The increase of consumer, medical and sensors electronics using radio frequency (RF) and microwave (MW) circuits has imp

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- Hans L. Hartnagel
- Rüdiger Quay
- Ulrich L. Rohde
- Matthias Rudolph

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*Table of contents : ForewordPrefaceContentsEditors and ContributorsThe Authors of this Book1 Resonant Circuits, One-Port Networks, Coupling Filters Made of Lumped, Passive Components 1.1 Vector Diagrams for Inductances and Capacitors with Losses 1.2 Parallel and Series Resonant Circuits 1.2.1 Lossless Resonant Circuits 1.2.2 Resonant Circuits with Resistive Losses 1.2.3 Resonant Circuits with Multiple Resistances 1.2.4 Multiple Feed Circuit Made of Lumped Elements 1.3 Coupling Band Filters in Transmission Systems 1.3.1 Two-Circuit Coupling Band Filters 1.3.2 Matching Circuits 1.3.3 Multicircuit Coupling Band Filters 1.3.4 Losses in Reactance Filters 1.4 Principle of Conservation of Energy, Impedance, Admittance and Quality Factor Definitions 1.4.1 The Principle of Conservation of Energy in Network Theory 1.4.2 Impedance and Admittance 1.4.3 Definition of the Quality Factor from the Phase Angle 1.4.4 Definition of the Quality Factor with the Aid of the Total Stored Energy 1.4.5 Definition of the Quality Factor from the Phase Slope 1.4.6 Definition of the Quality Factor from the Bandwidth at Resonance References2 Wave Propagation on Transmission Lines and Cables 2.1 Introduction 2.2 Propagation of Electromagnetic Waves on Transmission Lines 2.2.1 Equivalent-Circuit Representation of the Line and Derivation of the Telegrapher's Equation 2.2.2 Solution of the Telegraphers' Equation: Propagation Constants and Characteristic Impedance of the Line 2.2.3 Phase and Group Velocity 2.2.4 Exact Representation of the Attenuation and Phase Coefficients 2.2.5 Frequency Dependency of the Characteristic Impedance 2.3 The Reflection Coefficient 2.3.1 Chain Matrix Description of the Transmission Line 2.3.2 The Reflection Coefficient 2.3.3 Transformation of Reflection Factors Through a Transmission Line 2.3.4 Voltages and Currents on Transmission Lines and the Standing-Wave Ratio 2.3.5 Transmission Line Resonators 2.3.6 Reflection Coefficient, Transported Effective Power and Matching of Lossy Lines 2.4 Matching Techniques 2.4.1 Transmission-Line Charts 2.4.2 Narrow-Band Matching Techniques 2.4.3 Broadband Matching Techniques 2.4.4 Application Examples for the Smith Chart 2.5 Scattering Parameters 2.5.1 S-Matrix for Lossless Multiports 2.5.2 Deriving the S-Matrix of a Multiport 2.5.3 Wave Chain Matrix 2.5.4 Calculating Networks Based on S-Parameters 2.5.5 Example: FET and HBT Amplifier Matching References3 Impedance Transformers and Balanced-to-Unbalanced Transformers 3.1 High-Frequency Transformers Overview 3.1.1 Transformers for Impedance Transformation 3.1.2 Resonance Transformers Consisting of Lumped Elements 3.1.3 Line Transformers Consisting of Homogeneous, Low-Loss Lines 3.1.4 Line Transformation with Inhomogeneous Low-Loss Lines 3.1.5 Transformers in Microstrip Technology 3.2 Matching Between Balanced and Unbalanced Lines 3.2.1 Balancing Transformer 3.2.2 Baluns Consisting of Line Elements 3.2.3 Broadband Line Transformers for Transformation and Balancing Made of Lines and Ferrite Components References4 Properties of Coaxial Cables and Transmission Lines, Directional Couplers and RF Filters 4.1 Properties of Coaxial Cables and Transmission Lines 4.1.1 Concept of the Wave Impedance 4.1.2 Characteristic Impedance of a Line and Capacitance Per Unit Length 4.1.3 Characteristic Impedance of a Line and Inductance Per Unit Length 4.1.4 Power Transfer and Power Density 4.1.5 Voltage Loading, Line Attenuation and Heat Limitation in High Power Cables 4.1.6 Optimal Coaxial Cables 4.2 Striplines 4.2.1 Overview of Different Designs and Applications 4.2.2 Field Types in Striplines 4.2.3 Quasi-static Line Constants 4.2.4 Stripline (Triplateline) 4.2.5 Microstrip 4.2.6 Coplanar Waveguides 4.2.7 Coplanar Strips 4.2.8 Slotlines 4.3 Coupled TEM-Wave Lines 4.3.1 Line Differential Equations 4.3.2 Even- and Odd-Mode Excitation 4.3.3 Chain Matrix 4.4 S-Matrix for Matched Couplers and Power Dividers 4.4.1 Conditions for Non-dissipative Combiners and Dividers and the Even-Mode—Odd-Mode Analysis 4.5 Ring Couplers (180° and 90° Hybrid) 4.6 Directional Couplers 4.6.1 S-Matrix for Termination with the Characteristic Impedance of the Line 4.7 TEM Wave Directional Couplers 4.7.1 Definitions and Illustration of the Directional Effect 4.7.2 Spatially Dependent Coupling 4.7.3 Modified Coupling Sections for Attaining High Coupling 4.8 Matched Three-Port Network (Wilkinson Power Divider) 4.9 Microwave Filters Based on Lines 4.9.1 Richards Transformation 4.9.2 Bandstop Filter with Line Resonators, Circuit Transformations 4.9.3 Bandpass-Filters and Phase Shifters Made of Coupled Wave Lines 4.9.4 Interdigital and Comb Line Bandpass Filters 4.10 Tunable Filters 4.10.1 Impedance Matching 4.11 Surface Acoustic Wave Filters 4.11.1 Introduction 4.11.2 Interdigital Transducers 4.11.3 Interdigital Transducer Filters 4.11.4 Surface Acoustic Wave (SAW) Filters with Low Insertion Loss 4.11.5 Other SAW Devices References5 Field-Based Description of Propagation on Waveguides 5.1 Maxwell’s Equations 5.1.1 Wave Equations for E and H, the Electrodynamic Potentials A and φ 5.1.2 Maxwell’s Equations in Component Representation 5.1.3 Wave Equations for the Axial Components EZ and HZ and the Remaining Components 5.1.4 Boundary Conditions for the Electric and Magnetic Field Quantities 5.1.5 Poynting Vector and Poynting’s Theorem 5.2 Relationships Between Field Theory and Transmission Line Theory 5.2.1 TEM Waves 5.2.2 Consideration of the Conductor Losses 5.2.3 Comparison of Lecher, Transmission Line and TEM Waves 5.3 Plane Waves in an Infinite, Piecewise Homogeneous Medium 5.3.1 Homogeneous Plane Wave, TEM Wave 5.3.2 TE Waves and TM Waves 5.3.3 Laws of Reflection and Refraction 5.4 Dielectric Waveguides 5.4.1 Dielectric Slab Waveguides 5.4.2 Cylindrical Dielectric Waveguides 5.4.3 Optical Fibers 5.5 Surface Waveguides 5.5.1 Dielectrically Coated Metal Slab 5.5.2 Dielectrically Coated Metal Wire 5.6 Metallic Waveguides for Higher Order Modes 5.6.1 The Parallel-Plate Line 5.6.2 The Rectangular Waveguide 5.6.3 The Circular Waveguide 5.6.4 Generalized telegrapher’s Equations. Waveguide Equivalent Circuits and Attenuation of Waveguide Waves 5.6.5 Coaxial Line with Higher Modes 5.7 Components Used in Waveguide Technology 5.7.1 Junctions with Rectangular Waveguides 5.7.2 Metallic Irises and Posts in Waveguides 5.7.3 Waveguide Loaded with Inhomogeneous Dielectric Material 5.7.4 Cavity Resonators 5.7.5 Waveguide and Dielectric Resonator Based Filters 5.7.6 Waveguide Directional Couplers 5.8 Wave Propagation in Gyromagnetic Media (Directional Components, Ferrites and Yttrium Iron Garnet Garnets) 5.8.1 Basic Principles 5.8.2 Application in Nonreciprocal Components References6 Antennas 6.1 Introduction 6.2 The Hertzian Dipole 6.3 The Concept of Duality and the Small Loop 6.4 Antenna Parameters 6.4.1 Radiation Resistance 6.4.2 Directivity, Beamwidth and Equivalent Solid Angle 6.4.3 Efficiency and Gain 6.4.4 Near-Field and Far-Field 6.4.5 Polarization 6.4.6 Effective Length and Effective Aperture 6.4.7 Friis Transmission Equation 6.4.8 Effect of Earth’s Atmosphere and Radiation Power Exponent 6.5 Antenna Arrays 6.5.1 Image Principle and Monopole Antenna 6.5.2 The N-Element Linear Array 6.5.3 Beamforming Networks 6.5.4 The Two-Dimensional Array 6.5.5 Conformal Arrays 6.5.6 Mutual Coupling 6.6 Wire Antennas 6.6.1 Dipoles 6.6.2 Loop and Helix 6.6.3 Slot Antenna 6.6.4 Small Antennas 6.7 Aperture Antennas 6.7.1 Aperture Concept 6.7.2 Horn Antennas 6.7.3 Corrugated Horn und Dual-Mode Horn 6.7.4 Reflector Antennas 6.7.5 Lens Antennas 6.8 Patch and Planar Antennas 6.9 Antenna Measurement Techniques References7 Semiconductors and Semiconductor Devices and Circuits 7.1 Historical Approach to Physical Properties of Semiconductors 7.1.1 Conductivity of Semiconductors [80] 7.1.2 Intrinsic Conduction of Semiconductors (Ge, Si, GaAs, GaN) 7.1.3 Impurity Conduction (Doping) 7.1.4 Band Model of Semiconductors 7.1.5 Carrier Density as a Function of the Density of States and Fermi–Dirac Distribution 7.1.6 Electron Transfer Effect 7.2 Semiconductor Devices with Two Electrodes 7.2.1 The p–n Junction 7.2.2 The Metal–Semiconductor Junction 7.2.3 RF Diodes 7.2.4 Diodes for RF Oscillators 7.3 Bipolar Transistors 7.3.1 Manufacturing Techniques and Processing of Transistors 7.3.2 Current–Voltage Relationships (Ebers-Moll Equations) 7.3.3 Regions of Operation for Bipolar Transistors 7.3.4 Sets of Characteristic Curves for Bipolar Transistors 7.3.5 Bipolar Transistors as Amplifiers in Small-Signal Mode 7.3.6 Transfer Properties of Single—Stage Transistor Circuits 7.3.7 Temperature Dependency and Temperature Stabilization of Bipolar Transistors 7.3.8 Bipolar Transistors at Higher Frequencies 7.3.9 Bipolar Microwave Transistors 7.3.10 Heterojunction Bipolar Transistors (HBT) 7.4 Unipolar Transistors (Field-Effect Transistors) 7.4.1 Basic Principle, Embodiments and Characteristics 7.4.2 Small-Signal FETs 7.4.3 High-Power FETs 7.5 Analog High-Frequency Integrated Circuits (ICs) 7.5.1 Introduction 7.5.2 Monolithic Microwave Integrated Circuit Designs (MMICs) 7.5.3 Detailed Passive Components and Networks 7.5.4 Design Flow and Computer Aided Design (CAD) 7.5.5 Circuit Technology References8 Interference and Noise 8.1 Mathematical Description of Noise 8.1.1 Probability Density Function and Averages 8.1.2 Auto- and Cross-Correlation 8.1.3 Noise in the Frequency Domain 8.2 Physical Noise Sources 8.2.1 Shot Noise 8.2.2 Thermal Noise 8.2.3 1/f Noise 8.3 The Spot Noise Figure 8.3.1 Spot Noise Figure of Matched Cascaded Twoports 8.3.2 The Noise Measure and Its Significance in Cascade Connections 8.3.3 Spot Noise Figure of Matched Passive Twoports 8.4 Noise in Linear Multiports 8.4.1 Noise Parameters 8.4.2 Noise Circles 8.4.3 Noise Correlation Matrices 8.5 Noise in Transistors 8.5.1 Field-Effect Transistors 8.5.2 Bipolar Transistors 8.6 Antenna Noise References9 Amplifiers 9.1 Amplifier Characteristics in Complex Functions 9.1.1 Amplification and Gain 9.1.2 RF-Device Configurations 9.1.3 RF-Parameter Description of Small-Signal Amplifiers 9.2 RF-Feedback 9.2.1 Basic Principles 9.2.2 Basic Applications 9.2.3 Selective Amplifiers 9.3 Gain and Matching 9.3.1 Power Gain and Impedance Matching 9.3.2 Small-Signal Amplifier with Field Effect Transistors 9.3.3 Signal Flow Diagrams 9.3.4 Power Gain Definitions 9.3.5 Stability 9.3.6 Practical Stability 9.4 Amplifier Basics 9.4.1 Multistage Concepts and Interstage-Matching 9.4.2 Stability and Biasing 9.4.3 DC-/RF-Blocking 9.5 RF Small-Signal Amplifiers 9.5.1 High-Gain Amplifier 9.5.2 Low-Noise Amplifiers 9.5.3 Integrated Broadband Amplifier 9.5.4 Differential Amplifier 9.6 Nonlinear Effects and Large-Signal Behavior 9.6.1 Fundamental Device Limits 9.6.2 Large-Signal Characteristics and Nonlinear Distortions 9.6.3 Power Compression 9.6.4 Dependence of Gain on Impedances and Matching 9.6.5 Source and Load Reflection 9.6.6 The Generation of Harmonics 9.6.7 The Concept of a Loadline 9.6.8 Efficiency 9.6.9 Cascaded Intermodulation Products 9.6.10 The Frequency Pyramid 9.6.11 Linearity Concepts and Measures 9.7 Hybrid and Integrated Circuit Based Amplifiers 9.7.1 Lumped Elements and Hybrid Components 9.7.2 Integrated RF-Circuits 9.7.3 Passive RF-Components and Their Use for Matching 9.7.4 Transmission Lines and Parameters 9.7.5 mm-Wave and Sub-mm Wave Integrated Circuits 9.7.6 Cointegration of RF- and Digital-Functions 9.8 Design Rules and Layout 9.8.1 Design Rules 9.8.2 Layout 9.8.3 Thermal Limits 9.9 The ABC of Amplifier Classes 9.9.1 Classes-A, -B, -C 9.9.2 DC- and Load-Modulation 9.9.3 Class-D, Class-E, and Class-F Applications 9.9.4 General Harmonic Waveform Shaping 9.9.5 Continuous Modes 9.9.6 Switch-Mode Amplifiers 9.10 Problems References10 Oscillators and Frequency Synthesis 10.1 Oscillation Conditions and Stability Criteria 10.1.1 Linearized Time Domain Model 10.1.2 Feedback View of Oscillators 10.1.3 One-Port Negative Resistance Theory 10.2 Phase Noise 10.2.1 Effect of Phase Noise 10.2.2 Leeson's Empirical Phase Noise Model 10.2.3 Linear Analysis Approach 10.2.4 Mixer Analysis Approach 10.2.5 Hajimiri's Linear Time Variant Analysis Approach 10.3 Oscillators Using Negative Resistance Devices 10.3.1 Tunnel Diode Oscillators 10.3.2 Transferred Electron Devices (Gunn Elements) as Oscillators 10.3.3 Avalanche Transit Time Oscillators (Read and IMPATT Diodes) 10.3.4 Two-Terminal Oscillators with Transit-Time Tubes 10.4 Feedback Oscillators Using Two-Port Devices 10.4.1 General Considerations 10.4.2 LC Oscillators 10.4.3 RC Oscillators (Oscillation Condition) 10.4.4 Frequency Stability 10.4.5 Quartz Oscillators 10.4.6 Stabilization of the Oscillation Amplitude 10.5 Integrated-Circuit Oscillator Realizations Using GaAs-FET 10.5.1 Oscillator Circuits 10.6 Oscillators with Surface Acoustic Wave Resonators (SAW Oscillators) 10.6.1 Colpitts Oscillator Stabilized by SAW One-Port Resonator 10.6.2 Pierce Oscillator with SAW Two-Port Resonator 10.7 Voltage-Controlled Oscillators in CMOS Technologies 10.7.1 Ring Oscillators 10.7.2 LC Oscillators 10.7.3 Cross-Coupled Pair 10.7.4 Three-Point Oscillators 10.7.5 VCO Classes 10.7.6 Phase-Noise Optimization Techniques 10.7.7 Advanced Circuit Techniques References11 Frequency Synthesizer 11.1 Introduction 11.2 Building Blocks of Synthesizers 11.2.1 Voltage Controlled Oscillator 11.2.2 Reference Oscillator 11.2.3 Frequency Divider 11.2.4 Phase-Frequency Comparators 11.2.5 Diode Rings 11.2.6 Edge-Triggered JK Master–Slave Flip-Flops 11.3 Loop Filters—Filters for Phase Detectors Providing Voltage Output 11.4 Important Characteristics of Synthesizers 11.4.1 Frequency Range 11.4.2 Phase Noise 11.4.3 Spurious Response 11.5 Transient Behavior of Digital Loops Using Tri-State Phase Detectors 11.5.1 Pull-In Characteristic 11.5.2 Lock-In Characteristic 11.6 Loop Gain/Transient Response Examples 11.7 Practical Circuits 11.8 The Fractional-N Principle 11.9 Spur-Suppression Techniques 11.10 Digital Direct Frequency Synthesizer 11.10.1 DDS Advantages References12 Software Defined Radio, Receiver and Transmitter Analysis 12.1 Introduction 12.2 The Image Rejection Mixer/Quadrature Mixer 12.3 The Sampling Theorem 12.4 The AD-Converter 12.5 The DA-Converter 12.6 The Digital Down-Converter 12.7 The Digital Up-Converter 12.8 Demodulation Algorithms 12.8.1 AM Demodulator 12.8.2 FM Demodulator 12.8.3 Data Demodulators 12.9 SDR Realisation Example 12.10 Phase Noise, Desensitization 12.11 Filters 12.12 Noise Blanker 12.13 Automatic Gain Control 12.14 The S-Meter 12.15 Spectrum Monitoring 12.16 Adaptive Transmitter Pre-distortion References13 Mixing and Frequency Multiplication 13.1 Introduction 13.2 Theory and Applications of Mixing 13.2.1 Mathematical Model 13.2.2 Heterodyne Receiver 13.3 Combination Frequencies in Nonlinear Components 13.3.1 Small-Signal Theory of Mixing 13.3.2 Upconversion, Downconversion, Common Position, Inverted Position, Image Frequency 13.4 Realization of Mixers 13.4.1 Mixing with Semiconductor Diodes as Nonlinear Resistors 13.4.2 Mixing with Semiconductor Diodes as Nonlinear Capacitors 13.4.3 Mixing with Transistors as Nonlinear Element 13.4.4 Mixing with Active Transistor Multipliers (Gilbert Cell) 13.5 Frequency Multiplication 13.5.1 Frequency Multiplication by Transistor Circuits References14 Modulation Methods 14.1 Outline 14.2 Information Signals 14.2.1 Analog Signals 14.2.2 Digital Signals 14.2.3 The Signal Bandwidth 14.2.4 Shaping of Digital Signals 14.3 Carrier Signals 14.3.1 Manipulation of Carrier Parameters 14.4 Comparison of Analog and Digital Modulation Methods 14.4.1 Analog Modulations 14.4.2 Digital Modulations 14.4.3 Semantic Classification of Digital Modulations 14.4.4 The Modulations in Detail 14.5 The Amplitude Modulations 14.6 AM, DSB and QAM 14.6.1 The Amplitude Modulation in Time Domain 14.6.2 Block Diagram AM Modulator 14.7 Spectrum of Amplitude Modulation 14.8 AM Modulation Degree 14.8.1 Compatibility 14.8.2 Definition of the Degree of Modulation 14.9 Power of AM 14.10 AM Demodulation 14.10.1 Envelope Demodulator (Asynchronous Demodulation) 14.10.2 Synchronous Demodulation of AM 14.11 Demodulation of DSB 14.11.1 Carrier Recovery for DSB with Costas Loop 14.12 Quadrature Double Sideband Modulation QDSB 14.12.1 QDSB Modulation and Demodulation 14.13 Angle Modulation 14.13.1 The Angle Modulation in the Time Domain 14.13.2 Relation of Phase- and Frequency Modulation 14.13.3 Cosine Information Signal 14.14 The Angle Modulation in the Frequency Domain 14.14.1 Phase Modulation with a Frequency Modulator 14.14.2 Generation of FM with a Phase Modulator 14.15 Spectra of Angle Modulation 14.15.1 Classical Analysis of FM 14.15.2 Spectral Distribution of the FM Signal for Cos-Shaped Message Signal 14.15.3 Spectral Distribution and Bandwidth of the FM Spectrum for the General Case of the Message Signal 14.15.4 Narrowband Modulation Spectrum 14.16 Modulators and Demodulators for PM and FM 14.16.1 Generation of Phase Modulation with I/Q Phase Modulator 14.16.2 Generation of a Frequency Modulation 14.16.3 Demodulation of a Phase Modulated Signal 14.16.4 Demodulation of a Frequency Modulation 14.17 Noise in FM 14.18 Digital Modulations 14.18.1 Block Diagram of the Digital Modulator 14.18.2 Information Transmission Analog and Digital 14.18.3 Properties of Signals in the Physical Transmission Channel 14.18.4 Block Diagrams of the Digital Transmission System 14.18.5 Channel Capacity and Shannon Limit 14.19 Baseband Signals 14.19.1 The Baseband Channel 14.19.2 The Transmitter Side 14.19.3 The Receiver's Side 14.20 Spectra of Digital Signals in the Baseband 14.20.1 Data with Statistical Independence 14.21 Inter-Symbol Interference and Nyquist Condition 14.22 Nyquist Condition 14.22.1 Ideal Low Pass as the Simplest Form that Meets Nyquist Condition 1 14.22.2 Generalization of the Nyquist Condition 1 14.22.3 Cosinus Roll Off 14.22.4 Smoothing Filter with Cosine Roll Off 14.22.5 Nyquist Condition 2 14.22.6 Symbol Rate and Spectral Efficiency for Cosine Roll Off Rounding 14.23 Root Raised Cosine 14.23.1 The Eye Diagram 14.24 Digital Single-Carrier Modulation Methods 14.24.1 Model of the Digital Modulator 14.24.2 Systematics of Digital Modulations 14.24.3 Quadrature Modulation Method: Intervention into the Amplitude of the Carriers 14.24.4 Amplitude-Phase Modulation Method: Intervention in Amplitude and Phase of the Carriers 14.25 The Complex Envelope 14.25.1 Representation of Modulation Schemes with the Aid of Complex Envelopes 14.25.2 The Vector Diagram 14.26 Quadrature Carrier System 14.26.1 Higher Level QAM 14.27 Modulations with Constant Envelope 14.27.1 From QPSK to Offset QPSK (OQPSK) 14.27.2 From OQPSK to MSK 14.27.3 CPM Methods with Rounded Data Symbols 14.27.4 The Gauss Rounding 14.28 Demodulation Techniques for Single Carrier Modulations 14.28.1 Principle Structure of the Receiver 14.28.2 Equivalent Low-Pass Signals 14.28.3 Block Diagrams of the Digital Demodulator 14.28.4 Synchronous Demodulation of MSK Signals 14.29 Synchronization of the Digital Receiver 14.30 Multicarrier Modulation 14.30.1 Terrestrial Radio Channel 14.30.2 Channel Equalization Methods 14.30.3 Multicarrier Modulation 14.30.4 OFDM Time Curves 14.31 The OFDM in the Frequency Domain 14.31.1 Higher-Level Symbol Constellations in the Subchannels 14.31.2 Pilot Symbols 14.31.3 Time and Frequency Dependence of the Channel Transfer Function 14.32 OFDM Modulators and Demodulators 14.32.1 Why IFFT in the Transmitter and FFT in the Receiver? 14.33 Power Density Spectrum of the OFDM 14.33.1 Power Density Spectrum in the Receiver and Orthogonality 14.33.2 Synchronization 14.34 From OFDM to COFDM 14.34.1 The Need for Error Protection Coding 14.34.2 Two-way Path and Punctured Convolution Codes 14.34.3 Interleaving 14.35 Single-Carrier Modulation with Frequency Domain Equalization 14.35.1 Relationship to OFDM 14.35.2 SC-FDE Block Structure 14.35.3 Frequency Domain Filtering 14.36 3GPP-LTE Upstream 14.36.1 SC-FDMA as an Access Method 14.37 Spread Spectrum Modulations 14.37.1 Principle of ``Direct Sequence'' Spreading Technique 14.37.2 Features of the Spread Spectrum Modulations 14.37.3 Definition of Spread Spectrum Methods 14.37.4 Binary Pseudo-Random Signals 14.37.5 Cross Correlation of PN Sequences 14.37.6 Direct Sequencing Spread Spectrum 14.37.7 The Processing Gain 14.37.8 Frequency Hopping Method 14.37.9 Time Hopping 14.37.10 Chirp Procedure ReferencesAppendix AppendixA.1 Laws of Fourier TransformationA.1.1 Multiplication and ConvolutionA.1.2 Derivation of the Simplified Method of Convolution in the Time DomainA.1.3 Examples for ``Simplified Convolution''A.1.4 Forming of Data Symbols: Roll-OffA.1.5 RDS Symbol and SpectrumA.2 Frequency and Instantaneous FrequencyA.2.1 FrequencyA.2.1.1 Example: Vibrating Frequency MeterA.2.2 Filter BankA.2.3 The Time-Bandwidth LawA.2.4 Definition of the Term ``Frequency''A.2.4.1 Contradictions in Other Definitions of FrequencyA.2.5 Relationship with the Natural Oscillation of the Measuring Instrument; ResonanceA.2.6 Walsh Functions as Prototype for Orthogonal CodesA.3 The Instantaneous FrequencyA.3.1 The Frequency DeviationA.4 The Hilbert FilterA.4.1 Hilbert Allpass FilterA.4.2 Hilbert Lowpass FilterA.4.3 Hilbert Bandpass Filter ReferencesIndex*

