Ti E UNIVERSITY OF rC:. s ~ ENGINEERING LIBRAWI4' Technical Report EC OM- 0138-26- T Reports Control Symbol OSD- 1366 September 1971 DESIGN AND ANALYSIS OF A PARAMETRIC LOWER SIDEBAND UPCONVERTER C.E.L. Technical Report No. 211 Contract No. DAAB07-68-C-0138 DA Project No. 1H021101 A042.01.02 Prepared by W. A. Davis COOLEY ELECTRONICS LABORATORY Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan for U.S. Army Electronics Command, Fort Monmouth, N. J. DISTRIBUTION STATEMENT Each transmittal of this document outside the agencies of the U. S. Government must have prior approval of CG, U. S. Army Electronics Command, Fort Monmouth, N.J. Attn: AMSEL-WL-S

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ABSTRACT This report describes a method for synthesizing a parametric lower sideband upconverter (LSUC) in coaxial line and reports an analysis of a forward biased varactor diode as it would be used in a LSUC. In most analyses of parametric devices the circuit is assumed to consist of either ideal filters (total transmission for in-band frequencies and total rejection for out-of-band frequencies) or lumped-element filters. However, experimentally a LSUC requires the use of microwave circuit elements that are neither lumped nor ideal. The LSUC described here consists of a silicon varactor diode mounted close to the intersection of a coaxial tee junction where each port separately carries the signal, pump, and lower sideband power. These three frequencies are isolated from one another by making use of some nonideal properties of the coaxial tee junction and by using coaxial filters, which also serve an impedance matching function. The available theory for design of coaxial band-pass filters was found inadequate so an improved synthesis technique was developed to design filters with center frequencies from 1 to 20 GHz and bandwidths from 1 to 20 percent of the center frequency. This theory was then applied to the noniterative design and construction of a LSUC which exhibited 20 dB of power gain. As for the varactor diode itself, it has been known that increased capacitance variation can be obtained if the diode is forward biased for at least part of the pump cycle. This additional capacitance comes from diffusion effects. The theory for diffusion admittance is reviewed and extended to apply to the LSUC. An optimum forward bias level for maximum gain is found both theoretically and experimentally, and beyond this optimum bias, the LSUC performance rapidly deteriorates.

FOREWORD Microwave amplification and frequency conversion by parametric devices is a convenient way of obtaining low-noise performance without having to cool the device to cryogenic temperatures. The lower sideband upconverter (LSUC) is one such device, typically capable of at least 20 dB of stable power gain. This report describes a method of synthesizing a LSUC in coaxial line and an analysis of the varactor as it would be used in a LSUC while forward biased. Design of the upconverter requires band-pass impedance matching circuits which isolate the various frequencies so that only the desired frequency band can propagate through a particular port, and provide specific impedance levels at the diode chip at these various frequencies. A new filter design technique based on the impedance inverter concept is developed in Chapter 11 which eliminates the need for making a lumpedelement approximation. The formulas are simple to use and sufficiently exact so that no additional empirical modification is needed, while the filter itself, which consists of coaxial line and center mounted disks, is easy to make. Both of the above considerations are important in reducing fabrication costs. These filters are then employed in the LSUC design as discussed in Chapter III. However, special consideration must be given to the arrangement of these filters and the diode to provide the appropriate in-band and out-of-band impedances of the three ports. This is accomplished quite simply without recourse to complex iterative processes. Consequently a LSUC can be designed with readily available components using a one-step calculation. Subsequent numerical analysis and experimental work verified the LSUC design technique. The varactor diode itself for analysis purposes is normally assumed to be a reverse biased graded or abrupt pn junction. However, in practice the varactor is often forward biased for at least part of the pump cycle. Here a varactor diode model with both depletion capacitance (dominant under reverse bias) and diffusion admittance (dominant

under forward bias) is examined and applied to a simple LSUC circuit in conjunction with short-circuit assumptions. It is found both experimentally and theoretically that there is an optimum forward bias level for maximum gain. Although increasing the bias still further increases the diode capacitance, additional diffusion losses degrade LSUC performance. This report therefore provides a self-contained synthesis technique for coaxial band-pass filters (Chapter II), a synthesis technique for a LSUC with experimental results (Chapters III and IV), and an analysis of the effects of diffusion admittance in a LSUC (Chapter V). The author wishes to thank the U. S. Army Electronics Command for their support of this work under Contracts DAAB07-67-C-0278 and DAAB07-68-C-0138.

TABLE OF CONTENTS Page ABSTRACT iv FOREWORD LIST OF TABLES x LIST OF ILLUSTRATIONS Xii LIST OF SYMBOLS LIST OF APPENDICES xxix CHAPTER I: INTRODUCTION 1.1 Background 1.2 Literature Review 3 1.3 Topics of Investigation 5 CHAPTER II: COAXIAL MICROWAVE BAND-PASS FILTERS 7 2.1 Introduction 7 2.2 Review of Filter Theory 8 2.2.1 Derivation of the Lumped-Element Low-Pass Prototype Circuit 8 2.2.2 Band-pass Filter Networks 9 2.2.3 Immittance Inverters 10 2.2.4 Microwave Realization 13 2.3 Distributed Impedance and Admittance Inverter Design 17 2.3.1 Coaxial Impedance Inverter Design 18 2.3.2 Coaxial Admittance Inverter Design 22 2.4 Theoretical Results of the Distributed Design Method 23 2.4.1 Band-pass Filters Using K Inverters 23 2.4.2 Filter Using K and J Inverters 30 2.4.3 Impedance Transformer 30 2.5 Effect of Discontinuity Capacitance 32 2.6 Numerical Example and Experimental Results 40 2.7 Discussion 46 2.7.1 Filter Limitations Imposed by Choice of Bandwidth 46 2.7.2 Bandwidth Discrepancy 46 2.7.3 Low-frequency Characteristics 47 2.7.4 Effect of Filter Parameter Variations 49 2.8 Conclusions 49 vi

TABLE OF CONTENTS (Cont.) Page CHAPTER III: DESIGN THEORY FOR THE LOWER SIDEBAND UPCONVERTER 51 3.1 Introduction 51 3.2 Review of Existing LSUC Design Theory 51 3.2.1 Method of Harmonic Balance 52 3.2.2 Elastance Coefficients 55 3.2.3 Gain Relationship for the LSUC 56 3.2.4 Bandwidth and Gain-Bandwidth Product 60 3.2. 5 Noise Figure Relationship for the LSUC 61 3.2.6 Origin of Shot Noise 66 3.2.7 Summary of Existing Design Theory 67 3.3 Effects of Upper Sideband Reactance on the LSUC 68 3.3.1 Effects of X33 on Gain 68 3.3.2 Effects of X3 on Gain Sensitivity 74 3.3.3 Effects of X33 on Gain-Bandwidth Product 78 3.3.4 Effects of X33 on Noise Figure 80 3.3.5 Conclusions on the Effects of X382 3.4 Numerical Comparison of Three Possible LSUC Circuits 85 3.4.1 Diode Model and Desired Circuit Parameters 87 3.4.2 Gain for the Ideal Circuit with Unpackaged and Packaged Diode 88 3.4.3 Comparison of Three Circuit Designs Using Ideal Impedance Transformers 91 3.4.4 Effect of Signal Circuit Length 98 3.4. 5 Effect of Diode Parasitic Inductance 98 3.4.6 Effect of Shunt-Stub Tuners 102 3.4.7 Conclusions on the Use of the Three LSUC Configurations 102 3.5 LSUC Design with Coaxial Band-pass Impedance Matching Circuits 103 3.5.1 General Design Procedure for the Three LSUC Circuits 104 3.5.1.1 Pump Circuit 104 3.5.1.2 Lower Sideband Circuit 106 3.5.1.3 Signal Circuit 107 3. 5. 2 Special Design Considerations for LSUC1 109 3. 5.3 Special Design Considerations for LSUC2 109 3.5.3.1 LSUC2 with the Low-pass Signal Matching Circuit 111 3.5.3.2 LSUC2 with the Band-pass Signal Matching Circuit 111 3.5.4 Special Design Considerations for LSUC3 119 3.6 Conclusions 121 CHAPTER IV: EXPERIMENTAL EVALUATION OF THE LOWER SIDEBAND UPCONVERTER 125 4.1 Introduction 125 4.2 Diode Measurement 125 4.2.1 Diode Measurement Review 125 4.2.1.1 Method 1 127 4.2.1.2 Method 2 128 4.2.1.3 Method 3 129 4.2.1.4 Conclusions on the Three Methods 132 4.2.2 Experimental Diode Measurement Results 133 4.3 External Microwave Circuit Measurements 137 4.3.1 Band-pass Transformers 137 4.3.2 The Tee Junction 141 vii

TABLE OF CONTENTS (Cont.) Page 4.4 The Experimental LSUC Circuits 145 4.4.1 Gain 147 4.4.2 Bandwidth 147 4.4.3 Noise Figure 150 4.4.3.1 Circuit Losses 158 4. 4.3.2 Incorrect Value of Diode Bulk Resistance 1 58 4.4.3.3 Noisy Pump Source 160 4.4.3.4 Pump Heating of Diode 162 4.4.3.5 Shot Noise 1 62 4. 5 Conclusions 167 CHAPTER V: DIFFUSION ADMITTANCE 168 5.1 Introduction 168 5.2 Literature Review 169 5.3 Varactor Diode Model 173 5.3.1 Depletion Layer Capacitance 173 5.3.2 Diffusion Admittance 174 5.3.3 Alternate Formulation 178 5.3.4 Numerical Calculations of the Diffusion Admittance 180 5.3. 5 Varactor Diode Model with both Diffusion and Depletion Admittance 189 5.3.6 Comparison with Experimental Work 193 5.4 Effects of Diffusion Admittance on a LSUC 194 5. 4.1 Large Signal Analysis of the Diffusion Admittance in a LSUC 196 5.4.1.1 Incommensurate Frequencies in the LSUC 196 5.4.1.2 Phase of the Lower Sideband Voltage 197 5.4.1.3 Numerical Calculation of LSUC Currents 199 5.4.2 Small Signal Analysis of the LSUC 200 5.4.3 The LSUC with the Total Varactor Diode Model 209 5.5 Conclusions 218 CHAPTER VI: CONCLUSIONS AND SUGGESTION FOR FUTURE RESEARCH 219 6. 1 Introduction 219 6. 2 Summary of Original Contributions 219 6. 2. 1 Distributed Design of Coaxial Filters 219 6. 2. 2 Discontinuity Capacitance Compensation 219 6. 2. 3 Effect of Higher Order Sidebands in the LSUC 220 6. 2. 4 A Noniterative Design Algorithm for a LSUC 220 6. 2. 5 Verification of Noise Figure Dependence on Gain 220 6. 2. 6 Application of Diffusion Admittance to a LSUC 220 6. 2. 7 Modified Bessel Functions 221 6. 3 Suggestions for Future Research 221 6. 3. 1 Stripline Filters 221 6. 3. 2 A LSUC Made in Stripline 221 6. 3. 3 Double Sideband Operation 221 6. 3. 4 The Coaxial Tee Junction 221 6. 3. 5 Measurement of Diffusion Admittance 222 6. 3. 6 A LSUC with a Step Recovery Diode 222 6.4 Conclusion 223 viii

TABLE OF CONTENTS (Cont.) Page APPENDICES 224 REFERENCES 268 DISTRIBUTION LIST 273 ix

LIST OF TABLES Table Page 2.1 Disk parameters for three pole filters 27 2. 2 Disk parameters for five pole band-pass filters 27 2. 3 Filter parameters for three pole Chebyshev and 28 maximally flat filters where discontinuity capacitance has been neglected and all disk diameters are 0. 502 in. 2. 4 Band-pass filter using K and J inverters 31 2. 5 Disk parameters for band-pass impedance 36 transformer using three disks 2. 6 Experimental design parameters of the filter 42 shown in Figs. 2. 27 and 2. 28 3. 1 Values of X33 where the gain and noise figure 85 extrema occur 3. 2 Distances in cm between the diode and lower 94 sideband (LSB) or pump filters. The * values are not true minima as the slope in these cases is monotonically increasing 3. 3 Gain-bandwidth product when S1/S0 = 0. 35 103 3. 4 Gain and bandwidth of the LSUC2 using coaxial 119 band-pass filters in all three ports 3. 5 Design parameters for the LSUC when the diode 121 is in the lower sideband port 4. 1 References for measuring diode parameters 132 4.2 Noise figure of the LSUC 158 5. 1 Variation of the minimum loss with doping profiles 189 5. 2 Exact calculation of the current desnities 201 5. 3 Current densities caused only by nonlinear dep- 202 letion capacitance when normalized to give the same pump current IJ0 1 as the diffusion current 5. 4 Small signal method for calculating the current 203 densities

LIST OF TABLES (Cont.) Table Page 5. 5 Susceptance values in mhos needed to tune the 208 diffusion susceptance of the diode when it is biased at 0. 7 V D. 1 Laplace asymptotic method for calculating the 267 current densities xi

LIST OF ILLUSTRATIONS Figure Page 2. 1 Low-pass prototype circuit with shunt capacitor 10 adjacent to source 2. 2 Low-pass prototype circuit with series inductor 10 adjacent to source 2. 3 Band-pass prototype with shunt resonator adjacent 11 to the generator 2. 4 Band-pass prototype with series resonator adjacent 11 to the generator. The element values are given by the same formulas given in Fig. 2. 3 2. 5 Definition of an ideal K inverter and J interter 12 2. 6 Lumped band-pass impedance matching network 12 using K inverters 2. 7 Lumped band-pass impedance matching network 12 using J inverters 2. 8 Generalized impedance matching networks using 14 K inverters 2. 9 Generalized impedance matching networks using 14 J inverters 2. 10 Alternate K and J inverters 16 2. 11 Lumped-element K inverter realization 16 2. 12 Lumped-element J inverter realization 16 2. 13 (a) Disk impedance inverter, and (b) bisected 19 impedance inverter 2. 14 (a) Coaxial admittance inverter, and (b) bisected 22 admittance inverter 2. 15 A schematic diagram of a coaxial band-pass filter 24 with Teflon dielectric around the center two disks 2. 16 Theoretical comparison between the lumped and 25 distributed design for filters with bandwidth of (a) 1 percent, (b) 5 percent, and (c) 10 percent. The impedance of the 10 percent filter is shown in (d) xii

LIST OF ILLUSTRATIONS (Cont.) Figure Page 2. 17 Theoretical comparison between the lumped and 26 distributed design for 5 pole filters with bandwidth of (a) 5 percent and (b) 10 percent 2. 18 10 percent bandwidth band-pass filter with 0. 5 dB 29 ripple 2. 19 10 percent bandwidth, three pole, band-pass, maxi- 29 mally flat filter 2. 20 Band-pass filter with alternate K and J inverters 32 2. 21 Filters using alternate K and J inverters separated 33 by quarter wavelength lines with bandwidth of (a) 20 percent, and (b) 40 percent 2. 22 Theoretical comparison between the lumped and distri- 34 buted design for 50:1 impedance transformers with bandwidth of (a) 1 percent, (b) 5 percent, (c) 10 percent, and (d) 20 percent 2. 23 Theoretical comparison of the impedance between 35 the lumped and distributed design of the 10 percent bandwidth 50:1 impedance transformer 2. 24 (a) Length of transmission line representing disk, 39 and (b) length of transmission line representing disk with discontinuity capacitance 2. 25 7r equivalent circuit for a transmission line where 39 Xb= Z sin (jf) and Ba = Yo tan (j3//2) 2. 26 A 1 percent bandwidth band-pass filter for the ideal 41 case when there is no discontinuity capacitance (...), when the filter is compensated for discontinuity capacitance by method 1 ( — ), method 2 ( ---), and method 3 ( — ). The ideal case and the compensation by method 3 merge at the left side 2. 27 A 1 percent bandwidth band-pass filter designed by 43 the lumped method (- -), by the distributed method ( ---), by the distributed method with discontinuity capacitance accommodated by method 3 (-), and experimental results of the solid line filter (xxx). The ideal case when there is no discontinuity capacitance ( --- ) and the case where method 3 is used ( —) merge for the lower frequencies 2. 28 A 10 percent..band-pass filter designed by the distri- 44 buted method where discontinuity capacitance is accommodated by method 2 with experimental results (x). The insert is a portion of the characteristic drawn on an expanded scale. 2. 29 Photograph of 1 percent band-pass filter 45 xiii

LIST OF ILLUSTRATIONS (Cont.) Figure Page 2. 30 Low-frequency response of band-pass filter with 48 center frequency of 8. 5 GHz for bandwidth of (a) 1 percent, (b) 10 percent, (c) 20 percent, and (d) 10 percent bandwidth with modified diameters 3. 1 Noise equivalent circuit 63 3. 2 Transformed noise equivalent circuit 63 3. 3 Equivalent circuit of LSUC where Xc is the circuit 70 reactance and X33 is finite 3. 4 The impedances induced into the signal circuit by 72 the finite value of X33 3. 5 Midband transducer power gain when S2/S0 = 0. 044 76 3. 6 The transducer power gain when S2 = 0 77 3. 7 The gain-bandwidth product (G2w) when 79 S2/SO = 0. 044 where the dashed curve corresponds to the expanded X33 scale 3. 8 The noise figure when S /S = 0. 044 where the 83 dashed curve correspond2s to the expanded X33 scale 3. 9 The actual noise measure of the LSUC when 84 S2/S0 = 0. 044 3. 10 Drawings of LSUC1, LSUC2, and LSUC3 together 86 with the corresponding schematics of the circuits are shown in (a), (b), and (c) respectively. do, dl, and d2 are the distances between the diode and the corresponding impedance transformers respectively. Zs, Zp, and Zp-s are the impedances of the signal, pump, and lower sideband ports as seen by the diode 3. 11 Package varactor diode model 88 3. 12 LSUC transducer power gain with out-of-band circuits 89 completely isolated and the diode has no parasitic elements 3. 13 Variation of gain-bandwidth product (G2w) with 90 pumping level when out-of-band circuits are completely isolated and there are no diode parasitics, and Robinson's gain-bandwidth formula xiv

LIST OF ILLUSTRATIONS (Cont.) Figure Page 3. 14 LSUC transducer power gain with out-of-band circuits 92 completely isolated where the diode parasitic elements are as follows: L = 2 nH and S = 2. 856 1012 F-1 -, L= 1nH and S 1. 428 112 F-1 ---,- L 1 nH and SPp= 2. 856 1012 F-1 p 3. 15 LSUC gain-bandwidth product (G2w) with out-of-band 93 circuits completely isolated from one another where the diode parasitic elements are as follows: L = 2 nH and Sp= 1.428 1012 F-1, L = 2 nH and Sp = 2.856 1012 F-1 L = 1 nH and Sp =2. 856 1012 F --- —3. 16 Transducer power gain for the three LSUC circuits 96 when ideal impedance matching circuits and the 5 x 5 matrix approximation are used 3. 17 Noise figure for the three LSUC circuits when ideal 97 impedance matching circuits and the 5x 5 matrix approximation are used 3. 18 Variation of gain-bandwidth product (G2w) versus 99 pumping level with signal circuit distance as a parameter, using ideal impedance matching circuits, the 5 x 5 matrix approximation, and standard diode inductance of 2 nH 3. 19 Variation of gain bandwidth product (G2w) versus 100 pumping level with signal circuit distance as a parameter, using ideal impedance matching circuits, the 5 x 5 matrix approximation, and a diode package inductance of 0. 2203 nH 3. 20 Variation of gain-bandwidth product (G2w) with 101 diode package inductance for LSUC1, using the 5 x 5 matrix approximation and a signal port distance of X /4 3. 21 Pump circuit without (a) and with (b) ideal transformer 106 3. 22 Circuit model used to design the signal circuit 108 3. 23 Gain of LSUC when all filters in parallel with one 110 another 3. 24 LSUC1 circuit using a semilumped low-pass signal 112 circuit 3. 25 Gain of LSUC:-using coaxial filters with diode in 113 center conductor 3. 26 LSUC1 circuit using a distributed band-pass signal 114 circuit xv

LIST OF ILLUSTRATIONS (Cont.) Figure Page 3. 27 LSUC transducer power gain using the coaxial 115 microwave filters when the diode model has the following elements: Rs = 1 ohm, L = 2 nH, Cp= 0.7pF, S= 6.28 1011 F-1, and S /S0 = 0. 35 3. 28 LSUC transducer power gain using the coaxial 116 microwave filters when the diode model has the following elements: Rs = 1 ohm, L = 2 nH, Cp= 0. 5pF, SO= 6.28 1011 F-1, and S1/So = 0. 35 3. 29 LSUC transducer power gain using the coaxial 117 microwave filters when the diode model has the following elements: Rs = 1 ohm, L = 1. 428 nH, C p= 0.7pF, S0= 6.28 1011 F-1 and S1/S0 = 0. 35 3. 30 LSUC noise figure using the coaxial microwave 118 filters when the diode model has the following elements: Rs = 1 ohm, L = 2 nH, C = 0. 7 pF, S = 6. 28 1011 F-1, and S1/S0 =. 0.35 3. 31 LSUC3 using coaxial disk band-pass impedance 122 transformers 3. 32 LSUC transducer power gain using coaxial micro- 123 wave filters when the diode is mounted in the lower sideband port. The diode model elements are Rs =1.41 ohm, L= 1.834nH, Cp=0.1633pF, S = 1. 0456 1012 F-1, and S1/S0 = 0.35 4. 1 (a) The varactor diode equivalent circuit and (b) 127 the modified simple equivalent circuit 4. 2 Distributed diode model 134 4. 3 Houlding's varactor diode model 134 4. 4 Diode measurement mount with (a) the varactor 135 diode and (b) the short-circuit reference plane 4. 5 The measured diode impedance data points are 136 designated by o and the selected points obtained from analysis of the resulting diode model are designated by x 4. 6 Diode measurement circuit 137 4. 7 Comparison of the reactance for the measured 138 diode ~ ~ ~ and the diode model o o o 4. 8 Impedance of the signal frequency transformer 139 xvi

LIST OF ILLUSTRATIONS (Cont.) Figure Page 4. 9 Impedance of the lower sideband frequency trans- 140 former 4. 10 The impedance of the coaxial tee circuit as seen 142 from (a) the leg port, and (b) the arm port 4. 11 The impedance as a function of frequency in GHz 143 of a 7 mm coaxial tee junction as seen from the leg Za, -—, and the impedance as seen from the arm Zb,, when the other two ports are terminated with 50 ohms 4. 12 (a) The coaxial tee junction and (b) the equivalent 144 circuit of the tee junction at the reference plane 4. 13 Schematic diagram of the experimental LSUC2 146 including the final experimental dimensions. The transmission line is standard 14 mm 50 ohm coaxial line and the cross-hatched sections are Teflon 4. 14 Photograph of the LUSC 148 4. 15 Photograph of the LSUC when in operation 148 4. 16 Experimental power gain of LSUC3 when the midband 149 gain is 30 dB (A), 25 dB (o), 20 dB ([3), 15 dB (O), 10 dB (+), and 5 dB (X) 4. 17 Theoretical gain curves for LSUC3 when the circuit 151 is modified because of physical requirements 4. 18 Comparison of theoretical and experimental 152 gain-bandwidth product 4. 19 Noise figure measuring circuit 153 4. 20 Photographs of the noise figure measuring circuit 154 4. 21 Noise figure of LSUC2 156 4. 22 Noise figure of LSUC3 157 4. 23 Theoretical noise figure for LSUC3 159 4. 24 Thermal plus shot noise figure of a LSUC at 2900~K 163 when the diode is at 290 K and 300 K. The dashed curves have slopes 10 and 15 times larger than the noise figure curve 4. 25 Noise figure of a LSUC as a function of bias voltage 165 when the gain and pump power is constant xvii

LIST OF ILLUSTRATIGNS (Cont.) Figure Page 4. 26 Varactor diode dc current-voltage characteristics 166 with no RF power (o), with 71 mW RF (o), and with 0. 71 W (x). The applied RF is 9. 5 GHz, and the diode is series mounted in a terminated 14 mm coaxial line 4. 27 Mount for measuring diode current-voltage 167 characteristic under pumped conditions 5. 1 Normalized diffusion admittance when y = 1/2, 182 7' = 10- sec, and W/L = 0.1 5. 2 Normalized diffusion admittance when y = 1/3, 182 T = 10- sec, and W/Lp= 0.1 p p 5. 3 Normalized diffusion admittance when y = 1/ 5, 183 T = 10-7 sec, and W/L = 0.1 P P 5. 4 Normalized diffusion admittance when y = 1/3, 183 T = 10-7 sec, and W/L = 0.01 P P 5. 5 Normalized diffusion admittance when y = 1/2, 184 T =10-5sec, and W/L = 0.1 P P 5. 6 Normalized diffusion admittance when y = 1/3, 184 T = 10-5sec, and W/L = 0.1 P p 5. 7 Normalized diffusion admittance when y = 1/2, 185 T = 10-10 sec, and W/L = 0.1 P P 5. 8 Relative loss of an abrupt and graded junction 187 diode under forward bias conditons 5. 9 Relative loss of diffusion admittance with doping 188 profile N(x) = ax3 5. 10 Total varactor model of diode wafer 190 5. 11 The depletion and diffusion susceptance when the 191 minority carrier lifetime is chosen to minimize loss in the graded junction diode, i. e., T= 1.84 x 10-'10sec 5. 12 The impedance of the total varactor model when 192 the minority carrier lifetime is chosen to minimize loss in the graded junction diode 5. 13 The total diode-susceptance for T = 10 sec and 195 T = 1. 84 x 10-10 sec (minimum loss) when the bulk series resistance Rs = 1 ohm 5. 14 LSUC circuit model for the diffusion admittance 197 xviii

LIST OF ILLUSTRATIONS (Cont.) Figure Page 5. 15 A comparison of the sideband currents generated 203 by the nonlinear diffusion admittance and the nonlinear depletion capacitance when V = 0.01 V and V2= -0.1 V 5. 16 Maximum gain as function of bias voltage with 212 pump voltage = 0. 1 V when the optimum lifetime is used for both doping profiles. In each case 0v - 0.9 V, Si 102. 20 x 1011 W/L = 0. 1, and Pn(W) 1010 cm-3 5. 17 Noise figure as a function of bias voltage with 213 pump voltage = 0. 1 V when the optimum lifetime is used for both doping profiles. In each case 0v = 0. 9 V, S1 =2.20x 1011 F-1, W/L= 0.1, and pn(W)= 1010 cm-3 5. 18 Maximum gain as a fucntion of pump voltage at 214 zero bias voltage when the optimum lifetime is used for both doping profiles. In each case v = 0. 9V, S1 = 2. 20x 1011 F-1, and pn(W) = 1010 cm-3 5. 19 Noise figure as a function of pump voltage at zero 215 bias voltage when the optimum lifetime is used for both doping profiles. In each case v = 0. 9 V, S1 = 2. 20 x 1011 F-1, W/L = 0.1, and Pn(W) = 1010 cm-3 5. 20 Experimental gain of a LSUC as a function of bias 217 voltage on the varactor xix

LIST OF SYMBOLS Defined By Or Symbol Meaning First Used In A Diode area (5.22) A Voltage- current transmission matrix parameter Sec. 2.2.1 A k+m Terms used in recurrence relation (C. 2) a Parameter in diode doping concentration profile (3. 9) a Inner radius of coaxial line (2. 36) a = V0/(Vb- v) (5.41) ak Coefficients in infinite power series (C. 1) a Resistance slope Sec. 3.2.3 r a Reactance slope Sec. 3.2.3 x B Susceptance (2.9) B Voltage-current transmission matrix parameter Sec. 2.2.1 B Hole diffusion susceptance Sec. 5.2 p B Bandwidth (3.22) w B.. Admittance matrix susceptance for coaxial tee 1] junction Sec. 4.4o 3 b Susceptance slope parameter (2.9) b Exponent for diode profile power law (3. 9) bk Polynomial coefficient (C. 3) b Outer radius of coaxial line (2. 36) b' = (b+1)/2 Sec. 5.3.2 b" = (b-1)/2 Sec. 5.3.2 C Voltage- current transmission matrix parameter Sec. 2.2.1 Cb Zero bias varactor junction capacitance (5.6) XX

LIST OF SYMBOLS (Cont.) Defined By Or Symbol Meaning First Used In Cd Discontinuity capacitance (2.36) Cf Diode fringing capacitance Fig. 4.1 Cj Diode junction capacitance Fig. 401 Cp Band-pass filter capacitance Fig. 2.3 C Diode package capacitance Sec. 3.4.1 C.. Admittance matrix capacitance for coaxial tee junction Sec. 4.4.3 Cr. i Filter capacitance (2.7) c Velocity of light in a vacuum Sec. 20 6 D Voltage- current transmission matrix parameter Sec. 2.2. 1 Dp, D Diffusion constant for holes, electrons (5.5) d Transmission line length (2.32) d' = (w - w )/w Sec. 2.7.2 O O d Distance between waveguide adapter and last disk Fig. 3.31 E Electric field (5.12) E' Filter parameter (2.32) F Noise figure Sec. 4.4.3 F Average noise figure (4.9) F' Filter parameter (2.35) F. Noise figure of stage i Sec. 4.4.3 Fm Noise figure at output frequency w Sec. 3.2.5 FAM Additional AM upconverter noise from pump source Sec. 4.4.3.3 F Additional FM upconverter noise from pump FM source Sec. 4.4.3.3 fa Diode series resonant frequency Sec. 4.2.1.3 fb Diode parallel resonant frequency Sec. 4.2.1.3 f Diode cutoff frequency Sec. 4.2.1.3 ff Fundamental frequency Sec. 5.4.1.1 xxi

LIST OF SYMBOLS (Cont.) Defined By Or Symbol Meaning First Used In f fp' f Signal, pump, and lower sideband frequency s' pp-s G. Conductance shunting diode junction capacitance Fig. 4.1 G LSUC generator conductance Sec. 5o4.2 g Ge LSUC load conductance Sec. 5.4.2 G Transducer power gain with input at w and output at wm (3.11) Go Midband transducer power gain (4.11) GBW Voltage gain-bandwidth product (3.21) gi Lumped low-pass prototype circuit values Fig. 2.1 g', g', dc, cosine, and sine Fourier components of eaV Sec. 5.3.2 0 mc ms g, gn Hole, electron generation term (5.9), (5.10) H Coefficient of Fourier expansion of depletion layer capacitance (5.40) h(x) Function of x (D. 1) In (x) Modified Bessel Function i0 dc diode current (3.22) ik Current component at ok (3-.8) i Diode reverse saturation current Sec. 3.2.5 s < i2> Mean, square, fluctuation noise current (3.22) J, Ji i+1 Admittance inverter Fig. 2.5 J Current density Sec. 5.3.2 J ' Jn Current density due to holes, electrons (5.7), (5.8) J mc Jms dc, cosine, and sine Fourier components of J(t) (5.20), (5.21) J Current density at lower sideband frequency Sec. 5.4. 1.2 p-s J Small signal current density Sec. 5.2 ss K, K. Impedance inverter Fig. 2.5 K =1121 (3.15) K (z) Modified Bessel Function Sec. 5.3.2 k Boltzmann's constant (3.22) xxii

LIST OF SYMBOLS (Cont.) Defined By Or Symbol Meaning First Used In L Diode lead inductance Sec. 3.4.1 L2 Attenuation of second stage Sec. 4.4.3 L Admittance matrix inductance for coaxial tee junction Sec. 4.3.2 Le Band-pass filter inductance Fig. 2.3 Lp, Ln Diode hole, electron diffusion length (5.5) p n L i Filter inductance (2.7) r, 1 LSUC Lower sideband upconverter Sec. 1.1 LT Transforming inductance of diode package Fig. 4.3 Transmission line length Fig. 2.13 M Noise measure Sec. 3.3.4 Ma, Mb, etc. arbitrary constants mk Subscript for currents, voltages, etc. (5. 27) ms = S /S -C/C0 (537) N(x) Diode impurity concentration on the n side (3. 9) Nd, Na Diode donor, acceptor density (5.11) n Minority electron density Sec. 5.2 n' Transformer turns ratio Sec. 3.5 n Parameter in exponent of Shockley diode equation Sec. 5.3.4 n Equilibrium minority electron density Sec. 502 6(x) Order symbol (4.11) P Power Sec. 305.1.1 P Power at frequency mf + nf (3.1) mn p s PLR Transducer power loss ratio Sec. 2.2.1 p Minority hole density Sec. 5.2 Pn Equilibrium minority hole density Sec. 5.2 PWO mc, pms dc, cosine, and sine Fourier components of hole density Sec. 5.3.2 Q Charge (5.3)

LIST OF SYMBOLS (Cont.) Defined By Or Symbol Meaning First Used In Q Normalized charge on varactor Sec. 50 2 QA Resonated circuit Q Sec. 2.4.3 QB Charge on varactor at V = VB Sec. 5.2 QC Circuit quality factor Sec. 402.1.2 QV 0Charge on varactor at V = 0 v Sec. 5.2 Qij Varactor dynamic quality factor (3.12) q Electronic charge = + 1. 602 x 10 19 coul. (3.22) qb dc bias charge (5.3) q Charge at pump frequency Sec. 3.2.1 qs Charge at signal frequency Sec. 3.2.1 qs Signal plus sideband charge Sec. 3.2.1 qt Total charge (3.3) qu Charge at fp + fs (5.3) R Resistance Sec. 2.2.1 R Generator resistance Sec. 2.2.1 g Rgn LSUC generator resistance at wn (3.11) Rf, Rfm LSUC load resistance at wm (3.11) R Diode noise resistance Sec. 3.2o 5 x Rer Equivalent noise resistance at wr Sec. 3.2.5 Rind Resistance induced into the pump circuit (3. 41) Rs Diode spreading resistance Sec. 3.2.3 R ' Diode input resistance at signal frequency (3. 44) s RPA Reflection parametric amplifier Sec. 1.1 r Ratio of lower sideband to signal frequencies (3.17) r Radius of coaxial disk (2.36) r Critical r (3.30) c ropt Optimum r (3.31) xxiv

LIST OF SYMBOLS (Cont.) Defined By Or Symbol Meaning First Used In S(t) Diode elastance (3.3) Sb Zero bias elastance (3.10) SL Load SWR (4.4) S Measured SWR (4.4) m S Maximum SWR (4.4) max S Fourier component of diode elastance (3. 3) S Diode package elastance Sec. 3.4.1 T Absolute temperature (3.22) T Ambient temperature Sec. 3.2.5 Td Diode temperature Sec. 3.2.5 T Chebyshev polynomial of first kind (2.3) T * Shifted Chebyshev polynomial of first kind Appendix C t Time (3.5) USUC Upper sideband upconverter Sec. 1.1 U. Partial sum (C. 13) u Function of x Sec. 5.3.2 un Polynomial of degree n (C. 3) V(t) Voltage (3.10) V Normalized voltage on varactor Sec. 5.2 VB Diode breakdown voltage Sec. 5.2 Vb dc bias voltage Sec. 5.2 Vk Voltage at v k (3. 6) v Pump voltage Sec. 3.2.1 v ' Signal plus sideband voltages Sec. 3.2.1 vt Total voltage Sec. 3.2.1 W Diode depletion layer width Sec. 5.3.2 w Fractional bandwidth Sec. 2.2.2 xxv

LIST OF SYMBOLS (Cont.) Defined By Or Symbol Meaning First Used In Wk Weight function (C. 16) X Reactance Sec. 2.2.1 X.. Reactance matrix element (3.8) x Distance variable (3.9) x = x (Sec. 5.3.2 x Normalized generator resistance (3.14) x Normalized effective generator noise resistance (3. 24) Y, YO Characteristic admittance (Sec. 2.3.2 or Fig. 2.14) ~YI Image admittance Sec. 2.3.2 Y Admittance matrix element (3.11) mn yI Normalized image admittance Sec. 2.3o2 ye Normalized load resistance (3.15) y Normalized characteristic admittance Sec. 2. 3 2 Ys Normalized effective load noise resistance Sec. 3.2. 5 Zc' Z Characteristic impedance Fig. 2.13 Zd Diode impedance Sec. 3.5.1.3 ZI Image impedance (2.22) Z. Impedance of the lower sideband circuit at w20 1 required to make Z22 = R + R Sec. 3.4. 3 22 f S Zk Iteration characteristic impedances (2.41) Zkk Circuit impedance at frequency wk as seen by the diode Sec. 3. 2 1 Zs, Zp' Z LSUC circuit impedances at signal, pump, and lower sideband frequencies Fig. 3.10 Zt Impedance of the signal circuit at w10 required to make Z Rg +R Sec. 3.4.3 Zg -t R s Sec. 3.4.3 Z Impedance Sec. 3.2.3 ZvO Impedance Zv at w10 Sec. 3.2.3 z Complex variable Appendix C xxvi

LIST OF SYMBOLS (Cont.) Defined By Or Symbol Meaning First Used In ZI Normalized image impedance (2.24) z Normalized characteristic impedance (2.21) Zr = 1/z Appendix C a -= q/kT Sec. 5.2 fa Coaxial disk parameter i (2.36) am MDiffusion admittance parameter (5.19) Parameter used in gain-bandwidth expression (3. 20) a Modified parameter used in gain-bandwidth exw pression (3.42) Wave number (3. 20),M Diffusion admittance parameter (5.19) r Voltage reflection coefficient Sec. 2.2.1 r (x) Gamma function (5.40) y Doping Profile parameter (3.10) Decrement Sec. 2 4.3 5.. Kronecker Delta function Sec. 5.4.2 Permittivity Sec. 5.3.5 ee Error term (C.4) eo Free space permittivity (2.36) 0 Electrical distance (2.23) o0 Electrical distance between diode and pump filter (3. 42) 0 s Electrical distance between diode and signal filter (3. 43) K Constant coefficient (5.42) X Wavelength Sec. 3.4.4 Up, 'An Hole, electron mobility (5.7), (5.8) v Order of Bessel function (not necessarily integer) Appendix C = (1+ jmw p)/Lp (5.15) p Charge density (5.12) xxvii

LIST OF SYMBOLS (Cont.) Defined By Or Symbol Meaning First Used In dS ak k (3.37) dQk Q0 T Parameter in Lanzco's tau method (C.4) T Coaxial disk parameter (2.36) T Tn Hole, electron minority carrier lifetime (5.5) p n u Pass-band tolerance (2.2) Electrical length Fig. 2.12 0v Built-in potential (3.10) v X Reactance slope parameter (2.8) Range parameter Appendix C Electrical phase angle (5.3) wL, Radian frequency Sec. 2.2.1 a' Low-pass frequency variable Sec. 2.2.2 wc Filter cutoff radian frequency Sec. 2.2.1 ' Normalized filter cutoff radian frequency Sec. 2.2.2 c wf Fundamental radian frequency (5.28) Wk Radian frequency for k = 0, 1, 2... (3.7) WO Filter center radian frequency Sec. 2.2.2 w0 Pump radian frequency (5. 23) wp' ws Pump, signal radian frequency (3.1)

LIST OF APPENDICES Page APPENDIX A: PROGRAM LISTINGS 224 APPENDIX B: DETERMINATION OF PARAMETRIC CONVERTER 246 C IRCUIT IMPEDANCES APPENDIX C: CALCULATION OF Kv(z) BESSEL FUNCTIONS 255 WITH COMPLEX ARGUMENT APPENDIX D: LAPLACE ASYMPTOTIC METHOD OF INTEGRATION 265 xxix

CHAPTER I INTRODUCTION 1.1 Background In recent years much effort has been concentrated on the development of microwave receivers in response to increased activity in satellite communication, manned space flight, telemetry, radar, and radio astronomy. There are several low-noise microwave amplifiers available today, each having its own advantages and disadvantages; among these, listed in order of decreasing noise figure, are the traveling-wave tube, transistor, tunnel diode, parametric amplifier, and maser. Modern, ultra-low noise, traveling wave tube amplifiers are available today with a noise temperature as low as 4000K at 1 GHz, although most amplifiers of this kind are much noisier. At this same frequency noise temperatures range between 400~K and 800~K for transistor amplifiers, between 3000K and 4000K for tunnel diode amplifiers, between 800K and 1800K for uncooled parametric amplifiers, between 40~K and 65 K for parametric amplifiers cooled to 770K, between 100K and 300K for parametric amplifiers cooled to 200K, and between 50K and 100K for masers (Ref. 1). In applications where an ultra-low noise preamplifier is required, the choice is generally limited to either the parametric amplifier or the maser. The maser today offers the ultimate in low-noise temperature, although a cooled parametric amplifier has been built with a system noise temperature of 12.50K at 4 GHz, a value which is competitive with the maser at this frequency (Ref. 2). The maser has the advantages of not requiring a bias voltage and of being unaffected by changes in pump amplitude and frequency. However, usually the maser must be cooled to 40K for successful operation while the parametric amplifier may be operated at room temperature. Furthermore, the maser is very temperature-sensitive, has a long recovery time when overloaded (on the order of microseconds as compared to nanoseconds for the parametric amplifier), and has a narrower bandwidth than the parametric amplifier.

This thesis is concerned with the study of parametric devices. The parametric principle, which forms the basis for these devices, can be stated as follows: under certain circumstances, the energy of an oscillating system can be increased by supplying to the system energy the frequency of which differs from the oscillation frequency. The term "parametric" comes from the need to vary a system parameter to obtain gain. This principle was discovered over a century ago; Faraday (1831), Melde (1859), and Lord Rayleigh (1883-1896) all published calculations and observations on the principle as applied to mechanical systems. The most common form of electronic parametric device is the reflection parametric amplifier (RPA). In this device a high-power, high-frequency pump signal is reactively mixed with the incoming low-frequency signal. The resulting lower sideband signal is supported in a resonant circuit while the upper sideband signal is rejected. The lower sideband frequency mixes again with the pump frequency, providing an additional component to the low-frequency signal. The double reactive mixing process introduces a 1800 phase shift between the incoming and outgoing signal frequency, with the result that the device appears as a negative resistance. The upper sideband upconverter (USUC) suppresses the lower sideband and takes the output at the upper sideband frequency. The upconverter is a positive resistance device in which the power gain for the ideal lossless reactance is limited to the ratio of the output to the input frequencies (Manley-Rowe gain). The lower sideband upconverter (LSUC) suppresses the upper sideband signal and provides an output at the lower sideband frequency. The LSUC, like the RPA, is a negative resistance device; however, the gain of this amplifier includes a component due to the ratio of the output frequency to the input frequency. Within the family of parametric devices, a comparison between the widely used RPA and the LSUC shows the following differences: (1) Although the noise figures for both devices are identical at infinite gain, they differ for finite gain: the noise figure increases with gain in the RPA, but decreases with gain in the LSUC. (2) For gain below a threshold value, the gain sensitivity to pump amplitude variations is lower in the LSUC, since only part of its gain is derived from

the negative resistance. Above this gain value the RPA exhibits greater stability (Ref. 3, pp. 214-215). (3) The gain bandwidth product is larger for the LSUC than for the RPA, by a factor approximately equal to the square root of the output to input frequency ratio. (4) For optimum performance, the RPA requires a circulator at the input port while the LSUC requires an isolator; this provides an advantage to the LSUC since a circulator generally has higher insertion loss than an isolator. (5) The input and output of the RPA are at the same frequency, whereas the output of the LSUC is at a higher frequency. This is usually not a serious drawback since in both cases the signal must be downconverted, and the mixer loss and noise figure are independent of the frequency ratio. The aim of this thesis is to extend and develop further the design theory for the lower sideband upconverter, and to facilitate exploitation of the relative advantages of the LSUC, which have been enumerated above. 1.2 Literature Review Capacitive parametric amplifiers have been used successfully only since the advent of high-Q varactor diodes. The investigation into the properties of solid-state diodes necessary for parametric amplification can be considered to have been initiated in the mid-1940's by workers at General Electric, together with Torrey at the MIT Radiation Laboratory. They discovered that certain welded- contact mixer diodes exhibited negative IF conductance and gain, although noise figure was found to be higher than for other mixer diodes. The primary cause for this phenomenon was found to be the variable barrier capacitance of the diode (Ref. 4, Chapter 13). In 1948 A. van der Ziel (Ref. 5) derived relationships for the gain and impedances in terms of the conversion transconductance between the input and output, and thus generalized Torrey's work by giving a more unified treatment of the variable capacitance amplifier. In 1956 Manley and Rowe (Ref. 6) published their now famous work in which they derived the general power-frequency relationships for a nonlinear reactance. Practical application of this theory was dependent upon the development of a low-loss variable-capacitance diode (or varactor). Work on

this began at Bell Telephone Laboratories in 1954 and the successful results were published in 1959 (Ref. 7). The advent of low-loss varactor diodes motivated researchers to improve the theory and design of parametric amplifiers; many of the papers of that period are reviewed by Mumford (Ref. 8). Notable among these is the paper by Rowe (Ref. 9) who used the small-signal assumption and the method of harmonic balance to obtain expressions for gain, bandwidth, terminal admittances, and gain sensitivity to change in pump-power or terminal conductance for the RPA, the USUC, and the LSUC. Later Kurokawa and Uenohara (Ref. 10) derived design relations for achieving maximum gain (for the USUC) or minimum-noise figure for these parametric devices; this work was extended by Khan (Ref. 11) by eliminating the high-gain assumption which they had used. This extended theory has been found useful in the design of a LSUC and is discussed in more detail in Chapter III. Although historically the LSUC has not been as widely used as the other parametric devices, its special characteristics were found to be essential in a preamplifier designed for Project Echo (Ref. 12). The results reported in many of the later papers have recently been published in book form (Refs. 13, 14), providing an excellent introduction to the subject. Although much fundamental work has been carried out, parametric device theory remains in an incomplete form. Several deficiencies in the theory hinder designers in their attempts to apply these results directly to the synthesis or analysis of a parametric device. These inadequacies include the following. (1) The pumped varactor diode is represented by a sinusoidal time-varying elastance (or capacitance), while in a physical circuit it produces many harmonic components of the time-varying elastance. (2) The circuit is assumed either to short-circuit the higher order harmonic voltages or to open-circuit the higher order harmonic currents, so that there remains power flow at only two frequencies: that of the signal and that of either the upper sideband or the lower sideband. However, the impedance at a third frequency (such as the unwanted sideband) may drastically alter the performance of the device under study.

(3) The average varactor elastance (or capacitance) is assumed to be resonated by a single-tuned lumped circuit. However, in a physical microwave circuit the diode is tuned by a complex combination of lumped and distributed circuit elements resulting from the diode parasitic elements and the microwave transmission line. (4) Because of the small-signal assumption, the pump circuit elements are usually assumed to be completely isolated from the rest of the circuit. Physically the pump port must be considered part of the total circuit. (5) There is no adequate representation of the circuit behavior of a pumped forward-biased varactor in a parametric device. These assumptions have been made to describe, to a first approximation, the observed parametric effects; however, their effect has been to oversimplify the analysis to such an extent that application of the existing theory to the design of a parametric device must usually be supplemented by empirical tuning adjustments and circuit modifications. 1.3 Topics of Investigation The primary goals of this study are to refine the analysis, and to develop further the design theory for the varactor LSUC. With the aid of the studies reported by Oliver (Ref. 15), the effect of elastance harmonics, and hence higher order harmonic voltages and currents, may be included in the analysis. With an accurate synthesis technique and an accurate representation of the microwave circuit, the circuit environment in which the diode is embedded may be analyzed. Furthermore, by incorporating the diffusion admittance in the diode model, the effects of forward-biasing the diode in a LSUC can be found. The following chapters of this thesis describe how the goals of this study are achieved. In the course of the work, it was found necessary to design and build microwave impedance transformers in coaxial line, so that the required impedance transformation would be obtained as close to the diode as possible. Although much fundamental study has been carried out on microwave filters in the past, the existing theory had to be extended to achieve reasonable correlation between theory and experiment. It is now possible to design coaxial band-pass filters and impedance transformers with little or no experimental adjustment by use of the theory presented in Chapter II.

In Chapter III a detailed review of existing LSUC design theory is presented. This is followed by an extension to include the effect of the impedance at the upper sideband frequency, and a discussion of the effect of various circuits on the gain-bandwidth product of the LSUC. The chapter culminates with a realizable design of a microwave LSUC for which the experimental results are given in Chapter IV. Many varactor parametric amplifiers operate best when they are biased so that the diode is in the forward- conduction region during part of the pump cycle. This permits larger susceptance variation, which in turn may improve the gain stability and decrease the minimum noise figure. When the diode is forward-biased, the principal contribution to the capacitance is the diffusion susceptance. The theory for the diffusion admittance as it applies to a LSUC is outlined and extended in Chapter V. Finally, Chapter VI provides conclusions and recommendations for future work.

CHAPTER II COAXIAL MICROWAVE BAND-PASS FILTERS 2. 1 Introduction The design of a microwave lower sideband upconverter (LSUC) requires bandpass impedance transformers, to prevent unwanted frequencies from propagating in the signal, lower sideband, and pump ports, and to present to the diode the correct design impedance. Since the behavior of these impedance transformers has such a crucial effect on the operation of a LSUC, a detailed investigation of microwave filter design has been undertaken, and is reported here. Most of the theory developed here applies to filters terminated at both ends with loads equal to the characteristic impedance of the line; however it is shown how this theory may be modified for the design of impedance transformers in which the terminations are unequal. At low frequencies a very general and complete filter synthesis technique has been developed, utilizing lumped inductors and capacitors as the basic building blocks. However, filter synthesis is much more complicated at microwave frequencies since distributed-parameter elements must be used. Although these distributed elements often have a complex frequency dependence, the lumped techniques developed for low frequencies provide an invaluable guide to microwave filter synthesis. The following discussion presents the necessary equations for the design of band-pass filters in coaxial line. In Section 2.2, the conventional lumped-element filter synthesis theory is reviewed briefly; a lumped low-pass prototype circuit is developed, is transformed into a band-pass circuit, and is reformulated in terms of lengths of transmission line and impedance inverters. A new method of designing these impedance inverters is presented in Section 2. 3 and is shown, in Section 2. 4, to yield improved filter characteristics. Since a typical inverter utilizing a coaxial disk introduces discontinuity capacitance, the design must be modified to accommodate this reactance; three methods of so doing are reviewed in Section 2. 5. A detailed procedure,

together with some experimental results, is presented in Section 2.6. Section 2.7 follows with a discussion of the properties and limitations of this kind of filter, while Section 2.8 contains some concluding remarks. 2.2 Review of Filter Theory Two basic approaches may be used in designing a microwave filter: synthesis based on a direct transmission-line or waveguide formulation, or modification of lumpedconstant filter theory. The latter approach has been more successful, and has been used here to design coaxial band-pass filters. In the following sections are found a review of lumped filter synthesis (Sections 2.2. 1 and 2. 2.2), a description of immittance inverters (Section 2. 2. 3), and an indication of how this theory is applied to a microwave structure (Section 2. 2. 4). 2. 2. 1 Derivation of the Lumped-Element Low-Pass Prototype Circuit. The lumped-element filter synthesis technique is based upon derivation of the parameters of a network having a prescribed value of transducer power loss ratio PLR, which is defined by P _ available power to the network LR power delivered to the load For a lossless network with an input impedance Z = R + jX and a source impedance R, g LR 2r 1-iFi where F is the voltage reflection coefficient at the input given by R + jX - R r - g ' R + jX + R g Substituting for F, it is readily shown that (R - Rg) + X PLR = 1 + R (2.1) g

For a low-pass maximally flat or Butterworth filter PLR = 1/+ u ) (2.2) c and for a low-pass equal-ripple or Chebyshev filter PLR = 1+ T(/c) (2. 3) where w is the cutoff frequency, and U is the pass-band tolerence. T is the Chebyshev polynomial of degree n, defined by T(/ w) = cos [n Arccos(w/ c)] for < 1 = cosh [n arccosh(w/w)] for ct > 1c The polynomial T oscillates between + 1 for Iw/ow I < 1 and equals one at the band-edge lo/o3 = 1. To synthesize a Chebyshev filter, the two PLR expressions (2. 1) and (2. 3) are equated. This defines a load impedance in terms of the Chebyshev polynomial. The resulting polynomial can be realized as either one of the two dual prototype lumped ladder networks shown in Figs. 2. 1 and 2. 2. Although explicit expressions exist for the gk values of these filters for both the Butterworth and Chebyshev forms, tabulated values are readily available (e. g. Ref. 16) and are used in the filter designs studied here. 2.2.2 Band-pass Filter Networks. In building a lower sideband upconverter, band-pass impedance matching circuits are used rather than low-pass networks, because they provide the necessary filtering action and are easier to build at microwave frequencies. A band-pass filter is —easily derived from the low-pass prototype by use of the low-pass to band-pass frequency mapping: o' 1 /(_o o\ c \o

Rggo g2 g4 gn g o 4~gi Tg3 pg5 orE n+ RL=gn+l Fig. 2. 1 Low-pass prototype circuit with shunt capacitor adjacent to source Rg=go gl g3 g5 gn Vg 9 R L=gn+ Fig~ 2. 2 Low-pass prototype circuit with series inductor adjacent to source The low-pass frequency variable is o' and the band-pass frequency variable is co. The bandwidth of the band-pass filter is w2 - w1 and the center frequency is o0, such 2 o'2- W1 that o_2 = wl2: w =. This mapping transforms shunt capacitors into parallel resonant circuits and series inductors into series resonant circuits, both of which are resonant at wco. The circuits of Figs. 2. 1 and 2.2 are transformed into the band-pass circuits of Figs. 2. 3 and 2. 4. The steps are carried out in greater detail in several references, (e.g. Ref. 17, pp. 356-362). 2. 2.3 Immittance Inverters. Microwave low-pass filters can be constructed more conveniently if they can be built entirely from either capacitive or inductive elements. Similarly microwave band-pass filters can be constructed more conveniently if they can be built entirely from either series or shunt resonant circuits. This can be done by using impedance (or K) inverters or admittance (or J) inverters. Ideal

Cf= L =, (Shunt Rea(:tances) ~C~~L C g 91~C3 L 3 W C LW L (Series Reactan(:cs) Fig; 2. 3 Band-pass prototype with shunt resonator adjacent to the generator 1 C 1 L3 C Gg=g, g "" 5 C2 L2 gn+l Fig. 2. 4 Band-pass prototype with series resonator adjacent to the generator. The element values are given by the same formulas given in Figo 2. 3 immittance inverters are defined in Fig. 2. 5. One obvious example of a K inverter is a quarter-wavelength trans mission line. The lumped-element band-pass filters shown in Figs. 2. 2 and 2. 3 are equivalent to the circuits shown in Figs. 2. 6 and 2. 7, where series resonant circuits are separated by K inverters or shunt resonant circuits are separated by J inverters. Referring to Figs. 2.3 and 2. 6, the values of L Cri and Ki are found by equating a section r,i r' I i i+l of the filter in Fig. 2. 3 to the same section in Fig. 2. 6. This calculation (Ref. 18, pp. 415-422, Ref. 19, pp. 9-15) gives the following results:

K m 2 900 Image a Zb Phase Shift Zb J 900 Image a Phase Shift b b Fig. 2. 5 Definition of an ideal K inverter and J inverter L Cl Crl C L C ri ri r2 r2 rn r KA 01 12 Kn, n+ IRD Fig. 2. 6 Lumped band-pass impedance matching network using K inverters A 01 12 Fig, 2. 7 Lumped band-pass impedance matching network using J inverters o w IL (L o rLi(Lr i+l K. = ~. r~i = g1, 2,...n - 1 (2.4) Kil - ' gigi+l wwL R K r (2.5) K0, 1 = wg'Cc0gl K n+ o r,nB a (2.6) n,n+l L C = L. C. = c2 (2.7) r,i r,i 1 1 o

Similar expressions hold for the admittance inverters. 2.2. 4 Microwave Realization. Although the parameters for circuit of Fig. 2. 6 can be found explicitly, two problems still remain for realization of the filter in a microwave structure: (1) how are the series resonant circuits made, and (2) how are the K inverters realized. Before resolving the first question, it is convenient to generalize the expressions for the K's and corresponding J's to make them more compatible with o dX X = 2 dwd (2.8) 0o A shunt resonator, in which the susceptance is zero at wco, can be described in terms of its resonant frequency and a susceptance slope parameter b. o dB b 2 dw (2.9) For a series LC circuit X = w L, and for a shunt LC circuit b = woC. Thus the Q values for a circuit with resistance R in series with a series resonator or a conductance G in parallel with a shunt resonator are Qc = X/R and Qc = b/G respectively. The K and J values in terms of these slope parameters are given in Figs. 2. 8 and 2. 9, which should be used rather than Figs. 2. 6 and 2. 7 for distributed circuit design. The series resonant circuits Xi(w), shown in Fig. 2. 9 can be realized as halfwavelength transmission lines. Thus the choice of Li in the previous section is equivalent to the choice of characteristic impedance of the half-wavelength line. Often all the Xi can be made to have the same characteristic impedance of 50 ohms. In this case the 1

ggw K KjL1~!c K r K01!12 _ RA RB AwXl iw XjXj+ RBWX 0 gg 1 = - I J gog1 cX 0 1 j, j+l c gj+1 n, n+ ' Fig. 2. 98 Generalized impedance matching networks using K inverters i01 e"' — 12 B n _ Jn, n Fig. 2,9 Generalized impedance matching networks

reactance slope parameter is obtained from the transmission line equation in the following manner. The reactance of a transmission line of electrical length ~ and characteristic impedance Z terminated in a load RL is given by c L The slope parameter is readily obtained by differentiation of X, and for a transmission line resonant at q = X c c X - Z2C7. ( X ) (2. 11) 2 Z2 nator so that RL/Z << 1. In this case Z rT X - 2c (2.12) and in similar fashion Y 7T b 2 2 (2.13) When wavelengths are too long to feasibly permit spacing inverters of one kind every half wavelength, K and J inverters can be placed alternately every quarter wavelength as in Fig. 2. 10. The only difference between the values here and those in Figs. 2. 8 and 2. 9 is a difference in slope parameter. For a quarter wavelength line X = Z/4 and b = TrY 4. The series resonant circuits can be realized in a microwave structure by a half wavelength transmission line with a reactance slope parameter of Z c/2. The second question needing clarification is how the K inverters are realized. One inverter already mentioned is a quarter-wavelength line. Much broader bandwidth may be achieved using the K inverters shown in Fig. 2.11 or the J inverters shown in Fig. 2.12.

x/4 x/4 K01 J12 K23 Kn,n+l ~R I I 1 I ~~J I RB Fig. 2. 10 Alternate K and J inverters X > 0 X < 0 0<0 0>o (a) (b) Fig. 2.11 Lumped-element K inverter realization t 0/2 * *0/2- -0/2- t 0/2 -B< 0 B > 0 0 > 0 0 <0 (a) (b) Fig. 2.12 Lumped-element J inverter realization

In Fig. 2. 11(a) and 2. 12(b) the negative line length must be absorbed by available line length between inverters. The value of K for the inverters of Fig. 2. 11 is K = Z tan 10/21 (2.14) where =- Arctan (2X/Z) (2. 15) K/ Z X/Z =2 (2. 16) 1 - (K/ZC) The value of J for the inverters in Fig. 2. 12 is J = Y tan 10/21 (2.17) C where = - Arctan 2B/YC (2. 18) J/Y B/ Y = (2.19) c 1 - (J/Yc) Thus when K or J is calculated by the equations in Figs. 2. 8 or 2. 9, 0 and the reactance or susceptance can be found. The derivation of these formulas may be found in Cohn's work (Ref. 20) and is not reproduced here. A more general set of relations based on Fig. 2. 11(b) derived later for a distributed element in an inverter, gives the above results when specialized to a lumped capacitor. 2. 3 Distributed Impedance and Admittance Inverter Design Microwave filter design has been greatly facilitated by the development of directcoupled filter design techniques by Cohn (Ref. 20) described in the previous sections. Details have been published for a wide variety of microwave structures which provide immittance inverter realization over a range of frequencies in coaxial line, strip line, and waveguide (Ref. 18). In many cases these structures consist of a uniform section of transmission line loaded with a lumped reactive element. In particular, series

capacitances and shunt inductances were used by Cohn (Ref. 20) to fabricate immittance inverters in strip line and waveguide respectively. Coaxial-line filter design utilizing this method, has been based upon use of a short section of low-ZZ line to approximate a lumped shunt capacitance or a short section of high-Zo line to approximate a lumped series inductance (Ref. 21). Such elements are more easily obtained than lumped series capacitances or shunt inductances and are readily used in the design of coaxial-line immittance inverters. In addition the shunt capacitance which is realized by a low-ZZ line (disk) has been found to offer an important advantage for the coaxial line filter in that there is no dc path between the center conductor and the outer conductor. The lumped-element approximation to a short section of low —Z or high- Z line is shown here to yield significant error in the design of band-pass filters having bandwidths < 10 percent of the center frequency, and this error increases as the bandwidth is reduced. A more accurate immittance inverter realization technique is presented here, taking account of both the distributed line lengths and the reactive elements resulting from the discontinuity associated with the change in characteristic impedance. The accuracy of this technique has been ascertained through numerical analysis and experimental evaluation of filters having bandwidths as low as 1 percent. 2. 3. 1 Coaxial Impedance Inverter Design. A band-pass filter using impedance inverters requires that the parameter K, which determines the value of each impedance inverter, be specified. A network designed to realize such an impedance inverter must have an image impedance ZI equal to K and an image phase 0 which is some multiple of + v/ 2 (Ref. 20). Consider the coaxial impedance inverter shown in Fig. 2. 13(a), consisting of a short length f of line of low characteristic impedance ZO, together with two lengths 0/2 of line with characteristic impedance, Zc. The image parameters may be determined using the half-section ishown in Fig. 2. 13(b).

