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Microwave and RF Design Transmission Lines

Instrumentation, Electronics & Control Sensing Devices

Microwave and RF Design Transmission Lines
Volume 2
Third Edition
Michael Steer
North Carolina State University
304 Pages

Open: Microwave and RF Design Transmission Lines
Basic Membership Minimum Required

Preface:

This book begins with a chapter on transmission line theory and introduces the concepts of forward- and backward-traveling waves. Many examples are included of advanced techniques for analyzing and designing transmission line networks. This is followed by a chapter on planar transmission lines with microstrip lines primarily used in design examples. Design examples illustrate some of the less quantifiable design decisions that must be made. The next chapter describes frequency-dependent transmission line effects and describes the design choices that must be taken to avoid multimoding. The final chapter in this volume addresses coupled-lines. It is shown how to design coupled-line networks that exploit this distributed effect to realize novel circuit functionality and how to design networks that minimize negative effects. The modern treatment of transmission lines in this volume emphasizes planar circuit design and the practical aspects of designing around unwanted effects. Detailed design of a directional coupler is used to illustrate the use of coupled lines. Network equivalents of coupled lines are introduced as fundamental building blocks that are used later in the synthesis of coupled-line filters. The text, examples, and problems introduce the often hidden design requirements of designing to mitigate parasitic effects and unwanted modes of operation.