Hans L. Hartnagel Rüdiger Quay Ulrich L. Rohde Matthias Rudolph Editors

Fundamentals of RF and Microwave Techniques and Technologies

Fundamentals of RF and Microwave Techniques and Technologies

Hans L. Hartnagel · Rüdiger Quay · Ulrich L. Rohde · Matthias Rudolph Editors

Fundamentals of RF and Microwave Techniques and Technologies

Editors Hans L. Hartnagel Elektrotechnik und Informationstechnik Technische Universität Darmstadt Darmstadt, Germany Ulrich L. Rohde Brandenburgische Technische Universität Cottbus-Senftenberg Cottbus, Brandenburg, Germany Fakultät für Informatik Universität der Bundeswehr München Munich, Germany Rohde & Schwarz Munich, Germany

Rüdiger Quay Fraunhofer Institute for Applied Solid State Physics IAF Freiburg im Breisgau, Baden-Württemberg, Germany Fritz-Hüttinger Chair for Energy-Efficient High-Frequency Electronics Albert-Ludwigs-Universität Freiburg Freiburg im Breisgau, Baden-Württemberg, Germany Matthias Rudolph Fachgebiet Hochfrequenz- und Mikrowellentechnik Brandenburgische Technische Universität Cottbus-Senftenberg Cottbus, Brandenburg, Germany

ISBN 978-3-030-94098-0 ISBN 978-3-030-94100-0 (eBook) https://doi.org/10.1007/978-3-030-94100-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

The notion of microwaves has been generally coined in the twentieth century to represent very important radio frequency (RF) bands of about GHz range. The techniques developed were applied for the novel military systems of ground and airborne radars during World War II and applications related to wireless communications of radio links at the turn of the twentieth century with Marconi radios. However, the methodologies established at microwaves are applicable to any distributed circuits that offer physical circuit dimensions comparable to wavelength of the RF signals and hence resulting in phase variation along the path of signal flow. Fundamentals of distributed structures were applied to telegraphy and power transmission lines before finding importance for military and civilian system of radar or telecommunications in 1950s. Over a period of 50 years, many aspects of RF and microwaves have transitioned to many important commercial and scientific applications and now span areas of personal communications, Internet communications, and automotive radar for collision warning and their utility in self-driving vehicles, therapeutic and imaging application in medicine and biology, microwave oven appliances, energy and power transfer applications, Internet of things for home local area networks, exploration of deep space and radio astronomy, and weather and agricultural monitoring as remote sensing. In addition, ever-increasing demand for a higher data throughput and need for large volume manufacturing of miniaturized micro- and nano-electronics circuits have bridged a gap between microwaves and photonics, known as THz domain. The modern training of RF engineers requires teaching of fundamentals of electromagnetic fields, solid-state device physics, understanding of electronic and optical components, advanced knowledge of integrated circuit (IC) design for low power, and understanding of electron beams in high-power microwave systems. Understanding of many aspects of these foundational concepts is now augmented by a variety of computer-aided design (CAD) simulation tools that manage accurate solution by setting up complicated problems with good understanding of fundamentals. My first exposure to microwaves was in part due to performing well in my undergraduate electromagnetic fields course and given opportunity to assist with design of a microwave satellite receiver project. I was fortunate to be partnering in this project with my good friend, Prof. Kamal Sarabandi (now at the University of Michigan), v

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Foreword

as junior student in Arya-Mehr University of Technology (now Sharif University of Technology) under guidance of late Prof. Baghdesarian and two senior students that were heading to Stanford University for their PhD studies. The bug for the microwaves was planted, and later when I transferred to Case Western Reserve University in 1979, I got my further education under guidance of late Prof. Robert E. Collin. Through my graduate studies and meaningful interactions with many other great experts (Dr. Herbert Thal from GE-Aerospace and Dr. Arye Rosen from RCA Labs) and educators (Professors Beard, Rothwarf, Herczfeld, and Coren at Drexel University), I further got involved with many other aspects of microwaves. Since 1987 as an assistant professor of electrical and computer engineering of Drexel University, I have been involved with development of various courses in areas of microwaves and photonics to train undergraduate and graduate students with the essentials of modern microwaves in service of the regional aerospace, national security, and telecommunication industries. When I started with development of my introductory graduate courses in 1987 and later with a new senior sequence for meeting the employment opportunities for the telecommunication hardware engineers at microwaves starting in late 1990s, monolithic microwave integrated circuits (MMIC) were one of the most exciting topics in microwaves. I had structured my course sequence based on several great microwaves textbooks in English language to present a “complete” picture of the microwave field over three terms. Each book focused on different aspects of microwave engineering. I had used books of Jackson, Collin, Ramo/Whinnery/van Duzer, Watson, Rizzi, Liao, Kraus, Stutzman/Thiele, and then Balanis. I have settled in the last 15 years on the Pozar’s well-organized microwave engineering book, but I have continued to supplement it by handout and excerpts extracted from specific topic books by Kong, Bahl/Bhartia, Gonzales, Gupta, Ludwig/Bretchko, Vendelin/Pavio/Rohde, and Elliot. At any rate, I would have regularly started my sequence with electromagnetic field wave foundations and discussions on TEM and non-TEM transmission lines, then moved to passive circuits, and then end the three-term sequence with concepts of active microwave sub-systems, even though for undergraduate students, I would use the reverse order by starting from circuit perspective and then end in electromagnetic and radiating systems. In early 1990, CAD programs were becoming an integral part of the students training in microwave circuits and later in topics related to antennas and radiating systems, when computational tools of method of moments (MoM) and then finite element (FEM) and finite difference time domain (FDTD) techniques became available on desktop computers. These computational tools were introduced as supplementary aspect to solve more complicated circuit and electromagnetic field problems using first principles of electromagnetic field fundamentals and network theory concepts. As a graduate student in early 1980s, I was using Compact Software through modem link to David Sarnoff Research Laboratory to design and optimize performance of loaded line phase shifter to be better controlled using the recently developed optically controlled PIN diodes in Dr. Arye Rosen’s group. When I started developing my graduate courses at Drexel, I sought to include as part of my graduate education CAD tools and even advocated integrating concepts of high-speed

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optical transceivers as part of distributed microwave photonics-based optical distribution systems. I reached out to Dr. Ulrich Rohde, who had started Super-Compact Software Company by acquiring the rights to Compact Software from COMSAT Corp. and proposed to expand capabilities of Super-Compact with optical transmitter and receiver modules for microwave photonics sub-systems. Professor Rohde accepted my invitation to lecture to the IEEE AP/MTT-S Chapter in Philadelphia to educate our community about power of CAD modeling to a room filled primarily with eager graduate students of Center for Microwave/Lightwave Engineering and GE/RCA engineers in Delaware Valley region. Professor Ulrich Rohde due to his love of teaching and his generosity, donated a complementary package of SuperCompact and Harmonia to my group, which remained the best software package available at the time to accurately model nonlinearity, noise, and dynamic range of active microwave circuits and sub-systems. In one of his visits to Philadelphia and as part of his seminar presentation to students, he brought to my attention a great German bible of microwaves, titled “Hochfrequenztechnik 1 and 2” that had played a significant role in educating many successful German scientists. The seventh edition of this bible of microwaves is now translated into English by leading German microwave educators, professors Hans L. Hartnagel, Rüdiger Quay, Ulrich L. Rohde, and Matthias Rudolph. This new book is titled “Fundamentals of RF and Microwave Techniques and Technologies” is primarily driven upon recommendation of Prof. Ulrich L. Rohde to provide to English spoken readers a wealth of information on various aspects of microwave engineering. In this effort of organizing the seventh edition, valuable text and research material and novel ideas were added by these leading educators. The aim of this textbook is to provide a general knowledge on microwave engineering that includes various aspects of designs, from discrete electronic devices to planar modules for the benefits of readers who wish to master design methodology of practical circuits. I had privilege and pleasure of having opportunity to examine preliminary version of the textbook. I found the book completely ideal for my course instructions, as it provides educators to present a “complete” picture of microwaves. In this textbook, authors have outlined total of 14 chapters with emphasis on the fundamentals of RF and microwave techniques and technology, while the logical flow of text material will undoubtedly keep all readers—beginners and advanced—motivated from the first to the last chapter. Beginners in microwaves are encouraged to read systematically as the logical flow of chapters is laid upon concepts developed by the preceding chapters. Experienced readers in this field may find that navigation of the individual chapters is readily practicable. Chapter 1 provides a very detailed introduction to lumped resonant circuits as a one-port network with relevance to realization of coupling filters. “Lumped and Distributed Elements” pertains to the frequency range from RF to millimeter wave frequencies with introduction of important resonant structure relationship, such as Foster’s reactance theorem. Moreover, broadband tuning circuit components exhibit a continuous transition in behavior from that of lumped elements to distributed components. Filtering concepts represented from pole-zero and reactance/susceptance function and transformations using Richard’s transform, invertors, and Kuroda identities.

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Understanding this behavior is of particular importance in high-frequency distributed filter implementations. Chapter 2 describes the interesting subject of TEM lines and wave propagation of ideal and practical lines. It introduces concepts of reflection coefficient and its impact on power delivery. Various narrowband impedance matching methods are introduced using graphical method of Smith and its dual Carter charts. In addition, network parameters are introduced, particularly general S-parameters and their relevant representation of T for cascaded two ports. In addition, signal flow graph which has major utility in analysis of practical components is introduced. Chapters 3 focuses on transformers as impedance transformation techniques for narrow and broadband impedance matching. Both broadband multi-section quarter wavelength and tapered inhomogeneous impedance transformations are introduced. Moreover, some of the most elegant discussions of balanced–unbalanced transformers (balun) and their realizations at RF and microwave frequencies are presented in this book. Chapters 4 is a very important chapter for realization of printed circuits with introduction of popular TEM lines of coaxial cables, striplines, microstrip, coplanar waveguide, and coupled microstrip lines. Their utility in realization of multi-port networks and power dividers and couplers (Branchline as 90° and Rat-Race as 180° hybrids) is introduced. Filter realization using J/K inverters is also introduced for realizing filters. Finally, the concepts of interdigital and comb lines as filters are introduced with practical realizations as surface acoustic waves filters. Chapters 5 establishes fundamental of wave propagation in terms of electromagnetic fields rather than circuit theory. The fields for unbounded and bounded waves are introduced with emphasis on Helmholtz wave equation and its solution of E/H and energy stored and power flow, as Poynting vector. Polarization states of linear and circular are introduced and a general elliptical polarization representation of waves as practical case. The concepts of waves at interfaces are extended to waveguiding of non-TEM metallic and dielectric waveguides in terms of field profiles, excitation modes, and related wave attenuation rates. In addition, dielectric waveguides as rectangular cross section that are important for integrated optics and leaky wave antennas as well as circular cross-section optical fibers are discussed. Moreover, practical waveguide structures of power dividers (E, H, and Magic-T), and couplers (slotted line, Beth hole, and Schwinger couplers) are diligently introduced. Finally, ferromagnetic materials are introduced with applications as Faraday rotator and realization of isolator and circulators. A complete discussion of all these important topics is rarely seen in a single book. Chapter 6 describes the antennas and radiating systems as part of transducers relating electrical circuits to electromagnetic waves excitation in transmitters and reception in receivers. Once again, the depth of covered topics is truly outstanding. Amazingly, the topics covered are identical to elements that I cover in my antennas course, as I start with radiation fundamental of auxiliary potentials of A/F and their utility in solving for wire and aperture antenna structures. The basic properties of antennas from E&M (radiation pattern, polarization, power radiated), circuit perspective (radiation resistance/conductance, efficiency, and bandwidth from Q factor), and

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system level (directivity, gain) are also presented with discussions of near and farfield regions. Friis transmission is introduced as part of radio link discussions. The wire antennas of dipole, loop, and helical antennas are efficiently presented, before discussing aperture antennas of slot radiator based on duality (Babinet) principle. The concepts of horn and reflector antennas are introduced as alternative to linear and planar phased array antennas and their specific design of Luneburg and Rotman lenses. Moreover, practical patch radiators are introduced in terms without depth though a number of references are cited for further reading. Finally, antenna measurement systems using far-field and near-field measurements are introduced with a very basic overview. Chapter 7 is a comprehensive introduction to the physics of semiconductor devices, by introducing electron transport of doped semiconductors (Si, SiGe, GaAs, InGaAs, GaN, SiC) and I-V characteristics of both homo- and hetero-junctions of p-n and Schottky realizations. The p–n junctions are used for varactor diodes and bipolar transistors (BJT and HBT), while Schottky junction is employed for Schottky diodes and unipolar transistors (MESFET, MOSFET, LDMOS, and HEMT). Full physicsbased modeling of transistor dynamics is introduced. Moreover, small-signal equivalent circuit model is reviewed in terms of bipolar and unipolar devices for various operation points based on designed DC/AC biasing networks. The RF performance in terms of gain and noise is introduced for various device topologies of CE/CS, CB/CG, CC/CD configurations, and speed of devices (fT and fmax) is expressed in terms of equivalent circuit parameters. Both IC and hybrid realization of amplifiers modules are discussed. There is no microwave engineering book that emphasizes all physical and circuit parameters of modern electronic integrated circuits. Chapter 8 deals with the general introduction of various noise sources of semiconductor electronics. Physics-based modeling of amplitude (AM), phase (PM), and frequency modulation (FM) noise are presented with emphasis on the effects of interference and noise in modern communication systems. Noise circles are introduced to show dependence of noise figure in terms of source reflection compared against the optimum source reflection that leads to the lowest noise factor (Fmin). Noise of cascaded networks is calculated using Friis noise equation for both matched and unmatched stages. Moreover, sources of correlated and uncorrelated noise sources of bipolar and unipolar devices are discussed, and its contributions to the overall noise figure are estimated in terms of thermal, shot, Hooge, and spot noises. Finally, antenna noise temperature is introduced based on input cosmic noise fluctuations and emissivity of interconnects between antenna and receiver circuits. Such a detailed presentation is absent in most general microwave engineering textbooks. Chapter 9 describes the important concept of amplifiers, as a building block of signal conditioning in microwave circuits. Detail design of input and output matching circuits of various amplifiers (high gain, low noise, and integrated broadband, and differential amplifiers) is introduced in terms of stability, gain, noise, and VSWR circles that are calculated based on S-parameters of transistor with and without series or parallel feedback circuits. Z/Y/S and two-port parameters of ABCD, h, and p matrices are utilized to analyze the combined network parameter of gain (GT, GP, GA) and noise performance; of particular interest are multistage amplifier performance

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as inter-stage matching is employed for gain flattening versus frequency. Moreover, nonlinear characteristics of amplifier in terms of harmonic generation as result of compression and intermodulation distortion are explored, of particular interest are source and load pulling of power amplifiers using balanced amplifiers and power combining circuits. Finally, IC realization of amplifiers for various classes (A, B, C, AB, D, E, F) of operation is discussed with emphasis on IC layout and DRC evaluations. Such a comprehensive overview of amplifier designs is not seen in any microwave books. Chapter 10 presents oscillators design concepts of positive feedback in amplifiers from system perspective (Barkhausen), circuit perspective (Nyquist), and Sparameter modeling. Depending on the phase noise and tuning requirement, different oscillator topologies are discussed, validated, and discussed for the benefit of readers with emphasis on stable oscillation, output power, and pushing and pulling factors. Both fixed frequency and tunable oscillators based on lumped (Colpitts, Clap, Hartley, and YIG tuned) and dielectric resonator (DR) oscillators are discussed. Frequencystabilized oscillators based on quartz crystals are important parts of frequencystabilized oscillators using phase locking processes. In addition, high-power oscillators using electron beams tube technologies (klystron, magnetron, and TWT) are introduced in great depth. Such a feature is very unique and distinguishes this work from other microwave engineering textbooks. Chapter 11 deals with frequency synthesizer and uses the techniques of frequency multiplication, frequency division, direct digital synthesis, frequency mixing, and phase-locked loops to generate its frequencies. The progress and development modern wireless networks have encompassed new frequencies, driven efforts to transceiver architectures and frequency synthesizers, and explained in details to meet the criteria of standardize communication protocols and frequencies to enable seamless communication. Chapter 12 designates very exciting topic of software-defined radio (SDR) with emphasis on microwave receiver and transmitter analysis. This topic is the driving force behind the modern receiver development for advanced communication applications. Driven largely by fast, high-performance, application-specific integrated circuits, powerful microprocessors, and inexpensive memory, the promise of SDR is featured. In this chapter, important designs are covered for the implementation of SDR technologies which is presently used in a broad array of electronics and communication products. Chapter 13 introduces important topic of mixer circuits and frequency multipliers in detail, providing digital modulation scheme from RF perspectives. This scheme is arguably the most prevalent today and therefore receives the most thorough treatment. The details in modulation scheme are discussed for the completeness. While initially appearing to be not in the mainstream of microwaves, nonetheless this material is in fact essential to the concurrent design of microwave circuits and sub-systems. Chapter 14 is a comprehensive introduction to analog and digital modulation methods. Thoroughly basing the explanations on Fourier analysis, the chapter derives the various modern modulation schemes such as WCDMA and OFDM and enables the reader to understand the respective concept as well as similarities and differences.

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As a distinct difference to common introductions to modulation methods, the chapter always refers to circuit implementations and implications of a modulation scheme on microwave electronics. In my opinion, the main goal of this textbook for presenting all relevant information on microwave engineering has been met, as the latest revision includes many practical modern topics. This textbook genuinely addresses the needs of both students and the practicing engineers in ever-growing microwave engineering. Unlike other textbook available in the market, it provides a comprehensive treatment of the subject matter, while establishing strong foundation based on fundamentals. Authors have eloquently applied the acquired microwave field knowledge, while translating this manuscript and included many modern concepts. However, only a personal examination of the book will convey to reader the broad scope of its coverage and how well it succeeds in addressing the changing needs of the microwave techniques and technologies. The authors are to be commended for their efforts in this endeavor. This textbook will be a highly valuable resource for many designers. I look forward to having it on my bookshelf and using it for course instruction of my undergraduate and graduate students. January 2023

Afshin S. Daryoush, Ph.D. Fellow of IEEE Department of Electrical and Computer Engineering Drexel University Philadelphia, PA, USA

Preface

The subject of microwave engineering has been an important field for many years, and therefore, the two Professors Zinke and Brunswig from the Technical University Darmstadt in Germany wrote a book published by Springer in two volumes in the German language. They became a very successful document for both teaching graduates all over the German-speaking countries and as a reference for practicing engineers in industry and research. I remember on visits to industrial companies that I frequently saw a copy of these two volumes on the bookshelf of the engineers which they employed. The special feature of these books is that the various chapters are written by relevant experts in the particular fields. These range from components and circuit theory with the newest material and technology competence, via field theory with such areas as antennas, to applications such as telecommunication engineering. When I became the successor of Prof. Zinke in Darmstadt, I decided with great pleasure to reissue these two books a number of times, since the editions were regularly sold out within two to three years. This of course gave me the opportunity to bring the text fully up to date, since the field of microwaves has experienced a most dynamic and continuous development. With my extensive international experience (first in France, later on as Professor of microwave electronics in England and a number of work periods in Japan and the USA ), I had discussions with a number of influential microwave engineers regarding the question of publishing Zinke/Brunswig in the English language. In particular, I discussed also with Prof. Ulrich L. Rohde (whom I met first in 1982 when I was a consultant at the David Sarnoff Laboratory in Princeton, New Jersey, and Prof. Rohde in charge of the Government Radio business unit of RCA), how to republish this important and successful book in now English. We presented this plan to the Springer company and at the regular meeting of the German microwave professors in Munich, where we obtained excellent encouragement for this idea. I am now very happy to see that Prof. Matthias Rudolph, Fachgebiet Hochfrequenz- und Mikrowellentechnik, Ulrich-L.-Rohde Stiftungsprofessur at the Technical University of Cottbus, Germany and Prof. Rüdiger Quay, University Freiburg and Fraunhofer IAF, in collaboration with Prof. Rohde have brought together xiii

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an impressive range of experts in microwave engineering to write such a new book in the English language. Of course, many highly important microwave areas have since been opened up as, for example, Terahertz electronics in the upper frequency ranges of the microwave area or new materials such as graphene as an example of low-dimensional materials. The range of subjects covered in this new book has been highly (carefully) competently selected and treated. It is in the tradition of Zinke and Brunswig a high-quality book for both teaching of microwave engineering students and as a reference source for all those working in this profession. My sincere congratulations to Profs. R. Quay and M. Rudolph. Darmstadt, Germany

Prof. Hans L. Hartnagel

Contents

1

Resonant Circuits, One-Port Networks, Coupling Filters Made of Lumped, Passive Components . . . . . . . . . . . . . . . . . . . . . . . . . Renato Negra

1

2

Wave Propagation on Transmission Lines and Cables . . . . . . . . . . . . Matthias Rudolph

3

Impedance Transformers and Balanced-to-Unbalanced Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holger Maune

143

Properties of Coaxial Cables and Transmission Lines, Directional Couplers and RF Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . Matthias Rudolph

197

4

55

5

Field-Based Description of Propagation on Waveguides . . . . . . . . . . Holger Arthaber

335

6

Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jan Hesselbarth

485

7

Semiconductors and Semiconductor Devices and Circuits . . . . . . . . Rüdiger Quay

551

8

Interference and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matthias Rudolph

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9

Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rüdiger Quay

793

10 Oscillators and Frequency Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . Vadim Issakov and Ulrich L. Rohde

951

11 Frequency Synthesizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Ulrich L. Rohde

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12 Software Defined Radio, Receiver and Transmitter Analysis . . . . . . 1183 Ulrich L. Rohde and Hans Zahnd 13 Mixing and Frequency Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . 1241 Nils Pohl 14 Modulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297 Dietmar Rudolph Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1509 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1529

Editors and Contributors

About the Editors Hans L. Hartnagel received the Dipl.-Ing. degree in 1960 from the Technical University of Aachen, West Germany, and the Ph.D. and the Dr. Eng. degrees from the University of Sheffield, England, in 1964 and 1971, respectively. After having worked for a short period with Telefunken in Ulm, West Germany, he joined the Institute National des Sciences Appliquées, Villeurbanne, Rhône, France, and then the Department of Electronic and Electrical Engineering of the University of Sheffield as Member of staff. In January 1971, he became Professor of Electronic Engineering at the University of Newcastle upon Tyne, England. Since October 1978, he has been Professor of High Frequency Electronics at the Technical University of Darmstadt, Germany. He is Author of several books and numerous scientific papers on microwave semiconductor devices and their technology and circuits. He has held many consulting positions, partly while on temporary leave of absence from his University positions. In 1990, he was awarded the Max-Plack-Prize. In 1994, he received the Dr. h.c. from the University of Rome “Tor Vergata” in Italy, and in 1999, the Dr. h.c. from the Technical University of Moldova, Kishinev. He is at present Emeritus Professor at the T.U. Darmstadt. e-mail: hartnagel@imp. tu-darmstadt.de Rüdiger Quay received the Diplom-degree in physics from Rheinisch-Westfälische Technische Hochschule (RWTH), Aachen, Germany, in 1997, and a second Diplom in economics from Fernuniversität Hagen in 2003. He was at Los Alamos National Labs, New Mexico, and at the University of Illinois at Urbana Champaign, USA, as Visiting Researcher. In 2001, he received his doctoral degree in technical sciences (with honors) from the Technische Universität Wien, Vienna, Austria. Since 2001, he has worked at the Fraunhofer Institute of Applied Solid-State Physics (IAF), Freiburg, in various positions.

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In 2009, he received the venia legendi (habilitation) in microelectronics, from the Technische Universität Wien. Since 2020, he is also Fritz-Hüttinger Professor for energy-efficient highfrequency electronics at the Department for Sustainable Systems Engineering (INATECH), Albert-Ludwig University, Freiburg, Germany. Since 2022, he is Executive Director of Fraunhofer IAF. In 2012, Dr. Quay received the European Microwave Prize. Prof. Quay has authored and co-authored over 350 refereed publications, three monographs, and contributions to two further. Ulrich L. Rohde received his Dr.-Ing. degree from the Technical University, Berlin, and his Dr.-Ing. Habil degree from the Brandenburg University of Technology, Cottbus, Germany. He is a member of the Faculty of Technical Informatics as the Professor of Microwave Systems, Universität der Bundeswehr (Federal University of the Armed Forces), Munich, Germany. He also holds appointments at other universities worldwide. Dr. Rohde is Partner of Rohde and Schwarz, Munich, Germany, as well as Chairman of the Board of Synergy Microwave Corporation. Formerly Professor of Electrical Engineering at George Washington University and the University of Florida, Gainesville, Dr. Rohde has consulted on numerous communication projects in industry and government. He has authored over 350 papers, 6 books, and holds 35 patents. In 1983 with his team at the RCA Radio Business Unit he designed and proved the concept of a system that today is referred to as the software-defined radio (SDR). He later acquired Compact Software and introduced the ability to accuracy simulate noise contribution on signals in non-linear systems and its influence and distortion in oscillators, mixers, and amplifiers. Dr. Rohde has received numerous prestigious awards including honorary Ph.Ds degrees. He is a honorary member of the Bavarian Academy of Science and Humanity, a Life Fellow of the IEEE, and a Fellow of the Indian National Academy of Engineering (INAE). Matthias Rudolph received the Dipl.-Ing. degree in electrical engineering from the Berlin Institute of Technology, Berlin, Germany, in 1996, and the Dr.-Ing. degree from Darmstadt University of Technology, Darmstadt, Germany, in 2001. In 1996, he joined Ferdinand-Braun-Institut, Leibniz Insitut für Höchstfrequenztechnik (FBH), Berlin, where he was responsible for GaAs, InP, and GaN-based transistor modeling and later headed the low-noise component group. In October 2009, he was appointed Ulrich L. Rohde Professor for RF and Microwave Techniques at Brandenburg University of Technology, Cottbus, Germany. He authored or co-authored over 90 publications in refereed journals and conferences and five monographs on transistor modeling, circuit design, and noise.