Z Z z Z Z + 2A o 2 2 (a) (b) Fig. 2. 13 (a) Disk impedance inverter, and (b) bisected impedance inverter The general transmission circuit matrix (i. e., ABCD matrix) may be written for a lossless line of length 2, characteristic impedance ZO, and wave number 3, in the form: cos pQ j Z sin op (2.20) [ sin of cos opf o The transmission circuit matrix for the half-section of Fig. 2. 13b is readily found to be A B cos 2 cos2 j (cos 2 sin 2 1.. 1) - sin 2f sin +- cos sin z 2 2 z 2 0 0 C D c (sin 2 cos2 Os2 cos 2 +- Cos 2 z sin O sin z 21'":,: 2 0 2 2 (2. 21) where z = Zo/Z and Y = 1/Z. The image parameters ZI and 0 are given by (Ref. 16, p. 53)

20 z (AB~ 2(2. 22) ZI CD and 0 BC 2 j tan 0 ( 2 (2.23) where - is used because the inverter has been bisected. The conditions stated in (2. 22) and (2. 23) will be used to solve for the unknowns 2 and 0 when K and 0 are known. From (2. 21), A cot 2 - tan 0 c o 2 2 j Yc(z cot tan + 1) j jZ tan [- e + Arctan (zo cot 2)] and B i Zc(zo tan + tan 0) D 1 - z tan O tan 0 o 2 2 = Zc tan [2 + Arctan(z tan f)] so the image impedance is given by: 2 [ ] [_02 zI = tan [0 + Arctan(z tan tan - + Arctan(z cot (2. 24) where zI = Zi/Z. Using the impedance inverter requirement that 0 = 7r/2, together with the circuit reciprocity requirement, AD - BC = 1, (2. 23) may be written as 2AD = 1 (2.25) which yields upon substitution from (2. 21),

21 2,3( 2 o 2 1 j3Q 23 C Q.0 0 2(cos 2 cos - + silln -sin ) - 2(z + ) sin cos in cos = 1 22~~o z S 2 2 Using the double angle trigonometric identities, this can be reduced to 2z 0 tan h3 tan0 = (2. 26) 1 +z 0o Equations(2. 24) and (2. 26) provide a set of equations in the unknown lengths & and 0 in terms of ZI and Z. An expression for 0 can be obtained from (2. 26). 0 0 = Arctan 2) (cot - tan )] (2.27) 11 + z 2 22 Arctan(zo cot -) - Arctan(z tan -) (2. 28) Thus the image impedance can be written as: ZI = tan[ + Arctan(z tan 2)] tan 2 + z tan - 2 0 2 - 0 1f ~~~(2.29) 1 - z tan tan - o 2 2 which when solved for 0 yields z - z tan 2 01o 2 tan 2 - I 0 2 (2.30) i + ZozI tan 2 -Elimination of tan 0 between (2. 27) and (2. 30) yields a quartic which can be written in the following factored form: (t 2 n + 2) tan - 2E'tan - + 1) = O (2. 31) 2 \ 2 2

22 where ~I z -1 ( ) ( ) (2.32) The solution to this equation gives the following real values for tan - tan = E + (E'2_ 1)2 which also may be written as 22 1 tan Of = + (E- 1)2 (2.33) From this expression the value of tan 0 may readily be obtained using (2. 26). Although two solutions are given in (2. 33), the one with the lower sign has given better results. If the upper sign is used the impedance at the center frequency is correct but the pass-band ripple and the band edges are seriously distorted. Comparison of the above results with that of Cohn (Ref. 20) shows that when of << 1, (2. 27) is equivalent to his expression for K. The equations derived here do not take account of the discontinuity capacitance associated with the abrupt change in center conductor diameter. Design modification to accommodate this effect is discussed in Section 2. 5. 2. 3. 2 Coaxial Admittance Inverter Design. A suitable coaxial admittance inverter is of the form shown in Fig. 2. 14, consisting of a short length f of line of characteristic admittance Yo together with two lengths 0 of Y line. 0 2 c Yo Yc e t/ 1 + e2 e2 H 0/2 (a) (b) Fig. 2.14 (a) Coaxial admittance inverter, and (b) bisected admittance inverter

23 Derivation of expressions for Q and 0 in terms of the admittance inverter parameter YI follow the same procedure as used for the impedance inverter above. Imposing the inverter requirement 6 = 2 there result two analogous equations for and o: 2yo tan = ~ 2 cot (2.34) 1 + Yo tan f/3 = + (F'2- 1) 2 (2.35) where - yl F,=t 1 IY- 21 YI YI/Y c and Yo = Y/Y Y Other forms of the immittance inverter used by Cohn (Ref. 20) include a shunt inductance or a series capacitance together with a length of transmission line. These structures are not as readily analyzed on a distributed basis. 2. 4 Theoretical Results of the Distributed Design Method A new method for designing impedance inverters was derived in the previous section. In this section this method is compared with the lumped design method, and the distributed design is shown to be superior in (1) band-pass filters using K inverters, (2) band-pass filters using alternately K and J inverters, and (3) impedance transformers using K inverters. In the following theoretical curves, the effect of discontinuity capacitance has been neglected. However when a coaxial filter or impedance transformer is built, the design must be modified to account for the discontinuity capacitance as discussed below in Section 2. 5. 2. 4. 1 Band-pass Filters Using K Inverters. Several curves were derived for various band-pass filters of the form shown in Fig. 2.15 centered at 8. 5 GHz with

24 111-d, 22 Id2113 Id3 —424 Fig. 2.15 A schematic diagram of a coaxial band-pass filter with Teflon dielectric around the center two disks different fractional bandwidths, for the purpose of comparing the distributed design described in the previous section with the lumped design. The following filters were based on a Chebyshev low-pass prototype circuit with 0. 1 dB ripple. The design was carried out using 50-ohm coaxial line with an outer conductor diameter of 0. 5625 in. and a center conductor diameter of 0. 24425 in. In the theoretical curves which follow, the effect of discontinuity capacitance has been neglected. For a three pole Chebyshev filter Fig. 2. 16 shows that there is considerable improvement in using the distributed design. The distributed design filters are all centered at the design frequency although the bandwidth is narrower than the design bandwidth. The maximum ripple in the passband is smaller for the distributed case although it is still larger than the design ripple of 0. 1 dB ( F I = 0. 023). The design data for these curves are shown in Table 2. 1. Since each bandpass filter is symmetrical, the first two disks are identical with the last two; consequently only the values for the first two are given. Five percent and a 10 percent bandwidth 5-pole Chebyshev filters were designed using the distributed design technique. The resulting curves are shown in Fig. 2.17. The same improvements over the lumped design are evident for the fiveripple case. A comparison of the five-ripple case and the three-ripple case for the distributed design shows that the maximum ripple is higher for the five-ripple case. The design parameters for the filters are shown in Table 2. 2. Since the filters are symmetrical, the first three disks are identical to the last three.

25 ----- Distributed Design - Distributed Design -- Lumped Design i ~ ~-e — Luieeig - - Lum ed Design 1.0 0Il 0.9 0. 8 0. I 0.0 I ~~~~~~~~~~~~~~0.7 0.96 o o. - ~ 0 O I I 50.8 0. 0 ~~~~0.4 ~~~I I 0.4 I 0.4 - 0. 3 I! 0.3 )I 0. 3 0.2 0. I I o., ~ ~ ~ ~~~~~ii o.i 0' I..,I, { 7~~~~~...6 8.8 909.9. 7.6.8 8.0 8.2 8.4 8.6 8.8 9.0 9. 9.4 7.6 7.8 8.8 9.0 9.2 9.4 Frequency in GHz Frequency in GHz (a) (b) Distributed Design - Distributed Design --- Lumped Design 500 - Lumped Design 1.090.8 8.8 6/0 -~~ I \~~~~~~~~~~~~~ 1 ~~~~~~~~40 08.7 30 8. 1 3 20 u I 7.8 8.0 82 84 0.0 9.2 8.4 0.46 lI U~~~~ I 0 C 0~~~ 0.25 7.6~ 7.8 8.0 8.2 8.4 829.0 9..2 9.4 F'e enFrequncy in GiG -20 -0.4,11 30-C \ \,I 1 ~ ~~~~~~~~-40 -I ~~~~~-80 -o. I I -901 7.6 7.8 B. 8. 2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz (c) (d) Fig. 2. 16 Theoretical comparison between the lumped and distributed design for filters with bandwidth of (a) 1 percent, (b) 5 percent, and (c) 10 percent. The impedance of the 10 percent filter is shown in (d).

Distributed Design - Distributed Design Lumped Design -- — Lumped Design 1.0 1.0 f~~~ 0.9 0.9 I~~~~~~~~~~~~~~~I 0.8 — 0.8 0.7 I 0.76 0.6 I I 6 I I u1 ~~~~~~~~~~~~~~~o IC I 0.5 -- 0.5 c; o 0.4 - 0.4 - 3 0.3 0.3 0.2 I I 0.2 I V I 0.1 0.1 1 0 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 ~~~~~~~~~~~~Frequency in GHz ~Frequency in GHz Frequency in GHz (a) (b) Fig. 2. 17 Theoretical comparison between the lumped and distributed design for 5 pole filters with bandwidth of (a) 5 percent and (b) 10 percent

27 Lumped Disk Disk or Bandwidth K01 K12 Diameter Diameter 1 cm t cm 0 rad. rad. Distributed D1 in. D in. 1 2 1 2 L 1% 6.1701 0.72194 0.502 0.554 0. 3011 0.3496 0. 24556 0. 02888 ~ D,1% " ". 0. 3471 0.4302 0.15534 0. 01330 L 5% 13.7969 3.6097 0.502 0.532 0. 1263 0.2547 0.53847 0.14414 D. 5% f ". " 0.1298 0.2777 0.50663 0.10963 L 10% 19.5117 7.2193 0.502 0.502 0.0820 0.2559 0.74412 0. 28679 D 10% " " " " 0.0833 0.2816 0.72378 0. 21529 Table 2. 1. Disk parameters for three pole filters Lumped or Bandwidth K01 K12 K23 D1 in. D2 in. D3 in. Distributed w L 5% 13. 0849 3. 1316 2.3863 0. 502 0.532 0. 532 D 5% 1TI I T L 10% 18. 5049 6.2632 4.7726 0. 502 0.502 0. 532 D 10% Lumped cm cm cm or LmBandwidth 1 2 3 1 2 Distributed w radian radian radian L 5% 0. 1343 0.2940 0.3864 0. 51192 0. 12510 0. 09538 D 5% 0.1384 0.3328 0. 5462 0. 47793 0. 82929 0.01747 L 10% 0.0880 0.2965 0. 1919 0. 70893 0. 24923 0. 19033 D 10% 0. 0896 0.3399 0.2009 0.68706 0.16114 0. 16587 Table 2. 2 Disk parameters for five pole band-pass filters

28 Tables 2. 1 and 2. 2 show that the difference in disk lengths between the lumped and distributed case is not large. As would be expected, the difference in the phase angles 0 between the two cases is greatest when the disk length is large. The different disk diameters used in Tables 2. 1 and 2. 2 were the result of increasing the diameter to make E'> 1 in (2. 33). The above filter design examples were all designed from tables which specified 0. 1 dB ripple in the passband. Filters with other ripple factors as well as maximally flat filters may be designed by simply using the g values (Ref. 16, pp. 98-102) appropriate to these criteria. A 10 percent bandwidth Chebyshev filter with 0. 5 dB ripple and a maximally flat filter were successfully designed. The design parameters are shown in Table 2. 3, and the resulting characteristics are shown in Figs. 2. 18 and 2. 19. Igl g3 g2 K01 =K34 12=K23 01=04 02=03 Disk Length cm Radians Radians 1 4 = 2 -3 0. 5 dB ripple 1. 5963 1.0967 15. 685 5. 935 0. 5510 0. 1401 0. 2308 0. 3671 Maximally 1.0000 2.0000 19.817 5.554 0.7134 0.1124 0. 1685 0. 4075 Flat Table 2. 3. Filter parameters for three pole Chebyshev and maximally flat filters where discontinuity capacitance has been neglected and all disk diameters are 0. 502 in. For a 10 percent bandwidth when the center frequency is 8. 500 GHz the lower and upper band-edges should be 8. 075 and 8. 925 GHz respectively. For the Chebyshev filter, the band-edge is defined at the 0. 5 dB level ( I r12 = 0. 109) where the lower and upper frequencies are 8. 16 and 8. 89 GHz respectively. For the maximally 2 flat case, the band-edge is defined at the 3 dB point (Ir 12 = 0. 5). Here the lower and upper band-edge frequencies are 8. 15 and 8. 89 GHz respectively. Thus in both cases the bandwidth of the resulting filter is approximately 8. 5 percent which is slightly narrower than what was sought. As in the other examples the ripple for the Chebyshev filter exceeded the design ripple slightly. As would be expected the band-edges of the maximally flat filter were not as steep as those for the Chebyshev filter.

Reflection Coefficient I r 12 0 0 0 0 0 0 0 0 0O O - O I 0 c o 0 co cn c 1 0 0 0 00 COr? 00 Reflection Coefficient IF r1 CO I I0II I t ~oo ~~ ~o r~30 CD -tJ ~< O ~ cb_ a q

30 The distributed design procedure applies to any type of filter where the g values are known although it may be limited by practical considerations such as disk size and discontinuity capacitance. 2. 4. 2 Filter Using K and J Inverters. The possibility of designing a band-pass filter with center frequency of 1 GHz was investigated. The primary problem is that the wavelength at this frequency (30cm) is inconveniently long. To make the filter more compact, a filter consisting of alternate K and J inverters shown in Fig. 2. 20 was designed. Here the inverters are separated by a quarter-wavelength line rather than a half-wavelength line so the filter length is reduced by about one half. The J inverter can be either a series inductance or a series capacitance. For a series inductance J inverter, however, the diameter of the inductive wire would be impractically thin. Therefore a semi-lumped capacitance was used for the J inverter. Since the inverters are separated by a quarter-wavelength of line, the reactance slope parameter of the transmission line is half of that used previously. This means that filters can be built with approximately twice the fractional bandwidth. The lumped procedure and the distributed procedure were used to design filters with fractional bandwidths of 0. 20 (Fig. 2. 21a) and 0. 40 (Fig. 2. 21b). The design parameters for these four filters are shown in Table 2. 4. The filter characteristics again show that by using the distributed design procedure an improvement is obtained, although the difference becomes less significant at the broader bandwidths. The lower frequency end of the passband was disappointing, but this is probably due to the asymmetrical structure of the filter. This filter has a transfer impedance pole at dc in contrast with the filter with no series capacitances. The attenuation thus remains high from the lower end of the passband all the way down to dc. 2. 4. 3 Impedance Transformer. The distributed design approach was used to design an impedance transformer operating between 50 ohms and a complex load whose real part was 1 ohm. The load was resonated with an appropriate reactance so that the resonated Q was QA. The load was assumed to have a QA such that the decrement 6 was the same for all design bandwidths.

Distributed Diam Diam w K K23 J3 1 rad rad 0ra ad 1 cm C2 pF tm Cm C4 pF 1iaml I.am3 or Lumped 01 12 23 34 in. in. D 0.2 19.512 0.002888 7.219 0.007805 0.72378 -0.28679 0.21529 -0.74412 0.7085 0.4694 2.3939 1,4653 0.502 0.502 L 0o2 19.512 0.002888 7.219 0.007805 0.74412 -0.28679 0.28679 -0.74412 0.6967 0.4694 2. 1750 1.4653 0.502 0.502 D 0.4 27 594 0.005775 14.439 0.01104 0.99685 -0o56225 0.53214 -1.00856 0.4089 1.0028 1.0443 2.5260 0.502 0.502 L 0.4 27.594 0.005775 14.439 0.01104 1.00856 -0o56225 0.56225 -1.00856 0.4042 1 0028 1.0180 2.5260 0.502 0.502 ~~Table 2. ~4 Band-pass filter using K and J inverters~C3 Table 2. 4 Band-pass filter using K and J inverters

32 I I I t,, I '1 1 0.0571 In this case the complex load acts as the first series resonator; the first K inverter is KI which is given by K11 WRAX2 2 12 w I ) g1g2 6 The low-pass prototype g values can be obtained from a graph (Ref. 16, p. 128), and the values for K can be determined from the above expression and the formulas found in Fig. 2.t8. For a n-pole low-pass prototype circuit, only n K inverters are needed for the matching network whereas (n + 1) K inverters are needed for a 50-ohm to 50-ohm filter. The responses of the 50:1 impedance transformers are shown in Fig. 2. 22 for various bandwidths and the impedance in Fig. 2. 23 for the 10 percent transformer. Again the passband was centered around f for the distributed design case. The dis0 tributed design transformers generally had a flatter passband and a slightly narrower bandwidth than the design goal. "'Table 2. 5 summarizes the design data for these curves. 2. 5 Effect of Discontinuity Capacitance Discontinuities associated with an abrupt change in characteristic impedance of a coaxial transmission line result in the excitation of higher- order modes in addition to

1.0 1.0 /dDistributed disks - Distributed disks I 0.9 Lumped disks 0.9 Lumped disks I\ I I I 0.8 I 0.8 0.7 - 0.7 cu I ~~ ~ ~~Il I Cc I I~ ~~I Il I 0.6 0.6 I I I u I I It O~~~~~~~~~~~ I 0.5 0.5 0 I u.5 I u0 I S I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0.4 0.4 I I I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I~\ I 0.3 I I 0.3 C.', 0.2 'r ~ I II 0.2 0. ' ' I.V.. 0.7 0.8 0.9 1.0 1.1 1.2 1.3 0.7 0.8 0.9 1.0 1.1 1.2 1.3 Frequency GHz Frequency GHz (a) (b) Fig. 2. 21 Filters using alternate K and J inverters separated by quarter wavelength lines with bandwidth of (a) 20 percent, and (b) 40 percent

34 Distributed design Distributed Design - Lumped design -- Lumped Design 1.0 E.0 0.9 0.9 r AI I 0.8 I 0. 0.Li 0.8 - * 0 1 o.5 I 0. 55 n.ss 0.4 0.4 0.3 0.3 0.2 0.2 I 0. o. I 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 0 7.6 7.8 8.0 8.2 8.4 8.6 8.8 0.0 9.2 9.4 Frequency in GHz Frequency in GHz (a) (b) Distributed Design - Distributed Design. 1- -- Lumped Design f - Lumped Design 1.01.0 0.9 / 0.9 0.8 0.8 0.7 \ ooo 0.7 0.003 percent, and (d) 2oo c 0.002 0. 6 osm O 0.8 0. 001 0 0 I4 85 86 8.1 8.3 8. 5 8.7 C50.5 U L 0.5 a 0.4 n.4 0.3 0.3 I I / 0.2 /0.2 0.1 0. 1 0 I e I I. I yi I 1 I I Iu I ny 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 7.6 7.8 8.0 8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in Glz Frequency in GnHz (c) (d) Fig. 2.22 Theoretical comparison between the lumped and distributed design for 50:1 impedance transformers with bandwidth of (a) 1 percent, (b) 5 percent, (c) 10 percent, and (d) 20 percent

35 Distributed Design 1.3 1.3 — Lumped Design 1.2 1.1 1.0 0.9 0.8 8.0 8.2 8.4 -0.4- \2 0.6 -0.4 _ -0. 5 0.Fig. 2. 23 Theoretical comparison of the impedance between C) 0.2 Freque ncy in G hz the lumped and distributed design of the 10 percent bandwidth 50:1 impedance transformer

Band- Disk Diameters Lumped or width Distributed w K K 23 K34 D1 in. D2 in D3 in fl cm2cm cm rad ~~~~~~~~~~~~12 2m 3 34 1 21 ra.1 2 3 1r*2. L 1% 0.7338 0.6973 6.5221 0.554 0.554 0.502 0.3439 0.3620 0.2843 0.02935 0.02789 0.25942 D 1% | I. | " | 0.4183 0.4591 0.3215 0.01431 0.01100 9. 17667 L | 5% 1.6409 3.4864 14.5839 0.542 0.532 0.502 0.3747 0.2638 0.1184 0.06561 0.13923 0.56761 D 5% " " " "| " 0.4963 0. 2899 0. 1214 0.02006 0.10307 0.53787 L 10% 2. 3206 6.9727 20. 6247 0.542 0.502 0.502 0.2647 0.2653 0.0759 0.0276 0.27712 0.78246 D |o10% " " | " " " 0.2906 0 2944 0.0771 0. 06861 0.20206 0.76366 L | 20% 3.2818 13.9454 26.1678 0.532 0~502 0.502 0.2804 0.1248 0.0427 0.13108 0.54399 1.05618 D 20% " " " " " " 0. 3129 0. 1282 0.0432 0.09173 0.51256 1. 04568 Table 2. 5 Disk parameters for band-pass impedance transformer using three disks

37 the TEM mode. These higher-order modes are generally evanescent and thus contribute energy storage elements to the equivalent circuit of the transmission line. The discontinuities considered here give rise to electric energy storage which is modeled by capacitive elements in shunt with the transmission line. The low-frequency capacitance associated with a discontinuity remote from any other discontinuities may be determined by reference to the work of Whinnery et. al. (Refs. 22, 23) and Somlo (Ref. 24). The results of Somlo are particularly useful in filter design, since he has fitted an equation to the curves of discontinuity capacitance, Cd, as a function of line dimensions. For the case where the discontinuity occurs on the inner conductor, his expression gives 2be 1 + a 4 (2.36) C d 0 7 In 1 21n (2 36) a_ 2 + 0. 111 (1 - a) (F - 1) 27Tb 10 Farads where b = outer conductor radius, meters a = inner conductor radius, meters r = disk radius, meters a (b - r)/( -), = /a. The maximum error resulting from use of this expression is stated to be + 0. 03 (27rb) pF in the ranges 0.01 < a < 1. 0 and 1. 0 < T < 6.0. The capacitance value obtained by use of (2. 36) must be modified for the effects of frequency. At high frequencies the transverse dimensions of the coaxial line are of magnitude comparable to a wavelength, resulting in a transverse resonance effect. This series resonance in the shunt circuit may conveniently be accommodated by use of a frequency correction factor M' which modifies Cd. Although it is not possible to present a universally valid correction factor, sufficiently accurate values are obtainable by use of the curves of Somlo (Ref. 24). The discontinuity capacitance value Cd to be used in the filter design is thus given by

38 Cd = M'Cd. The filter disk dimensions determined in Section 2. 3 above must now be modified to take account of C', such that there is equivalence between the circuits shown in Figs. 2. 24(a) and 2. 24(b). Three methods are in common use for determining this equivalence. The first two methods retain the value Z for the terminated line and 0 use a modified line length 2'; the third method uses a modified value Z' in addition to 0 changing the disk length f to a different value 2'. Method 1 is due to Cohn (Ref. 25). The transmission line is replaced by a ff equivalent circuit of the form shown in Fig. 2. 25. The impedance Z' = Z while 2' O O is chosen such that Y tan - Y tan + WC o 2 o 2 d or 2' = 2 Arctan(tan 2 - Z (2. 38) Method 2 is due to Matthaei (Ref. 26). It follows that of Cohn with the additional assumption that the tangent functions may be replaced by their arguments, since 2 and 2' are small. 2' = 2 - 2Z C' 1 (2. 39) od 1d The principal inaccuracy with these two methods is that they take no account of the change in the value of Xb, which occurs when 2 is altered; this deficiency is avoided in the third method. Method 3 is due to Levy and Rozzi (Ref. 27). The transmission matrices of the two structures shown in Fig. 2. 24 are equated, yielding two equations in the two unknowns Z' and 2'. A difficulty arises in the solution of these equations, since the Cd in Fig. 2.24(b) has been computed for the original value Z, whereas the actual value of the discontinuity capacitance is determined by the revised disk size. Consequently recourse must be made to an iterative procedure to obtain an accurate solution. If O0,

39 O O l Zo CT Z, r --- e 1- < i Fig. 2. 24. (a) Length of transmission line representing disk, and (b) length of transmission line representing disk with discontinuity capacitance X T1 Fig. 2. 25. Tr equivalent circuit for a transmission line where Xb= Z sin (Of) and B = Y tan (of3/2) Z0, and CO = Cd are the length, characteristic impedance, and discontinuity capacitance for the original disk, then the new length and characteristic impedance are found by reapplication of Levy and Rozzi's formulas: cos(fk+) = cos(1fk) + W(Ck - Ck 1) Zk sin(fifk) (2. 40) sin(of k) Z - Z (2. 41) k+1 = k sin(fk+ (2. 41) where k= 0, 1, 2,... and C 0. Experience has shown that between three and eleven iterations are required to obtain a solution in which the discontinuity capacitance change between successive iterations is less than 1 percent. One sees from methods 1 and 2 that 2 must be sufficiently large and Cd sufficiently small that 2' > 0. A similar restriction holds for method 3 since the magnitude of the right hand side of (2. 40) must be less than 1. This implies the discontinuity

40 susceptance must be smaller than the susceptance of the transmission line equivalent circuit: B > wC' (2. 42) a d Equation (2. 42) imposes a more severe computational restriction than the limitations imposed by either methods 1 or 2. In practice, method 3 is not applicable for the broader bandwidth filters where the disk lengths can become rather short and the right side of (2. 40) becomes larger than 1. For the 1 percent bandwidth filter the relative accuracy of these various methods of accommodating the discontinuity capacitance is illustrated in Fig. 2. 26. On the basis of such comparisons the third method was selected for the narrow band filters; the less accurate methods 1 and 2 are used for the broader bandwidth filters. 2. 6 Numerical Example and Experimental Results The band-pass filter design consists of a series of impedance inverters each separated by a half wavelength line. A step by step design procedure for a filter is outlined here using the distributed design technique. 1. ) A low-pass prototype filter with the desired characteristics is chosen. From this a value of K = ZI is found from Fig. 2. 8 using a reactance slope parameter of Tc for the half wavelength line. 2 2. ) An arbitrary diameter is chosen for the disk. The larger the disk diameter, the shorter its length. 3.) The disk characteristic impedance ZO is found. 4.) The value for E' is found from (2. 32). 5.) A test is made to insure E'> 1. If E'< 1, the disk diameter is too small, and a larger diameter must be chosen. 6. ) The disk length is found from (2. 33). 7.) The length of line 0 on either side of the disk is found from (2. 30). 8. ) The discontinuity capacitance is found from (2. 36). 9. ) Finally the disk dimensions are modified in accordance with one of the three methods of Section 2. 5.

41 0. 20 Cq. 'o I'0 I' I I I I o. 100 Frqec in 0.05 i \'" /I 0 '.,. 8.44 8.46 8.48 8. 50 8. 52 8. 54 8. 56 Frequency in GHz Fig. 2. 26. A i percent bandwidth band-pass filter for the ideal case when there is no discontinuity capacitance (...), when the filter is compensated for discontinuity capacitance by method I ( — -), method 2 ( ---), and method 3 ( —). The ideal case and the compensation by method 3 merge at the left side This algorithm was used in computer program A listed in Appendix A which designs and analyzes these filters. As an example of the application of the design procedure, consider the design of a three pole Chebyshev filter with a center frequency of 8. 50 GHz, fractional bandwidth w = 0.10, and pass-band ripple of 0.1 dB. The prototype elements, which may be found in various references (e.g. Ref. 16), for this case are go =1. 0000, gl = 1. 0315, g2 = 1. 4740, g3 = 1. 0315, and g4 = 1. 0000. The filter is to be constructed in

42 standard 14 mm 50-ohm coaxial line, and each disk will be surrounded by Teflon material with a relative dielectric constant of 2. 03 (Ref. 28). Step 1 shows that ZI 19. 5117 for the first impedance inverter. A disk diameter of 1. 275 cm is assumed, giving a characteristic impedance of 9. 68 ohms. Step 4 shows E' = 2. 28 > 1 so the disk diameter is large enough. Since 3 = w(2. 03)2/c, (2. 33) gives the disk length as 0. 0833 cm. Step 7 indicates that the length of transmission line on each side of the disk is 0. 3619 radians. Since the network is symmetrical, the first and second disks are identical to the fourth and third disks respectively. The design must now be modified to accommodate the effect of discontinuity capacitance. This capacitance has a value of 0. 3306 pF for one side of the first disk. Method 2 is sufficiently accurate for this case, so the new disk length is found from (2. 39) to be 0. 03698 cm, giving a 50 percent reduction in disk length. The modified lengths of the remaining three disks may be found similarly. The above theory was experimentally verified with both 10 percent and 1 percent bandwidth filters. The preceding design procedure was used, with the exception that no Teflon material was used around the outer two disks, in order to make V' longer. Fig. 2. 15 shows a schematic and Table 2. 6 gives the final design parameters for these two filters. Figs. 2. 27 and 2. 28 show the experimental data obtained from slotted line measurements compared with calculations which accounted for the discontinuity capacitance. Method 2 was used for the 10 percent filter and method 3 for the 1 percent filter. The minimum insertion loss for the 1 percent filter was 2. 1 dB at center frequency. A photograph of the 1 percent filter is shown in Fig. 2. 29. Using this design procedure, several filters have been built with good results in 7 mm coaxial line, having center frequencies in the 14 - 20 GHz range, with bandwidths around 5 percent. Disk Diameters Fractional =D =d d Bandwidth 01 =04 02= 03 1 4 2 3 D1=D4 D2= d1d3 w radians radians cm cm inches inches cm cm 0.10 0.7019 0.2153 0.03698 0.21501 0.502 0.502 2.0209 1.8843 0. 01 0. 2067 0. 0133 0.21681 0.40795 0. 532 0. 554 1.8252 1.7710 Table 2. 6 Experimental design parameters of the filter shown in Figs. 2.27 and 2.28

43 1. G - 0.9 x I x x x 0.8 x~~~~~~~~~~~~ x I I 0.7 iI CN lx I I 0.6.Y I I X~~~~~~~~~~ 4-4 0Q I 0.5 / 0~~~~~~ / x / ', I 0.3 I \ i 0.2 0.1 K K 0 IKx 8. 35 8. 40 8. 45 8. 50 8. 55 8. 60 8. 65 Frequency in GHz Fig. 2. 27. A percent bandwidth band-pass filter designed by the lumped method (- -), by the distributed method ( ---), by the distributed method with discontinuity capacitance accommodated by method 3 (-), and experimental results of the solid line filter (xxx). The ideal case when there is no discontinuity capacitance ( ---) and the case where method 3 is used (-) merge for the lower frequencies

44 1.0 0. 9 0. 8 0. —o00 x x x x K 0)00 K 0). 7 x \0. 002- K 8.1 8.3 8.5 - 0.6 5? 0.4 0. 3 0. 2 K x KK 0. IK 6 7.8 8.i) 8.2 8.4 8.6 8.8 9.0 9.2 9.4 Frequency in GHz Fig. 2. 28. A 10 percent band-pass filter designed by the distributed method where discontinuity capacitance is accommodated by method 2 with experimental results (x ). The insert is a portion of the characteristic drawn on an expanded scale.

Fig 2. 29. Photograph of the 14

46 2.7 Discussion 2. 7. 1 Filter Limitations Imposed by Choice of Bandwidth. The filter design is limited to a maximum of approximately 20 percent bandwidth, for beyond this value disks of reasonable diameter are of such small length 2 that the modified disk length V' < 0. The filter is limited to a minimum bandwidth (somewhat less than 1 percent) by the requirement that E' > 1, i. e., the disk length must be shorter than a quarter wavelength. As the desired bandwidth decreases this inequality can be satisfied by increasing the disk diameter; consequently, the minimum bandwidth is restricted by the physical requirement that the difference between the disk diameter and outer conductor diameter not exceed the limits imposed by manufacturing tolerances. The lumped design approach places no a priori limitation on disk diameters, while the distributed design approach requires E'> 1, setting a lower limit on the disk size. When filters designed by the lumped approach use diameters below this boundary, the resulting filters fail to give even reasonably acceptable characteristics. When the disk diameters are above this boundary the distributed design method continues to give superior filter characteristics as the graphs have shown. 2. 7. 2 Bandwidth Discrepancy. The computed bandwidth of the filter obtained from numerical calculation, does not agree exactly with the design bandwidth. The calculated bandwidth is approximately 1 percent for the 1 percent filter design, 4 percent for the 5 percent filter design, 8. 5 percent for the 10 percent filter design, and 14. 5 percent for the 20 percent filter design. Young (Ref. 16, pp. 564- 566) listed two causes for bandwidth contraction with the direct coupled cavity filter designed using a quarterwave transformer prototype. They are (1) electrical separation between reference planes shrinks as frequency increases and (2) the electrical lengths (his 4' and 4") associated with the reactive discontinuity also shrink with increasing frequency. A similar argument can be used here, with reference to the impedance inverters and their associated angles '. Consider a shorted length of transmission line of reactance Xt, which is half-wave resonant at frequency w, and a series LC circuit of reactance Xc, which is also resonant at Ow. f the two circuits have equal values for the reactance slope parameter at resonance, the ratio Xt/X at a general frequency w will be given by

47 X (1 + 2 +d') [1/ + 3d)+ 2/15(7d').....] for Id' < 1/2 c where CO - co d'= and a series expansion has been used for the tangent function. Thus Xt > X c and the ratio increases rapidly with increase in Id']. Consequently the equivalence between the distributed-circuit filter and its lumped-element prototype might be expected to break down as the filter bandwidth is increased. 2. 7. 3 Low-frequency Characteristics. Since the filter structure discussed here is quite similar to the popular low-pass coaxial filter, it is expected that power transmission through the filter will occur at some lower frequency. Figures 2. 30(a) and 2. 30(b) show theoretical and experimental plots for the 1 percent and 10 percent bandwidth filters and Fig. 2. 30(c) shows the low-frequency response for a 20 percent bandwidth filter. All of these filters were designed to have a passband at 8. 5 GHz. The low-pass 3 dB point is 0. 750 GHz for the 1 percent filter, 3. 02 GHz for the 10 percent filter, and 3. 95 GHz for the 20 percent filter. Furthermore, as the fractional bandwidth increases, the lowfrequency passband increases and the low-frequency response looks more like a lowpass filter. These low-frequency characteristics will be seen to be of major importance in the design of a LSUC. However actually controlling these characteristics seems to be limited to the choice of design bandwidth. If the fractional bandwidth of the filter is specified, the only degree of freedom left is the choice of disk diameter. If the disk diameters are made smaller, and hence the disks made longer, some change in the lowfrequency response can be made as indicated in Fig. 2. 30(d). Although the low-pass 3 dB point is at a lower frequency when the disk diameters are smaller, the choice of disk diameters does not seem to have sufficient influence to make it a design parameter for the low-frequency characteristic. Figures 2. 30(a), 2. 30(b), and 2. 30(c) show that for a given passband ripple and a fixed number of impedance inverters, the bandwidth exerts the major influence on the low-frequency response.

48 Theoretical curve Theoretical curve o ---ro Experimental curve o- - — o Experimental curve with data points with data points 1.0 1.0 0.9 -0.9 0. 8t i P' 0.9 0.8? c 0.8/ /0.8 0.6 Il 0.6 -0.7 0.7 0.6 ~~~~.S ~~~~~~~~~~~~~0, 0.4 I L I' 0. 0.2 I0 II 0.2 0. -0t 0.1 0.2~ ~ ~~~~~~~~~~~~~~~~~. 0.I 0. I 0. 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 0 Fequency GMzI FreqoencyG~~~~~~~~a ~0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 (a) (a) ~~~~~~~~~~~~~~~~~~~Frequency GHz (b) All disk diameters 0. 502 in. Outer two disk diams 0. 452 i. and Inner two disk diams: 0.482 in. 1.0 1.0 ~~~0.09 ~0.9 $/ 0.9 —.1 0.8 0. I-0.8 0.7 0. 0.6 U~~~~~~~~~~~~~~~ 0.6 0 0.56 0.56 0.5 -4 M 0.5 -0.4 -0.4 0.3 0.3 /// - \o/ i ' 0. 2 0. 2 0.1 0.1 - 0 0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.1 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 FrequencyGHz Frequency G14z ~~~~~~(c). ~~(d) Fig. 2. 30 Low-frequency response of band-pass filter with center frequency of 8. 5 GHz for bandwidth of (a) 1 percent, (b) 10 percent, (c) 20 percent, and (d) 10 percent bandwidth with modified diameters

49 2. 7. 4 Effect of Filter Parameter Variations. Some knowledge of the mechanical tolerances is helpful in constructing a filter. When numerical calculations were made on a 10 percent bandwidth filter with all four disks shortened by 0. 02 cm the center frequency increased approximately 50 MHz, the high frequency ripple decreased from 2 2 I =0. 04 to 0, and the low frequency ripple increased from I =r 0. 04 to 0. 05. Lengthening all 4 disks by 0. 02 cm decreased the center frequency approximately 50 MHz, 2 increased the high frequency ripple to I r = 0. 12 and decreased slightly the low frequency ripple. In other words, lengthening and shortening the disk had the opposite effects. If the distances between disks are shortened 0. 04 cm, the center frequency increases only 20 MHz, which is less than was expected. Increasing or decreasing the disk diameter has a similar effect on the filter characteristic as increasing or decreasing the disk length. Thus the filter is relatively insensitive to errors in the distance between disks but moderately sensitive to errors in disk size, especially for a narrow-band filter where the ratio of disk to outer conductor diameters is only slightly less than one. The effect of using incorrect values for dielectric constant and discontinuity capacitance was studied. If the dielectric constant Er of the material surrounding the center two disks of the 1 percent bandwidth filter is reduced by 26 percent, from 2. 03 to 1. 50, the filter bandwidth increases to 1. 26 percent. Also when the filter response was computed using values for discontinuity capacitance which were 10 percent to high, the center frequency of the original 1 percent bandwidth filter decreased approximately 2 MHz and the bandwidth increased by only 2. 3 percent of its original value. Narrow-band filters generally have higher pass-band losses than wider filters. A lossy-line analysis of the 1 percent bandwidth filter using standard tabulated values for the conductivity of brass and silver, and the loss tangent of Teflon, and a roughness factor of 1. 8 indicated that bandwidth remained essentially unchanged. However the ripples near the band-edges started to deteriorate, as was confirmed experimentally. 2.8 Conclusions A design procedure has been developed for band-pass coaxial filters, which is based on the readily available low-pass prototype circuit, and is applicable for bandwidths between approximately 1 percent and 20 percent. Numerical and experimental

results have shown that the filter characteristics are greatly improved when the impedance inverters are synthesized as short sections of transmission line rather than as lumped capacitances. The emphasis has been on band-pass filters with 50 ohm terminations, but the design theory may readily be applied to impedance transformers. These impedance transformers are used in the design of the lower sideband upconverter.

CHAPTER III DESIGN THEORY FOR THE LOWER- SIDEBAND UPCONVERTER 3. 1 Introduction The analyses of parametric devices found in most textbooks and journal articles are performed with very simple lumped circuits. However, most LSUC's operate at microwave frequencies and therefore are built using distributed circuits. The analysis in this chapter will consider various distributed element circuits, which could be used in designing a LSUC. Section 3. 2 contains a review of the design and analysis relationships for LSUC's and includes a discussion of the cause for the nonresonant gain maximum seen in later calculations. A low reactance at the upper sideband frequency is shown in Section 3. 3 to produce deleterious effects on the gain and noise figure. Section 3. 4 includes a description of three LSUC circuit configurations. This is followed with numerical calculations of gain, gain bandwidth product, and noise figure for these three circuits when the diodes are tuned by constant impedances a specified distance from the diode, by single-stub tuners, and by double-stub tuners. The effects of diode parasitic elements and higher order sidebands are also explored. Finally in Section 3. 5 the filter theory developed in Chapter II is used to design several LSUC circuits with coaxial impedance matching transformers. 3. 2 Review of Existing LSUC Design Theory As mentioned in Chapter L, the progress in parametric amplifier design has been hastened by the development in the late 1950's of high quality varactor diodes. A reverse biased varactor diode chip now can be accurately modeled as a voltage dependent depletion layer capacitance in series with a small resistance. The basic design theory described below uses this model. The addition of diode parasitic elements and external circuit impedances can be easily accommodated in the design theory and therefore does not detract from its usefulness.

52 The fundamental relationships between power and frequency for lossless nonlinear reactances were derived by Manley and Rowe (Ref. 6) in 1956 and have since been named after these two authors. They are given by oo o mP E X mn = 0 (3.1) mw + nw m=O n=-oo p s L ~C nP = (3. 2) m=-co n=O mwp + nws mrn~~~n ~pp s lation the frequencies wp and s are either incommensurate or commensurate and incoherent (Ref. 29). These relations are useful in understanding the principles of such devices as frequency multipliers, reflection parametric amplifiers, frequency upconverters, and frequency down-converters. They also can be used to find the ultimate power gain and conversion efficiency of parametric devices. Although the Manley-Rowe relations illustrate the feasibility of frequency conversion in parametric (nonlinear reactive) devices, they do not contain detailed circuit design information. 3. 2. 1 Method of Harmonic Balance. Nonlinear circuits can be modeled by a series of nonlinear differential equations, which in general cannot be solved analytically. However, if a small signal assumption is made, they can be reduced to linear differential equations with time-varying coefficients. The circuit problem may then be solved either by direct solution of the differential equations or by use of the principle of harmonic balance. The latter approach has been the most successful in analysis of parametric circuits. In either case once the nonlinear differential equations are linearized, analysis of the ferroresonant effect as well as higher order instabilities occurring at certain harmonics and fractions of the pump frequency is no longer possible. However Oliver (Ref. 15) was able to explain these effects by exploring the original nonlinear equations. For the small signal assumption, the pump voltage (or charge) is much larger than the incoming signal voltage (or charge) or any components at the mgenerated sorideband frequencies. Thus the varactor elastance variation is caused solely by the pump source,

53 and the small incoming signal sees a time-varying elastance oscillating at harmonics of the pump frequency. The small signal (differential) elastance of a varactor diode can be written in terms of the total charge qt stored in the diode junction and the total voltage vt across the junction: dvt S(qt) = dqt (3. 3) The total applied voltage and charge is the sum of the pump (vp, qp) components, plus the signal and sideband components (vs, qs) vt = V + v' t p s qt = qp + qs where v >> v' and q >> qs Expansion of vt into a Taylor series yields an p s p incremental voltage-charge relationship vi = S(q ) q' (3. 4) 5 p s where co S(q%) = E Sm e pt (3. 5) m=-ec Applying the principle of harmonic balance (Ref. 30) to (3. 4), two infinite, but equivalent, sets of equations are obtained. One of these sets is written in matrix form in (3. 6). JS 1 -JS_i jS2 V Zw 1 11 co 2 W3 W4 41 ~2- w3 w4 I 2 3 31 3 64 -JS_, jS_ 1 s W3i W 2..." (x 3. 6) V2 | 19.2 iz4|

54 The notation used in (3. 6) is defined as follows: cok I21 )k+ 1)(37 + (- ) w (3. 7) where k = 1, 2, 3. [.,.[x] is the symbol for the greatest integer less than x, wp is the radian pump frequency, wIs is the radian signal frequency, Vk, Ik and Zkk are the -jS0 voltages, currents, and impedances at Wak respectively, and Zkk Wk + diode parasitic elements + circuit impedance. The time origin is ordinarily chosen so that S1 = S-i ' This matrix can be written more concisely as Zll -iX12 jX13 -jX14... I V2 iX21 22 jX23 -iX24 i2 V3 = jX31 -jX32 Z33 -jX34... i3(3. 8), x~l jX43 Z i mn Co n and j (_l)n 2n-1 ( lm 2m-1 4 4 Two additional assumptions are often invoked to simplify (3. 6): (1) S1 is assumed to be the only significant pumped elastance component, and (2) the open circuit assumptions are employed at the higher sideband frequencies. Most of the nondiagonal terms in (3. 6) can be set to zero by means of the first assumption. Removal of this assumption is discussed in the following section. The open circuit assumption is equivalent to saying that all currents above a given frequency are open-circuited by the external circuit. The simplest analysis of a LSUC would assume all currents at

frequencies > w2 are zero. This leaves two currents but an infinite number of voltages. However since no power is propagated at these higher frequencies, these voltages may be safely ignored and (3. 6) is reduced to a 2 x 2 matrix. In the numerical analysis presented in Sections 3. 4 and 3. 5 the first assumption is removed and the second relaxed. The same analysis can be performed to obtain an admittance matrix rather than the impedance matrix of (3. 6). The number of equations is reduced by using a short circuit assumption, i. e., voltages above a specified frequency are short-circuited by the external circuit. In general the open-circuit assumption is considered superior since the package lead inductance presents a high reactance to the diode chip at high frequencies, although if the lead inductance is resonated this argument fails. 3. 2. 2 Elastance Coefficients. The theory developed thus far for the LSUC can be applied only if the nonlinear characteristic of the varactor diode is precisely known. For a diode heavily doped on one side, and having a doping distribution of N(x) = axb (3. 9) on the other side, the nonlinear depletion layer elastance is S = Sb(1-V/0v)Y (3. 10) where y = 1/(b+2), Ov is the built-in potential, V is the applied voltage (V>0 for forward bias), and Sb is the zero bias elastance. One method of finding the harmonic elastance components is to expand (3. 10) in a Fourier series by assuming a sinusoidal voltage source and allowing harmonic currents to flow (short-circuit assumption). Alternately the expansion can be made by assuming a sinusoidal charge source and allowing harmonic voltages to exist (opencircuit assumption). These two cases have been studied by Oliver (Ref. 15, Appendix B). Leeson (Ref. 31) noticed this expansion could be expressed as a solution of the Laplace integral where the answer was found in terms of gamma functions and the associated Legendre function of nonintegral degree. However, a more accurate method has been found which is not subject to either of these assumptions. Oliver (Ref. 15) has solved

the nonlinear differential equation describing a varactor diode in a particular circuit to obtain a set of elastance components for (3. 5). In the numerical work which follows, Oliver's method is used to obtain the elastance components for a graded junction diode embedded in the particular circuit configuration under study. 3. 2. 3 Gain Relationship for the LSUC. The power gain of an amplifier can be expressed in at least five different ways, of which the transducer power gain has been found to be the most meaningful for negative resistance devices like the LSUC. In the following discussion the amplifier accepts a signal from a voltage source with internal impedance Zg and the amplified output power is dissipated in a load Z. 1. The power gain is defined as the ratio of power dissipated in the load to the power delivered to the input of the amplifier. This gain definition depends only on Zp, and since the properties of a LSUC depend strongly on Z and g Z, this definition is of limited use. 2. The available gain is the ratio of the amplifier output power to source available power. Since this gain depends only on Zg and not on Zl, this definition suffers a similar drawback to that of the first. 3. The exchangeable gain of an amplifier is the ratio of output exchangeable power to input exchangeable power. The exchangeable power of a source with internal impedance Z and voltage V is 2 P - v Re(Z) / 0 4Re(Z) which can be < 0 for negative resistance devices (Ref. 32). The exchangeable gain is an extension of available gain and suffers from the same limitation: it does not depend on Z 4. The insertion gain is the ratio of output power to the power dissipated in the load if the amplifier were not present. Exactly what replaces the amplifier in this definition is unspecified, and this becomes a major problem in frequency conversion devices such as the LSUC. Therefore this definition of gain is too vague to be useful here.

57 5. The transducer power gain is the ratio of the power delivered to the load to the source available power. This definition depends on both Zg and Z an advantage the other definitions lack. Also in contrast to the exchangeable gain, the present definition gives positive gain. When the transducer power gain is applied to frequency conversion devices it is given by G _ average power delivered to the load at Wvm mn average available power from the source at w n If the source voltage V is conducted through a source resistance R and the ren gn sulting output current im is absorbed by a load resistance Rpm the transducer power gain is I m I 2R Vn I 2/ gn Gmn = lYmnl 4Rm Rgn (3. 11) where Ymn is the (m, n) element of the inverse of the impedance matrix (3. 6). Synthesis of LSUC's to obtain a specified gain presently is limited to using only the signal and lower sideband frequencies. The open-circuit assumption is used for all frequencies > w2, so the infinite impedance matrix (3. 6) is reduced to a 2 x 2 matrix. Since there is only one voltage source V1, the matrix may be reduced to a lower triangular form even when more than two frequencies are considered. Thus a complete matrix inversion is unnecessary to find Y. After some striaght forward mn algebra the gain for the LSUC is 4Q 1R 2 (3. 12) Z11Z22 Q 1Q21 S. where Q.ij = i. R Rs is the diode series bulk resistance, and Zmn is a matrix element in (3. 6). At midband this expression can be written as

58 4Xgy rK (~g (YqKq 2 (3.13) [21 (l+xg)(l+y)- Kq] 2 where R1 x = Rg (3. 14) g R R R (3. 15) Kq Q21 (3. 16) co 20 r= (3.17) W10 and w10, w20 are the midband signal and lower sideband radian frequencies. When xg is set to a nonzero constant > (K -1), (3.13) shows that the gain is always finite. g q For a specified value of xg > (Kq-1), Ye can be varied to maximize the midband gain (Ref. 11) yielding QG1 Xg 21 (1+x )(l+x -K ) (3. 18) g g q K yf 1_ g (3. 19) g Since the gain expression (3. 13) is symmetrical in xg and ye, a similar expression for maximum gain is obtained for a specified value y > (Kq- 1). In reflection parametric amplifiers (RPA) and upconverters, the maximum gain often occurs slightly below the resonant frequency (Ref. 33). The reason for the nonresonant gain maximum can be seen more clearly by rearranging (3. 12): 4RglR,(Sl/w)1 21 - ZllZ2* -S2/(cwl w)2 I ~ll~22 1 1~)

59 As the frequency falls slightly below resonance, the denominator decreases because the parametrically induced negative resistance term has increased. The other frequency dependent terms in the gain expression are of less significance in many microwave structures. An analytical expression for the true maximum gain and the frequency shift can be found for the ideal LSUC in a fashion similar to that used for the ideal RPA (Refs. 34, 35). If the amplifier uses single-tuned signal and lower sideband circuits, then the transducer power gain near resonance is 4Rg1R 2S1 (1-26w/w10) G21 2 ( e 2 21 g aIZ22r 10 [(Rgl-Rv-arco)+ (aw) ] where o Co1 — 10. S2 1 Z * - jX1 co cW2Z 22 R + jX v v zvo V I c1= C0 I 10 dR v ar dw 1 = CO10 1 (20 - 10) 2 2 R22Wc10 20 dX a v x dwo ~i:;~ 1 o1 = 10 1= 10 S dX22 w10w20R2 C1 = 10 1 = 10

60 Maximizing the gain with respect to 6w gives an expression for the frequency deviation from resonance 5wm where the gain is maximum. After some algebra this gives m = 10 -\ /Rg RA/Rg Rv m - -v -at 0 ar+ a W10 \ 0 where the approximation X22 = 0 is used. The maximum gain can be found simply by using 6wm in the gain expression. This expression demonstrates the existence of the nonresonant gain maximum. However since the microwave circuits used in LSUC's are so complicated, accurate, tractable analytical expressions for the maximum gain cannot be found. 3. 2. 4 Bandwidth and Gain-Bandwidth Product. If the varactor diode contains no parasitic elements, and if the external circuit is assumed to consist of ideal filter elements, then approximate analytical expressions may be derived for bandwidth and gain-bandwidth product. Under high-gain conditions, the primary frequency dependence of gain is the reactive term in the denominator of the gain expression (Ref. 36). Thus when the frequency-dependent reactance increases to where the denominator is doubled, the gain is approximately halved, and the bandwidth as a fraction of the input frequency can be found: 1- oa w =Q10 a (3. 20) Q10 "w Q20 1+x r 1+ where K A q w (l+x)(l+yf ) The definition of gain-bandwidth product used here is the square root of the transducer power gain multiplied by the fractional bandwidth (3. 20). The square root of the gain is used since the frequency dependence of the gain-bandwidth product is minimized. The gain-bandwidth expression is GBW W(G1 )

61 1 4xg Y aOwr Q10 aw Q20 GBW = g lx10 + r l+y (3.21) I (lx)(1+yd I 1+x r 1+ L g ~ J g Because of the assumptions used, these two expressions are useful only in finding the approximate upper limits for a particular LSUC or for comparing different designs. 3. 2. 5 Noise Figure Relationship for the LSUC. Noise sources in an upconverter circuit can be broadly divided into two classes: thermal noise caused by random motion of electrons in a resistor and shot noise caused by random fluctuations of charge carriers in an electric current. The thermal noise in the LSUC is assumed to be caused entirely from Rrr = R + Rr, where R is the series bulk resistance of the diode and Rr is the circuit resistance at w. The shot noise is assumed to be caused by the charge carriers traversing the depletion layer of the varactor diode. Shot noise in varactor diodes is generally considered negligible in comparison with thermal noise in uncooled parametric amplifiers. This is based on the simple calculation made by Uhlir (Ref. 37) who showed that the noise current when both shot and thermal noise are present is 4kTB <i2> R W + 2qBw (3. 22) where i0 is the dc current, k is Boltzmann's constant, and T is the absolute temperature. Substituting numbers for a typical parametric amplifier into this expression shows that the first term is much larger than the second. Indeed, Uhlir has shown elsewhere (Ref. 38) that the ideal nonlinear capacitor (zero storage time) has no shot noise output. However the shot noise effect is more complicated than that implied by (3. 22) and cannot be automatically neglected. Even if the diode is open circuited as is the case of the LSUC discussed here, shot noise can still arise from forward-andback current (Ref. 38). Josenhans (Ref. 39) calculated the shot and thermal noise in a reflection parametric amplifier subject to several simplifying assumptions. The assumptions that will be retained for the calculation given here are the following. 1. The diode has a large dynamic Q at the signal and sideband frequencies.

