of the limits on the right hand side of the second equality vanishes individually.) Therefore, (D.52) can be reduced to + -G' d = GS(xd -( z')6(y- y') (D.54) a-O h z-a ZZ From which we obtain lim = |h+a — 6( - 6(Y - y'), (D.55) or (G - I)| =-6(X - z')S(y - y'). (D.56) O 9z /, Substitution in the above from (D.3) and (D.5), and simplifying yields ab {A(2) k( (2CO h' C) - (A(3),Je(ik2) - 3)-jk( } = - cos kz' sin ky' (D.57) where 2 for n = 0 (D8) 4 for n 0 The above represents the final equation needed to evaluate the complex coefficients.

194 EVALUATION OF THE COMPLEX COEFFICIENTS OF THE GREEN'S FUNCTION To evaluate the complex coefficients, we will make use of the equations derived above involving A M, Bn (i = 1,3), and C) ad D). Since we are only interested in A$,M B$, A(, and B$ these will be evaluated by eliminating the other complex coefficients. Now, recall that h' = h + Ah. If Ah -- 0 then h' -- h in equations (D.42), (D.44), (D.47), and (D.57). Starting with (D.42) with h' - h we can substitute from (D.30) to obtain B(1) cos k()h = B(2) cos k(2)(h - c). (D.59) Similarly, (D.28) and (D.44) yield A(2) sin k(2)(h - c) = A() sin k(1)h. (D.60) From (D.34) and (D.47) we get 1 F 1 1mm z B 2 sin k)(h - c) = k- [A ( - 1) k. sin k)h + n ' sin k()h. (D.61) From (D.32) and (D.57) b [A(3) k (3) cos k(3)(h' - c) - A() 10) cos kl)h] = -cos kS' sin ky'. (D.62) -n — n zmn' mPThe combination of (D.60) and (D.62) yields ab [Ai msink(h k2 ) cos )(h- c) - A(1)) cs k()h cos k,' sin ky' (Pn sin k(2)(h - c) n (D.63) Solving for A$ —, cos k.x' sin kyy' tan k(2)(h - c) ab dimn cos k') h

195 where d,,mn = k(2) tan k(')h - k() tan k(2)(h - c) (D.65) and p,n is given by (D.58). A(2 is found by substitution from (D.64) into (D.60) A(2) - -n cos kx' sin kyy' tan k()h 2 =mn - (D.66) a dimn cos (2)(h- c) Next, we combine (D.59) and (D.61) to get B( cos k()h sink(2)(h-c) 1 [A (1) cos k(h - c) k ) {,( + B(k) sin k1)h] (D.67) + J By substituting for A(1 from (D.64), the above can be rearranged to find B$M as ( -,n(l - e;)k, cos k.x' sin k,y' tan k(X)h tan k(2)(h - c) ab dnd2mn,, cos k( )h where d2m, = k(2) tan k(2)(h - c) - k) tan k()h. (D.69) Finally, if we place (D.68) in (D.59), B(2 can be expressed as: (2) -n,(l -;)k, cos kz' sin ky' tan k()h tan k(2)(h - c) ab din,,d2m cos k( )(h - c) We now have derived explicit relations for the desired complex coefficients A(, A(),, B() B2) It can be shown that the same relations can be obtained by moving the current source of Figure D.1 into the dielectric region and then bringing it back to the substrate surface.

196 APPENDIX E EVALUATION OF MODIFIED DYADIC GREENS FUNCTION For the purposes of evaluating the electric field (2.66), only the xx component of the modified dyadic Green's function i is needed. In this appendix, expressions for rl), and r( are derived. Then, each of these are evaluated at the air/dielectric interface (z = h) and shown to be equal. The modified dyadic Greens function was defined in (2.58) as r' — iu o 1 + 2v ) (i)T]. (E.1) From (2.35) ~,'= G ~xx + G,,x. The dyadic transpose is (G')T = G;. + G;,zi (E.2) Using (E.2) into (E.1) yields i = -jo G + a: + aZ, k?,\ ()i aZ + [| r, + aG=z (aGOx= 8G ) }( + Glzs 4 zx 1(E.3) Iz +z a O ] } Hence, the xx component of the modified Green's function is given by rw = -jwIo [Go + 1: z ') (E.4)

