NATL :NST of standards & TECH R.I.C. A11 100989945 /NBS monograph „ ^ QC100 .U5^ V137;1973 C.I NBS-PUB-C 1959 NBS MONOGRAPH 131 tare :rowaiie 8 » n 72o >c Applications of Waveguide and Circuit Theory r>o i57to the Development of Accurate Microwave Measurement Methods and Standards R. W. Beatty Institute for Basic Standards National Bureau of Standards Boulder, Colorado 80302 U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Director Issued August 1973 Library of Congress Catalog Number: 73-600158 National Bureau of Standards Monograph 137 Nat. Bur. Stand. (U.S.), Monogr. 137, 322 pages (Aug. 1973) CODEN: NBSMA6 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. C 13.44: 137). Price $5.20. Stock Number 0303-01169 Contents Page Abstract--'^ X Preface Acknowledgments xii 1. Introduction 1 1.1. General 1 1.2. Theory 2 1.3. Applications 4 1.4. Conclusions and References 6 2. Basic Theory of Waveguide Junctions 7 2.1. Definitions 7 a. Waveguide Junctions 7 b. Terminal Surfaces and Terminal Variables 7 2.2. V, i, a and b for Waveguide 8 a. Introduction 8 b. Basic Derivation of v, i, e" and h" 8 c. Complex and Vector Potential Functions 9 d. Power and Impedance Normalization 10 e. Examples of Waveguide v and i 11 (1) TEM-Mode in Coaxial Waveguide 11 (2) ^^10 Rectangular Waveguide 13 f. Traveling Wave Amplitudes a and b 15 g. Other Traveling Wave Amplitudes 15 2.3. Parameter Matrices 16 a. Impedance and Admittance Matrices 16 b. The Scattering Matrix 17 c. Power 17 (1) General -- 17 (2) Realizability Conditions 18 (a) Strict Realizability 18 (b) Semi -realizability 18 (c) Losslessness 19 d. Reciprocity 19 e. Sources; Joining Equations 20 (1) General 20 (2) Sources 20 (3) Joining Equations 21 3. Introductory Network Analysis : 24 3.1. Linear Network Parameters 24 a. Introduction 24 b. Terminal Variables 24 c. Network Parameters 24 d. Complex Wave Amplitudes a and b 25 e. Parameters Associated with a and b 25 iii Contents (Continued) Page f. Other Terminal Variables 26 g. Network Equivalent to a Waveguide Junction 26 3.2. The Scattering Matrix 27 a. General Remarks 2 7 b. Scattering Coefficients of a Two- Arm Waveguide Junction 2 8 c. Effects of Moving Terminal Surfaces 30 3.3. Reciprocity, Realiz ability , and Losslessness for 2-Ports 31 a. Reciprocity 31 b. Strict Realizability 32 c. Losslessness 33 3.4. Power and Efficiency 34 3.5. Representation of the Source 35 a. From Linear Relation for Source and Joining Equations- 35 b. From a Constant Voltage Generator 36 c. Summation of Wave Reflections 37 3.6. Net Power and Available Power 37 a. Net Power to a Waveguide Junction 37 b. Available Power from Generator 38 3.7. Mismatch Loss 39 a. Mismatch Loss in General 39 b. Meaning of Mismatch 39 c. Conjugate Mismatch Loss 40 d. The Zg Mismatch Loss 41 e. Difference Between Conjugate and Mismatch Losses 41 3.8. Transmission Properties of 2-Ports 41 a. Substitution Loss 42 b. Transducer Loss 43 c. Insertion Loss 44 d. Attenuation or Characteristic Insertion Loss 44 e. Components of Losses 45 3.9. Maximum Transmitted Power 47 a. Maximum Efficiency 48 (1) Gradient of Efficiency 48 (2) Maximum Efficiency from Maximum Power Considerations 50 b. Minimum Transducer Loss 51 c. Intrinsic Attenuation 52 3.10. Phase Shift 52 a. Relative Phase 52 b. Shift of Phase by a 2 -Port 52 c. Different Kinds of Phase Shift of 2-Ports 53 (1) Transmission Phase Shift 54 (2) Differential Phase Shift 54 (3) Substitution Phase Shift, Including Insertion Phase Shift-- 54 iv Contents (Continued) Page 3.11. Cascading 2-Ports 54 3.12. Cascading Coefficients 56 3.13. Transformation of Reflection Coefficients 57 a. Simpler Transformations 57 b. The Sliding Termination 5 8 c. Significance of the Radius of the r^-Circle 59 d. Displacement of Center from Origin 60 e. Locus of for Real 60 f. Other Circles- 61 3.14. The Linear Fractional Transformation 62 a. Constant |w|, variable ^, 62 b. Constant ii , variable Iwl 63 c. Invariance of cross-ratio 64 3.15. 3-Ports or Waveguide Junctions Having Three Arms 6 5 a. Introduction 65 b. Realizability Conditions 65 c. Conditions for Losslessness 67 d. Reciprocity 69 e. Symmetry 69 f. 3-Port, One Arm Terminated 70 g. Non-reciprocal 3-Ports 71 h. The Directional Coupler, 3 -Port 71 i. Solution of the Scattering Equations for b^ 73 i. Measurement of Reflection Coefficient V with General 3-Port 76 •' u 3.16. 