62 2. The charge carrier makes one pass across the space charge layer and recombines on reaching the far side. 3. The transit time of the carrier is short compared to the period of the pump. 4. The effect of any fast surface states is negligible. In addition to these assumptions, Josenhans assumed high gain and no higher order sideband frequencies. The first is unnecessary, as tractable results can be obtained for arbitrary gain while the second will be employed after the general expression is obtained. The noise equivalent circuit of the varactor chip shown in Fig. 3. 1 can be reduced to that in Fig. 3. 2 when the dynamic Q >> 1. In these figures Td is the diode temperature and Ta is the ambient temperature. The forward diode current i0 = is(eV/kT _ 1) is used to find the equivalent diode noise resistance: kT x q910 The shot noise source may be regarded as an equivalent resistance at w r given by er er 2 so that mathematically this noise source can be treated as a thermal noise source of mean square voltage < v > = 4kTB R w er There are several definitions of noise figure in the literature, but the "actual noise figure" proposed by Kurokawa (Ref. 40) is the most meaningful and the most convenient for negative resistance amplifiers. This definition includes the noise contribution from the load, so that two amplifiers which are not necessarily optimally loaded may be compared. The actual noise figure is defined as noise power delivered to the load at Wm 1 m available noise input power at wn G mn

63 1 s (-js/ r) = 2qB Rx x (wo S Fig. 3. 1. Noise equivalent circuit (-jS0/c) Px RX ~ (S/ W) rx(So/co)2 ~2 2 Rx + (So/w) 2P kT P x(So/w )2 ()<v' 2 2 x + (So/wr) R S <V2 = 4kTdBWRS Fig. 3. 2. Transformed noise equivalent circuit

64 The noise power delivered to the load can be analyzed as three components: 1. noise from sources at f (r:# m), 2. noise from internal sources of the amplifier at f, 3. and noise from the load itself which is amplified and returned to the load. The first noise component is N = IYmr Rm {4kBw[RsTd + Ta(R + R e)] r / m mr inn ~md a r er The second component is generated at f from the resistance R - R~m: m mm m Nmma = IYm 12 Rm 4kBw[RsT d T+R T +R T -Rk T ] mma mi Em w s d m a em a Am a The third component is the thermal noise arising from the load Rem and is obtained by multiplying the output reflection gain of the amplifier by the available noise of the load. 2 Rm- [(Y -R ] N kT B mmb = TaBw R + [(Ymm)- R ] =kT B 1- 2R m Y 2 a w km mm Summing these three noise contributions up gives the total output noise power Nm kkBW 4Rm Z IYmr 2 (RsTd + RTa + Rer Ta)- mmlRm w i d r a era mTI i~-~( + Ta 1- 2R mY where n - 1 is the number of sidebands considered. The noise figure is then N F m (3.23) m kTB G a w mn

64 The noise power delivered to the load can be analyzed as three components: 1. noise from sources at f (r#4 m), 2. noise from internal sources of the amplifier at f, 3. and noise from the load itself which is amplified and returned to the load. The first noise component is N = IYmr Rem {4kBw[RsTd + Ta(R + R e)] r / m mr inn ~md a r er The second component is generated at f from the resistance R - Rkm: m mm m Nmma = IYm 2 Rm 4kBw[RsTd+R T +R T -R T ] mma mi Em w s d m a em a Am a The third component is the thermal noise arising from the load Rem and is obtained by multiplying the output reflection gain of the amplifier by the available noise of the load. 2 Rm- [(Y -R ] N kT B mmb = TaBw R + [(Ymm)- R ] =kT B 1- 2R Ym 2 a w km mm Summing these three noise contributions up gives the total output noise power Nm = kB 4Rm Z IYmr 2 (RsTd + RTa + Rer Ta)- mmlRm + Ta 11- 2RmYm where n - I is the number of sidebands considered. The noise figure is then N F m (3.23) m kTB G a w mn

66 y( < 1, and for some fixed ye > (K -1) > 0, the gain is maximized for some xg < 1 The first maximum gain condition implies xg > y, and the second implies ye > xg Equation (3. 25) clearly shows that the first maximum gain condition (with large x ) gives superior noise figure and that noise decreases with increasing upconverter gain. When the noise figure is optimized with respect to the set of maximum gains of (3. 18), the minimum noise figure as determined by Khan (Ref. 11 ) is F2= = ~ 1+ Q+ (3. 26) and the resulting gain is G21 =r ( -- (3. 27) +Q1 q where xg= 1 (3. 28) K y, = 1 --- ~ (3. 29) 1+x g A 2 r>rc +- 1 (3. 30) If the diode Q11 is known and a specified gain is desired consistent with the minimum noise figure (3. 26), then the optimum frequency ratio is determined from (3. 27): rc(rc + 1) -1 ropt = r Lr- c+2)Gl (3. 31) The desired gain obviously must be sufficiently large to give a finite optimum frequency ratio. 3. 2. 6 Origin of Shot Noise. The measurements of shot noise made by Josenhans on gallium arsenide and germanium diodes were all in error by less than 46 percent when compared to his theory, but for silicon diodes the measurements differed from the

theory by a factor of 10 to 15. A similar descrepency was observed earlier by Uenohara (Ref. 41). He had ruled out circuit loss, pump noise, shot noise on the basis of (3. 22), microplasma noise, and higher order sideband noise. However he noted his theory is inadequate for diodes with a static Qc = SO/ (wRs) > 5. No quantitative reason is given for the discrepency, but he suggested that the diode Q deteriorates with increase in applied pump power. His assumptions of neglecting circuit loss, pump noise, and higher order sideband noise are verified later in Chapter IV as being reasonable, so the shot and microplasma noise were probably larger than he expected. Both Uenohara and Josenhans neglected carrier lifetime. Uhlir (Ref. 38 ) in discussing the noise contribution of the forward-and-back current still assumes wT7< < an invalid assumption for silicon at microwave frequencies. Although an analytical treatment of this large shot noise is not available, a qualitative explanation is. Siegel (Ref. 42) observed that when a diode was pumped by an RF source, the dc current-voltage characteristic exhibited a negative resistance region. He postulated that this was caused by some anomalous charge multiplication mechanism. Lindmayer and Wrigley (Ref. 43) verified that this is the case and described this effect in the following way. When a forward current step is applied to a p+n diode, holes begin to flow across the junction into the n region. At the same time an equal quantity of electrons flow from the n side ohmic contact, and since electron mobility is large in the n region, charge neutrality is established. When the junction is suddenly reverse biased, the faster moving electrons leave excess holes behind in the n region near the junction, thus initially giving the appearance of being heavily doped. The advance of the space charge is retarded, a high electric field is formed, and carrier multiplication occurs as the holes return to the p region. When all of the holes are returned, the electric field decays to that determined by the impurity concentration. The net result of this is that more charge is moved during the reverse transient than during the forward pulse. The same phenomena occur if a periodic signal is used rather than a voltage step. 3. 2. 7 Summary of Existing Design Theory. The fundamental design relationship is the midband gain expression (3. 13), which assumes a small input signal level and ideal circuit filters (i. e. no higher order sidebands propagate beyond f ) The same asp-s

68 sumptions are used in finding the midband noise figure (3. 25). These two expressions can be manipulated to find the conditions for maximum gain or minimum noise figure. The design procedure for a LSUC exhibiting minimum noise figure can be outlined as follows. 1. The input signal frequency is first specified. 2. The diode dynamic Q11 is then specified. This parameter increases with the pump power and should normally be as large as possible. 3. The optimum lower sideband to signal frequency ratio is determined from (3. 31) for a specified gain. 4. The generator and load resistances are found from (3. 28) and (3. 29). The optimum frequency ratio is not used in the designs of the following sections, because the ratio is so large that the required pump frequency is beyond the capabilities of the available equipment. Nevertheless low noise amplifiers with a specified gain are designed by choosing a large generator resistance and deriving the resulting low load resistance from (3. 13). Although the synthesis technique described here is based on only the 2 x 2 matrix approximation, analysis of the resulting design can be performed for a much larger number of frequencies. 3. 3 Effects of Upper Sideband Reactance on the LSUC The theory outlined in Section 3. 2 provides a method for designing the impedances at signal and lower sideband frequencies. However, impedances at the higher order sidebands may significantly change the upconverter performance. To show explicitly how the reactance at the upper sideband frequency X33 affects the upconverter, (3. 8) is truncated to a 3 x 3 matrix rather than a 2 x 2 matrix. With this more accurate approximation, various impedances, the gain, gain stability with variations in pump power, gain-bandwidth product, and noise figure can all be modified by X33 and the second harmonic of the pumped varactor elastance S2. 3. 3. 1 Effects of X33 on Gain. The effects of X33 on gain is more clearly seen if the 3 x 3 matrix derived from (3. 8) is reduced to a 2 x 2 matrix, because this shows what additional impedances are induced into the signal and lower sideband circuits. Since there is no applied voltage source at the upper sideband frequency, the term ig

69 can be eliminated from the first two equations resulting in x 31x13 x13x32 Z33 33 (3.32) X + X23X31 X32X23 jx + Z - Z^ i 21 Z33 22 Z33 2 If Z33 Rs + JX33' then Z = R sX31X13 X33X31X13 Z 2 = + Rs+ 2 2 21 Z 2 = + 11 +Xx Thus X3 3 s 33 = R + Rs + 2Rll- jaX11 12 2 2 2 + Rs + X33 Z R +sR -R X3Xj X331X23 Z22 = Rg+ Rs- 2 2 j s 33 s 33 - Rp + Rs +jAX, I _Z12Z21 Zin1~~ = ___ z22 Zin2 Zll Thus the equivalent circuit for the LSUC when X33 is finite is shown in Fig. 3. 3.

-jS R AR1 jAX -jS0/w1 R AR11 -jaX11 +jS0/W2 R -AR22 jAX22 V Rg V'I~~~~~~~~~~ R ( V2 2- V1? t Z22 Zl in. Zin2 Fig. 3.3 Equivalent circuit of LSUC where X is the circuit reactance and X33 is finite c

In the usual 2 x 2 matrix approximation at midband the only impedance induced into the signal circuit is Zinl = - S2/[WlW2(RQ + Rs)] the negative resistance arising from the double reactive mixing. However, for finite X33 the induced impedance Z inl is complex, and consists of terms which show the interaction between the circuits at a1 and w3' between co3 and w2 and between w2 and o. In addition there is the added impedance AZ11, which is dependent on the coupling between the circuits at W1 and 3. Similar remarks apply to the lower sideband circuit. Both AZ11 and AZ22 attain their maximum values when X33 = Rs The curves shown in Fig. 3. 4 provide further insight into the magnitude of the impedances that might be expected in a typical LSUC. The parameters used here to make the calculations are listed below: SO = 6. 283 10 1 F 1 S1/SO = 0. 35 S2/SO = 0.044 R = 100 ohms g R = 1. 3 ohms R = 1. 0 ohms f = 9. 000 GHz p f = 1. 000 GHz The large variation in these impedance values fortunately occur over a small range of X33 near zero. Except for Rin2 the corresponding impedances in the lower sideband circuit are negligibly small, all being less than 1 ohm. If S2 = 0 (or X23 = X32 = 0), Z in is equal to the negative resistance obtained from the usual 2 x 2 matrix analysis, while AZ1l varies with X33in the same manner since it is independent of S2. With The transducer power gain is given by (3. 11) where Y21 is the row 2, column 1 element of the inverse of either (3. 32) or the inverse of either (3.32) or the truncated 3 x 3 matrix derived from (3. 8). At midband where SO is resonated out, the gain is found to be

70 -4 60 -13 50 - 2 40 z-1 I-1 20 -~ ~ ~ ~ ~~~~~~~~ 30 I-0 0 10 -~~~~~~~~~~~~~~~~~~~~~~~6 120 - 40 - ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ 4 - 60 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~3 -2 0-18 -16 -14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 ohms Fig. 3. 4 The impedances induced into the signal circuit by the finite value ofX

74 X33 = 9 77 2 G21 = 21. 9 dB (Maximum gain) X -0. 538 Q G21 = 6. 48 dB (Minimum gain) 33 21 IX331: o Q G21= 20. 06 dB When S2 = 0, the minimum gain occurs at X33 = O. The midband gain plotted in Fig. 3. 5 for the typical LSUC described above clearly shows the minimum and maximum gain. In addition when the reactances Xin1, Xin2 AX, and AX22are tuned out by the external circuit, even larger variations in the gain occur (Fig. 3. 5). When S2 = 0, the midband gain curve shown in Fig. 3. 6 is symmetrical about X33 = 0. However the nonresonant maximum gain, also plotted for this case, does exhibit a peak for X3 < 0. 3. 3. 2 Effects of X33 on Gain Sensitivity. One possible source of gain instability in a LSUC is variation in the pump power source. The pump provides all the harmonic elastance coefficients S 1, S S,..., but to a first order approximation the gain sensitivity can be found by assuming that only the fundamental elastance coefficient is nonzero. Applying the formula 2 S1 3G21 PP G21 aS2 to (3. 35), the gain sensitivity is S G2 S = 1 + 2 M (3. 37) PP ()~20(RgR)2 S) where s = X33[ (Rg + Rs)(R + R s) - S /(wl0O20)] + [(R + R)(RQ + Rs)Rs + (RQ Rs)S1/ (o10Wo30)- RsS/ ( 10w20)] [s-R Rw2 3] Q +X32X3[(Rg + R~{)(R + R+) L~~~~g-~~) %) BWz/~]~(R2 ~~

69 can be eliminated from the first two equations resulting in x 31x13 x13x32 Z33 33 (3.32) 1X + X23X31 X32X23 jx + Z - Z^ i 21 Z33 22 Z33 2 If Z33 Rs + JX33' then Z = R sX31X13 X33X31X13 Z 2 = + Rs+ 2 2 21 Z 2 = + 11 +Xx Thus X3 3 s 33 = R + Rs + 2Rll- jaX11 12 2 2 2 + Rs + X33 Z R +sR -R X3Xj X331X 23 Z22 = Rg+ Rs- 2 2 j s 33 s 33 - Rp + Rs +jAX, I _Z12Z21 Zin1~~ = ___ z22 Zin2 Zll Thus the equivalent circuit for the LSUC when X33 is finite is shown in Fig. 3. 3.

25 24 induced reactances not resonated 23 - 22 -- induced reactances resonated / 21 20 19 a 16 o 15 5 14 L l I /l 12 10 9 200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180 200 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 X33 ohms Fig. 3. 5. Midband transducer power gain when S2/S0 = 0. 044

25 24 Midband Cain 23 Maximum Cain - - - 22 21 20 19 I 18 i S717 il II F 163 15 — 10 4 I -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180 200 x33, ohms Fig. 3. 6 The transducer power gain when S = 0 21

78 R < Rs(2o30/20 -1) (3. 41) This expression, being independent of X33, merely determines the boundary where gain sensitivity can always be improved. The amount of improvement is found from (3. 39). 3. 3. 3 Effects of X33 on Gain-Bandwidth Product. The gain-bandwidth product expression obtained in Section 3. 2. 4 was derived subject to a high gain assumption. A similar expression can be found when X33 is finite and S2 = 0, i. e., by using (3. 35). If AR11 and AX11 are assumed frequency independent, the gain-bandwidth product will be very similar to the previous value: GBW= x ( +x + (3. 42) (l+x + j +xx' x(y) L g r(l + y) ~AR K A A~11 q where x' = and a' = (1 + x)( Since both numerator and denominag R w +y) tor decrease by the factor 1 +x g 1~x +x' g g the gain-bandwidth product remains unchanged when X33 is finite, at least for this first order approximation. A more accurate picture of this parameter as X33 is varied can be obtained by removing the frequency independent assumption on the induced reactances, allowing S2 / 0, assuming X33 is a single tuned lumped reactance, and calculating the gain as a function of frequency when X33 is the desired reactance at midband. Using the same LSUC parameters defined in Section 3. 3. 1, the gain-bandwidth product was calculated and plotted in Fig. 3. 7. If the reactances induced into the signal and lower sideband ports by X33 are resonated by a single tuned reactance, the gain-bandwidth product is product increases for X33 < 0 and decreases for X33 > 0, but never by more than 0. 002. If S2 is increased rather than decreased, the gain-bandwidth product changes in the opposite direction.

7 6 5 'S / 0/ 4I 4 / Cd 2~~~~~~~~~~~~~~~~ 0~~~~~~~~~~~~~~ -200 -180 -160 -140 -120 -100 -80 -60 - 40 -2 20 40 60 80 100 120 140 160 180 2(0 - 10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 X33' ohms Fig. 3. 7. The gain-bandwidth product (Gcw) w hen S2/ So 0. 044 where the dashed curve corresponds to the expanded X3 scale 3 -~~~~~~~~~~~~~~~~~~3

80 The behavior of the gain-bandwidth product near X33 = 0 needs some explanation. When X33 < 0, the gain is low and bandwidth wide, so values for the product are difficult to obtain. When X33 > 0, the gain versus frequency curve has a large dip near the resonant point. For values of X33 in this range the maximum gain occurs at a frequency above resonance (rather than below as expected from the theory presented in Section 3. 2. 3), and the bandwidth is greatly reduced. Consequently the gain-bandwidth product is low for X33 > 0. 3. 3. 4 Effects of X33 on Noise Figure. The noise figure for the LSUC is obtained from (3. 23) where for these calculations X33 is finite, the shot noise is assumed to be zero, and the diode temperature is assumed to be equal to the ambient temperature. The noise figure is F 1 s+ 4RfRs(IY221 2+ IY231) 1 - 2RfY221 2 2 R G21 G21 where G21 is given by (3. 34). After some algebraic manipulation this can be reduced to a form similar to (3. 25): 2 ) 2 2 2 g 2:1 ( 9g)(21 ) [ (3123 - X21x33) + (X21s) ] + |s [X23 (R+ RS) + (X l21X13) + X13X31X 12X21] +i + R) [X23X32Rs(R + Rs) + X23X 32X13X31 - X33(X12X23X31 + X21X13X32)]| [(X31X23 - X21X33) + (X21RS) ] (3. 44) This equation by comparison provides explicit demonstration of the effect of X33 on the upconverter noise. When IX331 = oo, (3. 44) reduces to the simple expression (3. 25); when S2 = 0, (3. 44) reduces to

F1 1 1 X13X13 + X31 /r)45) 2= (1+-)( 1 ) + 2 Xg r G21 x4R + X33) where xg is the normalized generator resistance and r = 2/W1 = X21/X12. Just as the gain exhibited a minimum and maximum as a function of X33, so also does the noise figure. To find these extrema, (3. 44) may be written in the following form: R M +MbX + MCX F - S 1 +- a b33 c 33, 46) F2= 1 + R+ (3. 46) [(X31X23- X21X33) + (RsX21) ] where Ma =(+ )X 12X21R sxI23 R RS 1 13) + X 13X 31X 12X21 + ( 1 + )[X23X32RsRg + Rs) + X23X32X13X31] + g(Rg + s)( s)RRg+ g Rs) X23X32 + (R( + Rs)X13X31 + R X12X21] + [X21X13x32 + X23X31x12] }/(4R jRg + (++(% 2 Mb= + (-1223X31 X21X13X32) - (X21X13X32 + X23X31X12) [(R + Rs)(R + Rs)- X12X21] / (4RgR) c + RgX 12X21 + [(R + Rs)(Rl + Rs) - X12X21]2 / (4R R) Differentiating (3. 46) with respect to X33 results in the following quadratic equation in X33:

82 O=X [- 1 X 2M XX Rs 33 [-MbX1 McX31X23X 213 + X33[2M(X31X 23) 2MX21R 2MX2 1+R)XX2 2 + 2M X X31X23] a 21] + [Mb(X3lX23) + Mb(X2s + 2 23 which may be easily solved by the quadratic formula. When S2 = 0 (i. e., X23 = X32 = 0) the noise figure has only the one extremum at X33 = 0. Clearly from (3. 45) this extremum is a maximum since, when S2 = 0, the noise figure can only be increased by the finite X33. However, when S2 ~ 0, some improvement in the noise figure is possible. Using the LSUC parameters defined in Section 3. 3. 1, the noise figure extrema are found: X33 = - 0. 480 12 F = 1. 8124 (Maximum noise) X33 = 24. 50 Q2 F = 1. 1453 (Minimum noise) X 33 = F = 1. 1463 These extrema are also observed in Fig. 3. 8 which is a plot of (3. 44). It appears that the enhanced noise performance at X33 = 24. 50 is very small, although it could be improved by increasing S2 The "actual" noise measure, defined by Kurokawa (Ref. 40) as M F-1 M 1 1 21 is plotted in Fig. 3. 9. This parameter is particularly useful in assessing the noise characteristics of a low gain amplifier, a condition that occurs for X33 near zero. The noise measure parameter permits choosing between a low gain, low noise amplifier or a high gain, higher noise amplifier for the front end of a system. The noise measure in Fig. 3. 9 reached a maximum of 0. 791 at approximately the same place the maximum noise figure occurs. 3. 3. 5 Conclusions on the Effects of X33. All of the major effects on the LSUC that are caused by a finite value of X33 for the example chosen here occur for X33 < 20. However for any LSUC, the large variations between the 2 x 2 and 3 x 3 matrix approximations will diminish as X33 gets large. In a given upconverter design

2. 4 2.3 2.2 2. 1 2.0 1.9 a) 1.8 1.7 1.1 / I 1.5 I 1.4 1.200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180 200 - -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 X33, ohms Fig. 3. 8. The noise figure when S2/S = 0. 044 where the dashed curve corresponds to the expanded X33 scale

1. 4 1. 3 1. 2 1. 1_ 1. 0 0. 9 0. 8 02 0. 7 a) z 0. 6 0. 5 0. 4 0. 3 0. 2 0. 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 X3 'ohms Fig. 3. 9. The actual noise measure of the LSUC when S 2 S = 0. 044

85 the values of X33 where the minimum and maximum gain and noise figure occur are a function of the magnitude of S2 (Table 3. 1). II s2 = o S2/So0 0. 044 S2/S0 = 0. 1 Maximum Gain co 2 9. 77 12 5. 62 2 Minimum Gain 0 2 -0. 538 Q -1. 151 2 Min. Noise Fig. 0o 12 24. 50 ~ 15. 19 12 Max. Noise Fig. 0 2 -0. 480 2 -1. 062 Q2 Table 3. 1. Values of X33 where the gain and noise figure extrema occur Therefore measurement of X3 where the midband maximum and minimum gain occur offers at least the theoretical possibility of measuring S2 under operating conditions. The method would proceed by resonating the signal and lower sideband circuits for some large value of jX331. Then the values of X33, namely X33a and X33b, are found for which the gain is minimum and maximum respectively. Since these two values satisfy (3. 36), then - (X33a + X33b) equals the ratio of the coefficient of X33 to the 2 2 coefficient of X3 in (3. 36). Thus a linear expression in S2 is obtained in terms of the known circuit values and S1. Enhancement of S2 not only moves the extrema to lower X33 values, but also provides larger maximum gain and lower minimum noise figure. 3. 4. Numerical Comparison of Three Possible LSUC Circuits There are at least three possible circuits which could be used in fabricating a LSUC using a coaxial tee junction. If the diode is mounted at the tee junction between the center and outer conductors, then all three ports of the tee are in parallel with the diode (Fig. 3. 10a). This first circuit is designated LSUC1. If the diode is mounted completely within the center conductor with one end in the signal port, then the diode sees the signal port in series with the parallel combination of the pump and lower sideband ports (Fig. 3. lOb). This second circuit is designated as LSUC2. If the diode is mounted a short distance away from the tee junction and in the lower sideband port, then

f f f 11>S S S f - 1 1 I I fL fa ' dOt d2 ---- -- 0 d2 — 2 -diode diode diode d — diode.... Z I (a) (b) (c) Fig. 3. 10. Drawings of LSUC1, LSUC2, and LSUC3 together with the corresponding schematics of the circuits are shown in (a), (b), and (c) respectively. do d1, and d2 are the distances between the diode and the corresponding impedance transformers respectively. Zs, Zp, and Z are the impedances of the signal, pump, and lower sideband ports as seen by the diode

87 the diode sees the lower sideband port in series with the parallel combination of the pump and signal ports (Fig. 3. 10c). This third circuit is designated as LSUC3. In the analysis that follows, the tee junction for LSUC1 and LSUC2 is assumed to be an ideal junction. However at X-band frequencies and higher, this assumption is not valid. Hence in LSUC3, a more accurate assumption is used; the leg of the tee junction appears as an open circuit to power flowing between the two arms of the tee circuit for frequencies in X-band or higher. To make these circuit configurations into LSUC's, two additional factors must be considered: (1) the diode with its package parasitic elements, and (2) the external circuit which performs the impedance matching and filtering functions. A typical diode model, which is used throughout a large portion of this chapter, is shown in Section 3. 4. 1. In Section 3. 4. 2 the gain and gain-bandwidth product are found for a LSUC with frequency independent filters and impedance matching transformers using both an unpackaged and packaged diode. Under these circumstances all three circuit configurations give identical results. In the following sections each of the three LSUC circuits are examined under a variety of conditions: (1) ideal band-pass impedance transformers for the three ports are located a given distance away from the diode, (2) the diode package parasitic inductance is changed so that the diode is self resonant at f20, and (3) the ideal transformers are replaced by single- and double-stub tuners. Finally in Section 3. 4. 7 some conclusions are drawn, and the relative merits of these three circuit configurations are discussed. 3. 4. 1 Diode Model and Desired Circuit Parameters. The basic diode model is shown in Fig. 3. 11, with the typical parasitic values: R = 1 ohm L = 2 nH C = 0.7pF SO = 27r 10 F 1 The design of a LSUC must include these package parasitic elements since they have a major effect on the amplifier operation. To avoid needless repetition in later sections,

88 L S R s C p 00 = Sn exp(jnwpt) n= -oo Fig. 3. 11. Package varactor diode model this model with these particular parasitic element values is termed the "standard diode model. In the designs which follow, the signal center frequency f10 = 1. 000 GHz, the pump frequency f = 9. 500 GHz, and the lower sideband center frequency f20 = 8. 500 GHz. p The desired midband transducer power gain is 19 dB. From (3. 13) this value of gain can be obtained with small noise figure if R= 100 ohms and R = 1.3 ohms. g9 3. 4. 2 Gain for the Ideal Circuit with Unpackaged and Packaged Diode. The broadest bandwidth possible for a LSUC would occur with no diode package parasitic elements present. If the circuit presents to the unpackaged diode terminals a constant R = 100 ohms at all signal frequencies and a constant R 1.3 ohms at all lower g sideband frequencies, the transducer power gain as a function of frequency would appear as shown in Fig. 3. 12. This curve also provides a clear demonstration of the frequency dependence of the parametrically induced negative resistance. The maximum gain has increased by a factor of 1. 14 over the resonant gain while the frequency of maximum gain is 0. 96 of the resonant frequency. Subsequent analysis shows that when circuit Q increases, the frequency of maximum gain is closer to the resonant frequency. The gain-bandwidth product is shown in Fig. 3. 13 and is compared to the value given by Robinson in (3. 21). Clearly the gain-bandwidth product increases with pump level. Also, the high gain assumption made by Robinson is shown to be quite good for the normal pump level chosen here, i. e., S1/SO = 0. 35. However Robinson's expression is normally not of much help in determining the gain-bandwidth product since the diode

89 30 29 28 27 26 25 24 23 22 m 21 20 0 19 18 z 17 cJ 16 14 13 9 8 7 6 5 4 3 2 1 0 0.75 0.80 0.85 -J. 90 0.95 1.00 1.05 1.10 1.15 1.20 1.25 Frequency, GHz Fig. 3. 12. LSUC transducer power gain with out-of-band circuits completely isolated and the diode has no parasitic elements

90 4. 0 3. 8 3. 6 3. 4 3.2 3.0 2. 8 2. 6 2. 4 / 2.2 2. 0 1. 8 1.0 / 0.8 0.6 0. 4 0.2 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 Elastance Coefficient S1/S0 Fig. 3. 13. Variation of gain-bandwidth product (G2w) with pumping level when out-ofband circuits are completely isolated and there are no diode parasitics, and Robinson's gain-bandwidth formula ____

parasitic elements and complicated circuit arrangements seriously degrade upconverter bandwidth. Rather than trying to predict the gain-bandwidth product analytically, it is obtained by finding the half power points in the numerical gain curve. In actual upconverter circuits using packaged diodes and frequency sensitive circuits the bandwidth and gain-bandwidth product is reduced considerably from that indicated in Figs. 3. 12 and 3. 13. Here only the effect of package parasitic elements is considered while in the following sections various external circuit arrangements are analyzed. When the diode package shown in Section 3. 4. 1 is resonated by a single tuned circuit, the gain versus frequency and gain-bandwidth product are shown as the solid lines in Figs. 3. 14 and 3. 15 respectively. In particular the gain-bandwidth product has been reduced from 2. 5 for the unpackaged diode to 0. 19 for the "standard" packaged diode when it is pumped to give Si/SO = 0. 35. These curves also show that the bandwidth progressively increases when Cp is halved but L is the same, when L is halved but C is the same, and when both L and C are halved. P P 3. 4. 3 Comparison of Three circuit Designs Using Ideal Impedance Transformers. In each of the three LSUC circuits to be described here, the impedance presented by the signal port at all signal frequencies is a constant but is located a specified distance away from the diode. At the other frequencies, the signal port is assumed to be a short circuit at this same distance from the diode. Similarly the impedances presented by the lower sideband and pump ports are constant but are located specified distances away from the diode. Thus the frequency sensitivity of the upconverter model is caused by the line lengths and package parasitic elements. The actual impedance presented by the signal and lower sideband ports (Zt and Zi respectively) are different from the desired R and R. The impedances Zt and g Z must be chosen so that when they are combined with the reactances of the other two ports and the diode parasitic elements, the diode chip itself is presented with Rg at f10 and RQ at f20 The expressions for Zt and Zi as well as Z for all three SUC configurations are found in Appe t ndix B.

d d ----- T-J ZTOT 998' S put Hu T =I ' _ IOT 8' = dT S puB Hu T = =I ' -- ZTOT 998 ' =dS put HU Z = rI:sMolIOj st aox suauIala la]s~x d apoip atp axaqa pa Vos! AIalalduwoo sI!noxi; putq-Jo-Ino qLlt ui!~ JaMod jaonpsux oflSri 'PfI '-g '. zHD 'Aouanba~j Ijumis 9~'I O0'I c91 'I OI 9I0'1 00'i 96 'C 06 'O 98'0 08 '0 9L'0 0 / 9 \ 9 L // /- 8 / // 6 OT / / —0 11 / / / — II / /- 0 I / T 1 /i 91 / "':) 91 61 I/ zzl OC LZ 88 /9 f7:

------ I - ZTOT 998' = dS pue HuT =? - - -J ZTOT 958' = dS pUt' lu = 1 ZTOT 8Z 'T = dS Put HUg = dI:sMoloJ Se a se siuatuali oll~stle apolp aql a.aTqn aqlTouV auo tuoTj pal4IOS AlalaIdmuoo s4lno!xo putq-jo-Ino qjiA (MA3) jonpoLd q4pAMpuBq-uTV 3flSrI '1 ' g ' a T OS/ s lual3iJJooo aaueJseI L 'O 9 ' ' 'o0 0 Z 'o ' 0 — 'O z'O I I I I I | 8 I I 1-18- I I — 8'0 / I / / I M. /, Ic '0 / / /

94 The various circuit distances from the diode can be specified so that the upconverter bandwidth is optimized. However since the gain expression for these circuits is complicated, the optimization must be done numerically. The approximate technique used here is based on the assumption that the overall bandwidth is limited by the impedance at the signal frequency. For a given LSUC circuit configuration, diode model, and signal filter to diode distance, a search is made for the pump and lower sideband port distances which minimize the "slope function" dR11 2 dX 2 (347) do + ( 11 (3. 47) The procedure begins by choosing trial pump and lower sideband distances. Then Zt is found which is located at the plane of the signal filter. The impedance Zll is calculated at two closely spaced frequencies and the above slope function is obtained. The process is repeated for various pump and lower sideband distances until the minimum value for (3. 47) is found. The approximate optimum distances are shown in Table 3. 2 for the four cases when the signal port distances are s/4 and p/4, and when two values for diode inductance are used. These distances are used in the subsequent analys is. LSUC1 LSUC2 LSUC3 LSB Dist. Pump Dist. LSB Dist. Pump Dist. LSB Dist. Pump Dist. Signal Dist. = /4 6. 3 or 6. 0 4. 8 4. 8 3. 9 4. 2 L = 2 nH 6.0 6.3 nal Dist 5. =X7 5. 7. 5 1. 5 4. 5 1. 8 L = 2n.. Signal Dist. =X /4 6. 3 6. 0 4. 8 4. 8 4. 5 3. 0 L 0. 2203 nHs 6. or 6. Signal Dist. = / 4 L = 0.2203 nHP 5. 7 5. 7 1. 5 1. 5 4.8 1. 5 Table 3. 2 Distances in cm between the diode and lower sideband (LSB) or pump filters. The * values are not true minima as the slope in these cases is monotonically increasing

95 When the standard diode model of Section 3. 4. 1 is used (i. e. L = 2 nH) and the signal port is Xs/4 away from the diode, the transducer power gain for the three circuits is shown in Fig. 3. 16. In these calculations a 5 x 5 matrix approximation is used (signal frequency plus four sideband frequencies), and the pump modulation coefficient is S1/S0 = 0. 35. In each of these circuits the maximum gain occurs at resonance rather than at a frequency lower than resonance as was found in circuits containing less reactance (Figs. 3. 6 and 3. 8). Evidently the frequency sensitivity of the circuit masks the asymmetry of the negative resistance. The bandwidth of LSUC3 is larger than that for LSUC2 since the assumption of a nonideal junction in LSUC3 eliminates the frequency sensitivity of the signal port at the lower sideband frequency. When the gain is calculated there is only one applied voltage. This means a complete inversion of the impedance matrix is unnecessary. However, for noise calculations there are noise voltages at all the relevant frequencies, and thus in this case a complete matrix inversion is necessary. Although this impedance matrix is complex, it can be separated into its real and imaginary parts and manipulated as two real matrices (Ref.44 ). If the matrix is C = A + jB, then C = D- jDBA where D =(A + BA -1B)-1 Thus a computer program which inverts real matrices can be used to invert complex matrices. A program using this technique called CMINV, is listed in Appendix A. This technique was applied in calculating the noise figure from (3. 23) to obtain the plot of noise figure in Fig. 3. 17 for the 5 x 5 matrix approximation. These curves again show the inverse relationship between noise figure and gain. The gain-bandwidth product is defined as the square root of the maximum transducer power gain multiplied by the- fractional bandwidth f- f f max

96 30 29 -- - iLSUC1 28 28 --- LS UC2 27 - LSUC3 26 25 24 23 22 21 20 m 19 8 18 17 16 C\ 15 /~~ // 14 m 13 12 11 9 6 5 4 3 2 1 0. 997 0. 998 0.999 1. 000 1. 001 1. 002 1. 003 Signal Frequency, GHz Fig. 3.16. Transducer power gain for the three LSUC circuits when ideal impedance matching circuits and the 5 x 5 matrix approximation are used

97 2.0 - --- LSUC1 - -LSUC2 1. 9 LSUC3 1.8 1. 7 1. 6 1. 5 / 1.4 x 1.3 / 1.2 1. 1 0.997 0.998 0.999 1.000 1.001 1.002 1.003 Signal Frequency, GHz Fig. 3.17. Noise figure for the three LSUC circuits when ideal impedance matching circuits and the 5 x 5 matrix approximation are used

98 where fmax is the frequency of maximum gain and fa fb are the half power frequencies. The gain-bandwidth product is shown in Fig. 3. 18 for the standard diode model of Section 3. 4. 1 and in Fig. 3. 19 when the diode series inductance is reduced to 0. 2203 nH where the diode is series resonant at the lower sideband frequency. The gain-bandwidth product is plotted as a function of the elastance modulation coefficient. Since the maximum gains for all three circuits are almost the same (Fig. 3. 10) the gain-bandwidth product for LSUC3 is larger than that for the other two circuits when the standard diode model is used. 3. 4. 4 Effect of Signal Circuit Length. Shortening the signal circuit length and finding the corresponding approximate optimum lower sideband and pump port distances in the LSUC, increases the gain-bandwidth product. The curves of Figs. 3. 18 and 3. 19 show that when S1/SO = 0. 35 the gain-bandwidth product increases by a factor between 1. 22 and 1. 64 depending on the diode inductance, when the signal port distance is decreased from s/4 to Xp/4. Contrary to what is expected the gain-bandwidth product for LSUC2 decreases rather than increases. The reason for this behavior is unclear, but it may be related to the lack of a minimum in the "slope function" (3. 47) for LSUC2 when the signal port distance is X /4. The impedance slope for this case monotonically increases with pump and lower sideband distances, so that the values used in Table 3. 2 are actually the minimum practical distances. 3. 4. 5 Effect of Diode Parasitic Inductance. The diode parasitic element values were seen to have a major effect on the LSUC where the three circuit ports were completely isolated from one another (Figs. 3. 14 and 3. 15). For the three circuits discussed here, if the diode series inductance is reduced from 2 nH to 0. 2203 nH so that the diode is series resonant at f20 the gain-bandwidth curves are significantly changed. Comparison of Figs. 3. 18 and 3. 19 shows that each LSUC circuit has a larger gainbandwidth product when the diode parasitic inductance is decreased. The largest increase occurs with the LSUCj1 where the gain-bandwidth product increases by a factor between 15. 3 and 18 depending on the choice of signal circuit. The curve in Fig. 3. 20 shows explicitly the effect of diode inductance on LSUC1 when the signal port distance is Xs/4. Similar curves for LSUC2 and LSUC3 are difficult to obtain because the optimum pump and lower sideband port distances change when the diode model changes.

99 0. 040 0. 038 - - - LSUC1 - LSUC2 0. 036 -U LSUC3 0. 034 - 0. 032 - 0. 030 - 0. 028 - 0.026 - p/4 0. 024 - 0. 022- s/4 '/ p/4 ma3 0.018~ ~ / p 0.020 // 0-1 a / X, I/ 0.018 - 0.016. - 0.0140 /A/4! / s 0. 008 0. 004 0. 002 0.004 0.1 0.2 0.3 0. 4 0. 5 0. 6 0. 7 Elastance Coefficient S1/SO Fig. 3.18. Variation of gain-bandwidth product (G2w) versus pumping level with signal circuit distance as a parameter, using ideal impedance matching circuits, the 5x 5 matrix approximation, and standard diode inductance of 2 nH

HU g0gg '0 Jo oauVI -onput aoSeod apoip v put 'uoi.uai. xoLddt xalxtm g x g ayl 'sllnoXio 2umqyDltu aoutpadui. IeapT 2usn 'lalaxotad V st aouelsip I!n.xx.; Iu~.s qpiM laaal Suidund snsa3A (M.o) lonpoad qlpp.mpueq tuqe Jo uo~lW.leA '61 'g '.d T / Xt01 0S/ IS 4uaajjqoiw/ a3u, - js 'O 9'O 9 'O P 'O 0 'O V0 O 'O -90 o // /l t/d y T/ / /4 / / I I p/"/ / I: OZ 'O ~ I 89 '0 I -f r 9'0 I ~ *Q 001

0. 40 0.38 0.36 0.34 0.32 0.30 0.28 0.26 0.24 0.22 0.20 m. 0.18 \ Q 0.16 0.14 0.12 0.10 \ 0.08 - 0.04 _ 0. 02 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 Diode Lead Inductance, nH Fig. 3. 20. Variation of gain-bandwidth product (G2w) with diode package inductance for LSUC1, using the 5 x 5 matrix approximation and a signal port distance of Xs/4 S~~\

102 3. 4. 6 Effect of Shunt-Stub Tuners. Two common impedance matching mechanisms are the single-stub and double-stub tuners. In this section a comparison is made between the ideal impedance matching elements used previously and these two types of tuners. The single-stub tuner can match any impedance by choosing the proper stub position and stub length. The double-stub tuner can match a given range of impedances which is dependent on the distance between the two stubs. In the numerical work, a stub distance of t/ 8 at the center of the respective passband frequencies is used as this usually provides an adequate matching range. These tuners are placed at least the distance given in Table 3. 2 away from the diode. At out-of-band frequencies the ideal filters which act as short-circuits are located the distance specified in Table 3. 2 away from the diode. The change in the gain-bandwidth product is negligible when the ideal matching circuits are replaced by either single-stub or double-stub tuners. The largest deviation occurs with the LSUC3 circuit where the single-stub tuner degrades the gain-bandwidth product by only 8. 9 percent. The double-stub tuners failed to provide the required matching in LSUC1, but by changing the pump and lower sideband port distances a circuit was found in which the double-stub tuners work. In this revised LSUC1 circuit, the gain-bandwidth product is degraded much less than 8. 9 percent when the double-stub tuners are used in place of ideal matching circuits. 3. 4. 7 Conclusions on the Use of the Three LSUC Configurations. The results of the previous sections are briefly summarized in Table 3. 3 where each value uses the approximate optimum lower sideband and pump port distances shown in Table 3. 2. For the diode and frequencies used here, LSUC3 provided the best performance for all cases except one. The best value in Table 3. 3 however still falls short of the maximum obtainable gain-bandwidth product for theiunpackaged diode (Fig. 3. 13) by a factor of 7. Therefore, even though the simple theory presented in Section 3. 2 is useful for designing LSUC's an analysis similar to that presented here in Section 3. 4 is still needed to evaluate final upconverter performance.

103 L = 2 nH Ideal Circuits L = 2 nH Signal Dist. = X s/4 Signal Dist. = s/4 Signal Dist. = X /4 Single-stub Double-Stub Circuit L = 2nH L = 0. 2203 nH L = 2nH L = 0. 2203 nH Tuner Tuner LSUC1 0. 01446 0. 2205 0. 02037 0. 3616 0. 01357 LSUC2 0.01201 0. 0511 small small 0.01190 0. 01184 LSUC3 0. 02090 0. 06203 0. 02558 0. 1733 0. 01905 0. 02102 Table 3. 3. Gain-bandwidth product when S1/S0 = 0. 35 In practical situations, the choice of a circuit configuration is often dictated by criteria such as the frequencies involved or the type of transmission line that is to be used. The LSUC1 structure would probably not be useful in a coaxial tee or wye junction, because the diode would have to be mounted between the center and outer conductors. The fields surrounding the diode would be distorted in some unknown manner, making an analytical design difficult. However, analysis of this structure is performed because of the light it sheds on the circuit in which the diode is mounted between the two conductors in a microstrip tee or wye junction. Here a reasonable approximation could be made of the field configuration around the shunt mounted diode. The LSUC2 design is the obvious solution to the field distortion problem associated with a shunt mounted diode in a coaxial tee junction. Although an experimental LSUC was designed and built using this approach, several empirical modifications were required to obtain satisfactory operation. This deficiency is a result of the nonideal tee junction characteristics. The LSUC3 design attempts to use the nonideal tee junction characteristics advantageously. Table 3. 3 shows that the gain-bandwidth product for this design is superior to the LSUC2 design, as well as being the most practical in a coaxial system. 3. 5 LSUC Design with Coaxial Band-pass Impedance Matching Circuits In the previous sections the three LSUC circuits studied used ideal filters and either ideal impedance matching circuits or shunt-stub tuners. In the following sections these three LSUC configurations are investigated using the physically realizable coaxial

104 impedance matching transformers developed in Chapter II. These filter networks provide both the filtering and impedance matching functions and thereby replace the tuner and ideal filter used in the earlier models. The "standard" diode model described in Section 3. 4. 1, is used throughout the following discussion although the element values are sometimes changed. General design rules applicable to all three LSUC circuits are outlined in Section 3. 5. 1. The subsequent sections describe the individual design methods needed for each circuit configuration. 3. 5. 1 General Design Procedure for the Three LSUC Circuits. In designing each of the three LSUC's the pump circuit is designed first, then the lower sideband circuit, and finally the signal circuit. The design method described here does not use an iterative process so each circuit is designed independent of the other two. In the first two LSUC circuits described, this assumption means that ideal band-stop filters are needed. However, the third circuit, LSUC3, does not use any ideal band-stop filters but uses realizable distributed elements only. 3. 5. 1. 1 Pump Circuit. The pump circuit is designed first because, of the three, it is the least critical to the LSUC operation. The basic problem is to transfer power at the pump frequency to power at the signal and lower sideband frequencies. If the varactor were lossless, the power at w, which is converted to power at wl, can be modeled by the resistance induced into the pump circuit by the signal voltage. Even in a lossy varactor, pump power must be "absorbed" in this induced resistance in order to produce parametric gain. Perlman and Bossard (Ref. 45) obtained an approximate expression for this induced resistance when all voltages at frequencies > f2 were shorted. Since the open circuit assumption has been used here, an approximate expression for the induced resistance when all currents at frequencies > f2 are open circuited is given here. The varactor elastance as a function of stored chrage is S(Q) = S(0)[1 (1-y)s(o)Q] l)y (3.48) = S(O) + a1Q uQ+ 2Q2. (3. 49)

105 where dS S( 02 (1-y )S(0)Q 1-7 1 dQ Q 0 v OV The approximate voltage across the varactor is V = [S(O) + r1Q]Q If the charge is given by Q = 2Qp cos(Wpt)+ 2Q1 coS(w1t) + 2Q2 cos(w2t + '2) and assuming S(O)Q can be resonated, the voltages at wp, 10, and 0 are V = 2u11Q2 (3. 50) V1 = 2aQQp (3. 51) V2 = 2lQu1Qp (3. 52) where these voltages and charges are coefficients of ewt. Manipulation of these expressions give R P- -P - -- (3. 53) Rind i R (3. 53)2 In a typical LSUC where f = 9. 500GHz, = 8. 500GHz, R 3.1ohms, y= 1/3, 12 -1 -8 0v= 0. 7 volts, and S(O) = 10 F, the induced resistance is 4 x 10 8 ohms when V1 = 20 KV and 1 ohm when V1 =98 mV. If the pump circuit were as shown in Fig. 3. 21a, it would be unnecessary to match to R.ind In fact the smaller the pump source resistance, the more power there is available to the induced resistance. If however there is an ideal impedance transformer between Rind and the pump source (Fig. 3. 21b), then the power delivered to Rind is

106 Z Z o o 9 _ n':l V V P R. P R. (a) (b) Fig. 3. 21. Pump circuit without (a) and with (b) ideal transformer n 2R V ind p (n'2 R + Zo) ind 0 2 This is maximized with respect to the transformer ratio n' when Rind = Z/n'. As noted above, Rind is usually too small to be matched with available microwave impedance transformers, so a value of one ohm is used as the transformed pump source resistance. The design of the pump impedance transformer is based on the load being the varactor diode model including the parasitic elements (Fig. 3. 11) in which Rs is replaced by the one ohm resistor and the average elastance S0 is used. The resonated load Qc at f needed for the filter design procedure is obtained, and with the desired p fractional bandwidth, which in this case can be fairly narrow, the decrement 6 = 1/ (wQ) can be found. From 6 the low-pass prototype g values and the K values for the impedance inverters are found (Fig. 2. 8). The impedance transforming network is then designed using the distributed design procedure. 3. 5. 1. 2 Lower Sideband Circuit. The lower sideband circuit is designed so that the impedance presented to the diode variable elastance at the center of the lower sideband frequency band, f20, is Re + Rs The external circuit impedance is found by substituting the load resistance Rf for Rs in the diode equivalent circuit (Fig. 3. 11) and resonating this circuit at f20. In the LSUC1 and LSUC2 circuits this modified diode is resonated with two external circuit elements, while in the LSUC3 circuit the diode is resonated with only one external circuit element.

107 In the first two configurations the package capacitance is resonated by a shunt inductive reactance. This reactance is provided by adjusting the distance between the end of the pump circuit and the diode. Since f20 is an out-of-band frequency, the impedance of the pump circuit is almost entirely reactive. Using this approximation the angular distance between the pump circuit and the diode is readily obtained from the transmission line equation: zo[x- Xp(wo20)] 0= Arctan [ (3. 54) P [Z 2 + X X (W PI[ ZO +~X. X (w20)] where X = + W20Cp Although this procedure will change the impedance presented to the diode at f, pump power can be increased to overcome the additional mismatch. The actual resulting pump port impedance is obtained later for the LSUC3. The remaining diode reactance, which at these frequencies is inductive, is resonated by a semilumped series capacitance placed X s/2 away from the diode in p- the lower sideband port. The signal port impedance is eliminated from consideration by using ideal band-stop filters at f20 making Zs(f20) = co for LSUC1 and Zs (f20) = 0 for LSUC2. The diode now being resonated, the lower sideband band-pass impedance transformer can be designed using methods discussed in Chapter II. A slightly different method is used in the LSUC3 design. Here the total reactance of the diode (not just C ) is resonated by adjusting the pump circuit to diode disW20cp tance. This procedure is practical when the diode parasitic reactances transform the Rk to a value that can be easily matched. It will be shown later that the condition Zs(f 0) = oo can be obtained without using ideal band-stop filters, so the signal port need not be considered in the lower sideband port design. Having resonated the diode, the lower sideband circuit is designed as before. 3. 5. 1. 3 Signal Circuit. The signal circuit is designed so that the impedance presented to the diode variable elastance is R + Rs at the center of the signal frequency gs

108 band f10 ' The design of this circuit differs from the procedures used on the other two ports for two reasons: (1) f10 is the comparatively low frequency of 1 GHz, and (2) Rg is a relatively large value between 100 and 150 ohms. The low signal frequency implies that a filter consisting of a series of impedance inverters separated by half wavelengths of line would be impractically long. A better method is to use either alternate K and J inverters separated by X/4 lengths of line or to use a low-pass Chebyshev matching network (Ref. 46) based on semilumped inductors and capacitors. The relatively large value of R means that when Rs is replaced by R in the g g diode model (Fig. 3. 11), the resulting impedance can be easily resonated by simply inserting a length of transmission line between the diode and the signal circuit.. This approach differs from that used for the lower sideband circuit where an external reactance was used to resonate the diode. The procedure for designing the signal circuit begins with obtaining the diode impedance Zd = Rd + jXd as shown in Fig. 3. 22. Zd L S0 Zi = R' + jo R in s -g Fig. 3. 22. Circuit model used to design the signal circuit The diode impedance Zd seen a distance 0 away from the diode terminals is 0 d 2 - Z do(l + tan 0) + jZo[ (Z O Rd - Xd) tan 0+ ZOXd( -tan 0)] Zin 2 2 (Zo Xdtan 0) + (Rdtan e) The distance 0 =0 where Z. is resonate is s in 2 2 2 2 2 2 Z -R-X Z -R-X o d d o d d(3.55) tan0 22Zo X od od

109 and the corresponding input resistance at the signal frequency is RdZ (1 + tan2 s) R' = o 2 2 (3. 56) (Z - X tan )+ (Rd tan s) Once s is known, R' can be found. The two values of s in (3. 55) are a quarter wavelength apart. The value of 0 which gives R' < Z is the value chosen in the S S - 0 designs used here. When the values for the diode model of Section 3. 4. 1 and R = 100 ohms g are used, the resulting R? 11. 48 ohms and s = 1. 108 radians or 5. 38 cm at 1. 000 GHz. S S This value of Rs can be easily matched with either the band-pass or low-pass impedance matching circuits. In designing the signal circuit, it was assumed that the lower sideband and pump circuits could be ignored. This means that at f10 Zp - Zps = co for LSUC1, Z I]Z = 0 for LSUC2, and Z = oc, Z = 0 for LSUC3. Appropriately placed p p-s p p-s band-stop filters in these two out-of-band ports is an impractical method for obtaining these impedances because of the large size of the filters. However in Chapter II it was shown that the coaxial band-pass filters will pass low frequencies. This means that the positions of high-pass filters at the input of the pump port and at the output of the lower sideband port could be adjusted to provide the desired impedances at f0 ' Since both the pump and lower sideband frequencies are in X-band, these high-pass filters can be realized simply as X-band coax-to-waveguide adapters. 3. 5. 2 Special Design Considerations for LSUC1. The LSUC1 circuit was designed and analyzed using the band-pass circuit in the signal port. Fortuitously the signal circuit reactance at f20 is large, so the ideal band-stop filter in the signal circuit is not used. The resulting gain curve (Fig. 3. 23) shows this circuit has a bandwidth of approximately 4 MHz. 3. 5. 3 Special Design Considerations for LSUC2. For LSUC2 the circuit represented in Fig. 3. 10b must be designed so that (1) pump power can get into the diode Z s(Wp) = oo and Zs(Wp) = 0, (2) signal power can get into the diode Zp(W10) I I Zp s(l0) = 0, (3) lower sideband power can leave through the lower sideband circuit

30 29 28 27 26 25 24 23 22 21 20 19 18 ~* ~17 0 16: 15 14 13 d 12_ 11 10 9 8 7 6 5 4 3 2 0 0.98 0.99 1.00 1.01 1.02 Signal Frequency, GHz Fig. 3. 23. Gain of LSUC when all filters in parallel with one another

Zs(20) =, Zp(w20) =, and (4) the impedance seen by the voltage dependent capacitance at f10 is Rg + Rs and at f20 is Re + Rs No special consideration is given to requirement (1) since the amount of pump power can be changed to compensate for mismatch. If this assumption is unwarranted, additional band-stop filters may be needed to fulfill condition (1) more closely. Condition (2) is fulfilled by using the high-pass filters external to the pump and lower sideband ports. As for condition (3), Zs(w20) = 0 is accomplished by placing a band-stop filter Xp-s/4 away from the diode in the signal port, and Zp(a20) = ( is accomplished by resonating the diode package capacitance with the pump circuit. 3. 5. 3. 1 LSUC2 with the Low-pass Signal Matching Circuit. The signal circuit used in this model is a pseudo low-pass matching network described by Matthaei (Ref. 46). The circuit chosen for this LSUC is a 10 pole Chebyshev network which requires 5 shunt capacitors and 5 series inductors. They can be realized as coaxial disks and No. 20 wire (0. 03196 in. dia. ) respectively, and the resulting network is less than 2 inches long. The complete LSUC2 circuit is shown in Fig. 3. 24. From the numerical analysis shown in Fig. 3. 25 the bandwidth for the diode of Section 3. 4. 1 is only 9. 0 MHz. It is evident from Fig. 3. 25 that the diode parasitic elements have a strong effect on the amplifier bandwidth. Physically this low-pass circuit is difficult to realize because some of the series inductors are very short allowing the neighboring disks to introduce a sizeable series capacitance. 3. 5. 3. 2 LSUC2 with the Band-pass Signal Matching Circuit. The signal circuit used here in Fig. 3. 26 is a three pole band-pass impedance transformer discussed in Chapter II. The graphs in Figs. 3. 27 to 3. 29 and Table 3. 4 show the resulting gain, bandwidth, and gain-bandwidth product for this circuit. The different graphs indicate the variation in the characteristics when the LSUC is designed for diodes with different parasitic element values. The different gain characteristics for the 2 x 2 and 5 x 5 matrix approximations are clearly evident in these plots. The noise figure shown in Fig. 3. 30, which corresponds to Fig. 3. 27, resembles the reciprocal of the gain curve, and the minimum noise figure corresponds closely to the design objective expressed by (3. 25). The sharp dips in the gain curve illustrate the importance of including in the gain calculation the higher order sidebands. These calculations confirm the theory presented

f S Coax to Band-stop Coax to Waveguide Filter W Waveguide f / /4 ' l~p-s U~~~~~~~~~~~~~TP 1 Tuning Capacitance Fig. 3. 24. LSUC1 circuit using a semilumped low-pass signal circuit

30 29 C = 0. 7 pF, L =2 nH 28 - C = 0. 5 pF, L = 2 nH p 27 - C = 0.7 pF, L = 1 nH p 26 25 24 23 22 21 20 19 18 / \, \ / 13 c 2 _ 16 / o 15 14 / 13 2 C-,// 10, O. 98 -. 99 1. 00 1.01 1.02 Signal Frequency, I7Hz 8 -- \~~~~~~~~~~~~~~~ \. 0.98 0.99 1.00 1.01 1.02~~~~~~~~~~~~~~~~~~~~~~~N 7~~~~~~~inlFeuny ~ 6i.32.Gi fLU sn oxa ieswt id ncne odco

I IS Coax to Band-stop Coax to Waveguide Filter Waveguide Adapter Diode - p Adapter ~ /4 f f p p-s Tuning Capacitance Fig. 3. 26. LSUC1 circuit using a distributed band-pass signal circuit

30 29 2 x 2 Matrix 28 5 x 5 Matrix 27 26 25 24 23 22 19 18 a 5 \\ 14 13 12 11 / x15S i \\. 8.9 101 11 15 3 2 1 Signal Frequency, GHz =6.28 101 F-, and S = 0.35

30 29l 28 8 -— 2 x 2 Matrix 27- 5 x 5 Matrix 26 25 24 23 22 21 20 \ 19 I PI t go It It i: l l l l l l l l l I 18 17 15 13 / cd 14 12 / 11 / 10 - 9 I 8 \ 7 6 5 4 -3 2 0 0.85 0.90 0.95 1.00 1.05 1.10 1.15 Signal Frequency, GHz Fig. 3.28. LSUC transducer power gain using the coaxial microwave filters when the diode model has the following elements: Rs = 1 ohm, L = 2 nH, Cp = 0. 5pF, SO= 6.28 1011 F-1, and S/S= 0.35

30 29 28 — --- 2 x2 Matrix 27 5 x 5 Matrix 26 25 24 23 22 21 18 -~ 17 / 16 - 15 143 cl) 12 7 11 - 10 9 8 7 6 5 4 3 2 0.85 0.90 0.95 1.00 1.05 1.10 1.15 Signal Frequency, GHz Fig. 3. 29. LSUC transducer power gain using the coaxial microwave filters when the diode model has the following elements: Rs = 1 ohm, L = 1.428 nH, Cp = 0.7pF, S0 6.28 101i F', and S1/S0=0.35

2.0 2 x2 Matrix 1.9 _ _5 x5 Matrix 1.8 I 1.7 [ I 1. 6 1.2 -1.o. III I I I 1.3 / / 1.1 0.85 0.90 0.95 1.00 1.05 1.10 1.15 Signal Frequency, GHz Fig. 3.30. LSUC noise figure using the coaxial microwave filters when the diode model has the following elements: Rs = 1 ohm, L = 2 nH, Cp = 0. 7 pF, S0= 6.28 1011 F-, and S1/So = 0.35

in Section 3. 3 on the effects of the reactance at the upper sideband frequency on the midband gain. Matrix Size L nH C pF Max. Gain BW GHz G w Freq. of Max. dB Gain GHz 2x 2 2 0.7 21.04 0.031 0.349 0. 995 5x 5 2 0. 7 20.64 0. 032 0. 345 0. 990 2 x 2 2 0. 5 20.98 0.031 0. 348 0. 995 5x 5 2 0. 5 19. 23 0.022 0. 201 1. 015 2x 2 1. 428 0.7 21.36 0.086 1. 009 0. 990 5x 5 1. 428 0.7 21.44 0. 038 0.449 0. 985 Table 3. 4. Gain and bandwidth of the LSUC2 using coaxial band-pass filters in all three ports 3. 5. 4 Special Design Considerations for LSUC3. The primary reason for considering this configuration is that experimentally it was found (see Section 4. 3. 2) that at X-band frequencies most power entering one arm of the coaxial tee junction will go to the opposite port while most power entering the leg will be reflected back. Therefore, the diode is placed in the lower sideband circuit to avoid complications from the nonideal tee junction. The only unresolved design problem is the isolation of the signal port from the diode at the lower sideband frequencies. To insure this isolation the entire signal circuit may be moved toward or away from the tee junction without changing the electrical properties of the structure at f10 as long as the total distance between the diode and the signal filter remains constant. If, for example, the signal circuit is moved away from the junction, the diode and lower sideband circuit must be moved the same distance toward the junction. This degree of freedom allows one to choose the distance between the junction and signal circuit so that Zs(f20) = o. There are thus two mechanisms that isolate the signal circuit from the power at the lower sideband frequency -- the nonideal tee and the optimum placement of the signal circuit. The LSUC3 has been designed using only realizable distributed elements without recourse to an iterative process. Thus this circuit may be readily tested experimentally. Following the design procedure given in Section 3. 2. 6, the input signal frequency is

120 given as f10 = 1. 000 GHz. The diode used in this design was measured according to the method outlined in Chapter 4 with the following results: S = 1. 049x 1012 FL = 1. 840 nH Cp = 0.1675pF R = 1. 41 ohms The assumed pumping modulation coefficient is chosen as S1/SO = 0. 35 which from (3. 31) gives the optimum frequency ratio r t = 66. 72. This ratio is much too high for the chosen signal frequency, available circuit components, and available pump sources. Since the specifications for the 14-mm 50-ohm coaxial line are exceeded above 8. 50 GHz, the center lower sideband frequency is chosen at f20 = 8. 500 GHz. Although the minimum noise figure is therefore not achievable, a glance at (3. 25) F2 = (+ )( + 1) + (3. 25) g shows that the noise figure is insensitive to the choice of x as long as xg is sufficiently large. If xg is chosen to be 100, then from (3. 13), ye = 2. 201, or R = 141 ohms and Rf = 3. 104 ohms. When these parameters are used in the algorithm described here, the circuit specified in Table 3. 5 and Fig. 3. 31 results. The analysis of this design is shown in Fig. 3.32. When the diode is resonated at f20 by adjusting the distance between the pump circuit and the diode, a significant mismatch at f can occur, although the effects of this mismatch can be overcome by increasing the pump power. Analysis of this design shows that the impedance external to the diode at f is 0. 456 + j 49. 64 ohms and the P transformed impedance seen by the variable elastance of the diode is 1. 791 + j 191 ohms. The main computer program, used to generate this design and the accompanying analysis, is found in Appendix A as program B. The necessary subroutines required with this program are also listed there.