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Contributors Holger Arthaber TU Wien, Vienna, Austria Jan Hesselbarth University of Stuttgart, Stuttgart, Germany Vadim Issakov TU Braunschweig, Braunschweig, Germany Holger Maune Otto-V.-Guericke University Magdeburg, Magdeburg, Germany Renato Negra RWTH Aachen University, Aachen, Germany Nils Pohl Ruhr-Universität Bochum, Bochum, Germany Rüdiger Quay Fraunhofer-Institute for Applied Solid State Physics IAF, Freiburg im Breisgau, Germany Ulrich L. Rohde Brandenburg University of Technology, Cottbus, Germany Dietmar Rudolph Berlin, Germany Matthias Rudolph Brandenburg University of Technology, Cottbus, Germany Hans Zahnd Emmenmatt, Switzerland

The Authors of this Book

This book is the first edition of the Lehrbuch der Hochfrequenztechnik in English language, a book that immediately after its first publication in 1965 claimed its place as the leading textbook in German language known under the name of its two authors as the Zinke – Brunswig. A long list of new editions followed since then, and as early as 1973/1974, it became necessary to split the Zinke – Brunswig into two volumes. This edition provides the reader with a completely new and updated text. Some of the chapters required only minor revisions, others were rearranged or completely rewritten to reflect the advances of the state of the art within the past years. But we kept the tradition that a team of authors took over the task of revising the book, each of them responsible for a chapter that falls into his main area of expertise. The chapter authors are: Holger Arthaber, TU Wien authored • Chapter 5 Field-based description of propagation on waveguides on the basis of the 6th edition chapter authored by F. Arndt, A. Czylwik, R.W. Lorenz, B. Rembold, A. Vlcek, H. Vollhardt, O. Zinke, and U. Zwick Jan Hesselbarth, University of Stuttgart authored • Chapter 6 Antennas on the basis of the 6th edition chapter authored by G. Albert, H. Bottenberg, H. Brunswig, H. Heß, R.W. Lorenz, A. Vlcek, and O. Zinke Vadim Issakov, Braunschweig University of Technology authored • Chapter 10 Oscillators and Frequency Synthesis on the basis of the 5th edition chapter authored by H. Döring, K.-H. Gerrath, H. Heynisch, T. Motz, A. Müller, E. Pettenpaul, K.-H. Vöge, and O. Zinke

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Holger Maune, Otto-v.-Guericke University Magdeburg authored • Chapter 3: High-frequency transformers and balanced-to-unbalanced transformers on the basis of the 6th edition chapter authored by K. Mayer, R.W. Lorenz and O. Zinke • Section 4.9: Tunable filter Renato Negra, RWTH Aachen University authored • Chapter 1: Resonant circuits, one-portnet works, coupling filters made of lumped, passive components on the basis of the 6th edition chapter authored by H. Brunswig, G. Dittmer, R.W. Lorenz, A. Vlcek, K.H. Vöge, and O. Zinke Nils Pohl, Ruhr-Universität Bochum authored • Chapter 13 Mixing and frequency multiplication on the basis of the 5th edition chapter authored by H. Brunswig, K. Blankenburg, K.-H. Gerrath, K. Mayer, E. Pettenpaul, O. Zinke Rüdiger Quay, Fraunhofer-Institute for Applied Solid State Physics IAF and FritzHüttinger chair for energy-efficient high-frequency electronics, Albert-LudwigsUniversity Freiburg authored • Chapter 7 Semiconductors and semiconductor devices and circuits on the basis of the 5th edition chapter authored by G. Dittmer, H.L. Hartnagel, H. Heynisch, R. Losehand, K. Mayer, J.-E. Müller, E. Pettenpaul, A. Vlcek, and O. Zinke • Chapter 9 Amplifiers on the basis of the 5th edition chapter authored by H. Brunswig, G. Dittmer, H. Döring, H.L.Hartnagel, H. Heynisch, K.-H. Gerrath, A. Müller, J. E. Müller, E. Pettenpaul; A.Richtscheid, W. Welsch, K.-H. Vöge, and O. Zinke Ulrich L. Rohde, Brandenburg University of Technology, Universität der Bundewehr, Munich, and Rohde & Schwarz authored • Chapter 10 Oscillators and Frequency Synthesis • Chapter 11 Frequency synthesizer • Chapter 12 Software Defined Radio, Receiver and Transmitter analysis Dietmar Rudolph† authored • Chapter 14 Modulation Methods

The Authors of this Book

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Matthias Rudolph, Brandenburg University of Technology and Ferdinand-BraunInstitut, Leibniz-Institut für Höchstfrequenztechnik authored • Chapter 2: Wave propagation on transmission lines and cables on the basis of the 6th edition chapter authored by A. Vlcek and O. Zinke • Chapter 4: Properties of coaxial cables and transmission lines, directional couplers and RF filters on the basis of the 6th edition chapter authored by F. Arndt, R. Briechle, R. Dill, T. Motz, B. Rembold, H. Stocker, H. Vollhardt, and O. Zinke • Chapter 8 Interference and noise on the basis of the 5th edition chapter authored by A. Vlcek Clemens C.W. Ruppel authored • Section 4.11: Surface acoustic wave filters on the basis of the 6th edition section authored by H. Stocker and R. Dill

Chapter 1

Resonant Circuits, One-Port Networks, Coupling Filters Made of Lumped, Passive Components Renato Negra

Abstract This first chapter of the book is dedicated to resonant circuits. The frequency response and other important properties of inductances, capacitances and resonant circuits are introduced. Examples for applications in impedance matching circuits and filters will be provided. Understanding resonator properties like frequency response, quality factor and energy balance is instrumental as a basis for the subsequent chapters, since these concepts play an important role when wave phenomena are to be considered as well as in active microwave circuits such as amplifiers, mixers and oscillators.

This first chapter of the book is dedicated to lumped resonant circuits. A lumped component has characteristic spatial dimensions that are negligibly small with respect to its operating wavelength, i.e. wave propagation processes do not yet play any role in the description of its functioning. A component is passive if it is not capable of delivering more power than is supplied to it. Classic components in this category are the inductance, the capacitor and the resistor as well as semiconductor devices such as semiconductor diodes and transistors that are acting as such. In RF engineering, inductances and capacitors are interconnected to form resonant circuits which contribute to the selection of the desired frequency band to be transmitted either as parallel circuits (trap circuits) or series circuits (absorption circuits). Section 1.2 discusses the influence of the magnitude and location of the loss resistances on resonance curves and plots of the impedance. Section 1.3 covers analysis of two-circuit coupling band filters as well as synthesis of coupling band filters with more than two circuits using filter catalogues. This chapter examines the usage of circuits having capacitor, inductance and resistor components or equivalent circuits consisting of ideal C, L, and R values. In Sect. 1.1, simple equivalent circuits for inductances and capacitors with losses at a fixed frequency precede the observations on resonant circuits. R. Negra (B) RWTH Aachen University, Aachen, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. L. Hartnagel et al. (eds.), Fundamentals of RF and Microwave Techniques and Technologies, https://doi.org/10.1007/978-3-030-94100-0_1

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2

R. Negra

When applied in the form of equivalent circuits to microwave circuits, the insights explored in this chapter can also be useful in computer-aided design of microwave filters, for example.

1.1 Vector Diagrams for Inductances and Capacitors with Losses Figure 1.1 contains vector diagrams for a inductance with harmonic excitation at a frequency f = ω/2π . The phase offset between inductance voltage, V, and inductance current, I diverges from 90◦ by the loss angle δ L . The physical inductance is therefore defined as a circuit element by its complex impedance Z = jωL e− jδL .

(1.1)

The graphics in the centre show the division of Z into real part, Rs , and imaginary part, jωL s , for which tan(δ L ) = Rs /ωL s , or alternatively the division of the admittance, Y = 1/Z , into real part, G p = 1/R p , and imaginary part, 1/jωL p . Here, we have tan(δ L ) = G p ωL p . L s and L p are lossless inductances. Since it follows that L s = L cos(δ L ) and L p = L/ cos(δ L ) from the vector diagrams in the I V

dL Ls

V LS

Ls =LcosdL V

dL

Z-jwLe-jdL

jwLs Rs

dL

V RS

Rs =wLsindL

=wLstandL

V LS

RS Gp - 1 Rp

I

dL

I

dL 1 jwLp

V RS

IRp

LP =L/cosdL

p

Lp

Y = 1 Z

ILp

IL

Rp

V RP =

dL

= I

dL

IRp

Fig. 1.1 Voltages, currents, and resistances in a lossy inductance at a fixed frequency

wL sindL wLp tandL

1 Resonant Circuits, One-Port Networks, Coupling Filters …

3

centre, L s = L p cos2 (δ L ) = L p /(1 + tan2 δ L ) is slightly different from L. However, if tan(δ L ) ≤ 3%, the difference between L s and L p is equal to less than l%. From the diagrams, we can also recognize the quality factor of a inductance which is defined as the ratio of the reactive power to the effective power: QL =

Rp ωL s 1 1 = = = . Rs ωL p G p ωL p tan(δ L )

(1.2)

We can see that at a fixed frequency, we have the choice of representing a physical inductance either by its complex inductance L e− jδL or by the series connection of the small series resistance, Rs , and the lossless inductance L s or also by the parallel connection of the lossless inductance, L p , with a very large ohmic resistance R p . Figure 1.2 provides an analogous representation of the physical capacitor. We can characterise it in an equivalent manner as the complex capacitance C e− jδC with magnitude C and loss angle δC or by the parallel connection of C p = C cos(δC ) with the high resistance R p or also by the series connection of the lossless capacitance Cs = C/ cos(δC ) with the very small real resistance Rs . From the centre graphics in Fig. 1.2, we can conclude that the quality factor of the capacitor can be determined alternatively from the following relationships: QC =

ωC p 1 1 . = ωC p R p = = Gp ωCs Rs tan(δC )

(1.3)

V Cp=CcosdC dC -jdC Y= 1 - jwCe 2

VC

S

IC

jwCp

IR

p

p

Cp

dC

Cs

V Cs

Rs I

dC Z= IC

Rs =

V Rs

1 jwCs

1 Y

1 wCptandC

Cs = C/cosdC V

IRp

1 wCsindC

=

I

Rs

V RS

Rp = =

1 GP = Rp

I

V

Rp

p

SC

Fig. 1.2 Voltages, currents, and resistances in a lossy capacitor at a fixed frequency

Rs =

sindC wC tandC wCs

4

R. Negra

1.2 Parallel and Series Resonant Circuits 1.2.1 Lossless Resonant Circuits We will first consider lossless circuits with no effective resistance. Then, in the parallel circuit in Fig. 1.3a, the admittance is 1 Y p = j B p = jωC p + jωL p 1 , = j ωC p − ωL p

(1.4)

and in the series circuit in Fig. 1.3b, the impedance is 1 Z s = j X s = jωL s + jωCs 1 , = j ωL s − ωCs

(1.5)

The resonant angular frequency ωr = 2π fr

(1.6)

is defined by susceptance j B p = 0, 1 jω p C p + =0 jω p L p

(1.7)

or ω2p L p C p = 1 in the parallel circuit, and reactance

Fig. 1.3 Lossless parallel (a) and series (b) circuits

a

b

Ls Lp

Cp Cs

1 Resonant Circuits, One-Port Networks, Coupling Filters … Fig. 1.4 Curve of the susceptance of the parallel circuit (a) and reactance of the series circuit (b) as a function of angular frequency

5

a

b

B

X XL

BC

Xs

Bp wp

w

w

ws

XC

BL

j X s = 0, 1 jωs X s + =0 jωs Cs

(1.8)

or ωs2 L s Cs = 1

(1.9)

in the series circuit, or more generally in both cases by omitting the indices as follows: ωr = √

1 LC

,

ωr2 LC = 1.

(1.10)

Figure 1.4 shows the curves for B p and X s , respectively.1 At the resonant frequency, fr , the duration of the oscillation period is as follows: T =

√ 2π 1 = = 2π LC. fr ωr

(1.11)

Extending (1.4) with ωr , the parallel circuit has j B p = jω p C p

ωp ω 1 − ωp ω ωp L pωpC p

,

(1.12)

and the series circuit has j X s = jωs L s

1 ω ωs − ωs ω ωs L s ωs Cs

,

(1.13)

Combining this with Eqs. (1.6) and (1.8), we obtain the following for the parallel circuit: 1

“Thomson formula” according to William Thomson also known as Lord Kelvin.

6

R. Negra

ωp ω − ωp ω ωp ω 1 = j − ωp L p ωp ω

j B p = jω p C p

= j BK p ν p BK p

(1.14)

1 = ωpC p = = ωp L p

Cp . Lp

(1.15)

This is the “characteristic susceptance” of the parallel circuit. For the series circuit, it holds analogously that j X s = jωs L s

ω ωs − ωs ω

1 = j ωs Cs

ω ωs − ωs ω

= j X K s νs .

(1.16)

XKs

1 = ωs L s = = ωs Cs

Ls . Cs

This is the “characteristic impedance” of the series circuit. In both cases, we thus have the following: L 1 . = XK = Bk C

(1.17)

(1.18)

The characteristic impedance indicates the magnitude of the reactance of each of the elements at the resonant frequency. The quantity ν in (1.16) is known as the relative detuning: ν=

f ω ωr fr = − = . ωr ω fr f

(1.19)

Here, ω (or f ) is any frequency apart from the resonant frequency; for f > fr

we have v > 0,

f < fr

we have v < 0,

For small deviations from the resonant frequency, we can simplify (1.19) by introducing δω = ω − ωr as the deviation from the resonant frequency ωr . We then have

1 Resonant Circuits, One-Port Networks, Coupling Filters …

7

ω − ωr2 ω ωr (ω + ωr )(ω + ωr ) = − = ωr ω ωωr ωωr (2ωr + δω)(δω) = (ωr + δω)(ωr ) δω 1 + 2ω 2δω r = = . ωr 1 + δω ωr

ν=

(1.20)

If the deviation from the resonant frequency remains δω ≤ 0.1ωr , we have the following for an error ≤ 5%: ν≈

2δ f 2(ω − ωr ) 2( f − fr ) 2δω = = = . ωr fr ωr fr

(1.21)

1.2.2 Resonant Circuits with Resistive Losses Taking into account the circuit losses for the parallel circuit shown in Fig. 1.5a, Eqs. (1.4) and (1.16) become Y p = G p + jω C p −

1 ωL p

= G p + j BK p ν p .

(1.22)

For the series circuit in Fig. 1.5b, we have

a

b I

I0 IL p

tan d =

Gp wr Cp

IRp

Cs

ICp Ls

V

Lp

Gp

Cp

V Ls tan d =

V0

Rs

Fig. 1.5 Lossy parallel and series circuits

V Cs

V Rs

Rs wr L s

8

R. Negra

1 Z s = Rs + j ωL s − ωCs = Rs + j X K s νs .

(1.23)

With Eqs. (1.16) and (1.21), close to resonance we obtain the following: Y p ≈ G p + j2C p (ω − ω p ), Z s ≈ Rs + j2L s (ω − ωs ).

(1.24)

The imaginary parts go to zero at resonance, leaving Yp = G p Z s = Rs .

(1.25)

With the usual composition, the circuits represent (for the parallel circuit) a small conductance or a large resistance R p = 1/G p , and (for the series circuit) a small resistance Rs . These real components can be divided among the inductance and capacitance by introducing the loss angles δ L and δC as well as the loss factors tan(δ L ) and tan(δC ). 1 G p = G Lp + GCp = tan(δ L p ) + ω p C p tan(δC p ) ωp L p 1 XKs 1 = + = BK p QLp Q Cs Qs 1 Rs = R Ls + RCs = ωs L s tan(δCs ) + tan(δCs ) ωs Cs 1 XKs 1 = + . = XKs Q Ls Q C1 Qs

(1.26)

Here, Q is the “circuit quality factor” where 1 1 1 = + Q QL QC

or

Q=

Q L QC . Q L + QC

(1.27)

The loss factors for the inductor and capacitor yield the loss factor of the circuit tan(δ) = tan(δ L ) + tan(δC ).

(1.28)

If Y p or Z s from (1.23) is referred to the value at ω = ωr , the equations in the “normalised representation” are as follows:

1 Resonant Circuits, One-Port Networks, Coupling Filters …

9

For the parallel circuit BK p Yp =1+ j νp Gp Gp = 1 + j Q pνp = 1 + j Vp

(1.29)

and for the series circuit X Ks Zs =1+ j νs Rs Rs = 1 + j Q s νs = 1 + j Vs .

(1.30)

Here, the quantity D = Qν = Q

ωr ω − ωr ω

=Q

f fr − fr f

(1.31)

is known as the “normalised detuning”. The plots of the normalized complex impedance of the parallel circuit Z p /R p = G p /Y p or the complex admittance of the series circuit Ys /G s = Rs /Z s [Eqs. (1.23) and (1.30)] are shown versus frequency in Fig. 1.6a. Figure 1.6b shows the normalized representation vs. D. Here, Rs (or G p ) is assumed to be independent of frequency. In this case, the plots that come about are simple circles that arise through inversion of lines. Now of interest are the two frequencies at which the resistance and reactance (or conductance and susceptance) have an equal magnitude, i.e. the phase angle is equal of the impedance |Z p | or the admittance |Ys | has fallen to ±45◦ . Here, the magnitude √ to a value equal to 1/ 2 times its value at the resonant frequency, i.e. a drop of 3 dB. According to Eq. (1.30), the real part is equal to the imaginary part if |D| = Q|ν| = 1.

(1.32)

The magnitude for the parallel circuit is then obtained as Y p Gp

=

√ 1 + D 2p = 2.

(1.33)

and for the series circuit as √ |Z s | = 1 + Ds2 = 2. Rs

(1.34)

10

R. Negra

a

b

Im

Im fc 1

D=−1 0,5

0,5 f

f=0 f=∞

45˚

fr

45˚

1

0,5

V

Re

Zp Ys ; Rp Gs

−0,5

D=−∞ D=+∞

45˚ 45˚

−0,5

fc 2

D=0 0,5

D=+1

1

Re

Zp Ys ; Rp Gs

Fig. 1.6 Plots of Z p /R p and Ys /G s for the lossy parallel and series circuits a; graph b shows the normalized representation

Assuming (1.32) is fulfilled at the relative detunings νg1 and νg2 , i.e. at the frequencies f c1 and f c2 , we have νg1 = −νg2

and

|νg1 | = |νg2 | = |νg |

(1.35)

or f c2 fr f c1 − =− − fr f c1 fr 1 f c2 + f c1 = fr + fr f c1 Therefore, fr2 = f c1 f c2 ;

fr f c2 f c2 + f c1 1 = fr . f c2 f c1 f c2 fr f c1 = . f c2 fr

(1.36)

(1.37)

fr is thus not the arithmetic but rather the geometric mean of f c1 and f c2 . With (1.37), we finally obtain the following: f c2 fr fc fc − = 2 − 1 fr f c2 fr fr Lp f c − f c1 Δf Cp = 2 = = . fr Rp f c1 f c2

|νg | = νg2 =

(1.38)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

11

The difference frequency Δf for the 45◦ detuning serves as a measure of the width of the resonance curve, and Δf is known as the “bandwidth”. The narrower a circuit’s resonance curve, the smaller its bandwidth. Δf provides an indication of a circuit’s ability to separate closely spaced frequencies (“selectivity” of a receiver). Moreover, Δf is a measure of the quality factor, Q of the circuit since with (1.32) we have fr ωr 1 1 = = = = ωr C p R p |νg | Δf Δω tan(δ)

1 . 2π R p C p (1.39) In addition to the determination from the frequency change, the bandwidth can also be measured by detuning the capacitance, C, of a circuit. For the parallel circuit, we can rewrite (1.23) for the resonant frequency ω = ω p as follows: Q=

Yp = G p 1 + j

and

Δf =

1 2 (ω L p C p − 1) . ωp L pG p p

(1.40)

By applying 1 = C pr = capacitance in resonant case, ω2p L p we obtain

Yp = G p 1 + j

1 ωp L pG p

Cp −1 C pr

(1.41)

(1.42)

and analogous to (1.30) Yp = 1 + j Q p νc . Gp

(1.43)

Accordingly, we obtain νc =

Cp − 1. C pr

(1.44)

If C p1 and C p2 are the capacitances at which νc2 = −νc1 , then we have C p2 C p2 C p1 + C p2 , −1=− − 1 → C pr = C pr C pr 2 C pr is thus the arithmetic mean of C p1 and C p2 . For

(1.45)

12

R. Negra

ΔC p = C p2 − C p1

(1.46)

ΔC p 2

(1.47)

and C p = C pr ± it follows that |νc | =

ΔC p /2 . C pr

(1.48)

Moreover, if the relationship Q p |νc | = 1

or

Qp =

C pr 1 = |νc | ΔC p /2

(1.49)

is fulfilled, then C p1 and C p2 are the √ capacitance values at which the impedance of the parallel circuit has fallen to 1/ 2 times the value at resonance. We would now like to examine for the parallel circuit the ratio of the branch currents in the circuit to the current flowing into the circuit (or for the series circuit the voltages present on the individual circuit elements to the voltage on the entire circuit). For the parallel circuit in Fig. 1.5a, we have I0 = I L p + IC p + I Rp ,

(1.50)

and for the series circuit in Fig. 1.5b, we have V0 = VLs + VCs + VRs ,

(1.51)

As examples, we would like to compute |I L p | = |I0 |

ωL p Rp

2

1

,

(1.52)

+ (ω2 C p L p − 1)2

where I0 = constant current flowing into in the parallel circuit and 1 , |VCs | = |V0 | 2 (ωCs Rs ) + (ω2 Cs L s − 1)2 where V0 = constant open-circuit voltage in the series circuit.

(1.53)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

13

The maximum occurs if the root is at a minimum. Differentiation yields the angular frequencies Lp 1 ω Imax = 1− (1.54) 2C p R 2p L pC p

and ωVC max

1 =√ L s Cs

1−

2Cs Rs2 2L s

(1.55)

or applying the quality factors from (1.26), we obtain ω I L max = ω p 1 −

≈ ωp

ωVC max

1 2Q 2p

1 1− 4Q 2p

, (1.56)

1 = ωs 1 − 2Q 2s

1 ≈ ωs 1 − . 4Q 2p

The amount of the maxima is then equal to Qp I L p max = ≈ Qp |I0 | 1 − 4Q1 2

(1.57)

Qs VCs max = ≈ Qs . |V0 | 1 − 4Q1 2

(1.58)

p

or

s

For the parallel circuit, we also have IC p max I L p max = . |I0 | |I0 |

(1.59)

These currents in the inductive and capacitive branches are opposed to one another and can be significantly larger than the current inflow as a function of the quality factor of the circuit (resonance rise, current resonance). The greater the value of Q p , the less the maxima are offset with respect to ω p .