TOC

1 Introduction to Distributed Microwave Circuits . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Book Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Chapter Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Distributed Circuits . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5.1 Point Form of Maxwell’s Equations . . . . . . . . . . . 8
1.5.2 Moments and Polarization . . . . . . . . . . . . . . . . 10
1.5.3 Field Intensity and Flux Density . . . . . . . . . . . . . 11
1.5.4 Maxwell’s Equations in Phasor Form . . . . . . . . . . 14
1.5.5 Integral Form of Maxwell’s Equations . . . . . . . . . . 15
1.6 Electric and Magnetic Field Laws . . . . . . . . . . . . . . . . . 16
1.6.1 Ampere’s Circuital Law . . . . . . . . . . . . . . . . . . 16
1.6.2 Biot–Savart Law . . . . . . . . . . . . . . . . . . . . . . 16
1.6.3 Gauss’s Law . . . . . . . . . . . . . . . . . . . . . . . . 17
1.6.4 Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . 18
1.7 Electromagnetic Fields in Dielectrics and Metals . . . . . . . . 19
1.7.1 Electromagnetic Fields in a Dielectric . . . . . . . . . . 19
1.7.2 Refractive Index . . . . . . . . . . . . . . . . . . . . . . 19
1.7.3 Electromagnetic Fields in a Metal . . . . . . . . . . . . 20
1.8 Electricand MagneticWalls . . . . . . . . . . . . . . . . . . . . 21
1.8.1 Electric Wall . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.8.2 Magnetic Wall . . . . . . . . . . . . . . . . . . . . . . . 23
1.9 Fields in Lossy Mediums . . . . . . . . . . . . . . . . . . . . . 24
1.9.1 Lossy Dielectrics . . . . . . . . . . . . . . . . . . . . . . 24
1.9.2 Lossy Conductors . . . . . . . . . . . . . . . . . . . . . 26
1.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.12.1 Exercises By Section . . . . . . . . . . . . . . . . . . . . 32
1.12.2 Answers to Selected Exercises . . . . . . . . . . . . . . 32
1.A Mathematical Foundations . . . . . . . . . . . . . . . . . . . . 33
1.A.1 Angles of a Triangle . . . . . . . . . . . . . . . . . . . . 33
1.A.2 Trigonometric Identities . . . . . . . . . . . . . . . . . . 34
1.A.3 TrigonometricDerivatives . . . . . . . . . . . . . . . . 34
1.A.4 Complex Numbers andPhasors . . . . . . . . . . . . . 34
1.A.5 Vector Operators . . . . . . . . . . . . . . . . . . . . . . 36
1.A.6 Hyperbolic Functions and Complex Numbers . . . . . 39
1.A.7 Volumes and Areas . . . . . . . . . . . . . . . . . . . . 40
1.A.8 Series Expansions . . . . . . . . . . . . . . . . . . . . . 40
1.A.9 Trigonometric Series Expansions . . . . . . . . . . . . . 41
1.A.10 Special Polynomials . . . . . . . . . . . . . . . . . . . . 42
1.A.11 Matrix Operations . . . . . . . . . . . . . . . . . . . . . 43
1.A.12 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . 44
1.A.13 Circles on the Complex Plane . . . . . . . . . . . . . . 45
1.A.14 Bilinear Transform. . . . . . . . . . . . . . . . . . . . . 46
1.A.15 Quadratic and Cubic Equations . . . . . . . . . . . . . 48
1.A.16 Kron’s Method: Network Condensation . . . . . . . . 48
2 Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.1.1 When Must a Line be Considered a Transmission Line 52
2.1.2 Movement of a Signal on a Transmission Line . . . . . 52
2.2 Transmission Line Theory . . . . . . . . . . . . . . . . . . . . . 55
2.2.1 Transmission Line RLGC Model . . . . . . . . . . . . . 55
2.2.2 Derivation of Transmission Line Properties . . . . . . . 56
2.2.3 Relationship of RLGC parameters to Permittivity and Permeabilityof aMedium . . . . . . . . . . . . . . . . 59
2.2.4 Dimensions of γ, α, and β . . . . . . . . . . . . . . . . . 61
2.2.5 Lossless Transmission Line . . . . . . . . . . . . . . . . 62
2.2.6 Coaxial Line . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.7 Microstrip Line . . . . . . . . . . . . . . . . . . . . . . . 65
2.2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.3 The Lossless Terminated Line . . . . . . . . . . . . . . . . . . . 66
2.3.1 Total Voltage and Current on the Line . . . . . . . . . . 67
2.3.2 Forward- and Backward-Traveling Pulses . . . . . . . 69
2.3.3 Input Reflection Coefficient of a Lossless Line . . . . . 71
2.3.4 Input Impedance of a Lossless Line . . . . . . . . . . . 73
2.3.5 StandingWaves and Voltage StandingWave Ratio . . 73
2.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.4 Special Loss less Line Configurations . . . . . . . . . . . . . . . 78
2.4.1 Short Length of Short-Circuited Line . . . . . . . . . . 79
2.4.2 Short Length of Open-Circuited Line . . . . . . . . . . 79
2.4.3 Short-Circuited Stub . . . . . . . . . . . . . . . . . . . . 79
2.4.4 Open-Circuited Stub . . . . . . . . . . . . . . . . . . . . 82
2.4.5 Electrically Short Loss less Line . . . . . . . . . . . . . . 83
2.4.6 Quarter-Wave Transformer . . . . . . . . . . . . . . . . 85
2.4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.5 The Lossy Terminated Line . . . . . . . . . . . . . . . . . . . . 87
2.5.1 Input Reflection Coefficient of a Lossy Line . . . . . . 87
2.5.2 Input Impedance of a Long Lossy Line . . . . . . . . . 87
2.5.3 Input Impedance of a Lossy Line . . . . . . . . . . . . . 88