Relative d i Distance d. Length. Diameter Dielectric w inches inches inches Constant inches r. Pump Circuit 1 0. 6956 0. 1600 0. 542 2. 03 2 0. 6333 0. 08292 0. 532 2. 03 3 0. 6797 0. 03897 0. 502 1. 00 1. 833 Lower Sideband Circuit 1 0. 7394 0. 03791 0. 502 2. 03 2 0. 7500 0. 1446 0. 502 2. 03 3 0. 7778 0. 02022 0. 502 1. 00 5. 893 Signal Circuit 1 2. 8337 0. 6332 0. 502 2. 03 2 2. 6940 1. 1066pF ----- 3 3. 4221 0. 09781 0. 502 2. 03 dtv = 1. 7817 in., dst = 1. 0520 in. dl in the pump circuit is the distance between disk 1 and the tee junction. dl in the other two circuits is the distance between disk 1 and the diode. Table 3. 5. Design parameters for the LSUC when the diode is in the lower sideband port 3. 6 Conclusions This chapter contains a theoretical discussion on the design and properties of a LSUC. The simple design relationships reviewed in Section 3. 2 were applied to LSUC circuits that used a coaxial tee junction in three different ways: (1) the diode is mounted between the center and outer conductors, (2) the diode is mounted in the center conductor at the tee junction, and (3) the diode is mounted in the center conductor in the lower sideband arm of the tee. These three configurations were investigated using ideal filters for frequency isolation of~the separate ports, and ideal or shunt-stub tuners for impedance matching. Later these elements were replaced by the coaxial band-pass impedance matching transformers discussed in Chapter II. The methods employed here are valuable in designing and analyzing LSUC circuits with different diodes or tuning elements, but because of the large number of variables no general conclusions can be

f S Coax to Coax to Waveguide Waveguide Adapter d Diode Adapter St f i —bj402& P-s p3 p2 p1 Fig. 3. 31. LSUC3 using coaxial disk band-pass impedance transformers

123 30 29 28 --- 2 x2 Matrix 27 5 x 5 Matrix 26 25 24 23 22 21 20 19 18 17 ~d 16 15 14 - 13 ad 12 11 10 9 8 7 6 5 4 3 2 1 0.85 0.90 0.95 1.00 1.05 1.10 1.15 Signal Frequency, GHz Fig. 3. 32. LSUC transducer power gain using coaxial microwave filters when the diode is mounted in the lower sideband port. The diode model elements are Rs = 1.41 ohm, L = 1.834nH, Cp = 0. 1633 pF, S0 = 1.0456 1012 F-, and S1/SO 0.35

124 drawn as to which circuit is best. However the LSUC3 circuit is superior in coaxial systems for practical experimental reasons. The analyses of these circuits point out the importance the diode parasitic elements play in restricting the bandwidth of the amplifier, the importance the reactance at the upper sideband frequency has on the performance of the upconverter, and the effect of the nonresonant gain maximum. The work in this chapter culminated with a detailed design of a LSUC using coaxial filters, which, with a minor modification, accurately models the experimental LSUC3 discussed in Chapter IV.

CHAPTER IV EXPERIMENTAL EVALUATION OF THE LOWER SIDEBAND UPCONVERTER 4. 1 Introduction The previous chapters have dealt with design considerations for the LSUC and the practical problem of constructing band-pass impedance matching circuits in coaxial line. Since the circuit design is heavily dependent on the values of the diode equivalent circuit elements, these diode elements were measured first. The method and results of this measurement are outlined in Section 4. 2. Since three frequencies must enter or leave the amplifier through a coaxial tee circuit, the properties of the impedance transformers and the tee junction were also measured, and are reported in Section 4. 3. Using this information two upconverters were designed and built; detailed measurements on the second one showed a close correspondence to the design theory. The measurements of gain, bandwidth, and noise figure are presented in Section 4. 4. Since the measured noise figure is larger than that predicted from thermal noise sources alone, other sources of noise are considered in this section to explain the discrepancy. 4. 2 Diode Measurement The packaged diode is a complex circuit element that presents several measurement problems to the experimenter. The resolution of these difficulties has been the subject of numerous papers of which the most pertinent are reviewed below. From the three basic methods presented in these papers, the third is chosen to measure the diode used in the experimental LSUC design. 4. 2. 1 Diode Measurement Review. The equivalent circuit for the varactor diode shown in Fig. 3. 5 of the previous chapter is actually a simplified version of a more complex structure. The diode is considered to be physically much smaller than the wavelength at its highest operating frequency, so it can be represented by a lumped circuit. An equivalent circuit for a packaged varactor diode together with its more 125

126 common simplified circuit is shown in Fig. 4. 1. To be a valid equivalent circuit, the circuit elements must be independent of frequency. The depletion layer capacitance, lead inductance, and package capacitance are frequency independent for microwave frequencies through X-band. The bulk series resistance Rs is usually assumed frequency and voltage independent. However, careful measurements show this is not the case. At frequencies higher than 10 GHz, R increases with frequency because of the skin effect resistance (Ref. 47). The applied voltage across the diode can vary Rs by two mechanisms. First, when the applied voltage varies the depletion layer capacitance, it also changes the width of the epitaxial base material (Ref. 48, p. 69). This is sufficient to cause a large variation in the series resistance. For an abrupt junction diode this variation is given by R = p(W -W)/A s max where p is the resistivity, A is the diode area, and W is the depletion layer width. Second, in a packaged diode the fringing capacitance will cause the effective series resistance Rs to increase with applied voltage (Ref. 49). This can be seen by equating the impedances of Fig. 4. l(a) to that of Fig. 4. l(b). The effective resistance is RX s 2 R + (Xj + Xf) R 2 l>>wCfR [ 1 + Cf/Cj ()]2 where Xf and X. are the capacitive reactances of Cf and Cj respectively. For zero and reverse bias the junction conductance G. is very large and therefore can be safely neglected. The fringing capacitance commonly associated with the capacitance between the pillar and the top cap is also considered small enough to be neglected in most cases. However, as was just seen, this assumption may not always be warranted. One of the difficulties in accurately characterising a varactor diode is now already evident: the semiconductor region is accessible only through the equivalent components of the package. There are in addition other difficulties. Special care must be

127 L L ~R I ~~R C CT OpC S P f cj(v) I c(v) (a) (b) Fig. 4.1. (a) The varactor diode equivalent circuit and (b) the modified simple equivalent circuit exercised in measuring Rs as this is usually between 1 and 5 ohms; the high quality factor Qc of the diode implies that small losses in the measuring system can produce significant errors in the measurement of this important parameter. Also the diode is nonlinear, so low RF power must be used to avoid pumping the diode. In one method to be described, it is desirable to obtain the parallel resonant frequency of the diode. In some cases, this frequency is so high that TEM mode coaxial line is not available while impedance measurements in waveguide suffer from having a nonunique characteristic impedance. These difficulties have resulted in basically three measurement techniques: (1) low-frequency bridge measurement of the capacitance, (2) single-frequency measurement of the diode Qc at varying bias voltages, and (3) measurement of diode impedance including package parasitic elements, performed at a single bias voltage but over a range of frequencies. In the first case the package parasitic elements can be neglected; in the second case they are tuned out and the diode Qc is measured directly, while in the third case, the diode is measured with the package parasitic elements. 4. 2. 1. 1 Method 1. In the frequency range between 100 kHz and 10 MHz Rs and L can be safely neglected so that the total measured capacitance as found on a transformer ratio-arm bridge is CT = Cp+ Cf+ Cj(V) C + C(V) P

128 Crook (Ref. 50) assumes Cf is negligible and separates Cp from C(V) by an independent measurement of Cp on an empty diode cartridge. Roberts and Wilson (Ref. 49) describe an alternative way of separating these two capacitances. The capacitance law, i. e., y and the built-in potential can be found by measuring a large number of diodes at two bias values, e.g., 0 and -6 V. From the slope of the straight line plot of CT(-6) versus CT(O) - CT(-6), the capacitance law can be determined. Knowing this law, two bias levels enable finding Cp and C(V). Data given by Roberts and Wilson (Ref. 49), however, show these low-frequency measurements are in error by over 0. 1 pF in 0. 28 pF at -6V when compared to microwave measurements. Since the diode is to be used at microwave frequencies, the high-frequency measurement values are to be preferred. Therefore, method 1 is not recommended for diode characterization. 4. 2. 1. 2 Method 2. This method provides a direct method of finding the diode Qc = 1/[ wRsC(V)]. It is based on normalization of the measured impedance to Rs rather than Z, and impedance matching the diode at a particular frequency and bias with a lossless transformer. Impedance matching is not strictly necessary, but is readily accomplished and simplifies the mathematics. Houlding (Ref. 51) used this technique to find the diode Qc by determining the change in reactance with a change in bias. However, his method requires knowledge of the capacitance-voltage law for the particular diode. Harrison (Ref. 52) soon afterward described a method of finding the Qc without having to know the capacitance-voltage law of the diode. In this method the bias on the matched diode is varied, the impedance is plotted on a Smith chart, and this plot is rotated until it coincides with the unit circle. The location on the unit circle where the diode is a short circuit is numerically equal to the diode Qc ' Harrison showed that this short circuit can be provided either by replacing the diode with a shorted package or by forward biasing the diode until it is an effective short. This latter technique works well for silicon diodes, but in gallium arsenide diodes the effective short is obtained near maximum permissible power dissipation. Mavaddat (Ref. 53) extended Harrison's method by eliminating the need of a short-circuited diode. He shows that a plot of 1/AC versus 1/AX for various bias values gives 1/Qc as well as 1/C(V)

129 at the intersection of the curve with the 1/AX and 1/AC axes respectively. The value of C(V) so obtained does not include the fringing capacitance Cf. Criticism of this general approach was made by Hyde and Smith (Ref. 54) who showed, after analyzing several loss mechanisms, that the diode Qc found by this method may lead to erroneous results. Their conclusions may be summarized as follows. Additional series resistance whether within the diode cartridge or between the diode and the plane of measurement always reduces aQc (or aX). Distributed loss either inside or outside the cartridge causes the relative impedance locus to lie inside the unit resistance circle when reverse biased and outside it when forward biased. In addition AQc is reduced and the short-circuit point does not lie on the periphery of the chart. Shunt conductance loss, which occurs near the reverse breakdown voltage and the forward conduction region, results in an impedance locus that lies inside the unit circle in both regions but does not disturb the short-circuit point. When using method 2 it is necessary to plot experimental data on a Smith chart at an arbitrary reference plane. These data points are rotated to fit the unit circle as closely as possible. But it is clear that qualitative adjustment for the asymmetries that arise may be necessary, and the possible failure of the short position to lie on the periphery of the Smith chart could be caused by either an unknown loss mechanism or the lack of a good short. Since the mathematical representation of these alternate loss mechanisms is complicated, Hyde and Smith conclude it is not possible to extract lossless results from lossy data. Nevertheless, Sard (Ref. 55) does attempt to separate circuit losses from varactor losses by assuming the lossy matching network is a standard form. However, correcting for the circuit loss in series with diode bulk resistance requires extreme measurement care. 4. 2. 1. 3 Method 3. The first method is a low-frequency technique to find C(V), while the second is a high-frequency technique to find Qc ' The third method gives the diode equivalent circuit elements from impedance data as a function of frequency. This method can be divided into two techniques: transmission loss measurements of a diode shunt mounted in a waveguide and direct impedance measurement of the varactor diode when series mounted in a coaxial short-circuited line. The transmission technique was first proposed by DeLoach (Ref. 56) for the measurement of unencapsulated varactors. It requires measuring the ratio of the power

130 received by the load in the absence of the diode to that received when the diode is tuned to resonance, together with the two frequencies for which the transmitted power is doubled. With this information, the depletion layer capacitance, the series resistance, and the lead inductance can be found. This method has been extended to packaged diodes since at series resonance, the shunt package reactance is quite large. Roberts and Wilson (Ref. 49) use a variation of the transmission method to find Rs, L, and C(V). Since C(V) 2 4wr f2L a plot of 1/f2 versus bias gives a plot of MC(V) versus bias, where f is the series a a a resonant frequency and M is an unknown constant. With the frequency set to f at a a zero bias, the forward bias and reverse bias required to double the transmitted power are measured. The corresponding values of MaC(V) at these two bias values are read on the graph from which the Qc is readily calculated. This gives the cut-off frequency, fc = Qcfa f C(V), L, and the capacitance variation coefficient. The most obvious disadvantage of this method is that it provides no way of determining the package capacitance since the diode is assumed to be a series RLC circuit. Houlding (Ref. 57) has pointed out that the equivalent circuit of the more common diode packages (rather than the Sharpless package used by DeLoach) would be considerably more complex than a simple RLC circuit. Uncertainty in the exact transformation of the diode junction diode, compounded by the need to vary the frequency, would introduce further error in this method. In addition there is some doubt of the appropriateness of the choice of waveguide characteristic impedance used by DeLoach. The value he used was derived by Schelkunoff and is the impedance seen by a thin wire placed between two broad walls of a reduced height waveguide (small compared to X /4). Packaged diodes g usually cannot be considered as thin wires. Furthermore, a diode measured in a reduced height waveguide would be expected to operate in a similar diode mount, and this unduly restricts the circuit design. In none of the methods reviewed thus far is there a satisfactory method for obtaining the package capacitance C. Roberts (Ref. 58) has shown that all four elements P

of Fig. 4. l(b) can be obtained by series mounting the packaged diode in a short-circuited coaxial line and measuring the impedance over a broad range of frequencies. To obtain the three energy storage elements, the series resistance Rs is assumed negligible and three pairs of frequency-reactance variables are chosen. Roberts chooses the series resonant frequency fa fa/2, and the parallel resonant frequency fb. The corresponding reactances are X(fa) = 0, X(fa/2), and X(fb) = o. The corresponding parameters are then given by CP Tf X(f/22) [4 f)-' (4.1) a a a C(V) = Cp (f)1 (4. 2) 1 L = (4. 3) (2Tfa/) C (V) This method works well as long as fb < 18 GHz. Above this frequency, the unavailability of coaxial line and connectors makes this method cumbersome. Roberts and Wilson (Ref. 49) overcome this problem by recognizing that a minimum of transmission occurs at fb They, therefore, measure this frequency with the diode mounted in the inner conductor of the coaxial line between two coax-to-waveguide adapters. The series resistance may now be evaluated by one of the Q-measurement techniques described in method 2. This has the advantage of finding Rs at the operating frequency and the disadvantage of incurring additional circuit loss. Alternatively a straightforward SWR measurement at series resonance can be used. Account must be taken of losses in the measuring system, since these losses can easily mask the true diode resistance. Crook (Ref. 50) minimizes these losses by using a 10-ohm quarterwave impedance transformer between the slotted line and the diode. Bandler (Ref. 59) has shown for high standing waves

132 1 1 1 - = + (4. 4) m L max where Sm = measured SWR, SL = desired SWR of the load, and Smax = maximum SWR when the load is replaced by a short. The diode series bulk resistance is determined from SL rather than S L m 4. 2. 1. 4 Conclusions on the Three Methods. Table 4. 1 summarizes the references which should be consulted for measuring the various diode parameters. Of the three basic methods described above the first seems the least useful at microwave frequencies. In the design of a LSUC, the basic problem is knowing the values of the package parasitic elements, Rs, and the average depletion layer capacitance, while knowing Qc is of secondary interest. Hence, method 3 is the most appropriate. Author C(V) Rs L Cp Cf Qc Crook (Ref. 50) x x x Houlding (Ref. 51) x Harrison (Ref. 52) x Mavaddat (Ref. 53) x x x Roberts and Wilson (Ref. 49) x x x x x DeLoach (Ref. 56) x x x Roberts (Ref. 58) x x x x Bandler (Ref. 59) x Table 4. 1. References for measuring diode parameters Addition of a fifth element to the circuit in Fig. 4. 1(b) provides a more accurate representation of the physical diode. The fringing capacitance shunting the diode wafer and an extra inductance external to the diode package capacitance are two possibilities that have been suggested for a fifth element. Roberts and Wilson (Ref. 49) found that the fringing capacitance is larger than expected, and as has been shown, its presence can account for voltage dependence of the

133 effective series resistance. They found that the major part of the series inductance comes from the pillar on which the wafer rests (not the wires connecting the chip to the top cap) and that the magnitude of Cf cannot be explained only by the existence of capacitance between the top cap and the wafer. The pillar is actually a distributed line and the additional fringing capacitance can be explained by the capacitance from the sides of the pillar to the top cap through the ceramic ring (Fig. 4. 2). This distributed line can be approximated by an equivalent circuit that includes the fringing capacitance. An additional external inductance has been proposed by Houlding (Ref. 57) who criticized the four-element equivalent circuit for the error it produces, particularly at the higher frequencies. He proposed neglecting the fringing capacitance and including an extra inductance LT, which approximates additional transforming action of the package (Fig. 4. 3). Both of these five-element models give improved correlation with experimental measurements. However, the four-element equivalent circuit seems to be sufficiently accurate for the LSUC design. 4. 2. 2 Experimental Diode Measurement Results. The basic technique credited to Roberts (Ref. 38) has been employed to find the reactive elements of the circuit in Fig. 4. l(b). The transmission technique was not used because, in the LSUC, the diode in not shunt-mounted in a reduced height waveguide. Furthermore, the objection to Roberts' method which is based on a high parallel resonant frequency, does not materialize since the diode package used here had a parallel resonant frequency somewhere in the lower half of the X-band frequency range. The diode is therefore mounted in a shorted piece of 14-mm coaxial line as shown in Fig. 4. 4. Since the diode diameter is less than the coaxial center conductor, the latter is tapered to make a smooth transition from the 50-ohm line to the diode. Any error introduced by this taper is at least partially offset by a corresponding taper in the diode mount in the LSUC. The reference plane used in the measurement is taken at the edge of the diode ceramic nearest the generator. The diode was measured under self-bias conditions on the Hewlett Packard No. 8410A network analyzer using the No. 8413A phase-gain indicator from 1 to 11. 5 GHz. The results are plotted in Fig. 4. 5. Equations (4. 1) to (4. 3), when used in conjunction with this plot, give the following values for the reactive elements:

134 Wafer Ceramic Pillar Case Fig. 4. 2. Distributed diode model Lt L t C(V) Fig. 4. 3. Houlding's varactor diode model S = 6.12 x 10 F p 12 -1 S = 1.05x 1012 F L = 1. 83 nH. The series resistance was found using both Harrison's Q measurement technique and a standard slotted line measurement with Bandler's modification. The measuring system used for the first method is shown in Fig. 4. 6, and the resulting bulk resistance is found to be R= 1. 55 ohms. Using Bandler's method, the resistance is found to be R = 1. 41 ohms. Although both methods give approximately the same result, the first method is

135 1.444 DIODE MOUNT -0.55 0.5625 VL 0.2101 0.24425 m5 GR 900-AP 'REFERENCE CONNECTOR PLANE SHORT-CIRCUIT Fig. 4. 4. Diode measurement mount with (a) the varactor diode and (b) the short-circuit reference plane where all dimensions are in inches less accurate because of the additional losses in the double-stub tuner, the bias tee, and the additional connectors. The series resistance of the diode is therefore taken as 1. 41 ohms. The analysis of the resulting equivalent circuit is shown in Fig. 4. 5. The reactance of the measured diode and the equivalent circuit are shown in Fig. 4. 7. A comparison shows that the worst discrepancies occur at frequencies beyond parallel resonance, but these frequencies are also higher than the operating frequency of the LSUC, and so the discrepancies do not disturb the design.

136 IMPEDANCE OR ADMITTANCE COORDINATES 0.! 0.13~~~0.1 03% 0 ' ~ P~~~,, ZO m Ina op ~~~3 9:0 o~~~~~~~~~l.0t a ro o 4 R E S. 4.IST ANCE LOADPOWADGNENAT OR I I 112.01, I /7 /~ o:~~~~~~~~~~ o " 0 cJ~~~~~~~~~~~~~~~~~~~~~~~~~ ~'~ ~ ~~~~ ~ ~ ~~~~~~~~~~~~~ 20O.4 3. 2 51 el 6o 41. I TO AR LO A 's, TO2R GEERTO v" ''''' ' '9 ' ".cv cE~~,~~~~~~~~~~~~~~~~~~~~~~~G P a o,~~~~~~~~~~~~~~~~~~~CNE 0 2 I 1 LI~#LC-U.3.4.5.6.7 1.,.. I I 2 I 3.., 1,6 1.7 1.8 1. ORIGIN A MEGA-CHART ~ig. 4.5. The mesu~ed d~ode impeacedtpontaedsinedboadte seet e d pits otaie ~o na[~s~$~E~ o ~ _Lthe~M~`-f~:esu:n~d oe deinae by x

137 HP 8742A HP Network Generator Reflection Unit Analyzer - 8410A, 8414A Weinschel DS10 Double-stub Mosley Model Recorder GR 874 FBL Bias insertion unit Diode holder Fig. 4. 6. Diode measurement circuit 4. 3 External Microwave Circuit Measurements The two problems considered in this section are the measurement of the bandpass impedance transformers used in the signal and lower sideband ports, and the measurement of the equivalent tee junction. 4. 3. 1 Band-pass Transformers. In the design for LSUC3 discussed in Chapter III the signal transformer is designed to have a bandwidth of 40 percent of the center frequency and to transform the 50-ohm generator impedance to 7. 630 ohms (which normalized to 50 ohms is 0. 153) at the reference plane where the impedance is real. The lower sideband port is designed to have a bandwidth of 7 percent of the center frequency and to transform the 50-ohm load impedance to 30. 04 ohms (which normalized to 50 ohms is 0. 6). These two transformers were measured on the Hewlett-Packard No. 8410A network analyzer using the No. 8414A polar display and an X-Y recorder. The results shown in Figs. 4. 8 and 4. 9 indicate the bandwidths of these filters are somewhat narrower than the design bandwidth. However, this is in qualitative agreement with the analysis presented in Chapter II. Although the bandwidth is narrower than the design goal, the impedance transformers are sufficiently broad so as not to restrict the amplifier bandwidth.

138 500 450 400 - o 350 300 o 250 200 o 150 0 E 100 0 o O I o ~. 50 0 O Cd:3 0. 0 O 06.. -50-. a -100 o -150 -200 0 -250 -300 0 -350 -400 - -450 - I I I I! I I I 1, 1. 1 -5000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Frequency in GHz Fig. 4.7. Comparison of the reactance for the measured diode ' and the diode model o o o

139 IMPEDANCE OR ADMITTANCE COORDINATES 0.12 0.13 O.'38 0:3 o. 0 o~~~~~~~~~~~~~~~.? 90~~~~~~~~~~~~~~~~ 0~ oe~,~~ ~~~~~~~o In Io'~~~~~~~~~~~~~~~~~~~~~~~~~~~~J 0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1.~~~~~~~~~~~~~~~ o~~~~~~~,'s Po~~~~~~~ RADIALLY SCALED. PARAMETERS ~ - 0...8..6.. 4.~ '.[ '.I.0 0 I I:I4 I.!,,.5,~ '.?[ '1.8T.WARD LOA TOAR GENERATOR' ' ' 4* I 8 7 6 5 4, $ 2 I 99 I 0'"' ~.....................................................,.,,,............................. 5,,,.0,.,.,,,5, 0 J ~~~~~~~~~~1.2 0.~ 3.4 2.5 1.6 0.? 1.8 I.2I 1.3 1.4. 1.7 1.8 2.9 2''' Fi. 4.85.6. 04_ L.Imeacothsgafeunyrsomr o~~~~~~~~~~~01! '0v., 11 4 3 d5 6i '' j.' 'i4... j. 1.1 1. 2.5 `J ~ ~ ~ ~ >,)(~$3~cr.~(-~ A 0 T__ __3 - - - - — )~~;1~

140 IMPEDANCE OR ADMITTANCE COORDINATES 0.1 1 o~~~~~~~~~~~~~~~~~.~-..~,7.0~~~~~~~~~~~~~~~~~_ to 0~~~~~~~o 0.37~~~~~~~~~~~ o RADIALLY SCALE PARA jETERS O:IC 90 h\%1V0;4R02 2@0. | j~~~4 1 5. 4 1 11131 2 ~ 112.111.1 1 1;4 1 1. 11l@1 TOWARD LOAD TOWARD GENERATOR 9 22tC 4. 30. 20. I. IV 8. S. 5. 4. 3. 2. 1. 0 1. 1.1 1.2 1.3 1.4 1.6 1.8 2. 3. 4. 5. 10. 2Q Q OL% A,. 4.9. Imedn o f 2h oe iebn rqecyt sore ><9dM~~~~~~~~~~~~~~~~~~~~~~~~; 0 I I.'1 1. 1.. l.2' 1 i 4;1 5i1.1 6- ' 1~ II 1. 1 i4. I I I 2QIIli. a i 7 1 9.8.7 6.57 4..!.. 05,d,..99..... ~t1819i 25 0a5j 0 219 3.4,.8.7... s.,6.,.,5.........13....,.... I, 12..9,. 1,..95.. 1,...,8,, I.7,, 1..,.. 1*p, 4,7.}..,. 'e ORIGIN A MEGCA-CHfANT Fig. ~ 4.9 meac fth oe ieadfrqec rnfre

141 4. 3. 2 The Tee Junction. In much of the theoretical work described in Chapter III, it has been assumed that all three ports of the coaxial tee junction are symmetrical, so that power entering one port divides equally between the other two. However, empirically this assumption is found to be invalid at high frequencies. In connection with realizing multiplexing networks, Wenzel (Ref. 60) found that coaxial shunt stubs caused some difficulties at X-band and were very unpredictable at Ku-band. To gain further insight into the tee junction characteristics, two ports of the tee were terminated by 50-ohm loads, and the input impedance was measured at the third port with the reference plane at the center of the junction (Fig. 4. 10). These measurements indicate that the tee junction is not an ideal power splitter above 2 GHz for the 14 mm coaxial tee and above 4 GHz for the 7mm coaxial tee. The measurements of the 7 mm tee in Fig. 4. 11 show that beyond this border line frequency the input impedance Za looks increasingly reactive while Zb tends to look more like 50 ohms. Thus power entering the input port of Fig. 4. O10(a) is almost totally reflected at sufficiently high frequencies, while at these same frequencies most of the power entering the input port of Fig. 4. O10(b) is absorbed in the opposite arm. These characteristics are used to advantage in the LSUC3 design. To obtain a more quantitative model of the 14 mm tee junction used in the LSUC, parameters are obtained for a three port equivalent circuit shown in Fig. 4. 12. The matrix equation for the tee junction is i Yll Y12 Y V i2= Y Y21 Y22 Y23 V2 (4. 5) i3 1 Y32 33 1 V From symmetry, Y1 = Y22 and Y13 = Y23, and from reciprocity, the nine terms in (4. 5) can be reduced to only four terms: il Y Y Y V i2 = Y12 Y11 Y13. V2 31 L13 Y13 Y331 V3

142.50 5 50b zb- * L 1 5,;, Q Fig. 4. 10. The impedance of the coaxial tee circuit as seen from (a) the leg port, and (b) the arm port The measurement of Yl = Y22 and Y33 is easily accomplished by putting shorts at the reference planes of two ports and measuring the input admittance of the third. Measuring the off diagonal terms is slightly more involved. To measure Y12, for example, port 3 is shorted at the reference plane, reducing the 3 x 3 matrix above to r, i r Y..]. [ ]i (4.6)

143 IMPEDANCE OR ADMITTANCE COORDINATES 0.1 01 o.38 0.37 90 o 0.39 0.36 90 eh'..~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 0,, ~ ~ ~ L~, ~:-I '/ ~..cy- 02,o~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~g ~,, " X:-, 0~~~~~~~~~~~~~~~~~~ ~.\. '. ~ ' ' A-t A ~ ~ b-'",.i i ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,,~?~~~'/' 0r0.0 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Z i ~~~~~~~~~~~~~~~~~~~~~ p~~~~~~~~~ZI' ' ",~O~ _~OD 40 30 20. 1 5 10. 8. 6. S. 4 3. 2.. 01~ I.I. 1 i.2 1.3 1.4 1.6 1.8 2. 3. 4. 5., '0 20 50 O0 0~,'t~3 i S.3 4. 5. 6 7 129. 14. 2 0 30 OD 00 ~ ~ 4 61 Y L5 2. ' 3. 4 5. 6. 0 I Y%~/~ ~~~~~~~~~~~~~~~~~~~ '...... 7.....6..........o4.o, I: ',:"".''"'.'0 ' " 1.2...'3 '.'4 '.; 1.6............. ".9.8.~7.~6.5.4 '3.2.I O..99.95.9 '8..6.5.43.2.0'/r CENTER ~ 0..2. 3.4.5.6.7.8.9 I. I.I 1 2 i.3 1.,4 1.~5 1.6 1.7 1.8 1.9 2 ORIGIN A 1 Fig 4. 11. Theimednc as at fuionorqec in, fa7m c aia l te junction as sen from the legZa,...., ad teipdnea enfo the ar Zbl, ~::,whnteoer two po9~rts.- are~~ te::rmb ina t e w i t 50 ohms

144 Y12 Y11-Y12- Y 13 1 Y22-Y12-Y23 13 23 Y33-Y13-Y23 ~~~~~~~1 2 Fig. 4. 12. (a) The coaxial tee junction and (b) the equivalent circuit of the tee junction at the reference planes Equation (4. 6) can be solved for the unknown Y12 = Y21 in terms of the easily measured values Y11 Y22' Y lin = i1/V1, and Y2 = i2/V2 Y12 =(Ylin- Y1l) (Y2- Y22) V O. (4 7) Similarly Y13 is 2 Y13 (Ylin Y1l) (Y- Y33), V2 (4. 8) where Y3 = i3/V3 The value Ylin in both (4. 7) and (4. 8) is the input admittance looking into the junction from port 1 for some finite Y2 or Y3, while Y2 and Y3 are the

145 admittances looking away from the tee junction at the reference planes of ports 2 and 3 respectively. In the measurements reported here, a short was placed a quarter wavelength from the reference plane in either one of ports 2 or 3 so that Y2 or Y3 was -jYo cot (2) where X is the midband wavelength. The measured equivalent circuit for the 14 mm coaxial tee junction for frequencies between 8. 40 and 8. 60 GHz is given by B 11 = C11 - 0.148 mhos, Cll = 2.96pF B33 = -(wL33 - 0. 275) mhos, L = 55. 7 nH B12 = C12- 0. 38 mhos, C12 = 2.92pF B13 = wC13 - 0. 0915 mhos, C 13 = 2.00pF Although the coaxial tee junction is a common microwave component, no accurate analysis has been given for it. King (Ref. 61, pp. 389-397, 426-430) has analyzed the two wire transmission line tee junction and indicated how this method might be extended to give an approximate equivalent circuit for the coaxial tee junction. However, his approach is complex and would not cover the problem of higher order modes which are excited at sufficiently high frequencies. In the absence of an accurate theoretical description of the tee junction, the above empirical model is used in the analysis of the LSUC. 4. 4 The Experimental LSUC Circuits Two LSUC circuits were designed and tested: one based on the LSUC2 design and the other on the LSUC3 design. Although 20 dB of stable gain was obtained in both upconverters, the LSUC2 required several empirical modifications before this gain was achieved while the LSUC3 did not require these modifications. The theoretical LSUC2 used an ideal band-stop filter to help isolate the signal port from the lower sideband frequency. However, experimentally it was found expedient to not use a band-stop filter. This together with the effects of the nonideal tee junction made it necessary to modify both the signal and lower sideband circuits to obtain the desired gain. The final experimental amplifier is shown in Fig. 4. 13. In contrast to the preceding design, the LSUC3 required neither a band-stop filter nor an ideal tee

146 f s s3 p Ds3 ds3 ds2I I s2 Dsl — ~ A/DS1, Coax to waveguide Ds DD 1 < D adapter [ //_ZZDiode p 2- D D. _ 3 2 p3 [ p2: pl 2il Di2.i3 d.23 =p 0p. 2 -0011D3 Distances in. Disk Lengths in. Disk Diameters in. dpl = 0. 386 QPI =0.1544 D = 0.542 dp2 = 0.6338 p2 =0. 0817 Dp2 = 0. 532 dp3=0. 6804 Qp3 = 0.375 Dp3 = 0. 502 dil = 0.427 il = 0. 0103 Dil = 0. 499 di2 =. 8143 i2= 0. 0860 Di2 0. 502 doi3 = 0. 7966 t i3 = 0. 0111 Di = 0. 502a Qs = 0.1222 DsO = 0. 554 dSl = 2. 301 Qsl = 0. 952 Dsl = 0. 502 ds2 = 2. 599 s = 0. 020 ds3 = 3.424 Is3 = 0.1094 Ds3 = 0. 502 Fig. 4.13. Schematic diagram of the experimental LSUC2 including the final experimental dimensions. The transmission line is standard 14 mm 50 ohm coaxial line and the cross-hatched sections are Teflon

147 junction, so this upconverter provided the desired gain without recourse to empirical design modifications. A cut away view of the LSUC3 is shown in Fig. 4. 14, and the upconverter together with the associated equipment is shown in Fig. 4. 15. 4. 4. 1 Gain. Both amplifiers were designed to have a transducer power gain of 20 dB based on an elastance pumping coefficient S1/SO = 0. 35. When the input power was set to -80 dBm, the output power was 20 dB higher as measured on a calibrated spectrum analyzer. The gain can be increased or decreased by simply changing the applied pump power. The LSUC2 had a maximum gain of 37 dB, and the LSUC3 had a maxmum gain of 33 dB before oscillations occurred. 4. 4. 2 Bandwidth. If the amplifier bandwidth were limited by the pass-band of the signal or lower sideband impedance transformers, then its value would exceed 400 MHz. However both the measurements and the theoretical gain versus frequency calculations show the amplifier bandwidth to be much less than this. Evidently the complicated circuit impedance and the diode package parasitic elements, rather than the filters themselves, are primarily responsible for limiting the bandwidth. The 3 dB bandwidth of the LSUC2 was measured and found to be 3. 5 MHz. Because of the various above mentioned modifications in this circuit, a comparison with theory is not possible. The experimental gain curve for the LSUC3 is shown in Fig. 4. 16 for various values of pump power (or various values of maximum gain). The 3 dB bandwidth for the 20 dB curve is 2. 6 MHz. These curves cannot be compared directly with the corresponding theoretical calculation of the LSUC3 (Fig. 3. 22) in Chapter III because of some minor differences in the two amplifiers. These differences are listed below. 1. The experimental amplifier has a maximum gain of 20 dB while the theoretical amplifier has a midband gain of 20 dB. 2. The distance between the diode and the lower sideband filter is increased by X s/2 so as to make a gradual transition from the diode cartridge diameter to the coaxial center conductor diameter. 3. The distance between the tee junction and the pump filter is increased by p /2 so that the pump filter does not extend into the tee junction area itself.

148 Signal port Pump port Lower sideband port Varactor diode Fig. 4. 14. Photograph of the LSUC ~ i:': '............: ~'::::::;:: Signal source Pump port Lower sideband port Fig. 4. 15. Photograph of the LSUC when in operation

149 30 29 28 27 26 25 24 23 22 21-i 20 194 18 CtJ 17 1 6 a) o 1 5 14 -13 12 + 11 AE 1 + + + 9 7 {0) (A o} 3 + 0 2 (o 0 x A O x — ~~~~ O 1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 L (40 o ) /, I, I(<> 0.998 0. 992 0. 996 1.000 1.004 1. 008 1. 012 Signal Frequency, GHz Fig. 4. 16. Experimental power gain of LSUC3 when the midband gain is 30 dB (/x), 25dB (o), 20dB (o), 15dB (O), 10dB (+), and 5dB (X)

150 4. The distance between the lower sideband filter and the high-pass filter is approximately 3. 5 X because of the long line stretcher used. s 5. The distance between the pump filter and the corresponding high-pass filter is approximately 1. 5 Xs because of the long line stretcher used in the circuit. 6. The actual tee junction frequency sensitivity is neglected in the theoretical LSUC3 in Chapter III. All these differences do not change the basic design philosophy of the LSUC3, but they must be incorporated in the analysis of the amplifier to obtain correspondence between theory and experiment. When these changes are made the theoretical gain curves are shown in Fig. 4. 17 which at 20 dB gives a bandwidth of 4. 63 MHz. Shown in Fig. 4. 18 is a comparison of the gain-bandwidth product for various pump levels. The complex behavior of the element values for the equivalent circuit of the tee junction (the equations given in Section 4. 3. 2 are the best linear approximations to the actual susceptances), the connector reactances, the line stretcher reactances, and the SWR of the high-pass filters in the pass-band, all contribute to bandwidth reduction beyond that calculated for Fig. 4. 17. The major limitation to the bandwidth, however, seems to be the long line stretchers, for if they could be removed and the high-pass filters brought next to the pump and lower sideband filters respectively, a bandwidth of 21. 6 MHz would be expected. 4. 4. 3 Noise Figure. When the output impedance of an amplifier is negative, care must be taken in correctly measuring the amplifier noise figure. In systems using a LSUC an isolator is normally attached to the output of the amplifier, so the noise figure of the LSUC and output isolator was measured as a single unit. There are several definitions of noise figure, but the one most convenient when dealing with negative resistance amplifiers is the "actual" noise figure (Ref. 40). The actual noise figure is consistent with the definition of the transducer power gain and accounts for the noise generated in the load and generator impedances. A block diagram of the noise measuring circuit is shown in Fig. 4.19 while a photograph of the system is shown in Fig. 4. 20. The noise figure for a cascade of stages is

1 51 30 29 28 27 26 25 24 23 22 21 20 19 18 17 0 16 -16 15 14 Q ) 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0. 998 0. 992 0.996 1. 000 1.004 1. 008 1. 012 Signal Frequency, GHz Fig. 4. 17. Theoretical gain curves for LSUC3 when the circuit is modified because of physical requirements

152 0.06 0.05 0.04 0. 03 '0 3. k/ (E) (E) I002 0.02 0.0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Maximum Gain, dB Fig. 4. 18. Comparison of theoretical and experimental gain-bandwidth product

HP 349A UHF Noise Source Melabs 1 GHZ Isolator PRD 1203 HP X382A Varian AIL type 134 IIP 345A Waveguide. lPrecision V8302B: 60 MHz. ~Noise Figure LS-G I!Isolator ' Variable Hybrid Prea rnplifie- 7Meter o sol.ttenuator ixer 1341 Stage 1 Stage 2 X-Kl13stron Local Osc. Stage 3 Fig. 4.19. Noise figure measuring circuit

154 Noise source ':"- '' ''~"i " ~ Lower sideband output Pump power 60 MHz IF amplifier sE000000.: a.....-......:;..: Noise figure meter Local oscillator Lower sideband output Mixer Fig. 4.20. Photographs of the noise figure measuring circuit

155 F2 1 (F3 - 1)L2 F = F1 + G+1 1 G2 1 G21 where G21 is the gain of the LSUC (stage 1) and L2 is the loss in the variable attenuator (stage 2). If the ambient temperature is 2900K, then F2 = L2 and the total noise figure reduces to F 1 LL2F3 1 G21 G21 When the total noise figure is measured as a function of L2, the result is a straight line with slope = F3/G21 and intercept at F = F1 - 1/G21. Since F3 can be found by an independent measurement, G21 and finally F1 can be obtained, where F1 is the noise figure of the amplifier and isolator combination. This method gives accurate results since individual measurement errors can be averaged. The noise figure of both amplifiers was measured by setting the gain at 5 dB, 10 dB, and 20 dB by an appropriate choice of pump power when a sine wave was applied at the signal port. The noise figure of stage 3 was 10. 0 when the noise figure for LSUC2 was to be found, and 9. 5 for the LSUC3 measurement. The difference in F3 between the two cases is ascribed to the difference in frequency and output power of the local oscillator. The straight line graphs generated for these measurements are shown in Figs. 4. 21 and 4. 22 and the final results in Table 4. 2. These two figures experimentally confirm the result cited in (3. 25): F - 1/G21 is a constant. The results shown in Table 4. 2 indicate the upconverter amplifies noise power as much as 5 dB less than CW power. The reason for this discrepancy is unknown, but when the CW and noise gain of an Aertech TY538, X-band, tunnel diode amplifier were measured, they both came out to be 10 dB. The discrepancy therefore seems to be inherent in the upconverter itself.... The theoretical spot noise figure obtained from (3. 24) when the 5 x 5 matrix approximation is used is shown in Fig. 4. 23. To relate this with the measured value, a relationship between the spot noise figure F(f) at frequency f and the average noise figure F is needed. This expression is

156 25 24 - / CW Gain 23 / / 20 dB 22 - | / 10 dB -- - 1722 / 21 t / 19 17 I OI; 12 5 8 d 0/z~~~~~~~~~~~ 16 I I z / / 6 / 4 14 0 I2/ / 0m1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Attenuation Ratio L2 Fig. 4. 21. Noise figure of LSUC2

157 25 24 / CW Gain 20 dB 10 dB --- 5 dB - - - 22 21 20 19 / 18 / 17 x 16 / / 13 e 15 99~~~/ z 12 / / S 6 -0 I "/ ~{/ F / '8 /. 2 / 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Attenuation Ratio L2 Fig. 4. 22. Noise figure of LSUC3

158 CW Gain Slope Noise Gain Noise Figure F Excess Noise dB dB Temperature LSUC2 20 0. 326 14. 87 1. 58 168~K 10 1. 632 7. 87 1.71 2060~K 5 5. 362 2. 71 2.09 3160~K LSUC3 20 0. 236 15. 77 1. 88 2550~K 10 0. 931 9. 81 1.95 276~K 5 3. 210 4.44 2.21 3 510 K Table 4. 2. Noise figure of the LSUC _ F(f) G21(f) df F (4. 9) f G21(f) df where G21 is the upconverter gain (Ref. 62). When the data used in plotting Figs. 4. 23 and 4. 17 are numerically integrated, the theoretical average noise figure is found to be 1. 187: a value significantly lower than the measured noise figure. In the following paragraphs, various other sources of noise are investigated to explain this discrepency. 4. 4. 3. 1 Circuit Losses. Resistive circuit losses adds some thermal noise to the amplifier. However when the LSUC was numerically analyzed in a circuit taking account of this loss, the change in the noise figure was negligible. In the calculation the standard values for the resistivity for brass (6. 4 10-8 ohm-meter), the standard loss tangent for Teflon (3. 7 10 4), and a roughness factor of 1. 8 (Ref. 63) were used. The result is a change in the thermal noise figure of only 0. 0023: a value too small to be measured with a standard noise figure meter. 4. 4. 3. 2 Incorrect Value of Diode Bulk Resistance. One important source of noise is the series bulk resistance of the diode. If this resistance increases, the noise figure also increases. However when the noise figure was calculated for the LSUC with R= 1.41 ohms replaced by R = 7.71 ohms, the noise figure with the gain adjusted to give 20 dB increased to only 1. 19. Observation of the midband noise figure expression for the uncooled LSUC,

159 2.0 1. 9 1.8 1.7 1. 6 a) bf cD *- 1. 5 0 z — 4 H 1.4 -1.3 1.2 1.1 1. I I I I I I I I 0. 988 0. 992 0. 996 1. 000 1. 004 1. 008 1. 012 Signal Frequency, GHz Fig. 4. 23. Theoretical noise figure for LSUC3

160 1 1i 1 F = (1 + ) (1 + ) + (4. 10) x r G2 shows that when xg is large ( = 100 in this case), decreasing it by a factor of 5. 5 as done here does not greatly degrade the noise figure. 4. 4. 3. 3 Noisy Pump Source. The noise fluctuations of the pump oscillator are generally considered to have an insignificant effect on the noise figure of parametric amplifiers. Uenohara (Ref. 41) assumed this on the basis that for a properly adjusted amplifier, the gain is almost constant for small variations in pump power. More recently IMPATT diodes have been used for parametric amplifier pump sources. Although FM and AM noise of IMPATT diodes are considered to be from 20 to 25 dB higher than that found in klystrons (Ref. 64), Gray, Kikushima, Morenc, and Wagner (Ref. 65) found that the noise figure, bandwidth, and gain were the same for the klystron and IMPATT diode pump sources. A first order approximation to the FM noise introduced by the pump source can be obtained from (4. 10) which is valid for the 2 x 2 matrix approximation when the generator and load resistances are independent of frequency. The FM noise of the pump effects both the frequency ratio r and the gain G21, while the AM noise effects only the gain. The pump FM, which causes fluctuations only in w2, increases the frequency ratio term by the factor 1 + w20 where 6w = o2 - w20. Assuming single tuned circuit elements, the reactance at the lower sideband frequency is 2 S0 w 22 W20 W20 Using these expressions in (3. s2) and substituting into (4. 10), the modified noise figure

g xg r W20 (1+0 W 20 (1+Xg)(l + Y) Kq,,,,) \2 K4 + 4(1 + xg) Q20 g + 9 20(4. 11) \20/ [(1 + x )(1 + y)- Kq]2 where GO is the midband gain. The additional noise figure due to the FM noise alone FFM is readily obtained from this expression. The pump klystron used for the LSUC had an FM bandwidth of approximately 25 KHz at f = 9. 50 GHz. Using this value plus the LSUC parameters used in the design, FM 1 75x 10 +-7 1 (4. 86x 10-6) FM. x 0 2. 24 x 107 Thus the increase in the noise figure from frequency fluctuations in the pump source is negligible. An amplitude fluctuation in the pump source leads to AM noise in the upconverter through variations in S. The reciprocal of the gain under this condition is given by G = 1+ 1 q G2 1 GO 1+ S 2 -(1 1+ ( +x)(lx+y ) Kq [( 1 qg dCy 2 2 6S 2K K q q S (1 +Xg)(I + y Kq (l+x a+ K g23 of the LSUC are used, the additional AM noise in the upconverter is

162 F2 2 2 FAM 2 (4 31) +I' (6. 03)1 2A2 Assuming a gain of 20 dB and a pessimistic elastance variation of =S / Sl 0. 01, this AM noise reduces to -4 FAM 4.31x 10 Clearly both the AM and FM noise from the pump source add negligible noise figure to the upconverter. Thus from both experimental substitution of a noisy pump source for a quiet one and observation of (4. 10), the noise in the pump source can be neglected as a major contributor to the upconverter noise figure. 4. 4. 3. 4 Pump Heating of Diode. Garbrecht (Ref. 66) proposed that the heating effect of the pump power on the diode spreading resistance determines the lower limit of noise temperature for cooled parametric amplifiers. An expression which describes the effect of diode heating can be obtained from (3. 24) by setting the shot noise components to zero. Under large pumping conditions the diode temperature can be expected to rise approximately 40 K above ambient, which increases the noise figure by less than 0. 009. Although the pump heating effect is important for cooled parametric upconverters it is not important for room temperature upconverters. 4. 4. 3. 5 Shot Noise. In Chapter III an expression was derived for the shot noise in a LSUC caused by the dc bias current i0. This relationship, which is plotted in Fig. 4. 24 with diode temperatures Td = 290~K and 330~K, shows that the diode would have to draw an excessive current for the noise figure to be close to the measured value of 1. 9. However Josenhans (Ref. 39) noticed that the slope of the corresponding curve for his reflection parametric amplifier using silicon diodes was a factor of 10 to 15 higher than what his theory predicted. At the same time, when he measured germanium and gallium arsenide diodes, his theoretical and experimental values matched within the experimental error. If this same factor is applied here, only lO1A of dc current (or an equivalent forward-and-back current) is needed to give a noise figure of 1. 9. Apparently the zero minority carrier lifetime assumption used in deriving the noise figure

1.7 I I I. 5 I I I 1.6 4 I I II II II 1.5- i II Ii II ~ 1.4I bfl II ii o rI z 1.3 C) II 1.2 — 1. 1 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 DC Current, pA Fig. 4. 24. Thermal plus shot noise figure of a LSUC at 2900K when the diode is at 290 0K and 330 0K. The dashed curves have slopes 10 and 15 times larger than the noise figure curve

164 expression (3. 24) is in error for silicon diodes. The remainder of this section describes an experimental investigation of the effects of nonzero minority carrier lifetime. In the experimental LSUC3 described in this chapter, the diode circuit at dc is open. Thus any shot noise arising in the upconverter must be caused by forward-andback current. Unfortunately this current cannot be directly measured, but there are two indications that it is responsible for the high noise figure. The first is that the noise figure increases significantly as the diode bias increases, and the second is that the diode exhibits changes in the dc characteristic when pumped by an RF source, suggesting the existence of significant forward-and-back current. To verify the first of these indications, the noise figure of a LSUC as a function of bias voltage was measured. Since a change in bias changes the diode capacitance, variable tuning elements were needed to compensate for this change so that the gain and pump power could be kept constant. This is accomplished by replacing the impedance transformers in LSUC3 with numerous filters and variable tuning elements. The pump frequency in this case was 10. 59 GHz and the signal frequency 1. 843 GHz. The bias voltage was applied through a General Radio 874-FBL bias insertion unit placed in the signal port. The dc current was thus conducted through the center conductor to the diode and returned to the outer conductor by the shorted double-stub tuner in the lower sideband circuit. The noise figure was measured using the same techniques described in Section 4. 4. 3, and the results are shown in Fig. 4. 25. Although the magnitude of the noise figure for this empirically designed LSUC was poor (partly because of the low gain of this device) the noise figure clearly increased with increase in bias voltage. By allowing 7pA current to flow, the noise figure of the LSUC increased by 10 dB over the reverse bias configuration. (The break in the curve at Vb = 0 results solely from the change in scale of the abscissa. ) The existence of nonzero forward dc current, which is indicated near the appropriate data points, even when the dc voltage is zero shows that the pump source is producing some dc current. The second indication of significant forward-and-back current can be seen in the dc volt-ampere characteristics of the varactor when pumped by a large RF signal (Fig. 4. 26). These measurements were made by series mounting the same varactor

165 1000 800 600 500 400 300 200 100 80 6()0 4pA 50 - 40 0 30 3A 0 2 20 5 10 4 3 2 I I I i I I I I I I -3 -2 -1 0 0.05 0.10 0.15 0.20 Diode Bias Voltage Fig. 4.25. Noise figure of a LSUC as a function of bias voltage when the gain and pump power is constant

70 60 50 X 40 a 30 20 10 -10 I I I I -6 -5 -4 -3 -2 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Diode Bias Voltage Fig. 4. 26. Varactor diode dc current-voltage characteristics with no RF power (o), with 71 mW RF (o), and with 0. 71 W (X). The applied RF is 9. 5 GHz, and the diode is series mounted in a terminated 14 mm coaxial line

167 diode used previously in a 14 mm terminated coaxial line (Fig. 4. 27). This circuit, like Siegel's (Ref. 42), was used to avoid tuning the diode at a particular frequency. Although the measurement results found in Fig. 4. 26 failed to show any negative resistance region, the application of pump power clearly had a significant effect on the diode. Siegel found after careful investigation that the negative resistance he observed was not caused by the ac signal swinging from reverse breakdown to forward conduction. The same conclusion is valid here, since even under heavy pumping (0. 71 W), the voltage swing across the diode chip would be approximately + 0. 40 volts: a value much smaller than the reverse bias to forward conduction voltage range for the unpumped varactor. Thus the pump in the LSUC may well cause charge multiplication in the varactor which in turn contributes to the shot noise figure of the amplifier. 27 k62 RF- t F t 50 2 -Diode GR 874-FBL Bias Tee Fig. 4. 27. Mount for measuring diode current-voltage characteristic under pumped conditions 4. 5 Conclusions The results presented here verify the synthesis techniques developed in Chapter III for the LSUC. The additional adjustments in the analysis needed to account for the extra half wavelength line sections used in the experimental LSUC and the nonideal tee junction do not affect the basic design theory. The measured noise figure was however significantly larger than that calculated on the basis of thermal noise alone. Shot noise with an additional charge multiplication mechanism appear to be the primary cause of this extra noise.