14

R. Negra

For the series circuit, it holds analogously that VCs max VLs max = . |V0 | |V0 |

(1.60)

The voltage drops on the inductor and on the capacitor are opposed to one another and can be significantly greater than the source voltage as a function of the quality factor of the circuit (resonance rise, voltage resonance). The greater the value of Q s , the less the maxima are offset with respect to ωs . Figure 1.7 illustrates the frequency dependencies of the related currents, voltages as well as the overall impedances by magnitude and phase as a function of frequency referred to the resonant frequency. In the selected example, we have Q p = Q s = Q = 2.24. Graph (a) applies to the parallel circuit and graph (b) to the series circuit. In Fig. 1.8, the circumstances are plotted for Q p = Q s = Q = 2.24. The circuits

a

b 3

3 Q

ICp/I0

ILp/I0

Q

V Cs/V 0

V Ls/V 0

2

I/I0

V/V 0

2

1

1 IRp/I0 0 0,6

0,8

1,2

1,0

V Rs/V 0 0 0,6

1,4

0,8

1,0

1,2

1,4

w/ws

w/wp 2

2 2 Zp/Rp

1 1/ 2

1

90˚

90˚ Zs/Rs

0

0

0˚

jZp

0˚

45˚ jZs jZs

Zs/Rs

Zp /Rp

45˚ jZ p

−45˚

−45˚

Δwg /wp −1 0,6

0,8

1,0 w/wp

1,2

−90˚ 1,4

−1 0,6

Δwg/ws 0,8

1,0

1,2

−90˚ 1,4

w/ws

Fig. 1.7 Dependency of the currents, voltages, and input impedances by magnitude and phase on the angular frequency for the lossy parallel circuit (a) and series circuit (b). Circuit quality factor √ Q = 5 = 2.24

1 Resonant Circuits, One-Port Networks, Coupling Filters …

a

15

b 25

25 Q

Q

20

10

2 ICp/I0

ILp/I0

5

1

IRp/I0

0 0,6

1,0 w/wp

0,8

1,2

15

0

0 1,4

V Ls/V 0

V Cs/V 0

5

V Rs/V 0 0,6

0,8

1,0

1,2

1

0 1,4

w/ws 2

−1 0,6

0,8

1,0

Δwg/wp

−45°

1,2

−90° 1,4

Zs/Rs

0°

p

1/ 2

45°

0

90°

1

90° Zp/Rp

jZ

1

Zs/Rs

2

jZp

jZs 0

−1 0,6

w/wp

45° 0°

0,8

Δwg/ws

−45°

1,2

−90° 1,4

1,0 w/ws

jZs

2

Zp/Rp

2

10

IRp/I0

IRp/I0 ; ICp/I0

15

V Rs/V 0

V Cs/V 0 ; V Ls/V 0

20

Fig. 1.8 Dependency of the currents, voltages, and input impedances by magnitude and phase on the angular √ frequency for the lossy parallel circuit (a) and series circuit (b). Circuit quality factor Q = 10 5 = 2.24

behave thus at the following frequencies: ⎫ ω < ω p : Inductance ⎬ ω = ω: p Large real resistance Parallel circuit, ⎭ ω > ω p : Capacitance ⎫ ω < ω p : Capacitance ⎬ ω = ω p : Small real resistance Series circuit. ⎭ ω > ω p : Inductance The impedance of the complex parallel resonant circuit shown in Fig. 1.9 (approximate equivalent circuit for low capacitor losses) is obtained as follows: Z=

1 (R + jωL) jωC

R + jωL +

1 jωC

=

R + jω[L(1 − ω2 LC) − C R 2 ] . (1 − ω2 LC)2 + (ωC R)2

In the plot in Fig. 1.9b for Z /R, the imaginary part becomes zero at

(1.61)

16

R. Negra

Fig. 1.9 Circuit consisting of inductance with series resistance and parallel C (a). Plot of input impedance with phase resonance ω ph , magnitude resonance ωmax and circuit resonance ωr (b). √ Circuit quality factor Q = 5 = 2.24

ω ph = √

1

R2C 1 R 2 = ωr 1 − 1− = ωr 1 − 2 . L XK Q

LC

(1.62)

This frequency thus differs from the resonant frequency according to (1.9) by the attenuation correction (usually negligible for a circuit with low attenuation) of (R/ X K )2 since R X K . The resonance impedance is then obtained according to (1.61) as follows: Z ph ≈

X 2K L 1 = . ≈ CR R ω2ph C 2 R

(1.63)

The impedance, Z , has a maximum at

ωmax = ωr

1+2

R XK

2

−

R XK

2

R Xk

4

1 . 2Q 4 (1.64) and cannot be distinguished from ωr for ≈ ωr

ωmax is thus much closer to ωr than ω ph Q ≥ 10. Here, the quality factor Q is again defined as

1 1− 2

= ωr 1 −

1 Resonant Circuits, One-Port Networks, Coupling Filters …

Q=

17

L

1 XK ωr L C = = = R ωr C R R R

where ωr = √

1 LC

(1.65)

.

(1.66)

The magnitude and extreme values of the input impedance, Z , which is normalized to R are clearly determined by Q, as shown in the following (see also Fig. 1.9). The maximum impedance referred to R has the value |Z |max /R = Q / 2Q 2 ( 1 + 2/Q 2 − 1) − 1 ≈ Q 2 + 0.47 for Q ≥ 1. (1.67) 2

The plot has its maximum real part Remax /R = Q 2 /(1 − 1/(4Q 2 )) at ω/ωr =

1 − 1/(2Q 2 ).

(1.68)

In the first quadrant, the imaginary part, I m + /R, has the following value at Re/R = (Q 2 + 1)/2 ⎡ 2 1− 1 ⎣ Q 2 1+ I m + /R = 1− 2 2 Q Q 1+

1 Q2 1 Q2

⎤ +

1 ⎦ , 4Q 2

(1.69)

which provides a very good approximation for the maximum value. In the fourth quadrant, for the same value of the normalized real part (Q 2 + 1)/2 the imaginary part ⎤ ⎡ 1 2 1 − 1 Q 1 ⎦ 2 Q2 I m − /R = − 1 + 2 ⎣1 − + . (1.70) 2 Q Q 1 + Q12 4Q 2 lies in close proximity to its minimum which is reached for a real part between Q 2 /2 and (Q 2 + 1)/2.

1.2.3 Resonant Circuits with Multiple Resistances In practice, resonant circuits contain more resistances than are taken into account in the one-port networks in Figs. 1.5 and 1.9a. In network theory, it is common to use the complex angular frequency p = σ + jω instead of the real angular frequency. This abbreviation allows us to express the equations in a simpler manner. If we only consider harmonic oscillations, we have p = jω since σ = 0.

18

R. Negra

If we extend the resonant circuit in Fig. 1.5a with a series resistance, Rs (Fig. 1.10a), we obtain the impedance function for the resulting one-port network as follows: p 2 LC + p(L/Rs + L/R p ) + 1 1 = Rs . 1/R p + pC + 1/ pL p 2 LC + pL/R p + 1 (1.71) The plot remains a circle which, compared to the plot in Fig. 1.6, is shifted along the real axis by Rs (Fig. 1.10), as is also apparent from the√impedance function. In the case of phase resonance p = jω ph = jωr = j/ LC, we have Z ( p) = Rs +

Z ( jω ph ) = Rs + R p ,

(1.72)

whereas for p → 0 and p → ∞, we have Z (0) = Z (∞) = Rs .

(1.73)

A better approximation of real resonant circuits is given by the equivalent circuit in Fig. 1.10a. Its impedance function is p 2 LC + p(L/RC + R L C) + R L /RC (R L + pL)(RC + 1/ pC) . = RC pL + 1/ pC + RC + R L p 2 LC + pC(RC + R L ) + 1 (1.74) Here, we now have Z (0) = R L and Z (∞) = RC (see also the plots in Fig. 1.11b– d). Moreover, Z ( p) is real for p = jω ph (phase resonance) if the phase angles of the numerator and denominator are equal. It then follows from (1.74) that Z ( p) =

ω2ph =

1 − (R L / X K )2 1 L/C − R L 2 = ωr 2 . 2 LC L/C − RC 1 − (RC / X K )2

We then have Z ( jω ph ) =

L/C R L RC + . R L + RC R L + RC

(1.75)

(1.76)

In circuits with high quality factor, only the first summand is relevant. √ According to (1.75), phase resonance occurs only if the characteristic reactance L/C is less than the smaller of the two resistances R L and RC or greater than the larger of the two resistances. It is notable here that in the first case, the one-port network impedance passes through a minimum of the impedance at phase resonance, as if this were a series resonant √ circuit. Here, however, the quality factor Q is always < 1/2. In the second case ( L/C greater than the larger of the two resistances R L and RC ), the impedance with a maximum of the resonance impedance like in √ increase occurs √ (1.30). For L/C R L and L/C RC , we then obtain the resonance quality factor from (1.2), and (1.27) as follows:

1 Resonant Circuits, One-Port Networks, Coupling Filters … Fig. 1.10 Lossy parallel circuit extended with Rs (a) and plots of the impedance for (b) R p /Rs > 1, (c) R p /Rs = 1, (d) R p /Rs < 1

19

a Rs

L

C

Rp

b w

Rs

Rs+Rp

c w

Rs

Rs+Rp

d

w

Rs

Rs+Rp

√

Q=

L/C

1. R L + RC

(1.77)

For the impedance functions (1.71) and (1.74) plotted in Figs. 1.10b–d and 1.11b– d, four facts are relevant:

20 Fig. 1.11 Parallel circuit consisting of inductor and capacitor with losses (a) and plots of the impedance for (b) R L /Rs > 1, (c) R L /RC = 1, (d) R L /RC < 1

R. Negra

a

RL

RC

L

C

b

RC

RL

w

w

L/C < RC

L/C > RL

c w wph = 1/ L/C

RL RC L/C < RC

L/C > RC

d w w

RL L/C < RL

RC L/C > RC

1 Resonant Circuits, One-Port Networks, Coupling Filters …

21

1. The numerator and denominator polynomial of the impedance functions have the same degree with the highest power of 2. This corresponds to the number of reactances. 2. In the entire frequency range, the plot of Z ( jω) lies in the right half-plane. For ω → 0 and ω → ∞, it assumes real, finite values. 3. The maximum phase difference that Z ( jω) can assume at two different frequencies is less than in the circuits with only one resistance. 4. All of the poles and zeroes in the impedance function have a negative real part (in pi = σi + jωi , we have σi < 0). One-port networks in which, like in (1.71) and (1.74), the highest power of p agrees in the numerator and denominator polynomial of the impedance function and the coefficients of the lowest power are present are considered to be one-port networks with “minimal phase”. A larger variation in the plots is obtained by adding a third resistance, e.g. Rs as a series resistance to the one-port network in Fig. 1.11a. Applying a delta-star transformation on the three resistances, we obtain a one-port network with a bridge structure (see Fig. 1.12). A general representation of all one-port networks with an inductance and a capacitance is possible by formulating the impedance function in (1.74) as follows: Z 2 ( p) = R∞

p 2 + a1 p + a0 p 2 + b1 p + b0

(1.78)

The coefficients a1 and b1 can be construed as time constants that are easily determined, like the coefficients a0 and b0 , in the analysis of given one-port networks. R∞ is the real value of the impedance for p → ∞. For one-port network synthesis, it is important to be able to reduce the number of free parameters to be considered from five R∞ , a0 , a1 , b1 , b0 ) to three ( A0 , A1 , B1 ) by introducing a frequency and an impedance normalization:

Fig. 1.12 Series-parallel one-port network (a) and equivalent bridge one-port network (b)

a

b RS

RL

RC

L

C

22

R. Negra

A1/B1 1

0

1

A0

A1/B1 < 1 < A0

Wph < ∝

2

1 0

A0

A1/B1

1 A0

W

1 < A1/B1 = A0

1 < A1/B1 < A0 Wph imaginary

5

0

W

W

A1/B1 = 1 < A0

1 < Wph < ∝

1

A0

0 W

W

A0 > 1

1

A0

0

1 < A0 1 or

A1 /B1 < A0

and

A1 /B1 < 1.

These conditions were already discussed in connection with the example in Fig. 1.11a and (1.75). Figure 1.13 shows the different plot forms of the impedance function for P = jΩ and different value ranges for A0 and the ratio A1 /B1 . The outer columns contain the plots that exhibit a phase resonance. Plots 2, 5, and 8 represent cases for which Ω ph is either 0, ∞ or imaginary. Here, the plots

1 Resonant Circuits, One-Port Networks, Coupling Filters …

23

Antenna

Trap circuit for f1

Trap circuit for f2

Tuning unit for f2

Transmitter II (f2)

C'

Absorption circuit for f1

Tuning unit for f1

L'

Choke

f2 >f1

Transmitter I (f1)

Absorption circuit for f2

Fig. 1.14 Multiple feeding of an antenna, splitter circuit made of lumped elements

are restricted only to a single quadrant: Z 2N ( jΩ) is always capacitive for A0 > 1 and always inductive for A0 < 1. For A0 = 1, we obtain circles as plots with Ω ph = 1, as discussed previously in connection with Fig. 1.10a-d. The case in which A0 = A1 /B1 = 1 corresponds to the Boucherot circuit for which R L = RC = X K = √ L/C (Fig. 1.11a).

1.2.4 Multiple Feed Circuit Made of Lumped Elements An application example with resonant circuits is a switching arrangement that allows two transmitters to operate with the same antenna, as shown in Fig. 1.14. The function of the individual circuits should be clear based on the previously derived properties. The trap circuits for f 1 and f 2 represent a reactance for the passed useful frequencies f 2 and f 1 which is added to the antenna’s imaginary component present at the antenna base and tuned out using the tuning unit. For further suppression of the interfering voltage supplied by the opposite transmitter, absorption circuits are arranged parallel to the cable output for f 1 and f 2 , respectively, and their imaginary component is transformed into a parallel resonant circuit for the useful frequencies f 2 and f 1 by connecting a capacitance C or inductance L in parallel. The “static earth choke” Dr prevents static (i.e. atmospheric) charging of the antenna; it represents a very high impedance for RF.

1.3 Coupling Band Filters in Transmission Systems A radio frequency communications system must be capable of transmitting multiple messages in parallel without any mutual interference. One possible method of

24

R. Negra

differentiation2 involves assigning a separate frequency band to each message channel. Broadcast channels are an example of frequency multiplexing of this sort. Band filters are needed to select the channels; such filters are expected to pass all signals within a frequency band with no distortion while blocking signals at all other frequencies. We will first consider the coupling band filters used in RF engineering based on the analysis of two coupled parallel resonant circuits. For coupling band filters with more than two circuits, we will apply operating parameter theory to the synthesis of filters with predetermined operating characteristics. We will begin with some definitions from the theory of two-port networks. The effective gain factor A B of a two-port network connected to a generator (open-circuit voltage V0 , internal resistance R1 ) and a load (load resistance R2 ) is as follows: V2 AB = V0 /2

R1 . R2

(1.80)

|A B | is the root of the ratio of the power provided to the load P2 = |V2 |2 /(2R2 ) to the maximum available power of the generator, P1 max = |V0 |2 (8R1 ), which can be sourced from the generator in the case of impedance matching. For passive networks, we have |A B | ≤ 1. The logarithm of the effective gain factor is the effective gain level. The reciprocal 1/A B is the effective attenuation factor and accordingly the complex effective attenuation level is g B = ln (1/A B ) = − ln (A B ) = a B + jb B .

(1.81)

The real part is the effective attenuation level specified in nepers3 V0 aB R2 = − ln (|A B |) = ln , Np 2V2 R1

(1.82)

while the imaginary part is the effective phase level, or effective attenuation angle,

V0 b B = − arg{A B } = arg 2V2

2

R2 R1

.

(1.83)

For a broader discussion of multiplexing methods see Chap. 14. The Scottish mathematician John Neper, also written Napier, (1550–1617) discovered logarithms and published works on spherical geometry that were important in navigation.

3

1 Resonant Circuits, One-Port Networks, Coupling Filters …

25

The effective attenuation level is commonly specified in decibels4 instead of nepers. Then, the effective attenuation level is aB = −20 log(|A B |). dB

(1.84)

Instead of the effective phase level b B , the group delay time that a signal group requires to pass through the two-port network is commonly used in communications engineering as follows: (1.85) tg (ω) = db B /dω. The frequency dependency of the attenuation level and group delay time in a transmission channel results in distortion that corrupts the signal.

1.3.1 Two-Circuit Coupling Band Filters Figure 1.15 shows some possibilities for coupling two parallel resonant circuits via a reactance. In circuits (a) and (c), the sum of the voltages of both resonant circuits is present on the coupling element (voltage coupling), while in circuits (b), (d), and (f), the sum of the oscillating currents of both circuits flows via the coupling element (current coupling). The transformer in circuit (e) can be characterized using equivalent circuit (f). Circuit (c) is not used in practice due to the parasitic capacitance of the coupling inductance L 12 . However, we will initially assume ideal reactances.

1.3.1.1

Analytical Calculation of Two-Circuit Coupling Filters

Circuits (b) or (d) and (f) in Fig. 1.15 can be computed using a delta-star transformation from circuits (a) and (c), respectively. Accordingly, it suffices for our analysis to investigate the equivalent circuit in Fig. 1.16. The elements of the admittance equations I1 = Y11 V1 + Y12 V2 (1.86) I2 = Y21 V1 + Y22 V2 are Y11 = j (B1 + B12 ), Y12 = Y21 = − j B12 , Y22 = j (B2 + B12 ). Applying the equations for the termination of the two-port network V0 = V1 + I1 R1 , 4

I2 = −V2 /R2 ,

Physiologist Alexander Graham Bell (born 1847 in Edinburgh, died 1922 in Nova Scotia) invented an electromagnetic telephone in 1876 that was characterized by good electrical matching. Conversion: 1 dB ≡ 0.1151 Np or 1 Np ≡ 8.686 dB.

26

R. Negra C12

a

b

L1

C1

C2

R2

L2

~

V0

L12

c

C 0T

L2

L1T

L2T

d

C1

L1

L2

C2

R2

~

V0

M

e

R2

L0T

C1

~

f

R2

C2

L '1-M

L '2-M

M

C2

R1

R1 V0

L1

~

R1

R1 V0

C 2T

R1

R1 V0

C 1T

~

L '1

C1

L '2

R2

C2

V0

~

C1

R2

Fig. 1.15 Possible coupling types for two-circuit coupling band filters, a capacitive voltage coupling; b capacitive current coupling; c inductive voltage coupling; d inductive current coupling; e transformer coupling; f equivalent circuit for e Fig. 1.16 Equivalent circuit for two-circuit, voltage-coupled coupling band filters

jB12

I1

I2

R1 V1 V0

jB1

jB2

V2

R2

~

we obtain the following for the effective gain factor: 2V2 AB = V0 =

R1 R2

√ 2 j B12 R1 R2 2 R R − (B + B )(B + B )R R + j (B + B )R + j (B + B )R 1 + B12 1 2 1 12 2 12 1 2 1 12 1 2 12 2

(1.87) Analogous to (1.13)–(1.17), we can shorten this to B1 + B12 = B K p1

ω ωr 1 − ωr 1 ω

= B K p1 ν1 .

(1.88)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

27

B K p1 is the characteristic susceptance and ωr 1 the resonant angular frequency of the first circuit when the second circuit is short-circuited. Moreover, we have ω ωr 1 = B K p2 ν2 . B2 + B12 = B K p2 − (1.89) ωr 1 ω Combining with (1.31), the “normalized detuning” D becomes (B1 + B12 )R1 = B K p1 R1 ν1 = Q 1 ν1 = D1 , (B2 + B12 )R2 = B K p2 R2 ν2 = Q 2 ν2 = D2 .

(1.90)

Here, R1 and R2 correspond formally to the resonance resistance of a parallel resonant circuit. For the effective gain factor, however, it is essential for R1 to signify the signal source internal resistance and R2 the load resistance. Accordingly, we refer to Q 1 and Q 2 in (1.90) as the “external quality factor” as opposed to the “unloaded quality factor” of a lossy resonant circuit in (1.26). Further analysis relates to the important special case in which D1 = D2 = D (same resonant frequencies and loaded quality factors of the resonant circuits). With the “normalised coupling” (1.91) K = |B12 | R1 R2 we obtain the effective gain factor AB =

±2 j K . 1 + K 2 − D2 + 2 j D

(1.92)

The positive sign applies for capacitive voltage coupling, B12 = ωC12 , and the negative sign for inductive, B12 = −1/(ωL 12 ). The frequency dependency of A B in the vicinity of the resonant frequency is characterized primarily by D since K exhibits relatively little change with ω or 1/ω within the passband. As a first approximation, we will assume that K is independent of frequency. The magnitude 2K |A B | = 2 (1 + K − D 2 )2 + 4D 2

(1.93)

is plotted in Fig. 1.17a for different values of K . Extreme values are present for D = 0, i.e. at the resonant frequency of the circuits, and for Dh = ± K 2 − 1.

(1.94)

Real solutions for Dh exist only for K ≥ 1. The maxima (or “humps”) of |A B | lie at Dh . Applying (1.94) in (1.93) leads to

28

R. Negra 1.0

−Dh

Dh

ABH for K = 2

0.8 Dg

−Dg

AB

H

2

0.6 AB

Fig. 1.17 a Magnitude of the effective gain factor A B ; b normalized group delay tg N = tg /(d D/dω) as a function of the normalized detuning D. For K = 2, the hump detuning Dh , the mathematical limit detuning Dg and the practical limit detuning Dc are plotted for |A B |

Dc

−Dc

0.4

2 1

0.2 K = 0,5 0 1.6

tgN

1.2 0.8 2 1

0.4

0 −5

K = 0,5 −4

−3

−2

−1 D

0

|A B | D=Dh = 1

1

2

3

4

5

(1.95)

At the humps, R2 is thus optimally matched to the generator internal resistance R1 . We designate the different cases as follows: K = 1 “Critical coupling” (matching for D = 0), the maximum is flat (triple zero); K > 1 “Overcritical coupling” (matching for ±|Dh | in two places), |A B | D=0 = 2K /(1 + K 2 ) < 1; K < 1 “Undercritical coupling” (no matching is obtainable), an absolute maximum of |A B | < 1 for D = 0. For band filters with overcritical coupling, two more characteristic values are defined for the bandwidth. “Mathematical limit detuning” Dg : This is characterized in that for Dg the same effective gain factor is obtained as for D = 0, i.e. |A B | D=Dg = |A B | D=0 .

(1.96)

From (1.93), we thus obtain the conditional equation for Dg : 2 2 1 + K 2 − Dg2 + 4Dg2 = 1 + K 2 . Besides the trivial solution Dg = 0, it follows that

(1.97)

1 Resonant Circuits, One-Port Networks, Coupling Filters … Fig. 1.18 Effective attenuation as a function of the normalized detuning plotted on a logarithmic scale. Curve 1: single circuit; curve 2: two-circuit coupling band filter with K = 1; curve 3: two-circuit coupling band filter with K = 2.4

29

80 dB 70

60 40 dB/decade

GB

50

40

20dB

30 2

3

1

decade

20

10

0 0.1

10

1

100

D

Dg =

2(K 2 − 1) =

√

2Dh .

(1.98)

√ “Practical limit detuning” Dc : |A B | has dropped by the factor 1/ 2 (≡ 3 dB) with respect to the average value 1

|A B M | = 1 2

1 |A B |2D=0

It thus follows that Dc =

+

√ 2K .

1 |A B |2D=D

(1.99)

h

(1.100)

30

R. Negra

Fig. 1.19 On the graphical construction of the curve |A B | = f (V )

jD

a

P=jD g

P1 jDh K

S b

-jDh

P2

The benefit of the coupling band filter versus a single resonant circuit comes in the form of larger bandwidth and improved far selectivity. In the single circuit, the gain factor for D = 1 is already 3 dB below the value at resonance. Far away from the centre frequency, |A B | falls off proportional to 1/D in the single circuit and proportional to 1/D 2 in the two-circuit band filter. An edge steepness of 40 dB/decade is obtained compared to 20 dB/decade with the single circuit as shown in Fig. 1.18. The curves |A B | = f (D) can also be determined graphically from the position of the poles and zeroes in the complex P-plane. If we insert P = j D into (1.92),5 then we have AB =

±2 j K ±2 j K = P 2 + 2P + K 2 + 1 (P − P1 )(P − P2 )

(1.101)

with the complex poles P1,2 = −1 ± j K .

(1.102)

The two poles are plotted in the plane P = Σ + j D in Fig. 1.19. The form of the curves |A B | is dependent on the product of the path lengths a = |P − P1 | and b = |P − P2 |. const 2K = . (1.103) |A B | = |P − P1 ||P − P2 | ab As the point P = j D moves along the imaginary axis, the area of the shaded triangle P, P1 , P2 remains constant with base P1 − P2 and height, h = 1. The triangle area is Here, we use uppercase letters P = Σ + j D to indicate the normalized quantities as opposed to p = σ + jω.

5

1 Resonant Circuits, One-Port Networks, Coupling Filters …

FΔ =

ab sin(γ ) = |P2 − P1 |h/2 = const. 2

31

(1.104)

|A B | has a maximum at a minimum of ab and this is attained at a maximum of sin(γ ). Accordingly, the hump frequencies are obtained at the intersections of the Thales’ circle over the path |P2 − P1 |, where γ = π/2. (1.94) can then be gathered from the shaded triangle. Thanks to the graphical construction, we can immediately see the importance of the critical, overcritical, and undercritical coupling as well as the enlargement of the bandwidth with increasing coupling. Besides |A B |, we are primarily interested in the group delay (1.85) tg = −

Im{A B } d arctan . dω Re{A B }

(1.105)

Applying the transfer function according to (1.92), we obtain tg =

dD dD 2(D 2 + K 2 + 1) = tg N . 2 2 2 2 (D − K − 1) + 4D dω dω

(1.106)

Figure 1.17b shows the “normalized group delay” tg N as a function of D. For Dtm = ± 2K 1 + K 2 − (1 + K 2 ) (1.107) √ maxima of the group delay occur which converge for K = 1/3 = 0.577, i.e. for undercritical coupling. For K = 1, 20% delay fluctuations are to be expected in the passband, as seen in Fig. 1.17b, and for K = 2, the delay near the band edges is 160% greater than in the centre of the band. Due to the delay distortion, the value of K must not be chosen too large.

1.3.1.2

Dimensioning of Two-Circuit Coupling Filters

Based on our general analysis, we can derive a procedure for dimensioning twocircuit coupling band filters with specified properties. The following values are given: Lower cutoff frequency f c− and upper cutoff frequency f c+ at which |A B | has fallen by 3 dB with respect to the average value. The choice of K follows from the permissible ripple in |A B | or tg . R1 and R2 are known from the circuit in which the filter is to be installed. According to (1.90), we have the following for the normalized detunings at the band edges: Dc− = B K pi Ri ( f c− / fr − fr / f c− ), Dc+ = B K pi Ri ( f c+ / fr − fr / f c+ ),

(1.108)

32

R. Negra

This equation holds for resonator circuit 1 (i = 1) and resonator circuit 2 (i = 2). We then obtain the following with Δf c = f c+ − f c− : ΔDc = D

c+

−D

c−

= B K pi Ri

Δf c fr Δf c + fr f c+ f c−

(1.109)

Choosing the resonant frequency, fr to be the geometric mean of the two cutoff frequencies (1.110) f r = f c+ f c− , we then obtain ΔDc = 2B K pi Ri Δf c / fr .