197 Substitution from (2.36) and (2.37) into (E.4) results in 00 00 [) ( k2\ k _(l) r(l = -jwZo E E os kx sin ky sin kl) A I (1 - k - k 1) m=1 n=O [ I II1 If we use the expressions for A$), and B(), from (2.46) and (2.48) we may write, = abd E E {cos k () h abd n Imnd2mn COSz * k()e- 1 tan k()(h-c) - - ]tan )h} (E.5) where the expression for d2mn from (D.69) is used to combine t erms. Evaluation of (E.5) at z = h gives rT(z-A) =,-wof E {( ----c h)= 1 o (' abdlmnd2mn) m=1 n=O [cos k-x sin ksy cos ksx' sin kyy' tan k(A)h tan k()(h - c)] * [k(2); ( - )tan k)(h - c) -k() (1 Xk) tan k)h]} (E.6) Proceeding in a similar fashion for region 2, it can be shown that r = ( -.)jwjo co08 k: sin ky sin k 2)(z c) m -1 n=O b-n cos kx' sin ky' tan k()h (1 - k2 kok2 f-^(l - e,)ks cos kz' sin kyy' tan kf^)h tan k(2)(h - c)1 L abdlmnd2mn cos kW( h-c) J Evaluation of this expression at z=h and rearranging the result produces r(2)( = h) = jw, oE { (bd )

198 *[cos kjx sin kyy cos kx' sin ky' tan k()h tank(2)(h - c)] * [k(2)';(1-k) tan -(h-c) - ( kl ) tan kl) (E.7) Upon comparison of (E.7) with (E.6) we can readily see that rl(z = h) = r()(z = h) = r.(z = h). (E.8)

199 APPENDIX F INTEGRATION OVER SUBSECTIONAL SURFACES Consider the surface integral given by (2.71) Zqmn = / J/ cos kz' sin kyy' '(y)a,(x') dx'dy' (F.1) where from (2.7) ( 77v L-[ -Yol2I 0 From (2.8), for q # 1 WiK(x., -z')l OKq (Z') =(x'-Fp-i)] sin(K'ls) an from (29), for q 1 and from (2.9), for q = 1 (F.2) else. xq l< < xq+l q-1 / <s 2< x else, (F.3) sinK(t, -)] a, ( = in( s' 0 o < x' < I else. (F.4) In the above, s = Zq+l - Zq = z - Zq —1 and, for our purposes here1, we let xq = (q- 1)1/. Note that for strip geometries other than an open-end and a thru line, the position function Xq will be more complicated in general (see Section 3.1.4).

200 Figure F.1 illustrates the strip geometry used to determine the integration limits in equation (F.1). The boundaries of the qth subsection depend on q as follows: 0 < X' < Y, - < y < Yo + 2 for q = 1 Sq = (F.5) xq-1 < x' < zq+1 Yo- < V' < Yo + W else. With these subsection boundaries, Zq,, may be expressed as,qmn = z~'z (F.6) where z = w/ (y') sin kvy'dy' (F.7) o o 'cos.' cq(x')dZ' for q=1 (F.8) t Jt+f' cos kZx'acq(')dz' for q 1. INTEGRATION OVER y' From (F.7) and (F.2) we have t2 ^+W/ sin k,y = 2 JY+W/2 sinlk;y dy' (F.9) VfW Yo - W/2 2( '-a)2 Now, let sin = 2(-Yo) cos do = wdy' W W Y(F.10) y = sinO+Yo: dy' = wcos dk with these substitutions 1 sin[ sn Y)] (F.11) 2=- sin k,( -sin + Yo) d. (F.11)

201 y b y + W/2 0 y - w/2 0 I I I I I I o X X z X 3 q-1 ~ qol Figure F.1: Strip geometry used in evaluation of surface integrals