CHAPTER V DIFFUSION ADMITTANCE 5. 1 Introduction The work described in Chapters IL, III, and IV has dealt primarily with circuit aspects of the LSUC. Even the diode is considered as an equivalent circuit with a time varying elastance. Here the varactor diode chip itself is analyzed with special attention given to the evaluation and effects of diffusion admittance. Diffusion capacitance arises when a pn junction is biased in the forward conduction region for at least part of an RF cycle. Charge carriers driven across the junction become minority carriers and would recombine if they stayed there long enough. However if the lifetime of the carriers is much longer than the period of the applied ac voltage, most of the minority carriers will return across the junction before they recombine. This charge storage effect produces a circuit element known as diffusion capacitance. The diode elastance harmonics used in Chapter III were derived using the depletion layer capacitance alone. Since varactors used in parametric circuits are often forward biased during part of the pump cycle, diffusion admittance plays an important role in amplifier performance. The analysis presented here indicates how the diode characteristic can be optimized to enhance LSUC performance. A review of the literature in Section 5. 2 shows that overdriving the diode improves frequency converter and multiplier performance. This section also describes some work on diffusion admittance. In Section 5. 3 an expression for the diffusion admittance is obtained, unrestricted by small signal approximations, as a function of the doping profile of the diode. Previously this theory had been applied only in improving the design of a multiplier. Here a large signal analysis of a LSUC in conjunction with short circuit assumptions using only the diffusion admittance (no depletion capacitance) is presented in Section 5. 4. Following in Section 5. 5 is a small signal analysis of a LSUC when (1) the diode has only diffusion admittance, and (2) the diode has both diffusion 168

169 and depletion admittance. These calculations show the particular characteristics needed to improve LSUC performance when the diffusion admittance is significant. 5. 2 Literature Review Varactor diodes are used not only in small signal frequency converters, but in large signal frequency converters, frequency multipliers, reflection parametric amplifiers, and voltage controlled frequency tuners. Although varactor diodes manufactured specifically for parametric amplifiers, frequency multipliers, or voltage controlled tuners differ in such details as capacitance variation, cut off frequency, etc., the theory developed for a diode used in one application can often be transferred to a diode used in another application. Most analyses of either a LSUC or one of these other devices assume that the diode can be represented by a depletion layer capacitance under reverse bias, and by a short-circuit under forward bias. Mathematically this can be stated as follows: V - ( (5. 1) O Q>o, where - V - V Q0- Q Q Q= QO- QB VB = breakdown voltage QB = charge on varactor at VB Q0 =:;charge on varactor at V = 0 V > 0 for forward bias. This means that for Q > Q, the diode is assumed to act like an infinite capacitance, presumably a result of large diffusion effects.

170 Although this model assumes the minority carrier lifetime in the diode to be infinite and therefore does not specifically consider the effects of diffusion admittance, it has been used extensively in the analysis of large signal frequency converters and multipliers. Analysis of both multipliers (Refs. 67 to 70) and large-signal upper sideband upconverters (USUC) (Refs. 71 to 73) with this simple model show that increased power output and efficiency are obtained when the varactor "drive" > 1, where Qmax QB drive max B (5. 2) More specifically Davis (Ref. 68) calculated the output power and efficiency of varactor doublers based on the work of Penfield and Rafuse (Ref. 48). He found that overdriving a graded junction diode improves the doubler performance by a larger factor than overdriving an abrupt junction diode, although overdrive improves doublers using diodes of either profile. Burckhardt (Ref. 69) showed by a more complete numerical analysis of several multipliers, with and without idlers, that overdriven diodes with small y gives the greatest output power and efficiency. Similar conclusions have been reached with regard to y and drive for the USUC. Penfield and Rafuse (Ref. 48) assumed that if the total charge is Q = qb + qp COS(Wpt + p) + qs cs(wst + S) + qu COs( t + u) (5. 3) where qb is the bias charge and subscripts u, p, and s refer to the upper sideband, pump, and signal respectively, then the maximum charge for the fully driven diode is restricted by qp + qs + q u = constant.(5.4) Conning (Ref. 72) using this restriction compared the output power and efficiency for y = 0 and y = 1/2, and shows that output power is larger for drive > 1. 8 and efficiency is larger for drive > 1. 5 when y= 0. Nelson (Refs. 74, 75) showed that the estimate (5. 4) of Penfield and Rafuse was unduly restrictive and developed a more accurate and larger charge restriction. Grayzel (Ref. 72) generalized Nelson's results to include an

171 overdriven diode. He proceeded to show that the greatest efficiency and power handling capability occurs for a drive = 2. Also the maximum efficiency and maximum power handling capability increase with decreasing y. Gewartowski and Minetti (Ref. 73) pointed out that Grayzel had used the phase condition p = 4s =. Rather than making this assumption, Gewartowski and Minetti p s u choose 4'u to correspond to the maximum output power for a prescribed drive level, and thus they achieve as much as 16 percent more power output thereby making a further improvement in the maximum charge restriction. They found that the maximum output power is larger for the abrupt-junction than the graded-junction diodes for all drive levels between 1 and 2, but that for high drive levels, the power-impedance product is higher for the graded junction. The up conversion gain and efficiency for high drive levels were both larger for the smaller y varactors. For y = 1/2 the maximum efficiency and gain occured at a drive = 1. 4 to 1. 6 while for y = 1/3 the maximums occurred at drive = 1. 8. However they found the bandwidth is larger for the abrupt junction diode although the difference becomes negligible at drive = 2. These analyses show that the efficiency of these devices and the gain of the USUC increase when the varactor is overdriven and y is small. However, all of these results are based on the simple diode model described by (5. 1). If some forward conduction is so important to optimal use of the varactor, it appears that a better model of the varactor in the forward bias region is needed. If the depletion layer capacitance is neglected under forward bias conditions, an approximation for the diffusion capacitance can be found. The usual small signal approach for calculating the diffusion admittance (Refs. 76, 77, 78) assumes the voltage and current on the diode are V(t) =Vb + Vss et J(t) = Jb + Jss ejt where Vss << Vb and Jss << Jb ' Then the minority carrier densities are approximated as

172 eaV(t) aV p (1 + aV ejwt) e n n (1 + a Vss ejw ) e where pn and np are the equilibrium minority carrier densities, q is the electron charge, and a q/ kT. The small signal admittance found on the basis of these assumptions is Y V =( P n1 + j jT e (5 5) p n where T is the minority carrier lifetime, D is the diffusion constant, and L =./WDis the diffusion length. The most prominent feature of this expression is that the diffusion capacitance is proportional to e b, which explains why it is dominant in the forward conduction region. For high frequencies (Wco >> 1) the diffusion susceptance due to the holes only is B pn p b Bp L 2 e p /D a V qap e b Hence at high frequencies the susceptance is proportional to f/7 rather than w as in the ordinary parallel plate capacitor. Also Bp is independent of T. At low frequencies UqDppn ab B = p WT e p L p - Vb n VPP

173 which indicates that the susceptance is directly proportional to w and increases with ~/7p'. This analysis shows that at high frequencies, changing the minority carrier lifetime will not affect the susceptance since all of the minority charge is recovered. Increasing T- will not increase the charge storage because all of the charge is already p stored. Although this analysis displays several important characteristics of diffusion admittance, it is restricted to small signals. It therefore cannot be used with confidence to analyze large-signal frequency converters or multipliers; also there is no way other than by experiment to determine its degree of accuracy for small-signal converters. Although Shpirt (Ref. 79) calculated the diffusion currents of a forward biased diode without using small signal assumptions, his analysis was restricted to abrupt junction diodes. Parygin and Maneshin (Ref. 80) found the diffusion currents under large signal conditions for a wider range of doping profiles and applied the results to an analysis of a frequency multiplier. Later Romanova (Ref. 81) showed from low frequency experiments (1 to 60 MHz) that the varactor diode can be modeled by a depletion layer capacitance in parallel with the diffusion admittance. Soviet authors apparently have been more concerned than U. S. researchers in developing a more accurate model of the diffusion capacitance found in the forward biased diode. Of particular importance is the paper by Parygin and Maneshin, whose results are used extensively in this chapter. Since this study is largely unknown in the United States, the analysis is restated in Section 5. 3. 2. 5. 3 Varactor Diode Model The diode model described here allows for the depletion capacitance and diffusion admittance to occur in the same model. The varactor diode is characterized primarily by the. depletion layer capacitance when the diode is reversed biased and by diffusion admittance when it is forward biased. 5. 3. 1 Depletion Layer Capacitance. The depletion layer capacitance is found by solving Poisson's equation and is given by C = b (5. 6) (1- V/- v)

174 where Cb is the zero bias capacitance and 0v is the built-in potential. If the profile is N(x) = ax, then the exponent y in (5. 6) is y = 1/(b+2). Thus for a graded junction diode y= 1/3 and for an abrupt junction diode y = 1/2. 5. 3. 2 Diffusion Admittance. The diffusion admittance calculated by Parygin and Maneshin (Ref. 80) is rederived below. Since the diffusion admittance is proportional to the highly nonlinear exponential function, harmonically related currents are generated in aJ the diode. Harmonically related admittances m also result from the analysis. a v In general, a pn junction diode can be described by the following six basic equations. JP = q(lppE- D Vp) (5 7) P p = q(A nn+ Dn Vn) (5. 8) n n n ap P-Pn -V ~ J = q(~ T+ n (9 Pv ~Jat 7 g_ (5. 9) p0. at n-n n-n Vn = q(at-+ T - gn) (5. 10) n p = q(p-n +N Na) (5. 11) V (eE) = p (5. 12) The hole and electron minority carrier densities are p and n respectively, the equilibrium hole and electron minority carrier densities are pn and np respectively, J is the current density, Nd and Na are the donor and acceptor atom densities, p is the net charge density, and g is the generation term which will normally be neglected. The calculations that follow are based on a p+ n diode so that the depletion layer extends into the n region from x = 0 to x = W. At equilibrium the sum of the drift and diffusion electron currents at the edge of the depletion region is approximately zero, so Jn in (5. 8) is set to zero. If the impurity concentration is Nd(x) = axb, then from kT (5. 8) and the Einstein relationship, D = -q n, the electric retarding field in the n region is

175 kT 1 dNd kT b E (5. 13) q Nd dx q x The hole current is then given by (5. 7): Jp = -qD (+ ) (5. 14) P P x ax This can be combined with the continuity equation (5. 9) to yield D a2p +b ( p bp P- Pn ap 2 x ax 2 - a t p ax x p When a periodic voltage is applied to the diode, the hole concentration can be represented by a Fourier series: cc p(x, t) = PO + Z (Pmc cos mwt + Pms sin mwt). m=l ms If this is substituted into the above equation, and the harmonic balance technique is applied (coefficients of corresponding frequency terms are equated), the following two sets of equations are formed: d2P b dp b Pmc D Mc+ b- Mc b Mc mw p 2 x dx Pmc2 T mwms d2 dP p (mc+b d Pmc x) Pmc Pms b dPms b Pms Dx dx ms mp dx p Since Re [(p iPm) emwt = Pmc cos mwt + Pms sin mwt the above two equations can be combined into one by defining Pm Pmc JPms

176 where this is not to be confused with pn(W). 2 d PM dp M/ + jmwT 2 x vi + bx m- + L x) P 0 (5. 15) To transform this into the usual form for a modified Bessel equation the following substitutions are made: x = x -(l-b)/ 2 2 1 + jmcw7 where 2 (5.16) L p Using the first substitution, (5. 15) can be written as d2 -2 Pm dPm 2 2 + bx (b + x = P The second substitution transforms this expression into -2du du 2 2 x2 d + _ [(1+b) /4 + ]u = 0 dx- d This is the well-known modified Bessel equation, with the solution pm(X) = x(1-b)/2[ Mlb,(X) + M2Kb,(x)] where b' _ (b+1)/2. Two boundary conditions are now applied to evaluate the constants M1 and M2. Since pm(x) = 0 as x- co, M1 = 0. The constant M2 is found by using the hole concentration at the edge of the depletion region, x = W. If the resistance of the semiconductor is much less than the resistance of the depletion layer, the entire applied voltage, V(t), is across the junction. Therefore the hole density at the edge of the depletion region is

177 p(W) = Pn(W) e p (W) [ g+ ' 1 (g' cos mwt + g' sin mwt) -(W= Pn(W) M - ms where the g, gm and gm are the Fourier coefficients of e. If W is considered to be independent of the voltage, and Pm(W) Pn (W) (gme then Pn(W) (g' - jg') 2 (1b)/2Kb ( W) The final solution for pm(x) is then Pm~x) - n ) (W-b)/2) Kb,( x) 1 i eaeV (cos mwt - j sin mwt) dwt. (5. 17) Kb'(5W) ' 7T 0 The current can be expanded into a Fourier series just as was the hole density. J = J0 + k J cos mot + J sin mwot 0 mc ms m= 1 If J J - the expression for P(x) of (5. 17) can be substituted into (5. 14) m mc ms m to obtain an expression for the current. If b" (b-1)/2, then J(x) = (W)- Kb(x W ) fP S e~V (cos mAt - j sin mot) dot (5. 18) b 0 where the identity [Kn(Z)] = -znK (z) has been used. This expression gives dz n n-1 the harmonic current components due to the applied voltage V(t). If KbT(TW) am + j/m W, (5. 19) m m ~ Kb,(5W) then the coefficients for the cosine and sine terms for the currents are qDppn(w) 2e pn f e (c~ cos mwt + Pf sin mot) dwt (5. 20) mc 7wW fm m

178 qD p (W) 2ireCv PJ = eaV (a sin mwt- fm cos mwt) dwt. (5. 21) M s 7?W 0m m The acm should not be confused with a= q/kT If the admittance is defined as dJ Y A m m dV AaJ (5. 22) where A is the diode area, then there is clearly a resistive part (conductance) and a reactive part (diffusion capacitance). The conductance term is lossy and introduces thermal noise. Although the large nonlinearity of the exponential function is desirable, this loss term limits the usefulness of diffusion capacitance in low-noise amplifiers. 5. 3. 3 Alternate Formulation. The method used by Parygin and Maneshin consists of first finding the diffusion currents, and then by differentiation with respect to the applied voltage the diffusion admittance is found. In contrast to this, Sah (Ref. 82) first finds the total charge stored by the minority carriers and then differentiates this charge with respect to the applied voltage to obtain the diffusion capacitance. Thus using the second method, if the n region extends from x = W to the ohmic contact at x = d, the expression for the diffusion capacitance is c dQ dV d =qAd-vf SPn() dx Using the expression for pm(x) of (5. 17) the harmonic charge components stored in the n region are Pn (W)qA 2A (4) e (cos mwt- j sin mwt) dwt 7rW )/ Kb,( W) I ) f x(l-b)/ 21,(t x) dx W

179 This is the total charge for the p+n diode since the charge stored in the p region is negligible. If the second integral is designated by Mi, then M. = (b-3)/2 d (1-b)/2 Kb,( M. - f (x5) Kbf( x) d(5x) Using the identity K,(z) = K,(z) and the above cited derivative formula for a Bessel function, the expression for M. becomes 1=- Kb (~x)x(l-b)/2 d M = and when (d>> 1 W(1-b)/ 2 M = Kb ( W) A comparison of this value of Qm with the value of Jm in (5. 18) shows that J T A mp m 1 + jmwpT p Thus the diffusion capacitance defined by Sah, expressed in terms of the previously calculated diffusion admittance Ym is given by dQm C =dV oYJ TA m p 1 + jmwT Y T m p 1 + jmwT p

180 Sah's method yields a complex, frequency-dependent capacitance, and therefore it distorts the concept of capacitance. In fact it was shown previously in Section 5. 2 that the diffusion susceptance is not proportional to w for all frequencies; consequently, an attempt to determine a real and frequency independent diffusion capacitance is expected to fail. Thus the superior method is that of finding the admittance rather than trying to find the "capacitance" directly. The results obtained from the admittance method are used in the following analysis. 5. 3. 4 Numerical Calculations of the Diffusion Admittance. The numerical calculation of the diffusion admittance (5. 22) was performed for the doping profiles N(x) = axb where b = 0, 1, 3. These correspond to the abrupt junction y = 1/2, graded junction y = 1/3, and y = /5 respectively, when the diode is reverse biased. Since a p n diode is used, the current resulting from the diffusion admittance in the p side can be neglected. The calculations were performed using program C in Appendix A. The short circuit assumption is employed for all pump harmonics beyond the fundamental so the pump voltage is of the form Vb + V0 cos(w0t) (5. 23) Although in practice the external circuitry can be expected to impress other voltage harmonics on the diode, the short circuit assumption (on the basis of the results found in Chapter III) can be expected to give reasonably accurate results in most cases. When the pumping voltage is of this simple form, the integral contained in (5. 22) can be readily integrated by using Sonine's expansion z cos 0 e = 0(z)+ 2 Z I() cos n 0 (5. 24) n=l 1 where I (z) are modified Bessel functions. Using this expansion and making use of the orthogonality of the trigonometric functions, the m-th harmonic of the current is Kb,(4W) oVb J = pPn) Im(xV e (5. 25) m EKb,,(W) 2qDpPn(W) ImaVo)

where it should be recalled from (5. 16) that 5 is a function of w and T, and the diffusion admittance is found from (5. 22). Thus the large signal analysis shows the same low and high frequency dependence of admittance with w and T for the abrupt junction p diode as did the small signal analysis. The numerical calculation of (5. 25) requires the evaluation of the modified Bessel function K>(z) where z is complex. Since no computer program to carry this out was available in the IBM "Scientific Subroutine Package, " an algorithm was developed to perform this calculation. The details for this algorithm are found in Appendix C while the actual program listing is found in Appendix A under BESKC The diffusion admittance is found using (5. 25) and (5. 22) as a function of WTp and is plotted together with its harmonics in Figs. 5. 1 to 5. 7. Since the admittance is a function of the diffusion length L = (Dp Tp )2 the minority carrier lifetime must be specified independent of the frequency. In Figs. 5. 1 to 5. 7 the following parameters were used: Vb = 0 volt V0 = 0. 1 volt 2 = 401 cm /volt-sec a = 40. 1 There is a tendency, in all but the high frequencies of the abrupt junction case (Fig. 5. 1), for the harmonic of the susceptance to be larger than the fundamental. Also the ratio B1/G1 is always less than 1 for the abrupt junction case but can be larger than 1 for sufficiently large frequencies for the other two profiles analyzed. This indicates that the lower b (higher y) is, the more lossy is the diffusion admittance. The plots in Fig. 5. 4 show that decreasing the depletion layer width by a factor of 10 increases the ratio B1/G1 for the graded junction. For the abrupt junction, there is no change in the diffusion admittance. Mathematically the reason for this is that the parameter W/L enters only in the argument of the modified Bessel functions, which for the abrupt case

B1 / 2 B B~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~BB 0 G5 oO o~~~~~~~~ 3 - 3 B5~~~~~~~~~~~~~~~~~~~~~~ 2 42 '0 1 2 24 1 7 64 1011 1213 1 11 8 1718 1 37 2 27 3 240 1 3 4 6 18 9 011 2 131415 617 819 0212 222 ~~~~~~~~~~~~~a 5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ G BB I I I I I I I III I I I B G~~~~~~~~~~~~~~~~~~~~~~~~~~~G 5 0O 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 27 22 2 4 0 22 3 4 5 6 7 9 10 13 14 8 1 20 21 2 3 4 Normalized Freffusona wr Nal rmalized Frequency a mt 4 G 8 G,, ~~~~~~~~~~~~~~~~~~~C;2 G1 G2 Go Go ~~~~~~~~~~~~~C C~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3 - G~~~~~~~~~~~~~~~~~~~~~~~~~~~~~G "4~~~~~~~~~~~~~~~~~~~~~ C;, G4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Gg G5 G5G 0 1 2 3 4 5 6 7 8 9 10 I 1 12 13 14 15 1f 17 1 8 1 9 20 21 22 23 24 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 N —m~~~~~~~lized Freq-m-y ~~~~~~~~~~~~~~~~~~Normalized Frequency wT Fig. 5.-1. Normalized diffusion admittance when -y = 1/ 2, Fig. 5. 2. Normalized diffusion admittance when 1/3 T = 1 sec, and W/L =0. 1 T =O7sec, and W/L =0. 1 p p p p

17 15 - B 1 BI B2 B 3 14 -- GO G0 G0 G 13 - E 6 ~ E 12 -- u 5~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 101 9 - 8 G0 7C I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 4 B4 B4 3~~~~~~~~~~~~~~ //~~~~~~~~~~~~~~~~~~~ 2 B5 4 G 3 1 -2 3 4 5 6 7 8 9 10 11 12 13 14 15 18 17 1H 19 20 21 22 234B5 3~~~~~~~~~~~~~~~~~~~~~ Normalized Frequency u,Normalized Frequencywy' ~~~~~~~~~~~~~~~~~~~~~~~~~~~00 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 4. 0 3. 5 o 6~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 3.0 11 2 U Gt G [ 2. 5 23 4 5 0 1-3 24 51 G ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~G E 3/ 1 o 2.5 G1 -4 2 1.5 G' 3 G3 ~~~~~~~~~~~~~~~~~~~~~~~~~~G4 G GO 5 1 0. 5 G O:ILG: GO0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1[ — '- 11 1[11 13?4 1]516?7 18 19 20 21122 23 2~.. Normalized Frequency wr Normalized Frequency WT Fig. 5. 3. Normalized diffusion admittance when y 1/ 5, Fig. 5.4. Normalizeddiffusionadmittancewhen 1/3, - =107 -10 7 T = 10- sec, and W/Lp = = 10 s ec, and W/ L,=0. 01 ~~~~~~~~~~~P P

170 -40 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~160 -150 -B1 B 14 B 130- ~ /C 120 -30 E 110 -25 10 C~~~~~ ~~90 -vl I B~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ B3 3 B 20 'lo N 70 -i I. 60 -c 15 50 -B B4 10 -4 40 30 B B 5~~~~~~~~~~~~~~~~~~~~~~B 5 5 0 10~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0 1 0~~~0 1 2 3 4 56 01 1 31 5161 81 0 12 32 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Normalized Frequenicy ',IT all vausx 100) Normalized Freque,,cy 17 (all values 100) 170 - C 160 -40 35 G2~~~~~~~0140 - G1 G0 130 - G Gz 30 - 120 - C E r 110 -25C I ~~~~~~~~~~~~~~~~~~~~ ~~1003 8 "3 u~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 20 0 G x fu- Gg 60~~~~~~~~~~~~~~~~~~~~~~~~~ 15t 50 G4 G4 10 G ( 40- 30- G;5 G5 5 20 G0 G0 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0 1 23 4 5 6 7 891 11 31 51 71 92 12 32 Norrn-,!,'ed Freque,,cy,7- (all values x 100) Normalized Frequency,,, (all values. 100) Fig. 5. 5. Normalized diffusion admittance when y -- 1/ 2, Fig. 5. 6. Normalized diffusion admittance when 1/ 3 T 0 s ec, and AV/ = I 1 T: 17 sec, and W/L =0 p

185 1. 1.6 21.5 1 m 1.4 - 1.2 2 32 O 1.0 mB D.B o 0.78 m 0. 9-4 ~~~~~~0.3 ~/B 0.2 Go 0. 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Normalized Frequency waT (all values x 10-3) Fg1. 57. omlediuiadiacwe =1G1 1.5 1. 1.3 G2 0.2 0. 12 G 0 1.5 1. - o ] G2 1.2 - GO 0.9 0. 0.5 - 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Normalized Frequency ot (all values x 10-3)

186 Kb,,(W) K 1( W) -1 Kb, (W) - K1( W) The depletion layer width W entered the derivation for the admittance through the boundary condition p(W) = pn(W) e.V However, for the abrupt junction the donor concentration is constant throughout the n region, and hence the final solution for this case is independent of W. Figures 5. 5 to 5. 7 show the variation of diffusion admittance with minority carrier lifetime for both the abrupt and graded junction cases. It is clear from these graphs that increasing the lifetime of the minority carriers increased the diffusion susceptance. For the abrupt junction, the B /G1 ratio increased when 'P increased p although this ratio is still limited to a maximum of 1. For short lifetimes Fig. 5. 7 shows the conductance is independent of frequency, and the susceptance almost directly proportional to frequency. The plots of relative loss B1/G1 for various doping profiles in Figs. 5. 8 and 5. 9 indicate how the loss in the diffusion admittance can be controlled. The abrupt junction is seen to be inferior to the graded and cubic (b = 3) junctions at all frequencies. In general for a doping concentration profile given by N(x) = ax, as b increases, the loss in the diode decreases (B1/G1 increases). Also when the base width to diffusion W W length ratio decreases loss decreases. Furthermore as -- decreases, the angle P P WT where the minimum loss occurs increases. These facts are summarized in Table p 5. 1. An approximate calculation (Ref. 80) of the ratio B1/G1 for the graded junction where the arguments of the modified Bessel function are assumed small shows the minimum loss occurs at wT = 6 while the more exact analysis performed here shows the p minimum loss occurs at wT = 11. p In the preceding analysis, the celebrated Shockley equation for the ideal diode has been used. This is J(V) J (eqV/ nkT_ 1) where J is the thermal saturation current and n = 1. Experimentally this is valid 5

4. C 3.Abrupt junction - - - Graded junction W/L = 0.1 (Ref. 80) 3. 0 Graded junction W/Lp = 0. 1 - -- - Graded junction W/L =- 0. 01 2. 5 2.0 \ 1.6, 1. 6 c 1.24,-d 1. 0 0. 8 0. 6 0. 4 0. 2 0.1 1.0 10 100 1000 Normalized Frequency ce-, Fig. 5.8. Relative loss of an abrupt and graded junction diode under forward bias conditions

70 w/L = 0. 1 60W/L = 0. 01 60 B1 x 10 0 40 30r 0 0 20 10 0 0.1 1.0 10 100 1000 Normalized Frequency WT Fig. 5. 9. Relative loss of diffusion admittance with doping profile N(x) = ax3

189 b < Max. WT C at Max. b _____ B1/G B1/G1 0 1/2 ----- 1 cc 1 1/3 0.1 1. 8490 11 1 1/3 0. 01 3. 5264 30 3 1/5 0. 1 6. 5421 17 3 1/5 0.01 45. 198 100 Table 5. 1. Variation of the minimum loss with doping profiles for germanium, but for silicon 1 < n < 10. Several explanations have been advanced for this nonideal behavior. Sah (Ref. 83) analyzes four mechanisms which could account for values of n / 1: (1) bulk diffusion current at high levels gives n = 2, (2) bulk recombination-generation current in the transition region gives 1 < n < 2, (3) surface recombination-generation current gives 1 < n < 2, and (4) surface channel current gives 2 < n < 4. The actual value of n thus depends on many factors, some of which are not fully understood. However, for silicon diodes n = 2 is often used. The effect of using a = q/ 2kT = 20 (Volt)- 1 is a reduction in the magnitude of the diffusion admittance through the factor eVb Im (aV0) while the phase of the admittance remains unchanged. 5. 3. 5 Varactor Diode Model with both Diffusion and Depletion Admittance. The effects of both diffusion admittance and depletion admittance can be combined into a single diode model. At the edge of the depletion layer there are two current components. The first is the displacement current associated with the depletion capacitance and the second is the diffusion current which passes through the depletion layer. Since these two currents add, the depletion capacitance and the diffusion admittance are parallel elements as shown in Fig. 5. 10. The resistor Rs is in series since it represents the bulk resistance of the base region (the n side of the p n diode). Since the diffusion admittance, being proportional to e V, is larger than the depletion susceptance when the diode is forward biased, the latter does not short-circuit the former. In fact most diodes would burn out before the applied voltage reached the built-in potential.

190 Cdep Cdif Rs Gdif Fig. 5. 10. Total varactor model of diode wafer The impedance of the total diode model of Fig. 5. 10 was calculated for the representative parameters listed below: interrogation voltage V(t) = 0. 1 cos(27rfpt) frequency fp = 9. 5 GHz depletion width to diffusion length ratio = 0. 1 Lp minority concentration at depletion layer edge Pn(W) = 10 cm-3 -5 2 diode area A = 4. 41 105 cm minority carrier lifetime T = 1. 84 10 10 sec p dielectric constant of silicon e = 11. 8 e 0 built-in potential 0v = 0. 9 volt series bulk resistance R = I ohm exponential factor for silicon at room temperature a= 20 V-1 mobility p= 401 cm 2/ V-sec diffusion constant D = 10 p Except for the minority carrier lifetime, which was chosen to minimize the diffusion admittance losses, the above diode parameters are typical for silicon parametric varactor diodes. The relative magnitude.;of the depletion and diffusion susceptances shown in Fig. 5. 11 can be compared for various bias voltages for both the abrupt and graded junction diodes. Clearly the diffusion admittance is important only in the forward bias region. The impedance at the diode terminals is plotted in Fig. 5. 12. This plot shows that the real part of the diode impedance increases when the bias voltage is between 0. 4

191 100 80 - - Abrupt Junction 80 — Graded Junction 60 50 40 30 20 I B I 0.8 3i; 2;Bdif 0.6,,, 0.5 5 0. 4 0.8 I I I I 13 I 0.2 0.1 -5 -4 -3 -2 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Bias Voltage Fig. 5. 11. The depletion and diffusion susceptance when the minority carrier lifetime is chosen to minimize loss in the graded junction diode, i. e., T= 1.84x 10-1' 0sec

192 4. 0 3. 8 - -- Abrupt Junction Graded Junction 3.6 \ 3. 4 3.2 3. 0 2.8 \ -X 2.6 2.4 \ 2.2 - 2. 0 1.4 1. 8 0. 6 1. 4 0.2 2 0. -5 -4 -3 -2 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Bias Voltage Fig. 5. 12. The impedance of the total varactor model when the minority carrier lifetime is chosen to minimize loss in the graded junction diode

193 and 0. 8 volts. Outside of this range the parallel combination of the diode capacitance and diffusion conductance is negligible because of the low conductance at small voltages and the high susceptance at large voltages. If a different value of T is used, instead of the optimum value used above, the resistive part of the diode impedance can become -7 quite large. For example when T = 10 sec, this resistance has a maximum value p of 6. 5 ohms. 5. 3. 6 Comparison with Experimental Work. Several experimenters noticed some time ago that a forward-biased pn junction exhibits what is termed an "inductive effect. " They noticed that the diode capacitance increases with applied voltage according to the standard depletion capacitance law until the voltage reaches a value between 0. 60 and 0. 75 V, at which point the capacitance decreases rapidly with further increase in voltage - hence the term "inductive effect. " Sah (Ref. 82) made a thorough analysis of the graded junction diode which included the forward bias region. His capacitance measurements clearly show the capacitance has a maximum value, after which it rapidly decreases. However, his theory based on finding the charge storage and taking the derivative with respect to the applied voltage, does not show the capacitance maximum. He considered this discrepancy to be possibly caused by conductivity modulation, i. e., a change in conductance with applied voltage. dQ Wang (Ref. 84), following Sah's basic approach (i. e., obtain capacitance from d) showed theoretically that the capacitance would have a maximum and decrease toward zero at large forward bias. His analysis of an abrupt junction diode differs primarily from earlier approaches in that, instead of assuming the quasi- Fermi levels are constant, he takes these levels to be actually proportional to e V/ O'Hearn and Chang (Ref. 85) also show experimental evidence for the maximum in the capacitance versus voltage curve. In their theory an approximate expression for the diffusion admittance of an abrupt junction diode is found from i _ Aq ap V V at-dx

194 Analysis of this expression shows that, as the bias increases, the capacitance increases to a maximum, then falls to a local minimum; at voltages approaching the built-in potential, the capacitance starts to increase again. The maximum is here attributed to the conductance part of the admittance. This second increase in the capacitance predicted by O'Hearn and Chang is substantiated neither by their own experimental measurements nor by those of Sah. Furthermore, the existence of this local capacitance minimum is doubtful since it is unlikely that the diffusion capacitance would increase more rapidly than diode loss at high forward bias. The experimental data in Refs. 83 and 85 show that as frequency increases the maximum diode susceptance remains practically unchanged, and the voltage where this maximum occurs, decreases. Also, when the bulk resistance increases, the maximum susceptance decreases and the voltage where this maximum occurs, decreases. These characteristics are all consistent with the diffusion admittance theory presented in this chapter. The variation in the maximum diode susceptance with frequency and bulk resistance noted above can be predicted from (5. 22) and (5. 25). The curve shown in Fig. 5. 13 is the susceptance of the total diode model shown in Fig. 5. 10, calculated using the theory derived by Parygin and Maneshin (Ref. 80). It is seen here that there is one capacitance extremum as confirmed by the experimental points in Refs. 83 and 85. Figure 5. 13 also portrays the effect of changing the minority carrier lifetime, a parameter that was neglected in Ref. 84 and 85. The theory used here is valid for one sided abrupt and graded junction diodes as well as other doping profiles expressable as N(x) = ax and is not restricted to low frequencies as are the theories of Sah and Wang. Furthermore this analysis is not restricted by small signal assumptions, and it gives the admittance, rather than the less valid "complex capacitance". 5. 4 Effects of Diffusion Admittance on a LSUC The previous sections dealt with first finding the diffusion admittance of a varactor diode and then secondly with the total impedance of the diode when both diffusion admittance and depletion capacitance are combined in one diode model. A similar outline will be followed in evaluating the effects of diffusion admittance on the characteristics of a LSUC. First the diffusion admittance diode model will be used to analyze the LSUC

195 1.0 0. 9 - - -Abrupt Junction 0. 8 0.87 Graded Junction 0. 7 0. 6 T=l. 84 x 10 s 0. 5 0. 4 0.034 — I 0. 2 0 0.0 4 0.0.09 0.2. 08 ~ M 0. 07 ICQ o. 006 0. 05 0. 02 O 5 4 3 -2 1 0. 1 0. 2 0.3 0.4 0.5 0.6 0.7 0.8 0. 9 Bias Voltage Fig. 5. 13. The total diode susceptance for 7 = 10 7 sec and T= 1. 84 10O10 sec (minimum loss) when the bulk series resistance Rs = 1 ohm

196 circuit on both large signal and small signal bases. Then a combined diode model will be used to determine LSUC gain and noise figure from a small signal analysis. 5. 4. 1 Large Signal Analysis of the Diffusion Admittance in a LSUC. Although most parametric amplifiers are designed to operate with a reverse biased varactor diode, high gain can be achieved in practice if some forward conduction is allowed. The previous formulation of diffusion admittance will be applied to the LSUC. The assumed circuit model is shown in Fig. 5. 14 where the circuit elements are assumed to be ideal bandpass filters which act as short-circuits to out-of-band frequencies. This simple model can be contrasted with the more complex model used in Chapter III where the circuit impedances at out-of-band frequencies were neither necessarily open nor short circuits. However the results obtained from the model of Fig. 5. 14 will clarify the basic properties of a LSUC using a diffusion-admittance model, and the results obtained will give a first order approximation to the gain and noise figure of an actual LSUC when the diode is forward-biased or pumped into the forward region. In addition a direct comparison between the depletion- and diffusion-admittance models can be obtained from this analysis. For convenience, the following notation will be used for the different LSUC frequencies. wk2 wp + (-1l)l C (5. 26) u0 = p where k = 1, 2, 3,... and [a] is the symbol for the greatest integer less than a. The subscript k will be hereafter reserved only for the above definition. It should be noted that only the upper and lower sidebands of the pump are considered here, but that the following analysis, in principle, is not restricted to this choice of frequencies. Although several current frequencies will be generated, the voltage across the diode will consist of only the pump, signal, lower sideband, and bias voltages. Even with these assumptions two additional computational problems arise: the calculation of the gain when there are incommensurable frequencies, and the relative phase of the diode voltages. Both problems are discussed below. 5. 4. 1. i Incommensurate Frequencies in the LSUC. When the voltage applied to the diode chip is

197 Vo + - V + + iIi_ Vb Fig. 5. 14. LSUC circuit model for the diffusion admittance V(t) = Vb + V0 cos(w0t) + V1 cos(wolt + 41) + V2 cos(w2t + 2) (5. 27) then equations (5. 20) and (5. 21) must be modified because the frequencies used there were all harmonically related. The problem can be resolved by choosing the frequency wf in (5. 20) and (5. 21) so that the signal, pump, and lower sideband frequencies in (5. 27) are all harmonically related to wf. Such a choice can always be made if the frequencies are rational numbers, which is so in the physical world. This chosen frequency wof will hereafter be termed the fundamental frequency. The m in (5. 20) and (5. 21) therefore denotes the number of harmonics of the fundamental frequency and is related to the subscript k in (5. 26) by the following expression: Wk = mkof (5. 28) For the frequencies chosen for the LSUC under investigation, fl = 1 GHz, f2 = 8. 5 GHz, and f 9. 5 GHz. This means the fundamental frequency must be f =0. 5 GHz, and 0- f m0 = 19, m1 = 2, and m2 = 17. In the work that follows the subscript k on m will usually be implied. 5. 4. 1. 2 Phase of the Lower Sideband Voltage. The second problem which occurs in the LSUC is the calculation of the relative phases between pump, signal, and

198 lower sideband voltages. Since the pump and signal voltages are two independent input voltages, their relative phase will be arbitrary. However, the phase of the lower sideband voltage will be dependent on the signal and pump, and in order to find this relationship recourse must be made to a small signal assumption. If the varactor is pumped with V(t) = Vb + VO cos(w0t + 40), then (5. 18) should be modified so that the Fourier series is expanded in terms of the angle (mw0t + 0). The admittance then from (5. 18), (5. 19) and (5. 22) is 2qD p (W) V = (~k=0+ j ) Il(o) e b M (o0 + j0) Thus the cosine admittance is MCa0 cos(Woot + T/0) and the sine admittance is -Mc 0o sin(w0t + T/O) so that Yp = Mca0 cos (wt + 0o) -McPO sin(w0t + /O0) =iM a + 2 cos[ oot + Arctan ( 0/a ) + i0] c 0 0 0 0 0 The current at the lower sideband frequency is obtained by multiplying the admittance by the signal voltage, V1 cos(olt + 4s1) 2 2O J +J = Me c 2+ COS(ot + 0 + Arctan V1 cos(wlt + 4,1) McVi cos kw2t + 0-, + Arctan ) p-s 2 / + - If the lower sideband current flows through a real impedance, then the voltage is R Mcl + 1 p-s p-s p-s 2 cosw2t+ ~0" q1+Arctan-j

199 Since the lower sideband voltage is p-s = -V2 cos(w2t + '2) = V2 cos(w2t + g'2 + f) the relationship between the phases must be 0 ~C/2 ='O - + Arctan - IT. (5. 29) If the varactor were lossless, then cz0 = 0 and '2 = 0- /- 2 which agrees with the usual phase relationship in an upconverter (Ref. 86). 5. 4. 1. 3 Numerical Calculation of LSUC Currents. The numerical integration of (5. 20) and (5. 21) requires some care because the exponential term in the integrand varies over a wide range in the integration interval. The voltage applied to the diode chip, V(t) = Vb + V0 cos(W0t) + V1 cos(wlt + +1) + V2 cos(w2t+ 2) = Vb + V0 cos(coft — ) + V1 cos(ft- f i, + 1) + V2 cos(Wft + 'P2) "f CL~f f f has 19 maxima and 19 minima in the integration interval with the choice of frequencies made in Section 5. 4. 1. 1. Since this voltage appears in the exponential, it dominates the magnitude of the variation of the integrand. In order to obtain accurate results the total interval was divided into 40 sections: the division being made at the 38 extreme values of V(t) and the two end points. An IBM program which used a 12-point Gaussian quadrature formula was used one;each of these subintervals, and the total was added together to give each Jkc or Jks current component. The program listing is found in Appendix A under program D. The numerical integration was carried out using the same parameters for a p+n diode as were used in the calculation of the diffusion admittance in Section 5. 3. 4 except

200 now a = 20, W/L = 0., 1 1 = ~0 Vb = 0. 7, and V0 = 0. 6. Table 5. 2 summarizes the results for the abrupt and graded junction when V1 and V2 are varied. The table indicates that the graded junction has a higher current gain, IJ21/ IJ 1, than the abrupt junction. Decreasing the signal and lower sideband voltage by a factor of 10 decreased the output current, but the current gain remained approximately constant. In these calculations it was assumed that at t = 0, the relative phase between the pump and signal voltage is 00. When 41 is changed to 30~ and 60~ the computer program shows that the phase of the output currents change, but the current magnitudes remain unchanged. The Gaussian quadrature method used here to perform the numerical integration took approximately 15 seconds of execution time for each set of 10 current densities. A much faster but less accurate method is described in Appendix D. The currents shown in Table 5. 2 will be used later to gauge the accuracy of the small signal assumption employed in the next section. The importance of the diffusion admittance is best seen when compared with the depletion capacitance. The currents generated by a nonlinear depletion capacitance when the same voltage is applied to the diode chip as in the diffusion case is shown in Table 5. 3. These current densities were normalized so that the magnitude of the current at pump frequency IJ0 I is the same for both the diffusion and depletion admittances. In this way a direct comparison can be made between the two. In each case shown in Tables 5. 2 and 5. 3, the current sidebands arising from the diffusion admittance are much larger. This is further illustrated in Fig. 5. 15 where current magnitudes for the high voltage condition are plotted as a function of the frequency fk This behavior is expected since e is more nonlinear" than (1 - V/m0v)-Y This enhanced capability of producing a sideband current is useful in LSUC operation in producing more negative resistance at a given pump level. 5. 4. 2 Small Signal Analysis of the LSUC. In low-noise parametric amplifier analysis, the usual approach is:to make a small signal assumption for the signal and lower sideband voltages, and calculate an admittance or impedance matrix, from which an expression for gain can readily be found. The small signal assumption will now be applied to (5. 20) and (5. 21), leading to a diffusion admittance matrix and a gain expression.

201 Abrupt Junction Graded Junction V1= 0.01 k JkcA/cmA/c Jks/cm2 JkA/cm JA/cm2 JkA/cm2 IJkA/cm2 V -0. 10 0 660. 17 -670. 54 940. 98 600. 85 -664. 21 895. 65 1 220. 36 -16. 02 220. 94 189. 93 -27. 51 191. 92 2 42. 20 -667. 26 668. 59 -12. 73 -633. 94 634. 06 3 630. 87 - 53. 08 633. 10 600. 79 - 51. 65 603. 01 4 49. 29 - 92 5. 22 926. 53 -15. 63 - 891. 91 892. 0 5 5 706. 72 -67. 33 709. 92 681. 48 - 59, 39 684. 06 6 43. 80 -1000. 00 1000. 96 -21. 09 -969. 73 969. 96 7 641. 10 -68. 38 -644. 74 621. 78 - 58. 46 624. 52 8 32. 34 -931. 91 932. 47 -25. 27 -906. 67 907.01 9 507. 74 -60. 10 511. 28 494. 15 - 50. 96 496. 77 V1 = 0. 001 V2 =-0.01 0 267. 08 -267. 10 377. 72 244. 28 -265. 68 360. 91 1 13. 09 -891. 00 13.12 11.31 -1. 58 11.42 2 2. 52 -39.62 40.41 -0. 75 -37.73 37.74 3 37.45 -2. 79 37. 55 35. 76 -2. 72 35. 86 4 3. 21 - 54. 91 54. 99 -0. 64 - 53. 08 53. 08 5 41. 93 -3. 39 42. 07 40. 53 -2. 92 40. 64 6 3. 19 - 59. 29 59.38 -0. 65 - 57. 67 57. 68 7 38.01 -3. 32 38. 15 36. 95 -2.73 37.05 8 2. 74 - 55. 19 55. 25 -0. 66 - 53. 88 53. 88 9 30, 07 -2. 83 30.20 29. 33 -2. 28 29. 42 V1 = 0. 0001 2=-0. 001 0 264. 11 -264. 07 37 3. 42 241. 57 -262. 68 356. 87 1 1. 30 -0. 88 13.03 1.12 -0. 16 1. 14 2 0.25 -3. 94 4.02 -0.08 -3.75 3.75 3 3.72 -0.28 3.82 3. 55 -0.27 3. 56 4 0. 32 - 5. 46 5. 54 -0. 07 - 5. 28 5. 28 5 4.17 -0.33 4. 30 4.03 -0.29 4.04 6 0.31 -5. 89 5. 97 -0.68 -5. 73 5. 73 7 3.78 -0.33 3. 94 3.67 -0.27 3.68 8 0. 27 - 5. 48 5. 54 -0. 07 — 5. 35 5. 35 9 2.99 -70. 28 3. 13 2.92 -0.23 2. 92 Table 5. 2. Exact calculation of the current densities

202 Abrupt Junction Graded Junction V1= 0.01 k JJkcA/cm2 JksA/cm2 IJklA/cm2 JkcA/cm2 JksA/cm2 IJklA/cm2 V2 =-0.1 0 -0. 26 -941. 00 941.00 -0.16 -895. 65 895. 65 1 1. 64 -1. 68 2. 34 1. 03 -1. 58 1. 89 2 -142. 30 -1. 40 142. 30 -134. 66 -0. 88 134. 67 3 2. 52 -1.73 3.05 1.41 -1.09 1.78 4 -29. 52 -0. 42 29. 52 -18. 61 -0. 24 18. 61 5 0. 78 -0. 48 0. 92 0. 41 -0. 27 0. 49 6 -6. 61 -0. 10 6. 61 -3. 70 -0. 05 3.70 7 0. 20 -0. 12 0. 23 0. 10 -0. 06 0. 12 8 -1.45 -0.02 1.45 -0.76 -0. 01 0.76 9 0.04 -0.03 0.05 0.,02 -0.01 0.02 V1 = 0. 001 V2 =-0.01 0 0. 00 -377. 50 377. 50 0. 00 -360. 91 360. 91 1 0. 07 -0. 07 0. 09 0. 42 -0. 64 0. 08 2 -5. 71 -0.06 5. 71 -5. 43 -0.04 5. 43 3 0. 10 -0. 07 0. 12 0.05 -0. 04 0. 07 4 -1.18 -0.02 1.18 -0.75 -0.01 0.75 5 0. 03 -0. 02 0. 04 0. 02 -0. 01 0. 02 6 -0.26 0.00 0.26 -0.15 0.00 0.15 7 0.01 0.00 0.01 0.00 0.00 0.00 8 -0.06 0. 00 0. 06 0. 03 0. 00 0. 03 9 0. 00 0.00 0.00 0.~00 0.~00 0. 00 V1 = 0. 0001 V2 =-0. 001 0 0. 00 - 374. 00 374. 00 0. 00 - 356. 87 3 56. 87 1 0. 01 -0.01 0. 01 0. 00 -0. 01 0. 01 2 -0. 56 -0. 01 0. 56 -0. 53 0. 00 0. 53 3 0. 01 -0.01 0.01 0. 00 0. 00 0. 01 4 -0. 12 0. 00 0. 00 -0. 07 0. 00 0. 07 5 0.00 0.00 0.00 0.00 0.00 0.00 6 -0.03 0. 00 0.03 -0.01 0.00 0.01 7 0.00 0.00 0.00 0.00 0.00 0.00 8 -0.01 0.00 0.01 0.00 0.00 0.00 9 0.00.'0. 00 0.00 0.00 0.00 0.00 Table 5. 3. Current densities caused only by nonlinear depletion capacitance when normalized to give the same pump current iJ01l as the diffusion current

203 1000 800..~ ~Diffusion Current 50 400 300- \ 20 -— Abrupt Junction ---— Graded Junction 100 80 60 50 40 30 A Q It 20 I I, I ro~~~~~~~~~~ o8Z Ii~~~~~~~~~ 0.6 5 I I U II~~~~~~~~I I\ 1 4 0.8 0.36 0. 51 0.4 Il 0. 3~~~~~~~~~ 0.3I II V I~~~~ 0.2 0.1 0.I I I I I I II 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Current Sideband Number k in Jk Fig. 5. 15. A comparison of the sideband currents generated by the nonlinear diffusion admittance and the nonlinear depletion capacitance when V1 = 0. 01 V and V2 = -0. 1 V.

204 A comparison will be made between the current values determined by this method and by the exact method, to determine the applicability of the small signal method. If V1, V2<< V0, then eGV(t) ae (Vb V~ 0 Cos * wO0t) + 2 V cos([.t i ] Using Sonine's expansion this becomes eV(t) e b l(av) + 2 v I(aV cos(rwot)] [00 ri=1 1 + E V. CoS( w t + i) where Ir(cyV0) is a modified Bessel function of order r. When this is substituted into (5. 20) and (5. 21) an integrand is obtained with three factors each containing terms harmonically related to the fundamental frequency. Using the orthogonality of the trigonometric functions these integrals can be easily simplified and evaluated. Thus (5. 20) becomes caV qDpn (W)e 2b v pn }0f [I0(aTV) + 2 V Ir(aV0) cos(rwt) 0 r=1 cam Cos mwt + fm sin mwt] dwt + f 1 0(av ) + 2 l I(a) cos(rw t) [e Vi COS(-wit + ui] [Ym cons mt+ sin molft i dt ( becomes~~~r

205 After some straightforward but lengthy calculations, this reduces to qDpp(W) aV 2 J mc Pn(W) eb [2I (aV0) a 6 Mc W m m, rm0 + 6 aVIr (a (m cos 1 + 3m sin i1) m, rm0-m1 1Ir a0 m +0 + m 5rmI+m ~V(o) (aV)(a cos 4 - sini l) m, rm-m r 0 m 1 m +0 m, rm0-m2 aV2 r(aV0) (am cos g2 + Om sin 2) m, rm0+m2 aV2lr(Vi) (am cos gt2 O- m sin 2)(5 30) where 5 is the Kronecker delta function and r is the harmonic number of the pump m, n frequency. In similar fashion the sine component of the current density is qDDpn(W) aV ms e b [-21 (av) 6 W - Ir (aV0) m ms r m c +m, rm0-ml aVlIr(aV0) (am sin 4/l O tm cos zil) a a- 6) sin m cos m, rm0+ml + m, rm- m2 V2r (av0V) (a1msin ~2 - Om cos g/2) m, rm0+m2 aV2Ir(aV0) (am sin gi2 + Om cos /2)] (5. 31) It should be noted that (5. 31) can be obtained from (5. 30) by substituting -/m for +a, and +am for +m and all the unsubscripted m's above have an implicit subscript k in accordance with definitions (5. 26) and (5. 28). Table 5.4 shows the r:esults of the calculations based on the small signal assumption for the abrupt and graded junction. The program used to generate Table 5.4 is program E in Appendix A. As might be expected a comparison between the small signal and exact method of Table 5. 2 when V2 is comparable in magnitude to V0 indicates an

206 Abrupt Junction Graded Junction V1 0. 01 k JkcA/cm2 JksA/cm2 Jk A/c2 JkA/cm 2 JkA/cm2 iJklA/cm2 V2 =-0. 1 0 264.08 -264. 04 373.43 241. 54 -262. 65 356. 83 1 130. 11 -8. 86 130.41 112.46 -15. 74 113. 56 2 25. 05 -393. 93 394.73 -7. 41 -375. 18 375. 25 3 372.40 -27. 76 373.44 355. 54 -27.01 356. 57 4 31. 97 - 545. 94 546. 87 -6. 35 - 527. 77 527. 81 5 416.92 -33. 65 418.28 403.01 -29. 01 404.05 6 31. 84 - 589. 53 590. 39 -6. 39 - 573. 47 573. 50 7 377. 84 -32. 93 379. 27 367. 33 -27. 08 368. 32 8 27.37 - 548. 64 549. 32 -6.47 - 535. 63 535. 67 9 298. 88 -28. 06 300. 19 291. 56 -22. 64 292. 44 V1= 0. 001 V2 =-0.01 0 264. 08 - 264. 04 373. 43 241. 54 -262. 6 5 3 56. 83 1 13.01 -0.86 1.30 11.25 -1. 57 11.36 2 2. 51 -39. 39 39.47 -0. 74 -37. 52 37. 53 3 37.24 2.78 37.34 35, 55 -2.70 35. 66 4 3. 20 - 54. 59 54. 69 -0. 64 - 52.78 52.78 5 41. 69 -3. 37 41. 83 40. 30 -2. 90 40. 41 6 3.18 -58. 95 59.04 -0.64 -57.35 57. 35 7 37.78 -3. 29 37. 93 36.73 -2.71 36. 83 8 2. 74 - 54. 86 54. 93 -0. 65 - 53. 56 53. 57 9 29. 89 -2. 81 30.02 29. 16 -2. 26 29. 24 V1= 0. 0001 V2 =-0. 001 0 264. 08 -264. 04 373. 43 241. 54 -262. 65 356. 83 1 1. 30 -0. 09 1. 30 1.12 -0.16 1.14 2 0.25 -3. 94 3. 95 -0. 07 -3. 75 3. 75 3 3.72 -0.28 3.73 3. 56 -0.27 3. 57 4 0. 32 - 5. 46 5. 47 -0. 06 - 5. 28 5. 28 5 4.17 -0. 34 4. 18 4.03 -0. 29 4. 04 6 0. 32 - 5. 90 5. 90 -0. 06 - 5. 74 5. 74 7 3.78 -0. 33 3.79 3.67 -0. 27 3. 68 8 0.27 -5. 49 5. 49 -0.06 -5. 36 5. 36 9 2.99 -0. 28 3. 00 2. 92 -0. 23 2. 92 Table 5. 4. Small signal method for calculating the current densities

207 unacceptable error for the small signal method. However, when V1 and V2 are at least one order of magnitude lower than V0, the small signal method gives quite accurate results all the way to 4w + wo p s Recalling that the total current density is Jk = Jkc - jJks, the signal and lower sideband currents are i = K I0(aV0) (a + j31) cos 1/ + j(al + jl) sin kl V1 + (aV0 + ) C(+il) os 2 j(a1 + i3l) sin 42] V2 i2 = K 1(a(V0) a2 + j0 2) cos 1 - j(a2 + Jp2) sin V1 ] V1 + o0(aV) [ (2 ~ j2) cos +2 + i(a2 + P2) sin Y,2] V2 AqDppn(W)a aUVb where K = eW and A is the diode area. The subscripts are particular values of k rather than m, and therefore designate the sideband number according to definition (5. 26). The 2 x 2 admittance matrix is then i[ I[K0(V0) (al + ji1) KI(aV0) (l + JP1)1 i Vle (5. 32) i2,* K i1( (V0) (2 - jPi2) Ki0(a V0) (a2- j2i2)j (V2e ) The gain of a LSUC is G21= 4Z211GGg (5. 33) where Z21 is the (2, 1) element of the inverse of the admittance matrix Yll Y2 Y21 Y22 The symbol G is the generator conductance at signal frequency, and GQ is the load conductance at the lower sideband frequency. If the current is the forcing function, then i2 = 0 and

208 _ 21 21 Y1 Y2 - Y12 Y21 Therefore the gain for the diffusion matrix is 4K I2(2(+ 2i)G G G = 2 I 2 f2g (5. 34) 212_2,.,.2 22G 2 1 (KIo1 + G + jKI01)(KIOa2 + G j KI0I2) K I1 + j 1)(U2 j (2) where the argument of the modified Bessel functions is understood to be cVT0. If the input and output circuits are resonated by additional reactive circuit elements, the gain simplifies to 222 2 2 4K I1(a2+ 2) GGg G21 2F 12+g 2 2 (5. 35) LKIoa1 + Gg)(KIaO2 + G) - K 11(a1 2 + 2 +K I1312 21) The last term in the denominator does not have an analogous term in the usual reverse bias LSUC gain expression. This term is due to the real part in the off diagonal terms of (5. 32). Equation (5. 35) clearly shows how losses in the diffusion admittance can degrade LSUC gain and indicates the necessity for a high ratio of i3k/ k' The actual amount of additional circuit susceptance needed to tune the diffusion susceptance of the diode depends not only on the diode parameters and frequencies, but also on the bias and pump voltages. A list of these susceptance values is found below in Table 5.5 which is based on the diode parameters given in Section 5. 3. 5 when it is biased at 0. 7 volts. As the bias voltage decreases, these susceptance values decrease until at 0 volts, the susceptance is negligible. Diode Profile y= 1/2 1/3 1/5 f1 - 1. 000 GHz -0. 324 -0. 139 -0. 0356 f20= 8. 500 GHz - 1.328 -0. 835 -0. 296 Table 5. 5. Susceptance values in mhos needed to tune the diffusion susceptance of the diode when it is biased at 0. 7 V