4-Ports 77 a. The Directional Coupler 77 b. The Magic Tee 78 4. Power 80 4.1. Introduction 80 4.2. Mismatch Errors 80 a. Introduction 80 b. Calibration of Power Meters 81 (1) General Discussion 81 (2) Method I -- Alternate Connection to a Stable Power Source-- 82 (3) Method 2 -- Comparison Using T- Junctions 83 (a) Simultaneous Comparison , 83 (b) Alternate Connection to T-Junction 85 (4) Method 3 -- Comparison Using Magic T 85 (a) Simultaneous Comparison 85 (b) Alternate Connection to Magic T 87 c. Discussion of Calibration Methods-- 89 d. Use of Power Meters 90 (1) General Remarks 90 (2) Direct Measurement 90 V Contents (Continued) Page (3) Use of Calibrated Attenuator - 91 (4) Use of Directional Couplers 92 (a) Temporarily Inserted 92 (b) Permanently Installed 94 e. Measurement of Scattering Coefficients 95 4.3. Barretter Mount Efficiency Measurement 98 a. Introduction 98 b. Impedance Method 100 c. Improved Method 102 d. Discussion of Errors 107 e. Experimental Results 111 f. Conditions for Linear r^-Locus 113 5. Impedance-Reflection Coefficient 115 5.1. Introduction 115 5.2. Adjustable Sliding Termination 116 a. Introduction 116 b. Principle of Operation 117 c. Theory 118 d. Design 120 5.3. Quarter-Wavelength Short- Circuit 123 a. Introduction 123 b. Coaxial Line 124 c. Rectangular Waveguide 129 d. Errors and Design Information 136 5.4. Squared VSWR and Magnified Responses - 137 a. Introduction 137 b. Simplified Explanations 138 (1) Squared VSWR Response 138 (2) Magnified Response 138 c. Analysis 141 d. Means of Obtaining Various Responses 143 e. Measurements Using Squared VSWR Response 145 f. Measurements Using Magnified Response 146 g. Results 149 5.5. The Tuned Reflectome ter 149 a. Three -Arm Waveguide Junctions 149 b. Tuning for Squared VSWR Response 150 c. Tuning for Magnified Response 151 d. Tuning for Infinite Directivity 152 e. Analysis of Errors 152 vi Contents (Continued) Page 5.6. Hybrid Ref lectometer 155 a. Introduction 155 b. Basic Theory- 156 c. Experimental Equipment 15 7 (1) General 15 7 158 (2) Terminating Arrangements (3) Standard Reflections for (Air-Dielectric) Coaxial Line 160 (4) Sliding Loads 161 d. System Performance and Applications 162 e. Results and Discussion of Errors 164 f. Conclusions 166 5.7. Connector Reflections and Losses 166 a. Introduction 166 b. Preliminary Considerations 166 c. Brief Description of Method 169 d. Review of Ref lectometer Techniques 170 e. VSWR Determination 171 f. Efficiency Determination 173 g. Supplemental Techniques in VSWR Determination 177 h. Results 184 6. Attenuation 189 6.1. Introduction 189 6.2. Definitions of Attenuation 190 a. Introduction 190 b. Broad General Meaning 191 c. Restricted Meanings 192 d. IRE Definitions 193 e. Comparisons of Definitions 195 f. Terminology- 20 3 g. Precise Definitions 205 h. Future Trends 20 8 6.3. Cascade -Connected Attenuators 209 a. Introduction 209 b. Analysis 210 (1) Two Attenuators 211 (2) Three Attenuators 212 (3) Any Number (n) of Attenuators 214 6.4. Mismatch Errors in Attenuation Measurements 214 a. Introduction 214 b. Analysis of Case of Fixed Pad 215 c. Evaluation of Error 216 d. Effects of Attenuator Characteristics 218 e. Effect of Realizability Conditions 219 vii Contents (Continued) Page f. Variable Attenuators 219 (1) Introduction 219 (2) Expression for Mismatch Error 219 (3) Evaluation of the Mismatch Errors 221 g. Avoidance of Mismatch Error 22 2 h. Later Work 222 6.5. Effects of Realizability Conditions 222 6.6. Effects of Connectors and Adapters 224 a. Introduction 224 b. Previous Analyses - - 226 c. An Improved Representation 2 27 d. Substitution Loss 228 e. Standard Attenuation 230 f. Connector and Mismatch Errors 231 g. Same Fixed Attenuator in Two Systems 233 h. Adjusting Systems for Zero Reflections 237 i. Same Variable Attenuator in Two Systems 238 j. Standard Incremental Attenuation 242 k. Basic Insertion Arrangements 244 (1) Cases 2A and 3A -- Combining the Component with an Adapter- 244 (2) Cases 2B and 3B -- Substituting the Component for an Adapter 246 (3) Cases 2C and 3C -- Combining the Component with an Adapter and Substituting for Another Adapter 247 1, Conclusions - 248 6.7. Efficiency and Attenuation of 2-Ports from Reflection Coefficient Measurements 251 a. Introduction 251 b. 2-Port Having Terminal Pairs 253 c. 2-Port Having Waveguide Leads 255 d. Conclusions 256 6.8. Attenuation from Power Measurements 257 a. Introduction 25 7 b. The Measurement System 257 c. Theory of Measurement 259 d. Propagation of Error in Measuring d-c Power Differences 261 e. Mismatch Errors 261 f. Results 263 g. Analysis and Evaluation of Mismatch Errors 264 6.9. Two-Channel Nulling Method 266 6.10. Attenuation Divider Circuit - 272 viii Contents (Continued) Page Phase Shift 277 7.1. Introduction 277 7.2. Phase Shift Equations for 2-Ports 277 a. Introduction 277 b. General 278 c. Phase Shift Equations- 2 79 (1) Phase Shift of v 280 (2) Phase Shift of i 280 (3) Phase Difference Between and 281 (4) Phase Difference Between and 281 (5) Differential Phase Shift - 282 (6) Insertion Phase Shift 283 d.