202 The above may be rewritten as = = [ sin(k/ sin 2 ) cos kYo d0 + |L cos(kcy2 sin ) sin kyYo d 2 d?] MYl) r(2) +. (F.12) Consider the first term of (F.12): (1), 1 / - cos kYo sin(k- sin 'k) dO T Lf v 2 2 J = -cos kYo [ sin(kYT sin 4) d+ + L sin(k2 sin ) d* If, in the second integral above, we let ' = -; d+' = -dO then = - cos kyo sin(k2 sin ) d4 + sin(-k- sin )(-dk) j/ [: c /ok Jo = 0. Hence, (F.12) becomes 2' = = - sin kyYo cos(kcy sin k0) d4 0 w - sin kYo [ cos(ky- sin 2 ) do + cos(kIy sin) )d]. (F.13) If, in the first integral of (F.13), we let 4' = -4; da' = -d>

203 we obtain 2 - si =-sikyYo cos(ky- sin )d d. (F.14) By comparison of the Bessel function of the 277 order given by [65] 2 /2 J2,(z) = - cos 277; cos(z sin 0) dO with (F.14) we may readily see that = sin kVYoJo (k2 (F.15) This completes the y'-portion of the integration. INTEGRATION OVER x' From (F.8), - c = 1 cos k,'ag(x')dz' (F.16) First we consider the case for q 0 1. Substitution from (F.3) in the above yields = s in sin K(x' - xq-1) cos k.' dx' + | ^ sin K(x' + xq,-) cosc kx' dx'] 1 1), [I +?2)i-] ( 1) (F.17) For the first integral we have 1(-) = j sin K(x' - _,)cs kzx'dx' = 2 1 {sin [(If + k,)x' - Kxzql] + sin [(KI - k,)x' - IKx,]} dx' = - K + k [cos(KI, + k,.,) - cos kq,-] + K - k [cos(Il, - kzq) - cos ksxl]. (F.18) +\ K- k)kx.I

204 We can solve for Z12)i in a similar fashion to yield 2) I sin [(q+-)] cos kx'd' JqX [c { kcsx - cos(Kl, + kczx)] + + ^ C [cos k,,q+i - cos(KI - k,)]} (F.19) Substitution from (F.18) and (F.19) back into (F.17) yields 7 I — I —= -1 - -- = sin K (1K + kz + K - * [2 (cos k.,q+l + cos k,,z-) - COs KIl. cos k,]. (F.20) After some manipulation, this expression may be put in the following form = -4K cos ksxq sin [(k. + K)l] sin [(k.- I)l] q sin KI, (I + k) (IK - k,) = - sin Sinc -[(k, + K) I Sinc [(k -I)i, (F.21) where Sinc(t) = t (F.22) 1 t=0. Recall now the integral for the case q = 1 from (F.8) z= os kCsx'at (')dx'. Substitution from (F.4) for al(x) -sin K -- cos kz'sin(I - x)dz'. (F.23) Comparison of this expression to the integral of (F.19) shows that if we let Xq -+ 0 and zq+1 -- 1 in (F.19) we can obtain the solution for the integral in (F.23). The result is 2 sin Kl { K-1k [cos k:l[ - cos KIl,] + K + k [cos kl - cos Kl,] } 2 sin Kl, (K -k + kT ) [cos ksl -cos Kl]

205 The above can be rearranged to give = - Sinc [i(k + K)lr ' Sinc [(ks- K)l] (for q = 1) (F.24) s sin Kl4 Combination of this with (F.21) yields d =_-( J.. 4K sq inK [(k. + IC)l] Sinc [ (k - I)l] (for any q) (F.25) where 2 for q=l 1 (r = (F.26) 4 for q 1. Finally, substitution from (F.15) and (F.25) back into (F.6) yields q-mn- = J cos kx' sinkyy'ip(y')cq(x')dx'dy' = qKJKlcoskq.,. rik. r\, In Sincl [(k + K)l Sinc L(k, - )1] sin kYo Jo(ky). (F.27)