(1.111)

From (1.100) it follows that √ ΔDc = Dc+ − Dc− = 2|Dc | = 2 2K ,

(1.112)

and we can thus compute the characteristic susceptances of the two resonant circuits as follows: √ (1.113) B K pi = 2K fr /(Ri Δf c ). The coupling susceptance is determined from the definition of the normalized coupling in (1.91) as follows: B12 = K / R1 R2 .

(1.114)

For frequency-dependent coupling elements, (1.114) should be satisfied at the resonant frequency, fr . We then have the following for the circuits (a) and (c) in Fig. 1.15: or L 12 = 1/(ωr |B12 |). (1.115) C12 = |B12 |/ωr The elements in circuits (b), (d) to (f) in Fig. 1.15 can be computed using a deltastar transformation. The capacitances of the star in circuit (b) are then C0T = C1 + C2 + C1 C2 /C12 , C1T = C1 + C12 + C1 C12 /C2 , C2T = C2 + C12 + C2 C12 /C1 , and the inductances of the star in circuit (d) or (f) are

(1.116)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

33

Table 1.1 Numerical example for coupling band filters C1 and C2

a

C12 = 16 pF

C12 = 38.6 pF

C0T = 775 pF

C0T = 321 pF

c

C1 = C2 = 96.5 pF L 1 = L 2 = 2.27 µH C1T = C2T = 128.5 pF L 1 = L 2 = 2.27 µH C1 = C2 = 112.5 pF L 1 = L 2 = 2.64 µH

L 12 = 16 µH

L 12 = 6.62 µH

d

C1 = C2 = 112.5 pF

L 1T = L 2T = 1.98 µH

L 0T = 0.33 µH

L 0T = 0.8 µH

e

C1 = C2 = 112.5 pF

L 1 = L 2 = 2.31 µH

M = 0.33 µH

M = 0.8 µH

b

L 1 and L 2

Coupling values; K = 1 Coupling values; K = 2, 4

Circuit

L 0T = M = L 1 L 2 /(L 1 + L 2 + L 12 ), L 1T = L 1 − M = L 1 L 12 /(L 1 + L 2 + L 12 ), L 2T = L 2 − M = L 2 L 12 /(L 1 + L 2 + L 12 ).

(1.117)

Table 1.1 illustrates an example for the elements of the circuits in Fig. 1.15a-e which were determined from (1.114) to (1.117) with R1 = R2 = 1 kΩ, f c− = 9 MHz, f c+ = 11 MHz and K = 1. The transfer functions and delays are shown in Figs. 1.20 and 1.21 for K = 1. The curves with the coupling increased to K = 2.4 were also plotted according to √ (1.93). We then have |A B | = 1/ 2 for D = 0 without modifying the other circuit elements. The different behaviours of the curves for K = 1 compared to Fig. 1.17a and b can be explained as follows: 1. The normalized coupling K is frequency-dependent, and thus the cutoff frequencies are shifted to the right for capacitive current coupling and to the left for inductive current coupling. 2. The relationship between D and ω is nonlinear, (1.88) and (1.90), thus the frequency scale is more compressed for f < fr than for f > fr . In particular, this also causes the asymmetry of the delay curves for K = 1. For an increase in the coupling without changing the other elements, the resonant frequency decreases in circuits (a) and (d) and increases in circuits (b) and (c). For transformer coupling, the resonant frequency is unchanged but the humps drift apart more or less symmetrically. In the example in Figs. 1.20 and 1.21, the relative bandwidth Δf c / fr = 20% (for K = 1) is very large, while the asymmetries in the passband are less for smaller bandwidths.

1.3.2 Matching Circuits According to (1.113), the characteristic susceptance of the resonant circuits increases for smaller values of Δf c and Ri . Due to the losses and parasitic reactances of capacitors and inductors, the characteristic susceptances B K p can be produced satisfactorily only in a limited range (1 mS ≤ B K p ≤ 100 mS). A resistance transformation

34

R. Negra

a 1.00

b

0.75 K = 2,4

1

2,4

K=1

AB

0.50

0.25

0 0.40 µs

0.30

tg

0.20

0.10

0

6

8

10

12

MHz

14 6

8

10

12

MHz

14

f

f

Fig. 1.20 |A B | and tg for coupling band filters with the reactances in Table 1.1. a Capacitive voltage coupling; b capacitive current coupling

as depicted in Fig. 1.22 is commonly required. For coupling band filters, resonance transformers with two reactances are especially suitable for this purpose. The input admittance of the circuit according to Fig. 1.23 is Ye = j B0e + G 2 j B0 /(G 2 + j B0 ). It is necessary in the vicinity of fr that Re{Y } = G 1 = 1/R1 = t G 2 = t/R2 .

(1.118)

t = R2 /R1 = G 1 /G 2

(1.119)

(1.121) where

is the resistance conversion ratio, and Im{Ye } = j Be = 0.

(1.120)

We thus obtain B0 = ± t/(1 − t)G 2

and

B0e = ∓ t (1 − t)G 2 .

(1.121)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

c

35

d

1.00

0.75

AB

0.50 K=1

2,4

K = 2,4

1

0.25

0 0.40 µs 0.30

tg

0.20

0.10

0

6

8

10

12 MHz 14 6

e

8

10

12 MHz 14

f

f 1.00 0.75

AB

0.50

K = 2.4 1

0.25

0 0.40 µs 0.30

tg

0.20

0.10

0

6

8

10

12 MHz 14

f

Fig. 1.21 |A B | and tg for coupling band filters with the reactances in Table 1.1. a Inductive voltage coupling; b inductive current coupling; c transformer coupling

36

R. Negra

R1 U0

t1

~

R '1

Coupling band filters

R '2

t2

R2

RF - Transformers

Fig. 1.22 Matching of the generator internal resistance R1 and load resistance R2 to the characteristic resistance of the resonant circuits using high-frequency transformers with the resistance conversion ratio t1 = R1 /R1 or t2 = R2 /R2 Fig. 1.23 High-frequency transformer with two reactances

e

ye

Ge jBe

jB0

jB 0

Gr

The circuit in Fig. 1.23 therefore functions only for t < 1 since the root is imaginary otherwise. B0 and B0e are different types of susceptances, as follows from the different sign in Eq. 1.121. Since B0e is in parallel to the resonant circuit, B0e can also be replaced by a negative capacitance or a negative inductance if the element resulting from the parallel circuit remains positive. That is to say, matching circuits with elements of the same type and a different sign fulfil particularly (1.120) in a larger frequency range than matching circuits consisting of capacitances and inductances. For example, if the filter from Table 1.1 circuit (a) is to be matched to 200 Ω instead of 1 kΩ, then we obtain the following for t = 0.2 for the circuit according to Fig. 1.24: e = B0e /ωr = −31, 8 p F. C01 = B0 /ωr = 39, 8 p F and C01

(1.122)

e connected in parallel, and the resulting The capacitances C1 and C2 have C01 capacitance is thus 64.7 pF. The inductances and coupling capacitance do not change, and the filter has roughly the same transmission properties as circuit (a) in Fig. 1.15. Impedance matching is an important means to suppress reflections on transmission lines. It will therefore be discussed in greater detail in Sect. 2.4.

1.3.3 Multicircuit Coupling Band Filters There are two ways to achieve edge steepnesses greater than 40 dB/decade:

1 Resonant Circuits, One-Port Networks, Coupling Filters … C01

R1 U0

e

C01

37

C12

L1

C1

C02

C2

L2

e

C02

R2

~

Fig. 1.24 Two-circuit coupling band filter with high-frequency transformers containing negative capacitances

1. Decoupled with amplifier stages, multiple single circuits or two-circuit coupling band filters are connected in series6 or 2. Filters are built with more than two coupled circuits. Analytical calculation of multicircuit filters is very complex. Instead, we will consider a synthesis of multicircuit coupling band filters based on frequency transformation from the “normalized lowpass”.

1.3.3.1

The Normalized Lowpass

Using the cutoff frequency of the lowpass f g = ωg /(2π ), we can introduce the normalized angular frequency Ω = ω/ωg . The normalized angular cutoff frequency is then Ωg = 1. In order to simplify the characterization of the transmission properties, the mathematical functions are extended to negative frequencies. |A B | and tg are applied as even functions of Ω. In the ideal lowpass, for |Ω| < 1 (passband) we should have |A B | = 1 and tg = const (Fig. 1.25), and for |Ω| ≥ 1 we should have |A B | ≡ 0. These requirements cannot be fulfilled by a network having a finite number of elements. Like all network functions, it must be possible to express |A B | as the quotient of two polynomials in p = jΩ A B ( p) = g( p)/ h( p)

(1.123)

where the degree of the polynomials ensues from the number of independent reactances. Accordingly, only approximations of the curves in Fig. 1.25 can be realized. We will consider two classic approximation functions here: (a) The Butterworth approximation [2] |A B | = 1/(1 + α 2 Ω 2n )

(1.124)

provides a maximally flat approximation (Butterworth filter) 6

Since it is often not possible to miniaturize filters due to the required quality factor (in contrast to amplifiers), this approach is not always effective.

38

R. Negra

Fig. 1.25 Ideal curve of the effective gain factor magnitude |A B | and the group delay tg for the normalized lowpass

AB 1

−1

0

1

0

1

W = w/wg

tg

−1

W

(b) The Chebyshev approximation7 |A B | = 1/(1 + α 2 T2n (Ω))

(1.125)

allows optimum exploitation of a tolerance range within which |A B | may fluctuate arbitrarily. According to (1.131), the constant α is tied to the permissible reflection coefficient. Tn (Ω) is the Chebyshev function of the first order n. For |Ω| ≤ 1, we have |Tn (Ω)| ≤ 1, the function oscillates and attains (n + 1) times the values +1 or −1. For |Ω| = 1, we have |Tn (±1)| = 1. For |Ω| > 1, |Tn (Ω)| goes monotonically to infinity. The Chebyshev functions are expressed in parametric representation as Tn (Ω) = cos{n[arccos(Ω)]} Tn (Ω) = cosh{n[arccos(Ω)]}

for |Ω| ≤ 1 for |Ω| ≥ 1

(1.126)

These transcendental functions can be expressed as polynomials with a finite number of coefficients such that the approximation function in (1.125) can be realized as a network function. The polynomials are as follows for up to n = 6: T0 T2 T4 T6 7

= 1, T1 = Ω, = −1 + 2Ω 2 , T3 = −3Ω + 4Ω 3 , = 1 − 8Ω 2 + 8Ω 4 , T5 = 5Ω − 20Ω 3 + 16Ω 5 , 2 4 6 = −1 + 18Ω − 48Ω + 32Ω .

After the Russian mathematician Pafnuty Chebyshev (1821–1894).

(1.127)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

39

The curves of T0 to T2 are easily derived from the formulae, while the functions T3 to T6 are plotted in Fig. 1.26. In network theory, procedures have been developed to determine reactance circuits based on the specified curve of |A B (Ω)|. The key foundation of these procedures is the reduction of the two-port network problem to a one-port network problem. The power that is not consumed in the load resistance R2 (Fig. 1.30) is reflected into the generator since the reactance two-port network does not consume any power. Then, the input impedance Z e = V1 /I1 experiences a mismatch. The reflection coefficient 8 is defined as Z e − R1 . (1.128) Γe = Z e + R1 The power flowing into the two-port network is P1 =

1 |V1 I1∗ | = (1 − |Γe |2 )P1max , 2

(1.129)

where P1max = |V0 |2 /(8 R1 ) is the maximum power that can be delivered by the generator. Since the filter is considered to be lossless, we have |Γe |2 + |A B |2 = 1.

(1.130)

Based on the specified curve of |A B |, (1.130) is used to compute |Γe | and extend it into a complex, even function of p using the relationship |Γe |2 = Γe ( p) · Γe (− p), where the denominator polynomial of Γe ( p) must be a Hurwitz polynomial. From (1.128), Z e ( p) is then determined and this one-port network function is realized as a branch circuit by separating poles and zeroes [3, 4]. For normalized lowpasses with |A B | according to (1.124) (Butterworth filter) and |A B | according to (1.125) (Chebyshev filter), the circuits computed in this manner up to degree n = 9 are available in catalog form [5, 6]. If instead of the constant α the maximum reflection coefficient at the band edge is introduced. We then have 2 . α = Γmax / 1 − Γmax

(1.131)

The maximum return loss at the edge of the passband is α aΓ max 2 = −10 lg(1 − Γmax = 20 lg ). dB Γmax

(1.132)

Figure 1.27 shows the normalized, and thus dimensionless, elements of the normalized lowpass for Butterworth, P, and Chebyshev, T , filters of degree n = 3 (03..) with Γmax = 10% (..10), i.e. for P 0310 and T 0310. In Fig. 1.28, |Γe | and tg N are plotted as f (Ω) for six lowpass filters. 8

The reflection coefficient will be discussed in detail in Sect. 2.3.

40

R. Negra T3

T4

+1

+1

−1

0

+1 W

−1

T5

T6

+1

+1

0

−1

+1 W

−1

−1

−1

0

+1

W

−1

0

+1 W

−1

Fig. 1.26 Curves of the Chebyshev functions T3 (Ω) to T6 (Ω) in the passband (−1 ≤ Ω ≤ +1)

Further filter catalogs are available for the Cauer lowpass in which the reflection coefficient for |Ω| ≤ 1 does not exceed a value Γmax and poles of the stopband attenuation lie at finite frequencies. Here, the stopband attenuation does not fall below a minimum value asmin , whereas in the Chebyshev lowpass as tends monotonically to infinity for |Ω| > 1. For |Ω| = 1, the edge steepness of the Cauer lowpass is greater. The Fano lowpass achieves broadband matching of complex load impedances. The lowpasses considered hitherto are dimensioned to suit a desired curve of |A B |, but tg cannot be influenced. On the other hand, it is possible to dimension circuits to suit a specified curve of the group delay without the ability to influence |A B |. The problem of simultaneously approximating |A B | and tg to suit specified curves cannot be fulfilled by branch circuits because they always form a minimum phase system. Using cross-couplings as shown in Fig. 1.29, a lowpass is created in which |A B | and tg can be adjusted simultaneously. In comparison to the conventional method of connecting an all-pass filter as a delay equalizer after the filter dimensioned to suit |A B |, the expense is less for this cross-coupled filter.

1 Resonant Circuits, One-Port Networks, Coupling Filters …

a

g0

~

V0

a

g3

g1

b

g0

g1

g4

g3

g2

~

V0

g4

4

80 % 60

3 tgN = dbB/dW

Fig. 1.28 Magnitude of the input reflection coefficient |Γe | and the normalized group delay tg N . a Butterworth filters P 0310, P 0410, and P 0510; b Chebyshev filters T 0310, T 0410, and T 0510

g2

b

P 0510 40

2

Ge

Fig. 1.27 Circuits of the normalized lowpass of degree n = 3. Values of the normalized elements from [5, 6]: g0 = g4 = 1; P 0310: g1 = g3 = 0.4649; g2 = 0.9299; T 0310: g1 = g3 = 0.8534; g2 = 1.104

41

P 0410 1

20 P 0310

0

0

7

140 %

6

120 100

5 4

80

3

60

Ge

tgN = dbB/dW

T 0510

T 0410

2

40 T 0310 20

1 10 0

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

W

1.3.3.2

Denormalisation

The values of the circuit elements in the real lowpass are obtained through denormalisation. The reference resistance, Rr e f , is equal to the generator internal resistance R1 (for all normalized lowpasses, we have g0 = 1). The cutoff frequency, f g , of the real lowpass is the reference frequency fr e f . The reference inductance and capacitance are

42

R. Negra

1:ü L2

ü 2.5

C1

L4

ü 1.5

L6

C3 C5

Fig. 1.29 Lowpass of degree n = 6 with two cross-couplings and ideal 1 : N transformer

L r e f = Rr e f /ωr e f ,

Cr e f = 1/(ωr e f Rr e f ).

(1.133)

The elements of the denormalized lowpass are then R1 = g0 Rr e f = Rr e f , R2 = gn+1 Rr e f , Ci = gi C B (1 ≤ i ≤ n). L i = gi L r e f ,

(1.134)

The frequency and group delay are given by f = Ω fr e f ,

tg = tg N /ωr e f .

(1.135)

The conversion ratios for ideal transformers remain unchanged.

1.3.3.3

Lowpass-Bandpass Transformation

For a bandpass, the reflection coefficient should be less than Γmax in the frequency range f −D ≤ f ≤ f D and the effective attenuation should be as large as possible and f D in the stopband f < f −D and f > f D . f −D is the lower cutoff frequency √ is the upper cutoff frequency. The band centre-frequency is f m = f D f −D and the bandwidth is Δf = f D − f −D. The ratio Δf / f m is the relative bandwidth. The frequency transformation from the normalized lowpass to the bandpass is provided by the following equation: Ω = ( f m /Δf )( f / f m − f m / f ).

(1.136)

If we apply the abbreviation Q r e f = f m /Δf , the operational quality factor, and fr = f m in (1.136), then Ω is formally equal to the normalized detuning V from (1.31). We should keep in mind, however, that the operational quality factor Q r e f has a definition that differs from that of the unloaded quality factor in (1.26) or the loaded quality factor in (1.90). The reference quantities for denormalisation with the bandpass are Rr e f = R1 , fr e f = f m , L r e f = Rr e f /ωr e f , and Cr e f = 1/(ωr e f Rr e f ). Applying (1.136) in

1 Resonant Circuits, One-Port Networks, Coupling Filters …

43

(1.134) and (1.135) leads to jf f m gi fm 1 + , = jωC pi + Δf Rr e f f m jf jωL pi jf fm fm 1 gi Rr e f = jωL si + = + . Δf fm jf jωCsi

jΩgi ωr e f Cr e f = jΩgi ωr e f L r e f

(1.137)

A comparison with (1.22) reveals that due to the transformation the capacitance of the lowpass changes over to a parallel resonant circuit, the inductance to a series resonant circuit of the bandpass. We have Lre f gi fm 1 = Q r e f gi Cr e f , L pi = 2 = , Δf ωr e f Rr e f ωm C pi Q r e f gi Cr e f f m gi Rr e f 1 = Q r e f gi Cr e f , Csi = 2 = , L si = Δf ωr e f ωm L si Q r e f gi

C pi =

(1.138)

Figure 1.30 shows the bandpass that arises due to the frequency transformation from the lowpass in Fig. 1.27a. The characteristic reactances in (1.18) of the series and parallel circuits differ approximately by the factor Q r2e f because the normalized elements, gi , are on the order of magnitude of 1. For high frequencies and/or high operational quality factors, the circuit from Fig. 1.30 is not realizable in practical terms due to the parasitic reactances of the circuit elements. Here is an example: A bandpass should have the cutoff frequencies f −D = 97.5 MHz and f D = 102.5 MHz, we require R1 = R2 = 50 Ω and the normalized lowpass T 0310 should be used as the basis. For fr e f ≈ 100 MHz, we have Q r e f = 20, L r e f = 79.8 nH and Cr e f = 31.7 pF. According to (1.138), we then have C p1 = C p3 = 541 pF, L p1 = L p3 = 4.68 nH, L s2 = 1.76 µH and Cs2 = 1.44 pF. The lead inductance of the capacitors C p1 and C p3 is equal to ≈ 5 nH for a wire length of 5 mm, i.e. it is on the order of magnitude of the values of L p . An inductor with 1.8 µH has a parallel capacitance of ≈ 0.5 pF. This value is comparable to Cs2 . As a result, the electrical behaviour of the real-world band filter we build will differ significantly from the calculated behaviour.

1.3.3.4

Negative Gyrator and Development of Coupling Band Filters

In order to obtain only one type of resonant circuits, i.e. only parallel circuits or only series circuits, in the band filter, it is necessary to eliminate either inductors or capacitors in the lowpass. This is theoretically possible by inserting a two-port network that functions as a impedance inverter: The input admittance is the inverse to the output admittance

44

R. Negra

R1 V0

~

Ls2 Lp1

Cs2 Cp3

Cp1

Lp3

R2

Fig. 1.30 Bandpass derived from the third-degree lowpass from Fig. 1.27a

Yc =

G 2d . Y2

(1.139)

G d is the gyration conductance or dual factor. The gyrator defined by Tellegen [7] fulfils (1.139) for a real gyration conductance value. However, the gyrator cannot be represented by a network of passive one-port networks. It can be realized using Hall generators or two-port networks that exhibit asymmetrical transfer behaviour like transistors. The negative gyrator also fulfils (1.139) if the gyration conductance value is imaginary. The negative gyrator can be represented, for example, by the circuit in Fig. 1.31 containing negative circuit elements. Its input admittance is Ye = − j Bd +

B2 j Bd (Y2 − j Bd ) = d. j Bd + Y2 − j Bd Y2

(1.140)

For the circuits in Fig. 1.31b and c, respectively, the “dual factor” Bd = ωC

and

Bd = −1/(ωL)

(1.141)

is not constant but rather is frequency-dependent, which does not represent a problem for small relative bandwidths. Negative gyrators as illustrated in Fig. 1.31 (also known as admittance inverters [8]) lead to filters made of voltage-coupled parallel circuits by eliminating the inductors in the lowpass. Regarding the dual-structure T-circuits9 shown in Fig. 1.31, i.e. impedance inverters, filters made of current-coupled series circuits arise by eliminating the capacitors in the lowpass. By way of example, a multicircuit coupling band filter is developed in Fig. 1.32 with capacitive coupling of the generator and load resistance. The lowpass in Fig. 1.32a is transformed into the inductor-less lowpass in Fig. 1.32b with matching two-port networks (resistance conversion ratios t1 and tn ) and with admittance inverters (dual factors Bdi,i+1 ). Based on the selection of the dual factor, any arbitrary conversion ratio can also be set with the inverters; thus, the reference resistance Rr e f i = 1/(ωr e f Cr e f i ) is freely selectable from stage to stage, whereas in the lowpass in Fig. 1.32a or the bandpass in Fig. 1.30, the reference resistance Rr e f is 9

A dual-structure circuit is obtained by transforming each loop into a node and each node into a loop; inductances become capacitances and capacitances become inductances. Moreover, the dual-structure circuit fulfils (1.139), i.e. it is also electrically dual.

1 Resonant Circuits, One-Port Networks, Coupling Filters …

a

b

jBd

y1 −jBd

−jBd

y2

45

c

C

−C

−C

L

−L

−L

Fig. 1.31 a Negative gyrator as admittance inverter. Approximate realization: b with capacitances Bd = ωC; c with inductances Bd = −1/(ωL)

a

g2.LB

gn−1.LB

gn.LB

g0.RB

gn+1.RB oder g1.CB

gn.CB

gn+1.RB gn−1.CB

b R1

t1

Bd1,2 g1.CB1

tn

Bdn-1,n

g2.CB2

gn−1.CBn-1

R2

gn.CBn

c R1

t1

Bd1,2 Lp1 Cp1

C01

d

tn

Bdn-1,n

Lp2 Cp2 C12

Lpn−1 Cpn−1

R2

Lpn Cpn

Cn−1.n

Cn.n+1 R2

R1 e − C01

Lp1 Cp1 −Cp12 C12 Lp2 Cp2

Lpn−1 Cpn−1−Cn−1.n Lpn Cpn − C en.n+1

Fig. 1.32 Development of the multicircuit coupling band filter with capacitive voltage coupling and capacitive coupling from the normalized lowpass

equal for all stages. The values of t1 , tn and Bdi,i+1 (1 ≤ i ≤ (n − 1)) result from the fact that the reflection coefficients must be equal referred to the respective reference resistance in the circuits Fig. 1.32a and b. For the matching two-port networks, we have the following for Γ0 in Fig. 1.33a and b: Γ0 =

g0 Rr e f − Rr e f R1 /t1 − Rr e f 1 = . g0 Rr e f + Rr e f R1 /t1 + Rr e f 1

(1.142)

From this and from the analogous formula for the output two-port network, it follows that and tn = R2 /(R Bn gn+l ) (1.143) t1 = R1 /(R B1 g0 ) Combining with (1.134) and (1.139), we obtain the following:

46

R. Negra

a

RB

~ b

g0.RB

r0 RB1

~

c

t1

r0

R1

gi+1.LB

RBi

~

r1 gi+2.CB

d

RBi

~

r1

Bdi,i+1

Bdi+1,i·2

gi+1.CBi+1

gi+2.CBi+2

Fig. 1.33 For computation of the resistance conversion ratio t1 and duality factors Bdi,i+1 , Bdi+1,i+2

t1 =

R1 ωm C p1 g0 g1 Q r e f

and

tn =

R2 ωm C pn . gn gn+1 Q r e f

(1.144)

In order to determine Bdi,i+1 , the reflection coefficients Γi of the cutouts from the lowpass according to Fig. 1.33c and d are set equal. Since the short circuit of the capacitor gi+2 C Bi+2 is transformed into an open circuit by the second inverter, we obtain Γ1 =

2 jωgi+1 C Bi+1 /Bdi,i+1 − R Bi jωgi+1 L r e f − Rr e f = . 2 jωgi+1 L r e f + Rr e f jωgi+1 C Bi+1 /Bdi,i+1 − R Bi

and it thus follows with (1.134)

(1.145)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

Bdi,i+1

ωm = 1/(R Bi R Bi+1 ) = ωr e f C Bi C Bi+1 = Qr e f

47

C pi C pi+1 . gi gi+1

(1.146)

Based on a lowpass-bandpass transformation, we obtain circuit (c) in Fig. 1.32, and the resonant circuit elements L pi and C pi are computed from (1.138). With the matching circuit according to Figs. 1.23 and (1.119) and the admittance inverters according to Fig. 1.31b and (1.143), we obtain the circuit in Fig. 1.32d. It can be realized using positive elements if the capacitances resulting from the parallel circuits are positive. For small bandwidths (Δf / f m ≤ 25%), the circuit is practically always realizable. Due to the frequency dependency of Bdi,i+1 , (1.141), asymmetries arise for the multicircuit coupling band filters in the curves of |A B | and tg as discussed in Sect. 1.3.1.2.