2.5.4 Attenuation on a Low-Loss Line . . . . . . . . . . . . . 89
2.5.5 Power Flow on a Terminated Lossy Line . . . . . . . . 90
2.5.6 Lossy Transmission Line Dispersion . . . . . . . . . . . 91
2.5.7 Design of a Dispersionless Lossy Line . . . . . . . . . . 92
2.5.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 92
2.6 Reflections at Interfaces . . . . . . . . . . . . . . . . . . . . . . 92
2.6.1 Power Flow and Return Loss . . . . . . . . . . . . . . . 93
2.6.2 Maximum Power Transfer Theorem . . . . . . . . . . . 94
2.6.3 ASingle Interface . . . . . . . . . . . . . . . . . . . . . 96
2.6.4 Bounce Diagram . . . . . . . . . . . . . . . . . . . . . . 98
2.6.5 Theory of Small Reflections . . . . . . . . . . . . . . . . 102
2.7 ModelsofTransmissionLines . . . . . . . . . . . . . . . . . . . 105
2.7.1 Circuit Models of Transmission Lines . . . . . . . . . . 105
2.7.2 ABCD (Chain) Parameters of a Transmission Line . . . 108
2.8 Two-Conductor Transmission Lines . . . . . . . . . . . . . . . 110
2.9 Coaxial Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
2.9.1 Characteristic Impedance of a Coaxial Line . . . . . . . 111
2.9.2 Electromagnetic Fields on a Realistic Coaxial Line . . . 114
2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
2.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
2.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
2.12.1 ExercisesBySection . . . . . . . . . . . . . . . . . . . . 122
2.12.2 Answers to Selected Exercises . . . . . . . . . . . . . . 122
2.A Physical Constants and Material Properties . . . . . . . . . . . 123
2.A.1 SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
2.A.2 Greek Alphabet and Additional Characters . . . . . . . 127
2.A.3 Conductors, Dielectrics, and Magnetic Materials . . . . 127
3 Planar Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . 131
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3.2 Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.2.1 Dielectric Effect . . . . . . . . . . . . . . . . . . . . . . . 133
3.2.2 Dielectric Loss Tangent, tan δ . . . . . . . . . . . . . . 133
3.2.3 Magnetic Material Effect . . . . . . . . . . . . . . . . . 134
3.2.4 Substrates for Planar Transmission Lines . . . . . . . . 134
3.3 Planar Transmission Line Structures . . . . . . . . . . . . . . . 135
3.3.1 Microstrip . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.3.2 CoplanarWaveguide . . . . . . . . . . . . . . . . . . . 137
3.3.3 Coplanar Strip and Differential Line . . . . . . . . . . . 137
3.3.4 Stripline . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.3.5 Embedded Differential Line . . . . . . . . . . . . . . . 138
3.4 Modeling of Transmission Lines . . . . . . . . . . . . . . . . . 138
3.5 Microstrip Transmission Lines . . . . . . . . . . . . . . . . . . 139
3.5.1 Microstrip Line in the Quasi-TEM Approximation . . 139
3.5.2 Input Impedance of a Shorted Microstrip Line Modeled using a Field Solver . . . . . . . . . . . . . . . . . 140
3.5.3 Effective Permittivity and Characteristic Impedance . . . . . . . . . . . . . . . . . . . . . . . . . 143
3.5.4 Filling Factor . . . . . . . . . . . . . . . . . . . . . . . . 148
3.5.5 Microstrip Resistance . . . . . . . . . . . . . . . . . . . 148
3.5.6 Microstrip Conductance . . . . . . . . . . . . . . . . . . 149
3.6 Microstrip Design Formulas . . . . . . . . . . . . . . . . . . . . 150
3.6.1 High Impedance . . . . . . . . . . . . . . . . . . . . . . 150
3.6.2 Low Impedance . . . . . . . . . . . . . . . . . . . . . . 151
3.7 Stripline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.7.1 Characteristic Impedance of a Stripline . . . . . . . . . 152
3.7.2 Attenuation on a Stripline . . . . . . . . . . . . . . . . . 153
3.7.3 Design Formulas for a Stripline . . . . . . . . . . . . . 155
3.8 Coplanar Waveguide . . . . . . . . . . . . . . . . . . . . . . . . 155
3.8.1 Effective Permittivity and Characteristic Impedance of CPW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
3.8.2 Loss of CPW lines . . . . . . . . . . . . . . . . . . . . . 157
3.8.3 CPW Structures . . . . . . . . . . . . . . . . . . . . . . 157
3.8.4 Differential Line and Coplanar Strip . . . . . . . . . . . 158
3.8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 159
3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
3.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
3.11.1 Exercises by Section . . . . . . . . . . . . . . . . . . . . 164
3.11.2 Answers to Selected Exercises . . . . . . . . . . . . . . 164
4 Extraordinary Transmission Line Effects . . . . . . . . . . . . . . . 165
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.2 Frequency-Dependent Characteristics . . . . . . . . . . . . . . 166
4.2.1 MaterialDependency . . . . . . . . . . . . . . . . . . . 166
4.2.2 Frequency-Dependent Charge Distribution . . . . . . . 166
4.2.3 Current Bunching . . . . . . . . . . . . . . . . . . . . . 167
4.2.4 Skin Effect and Internal Conductor Inductance . . . . 169
4.2.5 Skin Effect and Line Resistance . . . . . . . . . . . . . . 171