209 5. 4. 3 The LSUC with the Total Varactor Diode Model. The admittance matrix for the LSUC when both depletion capacitance and diffusion admittance are considered is the following simple modification of (5. 32) KI0ol + j(W1C0 + KI0 1) KI1i1 + j(WlC1 + KIf1)1 [Y] = (5. 36) Llia2 + j(w2C_ + KI1f2) KI0o 2 - j(w2C0 + K1032) where C0 and C+1 are the average and first harmonic depletion layer capacitances. The impedance matrix for the total diode model of Fig. 5. 10 is obtained by inverting the Y matrix of (5. 36) and adding terms resulting from the series bulk resistance. The inversion is a lengthy but straightforward process, and the result is KIr0a2- j(w2C0+ KI0i2) -KIl1c- j(W1C1+ KI11ll)1 - L I1a2 + j(w2C_1+ K2 12) KI00al + j(WlC0 + KI01) 1 (5 Lo Rs where A is the determinant of the Y matrix. The gain for the LSUC then is obtained by resonating the reactance terms on the diagonal of (5. 37) and substituting Y21, the row 2 column 1 element of the inverse of the resulting matrix, in G 4 Y 2 R R (5. 38) 21 4 21 RRg. To display the effects of diffusion capacitance effectively, the gain and noise figure should be found as a function of bias or pump voltage. However if the voltage changes, then so does the depletion layer capacitance together with the relative magnitudes of the pumped elastance harmonics. In order not to mask the effects of diffusion admittance with the changes in depletion capacitance, the following numerical calculations of gain and noise figure were performed with a fixed value of S, the first harmonic of the nonlinear depletion layer elastance. To insure the circuit is tuned at midband the average varactor elastance and modulation coefficient are allowed to vary. This procedure means that if only depletion layer capacitance were considered, the midband gain would remain

210 constant for any bias or pump voltage, although the bandwidth would vary because of the different tuning reactances. However there are still two problems that must be resolved: (1) the circuit is described in terms of depletion layer capacitance rather than elastance, and (2) the bias and pump voltages give a dynamic capacitance different from the static capacitance. The first problem is easily resolved since the inverse of the 2 x 2 - capacitance matrix is an elastance matrix where 0 2 2 2 2 C (1 - C/C0) C0(1 - i) CO m S1 = 2 2 (5.40) C0(1 - m) o(1 - mi) and the modulation coefficient ms = S/S = - C /CO. When the rank of the matrix is larger than two however, the relationship between the elastance and capacitance matrices is not simple (Ref. 15, pp. 54-64). The second problem arises because the nonlinear depletion capacitance when expanded into a Fourier series will have an average capacitance different from the dc capacitance. The depletion layer capacitance is Cb C (t) = (5. 41) [1- V(t)/ v] where Cb is the static dc capacitance and V(t) = Vb + V0 cos(w pt). The Fourier series expansion of C(t) was carried out by Oliver (Ref. 15, Appendix B) and is shown below: (-1)a (2k+ )r(2k+~+y) cos(w pt) ~~H,~~~ ~~~~(a,) ~~~ v p ~(5. 42) k=0 r (y) 2 r +2k)F(+k+l)r (k+1) where A V0 v Vb- 0v

and r is the well known gamma function. Therefore the dynamic average capacitance is Cb H (av) C 0:1 __ )y > (5. 43) c~ -- (1-Vb/ ) ' the modulation coefficient is H1(av) M s H (a ' (5. 44) and the desired first harmonic capacitance is C1 = - msC0 (5. 45) Thus when the diode built-in potential 0v, the doping profile y, the desired elastance S1i and the applied voltage are known, ms is found from (5. 44), C0 from (5. 40), and C1 from (5. 45). These values of C0 and C1 can be used in (5. 35) to find the LSUC gain and noise figure. The numerical calculations of gain and noise figure were performed using program F in Appendix A, and the results are shown in Figs. 5. 16 to 5. 19. The diode model had the same properties listed in Section 5. 3. 4 with the additional property that the first harmonic of depletion elastance is S1 = 2. 20 x 10 F, and the diode area was not used explicitly. The circuit properties were chosen as follows: pump frequency f = 9. 500 GHz p signal frequency fs = 1. 000 GHz generator resistance R = 100 ohms load resistance R = 1. 3 ohms The maximum gain (not midband gain) and the minimum noise figure were calculated as a function of bias and pump voltage. If only the depletion layer capacitance were involved the transducer power gain would be 19. 12 dB and the noise figure would be 1. 1410 for all voltages. The graphs in Figs. 5. 16 to 5. 19 show the deviation from the depletion layer calculations for the LSUC. The diffusion admittance is deleterious to the operation of this LSUC when an abrupt or graded junction diode is used. However when y = 1/ 5

212 50 49 Doping Profile 48 W -— N =ax 47 3 N = ax 46 45 44 43 42 41 40 39 38 37 3d 36 35 34 33 - 32 31 30 29 28 ct 27 27 26 25 24 23 22 21 20 18 17 16N 15 \ 12 114 0. 1 0. 2 0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 Bias Voltaze Fig. 5. 16. Maximum gain as a function of bias voltage with pump voltage = 0. 1 V when the optimum lifetime is used for both doping profiles. In each case 0 = 0. 9 V, S = 2. 20 x 1011 F-1, W/L= 0.1, and pn(W)= 10' cm-3

A U D IIO=(M) d pue 'I =u/A 't-d TIOTtXOZ' Z S 'A 6 'o0 0 aso yoqe ul 'salojoid Su.dop qloq loj pasn sl aoiuilaJI tunnuzldo all uoaqI A I 0 = aet1IOA duind qtlM aS1tlIOA s.q Jo uo.jounf l se aS n.J as. ox 'LS ' '. aSelloA S1l[ aopola 80 L '0 9 0 '50 'O / '0 Z 0 '0 I I _ 9 / /!! - I ICD I Ir 18~~~ xe= N I E I xe= N -- - I 6 al.oad ~u.doa _ _, T01 IZ

214 50 49 48 Doping Profile 47 — N = ax 47 3 46 - N = ax 45 44 43 42 41 40 39 38 37 36 35 cm 34 33 32 0 31 a- 30 29 28 27 E 26.- 25 24 23 22 21 20 19 18 a 17 16 15 14 13 12 11 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Pump Voltage Fig. 5. 18. Maximum gain as a function of pump voltage at zero bias voltage when the optimum lifetime is used for both doping profiles. In each case 0v = 0. 9 V, S1= 2. 20x 101 F-1, W/L= 0.1, and Pn(W) = 1010 cm-3

215 10 Doping Profile 9 - N = ax 3 N =ax 8 -7 I I I 6 -- 5 I 4 - I I I k I I ) 3 I I 2 Fig. 5. 19. Noise figure as a function of pump voltage at zero bias voltage when the optimum lifetime is used for both doping profiles. In each case 0 = 0. 9 V, S1 = 2.20x 10F', W/L= O.1, and p_(W)= 1010 V-3

216 the diffusion admittance can substantially increase the gain or alternatively the bandwidth. The diffusion admittance increases the noise figure above that which would occur for the depletion capacitance, but noise figure improves as y decreases. Low loss diffusion admittance is a prerequisite for improved LSUC performance when the diode is forward biased. The general trends expressed in Table 5. 1 can therefore be used in judging how a varactor ought to be chosen if its diffusion admittance is to be utilized. The experimental existence of the optimum bias voltage for maximum LSUC gain shown in Fig. 5. 16 was observed in an empirically tuned LSUC (Fig. 5. 20). This circuit consisted of a silicon varactor diode series mounted in a length of 14 mm coaxial line, together with several standard filters and double-stub tuners. The input pump and signal frequencies (f = 9. 1 GHz and fs 0. 975 GHz) were combined through a coaxial p s wye junction and transmitted to the mounted diode. The output lower sideband frequency was observed at the opposite end of the diode mount. The bias voltage was applied through a General Radio 874- FBL bias insertion unit and monitered with a VTVM and a microampere meter. The variable tuning elements were needed to compensate for changes in diode capacitance when the bias voltage changed. Throughout the measurement procedure, the pump power remained constant, while the tuners were changed to achieve maximum gain at each bias voltage. Either because of parasitic circuit losses or lack of sufficient pump power, this LSUC did not oscillate when retuned for maximum gain. In both the theoretical and experimental curves the gain falls very quickly after achieving a maximum. This comparison together with the correlation of the diffusion susceptance curve in Fig. 5. 13 with the previously observed "inductive effect" adds to the confidence that can be placed in the diffusion admittance theory presented here. The present model provides more accurate description of the varactor diode than the model described in Section 5. 2 where the diffusion admittance was neglected. However a direct comparison cannot be made between the large signal USUC analysis reviewed there and the LSUC presented subsequently, since in the USUC power handling capability and efficiency are the primary parameters of concern, while in the LSUC gain by means of generating negative resistance is of primary concern. However, the additional diffusion admittance discussed here would affect the optimum drive level in the USUC since the additional capacitance would be helpful only as long as the diffusion conductance losses

217 12 10 --10 _ \ -20 -22 =8 -l -24 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Diode Bias Voltage Fig. 5. 20. Experimental gain of a LSUC as a function of bias voltage on the varactor were sufficiently small. TIhere appears to be no analytical discussion in the literature of the effects of forward bias on either the RPA or the small signal upconverter. The diffusion capacitance calculations here show why the capacitance-voltage curve has a maximum and how this can be utilized in a LSUC and presumably in other varactor devices. the effects of forward bias on either the RPA or the small signal upconverter. The dif

218 5. 5 Conclusions Calculations of the diffusion admittance showed that when the impurity distribution is N(x) = axb, increasing b, optimizing the minority carrier lifetime Tp, and decreasing the depletion layer width W decrease the losses in an overdriven diode. The current densities in the LSUC were calculated using the diffusion admittance theory. The small signal method is sufficiently accurate at all frequencies when the larger of the signal and lower sideband voltage is at least one order of magnitude lower than the pump voltage. Therefore this assumption was used to find an admittance matrix from which the gain and noise figure were obtained. The existence of an optimum bias where the gain is maximized was experimentally and theoretically verified.

CHAPTER VI CONCLUSIONS AND SUGGESTIONS FOR FUTURE RESEARCH 6.1 Introduction In the preceding chapters, the analysis of a LSUC was extended, and a design technique was developed and experimentally verified. The analysis consisted of both an investigation of the effect of the circuit and the effect of a forward biased diode in a LSUC. The synthesis made use of an improved design of coaxial filters and some high frequency properties of a coaxial tee junction. The original contribution of the previous chapters are summarized in Section 6.2, and several suggestions for related future research are given in Section 6.3. 6.2 Summary of Original Contributions 6.2.1 Distributed Design of Coaxial Filters. In Chapter II, two methods for designing coaxial band-pass filters were compared: the lumped method and the distributed method. In both cases a set of impedance inverters, each separated by a half wavelength of coaxial transmission line, have to be synthesized. In the lumped method, a capacitor is approximated by a disk whose size gives the correct capacitance at low frequencies. However, when the disk lengths in the filter are > 0. 08X, as often occurs when the bandwidth < 10 percent, the filter characteristic is seriously distorted. On the other hand, the distributed method which was derived here, gives much improved filter characteristics and is applicable to filter bandwidths down to 1 percent of the center frequency. 6.2.2 Discontinuity Capacitance Compensation. Three methods of accommodating the disk discontinuity capacitance associated with an abrupt change in the characteristic impedance of a transmission line were compared. Although all these methods were known, their relative strengths and weaknesses had not been analyzed previously. This comparison was made here when these three methods were applied to the coaxial filter design, in Section 2.6. 219

220 6.2.3 Effect of Higher Order Sidebands in the LSUC. The design of a LSUC is based on the assumption that only the signal and lower sideband frequencies exist, although in reality energy at many other sidebands often exists near the diode. The effects of as many as seven higher order sidebands were investigated in several LSUC designs. However, when the gain and noise figure were calculated, little change was observed between the approximation using four frequency sidebands and higher order approximations. Explicit expressions for the LSUC gain, gain sensitivity, gain-bandwidth product, noise figure, and noise measure were derived, which included the effects of the impedance at the upper sideband frequency as well as the second harmonic of the pumped elastance. These expressions show that a small impedance at the upper sideband frequency can seriously degrade the LSUC performance. Thus, while a LSUC can be synthesized using only impedances at the signal and lower sideband frequencies, an analysis including higher order sideband frequencies should be performed to insure the desired characteristics will be obtained. 6.2.4 A Noniterative Design Algorithm for a LSUC. A practical design for a LSUC which required neither trial and error nor iterative methods is described and experimentally verified. The design was accomplished by isolating each port of the amplifier from the out-of-band frequencies, thus making it possible to synthesize impedance transformers by a straightforward application of the coaxial filter theory. The nonideal characteristic of the coaxial tee junction was used advantageously to help isolate the different frequencies from the signal port. 6.2.5 Verification of Noise Figure Dependence on Gain. Most analyses of the noise figure of a LSUC show that noise figure is independent of amplifier gain. However the theory reviewed here showed that, in most cases, noise figure decreased with increasing gain (Ref. 11). The experimental measurement of noise figure clearly shows the relationship between noise figure and gain which this latter theory describes. 6.2. 6 Application of Diffusion Admittance to a LSUC. The theory for the diffusion admittance exhibited by a forward biased diode was developed by Maneshin and Parygin (Ref. 80). It was shown here with this theory that the diffusion admittance could explain the "inductive effect" observed by many workers. The diffusion admittance can improve

221 the bandwidth or gain of a LSUC when the impruity doping profile and depletion layer width W are appropriately chosen. However, some sacrifice in thermal noise figure must be made when diffusion admittance is used. In general a diode having small y and small W was found to give the best results. 6.2.7 Modified Bessel Functions. In the course of the work on diffusion admittance, numerical evaluation of the K modified Bessel function with complex argument z was required. The computer program which is developed here is based on economization of power series. It yields accurate values of the K Bessel functions for almost all values of z; the accuracy diminishes somewhat for z I > 1, arg(z) z 1800 6.3 Suggestions for Future Research In the course of the studies presented here, several related problems arose which would be fruitful areas of future research. 6.3.1 Stripline Filters. The theory developed in Chapter II on coaxial filters can be applied to synthesis of stripline and possibly microstrip band-pass filters. However, the range of characteristic impedances in stripline is more restricted than in coaxial line with the result that narrow bandwidth filters cannot presently be built in stripline. 6.3.2 A LSUC Made in Stripline. The filters described above could be used to design and build a LSUC in stripline. Some of the techniques used in the coaxial LSUC would not be applicable, however, since the nonideal tee junction and diode mount would be different in this case. 6.3.3 Double Sideband Operation. The theory for the LSUC described here can be extended readily to include a port at the upper sideband frequency as well as the lower sideband frequency. In this way a LSUC can be made more stable, and an upper sideband upconverter can be made to have higher gain. The resulting amplifier would provide output from both parts, but would differ from the phase- shift amplifier in having unequal loading at the two ports. 6.3.4 The Coaxial Tee Junction. The 7 mm coaxial tee junction is shown in Chapter IV to act as an ideal power splitter only below 4 GHz. No thorough analysis has been made of a coaxial tee junction above this limiting frequency, although it is a widely used circuit element. R. W. P. King (Ref. 61, pp. 389-397) has derived an equivalent circuit

222 for the two wire transmission line tee junction, and as a first approximation his technique could be applied to a coaxial tee. There are at least two other junctions which could be more useful than the standard coaxial tee: the coaxial wye and the stripline tee. A preliminary investigation of the first was performed by terminating two ports of a 7 mm coaxial wye junction with 50 ohm loads and measuring the input impedance of the third port referenced at the junction as a function of frequency. Instead of deteriorating above 4 GHz, the wye junction impedance stayed between 25 and 21 - j3 ohms up through 12 GHz and within 25 ~ (4 ~ j5) ohms from 12 to 17 GHz. The symmetrical power splitting capabilities of the wye junction stand in sharp contrast to the tee junction. The coaxial tee characteristics arise from its asymmetrical geometry, which results in a complicated electric field pattern in the neighborhood of the junction. However, in a stripline tee the electric field (to a first approximation) is always perpendicular to the center and outer conductors; consequently the high frequency characteristics of the stripline tee ought to be better than the coaxial tee. Further improvement would be expected from a stripline wye junction. 6.3. 5 Measurement of Diffusion Admittance. There has been no direct experimental verification of the theory presented in Chapter V on the diffusion admittance. To make such a correlation, the minority carrier lifetime, the depletion layer width, the diffusion length, the doping profile, and the minority carrier lifetime would have to be known. Measurement of the diode impedance as a function of applied bias voltage could then be compared to the theoretical prediction. The diode parasitic elements would of course have to be determined by a separate measurement. 6.3.6 A LSUC with a Step Recovery Diode. Step recovery diodes are characterized by their long minority carrier lifetimes and their ability to generate a large number of frequency harmonics. Since the lifetimes of commercial step recovery diodes are between 20 and 400 ns, the curves showing minimum loss for diffusion admittance would indicate these diodes are only useful in the lower microwave frequency range. However if one of these diodes were used in a LSUC, a large number of frequency sidebands would be generated. A detailed study of the step recovery diode used in a LSUC could yield

223 beneficial results with regard to either or both the minority carrier lifetime and the large number of frequency sidebands. 6.4 Conclusion The theory and experimental verification for the design of a parametric LSUC has been presented, together with a study of diffusion capacitance which could greatly improve the amplifier bandwidth. In this study several contributions to LSUC design have been presented; there also remain several topics which merit further investigation.

APPENDIX A PROGRAM LISTINGS The major programs and subroutines used in this report are listed here. There are six main programs A - F and seven subroutines which which are used in the main programs. These programs are summarized below and listed in full subsequently. The symbol SSP used below stands for IBM's scientific subroutine package. Main Programs Program Description Subroutines Required Referred From A Filter design and analysis DESIGN Sec. 2. 6 IMPST IMPM B Design and analysis of LSUC 3 with TGAIN Sec. 3.5.4 coaxial filters GAINST PROTOG DESIGN IMPST IMPM TRNR C Diode diffusion admittance BESI (in IBM SSP) Sec. 5.3.4 BESKC D Large signal currents with short DQG12 (in IBM SSP) Sec. 5.4.1.3 circuit assumptions in a LSUC BESKC using diffusion admittance DFT FC1 FC2 E Currents in a LSUC by small sig- BESI (in IBM SSP) Sec. 5.4.2 nal method and Laplace asymptotic metodBESKC Appendix D method F Gain of LSUC using combined BESI (in IBM SSP) Sec. 5.4.3 diffusion and depletion admittance GAMMA (in IBM SSP) diode model BESKC C MINV 224

225 Subroutines Name Entries Other Subroutines Called TGAIN GAINST IMPST IMPM TRNR CMINV TRNR TRNS IMPST IMPM TRNR DESIGN PROTOG CMINV MPRD BESKC

001 PROGRAM A 002 003 C IHIS PROGRAM SYNTHESIZES AND ANALYZES COAXIAL BAND-PASS FILTERS. 061 JJ=JJ+1 004 062 DO LO J=1,20,1 005 C SUBROUTINES USED ARE DESIGN,IMPSTIMPM. 063 FV=B+(J-1)*STEP 006 DIMENSION G( 10),K( 10),LNTH( 10 ) AN( 10),C(10),Z(42),FR(21 ),GAMA(22 ) 064 CALL IMPM ( FV,N,LD,DIST,CD,ZOC,ZO,O.,ROT,XUT 007 DIMENSION DI(10),EPS(10), LD(1C),DIST(1O0),CU(1O),ZOC(10) 065 Z(J*2-1)=ROJT 008 REAL K,LNTH,LL,LD 066 Z(J*2)=XO(JT 009 IN1EGER N,TEST,J,JJ,LIMIT,TESTL 067 FR(J)=FV 010 DATA EPS/10*17.97362/ 068 GAMA J)=( (RnJT-R ) **2+XOIJT*XXOUJT)/( (RO)JT+R )*;2+XOUtT*XOJT) 011 EPS(1) =8. 854 069 10 CONTINUE 012 EPS(4)=8.854 070 5 FORMAT (' '/(' ',3(F6.3,2X,F6.4,2X,E12.5,2X,E12.5,1X ))) 013 1PI=6.28318531 071 WRITE (6,5)(FR(J),GAMA(J, Z (J2-1), Z(J*2), J1,20,1 014 20.=50. 072 IF (JJ.GE.LIMIT) GO TO 40 015 VC=2.997925 E8 073 R=R+STEP*20. 016..-.- — R-Z- --— ' 074 GO TO 11 017 DELTA=1. 075 46 CONTINUE D — i —8 TL-=.5 5., 076 WRITE (6,90) 019 1 FORMAT:(Ill11,6E11.5/(7E11.5)) _77 ____G T 40 _ _ 020 2 FORMAT ('1DATA IS N,F,W,OA ',I113E13.5/(' G'7F13.5)) 078 45 WRITE (6,91) J 021 READ (5,1) LIMIT,STEP 079 91 FORMAT(' DISCI(NTINUITITY CAPACITANCE EXCEEDS DISTRILTJTED CAPACITANiC 022 40 CONTINUE 080 1 OF DISK ' I5) 023 READ (5,1) N,F,W,DA,(G(J),J=1,N,L) 081 GO TO 40 024 G(N+1)=1. 082 END 025 NN=N+1 026 WRITE(6,2) N,F,W,OA, (G(J),J=1,NN,1) 027 1EST=O 028 CALL 'ESIGN (R- G(N+-iF D I,W,nE LTA, NG,K,ANG, C, NTH, EPSTEST 029 IF (TEST E. 1 ) GO TO 46 030 TEST1=3...... 0316 021 CALL IMPST (DI,LNTH,EPS,N,TEST1,LD,CD,ZnC,F*1.E-9) 032 92 FORMAT (' DISCONTINUITY CAP = '4(5X,E14.6)) 033 WRITE (6,92) (CI)(J),J=1,4,1) 034 on 100 J=1,N,1 035 C_ DISTANCE AND LENGTH (FR) IN INCHES _ __ _____.__ 036 IF (J.EO.1) G) TO 47 0387 4 DIST( J )=( (ANG(J-1 )+ANG( J ) )/(2.*TPI )+TL ) VC / ( F-:.0254) 038 47 CONTINUE 039 FR.(J)= LD(J)/.0254 040 IF (FR(J).LT. O. )GO T7 45 041 100 CONTINUJE 042 DI ST (1)=0. 043 90 FORMAT (' THIS FILTER CANNOT BE BUILT') 044 30 FORMAT ('OF= 'E11.5,' O= ' F11.5,/' K INVERTER VALUES = ' 045 __14(5XE15.6)/' ANGLE PHI IN RADIANS = '4(5X,6E1.6)/)/ 046 2' CAPACITANCE IN PICO FARADS = ' 4(5X,F15.6)/' LENGTH IN METERS 047 3NEGLECTING DISCONTINUJITY CAPACITANCE ' 4(3X,F15.6)/' DISK DIAMETER 048 4S ARE ' 4(5XF15.9)/' I)ISTANCE BETWEEN DISKS IN INCHES '4(3X, 049 5F15.9)/' LENGTH IN INCHES OF DISK WITH DISC. CAP ' 4(3X,F15.9)) 050 WRITE (6,30) F,OA,(K(J),J=1,4,1),(ANG( J),;=1,4,1), (C(J),J=1,4,1), 051 1(LNTH(, ) =1,4,1 ), (I (J),J=1,4, 1) ( DIST(J)(,J=1,4,1, ( FR( J ), 052 2J=1,4,1) 053 DOI) 48.I=1,N,l 054- DbST(!Ji=DIST(J)*.0254 055 48 C NITIN NE 056 4 FORMAT (' ',3('F IN GC',' GAMA R',13X,'+JX',lOX)) 057 WRITE (6,4) 058 JJ=O 059 B=F*1. F-9-STEP*LI MIT.. 060 11 CONT INUF

001 PROGRAM P 061 403 FORMAT (' PUMP CIRCU IT CANNOT RE BUILT') 002 062 CALL IMPST (OIAP,LNTH,EP,N,LDISC,DLNTP,C,ZOC,F-P*1.F-9) 003 C THIS PROGRAM DESIGNS A LS1UC WITH DIODE IN LOWER SIDEBAND PORT. 063 DO NOD J=1,N,l 004 064 IF IJ.EO.11 GO TOI47 005 C THIS PROGRAM CALLS TGAIN,GAINSTPROTOG,DESIGNIMPST,IMPM,TRNR. DES DISTPIJI=IIANG 1J-11+ANG IJII/(2.ATPII+.S1UAC/FP OG C INPUT DATA CONSISTS OF DIODE PARAMETERS, OPERATING FREQUENCIES, DEE 47 CONTINUE 007 C DIELECTRIC CONSTANTS AROUND DISKS IN FILTERS, AND SOME AUXILIARY 047 IF IDLNTPIJI.LT.D.) GO TO 45 00D C INFORMATION. DEE 100 CnNTINIE 009 C PRT=1 MEANS NO PRINT: PRT=D MEANS PRINT 069 OISTP(I)=ANG(IN/(2.aTPI*FP)AVC 010 C LDISC DETERMINES METHOD FOR ACCOMODATING DISC. CAP. IN IMPST. 070 C DISTP(Fl WILL RE MODIFIED BELOW 011 C TESTS,TESTITESTP CHECKS FOR IINREALIZARILITY IN DESIGN PROI;. 070 C LOWERSIDEBAND PORT 012 C PRTT=D MEANS PRINT FULL GAIN CIURVE IN TGAIN. PRTT=1 MEANS PRINT 072 F=FI 013 C ONLY CENTER FREOUENCY GAIN. 073 LP=O. 014 C IF THE INPJT GA'S ARE 0. THEN PROGRAM FINDS GA. 074 RM=RL 015 C PRT=O MEANS PRINTDIAGNOSTIC INFORMATION 075 KEY=S 016 REAL LLLP 7LRK(1IO LNTH(10) 076 GO TO 72 017 IMPEDERPRTTESTSPRTTTESTTESTP,TESTI 077 77 CONTINUE 019 DIMENSION G(20),C1101,DIAS(INODIA11101,DIAP1101,ANGINO1),S11D, 078 RFI=DR 019 1EPSI101),ISTSI1O),DISTI(101,ISTP( 11),DLNTS(l 101,LNTI(101, 079 AFI=DA 020 2DLNTP110IZOC(1O1,DIST(10rDIA(IOODLNT(1O1,6S14),EI (4,EP(4) ORG IF (ARSIDAII LT.I.E-21 OAI=OA 02 1 - DIMENSION AREA(10) Dpi DELTA = 0./IWIA*AI 022 IPI=6.2R31R531 OR? CALL PROTOjG 1N,DELTA,TEST,GI 02 3 VC=2.997925ESE08D IF IPRT.EO.0) WRITE (6,311 FDELTAIG1JIJ=1JJl1 -024 9 FORMATI'/ (71111 ~084 IF 1TEST.EO.11 GO TO 21 025 10 FORMAT17E11.51 085 TESTI=D 026 7Z=O=0. O86 CALL DESIGN (RFI*G(N+1t),FI DIAIWIDELTANGKANGCLNTHEI,TESTI 027 21 CONTINJE 087 1 -029 READ 15,91 LDISCFIRSTNIJMPRTT PRTNNN N FIEST TSR 401 FORMAT 1' IDLER CIRCUIT CANNOT RE BUILT') 029 READ (5,10) STEP,QASOAIOAP 089 IF ITESTI.EO.1) WRITE (194011 030 READ 15,101 FSFPRG,RLRSLL SP, SDwS1J1,J=1NN,11,WSW IWP 090 29 FORMAT RESONATE VARACTOR SERIES REACTANCE WITH ELASTANCE = 031 1,I( ES(IJIJ=1,N,11,I( EI (J ),J=1,N,1,I(EP(J) -j=1,No,1 091 1E12.5,' INDIUCTANCE = 'E12.5) (122 20 FORMAT (I DATA IS LDISCFIRSTNIJMPRTTNNNFSFPRGRLRSLL, 092 IF (PRT.EO.O) WRITE (6,29) SRLR 033 __ ISP#SD'/613A,16)/RI3XE13.61/' WSW1,WP ='313AE13.61/ __ 093 CALL IMPST 1)IAILNTHEI,N,LnISCDLNTI,C,ZOC,FI*1.E-99) 03 4 2' 51... SINIESI... ESINIEIII...EIINIEPI... EP(N I'/ 094 0 DO 102 1=1,Nl1 035 31R13XE13.6111 _______ ______ 095 IF IJ.EO.1) GO TO 103 026 WRITE 16,201 LDISCFIRSTNUM,PRTTNNNFSFPRGRLRSLL,SPS0, 09 6 DI STI (J )=ANG.( J-1 )+ANG(J II/12.*TPI )+..RRVC/Fl 037 1WSWIWP,1S1J1,J=1,NN,11,IES~jIJ=1,N,11,IEIIJIJ=1,N,11 09 7 103 CONTINIUE 038 2(EP(JI,J=1,N,1 09R IF IOLNTIIJI.LT.D.) GO TO 45 039 C.05<DELTA<2.0 099 102 CONTINUE __ __ _ _ _ _ 046 C.MULTIPLY.THE MATCHING RESISTANCE RY 0(N+11 T(J GET RIGHT MATCH 100 DISTII1I=ANGI1LVC/(2.RTPIRFI1 041 C PPUMP PORT 101 C SIGNAL CIRCIJIT 042 F=FP 102- F=FS 043 FI=FP-Fg 103 RM=RG 044 C PUMP RESISTANCE ARBITRARILY SET TO 1. OHM. 104 LP=D 045 RM=1. 105 KEY=; 046 KEY=4 106 GO TO 72 -047 LP=SP/(TPI*FP*TPI*FPI _ _ _ _ _ _ __ _ _ _ _ _ _ 107 75 CONTINUE _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 64& G) IODE Z AND RESONATED 0, D lO RIJ=DR 049 GO TI 72 109 211=02 050 76 CONTINUE 110.KEY=2 051 IF I ABS( DAP).LT.l.E-10) _OAP=OA _- _ _ _ _ _ _ _ _ _1 1GO TO 4R8 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ DF~ OELTA=1./(WP*DAPi 112 52 CONTINIJE 053 CALL PROTOG (N,DELTA,TEST,GI _ _ __ _ _ __ _ _ __ _ _ 113 RFS=RF _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 31 FORMAT'F = '13.6,' DELTA =E13.6,'G1. N /4(3,E13.6)) 114 1HSM=BL*VC/(FSRTP I 05 5 JJ=N+1 115 IF (ABS(OAS).GT.1.E-10) GO TO 80 056 IF (PRT.ED.D1 WRITE 16,311 F,DELTAIG(J),J=1l.JJ,l1 116 C CALCULATE OAS 057- IF (TEST.ED.l11G TO 21. ~ ~117. F101F:058 IESTP=D 11R KEY=1 059 CALL DESIGN IG,(N+11,FP,DIAP,WP,DELTAN,G,,K,ANG,C,LNTH,EP,TESTP) 119 00 TOT?2 060 IF (TESTP.EQ.l1 WRITE 16,4031 120 73 Rl=[DR

121 Xi DX 181 BOT=RC*RC+(XC+XD)*(XC+XD) 122 F=.99*FS 182 VA=RC 1.23 KEY=2 183 RC=RC*XD*XD/ROT 124 GO TO 72 184 XC=(XD*(VA*VA+XC*XC)+XC*XD*XD)/BOT 125 74 CONTINUE ________ ___________185 WAVE=DIST (KJ)*F /VC 126 CALL TRNR IZX-O,I.OI*RL/TPI,Rl,Xl,RtU,XU) 186 CALL TRNR (ZO,WAVE,RC,XC,VA,VR) 127 CALL TRNR (ZO,.99*BL/TPI,L)R,DX,RL,XL) 187 IF (KJ.LE.1) GO TO 300 128 OAS=FS*ARS(XU-XL)/((R(J+RL)*(.O2*FS)) 188 C LNTHS(EVEN)=SERIES CAPACITANCE OF J INVERTER. 129 IF (PRT.EE1,2.O) R XLXLAS189 B=VBR-1.E12/(TPI*F *I)LNTS(KJ-1)) f130 1WRITE (6,12) RU, XURLXLOA S S190 WAVE=DIST (KJ-I)*F /VC 131 12 FORMAT (' RU = 'E13.5,' XU= 'E13.5,' RL 'E13.5,' XL = 'E13.5, 191 CALL TRNR (ZOWAVEVAVRCXC) 132 1' OA = ' E13.5) 192 VA=RC 133 80 DELTA=1./(WS*QAS) 193 VB=XC 134 CALL PROTOG (N,DELTA,TESTS,G) 194 KJ=KJ-2 135 IF (TESTS.EO.O) GO TO 6 195 GO TO 400 176 13 FORMAT (' NO G VALUIES FOR THIS W AND OAA ') — 96 300 COiNTINILE 137 WRITE (6,13) 197 RSC=VA 138 6 CON',NUE ---- 198 XSC=VB 139 IF (IPRT.EO.O) WRITE (6,31) F,DELTA,(G(J),J=1,JJ,1) 199 C FIND DISTANCE TO GET ZSIFI)=INFINITY. THIS COMES FROM: 140 C J INVERTER MEANS THAT TESTS=2.. THIS WILL BE CHANGED TO 0 200 C J*INF = J*ZO*(XL+ZO*TAN(PHI))/(ZO-XL*TAN(PHI)) 141 TESTS=2 201 TDDSTS=VC*ATAN(ZO/XSC)/(TPI*FI )+DIST(1) 142 CALL DESIGN (RFS*G(N+l),FS,DIAS,WS,DELTA,N,G,K,ANG,CLNTH,ES, 202 1DDSTI=DISTS(1)-TDDSTS 143 1TESTS) 203 C NOW FIND DISTP(1) TO RESONATE DIODE AT FI. GET C,ZOC. 144 VA=C(2) 204 CALL IMPST (DIAPDLNTPEPN,LDISCLNTH,C, ZOC-,Pi1.E-9) 145 404 FORMAT (' SIGNAL CIRCUIT CANNOT RE BUILT') 205 CALL IMPM (FI*I.E-9,N,DLNTPDISTPC,ZOC,ZOO,V A,VB) 146 IF (TESTS.EO.1) WRITE (6,404) 206 WAVE=TDDSTI*FI/VC 147 IF (TESTS.EO.1.OR.TESTI.EO.1.OR.TESTP.EO.1I) GU TO 21 207 CALL TRNR(ZOtWAVE,VAVB,RItXU) 148 CALL IMPST (DIASLNTH,ES, N,LDISCD)LNTS,C,ZUC,FS*1.E-9) 208 C IF X=ZO*(XL+ZO*TAN(PHI))/(ZO-XL*TAN(PHI)), 149 DLNTS(2)=VA 209 C THEN TAN(PHI) = ZO*(X-XL)/(ZO**2+X+XL). ASSUME RlJ=0. 150 DLNTS(1)=DLNTS(1)/.0254 210 IHPM=TPI*VC/( FI*4.I 151 DLNTS(3)=DLNTS(3)/.0254 211 BOT = (ZO*ZO-XU*XFI) 152 TL=.25 212 IF (ABS(BOT-.GT.-1.E-3) THPM=VC*ATAN(ZO*(-XFI -XU)/BOT)/(FI 153 DO 101 J=2,N,1 213 1*TPI) 154 lO11 DI-rSTS(J~)=(( ANG(J-1)+ ANG(J))/(2.*TPI)+TL)*VC/(FS*.0254) 214 DISTPI1I=THPM+DISTPIXI 214 DISTP(1)=THPM+DISTP(1) 155 DISTS(I):( ANG(I)*VC/(FS*TPI*2.)+THSM)/.0254 215 00110 J=1,Nl 215 DO 110 J:IN,1 -'-56 WRITE (6,34) FS,(DISTS(J),J=1,N,1),(DLNTS(J},J=1,N,1},(DIAS(J} 216 DISTIIJI=DISTIIJI/.0254 216 DIST~(J)=DISTI(J)/.0254 157 1,J=1,N,1I) 217 DISTP(J)=DISTP(J)/.0254 158 DLNTS(1)=DLNTS(1)*.0254 218 DLNTI(J)=DLNTI(J)/.0254 159 DLNTS(3)=DLNTS(3)*.0254. 219 110 OLNTPIJ)=DLNTP(J)/.0254 160 DO 19 J=I1,N,1 220 34 FORMAT i' F - ',E11.5/ DISTANCE RETWEEN DISKS IN INCHES' 161 DISTS(J)=DISTS(J)*.0254 221 13(3XE1l.5)/' LENGTH OF DISK IN INCHES WITH DISC. CAP. ' 162 - IST(JI=STS(JI 22-2 - 23(3X,Eil.5)/' DISK DIAMETERS I 3(3X,E11.5)) i63 DIST(J)=DISTS(J) 2 163 19 CONTINUE 223 WRITE (6,34) FII(DISTI(J),J=1,N,1I,(DLNTI(J),J=1,N,I), ( 164 C Al THIS POINT DISTP(I), TDDSTS,TDDSTI ARE NO1 KNOWN. 224 1DIAI(J),J=1,N,1) 165 DIST(1)=VC/FI 225 WRITE (6,34) FP,(DISTP(J),J=1,N,1),(DLNTP(J),J=1,N,l) 166 C FIND INPUIJT IMPEDANCE OF SIGNAL CIRCUIT. 226 1,(DIAP(J),J=1,3,1) 167 KJ=N 227 DO 18 J=1,N,1 168 VA=ZO 228 DISTI(J)=DISTI(J)*.0254 169 VB=O. 229 DISTPIJ)=DISTP(J)*.0254 170 C GEl DISCONTINUITY CAPACITANCE 230 DNTJLNTIJI=DLNTIJ(J )*.0254 171 CALL IMPST (DIAS,DLNTS,ES,N,LDISC,DLNT,C,ZUC,FS*1.E-9) 231 DLNTP(J)=DLNTP(J)*.0254 172 F=FI 232 18 CONTINUE 173 400 CONTINUE 233 35 FORMAT (' SIGNAL DISK TO JUNCTION DIST. = 'E13.5,' IDLER DISK TO J 174 XD=-1.E12/(TPI*F *C.(KJ)) 234 1UNCTION DIST. = 'E13.51 175 WAVE=DLNTS(KJ)*F *SORT(ES(KJ) /8.854)/VC 235 VA=TDDSTS/.0254 176 ROI =VA*VA+(VB+XD)*(VB+XD) 236 VR=TDDSTI/.0254 177 RC=VA ________ 237 WRITE (6,35) VA,VB 178 VA=RC*XD*XD/BOT 238 C CALCULATE DISTANCE OF COAX TO WAVEGUIDE ADAPlER FROM FILTERS. 179 VB=(XD*(RC*RC+VBVRB)+VB*XDRXD)/BOT 239 KEY= 180 CALL TRNR (ZOC (KJ),WAVE,VA,VB,RC,XC) 240 DO176J=1,N,1

241 DIA(N-J+1)=DIAI(J) 301 BLl=ATAN(VA+SORT(VA*VA+l.)) 242 DLNT(N-J+1)=DLNTI(J) 302 BL2=ATAN(VA-SORT(VA*VA+1.)) 243 EPS(N-J+1)=EI(J) 303 IF (BL1.LT.O.) BL1=BL1+TPI/2. 244 IF (J.EO.N) GO TO 177 304 IF (BL2.LT.O.) BL2=BL2+TPI/2. 245 176 DIST(N+l-J)=DISTI(J+l) 305 IF (BL1-BL2) 121,122,122 246 177 DIST(1)=O. 306 121 BL=BL1 247 WAVE = DISTI(l)*FS/VC 307 GO TO 23 248 CALL TRNR (ZO,WAVE,O.,O.,RC,XC) 308 122 BL=BL2 249 180 CONTINUE 309 23 CONTINUE 250 CALL IMPST (DIA,)LNT,EPS,N,LDISC, LNTH,C,ZOC,FI*i.E-9) 310 1N=TAN(BL) 251 CALL IMPM (FS*.E-9, LNT,DIST,C ZOCRC,XC,KOUT,XOUT) 311 RF=ZO*ZO*RU*( 1.+TN*TN) / ( ( ZO-XJ*TN)**2+(TN*RU) **2) 252 C ZOUJT IS THE CONJUGATE OF THE IMPEDANCE LOOKING AWAY FROM VARACTOR. 312 IF (RF.GT.ZO) GO TO 60 253 VA=TPI/4. 313 GO TO 52 254 IF (ABS(XOUT).GT.1.E-3) VA = ATAN(ZO/XO(UT) 314 60 CONTINUE 255 __ IF (VA.LT.O.) VA=VA+TPI/2. 315 IF (BL.GE.TPI/4.) GO TO 62 256 GO TO (174,175),KEY 316 BL=BL+TPI/4. 257 174 CO(NTINUE 317 GO TO 23 258 SIODST=VA*VC/(TPI*FS) 318 62 CONTINUE 259 C NOW FOR PUMP CITCUIT. 319 BL=BL-TPI/4. 260 DO 178 J=1,N,l 320 GO TO 23 261 DIA(N-J+1)=DIAP(J) 321 C GET Z AND 0 OF DIODE. LP IS AN EXTRA INDUtCTANCE IN PARALLEL WITH 262 DLNT(N-J+1)=DLNTP(J) 322 C SP. 263 EPS(N-J+1)=EP(J) 323 72 CONTINUE 264 IF (J.EO.N) GO TO 179 324 OMEGA=TPI*F 265 178 DIST(N+i-J)=DISTP(J+I) 325 VA=OMEGA*LL-SO/OMEGA 266 179 DIST(1)=O. 326 IF (LP.LT.1.E-17) GO TO 70 267 WAVE = DISTP(1)*FS/VC 327 VB=OMEGA/SP-1./(OMEGA*LP) 268 CALL TRNR (ZO,WAVE,O.,1.E15,RC,XC) 328 71 CONTINUE 269 KEY=2 329 ROT=( 1.-VA*VB)**2+(VB*RM)**2 270 GO TO 180 330 DR=RM/BOT 271 175 BSPDST=VA*VC/(TPI*FS) 331 DX=(VA*(1.-VB*VA)-RM*RM*VB)/BOT 272 43 FORMAT (' DISTANCE TO WAVEGUIDE IN LSB FILTER= 'E13.6,' METERS. 332 SR=O. 273 1DISTANCE IN PUMP FILTER= 'E13.6,' METERS') 333 LR=O. 274 WRITE (6,43) BSIDST,BSPDST 334 IF (DX.LT.O.) LR=-DX/OMEGA 275 SWT=O 335 IF (DX.GE.O.) SR=DX*OMEGA 276 C AREA USED FOR TGAIN. 336 VB=SP/OMEGA 277 AREA(1)=BSIDST 337 VD=VA-VB 278 AREA(2)=BSPDST 338 VE=OMEGA*LL+SO/OMEGA 279 AREA(3)=TDDSTS 339 BOTT = VB*VB*BOT 280 AREA(4)=TDDSTI 340 OA=ABS((VB/OMEGA)*((VA*VD+RM*RM-VE*VD-VA*(VE+VB))/BOTT+ 281 352 CONTINUE 341 1(VA*VD+RM*RM)*2.*VD*VD/(BOTT**2))+LR-SR/(OMEGA*OMEGA)) 282 IF (SWT.EO.1) CALL GAINST (41,2,NtJM,STEP,O,NFlEST) 342 OA=OA*COMEGA/(2.*DR) 283 IF (SWT.EO.O) CALL GAINST (1,2,NUM,STEP,1,NFTEST) 343 GO TO (73,74,75,76,77),KEY 284 CALL TGAIN (N,FS,FP,DLNTS,DLNTI,DLNTP,DISTS,D1STI,DISTP,DIAS, 344 70 CONTINUE 285 1DIAI,DIAP,ES,EI,EP,AREA,RS,LL,SP,SO,S,GAIN) 345 VB=OMEGA/SP 286. IF (GAIN.LT.O.) WRITE (6,420) 346 GO TO 71 287 420 FORMAT ( ' AMPLIFIER OSCILLATES') 347 45 WRITE(h6,91) J 288 IF (SWT.EO.1) GO TO 21 348 91 FORMAT ( ' DISCONTINUITY CAPACITANCE EXCEEDS DISTRIBUTED CAPACITAN 289 40 FORMAT (' GAIN CONVERGED TO ' E13.6) 349 1E OF DISK ' I5) 290 WRITE (6,40) GAIN 350 GO TO 21 291 IF (PRTT.EO.1) GO TO 21 351 END 292 SWI=l 29 3 GO TO 352 294 48 CONTINIUE 295- C GIVEN ZlJ, FIND RF AND PHI WHERE 296 C RF = ZO*(ZL+J-*ZO*TAN(PHI))/(ZO+J*ZL*TAN(PHI)) 297 IF (ABS(X[)).GT..5E-3) GO TO 26 298 BL=O. 299 GO TO 23 _ _ __ __ 300 26 VA=I(ZO*ZO-RU*RtU-XtJ*XU)/(2.*ZO*XU) )

001 SUBROUTINE TGAIN, ENTRY GAINST 061 F(7)=3.*FP+F(1) 002 062 F(8i=4.*FP-F(l) 003 C TGAIN AND ENTRY GAINST FINDS GAIN AND NOISE FI.GURE OF LSUC 063 DO 20 K=1,NUM,1 004 C WI1H DIODE IN LOWER SIDEBAND PORT. 064 IF (K.GT.2) GO TO 301 005 065 C FIND INPUIT IMPEDANCE OF SIGNAL CIRCUIT. 006 C 1HIS PROGRAM CALLS IMPST,IMPM,TRNR,CMINV. 066 KJ=NI 007 SURROUTINE TGAIN (NI,FS,FPtDLNTSDLNTI,DLNTPDISTSDISTI.,DISTP, 067 VA=ZO 008 1 DIAS,DIAI,DIAP,ES,EI,EP,AREARS,LL,SPSOSGAIN) 068 VR=O. 009 C PRT=O MEANS PRINT. PRT=1 MEANS DO NOT PRINT 069 C GET DISCONTINUITY CAPACITANCE 010 C WAVEGUIDE USED IN PUMP AND LSB CIRCUITS TO S1P SIGNAL FREQUENCY. 070 CALL IMPST (DIAS,DLNTSES,NI,LDISC,DLNT,CD,ZUC,FS*i.E-9) 011 C NFTEST =0 DO NOT FIND NOISE FIGURE. NFTEST=1 FIND NF. 071 C STORE DISTS(1) AND TRANSLATE TO JUNCTION 012 C. FREIUENCY FS,FP IS IN CYCLES/SEC. 072 RS1=DISTS(1) 013 C LENGTHS DLNTS...DISTP BSIDST,BSPDST ARE ALL IN METERS 073 DISTS(1)=TDDSTS 014 C DIAMETERS DIAS,DIAIDIAP ARE IN INCHES 074 400 CONTINUE 015 C ES,EI,EP,IS DIELECTRIC CONSTANT IN PF/M. 075 XD=-1.E12/(TPIF(K )K*CD (KJ)) 0'6 C RS IN OHMS, ELASTANCES SP,S ARE IN DARAFS 076 WAVE=DLNTS(KJ)*F(K)*SDRT(ES(KJ) /8.554)/C 017 C BSIDST AND BSPDST ARE THE DISTANCES TO WAVEGUIDE ADAPTERS IN METER 077 BOT=VA*VA+(VR+XD)A(VB+XD) 018 C NI I;: THE NUMBER OF DISKS IN BAND-PASS FILTERS. NN IN LP FILTER 078 RC=VA 019 DIMENSION CZRP(8),CZIP(8),CZRS()E,CZIS(8) 079 VA=RC*XD*XD/ROT 020 DIMENSION DISTS(10),DISTI(10),DISTP(10) 080 VR=(XD*(RC*RC+VB*VB)+VB*XD*XD)/HOT 021 DIMENSION DIAS(10),DIAI(10),DIAP(lO),DLNTS(lU),DLNTI(lO),DLNTP(10) 081 CALL TRNR (ZOC (KJ),WAVE,VA,VB,RC,XC) 022 — DIMENSION S(lO),F(IO),ZR(8,8),ZI(8,8),YR(E,8),YI(I,8) 082 BOI=RC*RC+(XC+XD)*(XC+XD) 023 DIMENSION ES(4),EI(4),EP(4),CD(10),DLNT(iO ),ZOC(1O ) 083 VA=RC 024 DOUBLE PRECISION DZR(8EE,DZI8) E,DYR(,8)R 8),DYI(8,8),DSA(8,8) 084 RC=RC*XD*XD/BOT 025 DOUBLE PRECISION DSB(8,8),DSC(,I8) 085 XC=(XD*(VA*VA+XC*XCI+XC*XD*XD)/ROT 026 DIMENSION IPERM(ID) 086 WAVE=DISTS(KJ)*F(K)/C 027 DIMENSION AREA(IO) 087 CALL TRNR (ZO,,WAVE,RC,XC,VA,VB) 025 REAL NFLLLR 088 IF (KJ.LE.1) GO TO 300 029 INTEGER J,NUM,LAST,K,M,N,L,TEST,FIRST,NO,MM,PRT 089 C LNTHS(EVEN)=SERIES CAPACITANCE OF J INVERTER. 030 RPARAL (VVA,VVB,VVC,VVD)=(VVA*VVC*(VVA+VVC)+VVB*VVB*VVC+ 090 VR=VB-1.El-2/(TPI*F(K)*DLNTS(KJ-1I)) 031 1VVD*VVD *VVA)/((VVA+VVC)**2+(VVR+VVD)*a*2) 091 WAVE=DISTS(KJ-1)*F(K)/C 032 XPARAL (VVAVVB,VVC,VVD)=(VVB*VVD*(VVB+VVD)+VRC*VVC*VVB+ 092 CALL TRNR (ZO,WAVE,VA,VBR C,XC) 033 1VVD*VVA *VVA)/((VVA+VVC)**2+(VVB+VVD)**2) 093 VA=RC 034 RSIDST=AREAI1i 094 VB=XC 035 BSPOST=AREA ( 2 ) 095 KJ=KJ-2 03'6 1DDSTS=AREA(3) 096 GO TO 400 037 TODSTI=AREA(4) 097 300 CONTINUE 038 IF (TEST.EQ.1) GO TO 15 098 DISTS(1)=RS1 039 NO=41 099 RS1=VA 040 FIRST=2 100 SS=VR 041 _TEP=1.E7 101 IF (K.EO.1) GO TO/149 042 NUM=5 102 301 RINI=ZO 043 PRT=O 103 XINI=O. 044 NF1EST=O 104 RINP=ZO 045 15 CONTINUE 105 XINP=O. 046 C LDISC - 1 TO AVOID CHANGING FILTER DISK DIAME1ERS IN IMPST. 106 GO TO 148 047 LDISC=1 107 149 CONTINUE 048 C=2.997925E8 108 RINI=O. 049 11 FORMAT ('0') 109 XINI=-ZO/TAN(BSIDST*TPI*F(K)/C) 050 1PI=6.28318531 110 - INP=O. 051 FI=FP-FS 111 X I NP=-Z O/TAN( BSPST*TPT I *F ( K )/C ) 052 ZO=50. 112 148 CONTINUE 053 DO 10 J=1,NO,1 113 CALL IMPST (DIAI,DLNTI,EI,NI,LDISC,DLNT,CD,ZUC,FI*I.E-9) 054 IF (PRT.EQ.O) WRITE (6,11) 114 CALL IMPM (F(K)*1.E-9,NI,DLNTI,DISTI,CD,ZOC,RINI,XINI,RI1,XI1) 055 F(1)=FS+ STEP*(J-(NO+1)/2) 115 CALL IMPST (DIAP,DLNTP,EP,NI,LDISC,DLNT,CD,ZUC,FP*1.E-9) 056 F(2)=FFP —F(l) 116 CALL IMPM(F(K)*1.E-9,NI,DLNTP,DISTP,CD,ZOC,RINP,XINP,RP1,XP1) 057 F(3)=FP+F(D) 117 12 FORMAT(' THE SIGNAL, IDLER, AND PUMP ARE '/(' FILTER OUTPUT ' 055 F(4)=2.*FP-F(I) 118 1612.5,' + J 'E12.5)) 059 F(5)=2.*FP+F(1) 9 IF (PRT.EO.1) WRITE (6,12) RSI,XSl,RIi,XI1,RPl,XPF 060 F(6)=3.*FP.-F(1) 120 IF (K.NE.1) GO TO 150

121 VA=RPARAL (RS1,XS1,RPI,XPl) 181 YI )LAST,N)=ZIHLASTN) 122 VR=XPARAL (RSIXS1,RP1,XPI) 182 60 CONTINUE 123 151 CONTINUE 183 81 CONTINUE 124 CALL TRNR (ZOTDDSTI*F(K)/CVAVgRCXC) 184 L=LAST-1.125 RCI=RPARAL (RC+RllXC+XI1,O.,-SP/(TPI*F(K))) 185 70 CONTINUE 126 XCI=XPARAL (RC+RI1,XC+XISO.,-SP/(TPI*F(K))) 186 IF (L-2) 100,80980 127 ZR(KK)=RS+RCI 187 80 CONTINUE 128 ZI(KK)=TPI*F(K)*LL-SO/(TPI*F()K)+XCI 188 RLAST=YR(L,) 129 C TAKE CONJUGATE 08 EVEN NUMBERED Z'S 189 XLAST=YI(LL) 130. IF (2*(K/2).EO.K) ZI(KK)=-ZI(KK) 190 MM=L-1 131 IF )J.EO.1) WRITE 86,12) VAVRRSI,XSI,RCIXCIZR(KK),ZI(KK) 191 DO 91 8=1,MM81 132 20 CONTINIJE 192 RM=YR(ML) 133 DO 31 M=1,NUM,1 193 XM=YI)ML) 134 DO 30 N=1,NUlM,1 194 00 90 N=1,Ll 135 IF )N.EO.M) GO TO 30 195 YR(MN)=YR(M,N)-)(RM*YR(LN)-XM*YI)LN))*RLAST+ 138 C SEE DAVE OLIVER, P.49. 198 1)RM*YI(LN)+XM*YR)LN))*XLAST)/)RLAST*RLAST+XLAST*XLAST I 137 L= )))-1)**N)*)2*N-1)+l)/4-())-l)**M)*(2*m-l)+l)/4 197 YI)MN)=YI(M,N)-((XM*YI)L,N)-RM*YR(LtN))*XLAST+) 138 IF(L) 23,24,21 198 IRM*YI(L,NI+XM*YR( LN))*RLAST)/)RLAST*RLAST+XLAST*XLAST 139 23 COET IINUE 199 90 CONTINUE 140 L=-L 200 91 CONTINUE 141 GO TO 21 201 L=L-1 142 24 ZI(M,N)=)-1I)**N)*SO /17P188(N)) 202 GD TO 70 143 GO TO 22. 203 100 CONTINUE 144 21 ZI(MN)=((-i)**N)*S(L)/(TPI*F(N)) 204 BOIR=YR)1,1)*YR(2,2)-YI)1,1)*YI(2, 2) 148 22 CONTINUE 205 BOTI=YRI1,1) *YI2,2)+YI 1,1) *YR2, 2) 148 ZR(MN)=0. 208 C AMPLIFIER OSCILLATES IF DENOMINATOR IS (0. IT OSCILLATES AT 147 30 CONTINIJE 207 C FREQUENCY WHERE REACTANCE IS ZERO. 148 i CONTINIJET I NU 208 IF (BOTR.GT.1.E-5.DR.ABS(B)TI).GT.1.) GO TO 93 149 40 FORMAT ('ODEGREE OF MATRIX = '14) 209 GAIN=-I. 450 41 EORMAT )'OZR MATRIX 1/()0lt83XtE13.6))) 210 GO TO 94 151 42 FORMAT (1OZI MATRIX '/('O'lR3X,,E3.6))) 211 93. CONTINUE' 152 18 )J.NE.)NO+1)/2) GO TO 50 212 VA=ZR)1,l)-RS 153 18 1887.80.1) GO 70 50 _____________213 VB=ZR)2,2)-RS 154 WRITE (6t4O) NOM ~~~~ ~~~~~~~~~214 GAIN=4.*VA*VB*)YR)2,11A*2+YI)2,1)**2)/(ROTR*AUTR+BOTI*BOTI) 15 WR ITE 146,41)_))ZR)(M,N),N= 1,8,1),M=1,8,1) 215 C GAIN IS ABS)Y21)**2A4.*VA*VB WHERE Y21=Z21/)Z22*Zll) 16.- _WRITE 14-,42) )Z I)(MgN),N=l,8,1),M=1,8,1) 2-18 C (HAT IS THE VARIABLES YR AND YI ARE REALLY IMPEDANCES. 157 50 CONTINUE 217 18 )NFTEST.EQO.) GO TO 94 158 C MAlRIX TRIANGULARIZATI).N. UPPER RIGHT HAND SIDE MADE ZERO. 218 CALL CMINV (LAST,8,DZRDZIIPERMDYRDYIDSAUSVISC) 159 IF )NFTEST.EO.0) GO TO 202 -219 00 204 M=1,LAST,1 160 DO 203 M=1,NOM,1 220 00 204 1=1,LASTl 161 DO 203 I=1,8)18,1_ 221 YR) IM)=SNGL)DYRI 1,M) 162 DZR)I,M)=DRLE)ZR(IM) 222 YI)1IM)=SNGL (DYI),M)) 163 DZI(IM)=DRLE)ZI(I,8)) 223 204 CONTINIE 164 203 CONTINIIE 224 VA=O. 165 202 CONTINUE 225 DO 201 I=1,LAST,1 188 DO 120 LAST=FIRSTNUM,1. 228 201 VA=)YR(2,I)**2+YI(29I)**2)*ZR()1,)+VA 187 - RLAST=ZR)(L.ASTLA S. _227 NF=)4.*V*V)VA-VBR*YR)(2,2)A*2+YI)2,2)*2)) 168 XLAST=ZI)LAST,LAST) 228 1+().-2.*VRAYR(2,2))**2+4.*)VB*YI)2,2))**2)/GAIN 169 MM=LAST-1 229 94 CONTINUE 170 GO 61 M=1,MM8, 230 110 FORMAT (I THE GAIN FOR THE 111,' 8Y '11,' MAIRIX IS 8F12.2, 171 RM=ZR)MLAST) _ 231 15X,' WITH SIGNAL FREQUENCY I E12.5 I' NOISE ))IG188 IS 8E12.5) 172 XM=ZI)MLAST) 212 IF )PRT.EOQ1) GO TO 120 173 00 60 N=1,LAST,1 233 WRITE (8,110) LAST,LASTGAINFl)l)NF 174 YR)-MN )=ZR)MN)-) (RM ZR)L ASTN)-XMV Z IT LAST,)) 234 120 CONTINUE 175 SARLAST+)RM*ZI)LASTN)+XM*ZR(LASTN)) 235 10 CONTINUE 178 2*XLAST)/)RLAST*RLAST+XLAST*XLAST) 238 RElURN 177. YI(M,N)=ZI)M,N)-))XM*ZI)LASTN)-RM*ZR(LAST,N) _ 3 150 VA=881___ 178 1)XATRMZLS,+XVRATN- 238 88=X11 L79 2)ARLAST)/ )RLAST*RLAST+XLASTXXLAST 239 GO TO 151 18A YR)LAST,PI)=ZR)LAST,N) 240 81188 GAINST )NOFIRSTNU)M,STEPPRT,NFTEST)

241 C NO IS THE NUMBER OF ITERATIONS. FIRST IS THE SMALLEST MATRIX 242 C APPROXIMATION AND FIRST >1. PRT=1 FOR NO PRINT OUT. 243 C STEP IS THE FREQUENCY INCREASE/ITERATION IN CPS. 244 C IF NFTEST=O, NO NOISE FIGURE: IF NFTEST=1 FIND NOISE FIGURE. 245 DATA TEST/O/ 246 1EST=1 247 RETURN 248 END

001 PROGRAM C 062 RCO=VA*P*BI*ALPH 002 003 C IHIS PROGRAM FINDS DIFFUSION DIODE IMPEDANCE WHEN APPLIED VOLTAGE 064 OO=VB/VA 004 C IS KNOWN. 005 066 201 CONTINUE 006 C THIS PROGRAM USES BESKC AND BESI WHICH IS AN IBM SSP PROGRAM. 067 YC(K)=VA*P*BI*ALPH 007 REAL LMU 068 YS(K)= VB*P*BI*ALPH 008 DIMENSION Q(10)YC(lO),YS(10),YCGO(lO),YSGO(10) 069 C IS(K)=-VB*P*BI*ALPH. BUT COS(A+90)=-SIN(A) lMEANS CURRENT LEADS 009 IN1EGER R 070 C VOLTAGE. HENCE CAPACITANCE AND POSITIVE SUSCEPTANCE. 010 COMPLEX ZBKOtBK1,ZA 071 YCGO(K)=YC(K)/RCO 011 AREA=4.41E-5 072 YSGO(K)=YS(K)/RCO 012 EPS=11.8*8.854E-14 073 O(K)=VB/VA 013 PHI=.9 074 200 CONTINUE 014 RS=1. 075 FV=OMEGA/TPI 015 N=2 076 K=O 016 C ELECTRONIC CHARGE 077 51 FORMAT (' FREQUENCY = 'E11.5, ' FOURIER COMPUNENTS OF ADMITTANCE 017 OE=1.60210E-19 078 1AND Q ARE '/( ' J= 'I2,' G = ' E11.5,' B = ' E11.5,' 0 = ' E11.5, 018 C OE/KT =40. 079 2' G/GO = 'E11.5,' B/GO = 'E11.5)) 019 ALPH=20 080 VA=1. 020 S0R2=-1.41421356 081 VB=RSO/RCO 021 1PI=6283185318531 082 NN=N-1 022 1 FORMAT (Ill6Ell.5/(7Ell.5)) 083 WRITE (6,51) FVK,RCO,RSO,OOVA,VB, (KYC(K),YS(K),O(K),YCGO(K), 023 READ(5,1) LIMIT,STEP 084 1YSGO(K),K=1,NN,1) 024 MUJ=401. 085 GO=RCO*AREA 025 DP=MU/40.1 086 BO=RSO*AREA 026 C PNW=PN/ IN CM**-3 SEE WATSON P 107 087 GI=YC(1)*AREA 027 25 CONTINUE 088 R1=YS(1)*AREA 028 READ (5,1) BRVOV1,TAUWL,PNW 089 BDEP=OMEGA*EPS*AREA/(W*(1.-VO/PHI)**GAM) o029 GAM=1./(B+2.) 090 BOT=Gl*Gl+(Bl+BDEP)**2 030 J=D 091 RD=RS+G1/BOT 030 J=O 031 50 FORMAT('ODIODE N= X**'I2, ' VO = 'E11.5, ' V = 'E11.5, 0932 XD=-(B5+RDEP)/BOT 032 1' HOLE LIFETIME = 'E11.5,' BASE WIDTH/DIFFUSION LENGTH='E11.5 / 093 55 FORMAT ('YDIF = 'E13.5,' +J 'E13.5,' BDEP = 't13.5,' ZDIODE 033, 1' ADMITTANCE IS IN MHOS/CM**2',' PN(W) = 'E11.5) 094 1E13.5,' +J 'E13.5) 034 C N=X**R, TAU=LIFETIME IN P MATERIALF=FRE.,VV=V OLTAGE WL=W/L RATIO 095 WRITE (6,55) Gl1BlBDEPRDXD 035 WRITE (6,50) B, VO,V1,TAUWLPNW 096 BtT=RD*RD+XD*XD 036 L=SORT(D)P*TAIU) 097 GD=RD/BOT 02 7 W=L*WL 09 8 BD=-XD/ BOT 038 P=QE*DP*PNW*2.*EXP(ALPH*VO)/W 099 WRITE (6,56) GD,BD 029 26 J=J+1 100 56 FORMAT (' DIODE Y = 'E13.6,' +J 'E13.6) 040 OMEGA=(J-1)*STEP 101 100 CONTINUE 041 DO 200 I=1,N,1 102 IF (J.GE.LIMIT) GO TO 25 042 K=I-1 103 GO TO 26 043 ALPHA=SORT(1.+SORT(1.+(K*OMEGA*TAUt)**2))*WL/SR2 104 39 FORMAT (' NEGATIVE ORDER FOR BESSEL FUNCTION') 044 BETA=SQRT(-1.+SQRT(1.+(K*OMEGA*TA(J)**2))*WL/SQR2 105 40 WRITE (6,39) 045 Z=CMPLX(ALPHA,BETA) 106 GO TO 25 044 IF (B.E(O.O) GO TO 30 107 41 FORMAT (' NEGATIVE ARGUMENT') 047 NU=(B-1)/2 108 42 WRITE (6,41) 048 CALL BESKC(Z,NUBKO,IER) 109 GO TO 25 049 GO TO (40,42,46,48),IER 110 43 FORMAT (' RESILT <1.E-69') 050 CALL BESKC (Z,NUJ+1,BKl,IER) 111 44 WRITE (6,43) 051 GO TO (40,42,46,48), IER 112 GO TO 25 052 ZA=Z*BKO/BK1 113 45 FORMAT (I ' ARGUJMENT>170. '1) 053 31 __ CONTINUE 114 46 WRITE (6,45) 054 VA=ALPH*V1 115 GO TO 25 055 CALL BESI)(VA,K,BI IER ) 116 47 FORMAT (' RESULT>1.E70') 056 GO TO (40,42,44,46),IER 117 48 WRITE (6,47) 057 VA=REAL(ZA) 118 GO TO 25 o058 VB=AIMAG(ZA) 119 30 CONTINUE 059 IF (K.GT.O) GO TO 201 120 ZA=Z 060 C D.C. TERM NEEDS FACTOR OF 1/2 121 GO TO 31 ~~~0613- VVA2122 END VA=VA/2.