206 APPENDIX G EVALUATION OF MAGNETIC FIELD COMPONENTS The magnetic field components anywhere inside the cavity are given by the surface integrals of (2.89) and (2.90) H' - -G) fibf()(')d.' =.- j, (y ')cq(x')ds'. (G.1) (G.2) We will evaluate Hi first. EVALUATION OF THE y- COMPONENT OF THE MAGNETIC FIELD From (2.36) and (2.37) 8Gz 9z aGM - x ax - E kA cos kx sin ky cos kl)z mm I n=O - E k^) cos &kz sin k,y cos k()z. mal n=O After appropriate substitutions and some manipulation we may write 8G( az 9G()Z 9x = c on, tank(2)(h-c) wml nO ab dImnd2mn cos k()i h * {k)k(2)e tan k2)(h - c) - [(k())2 + k2(1 - e)] tan k(')h} cos k=z sin kyy cos kl)z cos k=x' sin kyy' (G.3)

207 Similarly, substitution from (2.38) and (2.39) leads to G 2) aG 2 _ 82 -0 Co, tank(')h CZ CZ E z2) az m=1 n=O abdlmnd2mn cos kZ2 (h - c) {k~)ke tan lc~h -1 (2) - k2 (1 -r tan k(2) (h - c)} cos kx sin kI,y cos k(2)(z -c) cos kx' sin kLy'. (G.4) We are now ready to evaluate the y-component of the magnetic field. Substitution from (G.3) into (G.1) yields H(') = cptankAZ2)(h - c) qy(1 m= n=O ab dlmmd2mn COSO h * {k41)k2) tan k(2)(h - c) - (1))2 + k2(1 C ) tan k )h cos kix sin ky cos k-')z [Zqmn] (G.5) Replacement of qm,, with the expression from Appendix F, yields - K12 (0 h tan k(2)(h - c) 4ab sin K!: vnlo dlmid2mi cos k (1) k(1)k (2),r tan k(2)(h - c) - [(k))2 + k2(1 _ e,)I tan k0)h} -COS kxq Sinc [1.(k + I)l1] Sinc [( - K)l.] kyY0 COS kLx sin k~y cos k0)z. (G.6) Similarly, substitution from (G.4) into (G.1) and again making use of (F.27) yields H(1K 2 00 00 Wntank k1)h ) - ab sin K1. z= ini dimnd2mn cos k(2 (h - c) {kr)k(2) tan - [(k(2))2e - k2(1 - e)] tan k(2)(h - c)} 1i W Cos k,, Sinc -~(kz + K)!:] Sinc 1(k.T - K)!:] sin kvYo Jo(kyT) * cos kx sin ky cos k)(Z- c). (G.7) We now proceed to the evaluation of Hq5.., EVALUATION OF THE z- COMPONENT OF THE MAGNETIC FIELD

208 With the use of (2.36), we can write = X kA)cos kzx cos k~y sinkk') z = V=1 n= '3bdlimn COSk41)h cos k.,x cos k~y sin k(')z cos k~x' sin kyy'. G8 Similarly, from (2.38) -z kA2)cos k-x cos k,y sin k(2) (Z -C) C9Y m=l n=O - ~00 (pnk tan k(I~h m=1ln=o ab dimmn co kz (h - c) cos k x cos k,,y sin k2) (Z _C) cos k::' sin k~y'. (G.9) Substitution from (G.8) into (G.2) and using (F.27) yields = (~qKl2 000 npky tan Jk(2) (h - c) q z 4ab sinm I~ mii, dm ok'h *cos kzzq Sinc [( +Kl]Sinc [(km K)lI] * sin kvYo J0(kvY.) cos k~z cos k,,y sin k( 1)z (0.10) Lik-ewise, substitution from (G.9) in (G.2) yields = qKl2 000 (p,ik1 tan k(1)h qz 4ab sin Kl mvl n= dlmn COB k( )(h - c) * cos kz'Xq S inc -.(km + K)lI] Sinc k - K):] * sin ktY0 J0(kv2 ) cos kx cos k,y sin k,2( -c (0.1I1) In summary, the and Z' components of the magnetic field anywhere in the cavity may be expressed as follows: -V1 Hqo CnqC7, C05M kn i ~ cok1z (G.12) m=1 n=-O