1.3.4 Losses in Reactance Filters Inductors and capacitors always contain loss resistances. They cause an attenuation a Bv which is approximately equal to a Bv /d B = 4.3m Q B /Q 0 for coupling filters in the passband. (m is the number of resonant circuits, Q B = f m /Δf the operational quality factor and Q 0 the inherent quality factor of the individual resonant circuit). In practical instances, a Bv is often very much larger than the return attenuation aΓ max according to (1.132). The losses have practically no effect on the effective attenuation in the stopband, but they smooth the curve of |A B | or |Γe | at the edges of the band. The ratio of the stopband attenuation a Bs at a given frequency f s to the passband attenuation a Bv decreases with increasing losses. Accordingly, for the same quality factor per individual circuit a Bs /a Bv cannot exceed a maximum value, and any increase in the number of resonant circuits past an optimum value even degrades the ratio a Bs /a Bv [8].

1.4 Principle of Conservation of Energy, Impedance, Admittance and Quality Factor Definitions 1.4.1 The Principle of Conservation of Energy in Network Theory We will consider the one-port network shown in Fig. 1.11a that was previously examined in Sect. 1.2.3 and introduce the current and voltage designations according to Fig. 1.34. With the aid of Kirchhoff’s circuit laws

48

R. Negra

I0 = Ic + I L ,

(1.147)

V0 = VRC + VC = VR L + VL

(1.148)

we then obtain the following expression for the product of the terminal voltage V0 and the conjugate complex terminal current I0∗ : V0 I0∗ = VRC IC∗ + VR L I L∗ + jωL|I L |2 −

j |IC |2 . ωC

(1.149)

The complex power 21 V0 I0∗ that is fed to the one-port network from a source is tied by (1.149) to the sum Pv of the loss power converted into heat in the network PRC =

1 VRC IC∗ 2

and

PR L =

1 VR L I L∗ , Σ Pv = PRC + PR L 2

(1.150)

as well as to the magnetic and electric energies Wm =

1 L|I L |2 4

and

We =

1 C|VC |2 , 4

(1.151)

which are stored in the network averaged over time (I L and VC designate complex amplitudes). This equilibrium statement can be extended to arbitrarily complex networks made of lumped, linear, passive and time-invariant R, L, C elements; it is only necessary to sum over all of the loss powers and energies. (1.149) thus represents the principle of conservation of energy in network theory which has its counterpart in field theory in Poynting’s theorem (see Sect. 5.1.2): 1 V0 I0∗ = Σ Pv + 2 jω(Σ Wm − Σ We ). 2

(1.152)

1.4.2 Impedance and Admittance The formulation of the principle of conservation of energy according to (1.152) offers the possibility to still define the impedance or admittance of a one-port network even if no assessment of its R, L, C elements is possible, for example, because they can no longer be localized individually. The only prerequisite is that it must be possible to reasonably define at least one of the terminal quantities V0 or I0 . By dividing (1.152) by I0 I0∗ /2, we obtain the following expression for the one-port network impedance Z : Z0 =

V0 2Σ Pv + 4 jω(Σ Wm − Σ We ) = = R(ω) + j X (ω). I0 |I0 |2

(1.153)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

49

By dividing by V0 V0∗ /2, we can obtain a corresponding expression for the conjugate complex one-port network admittance Y ∗ . For the admittance Y , we then obtain Y =

I0 2Σ Pv + 4 jω(Σ We − Σ Wm ) = = G(ω) + j B(ω). V0 |V0 |2

(1.154)

For the magnitude of the phase angle ϕ of the impedance or admittance in the complex plane, we use (1.153) or (1.154) to obtain the relationship tan(φ) =

2ω|Σ Wm − Σ We | , Σ Pv

(1.155)

which can be employed advantageously in assessing the quality factor of components.

1.4.3 Definition of the Quality Factor from the Phase Angle Z and Y characterize a reactance or susceptance, respectively, having a gradually improved quality factor as the phase angle ϕ approaches π /2. Conversely, we can describe Z and Y as an improved ohmic resistance or conductance, respectively, the smaller the phase angle ϕ becomes or the closer the complement δ of the phase angle ϕ to π/2 approaches the value π/2. As a quality designation, the symbol Q ϕ is introduced for tan(ϕ) according to (1.155) and designated as the quality factor. We have Q ϕ = tan(ϕ) =

1 . tan(δ)

(1.156)

The index ϕ in (1.156) is intended to indicate that the quality factor according to (1.155) is defined by the phase angle ϕ. This definition of the quality factor was used extensively in the preceding sections. We consider Fig. 1.34 for RC → ∞ as the equivalent circuit of an inductor at frequencies below the first inherent resonant frequency and obtain the following in agreement with (1.2) for the inductor quality factor according to (1.155): Qϕ L =

2ω · 41 L · |I L |2 1 R 2 L

· |I L

|2

=

ωL 1 . = RL tan(δ)

(1.157)

Originally, the quality factor was defined only for the components inductor and capacitor. There, it has a useful meaning without any restrictions and can be determined, for example, with the aid of an impedance or admittance measuring bridge. However, it is obvious to extend the quality factor concept as an assessment criterion also for one-port networks that are capable of resonance of the type shown in Fig. 1.34. For the lossy resonant circuit that is shown, we are interested in a quality

50

R. Negra

Fig. 1.34 Equivalent circuit for a inductance with inductance L and inherent capacitance C and their loss resistances

I0

IC

IL

RC

V RC

C

VC

RL

V RL

V0

L

VL

assessment for the phase resonance ω ph . Phase resonance occurs if the phase angle of the resonance one-port network goes to zero, i.e. the magnetic energy stored in it is just equal to the stored electric energy. According to (1.155), the value zero is always obtained for the quality factor of the resonant circuit entirely independently of the magnitude or distribution of the losses. This nonsensical result makes it clear that the quality factor definition Q ϕ is unusable if applied to one-port networks that are capable of resonance. As long as the inductor and capacitor in the resonant circuit can still be identified as individual components, there appears to be a way out in that we can computationally determine the resonant circuit quality factor from the quality factors Q L and Q C according to (1.27). For the resonant circuit according to Fig. 1.34, we then obtain the following with ω02 = 1/LC:

Qϕ =

=

ωL · ωC1Rc RL ωL + ωC1Rc RL

√

√ =

ω=ω ph

L/C R L ωω0 + Rc ωω0

⎛

⎞

L/C ⎜ ⎝ Rc ω

ω ω0 2

ω0

.

+

RL Rc

⎟ ⎠ ω=ω ph

(1.158)

ω=ω ph

However, an approach of this sort is feasible and meaningful only under circumstances. As a prerequisite, it must be possible to characterize the resonant one-port network as least as an approximation using lumped components; this is purely a computational technique which cannot be reproduced using test equipment, and it delivers a meaningless result in certain cases. In order to allow us to recognize the latter, we insert the expression for ω ph according to (1.75) into (1.158) and observe the special case in which R L = RC = R with the result √

Qϕ =

L/C 1 = RC + R L 2

√

L/C , R

(1.159)

1 Resonant Circuits, One-Port Networks, Coupling Filters …

51

√ which is meaningful for L/C R in agreement with (1.77). A plot of the impedance √ for the special case that was discussed is shown in Fig. 1.11c. If we now set R = L/C, the plot converges to the point Z = R (see (1.74)), and logically we should obtain Q = 0. From (1.159), however, it follows that Q = 1/2.

1.4.4 Definition of the Quality Factor with the Aid of the Total Stored Energy The difficulties associated with the quality factor Q ϕ at phase resonance are avoided by introducing the stored total energy Σ Wm + Σ We with the quality factor definition Qw =

2ω(Σ Wm + Σ We ) . Σ Pv

(1.160)

For pure R L or RC one-port networks, it also has the benefit that it is equal to the quality factor Q ϕ . Disadvantages of the quality factor definition Q w are that a technically relevant parameter such as the phase angle ϕ is no longer evaluated and that Q w is purely a computational quantity that we have no means to measure at least directly.

1.4.5 Definition of the Quality Factor from the Phase Slope As a particularly suitable quality factor definition for one-port networks that are capable of resonance, we therefore introduce the quality factor Q s which evaluates the phase slope dϕ/dω of the one-port network. The phase slope is properly defined at all frequencies and can also be measured directly. The greatest phase slope always occurs at a phase resonance, and Q s is thus specified for this frequency dϕ . Q s = ω dω ω=ω ph

(1.161)

Based on (1.153) and (1.154), we obtain tan(ϕ) =

B(ω) X (ω) = R(ω) G(ω)

(1.162)

or in the implicit form F(ω, ϕ) = tan(ϕ) −

X (ω) = 0. R(ω)

(1.163)

52

R. Negra

We form dϕ/dω through implicit differentiation, ∂F dϕ = − ∂∂ωF = − dω ∂ϕ

−X (ω)R(ω)−R (ω)X (ω) R 2 (ω) 1 cos2 (ϕ)

.

(1.164)

and take the value at phase resonance from (1.164) specifically with ϕ = 0 and X (ω ph ) = 0, thereby obtaining 1 d X (ω) . Q1 = ω R(ω) dω ω=ω ph

(1.165)

An entirely analogous equation is obtained for the function pair B(ω), G(ω). Given that R(ω) =

2Σ Pv 4ω(Σ Wm − Σ We ) and X (ω) = 2 |I0 | |I0 |2

(1.166)

we obtain the following as a further form of the phase slope quality factor:

ω2 d |Σ Wm − Σ We | · Qs = 2 Σ Pv dω

ω=ω ph

.

(1.167)

Finally, comparison with the quality factor definition for Q ϕ reveals, as a 4th form for Q s , a reasonable relationship between the two quality factors: d Qϕ . Qs = ω dω ω=ω ph

(1.168)

1.4.6 Definition of the Quality Factor from the Bandwidth at Resonance For resonant one-port networks, there is another quality factor definition that is conventional which involves an evaluation of the resonance width. Conventionally, the resonance width encompasses the width of the resonance curve between the impedance values which differ exactly by 1/2 or 2 from the maximum or minimum value, respectively. This quality factor is distinguished by the index ω, and for Δω = 2π Δf (Δf = bandwidth) along with ωm as the angular frequency at which the impedance exhibits an extreme value, we have Qω =

ωm fm = . Δω Δf

(1.169)

1 Resonant Circuits, One-Port Networks, Coupling Filters … Table 1.2 Four quality factor definitions for one-port networks Evaluation parameter Defining equation Application area Phase angle

Q ϕ = tan(ϕ) =

1 tan(δ)

m +Σ We Average stored energy Q w = 2ω Σ WΣ Pv

Phase slope

dϕ Q s = ω dω

Bandwidth

Qω =

ω=ω ph

ωm Δω

=

fm Δf

Inductor, capacitor, reactance network

Resonance-capable components, microwave resonators

Resonance-capable components

Resonance-capable components

53

Comments Easy to measure. Meaningless for frequencies in the vicinity of a resonance Computational quantity, not directly measurable. For pure RL elements (We = 0) and pure RC elements (Wm = 0), we have Qw = Qϕ Directly measurable. ω ph is a phase resonance angular frequency. Meaningful even at frequencies in the vicinity of a resonance Allows good direct measurement. Also meaningful for magnitude resonance. The bandwidth Δf = Δω/2π is defined conventionally by the 3-dB drop-off

In the form of Q ϕ , Q w , Q s , and Q ω , we have examined a total of four differently defined quality factors. None of them provides consistently meaningful results in all imaginable application cases. Accordingly, we must always choose the most appropriate one for our particular application. In order to simplify the decisionmaking process, Table 1.2 shows these quality factor definitions along with the relevant assessment criteria.

References 1. Campbell, G.A.: Physical theory of electric wave filter. Bell Syst. Tech. J. 1, I (1922) 2. Butterworth, S.: On the theory of filter amplifiers. Wirel. Eng. 7, 536–541 (1930) 3. Weinberg, L.: Network Analysis and Synthesis, p. 628. McGraw-Hill Book Comp. Inc., New York (1962) 4. Darlington, S.: Synthesis of reactance-4-poles which produce prescribed insertion loss characteristics. J. Math. Phys. 17, 257–353 (1939) 5. Zverev, A.I.: Handbook of Filter Synthesis. Wiley (1967) 6. Saal, R.: The Design of Filters Using the Catalogue of Normalized Low-Pass Filters. Telefunken (1963)

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7. Tellegen, B.D.H.: Der Gyrator, ein elektrisches Netzwerkelement. Philips Technische Rundschau 18, 88–93 (1956) 8. Matthaei, G.L., Young, L., Jones, E.M.T.: Microwave Filters. Impedance Matching Networks and Coupling Structures. McGraw-Hill, New York (1964)

Chapter 2

Wave Propagation on Transmission Lines and Cables Matthias Rudolph

Abstract Once the geometric size of an electronic circuit with respect to the signal frequency gets very large, it is no longer possible to ignore the fact that an electrical signal travels no faster than the speed of light. It is observed, that changes do not take effect everywhere simultaneously in the circuit, which needs to be taken into account once the fastest signal transients happen to be on a similar time scale. As these effects first are observed on long cables – or transmission lines – this chapter is devoted to wave propagation on lines. After deriving how electromagnetic waves travel along lines, it will be addressed how reflections can be eliminated by proper impedance matching. In this course, the usage of the famous Smith Chart will be discussed in detail. It will also be highlighted, how to take advantage of wave-propagation phenomena in order to replace reactive components like capacitors by short lines. The chapter concludes by introducing scattering matrices, a generalized concept that allows for a holistic treatment of transmission lines and linear multiports.

2.1 Introduction The topic addressed in this chapter can be motivated through a simple gedankenexperiment of a resistive load R that is connected to a voltage source V0 with internal resistance Rs through a switch, as shown in Fig. 2.1. When the switch is open, no current flows through and no voltage drops across the load resistor. If the switch is closed, Ohm’s law applies, and the current through the resistor is found to be I L = V0 /(R + Rs ), the voltage at the resistor is given by VL = V0 · R/(Rs + R), and the power delivered to the load to be P = VL · I L . But this view only is valid if the transition period after closing the switch is neglected, which means that the distance l between voltage source and load is negligible, l ≈ 0. If source and load are located in a certain distance from each other (l = 0), we have to consider that after closing the switch at the side of the source, the power M. Rudolph (B) Brandenburg University of Technology, Cottbus, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. L. Hartnagel et al. (eds.), Fundamentals of RF and Microwave Techniques and Technologies, https://doi.org/10.1007/978-3-030-94100-0_2

55

56

M. Rudolph

a

b

Rs

Rs IL = V0 / (Rs + R)

I=0 R

V0

l

V=0

V0

R

VL = V0 · R R+ R s

0

Fig. 2.1 Short line section and equivalent circuits

to be delivered to the load will travel with a finite speed that can not exceed the speed of light c towards the load. Figure 2.2 sketches what will happen in this case. If the switch is open (Fig. 2.2a), no current flows and no difference to the previous case is observed. But shortly after the switch was closed, the source will start to deliver power P = V · I along the line towards the load (Fig. 2.2b). At this moment, we observe that the switch-on event travels towards the load with a velocity v ≈ c. Neither current nor voltage are constant along the line in this case. On the sourceside of the line, a voltage drops and a current flows, while on the load-side of the line, the electric potential and the current are still zero. At a certain point in time, current and voltage reach the load. What happens next is depicted in Fig. 2.2c. The resistance R locally enforces a certain ratio between voltage and current according to Ohm’s law. In the general case, the ratio of V to I as initially provided by the source does not match R = V /I , since the source has no a priori knowledge of the load impedance attached to the other end of the transmission line. As a consequence, part of the power delivered by the source gets reflected, so that the impedance level at the load and power conservation can be guaranteed at the same time. In the following, which is not shown in the figure, the system will approach the static solution with VL = V0 · R/(Rs + R) and I L = V0 /(R + Rs ) through a series of reflection and rereflection events like the one discussed. This chapter will introduce to transmission-line theory, which allows for an understanding and engineering of this type of wave-propagation on electrical lines. Wave propagation on electrical cables was first described as early as 1890, and the waves are also referred to as Lecher waves [1]. The first topic to address is how to understand and predict the transient and wavepropagation effects on transmission lines. The discussion so far did not answer what magnitude the current initially had. Apparently, it is not determined by the source or by the load. Thus it must be a property of the transmission line that determines the initial ratio of current and voltage. It is certainly also a property of the line at which velocity the switch-on event travels towards the load. Therefore, the theory of wave-propagation on transmission-lines will be addressed, which is expressed through the transmission-line, or telegrapher’s, equations. The second topic concerns the reflection effects at the load-side of the line, which leads us to the definition of the reflection coefficient, and how it relates to load impedance and line properties. It will also be shown how connecting a line to a

2 Wave Propagation on Transmission Lines and Cables

a

57

Rs I=0

l

b

V=0

R

V0

Rs

0

c

I =?

Rs

I =? I =?

I=0 V0

V=0

R

V =?

I, V V =?

V =?

V0

I, V I =?

V =?

R = V/I

c

V =?

c x

x

Fig. 2.2 Short line section and equivalent circuits

load impedance alters the impedance observed at the line input in case of harmonic (sinusoidal) signals. In many applications, RF engineers exploit this impedancechanging property of lines. The third topic of this chapter addresses the question of how to avoid reflection effects that are in general unwanted and destructive in high-speed and RF applications. This can be achieved through lossless passive so-called matching circuits, and a graphical way of constructing these is by using the iconic Smith-Chart. The final topic introduces a generalized concept of linear network analysis that allows to take the wave propagation effects into account. It is assumed that the reader is familiar with network description through Y- and Z-matrices that relate port voltages to port currents. We will extend this concept by introducing the Smatrix that relates the incoming and outgoing electromagnetic waves at the ports, and show how to transform Z-, Y- and S-matrix representations into each other. The properties of transmission lines play an important role in RF electronics. In RF electronics, one usually reverts to well-established transmission-line types, depending on the application. Therefore, Chaps. 4 and 5 will address certain types of cables and waveguides in addition to the information provided within this chapter. In general, a transmission line needs to provide low losses and frequency-independent line parameters in order not to distort broadband signals, and it must not radiate its fields or be susceptible to external RF fields. In addition to that, it needs to be manufacturable with the least effort possible. Examples for common transmission lines are, as shown in Fig. 2.3: • Coaxial cables, consisting of a center conductor and an outer conductor, that encloses the center conductor with a defined radius. These cables are commonly used in measurement systems or to connect antennas or other RF components that are located rather close. TV antenna cables are an example in consumer electronics. These cables are used up to frequencies in excess of 100 GHz.

58

a

d

M. Rudolph

b

c

e

Fig. 2.3 Cross-sections of typical RF transmission lines. Dashed areas are conductors, dielectric is drawn in white. Shielded twisted-pair (a), coaxial cable (b), rectangular waveguide (c), microstrip line (d), and coplanar line (e)

• Twisted pair cables consist of a cable pair that is twisted in order to cancel out magnetic coupling through the area between them. These cables are, for example, the standard for LAN connections. The twisted pair commonly is enclosed in a shielding realized through a conductive foil or mesh wire. The upper frequency limit is with a few 100 MHz much lower than for coaxial cables. • For highest frequencies or highest powers, rectangular waveguides are used. These resemble rectangular pipes, conducting the power in the form of an electromagnetic wave propagating within the waveguide. Since currents flow only in the enclosing metal, losses are lower than for other types of cables, and also shielding is perfect. The drawback of rectangular waveguide is that they are not flexible. Since the metallization of the waveguide in its cross-section is closed, rectangular waveguides can not transport DC currents. They show a cutoff-frequency towards lower frequencies depending on their dimensions. • On circuit boards and in integrated circuits, planar transmission lines are used. The most prominent is the microstrip line, consisting of a conductor line that is separated from a ground plane by an insulated layer. The microstrip line transports an electromagnetic wave mostly underneath the top conductor. Also coplanar waveguides are used, where the conductor line lies on the same level as the ground plane, separated by gaps on either side. The electromagnetic energy of the transmitted wave is concentrated within these gaps. • For frequencies in the optical spectrum, dielectric optical fibers are used that provide extremely low losses for transmission over long distances. What is common for these types of transmission lines is that they are well suited for RF transmission, and also well studied so that the impact of the transmission lines can be accounted for in RF circuit and system design. But knowledge of transmission line theory is also important outside the classical RF area. In digital electronics, ill-designed connections lead to transient reflection effects that restrict the maximum

2 Wave Propagation on Transmission Lines and Cables Fig. 2.4 Short line section and equivalent circuit

59

a i2

i1

v1

v2

i1

i2

dz

b i1

v1 i1

L'

R'

i2

C'

G'

v2 i2

clock rate. In digital electronics, one speaks of signal integrity, meaning that a bit transmitted through an interconnection still is recognized at the receiving side. The application and the technology applied to solve it might be different, but the underlying effects are the transmission of electromagnetic waves on lines, which is treated in the following. In electronics, when signal frequencies are low and electrical connections are short, it is appropriate to describe circuits in terms of currents and voltages only. For RF electronics, the description has to be expanded in order to account for the fact that currents and magnetic fields always occur together, and the same holds for voltage and electric field. With increasing frequency, our interpretation of cause and effect shift from voltage and current causing electric and magnetic fields to the fields causing currents and voltages. This paradigm shift is implied in the list of transmission-line examples given in the previous section, where twisted-pair lines for moderately high frequencies stand in contrast to rectangular waveguides used for highest frequencies. Transmission-line theory as discussed in the following is assuming homogeneous lines, meaning that the line properties are constant over the whole line length. It will also be abstract in a way that geometry is not yet considered. There will be a current i and a voltage v, while the concrete discussion of different types of transmission lines will be relying on this theory in Chaps. 4 and 5. Another assumption is, that radiation and cross-coupling is not taking place yet. The fields associated with i and v are confined to the line and do not interact with the environment. Cross-coupling of adjacent lines, and how to take benefit from it, is addressed in Chap. 4, while radiation is covered in Chap. 6 addressing antennas.

60

M. Rudolph

2.2 Propagation of Electromagnetic Waves on Transmission Lines 2.2.1 Equivalent-Circuit Representation of the Line and Derivation of the Telegrapher’s Equation Transmission-line theory assumes that the current i = f (z, t) and voltage v = f (z, t) can vary along the line, where coordinate z denotes the observation point at the line, and be dependent of time t. In order to stay within an equivalent-circuit based description of the line, electrical fields are lumped into a capacitance per line length C . In an analog way, magnetic fields are described through an inductance per line length L . Cable losses and dielectric losses are described similarly through resistance and conductance per line length, R and G , respectively. The derivation of the equivalent circuit description of an infinitesimally short line section, with length l = dz is shown in Fig. 2.4. In order to derive the dependency of the current i and voltage v on the spatial coordinate z, it suffices to apply Kirchhoff’s two laws to the transmission-line section of length dz as shown in Fig. 2.4. In order to simplify the nomenclature, i 1 shall denote the current i = f (z 1 , t) at an observation coordinate z 1 , while i 2 shall denote the current i = f (z 1 + dz, t), an infinitesimally short distance dz apart from coordinate z 1 . No index will be written when an equation applies to an arbitrary observation point along the line. The same nomenclature will be applied to voltage v. Under the effect of the voltage v which should be construed as the arithmetic mean of the nearly identical voltages v1 and v2 , leakage currents (insulation currents) and displacement currents flow between the two conductors. The total leakage current of the section is G dz v and the total displacement current is ∂q/∂t = C dz(∂v/∂t), as denoted in the equivalent circuit in Fig. 2.4b. Approximating the voltage between the conductors by v ≈ v1 enables us to write the equation for the currents through the infinitesimally short section of the line: i 1 = i 2 + G dz · v1 + C dz · and with i 2 = i 1 + di;

di =

∂v1 ∂t

∂i dz ∂z

(2.1)

(2.2)

allows us to determine the change of the current over the line segment dz as a function of voltage v: ∂i ∂v − = G · v + C · . (2.3) ∂z ∂t

2 Wave Propagation on Transmission Lines and Cables

61

The change in voltage over the line segment can be determined in a similar fashion. The voltage v2 will be reduced with respect to v1 by ohmic losses in the conductors, represented by R dz, and the voltage drop caused by line inductance L dz. The equation reads: ∂i 1 (2.4) v1 = v2 + R dz · i 1 + L dz · ∂t and with v2 = v1 + dv;

dv =

∂v dz ∂z

(2.5)

allows us to determine the change of the voltage v over the line segment dz as a function of current i: ∂v ∂i = R · i + L · . (2.6) − ∂z ∂t One of the two unknown quantities v or i can be eliminated by combining (2.3) and (2.6). For example, we obtain the following by differentiating (2.6) by z: ∂ 2v ∂i ∂i ∂ +L − =R − ∂z 2 ∂z ∂t ∂z

(2.7)

∂i Moreover, by applying − ∂z from (2.3), we obtain 2 ∂ 2v ∂v ∂ v + R G · v = L C · + (R C + G L ) · ∂z 2 ∂t 2 ∂t

(2.8)

A very similar equation is obtained if the voltage is eliminated instead: 2 ∂i ∂ 2i ∂ i + R G · i = L C · + (R C + G L ) · 2 2 ∂z ∂t ∂t

(2.9)

These equations can be rewritten, considering that a good transmission line should show low losses, represented by R and G : 2 ∂v R G R G ∂ 2v ∂ v + ·v . =LC + + · ∂z 2 ∂t 2 L C ∂z L C

(2.10)

2 R G ∂i ∂ 2i R G ∂ i + ·i . =LC + + · ∂z 2 ∂t 2 L C ∂z L C

(2.11)

62

M. Rudolph

This partial differential equation of second order is known as the telegraphers’ equation. It allows for the calculation of transient effects like switching processes and of travelling waves along the line.