001 PROGRAM D 061 GO TO (40,42,46,481,IER 002 062 CALL RESKC (ZNU+1,RKSIER) 003 C THIS PROGRAM CALCULATES THE LARGE SIGNAL CURRENTS DEVELOPED IN A 063 GO TO 140,42,46,481, IER 004 C LSUC WHEN DIFFUSION ADMITTANCE IS USED. 064 ZA=Z*BKO/BK1 005 065 31 CONTINUE 006 C IHIS PROGRAM CALLS BESKC. IT ALSO CALLS DOG12 WHICH IN TURN 066 ALPHA=REAL(ZAI 007 C USES THE EXTERNAL FUNCTION OFT WITH ENTRIES FC1, FC2. 067 BETA=AIMAGIZA) 008 C DOG12 IS AN IBM SSP ROUTINE WHICH IS A GUASSIAN DIJADRATURE 068 011)= BETA/ALPHA 009 C INTEGRATION PROGRAM. 069 ALPHAV( I )=ALPHA 010 IMPLICIT REAL*8 (D) 070 200 CONTINUE 011 REAL*B FC1,FC2 071 PHI2=ATAN(Q(1))-PHIl-TPI/2. 012 EXTERNAL DFT 072 C IHIS PHASE CONDITION USED SMALL SIGNAL ASSUMPIION 013 Hl(THET)=-VP*VARO*SIN(THET*VARO)-VARI*Vl*SIN('HET*VAR1+PHII) 073 VARO=FP/F 014 1-VAR2*V2*SIN(THET*VAR2+PH12) 074 VAR1=FS/F 015 HSIITHET1=-VP*VARO*VARO*COS(THET*VARO)-VAR1*VARI*COS(THET*VAR1 075 VAR2=VARO-VAR1 016 1+PHI1 )*V1-VAR2*VAR2*V2*COSiTHET*VAR2+PH12) 076 DALPH=DPLEiALPH) 017 REAL*4 OP 077 DVP=DRLE(VPI '01( REAL LMUJ 078 DVI=DBLE(VI) 019 REAL JS(10),JC(10) 079 DV2=DBLE(V2) 020 DIMENSION 0(10),ALPHAV(10),THETA(60) 080 DVARO=DRLE(VARO) 021 INTEGER R 081 DVAR=DBRLE(VARSI 022 'COMPLEX ZRKOBK1SA 082 DVAR2=DBLE(VAR2) 023 N=S0 083 DPHI=DRLE(PHIl) 024 C ELECTRONIC CHARGE 084 DPHI2=DBLE IPHI2) 025 QE=1. 60210E-19 085 C NEWTON METHOD 026 C OE/KT =40. 086 IMAX=O. 027 C ALPHA/N WHERE N.NE. I IMPLIES TRAPS IN FORHIDDEN RAND 087 BH1=Hl(0.I 028 ALPH=20. 088 M=0 029 SDR2=1.41421356 089 V8=0. 030 IPI=6.28318531 J= 031 1 FORMAT (I11,6El1.5/(7E11.5)) 091 THETA(l)=0. 032 MUJ=401. 092 211 CONTINUE 033 P=M/U/40.1 093 C FIND ROTH MINIMUM AND MAXIMUM OF Hi 034 C PNW=PN/ IN CMAA-3 SEE WATSON P 107 094 M=M+1 035 PNW=1.El7 095 VA=M*TPI*F/(FP*6.) 036 EPS=1.E-4 096 VC=Hl(VA) 037 25 CONTINUE 097 IF (VCARHI.LE.0.) GO TO 217 038 50 FORMAT('1DIODE N= s**'12,' VO = 'E1.5,1' VP = 811.5,1 V = ' 098 BHS=VC 039 1E11.5, ' V2 = '111.5, I' HOLE LIFETIME = 'Ell.59, EASE WIDTH/DIFFUS 099 VB=VA 040 2ION LENGTH = 'Ell.5/' FUNDAMENTAL FREDI E11.5,' PS ='E11.5, SOD GO TO 211 041 4' PP = E11.5,' PHIO = 0 PHIl = '811.5) 101 217 CONTINUE 042 READ (5,1) R,VOVPFV,V2,TAU,WL,F,FS,FP,PHII 102 C iRIAL THETA VALUE. LINEAR INTERPOLATION 043 WRITE (6,50) BVOVPVl1V2,TAUWLFFSFP,PHII 103 THETR=VR+(VR-VAURRHJ/(VC-BH1U 044 C N=XRRB, TAU=LIFETIME IN P MATERIALF=FRED.,V=VOLTAGE WL=W/L RATIO 104 RHI=VC 045 L=SDRT(DPRTAtJ) 105 VR=VA 046 W=L*WL 106 11=0 047 PP=OE*DP*PNW*2./(W*SQRT(TPI1) 107 218 CONTINUE 048 PSS=OERDP*PNW*EXP(ALPH*VO1/W 008 FC 3=H iTHETR) 049 C CHANGE FROM DEGREES TO RADIANS 109 THET1=THETB-FCT/Hll(THETRB 050 PHII=PHII*TPI/360. 110 1OL=EPS 051 DO 200 I=1,N,1 111 IF (ABS(THET1-1.) 4,4,3 052 =1-1 112 C SEE SSP (UNDER NEWTON METHOD 053 OMEGA=TPI*((K/2U*FP-FSAU-SUA*KU 113 3 TOL=TOL*ABS(THETlU 054 --- —-~ IF (K.EO.O) OMEGA=TPI*FP 114 4 IF(ABS(THETI-THETRU-TOL) 5,5,6 055 AV=SORTU1.+SDRTUS.+(OMEGA*TAU))**2)U*WL/50R2 115 5 IF (ABS(FCTU-100.AEPSU 7,7,6 056 BV=SDRTU-1.+SORTUI.+(OMEGA*TAUU)**2))*WL/S0R2 116 6 CONTINUE 057 Z=CMPLX(AVVV) 117 THETR=THETI 058 IF 1)5.60.0) GO TO 3D - 118 11=11+1 059 NlO=UB-1) /2 _____ ___ ____________ 119 IF U1I.GT.101 GO TO 220 00 CALL RSCZNJRDIR 120 DO TO 218

121 7 IF ((THET1-TMAX).GT.1.E-4) GO TO 8 181 45 FORMAT ' ARGUMENT>170.' 122 GO TO 211 182 46 WRITE (6,45) 123 8 CONTINUE 183 GO TO 25 124 IF (THETi.GE.TPI) GO TO 221 184 47 FORMAT (' RESULT>1. E70') 125 VHI1=HI(ITHETI) 185 48 WRITE (6,47) 126 1MAX=THET1 186 GO TO 25 127 J=J+1 187 30 CONTINUE 128 1HETA(J)=THET1 188 ZA=Z 129 GO TO 211 189 GO TO 31 130 220 CONTINUE 190 END 131 56 FORMAT (' FUNCTION = 'E11.5,' DERIVATIVE = 'E11.5,' THETA = 'E11.5 191 REAL FUNCTION DFT*8(THET) 132 1) 192 IMPLICIT REAL*8 (A-HO-Z,$),INTEGER(I-N) 133 VH1=H11(THET1) 193 C FOR USE WITH SSP GUASSIAN QUADRATUIRE. CALL FCI,FC2 BEFORE FCT. 1i4 FCI=H1(THET1) 194 GO TO 11,12),I 135 WRITE (6,56) FCT,VHll,TH 1 195 RETURN 136 GO TO 7 196 11 VA=DCOS(AM*THET)+O*DSIN(AM*THET) 137 221 CONTINUE 197 GO TO 13 18 57 FORMAT (' THETA(J)= '/(' '8(Ell.5,3X))) 198 12 VA=DSIN(AM*THET)-O*DCOS(AM*THET) 139 WRITE (6,57) (THETA(JJ),JJ=1,J,1 ) 199 13 CONTINUE 140 1HETA(J+1)=TPI 200 DF 1=VA*DEXP ( ALPH* ( VP*DCOS ( THET*VARO ) +Vl*DCOS( VAR1*THET+PHI 1)+ 141 DOl-10 I=1,N,1 201 lV2*DCOS(VAR2*THET+PH I2 ) )) 142 JC(I)=O. 202 RETURN 143 JS(I)=O. 203 ENTRY FCI (AM,Q,ALPH,VP,Vl,V2,VAROVARO,VVA R2,PHI1,PHI2) 144 K=I-1 204 FC1=1. 145 AM=( (K/2)*FP-(-1)**K*FS)/F 205 RETURN 146 IF (K.EO.O) AM=FP/F 206 ENlRY FC2(I) 147 DAM=DBLE(AM) 207 FC2=1. 148 DO=DBLE(O(I)) 208 RETIURN 149 DV= FC1(DAM,DO,DALPH,DVP,DV1,DV2,DVARO,DVARI,DVAR2,DPHI,DPHI2) 209 END 150 VA=ALPHAV(I)*PSS*2./TPI,151 DO 226 JJ=,J,l 152 DXL=DBLE(THETA(JJ)) ) 153 DXH=DBLE(THETA(JJ+1)) 154 DV= FC2(1) 155 CALL DOG12 (DXL,DXH,DFT,DJC) 156 DV= FC2(2) 157 CALL DQG12 (DXL,DXH,DFT,DJS) 158 JC(I)=VA*SNGL ( DJC)+JC(I) 159 JS(I)=VA*SNGL(DJS)+JS(I) 160 226 CONTINUE 161 210 CONTINUE 162 53 FORMAT(' PHI1= 'El1.5,' PHI2 = 'E11.5) 163 52 FORMAT (' (',I2,') = 'E11.5 ' JC' I2,'I = 'E11.5,' JS('12, 164 1!) = 'E11.5,' J(I12,') = 'E11.5) 165 WRITE (6,53) PHI1,PHI2 166 DO 216 I=1,N,1 167 K=I-1 168 VA=SQRT(JC(I) **2+JS( I )**2) 169 WRITE (6,52) K,Q(I),K,JC(I),K,JS(I),K,VA 170 216 CONTINUE 171 GO TO 25 172 39 FORMAT (' NEGATIVE ORDER FOR BESSEL FUNCTION') 173 40 WRITE (6,39) 174 GO TO 25 175 41 FORMAT (' NEGATIVE ARGUMENT ' ) 176 42 WRITE (6,41) 177 GO TO 25 178 43 FORMAT (' RESULT <1.E-69' ) 179 44 WRITE (6,43) 180 GO TO 25

001 PROGRAM E 061 31 CONTINUE 002 062 ALPHA=REAL(ZA) 003 C IHIS PROGRAM CALCULATES THE CURRENTS DEVELPED IN.A LSUC WHEN 063 BElA=AIMAG(ZA) 004 C DIFFUSION ADMITTANCE IS USED BY MEANS OF THE LAPLACE METHOD AND 064 O(1)= BETA/ALPHA 005 C SMALL SIGNAL METHOD. 065 ALPHAV(I)=ALPHA OOh 066 200 CONTINUE 007 C IHIS PROGRAM CALLS ON BESKC AND BESI, THE LAlER BEING AN IBM 067 PHI2=ATAN(O(1))-PHI1-TPI/2. 008 C SSP SURROUTINE. 068 C THIS PHASE CONDITION USED SMALL SIGNAL ASSUMPTION 009 HI(THET)=-VP*VARO*SIN(THET*VARO)-VARI*Vl*SIN(1HET*VAR1+PHI1) 069 VARO=FP/F 010 1-VAR2*V2*SIN(THET*VAR2+PHI2) 070 VAR1=FS/F 011 H11(THET)=-VP*VARO *COSTHETVARO*COSTHETVARO)-VAR1*VARI*COS(THET*VAR1 071 VAR2=VARO-VAR1 012 1+PHI1)*V1-VAR2*VAR2*V2*COS(THET*VAR2+PHI2) 072 C NEWTON METHOD 013 REAL*4 DP 073 IMAX=O. 014 REAL L,MU 074 BH1=H1(0.) 015 REAL JS(10),JC(10) 075 M=O 0 ---0-16 _ TMENSION 0(10),ALPHAV(10),THETA(30) 076 VB=O. 017 IN1EGER B 077 J=O 018 COMPLEX Z,BKOBK,ZA 078 211 CONTINUE 019 N=10 079 M=M+1 020 C ELECTRONIC CHARGE 080 VA=M*TPI*F/(FP*4.) 021 OE=1.60210E-19 081 VC=H1(VA) 022 C OE/KT =40. 082 IF (VC*BH1.LE.O.) GO TO 217 023 C ALPHA/N WHERE N.NE. 1 IMPLIES TRAPS IN FORBIDDEN BAND 083 BH1=VC 024 ALPH=20. 084 VB=VA 025 SOR2=1.41421356 085 GO TO 211 026 TPI=6.28318531 086 217 CONTINUJE 027 1 FORMAT (I11,6E11.5/(7E11.5)) 087 C 1RIAL THETA VALUE. LINEAR INTERPOLATION 028 MU=401. 088 THETB=V+(VB-VA)*BH1/(VC-BH1) - 029 _ DP=MU/40.1 089 BH1=VC 030 C PNW=PN/ IN CM**-3 SEE WATSON P 107 090 VB=VA 031 PNW=1.E17 091 II=O0 *032 EPS=1.E-4 092 218 CONTINUE 033 25 CONTINUE 093' FC1=H1(THETB) 034 50 FORMAT('1DIODE N= X**'12,' VO = 'E11.5,' VP = i E11.5,' V1 = ' 094 THETI=THETB-FCT/H11(THETB) 035 1E11.5,' V2 = 'El1.5, /' HOLE LIFETIME = 'Ell.5,' BASE WIDTH/DIFFUS 095 TOL=EPS 036 2ION LENGTH = 'E11.5/' FUNDAMENTAL FREO ' E11.5,' FS ='E11.5, 096 IF (ABS(THET1)-1.) 4,4,3 037 4' FP = ' E11.5,' PHIO = 0 PHI1 =-'E11.5) 097 C SEE SSP UINDER NEWTON METHOD 038 READ (5,1) BtVOtVP,V1,V2,TALJ,WL,F,FS,FP,PHI1 098 3 TOL=TOL*ABS(THET1I) 039 WRITE (6,50) B,VO,VP,V1,V2,TAU,WL,FFS,FP,PHI1 099 4 IF(ABS(THET1-THETB)-TOL) 5,5,6 040 C N=X**B, TAU=LIFETIME IN P MATERIAL,F=FREO.,V=VOLTAGE WL=W/L RATIO 100 5 IF (ABS(FCT)-100.*EPS) 7,7,6 041 L=SORT(DP*TAlU) 101 6 CONTINUE 042 W=L*WL 102 THETB=THET1 043 PP=OE*DP*PNW*2./(W*SORT(TPI)) 103 II=II+1 044 PSS=OE*DP*PNW*EXP(ALPH*VO)/W 104 IF (II.GT.10) GO TO 220 045 C CHANGE FROM DEGREES TO RADIANS 105 GO TO 218 046 PHI1=PHI1*TPI/360. 106 7 IF ((THET1-TMAX).GT.1.E-4) GO TO 8 047 DO 200 I=1,N,1 107 GO TO 211 048 K=I-1 108 8 CONTINUE 049 OMEGA=TPI*((K/2)*FP-FS*(-1)**K) 109 IF (THETi.GE.TPI) GO TO 221 050 IF (K.EO.O) OMEGA=TPI*FP 110 VH11=H11(THET1) 051 AV=SORT(1.+SORT(1.+(OMEGA*TAUJ)**2))*WL/SOR2 111 IF (VH11.GT.O.) GO TO 211 052 BV=SORT(-1.+SORT(1.+(OMEGA*TAU)**2))*WL/SOR2 112 TMAX=THET1 053 Z=CMPLX(AV,BV) 113 J=J+1 054 IF (B.EO.O) GO TO 30 114 THETA(J)=THETi 055 N/J=(B-1)/2 115 GO TO 211 056 CALL BESKC(Z,N(U,BKO,IER) 116 220 CONTINUE 057 GO TO (40,42,46,48),IER _ 117 56 FORMAT (' FI)NCTION = 'E1I.5,' DERIVATIVE = 'Eil.5,' THETA = 'E11.5 058 CALL BESKC (Z,NlU+1,BK1,IER) 118 1) 059 GO TO (40,42,46,48), IER 119 VH11=Hl(THET1) 060 ZA=Z*BKO/BK1 120 IF (VH11.GT.O.) GO TO 211

121 FCT=H1(THET1) 122 WRITE (6,56) FCT,VHll,THETI 123 GO TO 7 181 1-KD2*Vl*BRI*(SN1+0(I)*CS1) 124 221 CONTINUE 182 1+KD3*V2*BR2*(SN2-O(I)*CS2) 125 57 FORMAT (' THETA(J)= '/(' '8(Ell.5,3X)) 183 4-K04*V2*BR1*(SN2+()I)*CS2)) 126 58 FORMAT ('OLAPLACE ASYMPTOTIC METHOD' 184 300 CONTINUE 127 59 FORMAT ('OSMALL SIGNAL METHOD' ) 185 DO 303 I=1,N,l 128 WRITE (6,58) 186 K=I-1 129 WRITE (6,57) (THETA(-JJ),JJ=1,J,1 187 60 FORMAT (' JC('12,') = 'E11.5,' JS('12,') = 'I11.5,' J('12, 130 DO 210 I=l,N,1 188 1') = 'E11.5,' PHI('I2,') = 'E11.5) 131 JC(I)=O. 189 CJ=JC(I) 132 JS(I)=0. 19 0 SJ=JS(I) 133 K=I-1 191 VA=SORT(CJ*CJ+SJ*SJ) 134 FK=((K/2)*FP -FS*(-1)**K) 192 IF (ABS(SJ).LT..001) GO TO 305 135 IF (K.E.O0) FK=FP 193 PHI=ATAN(CJ/SJ)*360./TPI 136 DO 222 JJ=1,J,I1 194 GO TO 306 137 VA=ALPHAV(I)*PP*EXP(ALPH*(VO+VP*COS(THETA(JJ)*VARO)+VI*COS( 195 305 PHI=90. 138 1THETA(JJl)*VARl+PHI1)+V2*COS(THETA(JJ)*VAR2+PHI2)))/SORT(-Hll( 196 306 CONTINUE 19: 21HETA(JJ))*ALPH) 197 WRITE (6,60) K,CJ,K,SJ,K,VA,K,PHI 140 JC(I)=VA*(COS(THETA(JJ)*FK/F )+O(I)*SIN(THETA(JJ)*FK/F ))+ 198 303 CONTINUE 141 lJC(I) 199 GO TO 25 142 JS(I)=VA*(SIN(THETA(JJ)*FK/F )-O(I))COS(T HETA(JJ)*FK/F )) 200 39 FORMAT (' NEGATIVE ORDER FOR BESSEL FUNCTION') 143 1+JS(I) 201 40 WRITE (6,39) 144 222 CONTINUE 202 GO TO 25 145 210 CONTINUE 203 41 FORMAT (' NEGATIVE ARGUMENT') 146 53 FORMAT(' PHI1= 'E11.5,' PHI2 = 'Ell.5) 204 42 WRITE (6,41) 147 52 FORMAT (' 0(',I2,') = 'E11.5,' JC('I2,') = 'E11.5,' JS('I2, 205 GO TO 25 148 1') = 'E11.5,' J('I2,') = 'E11.5) 206 43 FORMAT (' RESULT <1.E-69') 149 WRITE (6,53) PHI1,PHI2 207 44 WRITE (6,43) 150 DO 216 I=1,N,1 208 GO TO 25 151 K=I-1 209 45 FORMAT (' ARGUMENT>170.') 152 VA=SORT(JC(I)**2+JS(I)**2) 210 46 WRITE (6,45) 153 WRITE (6,52) K,Q(I),K,JC(I),K,JS(I),K,VA 211 GO TO 25' 154 216 CONTINUE 212 47 FORMAT (' RESULT>1.E70') ) 155 C SMALL SIGNAL, NONDEGENERATE CASE 213 48 WRITE (6,47) 156 WRITE (6,59) 214 GO TO 25 157 VA=ALPH*VP 215 301 CONTINUE 158 CS1=COS(PHII) 216 KD1=O 159 SNI=SIN(PHI1) 217. KD2=1 160 CS2=COS(PHI2) 218 KD3=1 161 SN2=SIN(PHI2). 219 KD4=0 162 CALL BESI(VA,I,BRI,IER) 220 IN=IN+1 163 GO TO (40,42,44,46),IER 221 BR1=BR2 164 JC(1)=PSS*BRi*ALPHAV(1)*2. 222 CALL BESI (VA,IN,BR2,IER) 165 JS(1)=-JC(1)*( ) _ 223 GO TO (40,42,44,46),IER 166 CALL BESI (VA,O,BR2,IER) 224 GO TO 302 167 GO TO (40,42,44,46),IER 225 30 CONTINUE 168 IN=O 226 ZA=Z 169 Dn 300 I=2,N,1 227 GO TO 31 170 IF ((I/2)*2.E0.I) GO TO 301 228 END 171 KD1=1 172 Kl2=0 173 KD3=0 174 KD4=1 175 302 CONTINUE 176 JC(I)=PSS*ALPHAV(I)*ALPH*(KDi*Vl*BR2*(CSI+0(I)*SN1) 177 +KD2*Vl*BR1*(CS1- I )*SN1) 178 1+KD3*V2*BR2*(CS2+O(I)*SN2) 179 1+KD4*V2*BR1*(CS2-0( I )*SN2)) _ __.___.180 JS(I)=PSS*ALPHAV(I)*ALPH*(KDI*Vl*BR2*(-SNl-O(I)*CS1)

001 PROGRAM F 061 221 VKSAO=VA 002 002 ~~~~ ~~~~~~062 JO=JJ 003 C LSUC GAIN AND NOISE FIGURE USING DEPLETION AND DIFFUSION 063 GO TO 223 004 C ADMITTANCE IS CALCULATED HERE. 064 222 VKSAD=VA 005 065 Jl=JJ 006 C THIS PROGRAM USES RESKCCMINV, AS WELL AS HESI AND GAMMA 066 223 CONTINUE 007 C WHICH ARE SSP ROUTINES. 067 VM=VKSAl/VKSAO 009 REAL LMUJLRILR2,NF,NFO 069 CO=VM/(Sl*(1.-VVM*VMI 009 DIMENSION IPERM(4),YR(2,2),YI(2,2)..ZR(2,2)tZI(2,2),ALPHA(3), 069 C(1)=VM*CO 010 IRETA(3),F(8),C(R) 070 COS=CO*I1.-VB/PHI)**GAM/VKSAO 011 DOURLE PRECISION DYR(2,21,DYI(2,2), DZI(2,2),DSA-(2,2),DSR(2 071 CB=CO/VKSAO 012 1,2),DSC(2,21,RSOZR(2,2) 072 WRITE (6,55) COJOC(l),JlVMCOSCR 01 2 COMPLEX ZRK0,BK1,ZA 073 55 FORMAT I' CO 'E13.5, ITERATIONS = '13,' Cl 'E13.5, ITERATION 014 TPI = 6.29319531 074 iS = '13,' H = 'E13.5,1 COS =E13.5,' CB ='13.5) 015 INTEGER R 075 IF (TA)).GT. 0.) GO TO 216 016 EPS 11.8*9.R54 E-1401 017 C OR = ELECTRONIC CHARGE 076 00P217J=DN 077 ALPHAIJ)0., 019 OE=1.60210E-19 079 RETA(J)=0. 019 A L P H.2 0. 079 217 CONTINUE 020 SOR2=1.41421356 080 PSS=0. 021 N=2 091 RR1=0. 022 10 FORMAT(2I11,5E11.5/(7Ell.51 082 RRO=0. 023 READ (5,10) NO,JSTEP 093 GO TO 219 024 M(J=401. 054 216 CONTINUE 025 DP= MU/40.1 055 L=SORT(DP*TAtJ) 026 C PNW=PN IN CM**-3 SEE WATSON P. 107 096 W=L*WL 027 C PHI = RIJILT-IN POTENTIAL 087 AREA = CO*W/EPS 029 PHI=.9 088 PSS=OE*DP*PNW*EXPiALPH*VRB/W*AREA*ALPH 029 25 CONTINUE 089 VA=ALPH*VO 030 C OLIVER P.54 CO=SO/(SO*SO-Sk*S1l. IF SO=6.2853E11, THEM CO=1.8079 -6090 CALL RESI(VA,1,BR1,IER) 031 C - IAU=LIFETIME, WL=DEPLETION WIOTH/DIFFUSION LENGTH, 091 GO TO (40,42,44,46),IER 032 C VB=BIAS VOLTAGE, VO=PUJMP VOLTAGE. 092 CALL.RESI (VA,0,RROIER) 033 READ (5,10) BNNFSFPRSRGRLTAUWLPNWVbV0,Sl - D93 GO TO (40,42,44,46),IER 034 WRITE (6,3) BNNFS,FP,RS,RG,RL,TA)(,WL,PNWVWVOS1 09 4 215 CONT INUE 03_5 3 FORMAT (I' N = 39*II2,' N = ' 12,'FS,FPRSRGRLTAIJWL,PNWVrV 095 JL=0 036 1,11 = ' /(9(3X,E13.5))) 096 037 GAM=1./(B+2) 097 F(1)=FS 039 A=VO/(VR-PHI) 099 211 CONTINUE 039 DENG=GAMMA(GAM) 099 F(2)=FP-F(1) 040 DO 223 MM=1,2,1 100 IF (TAU.LE.0.) GO TO 219 041 VA=O. 101 DO 200_K=1,N,1 042 H=MM-1 102 OMEGA = TPI*F(K) D43 DO 220 JJ=1,20,1 103 AS =SORT(1.+SORT(1.+( OMEGA*TAt)j**2))*WL/SQR2 044 J=JJ-1 104 BV =SORT(-1.+SORT(S.+( OMEGA*TAU)**2))*WL/S QR2 045 IF1IJ.GT.0.OR. M.GT.O) GO TO 225 105 Z=CMPLX(AVrBV) 046 VC=1.. 106 IF (B.EO.O) GO TO 30 047 G___GO -TO -' 224 107 NU=(B-1)/2 049 225 VC=GAMMAI2*J+M+GAM)/DENG 109 CALL RESKCZN(JBKOIER) 049 224 CONTINUE 109 GO TO (40,42,46,48),IER 050 VD=VA 110 CALL BESKC (ZNU+1,BK1,IER) 051 IWO = 2.**(M+2*J-1) 111 GO TO (40,42,46,491, IER 052 C SEE DAVE ((LIVER APPENDIX 9 ED. B.11 112 ZA=Z*BKO/BK1 053 VA=A**(2*J+M(*(-1I**M*VC/(TWO*GAMMA(M+J+1.)*GAMMA(J+1.I)+VA 113 31 CONTINUE 054. IF (J.LE.0 'GO TO 220 114 — ALPHAI(-=REALIZA) 055 VC=ARSI(VO-VAI/VD) 115 200 REIA(K(=AIMAGIZAI 056 IF (VC.LT..0001l GO TO (221,222IMM 116 219 CONTINUE 057 220 CONTINUE OSS 13 FORMAT (I PERCENT CHANGE = E13.5) 19 DRJJIDEPS*RALHJI 059 W-R-ITE_______6__13___________ 119 201 DYIIJJ)= DBLE((PSS*BRO*BETA(JI+CO*TPI*F(J))*(-lI**(J+1(I (60 GO TO (221,222(,MM 120 - -DY R (I1,21DR L E P S S*BRR1 *AL PHA (1(

121 OYI)1,2)=DBLE(PSS*RR1*BETA)+TPI*F)1)*C)1)) 122 DYR(2,1)=DBLE(PSSBRR1*ALPHA(2)) 123 DYI(2il)=-ORLE(PSS*BRIBETAT)2)+TPI*F(2)*C)1)) 181 NF=(4.*VC*(VA-VC*(YR(2,2)**2+YI)2,2)**2)) 124 IF (JL.GT.0) GO TO 215 182 1+(1.-2.*VC*YR(2,2))**2+4.*(VC*YI(2,2))**2)/GAIN 125 DO 214 J=1,N,1 183 IF (ITEST.EQ.1) GO TO 209 126 DO 214 K=1,N,1 184 213 WRITE (6,4) NNGAINF(l),NF 127 YR(JK)=SNGL(DYR(J,K)) 185 4 FORMAT I THE GAIN FOR THE 'Ilt' BY 'Ii,' MAIRIX IS 'F12.2, 128 YI(JK)=SNGL(DYI(JK)) 186 15X,' WITH SIGNAL FREQUENCY ' E12.5,5X1' NOISE FIGURE ='12.5) 129 214 CONTINUE 187 210 CONTINUE 130 WRITE (6,51) ((YR(MJ),J=1,2,1),M=1,2,1) 188 JL=JL+1 131 WRITE (6,52) ((YI(MJ),J=1,2,1),M=1,2,1) 189 F)1)=FS+STEF*(JL-)NO+1)/2) 132 BDIFI =FSS*BROR BETA1) 190 IF )JL.GT.NOI GO TO 25 133 RDIF2=FSS*BRO*BETA(2) 191 IF )JL.NE.) NO+1)/2) GO TO 211 134 BDEP1=CO*TPI*F(1) 192 NF=NFO 135 RDEP2=CO*TPI*F(2) 193 GAIN=GAINO 136 RDIF12=FSS*RR1*BETA)1) 194 00 TO 213 137 RDEP12=TPI*F)1)*C1) 195 207 CONTINUE 138 WRITE (6,53) F(l),RDEPFI,BIFEF(2),BD EP2,RGIF2,F~l)1), EP12,RGIF12 19. — LR1=0..- - 139 53 FORMAT (I' F = '13.5,' 8DEP = I E13.5,' bUIF = '813.5) 197 SR1=LR1 140 215 CONT:INUE 198 LR2=LR1 141 CALL CMINV (N,2,OYRDYI,IFERMGZROZI,DSA,0SbDSC) 199 5R2=LR1 142 IF (IPERM)2*N-1).NE.0) GO TO 230 200 VA=SNGL(DZI(Ll) 143 DOURLE PRECISION DZIE(4) 201 IF )VA.LT.0.) LR1=-VA/(TPI*FS) 144 EQUIVALENCE (DZIE(l),ZI(l1,)) 202 IF )VA.GT.Q.) SRI=VA*TPIAFS 145 CALL CMINV (N,2,OYI,DYR,IPERMODZIZRDSA,DSbOSC) 203 VA =-SNGL (DZI(2,2)) 146 IF (IPERM(2*N-1).EO.O) GO TO 25 204 IF )VA.LT.O.) LR2=-VA/)TFTI*F 22 147 M=2*N 205 IF (VA.GT.O.) SR2=VA*TPI*F)2) 148 00 231 J=1,M,1 206 WRITE (6,5). LRISR1,LR2,SR2 - 149 DZIE(J)=-OZIE)J) 207 5 FORMAT I DIODE RESONATED WITH LR1= 'E13.5,' SRI I E13.5,' LE2 150 231 CONTINUE 208 1= '813.5,' SR2 = ' E13.5) 151 230 CONTINUE' 209 GO TO 208 152 IF )ITEST.EQ.1) GO TO 207 210- 209 CONTINUE 153 208 CONTINUE 21.1 GAINO = GAIN 154 DRS=DRLE)RS) 212 NFO=NF 155 DZR(1,1)=DZR)1,1)+DRS+DBLE(RG) 213 IlEST=0 156 DZR(2t2)=DZR(2,2)+DRS+DRLERL) -- 214 GO TO 210 157 0ZI)1,1)=DZI)1,1)+DBLE(F)1)*TPI*LR1-SR1/(F()1)TPI)) 215 39 FORMAT (' NEGATIVE ORDER FOR BESSEL FUNCTION') 158 DZI)2,2)=DZI(2,2)-OBLE(F(2)*TPI*LR2-SR2/)F(2)*TPI)) 216 40 WRITE (6,39) 159 CALL CMINV (N,2,DZRDZIIPERMDYR,0DYISA, 0S6,oSC) 217 GO TO 25 160 00 203 )=IN,1 218 41 FORMAT (' NEGATIVE ARGUJMENT') 161 DO 202 K=1,N,1 219 42 WRITE (6,41) 162 ZR(JK)=SNGL)DZR(J,,K) 220 GO TO 25 163 ZI(JK)=SNGL(DZI(J,K)) 221 43 FORMAT)' RES!JLT <1.E-69') 164 YR(JK)=SNGL(DYR)J,K)) 222 44 WRITE (6,43) 165 YI(JK)=SNGL(DYI)J,K)) 223 GO TO 25 146 202 CONT INUE 224 45 FORMAT I' ARGUMENT>170.') 167 203 CONTINUE 225 46 WRITE (6,45) 168 IF )JL.GT.0) GO TO 212 226 GO TO 25 169 51 FORMAT ('OCR MATRIX '/)'Q',2)3XE13.6))) 227 47 FORMAT 1' RESUJLT>i.E70') 170 52 FORMAT ('OZI MATRIX '/('0',2(3XE13.6))) 22R 4'1 WRITE (6,47) 171 WRITE (6,51) ((ZR)M,J),J=1,2,1),M=1,2 i) 229 GO TO 25 172 WRITE (6,52) ((ZI(MJ),J=1,2,1),M=1,2 11) 230 30 CONTINUE 173 __ WRITE (6,51) ((YR)M,J),J=1,2 1i),M=1,2 I) 22?1 ZA=Z 174 WRITE (6,52) ((YI)M,J),J=1,2 1i),M=1,2,i) 232 GOTO 31 175 212 CONTINUE 233 FND 176 GAIN=4.*RG*RL*)YR)2,1)**2+YI (2,11**2) 177 VC=ZR)2,2)-RS 178 VA=0. 179 D0 206 I=1,N,1l_ _ _ _ _ _ __ _ _ _ _ _ _ _ 1-80O 2~06 V A=)Y R)2,I)*2 +Y I2,-I) X2)Z R)(I,I)+VA - -

001 SUJBRRnUIINE IMPST, ENTRY IMPM 001 SUBRRuTINE TRNR 002 002 005 C IMPST FINDS CHARACTERISTIC IMPEDANCES AND ACCUUNTS FOR 003 C TRNR TRANSLATES RECEIVING END IMMITTANCE TOWARD GENERATOR. 0 043_ C TENRY TRNS TRANSLATES AWAY FROTESRM TGENERATOR. DDERAR.004 C DISCONTINUIITY CAPACITANCES OF DISKS IN COAXIAL FILTER. DDS DDS004 C E~lRY TRNS TRANSLATES QWAY GENERATOR.005 C ENlRY IMPM FINDS INPUJT IMPEDANCE [IF FILTER WHEN FILTER TERMINATION 006 C IS KNOWN. o006 SUBROUTINE TRNR (YO,D,GRBR,GS,BS) 007 C Z = (ZR+J*ZN*TAN(PHI))/(ZO+J*ZR*TAN(PHI)) 007 008 C D IS DISTNACE TRANSLATED iN NUJMBER OF WAVELENG(THS D08 C THIS PROGRAM USES TRNR. IMPST MUST BRE CALLED FIRST TO SET 009 C GR IS THE REAL PART OF THE RECEIVING ADMITTAmCE 009 C VARIOUS PARAMETERS IN IMPM. Ol( C MR IS THE IMAGINARY PART OF THE RECEIVING ADMITTANCE 010 SUBROUTINE IMPST (DIA,LNTHEPS,N,TEST,SIZE,CU,ZOC,FN) 011 C THIS FUNCTION RETURNS SENDING END ADMITTANCE GS AND BS 011 RPARAL (VVA,VVB,VVCVVD)=( (VVA*VVA+VVB*VVB)*VC+(VVC*VVC+ 012 C THE CHARACTERISTIC ADMITTANCE IS YO MHOS - — 012 lVVD*VVD)*VVA)/( (VVA+VVC)**2+(VVR+VVD)**2) 013 THETA=6.2RT8318531*A)D 0()13 XDPAR*VAL( VVA,VVB,VVCVVD)( (VVA*VVA+VVB*VVBR)A*VMU+(VVC*VVC+ 014 10 CONTINUE 014 1VVD*VVD)*VVB)/((VVA+VVC)**2+)VVB+VVDU**2) 015 BOT=(YO*COS(THETA) -BRRSIN(THETA ) ) **2+ ( GRTS I N ( THE TA ) )-2 I R J NTEST TEST1,NV 016 - IF (ABS(BOT).GT.1.E-20) GO TO 20 01 )LNTLN-_ __ ______ ~001~~~7 GR-s-'~~~~= ~. ~017 DIMENSION LNTH(10),DIA(10),DIST ( 10 ),LN(N1C0) D10,ZOC 10 ), 018 15IZE(10),EPS(10) 018 RS=1.E20 019 C TW7SO. E20S 019 DIMENSION CDO(10) 020 C YO =RCTAN(THETA) AND GERSIN(THETA) THEREFORE GR=O 020 C LNTH IS THE LENGTH SIZE OF THE DISK IN METERS 021 C OR GR:O AND BR = 0 ---02 < 1 C OR Gr D- ANK1 D ER _ r 021 C DIST IS THE DISTANCE BETWEEN DISKS IN METERS 022 C F IS THE FREQUENCY IN DC/S 023 2 TIURN 022 RE320 CONTRN 023 C DIl IS DIAMETER OF OUTER CONDUCTOR OF G R LINE 023 20 CONTINUE 24_ ___ - __ _ __ S=YYOGR/OT 024 C DIST(J) IS DISTANCE BETWEEN DISK(J-1) AND DISK (J) 024 GS=Y5*YO*GR/,0T 025 rS=YlO*(BR*Yn*(1.-2.*SIN(THETA)*SIN(THETA))+(YU*YO-BR*BR 025 C SIZE IS THE MODIFIED DISK LENGTH IN METERS ~026 MS-YO*IRREYOSIN(TH OSINTHETAIINBOTH026 C CD IS THE D)ISCONTINUITY CAPACITANCE IN PF 02h6 -GR*GRIESINITHE-TAECOSITHETAI/EOT 027 C TEST=1 SEE REF. MATTHAEI MTT At(G. 1966 PP.372-383 027 RETURN!B C -- ___________ _____ ---- _ 028 C TEST=2 MODIFY MATTHAEI S METHOD BY UJSING DISTRIBUTED CAPACITANCE 028 C Z=(ZS-ZO*TANH(A))/(ZO-ZS*TANH(A)) 028 ICZENTRY ETANHUAUrZRO-DSE*TANH(A)I )029 C lEST=3 SEE REF. LEVY & ROZZI, MTT MARCH 1968 PP.142-147. _ 029 ENTRY TRNS(YOD,GR,BRGSBS) -DAD~~~~~~ THETA=~~~~~~ ~~ --- —--— A DAD --- —— ~-'O~ 3 C TEST=4,5,6 CORRESPOND TO TEST =.I2,3 RESPECTIVELY BUT WITH 030 1HETA=-6.28318531*iD 3... A31 GO TO 10 031 C LOW-PASSMATCHING NETWORK. SEE MATTHAEI, AUG. 1964 PROC. IEEE ~~~~~~~~~032 END ~032 TPI=6.28318531 -- OA T_ 3.3_ T IEST1-=0 ______o __ ___ 034 IF (TEST.LE.3) GO TO 21 035 TEST1=1 03h TFST=TESTEST-3 0_7 21 CONTINUE 038 DO=.5625 039 ZO=50. — --------------- 040 CN=. 299 7925 041 BETA=TPI*FN/CN 042 JJ=O 043 KFY=O 044 C JJ COUINTS ITERATIONS FOR TEST=3, KEY IS UUSEI) FOR CONDITIONAL BRNCH 045 DO 11 J=1,N,. 046 CDO(J)=O. 047 11 SIZE( J =LNTHU(JU. _ 048 ZOC ( N+ ) =0. 049 IF (TEST1.ED.O) GO TO 14 050 LN(1) =ALOG()UO/DIA(1)) 051 ZOC(N+1)=100.*LN lS( ERT_ 20./(TPI *.8.8.54) ) _ 052 14 C(TNT INIJE 053 12 CONTINUJE __. _____. _.__._.._ __ _ 054 nn 10 J=1,N,1 055 THFTA=BETA*SIZE(J) 056 ALPHA =(I)O-DI-A(J) )/(D(-.24425) 057 AUJ=Dl/.24425 058 IF (TEST1.EQ.O) G() TO 13 059 ALPHA=(DO-DIA (2,*I))/(DO-DIA(1 ) 060 TAU=DO/DUI A( 1 )

061 C DIA(l)=.03196 INCHES OR #20 WIRE. 121 RA=R2 062 13 CONTINUE 122 XA=X2 063 C.24425 IS DIAMETER IN INCHES OF INNER CONDUClOR OF GR LINE 123 R2=RPARAL (RA,XA,-XD*EPS(5),XD)) __ 064 C CD IS THE DISCONTINUITY CAPACITANCE 124 X2=XPARAL (RA,XA,-XD*EPS(5),XOI 065 C REF. P. 1. SOMLO MTT-15,NO. 1 P.48 125 WAE=ISTJF/CN 126 CALL TRNR (ZCWAVER2,X2,RI,X1) 066 CDO(J ) =. 54*0D0.*0254*I( ALPHA *ALPHA+1.)/ALPHA) )*ALOG((1.+ALPHA) 126 CALL TRNR 127 J:J-1 067 1/(l1.-ALPHA) -2.*ALOG(4.*ALPHA/(.-ALPHA*ALPHA))) +.111* 127 12R Gf TO 20 068 2(1.-ALPHA)*(0t/.24425-1.) *TPI*DO/2. *.0254 12 T 069 IF (KEY.EO.1) GO TO 10 129 30 COnTINUE 070 0'I-(J)=ALOG(DO/DIA ((TESTI+1)*J)) 131 XOUT=X1 071 ZOC(J)=100.*LN(J)*SORT (20./(TPI*EPS(J)I) --- 07 2 GO TO 141042,431, TEST 132 RETIIRN 073 41 CONTINUE 133 END 074 SIZE(J):LNTH(J)-2.*CD(J)*LN(J)/(TPI*EPS(J) 075 GO TO 10 076 42 CONTINUE 077 SIZE(J)=2.*ATAN( TAN(THETA/2.)-ZOC(J)CD(J )*TPI*FN*1.E-3)/BETA 078 GO TO 10 679 43 CONTINUE 080 VA=COS (THETA) +TPI*FN*(CD( J )-CDO( J ) )*ZOC ( J )*SIIN( THETA ) 1.E-3 081 IF (ABS(VA).GT.1.) GO TO 44 082 SIZE(J)=ARCOS(VA)/BETA 083 ZOC(J)=Z)OC(J)*SIN(THETA)/SIN(BRETA*SIZE(J) ) 08R4 DIA(J)=DO*EXP(-ZOC(J)*.01*SORT(TPI*EPS(J)/20.)) 085 IF (JJ.EO.O) CDO(J)=CD(J) 086 10 CONTINUE 087 IF (TEST.NE.3) RETURN 088R IF (KEY.EO.1) RETURN 089 JJ=JJ+1 090 C AT LEAST TWO ITERATIONS REOUIRED. 091 IF (JJ.EO.1) GO TO 12 092 VA=(CD- 1)-CDO( 1))/CDO(l) 093 31 FORMAT (' PERCENT CHANGE = 'E13.5,' JJ= 'I11) 094 WRITE (6,31) VA,JJ 095 IF (ABS(VA).LT.. 01.OR.JJ.GT.10) GO TO 46 096 00 45 J=l,N,1 097 45 CDO(J)=CO(J) 098 GO TO 12 099 44 SIZE(J)=-1. 100 KEY=1 101 GO TO 10 102 46 KEY=1 103 GO TO 12 104 ENTRY IMPM,(F,N,SIZEDIST,CD,ZOC,RIN,XIN,ROUJT,XOtJT) 105 R1=RIN 10 X1=XIN 107 J=N __ lO8 ZC=ZO 109 IF (ZOC(N+1).GT.ZO) ZC=ZOC(N+1) 110 20 CONTINUE 111 IF (J-1) 30,40,40 112 40 CONTINUE 113 XD=-l.E3/(TPI*F*CD(J) ) 114 WAVE=S IZE( J *F*SORT (EPS ( J )/8.854) /CN 115 RA=R1 l16 XA=X[ 117 C EPS(5) = FACTOR TO VARY DISCONTINUITY LOSS 118 R1i=RPARAL ( RA,XA,-XD*EPS ( 5 ),XD ) 119 Xl=XPARAL(RA,XA,-XD*EPS(5),XD) __ _120 CALL TRNR (Z(C(J),WAVE,Rl,Xl,R2,X2)

001 SURROIIINE DESIGN, ENTRY LUMP 061 V=ATAN(V) 002 062 LNTH(J)=2.*VC/(TPI*F*SORT(EPS(J)/8.854) )VB 003 C DESIGN SYNTHESIZES COAXIAL BAND-PASS FILTERS USING DISTRIBUTED 063 ANG(J)=2.*ATAN(Z*(K(J)-ZC*VD)/(ZO*Z+K(J)*ZUC*VD) 004 C METHOD. ENTRY LUMP USES THE INFERIOR LUMPED METHOD. 064 33 CONTINUE 005 oos ~~~~ ~~~~~~~~~~~~~~~~~~065 IF (JINV.GE.1) GO TO 35" 006 C NOTATION USED FOLLOWS COOLEY TECH. MEMO. 100, BUT IS EQUIVALENT 007 C 10 NOTATION USED IN CHAPTER 2 HERE. NUMBERS 1N THIS PROGRAM 067 41 DI(J)=.532 008 C ARE FOR 14 MM LINE. FOR DIFFERENT LINE DODI(J), AND POSSIBLY 06 GO TO 21 069 4? DI(J)=.542__ _ _ _ _ _ _ ____ 009 C ZO WOULD HAVE TO BE CHANGED. 0 2 010 SBJROUTINE DESIGN (RFDIWDELTA,N,G,K,ANG,C,LNTH,EPS,TESTI) 071 4 O 2 1 011 REAL KyLNLNTHLLJ2 012 DIMENSION G(10),K(10),C(10),LNTH(10),ANG(10),X(10),DI(10) 07 GTO 013 DIMENSION EPS(10) 07 4 T 014 C F- IS IN CPS074 GO TO 21 015 C 1ESTI=O OUTPUT MEANS GOOD DESIGN: TESTI=1 MEANS DESIGN WAS NOT 075 45 1ESTIS 0 6 C POSSIRLE SINCE V.1 WHICH MEANS THAT K IS PROBABLY LESS THAN ZOC 076 IF (JINV.GE.1.AND.J.E0.2) TRSTI=0 017 C IF TESTI=2 IS PUT IN THIS PROGRAM, THE MIDDLE INVERTER IS A J 077 GO TO 33 018 C: INVERTER AND THE J INVERTER IS A SERIES CAPACITOR. 078 35 CONTINUR 019' C IF TESTI =3, THE INVERTER IS A SERIES INDUCTOR (DISTRIBUTED DESN.) 079 REAL J2 020 C V IS FRACTIONAL RANDWIDTH 090 C J INVERTER IS 1/4 WAVE LENGTH FROM K INVERTEKS 021 C G ARE THE PROTOTYPE G VALUES D81 J2=W*TPI/(Zn*R.*SORT(G(2)*G(3)) 082 K(2-)=I~J2 022 C LNTH ARE THE DISK LENGTHS AND ANG ARE THE K INVERTER LENGTHS 082 K(2=1/2 023 INIEGER JN,NNI 083 R2=J2/(1.-ZO*Z*J2*J2)_ 023 INTEGER JNN ~ ~ ~ ~ ~ ~ ~ 84 C(2=1E3B2ngIA 024 INTEGER TESTI,JINV 04 C(=.3*2/OMGAN ~~~~~~~025 JINV=O ~~~~~~~~~~~E085 IF (JINV.EO.2) GO TO 50 02 INV= O__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 086 AN(G(2)=-ATAN(2.*B2*ZO) 026 IF(TESTI.GE.2) JINV=TESTI-1 087 LNIH(2)=EPS(2)*TPI*(.24425*.0254)**2/(9.*C(2)) 027 1ESTI=O 029 TPI=6.2E318531 ORE RETURN 029 Z O=50. 029 ZO=50.89 ENIRY LUMP (C) 030 VC=2.997925 E8 O -4 031 OMEGAN=TPI*F*1.E-9 091 ANG(J)=ATAN(2.*X(J)/ZO) 032 C REACTANCE SLOPE PARAMETER FOR 1/2 WAVE K INVERTER IS ZO*PI/2 092 LN=ALOG(DO/I(J)) - 033 K(1)=SORT(W*R*TPI*ZO/(G(1)*G(2)*4.*DELTA)) 093 LNIH(J)=C(J)*LN/(TPI*EPS(J)) 034 NN=N-1 094 34 CONTINUE 035 DO 31 J=2,NN,1 095 REIURN 036 31 K{J)=(TPI*ZO/4.)/(SORT(G(JI*G(J+1)))*W 094 50 CONTINUE 0?7 K(N)=SDRT(ZO*ZO*TPI*W/(G(N)*G(N+1)*4.I) 097 C ASSUME #20 WIRE (31.96 MIL DIAMETER) FOR J INVERTER 039 C OUTSIDE DIAMETER OF COAXIAL LINE IN INCHES 099 LN=ALOG(DO/.03196) 039 00=.5625 099 YOC=.O1*SORT(TPI*8.854/20.)/LN040 IF (JINV.EO.O) GO TO 25 100 YO=i./ZO 041 K(1)=K(1)/1.41421356 101....P=(YO*YO-YOC*YOC)/(YO*YO-J2*J2) 042 K(3)=K(3)/1.41421356 102 V2)2PP/YOC 043 25 CONTINUE 103 IF (ABS(V).LT.1.) GO TO 51 044 D0) 33 J=1,N,1 104 VO=V-SDRT(V*V-1.) 045 DI(J)=.502 105 VR=ATAN(VD) 046 C X IS REACTANCE, C IS CAPACITANCE 106 LNTH(2)=2.=*VCV/)TPI FF 047 X(J)=K(J)*ZZO*ZO/(ZOEZO-K(J)*K(J) )107 ANG(2)=2. ATAN(YO(J2-YO C*VD)/(YOYOYO+J2YOC\VU)) 509 GO TO 34 048 C(J)=1.E3/(X(J)*OMEGAN) 10 5 ET 049 C LENGTH OF PHI OF INVERTER IN RADIANS 050 I=O 110 GO TO 34 051 21 CONTINUE END 052 I=I+1 053 LN=ALOG(DO/DI(J)) 054 ZOC=100.*LN*SORT(20./(TPI*EPS(J))) 055 P=(ZO*ZO)-ZOC*ZOC)/(ZO*ZO:K(J)*K(J)) 056 V=K)J)*P/ZnC 057 C IF V(<, THEN INCREASE DIAMETER. 059 IF )ABS)V).LT.1. ( GO TO (41,42,43,44,45),l 059 20 CONTINUE 060 VD=V-SORT(V*V-1)