4.2.6 Dielectric Dispersion . . . . . . . . . . . . . . . . . . . . 173
4.2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.3 High-Frequency Properties of Microstrip Lines . . . . . . . . . 174
4.3.1 Frequency-DependentLoss . . . . . . . . . . . . . . . . 175
4.3.2 Field Simulations . . . . . . . . . . . . . . . . . . . . . . 176
4.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 178
4.4 Multi moding on Transmission Lines . . . . . . . . . . . . . . . 178
4.5 Parallel-PlateWaveguide . . . . . . . . . . . . . . . . . . . . . 179
4.5.1 Electromagnetic Derivation . . . . . . . . . . . . . . . . 180
4.5.2 Multimoding and Electric and MagneticWalls . . . . . 182
4.6 Microstrip Operating Frequency Limitations . . . . . . . . . . 183
4.6.1 Microstrip Dielectric Mode (Slab Mode) . . . . . . . . 185
4.6.2 Higher-Order Microstrip Mode . . . . . . . . . . . . . 186
4.6.3 Transverse Microstrip Resonance . . . . . . . . . . . . 187
4.6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4.7 High-Frequency Considerations for CoplanarWaveguide . . 189
4.8 High-Frequency Considerations for Stripline . . . . . . . . . . 190
4.9 Power Losses and Parasitic Effects . . . . . . . . . . . . . . . . 191
4.10 Lines on Semiconductor Substrates . . . . . . . . . . . . . . . . 192
4.10.1 Modes on the MIS (MOS) Line . . . . . . . . . . . . . . 194
4.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
4.12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
4.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
4.13.1 Exercises by Section . . . . . . . . . . . . . . . . . . . . 201
4.13.2 Answers to Selected Exercises . . . . . . . . . . . . . . 201
5 Coupled Lines and Applications . . . . . . . . . . . . . . . . . . . . 203
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.2 Physics of Coupling . . . . . . . . . . . . . . . . . . . . . . . . 204
5.3 Coupled Transmission Line Theory . . . . . . . . . . . . . . . 208
5.3.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 210
5.4 Low-Frequency Capacitance Model of Coupled Lines . . . . . 210
5.4.1 Capacitance Matrix Extraction . . . . . . . . . . . . . . 212
5.5 Symmetric Coupled Transmission Lines . . . . . . . . . . . . . 213
5.5.1 Odd-Mode and Even-Mode Capacitances . . . . . . . 215
5.6 Formulas for Characteristic Impedance of Coupled Microstrip Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
5.6.1 Even-Mode Coupled-Line Parameters . . . . . . . . . . 218
5.6.2 Odd-Mode Coupled-Line Parameters . . . . . . . . . . 218
5.6.3 System Impedance of Coupled Lines . . . . . . . . . . 219
5.6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 223
5.7 Terminated Coupled Lines . . . . . . . . . . . . . . . . . . . . 223
5.7.1 Reflection less Condition . . . . . . . . . . . . . . . . . 223
5.8 Directional Coupler . . . . . . . . . . . . . . . . . . . . . . . . 227
5.8.1 Design Equations for a Directional Coupler . . . . . . 229
5.8.2 Directional Coupler With Lumped Capacitors . . . . . 230
5.8.3 Physical Design of a Pair of Coupled Lines . . . . . . . 230
5.8.4 The Lange Coupler . . . . . . . . . . . . . . . . . . . . . 234
5.9 Models of Parallel Coupled Lines . . . . . . . . . . . . . . . . 235
5.9.1 Chain Matrix Model of Coupled Lines . . . . . . . . . 235
5.9.2 ABCD Parameters of Coupled-Line Sections . . . . . 236
5.9.3 Synthesis of Specific Coupled-Line Connections . . . . 236
5.9.4 Model Using Even- and Odd-Mode Lines . . . . . . . 238
5.9.5 Alternative Coupled-Line Model . . . . . . . . . . . . 240
5.9.6 Approximate Synthesis from the Coupled-Line Model 242
5.10 Differential and Common Modes . . . . . . . . . . . . . . . . . 243
5.11 Common Impedance Coupling . . . . . . . . . . . . . . . . . . 246
5.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
5.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
5.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
5.14.1 Exercises by Section . . . . . . . . . . . . . . . . . . . . 251
5.14.2 Answers to Selected Exercises . . . . . . . . . . . . . . 251
6 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.2 The Rectangular Wave Equation . . . . . . . . . . . . . . . . . 254
6.3 Parallel-PlateWaveguide . . . . . . . . . . . . . . . . . . . . . 257
6.3.1 TEM Mode . . . . . . . . . . . . . . . . . . . . . . . . . 257
6.3.2 TM Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 259
6.3.3 TE Mode . . . . . . . . . . . . . . . . . . . . . . . . . . 261
6.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 262
6.4 RectangularWaveguide . . . . . . . . . . . . . . . . . . . . . . 262
6.4.1 TM Modes . . . . . . . . . . . . . . . . . . . . . . . . . 263
6.4.2 TE Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 265
6.4.3 Practical Rectangular Waveguide . . . . . . . . . . . . 266
6.4.4 RectangularWaveguide Components . . . . . . . . . . 268
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
6.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
6.7.1 Exercises by Section . . . . . . . . . . . . . . . . . . . . 277
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

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