2.2.2 Solution of the Telegraphers’ Equation: Propagation Constants and Characteristic Impedance of the Line In the following, the telegrapher’s equation will be solved in the frequency domain, i.e., assuming signals that show a sinusoidal time-dependence with an angular frequency ω = 2π f . The reason to switch to the frequency domain is threefold. First, the mathematical treatment of the differential equations is simpler. Second, traditional RF signals tend to be rather narrow-band and in many cases can be approximated by an investigation of the center, or carrier, frequency. Third, it is always possible to use the Fourier transform of a non-sinusoidal excitation signal to calculate its representation in the frequency domain. As we assume the transmission line to be a linear system, it is possible to determine the response of the transmission line to this excitation in the frequency domain and finally use the inverse Fourier transform to calculate the output signal in the time-domain. The following treatment is therefore easily expanded to cover non-sinusoidal signals. We will be using complex notation for the instantaneous values of voltage and current, as it is common in AC circuit theory: v = {V e jωt } we obtain the equations

and

i = {I e jωt }

(2.12)

−

∂V = (R + jωL )I ∂z

(2.13)

−

∂I = (G + jωC )V. ∂z

(2.14)

and

From the telegrapher’s equations (2.10) and (2.11), we obtain ∂2 V = (R + jωL )(G + jωC )V. ∂z 2 ∂2 I = (R + jωL )(G + jωC )I. ∂z 2

(2.15) (2.16)

2 Wave Propagation on Transmission Lines and Cables

63

Since (2.15) and (2.16) are differential equations with constant coefficients, we expect exponential functions as our solution. We thus apply V = V0 eγ z and I = I0 eγ z as our educated guess. The important quantity γ is known as the “propagation constant” or “propagation coefficient” and is obtained in conjunction with Eqs. (2.15) and (2.16); V0 γ 2 eγ z = (R + jωL )(G + jωC )V0 eγ z . 2 γz

I0 γ e

γz

= (R + jωL )(G + jωC )I0 e .

(2.17) (2.18)

Thus, we obtain γ12 = ± ( jωL + R )( jωC + G ) √ R G 1− j 1− j . = ± jω L C ωL ωC

(2.19)

It is fair to assume that a good transmission line provides low losses. At frequencies above 10 kHz, the loss factor R /ωL for the current ohmic losses is 1, whereas the loss factor G /ωC for the dielectric losses already drops below a value of 1 at frequencies of only a few hertz. Thus, we can easily approximate the second root through a first-order Taylor-series expansion and obtain γ12 ≈ ± jω

√

L C

R 1− j 2ωL

G 1− j . 2ωC

(2.20)

Since γ12 is complex, we set γ12 = ±γ = ±(α + jβ).

(2.21)

The real part α serves as a measure of the attenuation and is thus known as the attenuation coefficient or attenuation constant. From (2.20) it follows that G R α≈ + 2 2 CL

L . C

(2.22)

The imaginary part β takes into account the phase-shifting characteristic of the transmission-line and is thus known as the phase coefficient or phase constant. β=

√ 2π ≈ ω L C . λ

(2.23)

Here, λ is the wavelength on the transmission-line at frequency f . The significance of the attenuation and phase coefficients is depicted in Fig. 2.5, for a wave travelling in positive direction of z at a certain time t = t0 . The sinusoidal shape of the wave is governed by β, which is directly related to the wavelength.

64 Fig. 2.5 Sketch denoting the significance of phase coefficient β and attenuation constant α

M. Rudolph

V V0 e–az

V0

z V0 sin(bz) e–az l =

2π b

The attenuation coefficient is not affecting the periodic nature of the curve, it merely describes the exponential decay of the amplitude, indicated by the dashed line. So far, we were able to find a solution of the telegrapher’s equation for current and voltage that shows that both lead to the description of a wave with identical propagation constant. But since power is delivered according to the product of current and voltage, the ratio of the two quantities is to be established. It will be shown that the line properties determine this ratio, which carries the unit Ohm and therefore is called the characteristic impedance Z 0 of the line. Equations (2.13) and (2.14) establish the connection between line current and voltage. Starting from (2.13) I =−

∂V 1 . R + jωL ∂z

(2.24)

By inserting V = V0 e±γ z , we obtain the following: γ · V0 e±γ z R + jωL R + jωL = · V0 e±γ z G + jωC

I =

(2.25) (2.26)

Since the z-dependence of I is governed by the same propagation constant as in case of the voltage, with I = I0 e±γ z , it follows for the ratio Z 0 = V0 /I0 , which is referred to as the characteristic impedance of the line: Z0 =

R + jωL . G + jωC

(2.27)

The characteristic impedance Z 0 is an important concept in the transmission line theory. It states that current and voltage of a wave travelling along a line always

2 Wave Propagation on Transmission Lines and Cables

65

stand in a certain ratio of each other and that the properties of the line (like geometry, dielectric constant and conductance) determine the relation of the amplitude and phase. This characteristic impedance is therefore the answer to the question that arose from the introductory gedankenexperiment at the beginning of the chapter: If a line is excited at one end with a certain voltage V , we now know that the current of the wave travelling towards the load is given by I = V /Z 0 . In case of a low-loss line, the characteristic impedance can be approximated by: Z0 ≈

L . C

(2.28)

The characteristic impedance must not be confused with an ohmic resistance or a lumped impedance in general. Together with the transmission coefficient γ , it defines how electromagnetic power is delivered along the line in the form of a guided wave, while a lumped impedance value denotes the amount of energy that is dissipated or stored in this network branch. How to determine input impedances in networks containing transmission lines will be discussed further on in this chapter, after some more basics are explained. Electromagnetic waves can propagate in positive as well as in negative direction of z, with propagation coefficients γ and −γ , respectively. In positive direction, current and voltage will be denoted as It and Vt , while in negative direction, it will be Ir and Vr . The characteristic impedance can be written as: Z0 =

Vt Vr = , It −Ir

(2.29)

as the current Ir is flowing into the negative direction.

2.2.3 Phase and Group Velocity In order to determine the velocity of the wave travelling along the line, we are required to express the harmonic signal in time domain. For the forward-propagating wave, we get: (2.30) Vt (t, z) = Vt0 · sin(ωt − βz + ζ ) · e−αz where ζ accounts for an arbitrary phase shift of the wave, ω denotes the angular frequency, β propagation constant and α the attenuation constant. The phase velocity is defined as the speed that a certain phase ξ is travelling along the line, e.g. ξ = π/2: π = ωt − βz + ζ (2.31) 2 Differentiating with respect to time t, one obtains:

66

M. Rudolph

dz dt ω 1 dz = =√ = , dt β L C

0=ω−β v ph

(2.32) (2.33)

√ with β = ω L C . This definition is referred to as the phase velocity, since the reference is the propagation of a certain phase angle of the wave. But the phase velocity in steady state is not very meaningful when it comes to the question at which velocity information is transported along the line. Transporting information requires to change between different states in magnitude and/or phase, as will be discussed in the final chapter of the book, on signal modulation techniques. If we consider, for example, a pulse where the signal is switched on only for a short period of time, it is known that the respective spectrum spreads over all frequencies and can be described by a sinc function. In order to determine how fast information – or energy – travels, it is therefore required to investigate a certain spectrum, or a group or frequencies. The so-called group velocity describes the velocity of a narrowband spectrum, that could for example be an amplitude-modulated signal. In order to simplify the analysis further, only the two frequencies ωc ± δω defining the bandwidth around the carrier frequency ωc are considered. This signal has the form of: Vt (t, z) = Vt0 · {cos[(ωc − δω)t − (βc − δβ)z + ζ ]+ + cos[(ωc + δω)t − (βc + δβ)z + ζ ]} · e−αz = Vt0 · 2 · {cos[ωc t − βc z + ζ ] · cos[δωt − δβz + ζ ]} · e−αz

(2.34) (2.35)

where the relation cos(α) · cos(β) = 1/2[cos(α − β) + cos(α + β)] was applied. The term δβ was introduced in order to account for the fact that the propagation constant, in the general case, is not independent of frequency. Considering that the first term in the brackets describes the carrier signal at ωc , i.e. a steady-state signal merely defining the center frequency, it is safe to ignore it in the determination of the group velocity. The second term at frequency δω, on the other hand, represents the information modulated onto the carrier signal. The group velocity is now determined by calculating the speed of a constant phase, e.g. π/2 of the information signal: π = δωt − δβz + ζ (2.36) 2 Differentiating with respect to time t, one obtains: 0 = δω − δβ dz δω = dt δβ

dz dt

(2.37) (2.38)

2 Wave Propagation on Transmission Lines and Cables a

b

67

t = t0 + Δt

c

z'

c

t = t0 f zero phase

E propagation direction

ugr = c . cos(f)

E

c f

z''

uph = c cos(f)

Fig. 2.6 Sketch explaining the difference of phase velocity v ph and group velocity vgr (a) a wave guided between conducting walls by reflection (b) phase of the wave as superposition of two plane waves (c) observed group vgr and phase v ph velocities

⇒ vgr =

dω dβ

(2.39)

The group velocity therefore denotes the velocity of an infinitesimally narrowband signal. For broadband signals, it keeps it importance, since constant group velocity over the full bandwidth is the condition for undistorted transmission. In case of non-dispersive lines, meaning lines that feature constant line parameters L , C , R , and √ G , there is no difference between the two definitions. In this case, with β = ω L C , it follows β/ω = dβ/dω. Different types of lines, and to what extend dispersion has to be considered will be covered in the following chapters. In general, all types of lines show dispersion, but technically relevant types of cables and lines also provide frequency ranges that can be considered to be free of dispersion when selected carefully to match the application. The treatment of phase and group velocities would not be complete without mentioning that phase velocity is able to exceed the speed of light, while group velocity is rather lower than the speed of light. This is very well visible in the case of rectangular waveguides which will be addressed in Chap. 5. This phenomenon can be explained at the example of a wave guided between two conducting planes, as shown in Fig. 2.6a. The wavelength is short with respect to the distance between the plates, so that the wave propagation can be understood similar to light beams being reflected between the planes. Although the direction of propagation is in the horizontal direction relative to the plates, the electromagnetic wave travels on a zig-zag path. The observed group velocity in the propagation direction is thereby slower than the speed of the reflected and re-reflected beam, depending on the angle between beam and total propagation directions. If the individual beam travels with the speed of light c, a group velocity vgr = c · cos φ ≤ c is observed, see Fig. 2.6c.

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M. Rudolph

The phase velocity in this case has to consider the speed of a constant phase, e.g. the point where sin(ζ ) = 0 holds. This point depends on the superimposed waves, as indicated in Fig. 2.6b. The figure depicts the zero-crossings at ζ = π for the two planar waves. Only at the crossing of the lines, the superimposed waves show a zerocrossing. These phase conditions travel in the propagation direction of the wave, but, as depicted in Fig. 2.6d, at a phase velocity of v ph = c/ cos φ ≥ c. This example intends to motivate the phase velocities determined when calculating e.g. the electromagnetic field patterns in rectangular waveguides. The closed-formula results tend to disguise why phase velocity could exceed the speed of light, and why the transmission speed of energy and information is much lower.

2.2.4 Exact Representation of the Attenuation and Phase Coefficients In the upcoming sections, we will rely on the approximated formulae for low-loss transmission lines as derived above. The reason is that providing low losses is one of the key properties of transmission lines. Nevertheless it might be interesting in specific cases to determine the exact values so we can examine the errors in the approximate values at any time. We begin with (2.19) R G 1− j 1− j γ = −ω L C ωL ωC

(2.40)

R G = sinh(δ ) and = sinh(δG ). R ωL ωC

(2.41)

2

and set

2

Thus, we have γ 2 = −ω2 L C {1 − sinh(δ R ) sinh(δG ) − j[sinh(δ R ) + sinh(δG )]}.

(2.42)

Now we have 1 [cosh(δ R − δG ) − cosh(δ R + δG )] 2 2 δ R − δG 2 δ R + δG − sinh = sinh 2 2

− sinh(δ R ) sinh(δG ) =

and moreover

δ R + δG sinh(δ R ) + sinh(δG ) = 2 sinh 2 Thus, we have

δ R − δG cosh 2

(2.43)

.

(2.44)

2 Wave Propagation on Transmission Lines and Cables

69

δ R − δG δ R + δG − sinh2 γ 2 = −ω2 L C 1 + sinh2 2 2 δ R − δG δ R + δG cosh −2 jsinh 2 2

(2.45)

or

δ R − δG δ R + δG 2 − j sinh γ = −ω L C cosh , 2 2 √ δ R + δG δ R − δG + cosh γ = α + jβ = jω L C − j sinh 2 2 2

2

(2.46)

which is equivalent to √ δ R + δG , α = ω L C sinh 2 √ δ R − δG . β = ω L C cosh 2

(2.47)

The upper equation can be written even more clearly. According to the relationship

δ R + δG sinh 2 we have

⎛

R G α=⎝ + 2 2 CL

⎞ 1 L ⎠

δ R −δG C cosh 2

and

β = ω L C cosh as well as v ph =

R G sinh(δ R ) + sin(δG ) + = .

δ R −δG = 2ωL δ R 2ωC −δG 2 cosh cosh 2 2

δ R − δG 2

(2.48)

(2.49)

1 1 ω =√

δ R −δG . β L C cosh 2

(2.50)

(2.51)

Equations (2.49)–(2.51) hold strictly for any frequency and any values of R /ωL or G /ωC . With (2.49), the correction of the approximate values for α and β is reduced to a single correction function cosh(δ R − δG )/2 which deviates from 1 only by a maximum of 2% for δ R − δG ≤ 0.4. The correction factor is exactly equal to 1 for δ R = δG or

R G = . L C

(2.52)

70

M. Rudolph

10

√ km

10 NA/km

β

1

α

1

10-1

— 1 10-1 √f

10-2

2

β

α

10-2

10-3 10-4

—— √R′G′

10-3

α

10-5 1.5·10-6 1.0·10-6 0.5·10-6 0 a

10-4

β0 − 2π ~f λ0

10-5

β

10-3 10-2 10-1 0 0,5·10-3

100

101 f

102

103

104

105 Hz 106

Fig. 2.7 Attenuation coefficient α and phase coefficient β as a function of frequency R0 = 43.6 µ /cm; L 0 = 3.126 nH/cm; G 0 = 485 fS/cm; C0 = 0.46 pF/cm

According to Heaviside,1 a line with this property is said to be distortionless, or non-dispersive. In general, we have δ R > δG . However, (δ R − δG )/2 assumes values only in the range to a few kilohertz such that cosh(δ R − δG )/2 > 1. Practically speaking, we can assume the following starting from 10 kHz for all existing line types:

δ R − δG cosh 2

1 ≈1+ 8

R G − ωL ωC

2 ≈ 1.

(2.53)

Figure 2.7 shows the attenuation coefficient, α, and phase coefficient, β, according to (2.49) as a function of frequency (Fig. 2.8).

2.2.5 Frequency Dependency of the Characteristic Impedance The characteristic impedance of a transmission-line follows from (2.27): 1

O. Heaviside was an English physicist who lived from 1850 to 1925. He mathematically analyzed the propagation of waves on transmission-lines and cables for the first time (1893) with the aid of the transmission-line equations, simultaneously developed the basic principles of vector calculus and explained concurrent to Kennelly the propagation of long waves around the earth due to the presence of a high-lying ionosphere (Kennelly-Heaviside layer) [2].

2 Wave Propagation on Transmission Lines and Cables Fig. 2.8 Plot for γ = α + jβ between 10−2 Hz and 1 MHz; R0 = 43.6 µ /cm; L 0 = 3.126 nH/cm; G 0 = 485 fS/cm; C0 = 0.46 pF/cm

71

102 1/km 106 10

Hz 105

1

104 10–1 b

103 10–2

102 10

10–3

10–4

1

10–1

10–2 Hz 10

–5

10–4

10–3

10–2

10–1

Np/km 1

α

Z0 =

R + jωL = G + jωC

R L 1 − j ωL . G C 1 − j ωC

(2.54)

(R /ωL ) is the share of the loss factor for the cable which corresponds to that of a coil and (G /ωC ) is the share that corresponds to the losses of a capacitor. We will now examine the frequency dependency of Z 0 based on three important special cases.

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M. Rudolph

For direct current ( f = 0),

Z 0 |dc =

R G

(2.55)

is real and determined solely by the effective resistances. In ordinary cables with √ insulating material having a resistivity ρ = 1013 to 1017 cm, R /G is on the order of magnitude of several times 1000–100,000 . At frequencies between 1 Hz and 1 kHz, we have the loss factor of the insulation G /ωC 1 but R /ωL 1 given ρ ≥ 1013 cm. Within this low-frequency range, e.g. at 50 Hz, R is about 100 times as large as ωL such that in the lowfrequency range up to a few kHz we can neglect ωL versus R . We thus obtain Z 0 |n f ≈

R 1− j = √ jωC 2

R . ωC

(2.56)

For low-frequency applications, the magnitude of the characteristic impedance √ |Z 0 | thus decreases with 1/ f . The characteristic impedance is not real but instead has a phase angle at low frequencies of almost −45◦ , as seen from the factor 1 − j in the above formula. (2.56) holds only from ≈ 3 to ≈ 100 Hz. Above 10 kHz we enter the RF regime and we also have R /ωL 1, i.e. Z 0 |r f ≈

L C

(2.57)

√ range, Z 0 is again real but the values L /C are much smaller than √ In this R /G ; they are between 10 and 1000 , and for coaxial cables between 50 and 75 . In this manner, we can plot the characteristic impedance Z 0 (Fig. 2.9) in a qualitative manner with numerical values calculated according to (2.54). The values are calculated for the example of a coaxial cable (Fig. 2.10). A more precise representation must take into account the frequency dependency of the line equivalent-circuit elements. R , G , L , and C can be considered to be constant only within a certain frequency range. The skin effect, for example, leads to a concentration of the current on the conductor surface. As a consequence, R increases. C and L depend on the electromagnetic field patterns that are in general frequency dependent, albeit to a different extent for the different types of lines.

2.3 The Reflection Coefficient 2.3.1 Chain Matrix Description of the Transmission Line We would now like to relate the equations derived in Sect. 2.2.2 to the terminating impedance as well as the current and voltage at the end of the line. With waves travelling in positive and negative directions, the voltage at a certain point z is given

2 Wave Propagation on Transmission Lines and Cables Re(ZL) 0

25

50

Re(ZL)

L '/C ' 75 100 Ω 125

L '/C ' 0

10kHz 5kHz

Ω

kΩ

–25

–1

1kHz

–75

2

1

3

4

5

6

7

102Hz 101

R '/GI ' 8 10–2Hz 10–1

f

3

9

10

R '/GII' R '/GIII ' 11 kΩ 12 10–2Hz 10–2Hz

f

f

I

–2 Im(ZL)

Im(ZL)

2kHz –50

73

1Hz 1Hz

3.10–1

II 10–1

III

–3 1Hz

0.5kHz –100

3.10–1

–4

3.10–1

10–1

–5

Fig. 2.9 Plot of the characteristic impedance Z 0 of a coaxial cable with R0 , L 0 , C0 as in Fig. 2.7; G 0 = 4.85 · 1/ρe f f ; ρe f f = Average value for supporting ring and air I: ρe f f = 5 × 1012 cm; II: ρe f f = 1013 cm III: ρe f f = 1.5 × 1013 cm D

a

d S

Fig. 2.10 Diagram of the CCI standard coaxial pair, cross-section and longitudinal section. Ring spacing α ≈ 29 mm is standardized: diameter of inner conductor d = 2.6 mm; inner diameter of outer conductor D = 9.5 mm; thickness of outer conductor s = 0.25 mm

by the superposition of the voltages of the waves according to: V (z) = Vt e−γ z + Vr e+γ z

(2.58)

Also the current can directly be given, if the voltage waves and the characteristic impedance Z 0 are known: I =

Vt −γ z Vr +γ z e − e . Z0 Z0

(2.59)

In order to describe the impact of the line in terms of a linear two-port matrix, the relation between current and voltage at the input of the line (z = 0) to current and voltage at the output of the line (z = l) has to be derived. At the output of the line, we get: (2.60) V (z = l) = Vt e−γ l + Vr e+γ l and

and thus

Z 0 · I (z = l) = Vt e−γ l − Vr e+γ l

(2.61)

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M. Rudolph

V (z = l) + Z 0 · I (z = l) = 2 · Vt e−γ l , V (z = l) − Z 0 · I (z = l) = 2 · Vr e+γ l .

(2.62)

considering that at the input of the line, at z = 0, the voltages read Vt (z = 0) = Vt and Vr (z = 0) = Vr , and denoting the quantities at the input, z = 0, by the index 1 and at the output, z = l by the index 2, we obtain: V2 + Z 0 · I2 +γ l V2 − Z 0 · I2 −γ l e e , + 2 2 V2 + Z 0 · I2 +γ l V2 − Z 0 · I2 −γ l e − e . I1 = 2Z 0 2Z 0

V1 =

(2.63)

After combining the factors associated with V2 or I2 , this relation can be written in form of a chain matrix: cosh(γ l) Z 0 sinh(γ l) V2 V1 = 1 (2.64) · sinh(γ l) cosh(γ l) I1 I2 Z0 In many cases it is possible to neglect the attenuation on a low-loss line. The chain matrix simplifies the to:

V1 I1

=

cos(βl) Z 0 sin(βl) 1 sin(βl) cos(βl) Z0

V2 · I2

(2.65)

This chain matrix describes the line as a two-port and is in principle sufficient for network calculations. We will take advantage of this relation in the subsequent analysis of wave propagation and reflection effects.

2.3.2 The Reflection Coefficient So far, it was established that current and voltage of a wave travelling along a line are related to each other through the line’s characteristic impedance. If the line is terminated by a load of a different impedance level, a part of the transmitted power will get reflected, according to the different impedance levels. Figure 2.11 sketches a terminated transmission line. Assume that the source generates a wave transmitted towards the load with voltage Vt (z) = Vt · e−γ z , current It (z) = Vt (z)/Z 0 and power Pt = 1/2 · (|Vt |2 · Z 0 ). At the load side of the line, at z = 0, the load impedance requires I L = VL /Z L which differs from the currentto-voltage ratio on the line if Z L = Z 0 . As a consequence, a reflected wave will be observed, so that the superposition of the two waves fulfills Ohm’s law at the load: ZL =

VL Vt + Vr Vt + Vr = = Z0 · IL It − Ir Vt − Vr

(2.66)

2 Wave Propagation on Transmission Lines and Cables transmitted wave

reflected wave

It (z)

Z0

Z0, b, a It (z)

IL

Ir (z)

Vt (z) Vt (z)

75

ZL

Vr (z) Vr (z)

= Z0

–Ir (z)

VL = ZL · IL

= Z0

z=−l

z=0

Fig. 2.11 Transmission line terminated by a load impedance Z L

This relation allows us to determine the ratio between reflected and transmitted wave: Vr Z L − Z0 = (2.67) ΓL = Vt Z L + Z0 Γ L , or Γ in general, is called the reflection coefficient. It is a dimensionless quantity defined in terms of voltage, with 0 ≤ |Γ | ≤ 1 for passive loads. In terms of power, we get Pr = Pt · |Γ L |2

(2.68)

PL = Pt · (1 − |Γ L | ) 2

(2.69)

where Pt is the power of the wave emitted by the source, Pr is the power of the reflected wave, and PL is the power delivered to the load. The quantity Pt is also called the available power of the source (Pav ) that would be the power the source could provide to a matched load with Γ L = 0. From (2.67), it is also possible to determine the load impedance when the reflection coefficient and the characteristic line impedance are known: 1 + ΓL ZL = Z0 1 − ΓL

(2.70)

Which again highlights that the reflection coefficient of a load is defined relative to the characteristic line impedance, in contrast to the impedance value that is an independent property. Nevertheless, it is common to characterize RF devices rather in terms of their input- and output reflection coefficients as through the respective impedances. This custom is justified, since Z 0 = 50 is the industry standard, for which virtually all packaged components, cables, connectors and measurement equipment are designed. In all other cases, it is required to state the value of Z 0 .