2W0 9 00 00 HO~) - q~~Cq~cos k.,x cos k,,y sin k~1~z 1.13) m=l n0O 00 00 Hqy~ Hqo CqC,,c(2m, cos k,,x sin k,,yj cos k2(z-c) (.) m=1 n0 -(2) Hqo 00 00 ()k2( ffq) - o E CmqCzgn coskx cos kyg~ sin k _)( -C) (G.15) m1l n0 -where Hq~=4ab sin Kl,, Cq= Cos kzxq Sinc + K)l.] Sinc -K)l] and - {k(1)k(2)ie* tank k~)(h - c) kitd2mn [ (k(1) )2 + k2 (1 _;)] tan kO)h} (G.16) V~ tan k(2)(h - c) W (.7 C(I) '4m co14 sinkYYoJO(ksfT) (~ -(2 {kn )k (2) tan kO1h k [(k2) )2 Cm - k2(l _ fr)I tan k(2)(h - c)} (G.18) S2)(mk. tan k~l) h w dimm cos kW (h - c) si.0J~,) (G29

210 APPENDIX H MICROSTRIP CONNECTION REPEATABILITY STUDY This appendix includes a description of the microstrip connection repeatability study carried out by the author while at Hughes Aircraft Company. In this study, key repeatability issues related to the measurement of microstrip discontinuities with the TSD (thru-short-delay) technique [44,49], are explored experimentally. These include the repeatability of coax/microstrip connections, rnicrostrip/microstrip interconnects, microstrip line fabrication and substrate mounting, and the electrical characteristics of coax-to-microstrip transitions (launchers). Each of these issues has been explored experimentally and the results are presented for two types of coaxial-to-microstrip test fixtures: one usable to 18GHz, and the other to 40GHz. The objectives met by this study are two-fold. First, trade-offs were explored for choosing between different connection alternatives for de-embedding in microstrip with the (TSD) technique Secondly, data was obtained to assess the uncertainties in microstrip fixture measurements due to the non-repeatability of various microstrip connections. Finite measurement uncertainties, due to the repeatability issues described above, are inherent in each fixture measurement made on the TSD standards (during fixture characterization), and the D.U.T.. The measurements described next illustrate how experimentation can be used to determine the magnitudes of

211 Figure H.I: K-connector (2.9mm) coaxial/microstrip test fixture. these uncertainties. DESCRIPTION OF CONNECTION REPEATABILITY EXPERIMENTS Two types of coaxial test fixtures were used in the experiments. The first, usable to 18 GHz, consists of a pair of 7mm "Eisenhart" launchers [57]. The test circuit used here consists of a 1" section of 50 ohm microstrip line on an alumina (h =.025") substrate (Figure 4.4). The other fixture, operable to 40 GHz, uses a pair of Wiltron K-connector (2.9mm) launchers and a.40" section of 50 ohm microstrip line on a quartz (h =.01" ) substrate (Figure H.1). The repeatability experiments performed are summarized in Table 1. For our purposes, "cycling" a connection refers to disconnecting and reconnecting both the input and output connections simultaneously and repeating the measurement. For the coax/coax repeatability experiment, each fixture was connected to and

212 Table H.1: SUMMARY OF REPEATABILITY EXPERIMENTS EXPERMWNT ( TRIALS PAN IOSRPTiOt4 1. COAXCOAX A) 7 mmFIXTURE U) K.CONN. FIXTURE 2. COAX/MICROSTRIP A) 7 me FIXTURE U) K-COHN. FIXTURE K-COHN. FIXURE 4. MiNROSTW PA&FASRCAT1OW MOUNTING K-COHNK MXURE A) 7 mm FIXTURE 5) K-CON. FIXTURE a0 a0 I$ I0 I 0MEUPS GM CYCLED CONNECTIONS AT 7 mum COAXIAL EASURWPORITS CYCLED CONNECTIONS AT K-CONN. COAXIAL CYCLED PRSUECONTACT MADE FROU LANHRTO MOCROSTRPu LIN CYCLED GAP WELD CONNECTION MADE PROM TAS ON K-COHN. ELIAN CONTACT TO CYCLE TWO GAP WELDE RISSONS USED TO CMONNEC ThRES M tOFI IE MEASURED FIVEI SEPARATER KMCOSTRIP LINES WITH SAME LAN R MASUREDSANE LIN WITH DIFFERNT PAIR OP 7m ANHR MESRDSANES LINE WITH DIFFERNT PAIR OF K-CONN. LANHR I S 3 GAM&MI am 0MB-aS GM I