001 StBROUINE PRTOG 063 G( 1+3)=1.62 005 C THE NUMBERS HERE ARE FOR A 3-POLE CHERYSHEV MATCHING CIRCUIT 067 DO TO 351 007 C IN MATTHAEI, YOUNG, AND JONES P.128. DEL1-.5 008 SUBRIOUTINE PROTOG (N,DELTAKG) 071 DEL2=.7 009 DIMENSION 0 (20. GG(40) 0712 351 GG(I+1 DEL2=. 27 0 010 INTEGER N,K,I 072 351 00(1+11=2.0 011 K=0 073 GG(1I+2=.76 012 1=0 074 GG(I+3)=1.35 013 IF (N.NE.3.OR.DELTA.LT.. 05) GO TO 90 075 GG(I+4=.60 014 IF (DELTA.GT.. 1) GO TO 310 06- 1=1+4 ~~~~~~~~015 DEL1~=.0S~ ~077 IF (I.GE.2*N+2( GO TO 91 015 DEL2=.1 078 GO TO 371 017 G011+1=20. 1079 370 CONTINUE 017 GG(I+2)=120. 080 IF (DELTA GT.1.00) GO TO 3100 018 GG(SI+2)=.10 OR1DEL1=K7 019 GG( I+3)=12.2 081 DEL2=.7 020 GG(I+4)=.0715 0O2 DEL2=1.00 021 1=1+4 022 IF (I.GE.2*N+2) GO TO 91 094 GG( 1+2=.90 023 GO TO 311 085 GG(I+3)=1.0 024 310 CONTINUE 086 GG(I+4)=.74 025 IF (DELTA.GT..2) GO TO 320 08F (I.GE.N+) G T 91 026 DELI=.l1 - 089 IF (I.GE.2*N+2) GO TO 91 027 DEL2=. l2.089 GO TO 3101 021_ DEL2=.2 090 3100 CONTINI)E 029 311 GG(I+1)=10. 0903100 CONTINE 02R9 1 GG(0 I+2)=.102 091 IF (DELTA.GT. 1.50) GO TO 3150 02930 GG(I+)=.182 092 DEL=1.00 0301 GG(I+3)=h1.3 093 DEL2=1.50 093 DEL2=1.0 031 GG(I+4 094 3101 GG( 1+1)=1.0 095 GG(I+2)=.965 033 IF (I.GE.2*N+2) GO TO 91 ( I + 3 7 034 GO TO 321 1 095 GG 035 320 CONTIN()E 097 GG(1 I+4)=.865 036 IF (DELTA GGT.31 0O TO 330 099 F I=+4 037 DEL1=.2 09IF (I.GE.2*N+2- GO TO 91 038 DEL2=. 3 100 GO TO 3151 0398 321 00(1+11=5. _________ ___ _________________________________ 101 3150 CONTINUE 049 321 GG(I+1)=5. 102 IF (DELTA.GT. 2.0 ) GO TO 90 040 GG41+2)=.360 041 GG(I+3)=3.2 103 DEL1=2.(0 042+4=.272 104 DEL2=2.02 043 2 =1+4 105 3151 GG(I+1)=.66 044 IF (I.GE.2*N+2F GO TO 91 106 GG(I+2)=.90 049 _ _ _ 0 T 331. _ ____ ____ ___ ____ ___ ____ _107 G G (1I+3 =.55 0646 330 C( ONIN kUE 101 GG(I+4:=.95 047 IF (DELTA.GT..4) GO TO 340 109 1=1+4 048 DEL1=.3110 IF (I.GE.2*N+2) GO TO 91 049 DEL2=.i4 - 111 GG(I+1 ).51 050 331 GG(1+11=3.3 112 GG(I+2(.79 05 1 CG, ( I + ) -. 5 2 __ __ __ __ _ _1 43 --- —-- - -— 09GG(I+23)=78.52 052 GG(I+3)=2.1 115 91 CONTINUE 053 7 GG(I+4)=.405 053 _S~rf I+4)=! _ __ _____ -h116 NN=N+1 054 1=1+4 056 IF (3I.GE.2N+) G Tn 91 92 G(I)=()ELTA-DELI)/(DEL2-OEL1)*(GG(N+I+1)-GG(l()+GGI) 057 340 C(ONTINIIE ___- 1 RE —_R_ 05-9 IF (DELTA.0.GTo.3... ) GOI0 TO 350 120 C NO 0 VALUES FOR THIS DATA 059 DELl=.4 ___ 12 0 -060 DEL2=. S 061 341 GG(1+1)=2.5 123 END 062 GG(I+2)=.65

001 SUBROUlINE CMINV, ENTRY MPRD 061 IR=IK 002 062 C THIS GETS A PARTICULAR COLUMN. 003 C CMINV INVERTS COMPLEX MATRICES, AND ENTRY MPRD MULTIPLIES REAL 064 IR=IR+D 004 C MATRICES. 064 IR=IR+1 005 065 JI=J-LEN 006 C CMINV CALLS MPRD AND INV. INV INVERTS REAL MATRICES AND IS 066 IB=IK 007 C AVAILABLE IN THE UJ. OF M. LIBRARY. 067 R(IR)=O. 008 SUBROUTINE CMINV(N,LEN,A,B,IPERM,CR,CI,RAA,BB) 068 DO 10 I=1,N,1 009 C A + J B = INPUT MATRIX. CR + J CI = OUTPUT INVERTED MATRIX 069 _ JI=JI+LEN 010 C IPERM IS INTEGER VECTOR OF LENGTH 2*N 070 IB=IR+1 011 C - LEN X LEN IS DIMENSION OF MATRICES 071 10 RIRR AAJ _ 012 C N X N IS THE NUMBER OF ELEMENTS IN THE MATRICES. 072 GO TO (31,32,33),KEY 013 C R,AA,BB,T ARE SCRATCH MATRICES. 073 RE1URN 014 C REF. C.LANCZOS, APPLIED ANALYSIS, P. 137. 074 END 015 C A' MEANS INVERSE OF A 016 C IF C = A + J*B, AND D = (A + B*A'*B)I, THEN C' = D -J*D*B*A' 017 IMPLICIT REAL*8(A-H,O-Z,$),INTEGER UI-N) 018 R: DIMENSION A ( 1 ), B ( ),CR(1), C I ( 1 ),R( 1 ),AA( 1 ), B ( 1 ), IPERM(1) 019 CALL INV (NLEN,A,IPERMLEN,BB) 020 C RETURN IF MATRIX IS SINGULAR. HENCE THIS PROGRAM WILL NOT WORK FOR 021 C PURE IMAGINARY MATRICES. 022 IF (IPERM(2*N-1).EO.O) RETURN 023 LEN2=LEN*LEN 024 DO 100 I=1,LEN2,1 025 100 AA(I)=B(I) 026 KEY=1 027 GO TO 20 028 31 CONTINUE 029 DO 110 I=1,LEN2,1 030 CIII)=R(I) 031 AAI )=R{I) 032 BB(I)=B(I) 033 110 CONTINUE 034 KEY=2 035 GO TO 20 036 32 CONTINUE 037 00 120 I=1,LEN2,1 038 AA(I)=A(I)+R(I) 039 120 CONTINUE 040 CALL INV (N,LEN,AA,IPERM,LEN,CR) 041 IF (IPERM(2*N-1).E.0O) RETURN 042 DO 130 I=1,LEN2,1 043 AA(I)=CR(I) 044 BB(I)=CRII) 045 130 CONTINUE 046 KEY=3 047 GO TO 20 048 33 CONTINUE 049 DO 140 I=1,LEN2,1 050 CI(I)=-R(I) 051 140 CONTINUE 052 RETURN 053 ENTRY MPRD(AA,BB,R,N,LEN) 054 C AA*BB=R 055 C SUBROUTINE BASED ON GMPRD ON PAGE 99 OF SSP. 056 KEY=O 057 20 CONTINUE 058 IK=-LEN 059 DO 10 K=1,N,1.... 060 IK=IK+LEN

001 SUBROUTINE BESKC 061 G00=.90133333E3*T(S(+.80759467E5*T(2)+.13536825E7*T(3)+ 00? 062 1.825l017lE7*T(4)+.233171599EB*T(5)+.347855076b*T(6)+ 003 C BESKC FINDS THE K BESSEL FUNCTION OF COMFLEX ARGUMENT. 063 2.29579B43EB*T(7(+.S499069EB*T(R(+.46204534E7~T(9(+.86319307E6* 004 064 3 7( 10) +. 9440 2 52 3E 5 *T (I+.5 50 3R82 1E 4* T ( 12(+1 N098 72 E3XT(13 005 SUBROUTINE RESKC(ZN,RK,IER( 065 GI=GON D 006 C COMFUTE THE K BESSEL FUNCTION FOR A GIVEN ARGUMENT AND ORDER 066 IF (N-i)?0,30,31 007 C USAGE IS CALL RESKC(ZN,BKIER( 067 30 BK01 008 C Z = THE ARGUMENT OF THE K BESSEL FIUNCTIUN DESIRED 068 RETURN 009 C N = INTEGER ORDER OF THE K BESSEL FUNCTION DESIRED 069 C FROM KDKD COMFUTE KN (SING RECURRENCE RELATIUN 010 C BK = THE RESULTANT K BESSEL FUNCTION 070 31 DO 35 J=2,N Oi C IER = RESJLTANT ERROR CODE WHERE 071 GJ2.*(FLnAT(J(1.)*G1/Z+GO 012 C IER=0 MEANS NO ERROR, IER=1 N IS NEGATIVE, IER=2 X IS0 OR <0 013 C IER=3 Z.GT. 170, MACHINE RANGE EXCEEDED, IER=4 BK.GT.10a*70 073 32 IER4 014 C N MUST BE >0 AND NO OTHER S(JBRO(TINES ARE REQUIRED 074 Gn TO 34 015 C THE METHOD INVOLVES COMPUTING THE ZERO ORDER AND FIRST ORDER 075 33 00=01 016 C BESSEL FIUNCTIONS ((SING SERIBS AFFROXIMATIONS AND-THEN COMFUTES 076 3-5 G1GJ 017 C ORDER FU(NCTION USING RECURRENCE RELATION. 077 34 BK=GJ 018 C:OMPLEX Z,AR,CTG0,GlGJBKZ2J vGONGOD 078 RETURN 019 DIMENSION T(13( 079 36 CONTINUE 020 BK=(O.,0.( OBO VAREAL IX) 021 PI=3.1415927 ORS VB=AIMAG(Z( 022 SFI=SORT(FI/2.( 082 XCMFLX(VAVB( 023 IF (NI 11,11,11 083 B=Z/2. 024 10 IER=1 084 A=.57721566+CLOG(B( 025 RETLIRN 085 C=B*B 026 11 CONTINUE 086 IF (N-1) 37,43,37 027 20 IF(CABS(Z(-170.(22,22,21 087 C COMFUTE KO USING SERIES EXPANSION 02R 21 IER=3 0P8 37 00=-A Q29 RET((RN 089 Z2J(l.,0.) 030 22 IER=0 090 FACT=D. 031 VA=REAL(Z( 091 HJ0. 032 VR=AIMAG(Z) 092 DO 40 J=1,6 033 IF (CABS(Z)-l.) 36,36,25 093 RjD./FLOAT(J( 034 25 A=CEXF(-Z( 094 Z2JZ2J*C 035 C=1./CSORT(Z) 095 FACT=FACTRRJ*RJ 036 1(1)=Z 096 HJ=HJ+RJ 037 DO 26 L=2,13 097 40 G0=GO+Z2J*FACTR(HJ-A( 038 26 l(L)=T(L-1)*Z 098 4FN(4342, 43 039 IF (N-lI 27,29,27 099 42 RKGO 040 C COMFUTE KO USING POLYNOMIAL APPROXIMATION 100 RETURN 041 27 CONTINIE 101 C COMFPTE KI USING SERIES EXPANSION 042 GONFA*C*SFI*(.93444970E3*T(1(+.74419710E5*T(2)+.13370803E7T(310 043 i+.R5699270E 7RT (4(4.249 27645E8T (5+.37746R65*T (6(+.32310445E8 103 FACT=1. 044 11(71 104 HJl. 045 2+.16404092E8*T(RI+.50516268R7*T(9(-e.94154163E6*T(10+.10265537E6 105 G11./Z+Z2J*(.5+A-HJI 046 3*1(111+.59648092E4*T(12)+.14137062E3*T(13) ) 106 DO 50 J=2,R 047 GOD=.2704E4*T(11+.134599llE6*T()1+.18951555E7*T(3) 107 Z2JZ2J*C 048 1+.1060845lERRT(4(+.2849875lERRT(5 (+.41110144I8*T16+.34130589ER 108 RJ=i./FLOAT(J) 049 2T(17)+. 16989,449E8*T8()+.51640362E7T(19)+.95405550E6*T(10(+ 109 FACT=FACTRRJRRJ 050 1.10339324E6*T(11)+.59824805E4*T(12)+.141370617E3*T(13) 110 HJ=HJ+RJ 051 GO=GON/GOD II1 50 Gl=Gl+Z2j*FACT*I.5+IA-HjI*FLOAT(JI( 052 IF (NI 20,28,29 2 IF IN-11,52,31 053 28 BK=GO 113 52 BKGl 054 RETURN 114 RETURN 055 C COMFUTE KI USING FOLYNOMIAL APPROXIMATION 115 END 056 29 CONTINUE 057 GON=A*C*SPFI*I.9191825E4*T I(+.3155701RE6*T(2(+.32568246E7*T3(3 08 1+.14642396ER*T(4(+.3395202lE8*T(5(+.4449327668*T(6 (+.3473623lER 059 2*T171+.16631127ERRT(R(+.49336335E7*T(9(+.R9X96245E6*T(10(+ 050 3.96451139E5*T(111(+.55529691E4*T(12)+.13089R72E3*T(I13)

APPENDIX B DETERMINATION OF PARAMETRIC CONVERTER CIRCUIT IMPEDANCES B. 1 Introduction The optimum load and generator impedance levels for a lower sideband upconverter were found in Chapter II. However, the actual impedances of the signal and lower sideband circuits are far different, because the diode parasitic elements and the characteristics of the out-of-band ports modify these impedances. First the impedance of the diode parasitics plus circuit are found for signal and all sideband frequencies Then the signal and lower sideband port impedances are found which give the desired impedance level at the diode chip. The various impedances at terminal A-B are found in terms of the unknown and Z, after which the Zt at 10 and Zi at 20 are found. In all cases the circuit impedances Z p, and Zps at out-of-band frequencies are assumed to be purely pp-s reactive In the computer calculations using this analysis, it was generally assumed that these reactances were in the form of short circuited quarter wave transmission lines so that, or exaple, X ~ (wkZ tan(J2 k = 1, 2, 3,. for out- of-ba nd fr equencies.- However, in this analysis these reactances are left in terms of X, X, and X so that any reactance function desired may be used. B. 2 LSUC1I All three ports are in parallel in an ideal tee junction or wye junction if the diode is mounted between the center and the outer conductors.- The diode and circuit can be modeled as shown in Fig. B.1I The input impedance Z at terminal A-B at frequency W1 is obtained in the following 11 - manner. 7Z =R ~ji(w, L. - S/w)4+ 1(B)

247 S0 L R s~~~~~~~~~~~ p S S Fig B.1. Equivalent circuit when all ports are in parallel if R M (B2) g 2 R 2 + Xt2 t t and t~~~X M A 3l 1 1 ) b= S XX - 2X p p p-s R~X t t then M ~IM Z =R + g + j w L- S jw- b (B.4) Z is obtained in the same way except that X replaces X, Z. replaces 22 s p-s~ 1 Zand w2 replaces w. For higher order sidebands Zkk R [kL - s/k- (5 - R- X -+ ] BS The desired impedance at signal frequency is Z R 9+ R. If this is substituted into (B. 1), the desired value for Z can be found. t lw I I-1 Rg9 +R = R s+ j(W 10 L - S0/w 10 ) +j [(s' Y-+ - g- + Z7 (B.6)

248 Rg j(wo10L - S0/w10) 10- So10) -; x -X ) -0 0g( - if Mr 1- (wo 10L - So/O10)(10 _ X )9) I 0 ' ' p p -sp Mx Rg( S X X (B 1) p p p-s then R M R M + -m2 heg -j x g r imped/nce (Bb1u) Zt...M2+M2 M2+M2 r x r x Similarly the lower sideband impedance Z. can be found at w 20 P 1~~~~~~~ which gives the desired impedance Z 22= R f+ R. The result is the same as (B. 1 except R2 replaces R,) X replaces X,_ and w0 replaces w10 B.3 LSUC2 This circuit is obtained when the diode is mounted in a tee junction with one end in the signal port and the other between the, pump and lower sideband ports. The input impedance Z of the circuit of Fig.- B.2 at terminals A- B is found below. S L R 0 s z 5 S z z Fig.B. 2 Equvalet cicuitwhe ZPsi eiswt Z iI

249 [(jX + Rt) + jXt IIX ] (P) t ( t p p-s w1 R + R P- S ) z R t + j L + (B1 X - 12)~ ) + j '( - - -~~ P R + 2+ X Xll Rt+XIX (X +X II X - )p p p-s \21 ) P LRt2+k(Xt +XpI p x w- R) ( 1 The ipedane Z, is found by setting Zp-s(w20) Ri + jX1i +~~~~ ~~~~~~~~~~~~ jS( ( 0 ) ~[jX5 + jXp 11(R + iXi)] 'jj2 z = I+ S a- R - a Ri + + Xl i) x x R) p s 1 1 p ~~1 R.X2 X R.2X X.(X +X.) ___ __ 1 jp1 pi1p 1 1 p 1 1 p 1 -a Rt2~[a + jX + XR + Xp) P p- s(W i i ' 2 22 Rs J ( 2L- 2 S R~j Xs + jXp (jXi+ R.)- PI jA j(Ri + JXi R~~ R+ (Xp+Xi a a R. +jX 2 R Xp2 XpR i2 + X/xi (Xp+ Xi) R~ ~~~ + (X X i R1 (p Xi " a a R.2+ (X +X 2 R 2 + X+.2 P + -iSn

250 s 2 Ra z22 R, s \ 2 jW L R2+ PX +X) a s co2 a S S S R _P +Pp,cr\Irr P + X Ra co U 2 oa -'2 a a w2 ~~~~~25ct\5W2 'I ~~~(B.14) R 2 + - P+X and for k > 2 Rs s+ - x ~ 0 s P- Xpk~~ Z = R +j IL- - (B15) kk s kw S k ~~~~~p ~) ~ x +x IIx J s p p-s Wk For the simpler diode model when C =0, the above equations apply by simply p taking the limit as S- oc. The results of doing this are shown below. p~~~~~~~ Z1 =R ~Rt+j(X - wL +xnx 1 (B. 16) R.X 2 [S0 R.2X ~ X X(X + X.) = R + ~~ + j X + w L — + 1p p1p 1 (B.17) 22 s R. 2 (X +X.)2 [ 2 W2 R.2 +(X +X.)2 I 1 p 1 1 p 1 kk s k ~ - SO/Wk+ X ~ X liX ) k> 2 (B.18) kk s k 0 k s p P- s In all cases XS X and X p are evaluated at the frequency of interest. For example in the Z term, X = W and X =X (w) The impedance, Zt is found by 11 p X-cv1) S p-s 1 setting Z R + R and solving for Z~ (Fig.- B. 3) at w10 g1 5 0

251 L R 0 ~~~s B? ~~~~~~~S.x xp Splo Fig. B.3. Circuit used for the determination of the signal impedance Zt / 0o -jSp Z R+R =s Rs L + i + + ' + Xp 11 g 5 5 \10 ~wj (Zt iXpIX5 1 16 1 W _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ i Sco )+ j l 10 0 z + jxilx R IwL- co t i r5 P g 10 1 p-s - [Rg j(w10L - 0/wO10)] 1 p S F/s\ p IR -iwL -0 w I 0~).10 Lc \1 I (Sp + S0 )/w 10 - w0L - jRg Rg P) ~Rg+ (0L - ~5P(W1L -~ Rg2 + [w 10 L - (S+ Sp)/Wo10] 2 + Rg92+ [w10L - (o+ Sp)/w 10]12 - x IIxp-s (B.19) Similarly Z.i is f ound by setting Z2Rf + R sand solving for Z. (Fig. B.4) at W 20 Ow -w 10'

252 S L R 0 s S~ t h ~~gH jX5 ~~~~SjX p ~~~~p B.. lo. Fig. B.4. Circuit used for the determination of the lower sideband impedance Z. 1 20 R = HR + i ( pL - 20 f ~~s o s 2 20! 20 if ~0 +S A S0 X ~ =wL-L p and X = wL - ~b 20 c 20 Wt 1 20_ 1 R&.-jX ixp js +xp1 Zi S jX lIZ.= -jX co 20 R- f Xjx Z. -jX j 1 p p- (R, -jX)jX (R -jX) W f ~c 5 S 1 j Xp(Rf jXb)~ijW2 (R f -jXc )iXs(R2 jX b) Sx~~~~~~~ 20- /. + ixP(Xi- cox

253 Now if M is defined as the magnitude squared of the denominator, pX c 2 S p 2 ~~~~~~p s Md X (X +X )]+R 2 X - X21) = [ 20 b p s P 0 P s then 2 X 2 X ~2(~5) (Xp - ) + xP (xsxb - ' 2)(0 Xc) Xb) 2x) Md Agan fr te dodemodel with Cp = 0, the impedances at the signal adlwrsdbn pors a thir espctive pass-band frequencies are obtained by letting c h e sult s arehowXbeows. p =2g 20wlo~ ) (BM24 R X2 1) R~(x sx Xp( L-S) Md = -X~~~~~~~~~~~ / ~~~~~~(B. 26) Aanfrthe eqivalen circuith when0 the diodedismonted in the lowerl sidloebadprt (Fig. p ~ ~ ~ - chrts ang thed. respective pass-band frequencies are obtained by lettin sults are shWn.eo

254 S L R 0 s! Ah - S Z z P P X B? Fig B 5. Equivalent circuit when diode is in the lower sideband port The difference in these two situations is not evident from a comparison of the two circuits in Figs B 2 and B 5 since now the tee junction is assumed to be nonideal In Fig. B. 5 at all frequencies above the signal frequency Z is an open circuit. s Making use of the similarities of these two circuits Z is given by the 11 term in (B. 14) of the previous section except wvI replaces w 2, X P-sreplaces X Si and Z t replaces Z1.- Also, Z 22 is given by the Z 11 expression (B. 13) of the previous section except w 2 replaces wl1, X Preplaces X P IXp sand Z.i replaces Z For higher order sidebands (k > 2): S (X +X) __ P- s P (wL -kOwk J(.7 The values for Z and Z. can also be found from the previous section by the t 1 appropriate substitutions. The impedance Z t is given by (B. 22) and (B. 23) whenco1 replaces w 20, xp~ replaces X 5, and R replaces R,, while Z.i is given by (B.-19) when w2 replaces w10 X~ replaces X~ 1ix~ and R~ replaces Rg

APPENDIX C CALCULATION OF K (z) BESSEL FUNCTIONS WITH COMPLEX ARGUMENT The general method for calculating the K (z) Bessel function is first to find K0(z) and Kl(z) and use the recurrence formulas to find K (z). Values for the Bessel j) functions when I z I < 1 can be found by straightforward substitution into the power series expansion. When I zI > 1, an expansion of the Bessel function in Chebyshev polynomials is needed to insure reasonably fast convergence, although care must be taken when 1 < Izl < 5 and 170~ < arg z < 190 (i.e., near negative real axis for not large z) since the Chebyshev approximation is poor in this region. Following is a detailed description of the method of economization of power series due to Lanczos (Refs. 44 and 87), and its specific application to K (z) The Chebyshev polynomials are a special case of the ultraspherical polynomials with the important property that an expansion of a function into Chebyshev polynomials has the smallest error of any of the other ultraspherical polynomials. Expansion by Taylor series yields the slowest convergence while expansion in the Chebyshev functions, Tn(x), n yields the fastest convergence. The Chebyshev polynomial of the first kind is T (x) = cos (n arccos x) n where ITn(x)l < 1 for x in the range [-1,1]. However for many problems a more appropriate range for the argument is [0, 1]. One example where this is useful occurs when a function is evaluated for large values of x by taking powers of the reciprocal of x. This will be the case for the evaluation of K (z) for large argument. The new shifted Chebyshev polynomial is Tn(x) = Tn(2X-1) n n The normalization used by Macdonald is assumed here. 255

256 which will be used for the remainder of this discussion. In addition the range of x may be normalized to x/T to extend the polynomials over the range [0,4] The formal solution of a differential equation about an ordinary point of the equation is = kxk. u = a x (C.-1) k=Ok This formal solution is substituted into the given differential equation to give in general an m1 term recurrence relation: 1 2 m+1 Aka + A2 Am+. a = 0. k k k+1 k+1 + k+m k+m However, the final numerical solution of the equation will involve a finite number of terms, so a finite expansion, un(x), will be used which approximates u. n k u bx.C 3) n k=O If un is substituted back into the given differential equation, there will be m more equations than unknowns. A sacrifice in accuracy must be made somewhere, and this can be accomplished best by putting an appropriate error term on the right side of the differential equation. One procedure is to use some fraction of the next term in the power series c (x) = TX1 (C.4) e as the error term, which when m=1, makes the power series in b k uniquely determinate. However any polynomial of order n+1 will also work as an error term, and using e (x) n TT1(;~-) 0<<'(C 5) the error will be distributed more evenly throughout the interval. If the coefficients of thie shifted polynomials are T*() then n n T*(x) = ~ ~ T(k) xk.(C6 n Y k= n

257 Suppose the given differential equation is ~u = ~-T*n +) where J? is the differential operator, and the recurrence formula for the finite term solution is T*(k+l) 1 2~~~~A~ _ nTT AIb +A2 b + + AM+Ib n+ 1 -.( k + k+ lbk+1 + k+m k+m k+m (kt81 If the degree of approximation n is changed, the coefficients of T* (x) are also changed, n+ 1 and for successive recursions, an entirely new set of coefficients are needed for each n To avoid having to recalculate all the coefficients when n is changed, a fixed set of canonical polynomials, Q.(x), are found based on the solution of?u = xj+1 j = 0,1,2,... n.(C.9) These polynomials are generated by replacing the right side of (C.8) by k the Kronecker delta function. _1A+b (C.1O) = k~i k~i kj These equations are solved by successive recurrences starting with the last term b. 1 (since all b's with higher subscript are zero) and working backward toward b 0. The se bk coefficients define the canonical polynomial Q.(x) = b0 + b x b x 2 + ~ 0 1 2 which is the approximate sohition of fu = Tobtithsluono

258 the linear superposition principle is used. The solution is n T * (k+ 1) n+lI u (x) = T k+ Qk (C.12) k=O 4,k+1 k(X) Still needed are the m initial conditions to make the problem determinate One term is not enough, so an error term of the form = TTn+l(x/) + T2Tn+2(x/I) + 'm 'n+m is needed which gives the final solution UT~ (x) = Yn(X) + yn(x) + + y (X). n+ m- 1 Yn+l 1 n+m The undetermined T factors are found by satisfying the m initial conditions For many physical problems m =1 as will be found for the case K (z), and a two term recurrence relation is obtained. In this special case U. (x Q.(x) I (C.13) A la. where U.i(x) is the partial sum k U (x) = ax The proof of this follows. The infinite series solution gives the two term recurrence relation Aa +A 2a =0 k =O,11, 2. (C.14) kak +k+ lk+ 1 If a0 ~1,the ak s are

259 At 0 a 1 2 A1 A1A11 01 a2 = 2 2 2AA 1 2 A1...A1 a 1)n 0 n-1 n — 1 2 A2 1" n For the corresponding truncated solution, the recurrence relation 1 2 A1b + A2 b = k k ok k+1 k+1 knt yields the following polynomial coefficients. bn 1 nA n A 2 n-i AIA1 n n-i b n n- I n- 2 = AIAI A'I n n- I n- 2 2 2 b = 1)n n 1 0 An..A0 Using these factors, the expression (C.i13) is proved.

260 1 1 A = Wnx QED If he eigt fctos are defined as ~~~~~~A0 AA"A 0* 2x(k+ ()1) A 22 '' 2 2 n~~n w A2 _ +1... Ak+Iax k+. If he eigt fctos are defined as T* (k+ 1) n+ — 1 Wk= k+l A2 l a k k~ k~ ' -T* (k(i ) 0 n+ 1 n+1 1 ~~(k=0,... n)(C16 iezrt ewu C'cerzu reccwre ace re ~ki O a 9( o6 a fnec( (r orn (C. f I' (C. VI 5 and (C. 16). n u~ (x = T Z WkUk (x) If the given differential equation is homogeneous, T can be found by using the boundary condition, u(0) 1 I, which implies Uk(0) =1 n un(O) =1 T( wk For an inhomogeneous equation, where the right side is 1 and a 1 b 1+ = T1-O Tw 0 ~n+i Uk(0) =a0 k 0~~k=

261 T n Wo'+ Wk k=O Therefore the solution of the differential equation with the two term recurrence formula is n Z wkUk(x) u (x) k=O (C. 17) n n I w0 + ~ wk k=O where w 0 for the homogeneous case. When the solution of an equation is complex, the above theory must be modified The parameter 'I' in this case can be adjusted to make possible approximations to arbitrary complex values of z. However, now instead of having a polynomial in x, rational functions of z are obtained. If the given differential equation is solved along the ray [0, z], then can be set to z, and the Chebyshev polynomials of the form T*(z,) can be used. To avoid n z stretching the range of convergence further than necessary and thereby increasing the error, set z ='' Then z' can assume any complex value in which u(z) is analytic along the ray that connects the origin to z, and the range parameter 'I' is absorbed in z. Thus the modified solution when m = 1 is k1 wk Uk(Z) z — u(z) n= (C.18) 0k=0 where here wk T T(l)/(1 Since most asymptotic series are expanded in descendk n+1 /(kak ing powers, it is often more convenient to give the solution as 1 k=Owk kk1 (4 un(i ) k=n kz(C.19) n z +n3wk

262 The evaluation of K (z) is now done by using the above theory An asymptotic form of the Bessel function is Kz- -z 1 12(Z) = - e u(), which when substituted into the modified Bessel equation gives z2 du 2z2 du - (2 + 1/4) u - 0. dz2 dz If zr /z, then this equation is transformed into 2 d2u du z + 2(1~d - (92 - 1/4) u = 0 r dz2 r dzr r The recurrence formula for this last equation is [(k+1/2)2 - v2 ] ak + 2(k+1 ) ak+ 1 = 20) 1 and the resulting solution for u ()can be written as z 1 2~~v~ u(-) = k k( iarg z1 — ) (C.21) Z k=o rvk 2kk! zk The weight factor then becomes T* (k~1) n+1 wk~~~+ T*(k+l)ak! k1 k /2 n-f1 Pr(3/2 +12~k) 'o 0 and the solution for the nth appiroximation is found from (C. 19). The numerator of (C. 19), desgntedbyM a' be expressed as polynomial in z. a k=1 k=0~a

263 M (zw0 a + z2w(a0 + alz ) -1+ z w2(a + a 2z Z-) +... 1 ~~-n) nZ ~1 +... +a z+z n (0 ~ an M ~z~(wa ~wa '' +. wa ) + z2(w a + a. w a +... + w a a 0 0 1 0 nf 10 2 n ni nThe series in cis therefore n 1 k=O kwk n-i 2 k=O k k+l n-m or n- m+i c akw 1 1<m<n~1l (C. 22) = ~kWk+l- k=O The final solution for the K Bessel function is therefore given by 3 Ck Z K (z) =ej.e - (C. 23) This was evaluated for n=12, v0 1weevlsfoT13 were obtained from Ref.- 87. The coefficients of the two polynomials in (C.- 23) are listed below.

264 0~k ~ Ck Wk 0 0 -2.7040000 000000 13 1 -9.3444970 446304 102 -1.3459911 111111 10 2 -7. 4419709 831272 104 -1.8951554 844444 6 6 7 3 -1.3370803 316846 10 -1.060850 525170 10 4 -8. 5699270 111211 106 -2.8498751 040457 10 5 -2.4927644 878595 107 -4.1110144 250697 7 6 -3.7746864 595237 107 -3.4130588 581370 17 7 -3.2310444 731793 107 -1.6989448 538282 17 8 -1.6404092 135688 107 -5.1640361 931952 6 9 -5.0516267 780907 10 -9.5405549 847164 10 5 10 -9.4154163 240077 10 -1.0339323 951985 10 11 -1.0265536 957496 10 -5.9824805 311068 13 12 -5.9648092 039995 1 3 -1.4137061 685815 2 13 -1.4137061 685815 10 2 0 V=1 0 0 +9.0133333 333333 1o2 1 +9.1918251 276379 tO3 +8.0759466 666667 tO4 2 +3.1557018 013485 10 5 +1.3536824 888889 to6 3 +3.2568245 707891 10 +8A.2 510170 7 513 23 to6 4 +1.4642396 054591 10 7 +2.3317159 942192 tO7 77 5 +3.3952020 559774 10 +3.4785506 673666 tO7 77 6 +4.4493275 831754 10 +2.9579843 437188 tO7 7 7 7 +3.4736230 524404 10 +1.4990689 886720 10 8 +1.6631127 132600 10 +4.-6204534 360167 106 9 +4.933633 5 295110 10 6 +8.6319307 004577 105 10 +8.9796245 223542 10 5 +9.4402523 039865 tO4 11 +9.6451139 129428 tO 4 +5.5038820 886182 tO3 12 +5.5529691 083607 10 3 +1.3089871 931311 to2 13 +1.3089871 931311 10 2 0

APPENDIX D LAPLACE ASYMPTOTIC METHOD OF INTEGRATION Numerical integration of (5. 20) and (5.21) by means of a Gaussian quadrature formula is described in Section 5.4.2. A much faster integration method, though less accurate, is the Laplace asymptotic method which is really a special case of the saddle-point method. This method is well suited for this kind of integral and has been found to be approximately 15 times faster than the Gaussian quadrature method. If the integral to be solved is 30 F(t) = f u(x) eth( dx large t > 0 (D1) -acc the solution is approximately (Ref. 88, p. 65) U(Xa) th(xmax ) j- th" (x ) maxma where x is the value of x where h(x) is maximum, i.e., h'(x m 0 and max max h(x ) < 0 This is valid when there is only one maximum of h(x), so to apply this max technique to (5.20) and (5. 24), the contributions from each maximum in the integration interval have to be added together. If 0 = wt are the values of wot where V(t) = h(t) are maximum, then the solutions to (5. 20) and (5. 21) are qDppn(W) V"I J c- = _ _ _ w kc frw exp)a[V b + V0 Cos( i2) V1 Cos( i+~) + V2 Cosoi~42)1 max V(9) J V( )Cos ( 0o 79) +Cos 5+ + VCos 5 2J [c W mkcos fmfO)+2mk sifm j(D 3)2

266 qDppn( W) and ~~and ~~Jks = 7rW expL [vb + VO cos ( i)+ V1 cos ( +i + V2 cos 2 max V(O) 2) + V1 Cos + 2 s 2 L 0O ~ g o fO 1 ~ f: w2 ff 2 [a sin(mk ) - cos(mkO)] (D.4) m si ~k0) mkc ~k9) The results of using thebabove formulas are shown in Table D. 1 for both the abrupt and graded junctions for various voltage levels in the diode. The calculations show fairly close agreement with the Gaussian quadrature method summarized in Table 5.2 up to about J6' 3f - fs and at the highest frequency calculated, 4fp +fs, this method gives currents less than twice the correct value. If V1 = V2 =0, all currents for k ' 1 should be zero, whereas the Laplace asymptotic method gives values of I Jkl as high as 42 A/cm2.

267 Abrupt Junction Graded Junction V1 = 0.01 k JkcA/cm JkA/cm2 IJk I A/cm2 JkcA/cm2 JksACM2 j IA/cm2 0 680.32 -690.89 969.62 619.19 -684.37 922.91 1 218.41 -15.87 218.98 188.25 -27.26 19022 2 43.22 -681.44 695. 01 -12.87 -64742 647.54 3 655.30 -55.20 673.35 624.06 -5372 626.3 4 56.72 -1052.20 1053.73 -17.10 -101440 1014.50 5 829.30 -79.70 833.11 799.67 -7040 802.76 6 61.45 -1369.00 1370.38 -27.37 -1327.60 1327.90 7 913.70 -99.79 919.13 886.20 -85.66 890.3 8 60.13 -1658.60 1659.69 -42.32 -1613.80 161440 9 941.96 -116.72 949.16 916.70 -99.77 922.20 75.95 -275.96 390.26 252.4001 1 V =0. 01 2 0 275. 95 -275. 96 390. 26 2 52. 40 -274. 50 372. 90 1 12. 96 - 0.88 12. 96 11. 20 -1.57 11.31 2 2. 58 -40. 50 41. 32 - 0. 76 -38. 57 38. 58 3 88. 98 - 2. 92 40. 06 37. 22 - 2.84 3 7.33 4 3. 71 62. 89 63. 00 - 0. 70 - 60.80 60. 80 5 49.64 -4. 06 51. 35 47. 98 - 3.51 48.10 6 4. 55 - 82.61 82. 74 - 0.81 -80.37 80.38 7 55. 22 -4. 97 55.4 5 53.-67 - 4.14 53. 83 8 5. 23 -101. 50 101. 59 -1. 04 - 99. 05 99. 05 9 57. 67 5. 77 57. 95 56. 25 - 4. 75 56.4 5 V1 = 0. 0001 V 2 = -0.001 0 272. 88 -272. 83 38 5.86 249. 60 -271. 40 3 68. 70 1 1.29 - 0.09 1.29 1.11 - 0.16 1.13 2 0.26 - 4. 03 4.04 - 0.08 -3. 84 3. 84 3 3.-88 - 0. 29 3. 88 3. 70 - 0.28 3.71 4 0.37 - 6.2 5 6.3 5 - 0. 07 -6. 04 6. 05 5 4. 94 - 0.41 4. 95 4. 77 - 0.3 5 4. 78 6 0.46 - 8.23 8.24 - 0. 08 - 8. 01 8. 01 7 5.49 - 0. 50 5. 53 5.33 - 0.40 5. 34 8 0. 52 -10. 09 10.10 - 0. 09 - 9. 87 9.87 9 5. 74 - 0. 57 5. 77 5. 61 - 0.47 5. 62 Table D. 1. Laplace asymptotic method for calculating the current densities

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269 REFERENCES (Cont.) 18. R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1966. 19. W. A. Davis, "Coaxial Microwave Bandpass Filters," (Technical Memorandum No. 100), ECOM-0318-5, prepared for USAEL, Fort Monmouth, New Jersey, Contract No. DAAB 07-68-C-0138, Cooley Electronics Laboratory, The University of Michigan, Ann Arbor, May 1969. 0. S. B. Cohn, "Direct Coupled Resonator Filters," Proc. IRE, Vol. 45, February 1957, pp. 187-196. 1. S. B. Cohn, "Principles of Transmission Line Filter Design," Very High Frequency Techniques. Vol. II, New York: McGraw-Hill, 1947, Ch. 26. 2. J. R. Whinnery and H. W. Jamieson, "Equivalent Circuits for Discontinuities Proc. IRE, Vol. 32, February 1944, pp. 98-114. 3. J. R. Whinnery, H. W. Jamieson and T. E. Robbins, "Coaxial-line Discontinuities, Proc. IRE, Vol. 32, November 1944, pp. 695-709. 24. P. I. Somlo, "The Computation of Coaxial Line Step Capacitances," IEEE Trans. Microwave Theory Tech., Vol. MTT-15, January 1967, pp. 48-53. 5. S. B. Cohn, "Design of Transmission-line Filters," Very High Frequency Techniques, Vol. II, New York: McGraw-Hill, 1947. 26. G. L. Matthaei, "Short-step Chebyshev Impedance Transformers," IEEE Trans. Microwave Theory Tech., Vol. MTT-14, August 1966, pp. 372-383. 7. R. Levy and T. E. Rozzi, "Precise Design of Coaxial Low-pass Filters," IEEE Trans. Microwave Theory Tech., Vol. MTT-16, March 1968, pp. 142-147. 28. S. B. Cohn, "Confusion and Misconceptions in Microwave Engineering," Microwave Journal, Vol. 11, September 1968, p. 20. 9. B. D. 0. Anderson, "When do the Manley-Rowe Relations Really Hold?, Proc. EE Vol. 113, April 1966, pp. 585-587. 30. A. P. Bolle, "Application of Complex Symbolism to Linear Variable Networks," IRE Trans. on Circuit Theory, Vol. CT-2, March 1955, pp. 32-35. 31. D. B. Leeson, "Capacitance and Charge Coefficients for Varactor Diodes," Proc. IRE, Vol. 50, August 1962, p. 1854. 32. H. A. Haus and R. B. Adler, "An Extension of the Noise Figure Definition, " Proc. IRE, Vol. 45, May 1957, pp. 690-691. 33. P. J. Khan, "Parametric Amplifier Nonresonant Gain Maximum, " Proc. IEEE, Vol. 56, January 1968, pp. 99-100. 34. P. J. Khan, "Determination of Parametric-Amplifier Nonresonant Gain Maximum, IEEE J. Solid State Circuits, Vol. SC-5, April 1970, pp. 79-81.

270 REFERENCES (Cont.) 37 A Uhlir, "High-Frequency Shot Noise in P-N Junctions," Proc. IRE, Vol 44, April 1956, pp. 557-558. 38. A. Uhlir, "Shot Noise in p-n Junction Frequency Converters," Bell System Technical Journal, Vol. 37, July 1958, pp. 951-988. 39. J. G. Josenhans, "Forward Bias Shot Noise in Varactor Diodes," in Microwave Diode Research Report No. 16, Seventh Quarterly Progress Report, Contract DA 36- 039 SC-89205 Bell Telephone Laboratory, 1964. 40. K. Kurokawa, "Actual Noise Measure of Linear Amplifiers," Proc. IRE, Vol. 49, September 1961, pp. 1391-1397. 41. M. Uenohara "Noise Consideration of the Variable Capacitance Parametric Amplifier, Proc. IRE, Vol. 48, February 1960, pp. 169-179. 42. K. Siegel, "Anomalous Reverse Current in Varactor Diodes," Proc. IRE, Vol. 48, June 1960, pp. 1159-1160. 43. J. Lindmayer and C. Wrigley, "A New Aspect of the Semiconductor Diode Journal of Electronics and Control, Vol. 14, March 1963, pp. 289-301. 44. C Lanczos, Applied Analysis. Englewood Cliffs, N. J.: Prentice Hall, 1956. 45. B. S. Perlman and B. B. Bossard, "Efficient High Level Parametric Frequency Converters," 1963 IEEE International Convention Record Part 3 - Electron Devices; Microwave Theory and Techniques, March 25-28, 1963. 46. G. L. Matthaei, "Tables of Chebyshev Impedance-Transforming Networks of Low-pass Filter Form, " Proc. IRE, Vol. 52, August 1964, pp. 939-963. 47. J.- C.- Irvin, T.- P.- Lee, and D.- R.- Decker, "Varactor Diodes, " Microwave Semiconductor Devices and Their Circuit Applications, ed. by H. A. Watson, New York: McGraw-Hill, 1969, p. 159. 48. P.- Penf ield, Jr.- and R.- P.- Rafuse, Varactor Applications.- Cambridge, Massachusetts: M.I.T. Press, 1962. 49. D. A.E. Roberts and K. Wilson, "Evaluation of High Quality Varactor Diodes, " The Radio and Electronic Engineer, Vol.- 31, May 1965, pp.- 277- 285. 50. D.- E. Crook, "A Simplif ied Technique f or Measuring High Quality Varactor Parameters," Solid State Design, VoL. 6, August 1965, pp. 31- 33. 51. N. Houlding, "Measurements of Varactor Quality, " Microwave Journal, Vol.- 3, January 1960, pp. 40-45. 52. R. Harrison, "Parametric Diode Q Measurements, Microwave Journal, Vol. 3, May 1960, pp. 43-46. 53. R. Mavaddat, "Varactor Diode Q-factor Measurements, " Journal of Electronics and Control, Vol. 15, July 1963, pp. 51-54.

271 REFERENCES (Cont. ) 6. B. C. DeLoach, "A New Microwave Measurement Technique to Characterize Diodes and an 800 Gc Cutoff Frequency Varactor at Zero Volts Bias," IEEE Trans. Microwave Theory Tech., Vol. MTT-12, January 1964, pp. 15-20. 7. N. Houlding, "Varactor Measurements and Equivalent Circuits," IEEE Trans. Microwave Theory Tech., Vol. MTT-13, November 1965, pp. 872-873. 58. D. A. E. Roberts, "Measurements of Varactor Diode Impedance," IEEE Trans Microwave Theory Tech., Vol. MTT-12, July 1964, pp. 471-475. 59. J. W. Bandler, "Precision Microwave Measurement of the Internal Parasitics o Tunnel-Diodes," IEEE Trans. on Electron Devices, Vol. ED-15, May 1968, pp. 275 -282. 60. R. J. Wenzel, "Compact Multiplexing Networks for L-band through Ku-band," Digest of the 1970 G-MTT International Symposium, May 11-14, pp. 85-89. 61. R. W. P. King, Transmission-line Theory. New York: Dover Publications, Inc. 1955. 2. 59 IRE 20. S1, "IRE Standards on Methods of Measuring Noise in Linear Two Ports, 1959," Proc. IRE, Vol. 48, January 1960, pp. 60-68. 3. R. D. Lending, "New Criteria for Microwave Component Surfaces," Proc. of th National Electronics Conference, Vol. 11, 1955, pp. 391-401. 4. R. J. Josenhans, "Noise Spectra of Read Diode and Gunn Oscillators," Proc. IEEE, Vol. 54, October 1966, pp. 1478-1479. 5. W. M. Gray, L. Kikushima, N. P. Morenc, and R. J. Wagner, "Applying IMPATT Power Sources to Modern Microwave Systems," IEEE J. Solid State Circuits, Vol. SC-4, December 1969, pp. 409-413. 6. K. Garbrecht, "Lower Limit of Paramp Noise due to Pump Heating," Digest of Technical Papers: 1965 International Solid-State Circuits Conference, 1965, pp. 22-23. 7. J. A. Morrison, "Maximization of the Fundamental Power in Nonlinear Capacitance Diodes," Bell System Technical Journal, Vol. 41, March 1962, pp. 677-721. 68. J. A. Davis, "The Forward-driven Varactor Frequency Doubler," M.S. Thesis, Dept. of Electrical Engineering, M.I.T., Cambridge, Massachusetts, May 1963. 9. C. B. Burckhardt, "Analysis of Varactor Frequency Multipliers for Arbitrary Capacitance Variation and Drive Level," Bell System Technical Journal, Vol. 44, April 1965, pp. 675-692. 70. J. I. Smith, "Practical Analysis of the Overdriven Varactor Multiplier with a Single Idler," Proc. IEEE, Vol. 55, April 1967, pp. 575-576. 71. W. Conning, "An Analysis of the High-power Varactor Upconverter, Proc. IREE Australia, Vol. 27, July 1966, pp. 163-173. 72. A. I. Grayzel, "The Overdriven Varactor Upper Sideband Upconverter," IEEE rn Microwave Theory Tech., Vol. MTT-15, October 1967, pp. 561-565. 3. J. W. Gewartowski and R. H. Minetti, "Large-Signal Calculations for the Overd Varactor Upper-Sideband Upconverter Operating at Maximum Power Output, " Be System Technical Journal, Vol. 46, July-August 1967, pp. 1223-1242.

272 REFERENCES (Cont.) 74." C. E. Nelson, A Note on the Large-Signal Varactor Upper-Sideband Upconverter, Proc. IEEE, Vol. 54, July 1966, pp. 1013-1014. 75. C E Nelson, "The Theoretical Approximate Maximum Output Power of Varactor Upper-sideband Upconverters," General Electric Co., Syracuse, NY, Internal iReport TIS R66ELS-46, June 1966. 76. J. L. Moll, Physics of Semiconductors. New York: McGraw-Hill, 1964. 77.. M. Sze, Physics of Semiconductor Devices. New York: Wiley-Interscience, 1969. 78. T. P. Lee, p-n Junction Theory," Microwave Semiconductor Devices and Their Circuit Applications, ed. H. A. Watson. New York: McGraw-Hill, 1969, Ch. 5. 79. V. A. Shpirt, "The Properties of p-n Junctions Under the Influence of a Sinusoidal Voltage of Arbitrary Amplitude (Low Injection Level)," Radio Engineering and Electronic Physics, Vol. 11, December 1966, pp. 1947-1953. 80. V. N. Parygin and N. K. M~neshin, "Frequency Multiplication Using the Diffusion Capacitance," Radio Engineering and Electronic Physics, Vol. 11, July 1966, pp. 1111-1118. 81. R. M. Romanova, "Equivalent Circuit of a Turned-on Varactor, " Radio Engineering, Vol. 23, October 1968, pp. 111-115. 82. C. T. Sah, "Effects of Electrons and Holes on the Transition Layer Characteristics of Linearly Graded P- N Junctions, " Proc.- IRE, Vol.- 49, March 19 61, pp.- 603- 618. 83. C. T.- Sah, "Effect of Surface Recombination and Channel on P- N Junction and Transistor Characteristics, " IRE Trans.- Electron Devices, Vol.- ED- 9, January 1962, pp. 94-108. 84. C.- C. Wang, "Space-Charge- Layer Width and Capacitance of Symmetrical Step Junctions," Proc. IRE, Vol. 50, August 1962, pp. 1838-1839. 85. W. F. O'Hearn and Y. F. Chang, "An Analysis of the Frequency Dependence of the Capacitance of Abrupt P-N Junction Semiconductor Devices, " Solid-State Electronics, Vol. 13, April 1970, pp. 473-483. 86. D.- B.- Anderson and J.- C.- Aukland, "Transmission- Phase Relations of Four-Frequency' Parametric Devices," 1-IRE Trans.- Microwave Theory Tech., Vol.- MTT- 9, November 1961, pp. 491-498. 87. C.- Lanczos, Tables of Chebyshev Polynomials Sn_(x) and C (x) Washington, D.C.: U.S. Department of Commerce, National Bureau of Standards,7T-52, pp. V-XXIX. 88. N. G. DeBruijn, Asymptotic Methods in Analysis. New York: Wiley- Interscience, 1961.

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UNCLASSIFIED Set c urity Classific(ation.... i{ i -. i ~ Ii DOCUMENT CONTROL DATA- R & D N'crttv ctzs..ifir,,tioti of titlt', hody of abstract and indexing annotation must be entered when the overall report i c;ified) O. VF` IGINA TING ACT I VI TY (Corporate author) 2a. REPORT SECURITY CLASSICATION Cooley Electronics Laboratory UNCLASSIFIED University of Michigan 2b. GROUP Ann Arbor, Michigan 48105 3. R PORT I ITLE DESIGN AND ANALYSIS OF A PARAMETRIC LOWER SIDEBAND UPCONVERTER 4. DESCRIPTIVE NOTES (7)'pe of report and.inclusive dates) Cooley Electronics Laboratory Technical Report No. 211, September 1971 5 AU THOR(S) (First name, middle initial, last name) Wendell A. Davis 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS September 1971 303 8a. CONTRACT OR GRANT NO. 98. ORIGINATORS REPORT NUMBER(S) DAAB 07- 68- C - 0138 014820-26-T TR211 b. PROJECT NO. 1H021101 A042.01.02 C. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) d. ECOM-0138-26- T 10. DISTRIBUTION STATEMENT Each transmittal of this document outside the agencies of the U.S Government must have prior approval of CG, U.S. Army Electronics Command, Fort Monmouth, N.J., Attn: AMSEL-WL-S. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U.S. Army Electronics Command Fort Monmouth, New Jersey 07703 Attn: AMSEL-WL-S 13. ABSTRACT Most analyses of lower sideband upconverters (LSUC) or other parametric devices consider the circuit to consist of ideal filters and impedance transformers. In the analysis of the varactor diode used in the LSUC, it is usually assumed to be a reverse biased graded or abrupt pn junction. However, in designing a LSUC, microwave circuit elements must be employed which are not ideal, and furthermore the varactor diode is often biased so that during part of the pump cycle it is in the forward conduction region. Here a noniterative design technique for a LSUC is f ormulated using readily analyzable microwave filters.- In addition the analysis of the varactor in the LSUC is extended to include forward bias effects.- The LSUC consists of a coaxial tee junction where each port separately carries the signal, pump, and lower sideband power.- The impedance matching and filtering functions required in the amplifier are performed by coaxial band-pass impedance transformers. A new synthesis technique is developedi for these filters and has been used to design filters with center frequencies from 1 to 20 GHz and bandwidths from 1 to 20 percent of the center frequency. These coaxial impedance transformers and the varactor diode

UNCLASSIFIED Se 'lrit' Classification 14 LINK A LINK B LINK C KEY WORDS ROLE WT ROLE WT ROLE WT Lower Sideband Upconverter Varactor Diode Forward Bias Effects Band-pass Filter Microwaves Diffusion Admittance Parametric ~Parametric THE UNIVERSITY OF MICHIGAN DATE DUE _ UiVERSITYr WIF8 MCIAN

UNCLASSIFIED Set c urity Classific(ation.... i{ i -. i ~ Ii DOCUMENT CONTROL DATA- R & D N'crttv ctzs..ifir,,tioti of titlt', hody of abstract and indexing annotation must be entered when the overall report i c;ified) O. VF` IGINA TING ACT I VI TY (Corporate author) 2a. REPORT SECURITY CLASSICATION Cooley Electronics Laboratory UNCLASSIFIED University of Michigan 2b. GROUP Ann Arbor, Michigan 48105 3. R PORT I ITLE DESIGN AND ANALYSIS OF A PARAMETRIC LOWER SIDEBAND UPCONVERTER 4. DESCRIPTIVE NOTES (7)'pe of report and.inclusive dates) Cooley Electronics Laboratory Technical Report No. 211, September 1971 5 AU THOR(S) (First name, middle initial, last name) Wendell A. Davis 6. REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS September 1971 303 8a. CONTRACT OR GRANT NO. 98. ORIGINATORS REPORT NUMBER(S) DAAB 07- 68- C - 0138 014820-26-T TR211 b. PROJECT NO. 1H021101 A042.01.02 C. 9b. OTHER REPORT NO(S) (Any other numbers that may be assigned this report) d. ECOM-0138-26- T 10. DISTRIBUTION STATEMENT Each transmittal of this document outside the agencies of the U.S Government must have prior approval of CG, U.S. Army Electronics Command, Fort Monmouth, N.J., Attn: AMSEL-WL-S. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY U.S. Army Electronics Command Fort Monmouth, New Jersey 07703 Attn: AMSEL-WL-S 13. ABSTRACT Most analyses of lower sideband upconverters (LSUC) or other parametric devices consider the circuit to consist of ideal filters and impedance transformers. In the analysis of the varactor diode used in the LSUC, it is usually assumed to be a reverse biased graded or abrupt pn junction. However, in designing a LSUC, microwave circuit elements must be employed which are not ideal, and furthermore the varactor diode is often biased so that during part of the pump cycle it is in the forward conduction region. Here a noniterative design technique for a LSUC is f ormulated using readily analyzable microwave filters.- In addition the analysis of the varactor in the LSUC is extended to include forward bias effects.- The LSUC consists of a coaxial tee junction where each port separately carries the signal, pump, and lower sideband power.- The impedance matching and filtering functions required in the amplifier are performed by coaxial band-pass impedance transformers. A new synthesis technique is developedi for these filters and has been used to design filters with center frequencies from 1 to 20 GHz and bandwidths from 1 to 20 percent of the center frequency. These coaxial impedance transformers and the varactor diode