213 GAP WELDED - SLIDING CONTACT Figure H.2: Coax/microstrip connection technique used with Kconnector (2.9mm) launchers. removed from the coaxial measurement terminals several times, taking care not to disturb the coax/microstrip connection. The 7mm coax/microstrip connection test was performed by cycling pressure contacts made between the wedge shaped center conductors on the Eisenhart launchers, and the microstrip line. For the K-conn. fixture, gap welds used to connect the.018" tab on the K-conn. "sliding contact" (a small gold plated tab with a sleeve that fits over the launcher's center conductor) to the microstrip line (Figure H.2) were cycled. In order to preserve the microstrip metalization, the minimum amount of weld voltage and pressure needed to secure the tab was used. This made it possible to use the same microstrip line and sliding contacts for all 10 trials. A similar connection approach was used for the microstrip/microstrip interconnects. Two.020" x.025" Gold ribbon straps were gap welded across the connection interfaces between three microstrip lines (Figure H.3). To cycle this connection,

214.. Is -J *-: 31F Figure H.3: K-connector multi-line test fixture used for testing microstrip/microstrip interconnects. the straps were removed and then replaced with new ones. A minimum amount of weld voltage and pressure were used. Note that none of the repeatability cycling could be performed without removing the fixture from the coaxial measurement ports; hence, coax/coax connection uncertainties are included in all the results. In the microstrip fabrication/mounting experiment, three uncertainty factors are present simultaneously: coax/coax connections, coax/microstrip connections, and the variations in the carrier mounted microstrip lines. Two automatic network analyzers were employed for the testing; an HP8510 ANA for.045 to 26.5 GHz measurements, and an HP8409 ANA with an in-house frequency extension system for Ka-Band (26.5 to 40GHz) measurements. The HP8510 ANA was operated in step mode with 201 calibration points, and the Ka-Band ANA was operated in phase-locked mode with 51 calibration points.

215 Calibration was achieved using either 7mm or K-conn. coaxial standards, depending on the fixture. After testing, the S-parameters for each of the measurement trials were stored in separate files so that data processing could be carried out later. DISCUSSION OF RESULTS Due to the large volume of data generated, only a sample of the results can be presented. Therefore, we will limit the discussion to the uncertainties in S21 (the forward transmission coefficient) due to the repeatability issues discussed above. Figure H.4a shows the coax/microstrip repeatability measurements made with the 7mm test fixture. At a glance this connection looks very repeatable, yet with some data processing we can get a closer look. A program was written that allows for statistical computations to be made using the data stored on file. With this program, an average set of S-parameters was computed for each experiment. Figure H.4b shows the results of normalizing the S21 data to the average S-parameters for the coax/microstrip test. This was achieved by performing a complex division between the S-parameters from e trial and te average S-parameters. After normalization, it is easy to see the variations in phase and magnitude resulting from the repeated connections. For the 7mm coax/microstrip connection, the repeatability in S21 is quite good and can be held to within a range of.ldB(+/-.05dB) in amplitude and 1 degree (+/-.5deg.) in phase. Standard deviation data for each experiment was also computed as a function of frequency. This was done separately for the magnitude and phase angles of the S-parameters using the following formula: Std.Deviation = s = N N (H.1) where

216 * 1 2 3 4 5 d 7 * 9 O 1 11 12 13 14 Is 1s 17 1I1 a) Amplitude measurements I I 2 2 4 5 6 7 I S 15 Li 12 12 14 15 16 17 IC~ PN3KKv aO) b) Su results normalized to average S-parameters. Figure H.4: 7mm Coax/microstrip connection repeatability measurements.

217 - a 3 /' la p % Y14V: Id / I %/ ~,. 44 Iv I I IdI n // PHASM OF S21 j^N lam ^ M-i _______ _ _^^^^ I I..W UUl — - -% - l I 1 2 3 4 5 8 7 1 3 11 12 13 14 15 17 I U C CR D Oh) Figure H.5: Standard deviation data for experiments with 7mm fixture N = number of trials Xi = magnitude or phase of a particular S-parameter (at a given frequency) for trial i The standard deviation data on S21 for the three experiments performed with the 7mm fixture(Figure H.5), shows clearly that electrical variations between Eisenhart launchers can be significant. This is not surprising, since the center pin on each is individually tuned to achieve good performance. On the other hand, the standard deviation resulting from the coax/microstrip test was almost as good as that from the coax/coax test; thus, the uncertainty contribution to S21 caused by the 7mm coax/microstrip connection errors is finite but minimal. The normalized S21 data for the k-connector coax/microstrip experiment (Fig

218 ure H.6) displays similar repeatability performance to the 7mm coax/microstrip connection to 18 GHz. The approximate range in amplitude is.1dB to 26.5 GHz and.2dB to 40 GHz, and the range in phase is 2 degrees to 26.5 GHz, and 8 degrees to 40GHz. For the microstrip fabrication/mounting experiment (Figure H.7) the amplitude and phase deviation were about the same if one apparently erroneous, measurement in the.045 to 26.5GHz band is neglected. The connection repeatability of the microstrip/microstrip interconnects (Figure H.8) was considerably worse than the coax/microstrip connections, with ranges in amplitude of.5dB to 26.5 GHz and.6dB to 40 GHz, and in phase of 10 degrees to 26.5 GHz and as much as 15 degrees to 40 GHz. A good summary of the K-conn. fixture repeatability experiments is provided by the standard deviation plots of Figure H.9. Admittedly, some of the results (e.g. launcher-to-launcher uniformity) are not entirely conclusive due to the limited number of trials performed, however, some definite trends are apparent. The erroneous measurement in the fabrication/mounting experiment (Figure H.7) caused the standard deviation in phase below 26.5 GHz to appear much worse than it should. This is supported by the comparatively good standard deviation observed in the 26.5 to 40 GHz band, which follows the coax/microstrip repeatability curve closely. Hence, the measurement uncertainties due to microstrip fabrication/mounting variations are not believed to be a major concern. On the other hand, although the K-conn. launchers are not individually tuned, Figure H.9 indicates significant deviations in S21 between the three sets of launchers tested. Consequently, for accurate de-embedding work, launcher-to-launcher uniformity should not be assumed. The standard deviation data for the coax/microstrip and microstrip/microstrip experiments reinforces the S21 data of Figures H.6 and H.8. In terms of the TSD connection alternatives discussed in Section 5.3 these results clearly favor an approach relying on repeatable coax/microstrip connections rather

219 O AD AirTU 12r O1 dT r a' m? M CT ~! _1 - P r ' _. - Aid I 2 4 Ui 1s2 14 1s 1is D 2 24 a2 a) Measurements from.045-26.5 GHz A -L; a 94 Ad I 33 Mr a - L:, =* d 'P~bl;-~ChV f v rrpr u~ur~r C~ll Q I In - 11 31 a ati 3a m X X m U 3 m 4a b) Measurements from 26.5-40 GHz Figure H.6: K-connector coax/microstrip repeatability measurements normalized to average.

220 MLOr AMP.n.fL O S2l 9 i a) Measurements from.045-26.5 GHz umwAz M' mm w=I 9 I1 C-r b) Measurements from 26.5-40 GHz Figure H.7: Microstrip fabrication/mounting repeatability measurements normalized to average.

221 r APLUE OF S2 i a) Measurements from.045-26.5 GHz r ii I 1 I In Is - S - t -4X PH -E OP I ^ b) Measurements from 26.5-40 GHz Figure H.8: Microstrip/microstrip interconnect repeatability measurements normalized to average.

222 AMPLMME OF S21 *...-..-...~... —.. MIn:C TFA LMJC q Fad aw a) Results for expements conducted om.045-26.5 GHz I 4 9 I 11 12 14 16 11 2 i2 24 2 a) Results for experiments conducted from.045-26.5 GHz Mmji OF 821 5 a PNLAsIaru Nin. 4tNWWinTRW towe X X 31:i X 34 S X 37 X S b) Results for experiments conducted from 26.5-40 GHz Figure H.9: Standard deviation data for experiments with Kconnector fixture.

223 than microstrip/microstrip interconnects for the hardware tested.

BIBLIOGRAPHY 224

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