76

M. Rudolph

After the derivation of the reflection coefficient in general, we should have a closer look at four typical types of load resistances: matched, open circuit, short circuit and reactive loads. Matched Load means the case where load and characteristic line impedances are equal, Z L = Z 0 . For the reflection coefficient, it follows: Γ Lmatched =

Z0 − Z0 =0 Z0 + Z0

(2.71)

consequently, since no power gets reflected, the available source power is fully delivered to the load. Open-circuit load provides Z L → ∞ to the end of the line. The respective reflection coefficient is: Z L − Z0 Γ Lopen = lim =1 (2.72) Z L →∞ Z L + Z 0 No power is delivered to the load, since the open-circuit condition prevents that current flows into the load terminals. Instead, the power transmitted from the source towards the load gets reflected. Short-circuit load provides Z L = 0 to the end of the line. The respective reflection coefficient is: −Z 0 Γ Lshort = = −1 (2.73) Z0 As in case of the open circuit, no power is delivered to the load, since the shortcircuit condition prevents that a voltage exists at the load terminal. The power gets reflected, but with a 180◦ phase shift. Reactive load provides a purely imaginary impedance Z L = j X to the end of the line, which stores energy. The respective reflection coefficient is: Γ Lreactive load =

j X − Z0 = e jψ j X + Z0

(2.74)

with ψ = 2 arctan(Z 0 / X ). Since the reactive load is not dissipating any power, it is also in this case fully reflected back towards the source. The reflection coefficients determined for these four load impedances are drawn into the complex reflection coefficient plane in Fig. 2.12. In addition to these cases, it can be stated that for any passive load providing (Z L ) > 0, one obtains reflection coefficients within the unity circle. This can be expected from the fact that a mismatched load always yields a reflected wave, and from the fact that a passive load does not emit power but dissipates it. Therefore, the reflected power will be less than the available source power, and if follows |Γ | < 1. The mathematical transformation of an impedance into the reflection coefficient graphically maps any value providing a positive real part into the unity circle. We will come back to this property when

2 Wave Propagation on Transmission Lines and Cables

77

(Γ) ZL = jX Γ =1

ZL = 0 Γ = –1

ZL Γ=1

∞ (Γ)

ZL = Z0 Γ=0

Fig. 2.12 Location of matched, open circuit, short circuit and reactive loads drawn in the complex reflection coefficient plane. Passive load impedances yield reflection coefficients |Γ | < 1 within the unity circle

the mapping of impedances into the complex reflection factor plane is addressed in order to derive the so-called Smith chart.

2.3.3 Transformation of Reflection Factors Through a Transmission Line Once that the reflection coefficient at the end of the transmission line is defined, we would like to determine the reflection coefficient at the input of the line. This would be the reflection coefficient presented at the source side, which can then be used to derive an input impedance. This question is of high practical relevance, as it addresses the question to what extent a known input impedance is altered if a cable of a certain length l is connected to the respective port (Fig. 2.13). Considering that the voltage along the line is given by the superposition of transmitted and reflected waves, which can be expressed by V (z) = Vt e−γ z + Vr e+γ z ,

(2.75)

78

M. Rudolph L

in

Zin

Z0 Z0, b, a

z=–l

ZL

z=0

Fig. 2.13 Sketch of a transmission line terminated by a load Z L , indicating input impedance Z in and reflection coefficient Γin

it is possible to derive the reflection coefficient from the ratio of the two waves. If we consider that the load reflection coefficient Γ L is defined at z = 0, where the exponential terms assume the value 1, we obtain: Vr e−γ l Vr (z = −l) = Vt (z = −l) Vt eγ l = Γ L · e−2γ l = Γ L · e−2αl e− j2βl

Γin =

= ΓL · e

−2αl − j4π λl

e

(2.76) (2.77) (2.78)

Inserting a transmission line therefore shifts the phase of the reflection coefficient according to the phase constant β, and reduces its magnitude according to the attenuation constant α. On a lossless line, |Γ | would remain constant, while the phase shifts clockwise according to 4πl/λ. Figure 2.14 shows the example of an arbitrary load reflection coefficient Γ L , and the corresponding input reflection coefficient Γin observed at the input of a quarter-wavelength (l/λ = 1/4) transmission line. Lines with a length of a quarter wavelength shift the phase of a reflection coefficient by 180◦ . This property is beneficial in many respects, a few examples will be given after the transformation of the load impedance through a transmission line is introduced. The transformation of the load impedance Z L to the source side of the line can be derived analogously: 1 + Γin 1 − Γin 1 + Γ L e−2γ l . = Z0 1 − Γ L e−2γ l

Z in = Z 0

(2.79) (2.80)

It is advantageous to substitute the reflection coefficient by the respective expression relying on the load and characteristic line impedance only:

2 Wave Propagation on Transmission Lines and Cables

79

(Γ)

Fig. 2.14 Example of an arbitrary load reflection coefficient Γ L transformed by a line of length l = λ/4

ZL = jX Γ =1

ΓL l

l 4

1 18 0º Γin

(Γ)

losses: e–2al

(Z L + Z 0 )e jγ l + (Z L − Z 0 )e− jγ l (Z L + Z 0 )e jγ l − (Z L − Z 0 )e− jγ l Z L cosh(γ l) + j Z 0 sinh(γ l) = Z0 Z 0 cosh(γ l) + j Z L sinh(γ l) Z L + j Z 0 tanh(γ l) = Z0 Z 0 + j Z L tanh(γ l)

Z in = Z 0

(2.81) (2.82) (2.83)

Transformation of a load impedance or reflection coefficient through a quarterwave line is of high practical relevance especially in case of open or short circuit terminations, or of purely resistive loads. In order to take advantage of the transformation, the line needs to provide very low attenuation, and therefore the following examples are discussed with the approximation of a lossless line. A matched load with Z L = Z 0 and Γ L = 0 is not altered at all, if a line is inserted, simply since the load does not cause a reflected wave. A resistive load R L will be transformed into another resistive input impedance Rin . Equation (2.83) for the lossless case yields: Z in = Z 0 ·

Z L + j Z 0 tan(βl) Z 0 + j Z L tan(βl)

(2.84)

and for a quarter-wavelength line, βl = π/2, so that we obtain, considering that limx→π/2 tan(x) = ∞: Z2 (2.85) Z in = 0 ZL

80

M. Rudolph

Fig. 2.15 Using a quarter-wave transmission line to convert the value of a load impedance

l/

= 1/4 Z0 = 70

100

50

The impedance observed at the input of the line thereby is again purely resistive. It is either higher than Z L , if Z 0 > Z L , or lower than Z L , if Z L < Z 0 . The quarterwavelength line therefore serves as an impedance transformer. Microwave circuits, where Z 0 of microstrip lines can be varied in a broad range of impedances, use this property regularly to transform an impedance to attain a desired level, by proper choice of Z 0 . Example: • Consider that an amplifier with an output impedance of 100 should be connected to an antenna with an input impedance of 50 . We can use a quarterwavelength line to connect the two components, see √ Fig. 2.15. A characteristic line impedance according to Eq. (2.85) of Z 0 = 100 · 50 = 70 will do the job. In real life, the antenna might be farther apart than exactly λ/4. In addition, commercial cables provide Z 0 = 50 . In this case, we include the impedance-transformer line in the amplifier, maybe even on the circuit board, so that the 50- impedance level is already reached at the amplifier output port, and the distance to the antenna can be covered by a regular cable. An open circuit with Z L → ∞, gets transformed by a quarter-wavelength line from its reflection coefficient Γ L = 1 by 180◦ to Γin = −1. As already introduced, this reflection coefficient corresponds to a short circuit. While this finding might sound strange at first, just means that we moved our point of observation of the standing wave caused on the line by the full reflection from the open-circuit end (where the current is zero while voltage is maximum) a quarter-wavelength further, where the voltage is zero, while the current is maximum. Between these extreme cases, Z in can mimic other reactive values. Inserting Z L → ∞ into Eq. (2.84), we get: Z in =

Z0 = − j cot(βl)Z 0 j tan(βl)

(2.86)

Figure 2.16 shows the value the of the input impedance (a) and admittance (b) as a function of line length. We observe that the open circuit providing Z in → ∞, Yin = 0 is transformed into a short circuit by a quarter-wavelength line, providing Yin → ∞, Z in = 0. This transformation from short to open and back continues according to the − cot(βl) function, since line losses are neglected. This transformation of a short has a number of applications. For example:

2 Wave Propagation on Transmission Lines and Cables

81

• Realize a capacitance to ground. For line length of less than l/λ = 1/4, the input impedance of a lossless line terminated by an open circuit only provides a negative imaginary part, which can be interpreted as the equivalent to the impedance of a capacitance providing − j/(ωC). This analogy is true even within a certain√bandwidth. For βl 1, it holds that tan(βl) ≈ βl. Considering that β = ω L C , we see that changing the frequency has the same effect on Z in as changing line length. Figure 2.17 shows the input admittance of a short lossless line terminated by an open circuit as a function of frequency. At millimeter-wave frequencies (30 GHz and beyond), wavelengths become very short and parallel capacitances are commonly realized by short open-circuit stub lines. The advantages are clear: realizing a short line on a circuit board or in an integrated circuit is easy and cheap, and line dimensions are commonly very well controlled, so that the capacitance can be realized with very low parameter spread. At lower frequencies, the length of the required line and the losses associated with it commonly render the open stub less attractive than a lumped capacitance element. • Realize a series resonant circuit. A quarter-wavelength line transforms the open circuit termination into a short-circuit. The impedance crosses (Z in ) = 0 just for one single frequency, showing a positive slope resembling the curve a series resonance circuit would show around its resonance frequency. Figure 2.18 compares the two impedances, where ω0 denotes the resonance frequency of the resonator, and the frequency where l/λ = 1/4 holds for the line. • Realize a transmission-line resonator. A transmission-line resonator is a line of length l/λ = 1/2 terminated either by a short or an open circuit on both ends. On circuit boards and integrated circuits, lines losses prevent high-quality line resonators, but the low losses in rectangular waveguides yield extremely good quality factors. This example is not really fitting here and just mentioned to support the previous example. The transmission-line resonator will be addressed when voltage and current distribution on the line are discussed, and in Chap. 6. A short circuit gets transformed, according to Eq. (2.84) and inserting Z L = 0 , into: (2.87) Z in = j Z 0 · tan(βl) Its behavior is analog to the case of the open circuit. For l/λ = 1/4, the short gets transformed into an open circuit. The reflection coefficient Γ L = −1, is transformed into Γin = 1 by the 180◦ phase shift, and in terms of impedance, we observe that the tanh has a pole for a quarter wavelength line, leading to Z in → ∞. The overall behavior of the short circuit resembles that of the open circuit transformed through a line, but either with inverted values (i.e., the impedance of the transformed short resembles the admittance of the transformed open circuit), or the short circuit is interpreted as an open circuit that was transformed to a short by a quarter-wavelength line. Transformation of a short circuit has a number of practical applications, for example:

82

M. Rudolph

a (Zin) – cot(bl ) Z0

1/4

1/2

3/4

1

5/4

3/2

7/4

l /l

3/4

1

5/4

3/2

7/4

l /l

b (Yin) tan(bl ) / Z0

1/4

1/2

Fig. 2.16 Impedance of an open circuit transformed by a line, depending on line length. a: input impedance Z in , b input admittance Yin = 1/Z in

2 Wave Propagation on Transmission Lines and Cables Fig. 2.17 Admittance of an open circuit transformed by a short lossless line, as a function of frequency, compared to the admittance of a capacitance

83

(Yin)

tan(bl )/Z0 = tan(w L'C'l )/Z0 wC

C

w

Fig. 2.18 Impedance of an open circuit transformed by a quarter-wavelength lossless line, as a function of frequency, compared to the impedance of a series resonant circuit around its resonance frequency

(Zin)

–cot(bl )Z0 = –cot(w L'C'l )Z0

1 wC

wL w0

w C L

• Realize a parallel inductance. For short line-lengths βl 1,√it holds that tan(βl) ≈ βl, and the input impedance becomes Z in ≈ jωZ 0 · L C l, which acts in the same fashion as a parallel inductance. • Providing DC bias to a transistor in a microwave circuit. The task is to decouple the DC path from the microwave path, which is usually achieved by applying the DC voltage through a choke inductance, which provides a very high impedance ωL ∞ at microwave frequencies. A choke inductance is increasingly difficult to realize at higher frequencies, and impossible to integrate on a chip or a circuit board. On the other hand, the DC voltage source can be brought to providing a short to ground for all higher frequencies, e.g. through a capacitor. This short can then be decoupled at the signal frequency through an open- to short-circuit transformation, see Fig. 2.19. • Realize a parallel resonance circuit. While a quarter-wavelength transformation of an open circuit was shown to provide the equivalent to a series resonance circuit, we observe the analog behavior of the transformed short circuit that provides the equivalent to a parallel resonance circuit. The examples showed a number of practical applications of the transformation through transmission lines. We will take advantage of these results later on, when impedance matching is discussed. Using short or quarter-wavelength lines, as we

84

M. Rudolph

Fig. 2.19 Using a quarter-wave transmission to provide drain bias to a transistor

VDD

RF short l / l = 1/4

C•

RF open

have seen, allows us to replace lumped elements like capacitors or inductors. Relying on lines is commonly advantageous at frequencies beyond 10 GHz on circuit boards or in integrated circuits, when wavelengths become short and lines are easily realized. But it needs to be considered, that the line properties are periodic with frequency, which is not the case for lumped element realizations. Bandpass-filters therefore will provide additional pass-bands at multiples of the center frequency, which needs to be considered in the design process.

2.3.4 Voltages and Currents on Transmission Lines and the Standing-Wave Ratio The superposition of transmitted and reflected wave on a line results in a standing wave. So far, we discussed the reflection coefficient, which is the ratio of the wave amplitudes, and the input impedances, which is defined by the ratio of the superimposed voltages and currents. But it also is important to know the evolution of voltage and current standing wave maxima and minima. The voltage maximum, for example, can exceed the maximum voltage estimated from the load impedance alone. Recalling that the voltage and the current along a transmission line is given by V (z) = Vt e−γ z + Vr e+γ z Vr +γ z Vt −γ z e − e I (z) = Z0 Z0

(2.88) (2.89)

2 Wave Propagation on Transmission Lines and Cables

85

it is to be expected that the amplitude of a standing wave resulting from a reflected wave caused by a reflection coefficient Γ L provides a maximum if both waves superimpose constructively, and a minimum, where the reflected wave is 180◦ phase shifted with respect to the transmitted wave. For sake of simplicity, we will restrict the following discussion to lossless lines. The impact on line losses can be included later after the concept is established. The maximum voltage Vmax on the line will be observed, when the maximum amplitudes of transmitted wave Vt (z) and reflected wave Vr (z) add up: Vmax = |Vt | + |Vr | = |Vt | · (1 + |Γ L |)

(2.90)

Likewise, the minimum voltage is reached, when the two waves are destructively superimposed: (2.91) Vmin = |Vt | − |Vr | = |Vt | · (1 − |Γ L |) The standing wave is commonly characterized the ratio of the maximum voltage to the minimum voltage, called the voltage standing wave ratio (VSWR): VSWR =

1 + |Γ L | Vmax = Vmin 1 − |Γ L |

(2.92)

The VSWR ranges from 0 for a perfect match to ∞ for |Γ L | = 1. The standing waves for short-circuit and open-circuit terminated lines are shown in Fig. 2.20. Since |Γ L | = 1 in both cases, Vmin = 0 and Vmax = 2 · Vt , i.e. the maximum voltage exceeds the voltage observed at a matched load by a factor of two. The minima and maxima of the current are obtained in an analog manner. For these special cases, the location of the maxima and minima are easily determined: For the short circuit V (z 0 ) = 0 is enforced at the line end (z 0 ). At the same time, the shortcircuit termination yields a current maximum at the line end, see Fig. 2.20a. For the case of an open-circuit termination, shown in Fig. 2.20b, the conditions for current and voltage are reversed, leading to current and voltage switching their behavior. The graph shows clearly, how the short-to-open circuit transformation (and vice versa) through a quarter-wavelength line works, and that the termination is transformed into itself by a half-wavelength line. In case of an arbitrary reflection coefficient Γ = |Γ |e jφΓ , with |Γ | < 1, the values for VSWR, Vmax and Vmin are easily calculated, but the location Vmax is still to be defined. From |V (z)|2 = V (z) · V ∗ (z), we obtain:

|V (z)|2 = e−2αz + |Γ |2 e2αz + 2 Γ e j2βz |Vt |2 = e−2αz + |Γ |2 e2αz + 2|Γ | cos(φΓ + 2βz) |Vt |2

(2.93) (2.94)

The formula shows, that Vmax is observed at the end of the line (z = z 0 = 0) if Γ is real and positive (∠Γ = 0), which requires a load resistance that is real and larger than Z 0 . For the general case of a complex reflection coefficient, the first maximum

86

M. Rudolph

a V (z) I (z)

I (z)

V (z)

short circuit I (z0) = 2 · It

V (z0) = 0

l/4

l/4

z0

z

b V (z) (z)

V (z)

I (z)

open circuit V (z0) = 2 · Vt

I (z0) = 0

l/4

l/4

z0

z

Fig. 2.20 Standing wave amplitudes on a short (a) and open-circuit (b) terminated line

is observed at z = −(φΓ /4π ) · λ. The periodicity of voltage and current maxima and minima with respect to each other is the same as for the open and short terminations. Figure 2.21 shows examples for a lower and a higher complex reflection coefficients. In case that line losses can not be neglected, transmitted and reflected waves experience attenuation, and the value of the VSWR gets reduced with increasing distance from the load side of the line. The standing wave amplitude on a very lossy line is shown in Fig. 2.22. The VSWR and the wave amplitudes for lossy lines can be calculated by the formulas given above, considering that the magnitude of a reflection coefficient at the end of a line, |Γ L | gets transformed through the line according to |Γ (z)| = |Γ L |e−2αz . If the line is long and losses are high, the reflected wave can get completely attenuated, in which case VSWR approaches unity. This case is not very advantageous, since all the reflected power simply got converted into heat by the line. Besides of providing an alternative measure to quantify the reflected wave, the VSWR highlights the issue that voltages and currents on transmission lines can exceed currents and voltages at the load significantly. For very high powers, standingwave current maxima can lead to significant local losses even in low-loss lines, and voltages might locally exceed the breakdown voltage. But also at moderate power levels, the standing wave issue is important. For example, the transistor in a power amplifier is probably dimensioned to provide the maximum power it can, which means that it provides almost its maximum current, and the maximum voltage gets close to the breakdown voltage. It is required to connect the power amplifier to a

2 Wave Propagation on Transmission Lines and Cables

87

a V (z), I (z) I (z)

V (z)

Vmax, Imax Vt , It Vmin, Imin fΓ 4p

l

z=0

z

b V (z), I (z) I (z)

V (z)

Vmax, Imax

Vt , It

Vmin, Imin

fΓ 4p

l z=0

z

Fig. 2.21 Standing wave amplitudes on a lossless line terminated by Γ L = 0.3∠π/9 (a) and Γ L = 0.7∠π/3 (b) Fig. 2.22 Standing wave amplitudes on a lossy line terminated by Γ L = 0.7∠π/3

88

M. Rudolph

matched load in order to keep the voltage and current amplitudes within the safe operation area of the device. A standing wave causing, e.g., a 1.5 times higher output voltage swing is in general harmful and might cause the transistor to break down and melt. Therefore, the maximum safe VSWR is commonly given in power amplifier data sheets.

2.3.5 Transmission Line Resonators In the form of so-called λ/4 lines, open-circuited and short-circuited lines play a special role in RF engineering. They are called “line resonators” because they exhibit resonant circuit behavior if they have a length l = λ/4. They are used at higher frequencies where it is no longer possible to build resonant circuits using lumped circuit elements. We will first consider the line that is open at the end. Since the quality factor Q is an important quantity in resonators, Q describes the ratio of stored energy to energy that is dissipated by the resonator losses, it is required to consider the attenuation coefficient α of the line, even though we are assuming low-loss lines. The input impedance is given by: Z1 =

cosh γ l cosh αl cosh βl + j sinh αl sinh βl V1 = Z0 = Z0 I1 sinh γ l sinh αl cosh βl + j cosh αl sinh βl cosh βl + jαl sinh βl 1 + jαl tan βl ≈ Z0 = Z0 . αl cos βl + j sin βl αl + j tan βl

(2.95) (2.96)

If we insert the value l = λ/4, we obtain the resonance impedance λ R λ Z 1 ≈ αl Z 0 = α Z 0 ≈ . 4 2 4

(2.97)

This value of Z 1 is small with respect to Z 0 since αl 1. Due to the quarterwavelength transformation, the open circuit gets transformed into a short circuit, resembling a series resonant circuit behavior, as discussed in Sect. 2.3.3. We would now like to compare a series circuit as shown in Fig. 2.23 with the quarter-wavelength line terminated with an open circuit. Here, the series circuit and λ/4 line should exhibit the same behavior at their resonant frequency and in the immediate vicinity thereof. For the impedance of the series circuit, we have 1 . Z = R + j ωL − ωC

(2.98)

At the resonant frequency ωr , we have LC = 1/ωr2 . In this case, the impedance Z is equal to the effective resistance R. Comparing this with Eq. (2.97), we then obtain the following:

2 Wave Propagation on Transmission Lines and Cables

a

89

b

c L'

L

R'

l 2

C 'l

R

L'

l 2

G 'l

8 p2

R'

l 2

L'

l 2

l 2

8 p2

R'

l 2

C 'l

8 p2

C

Fig. 2.23 Series resonant circuit and λ/4 line open at end. a Series resonant circuit; b T -element of open λ/4 line; c comparable equivalent circuit

R ≈ αl Z 0 ≈

R l Rλ ≈ . 2 2·4

(2.99)

So far we only know the product of L and C. Their individual values can be determined by considering the circuits in the vicinity of their resonant frequency. For a small detuning, we can write: ω = ωr + dω

(2.100)

√ Given β = ω/v ph , with v ph = 1/ L C , we can use Eq. (2.96) to obtain the input impedance of the line that is open at the end Z1 ≈ Z0

ω l+ v ph cot vωph l

cot

jαl

αl

+j

= Z0

ωs +dω l+ v ph cot ωsv+dω l ph

cot

jαl

αl

+j

.

(2.101)

where l = λ/4 and (ωr /v ph )l = π/2. Developing the cot at the position π/2, we obtain − vdω l + jαl ph Z 1 ≈ Z 0 dω . (2.102) − v ph lαl + j Neglecting second-order quantities in the denominator, we obtain dω l . Z 1 ≈ Z 0 αl + j v ph

(2.103)

√ √ With v ph = 1/ L C and Z 0 = L /C , we then obtain Z 1 ≈ αl Z 0 + j L ldω ≈

R l dω + jωr L l . 2 ωr

(2.104)

The real part of Z 1 agrees with the resonance impedance of the line according to Eq. (2.97). It remains unchanged for small deviations from the resonant frequency. For a series circuit, we have the following in the vicinity of the resonant frequency

90

M. Rudolph

where ωr LC = 1 and ω = ωr + dω: Z ≈ R + jωr L

2dω . ωr

(2.105)

Comparing the real and imaginary parts of Eqs. (2.104) and (2.105), we obtain R ≈ αl Z 0 ≈

R l , 2

1 L = L l 2

(2.106)

Using L, we can now also calculate the capacitance C from the resonance condition. We find that 8 (2.107) C = 2 C l. π Figure 2.23 helps to illustrate the result of Eqs. (2.106) and (2.107). Although the line is a distributed element, it suffices to describe the λ/4 line that is open at the end using a single T -element with two series inductances L l/2 and two series resistances R l/2 as well as a parallel element. The parallel element contains the cross-capacitance (and the cross-conductance) of the line section with a correction factor 8/π 2 which takes into account the uneven voltage distribution. Since the open λ/4 line and the series circuit exhibit the same behavior in the vicinity of the resonant frequency, we can assess the quality factor of the line circuit. The quality factor is defined as follows: Q=

ωr L 1 Reactive Power = = . Active Power R ωr RC

(2.108)

If we apply these quantities according to Eqs. (2.99) and (2.107), we obtain √ ωr L ωr L l β π ωr L C Q= ≈ = = . = R 2αl Z 0 2α 2α αλ

(2.109)

According to the equation, α increases above the limit frequency √ proportional √ to f ; since λ decreases with 1/ f , the quality factor increases with f . At high frequencies, line circuits are superior to resonant circuits made of lumped elements. We can analyze the short-circuited line in the same manner. If we switch to the input admittance, the frequency-dependent factor is the same as for the input impedance of the open-circuited line. We thus obtain the following: Z1 = Z0

sinh γ l sinh αl cos βl + j cosh αl sin βl = Z0 . cosh γ l cosh αl cos βl + j sinh αl sin βl

For low attenuation (αl < 0.005), the formula can be approximated by:

(2.110)

2 Wave Propagation on Transmission Lines and Cables

a

b

c L'l

C

91

p

L

R C'

l 2

R 'l

8 p2

8 p2

G' 2l

G' 2l

L'l

C'

l 2

C ' 2l

8 p2

R 'l p 2 8

G' 2l

Fig. 2.24 Parallel circuit and λ/4 line short-circuited at end. a Parallel circuit for which the series losses of L and C are converted into a parallel resistance R p ; b π -element of the λ/4 line with short-circuit at end; c comparable equivalent circuit

Z1 ≈ Z0

αl cos βl + j sin βl αl + j tan βl = Z0 . cos βl + j α sin βl 1 + j αl tan βl

(2.111)

Which can be expressed in terms of an admittance according to: 1 1 cot βl + jαl Y1 ≈ ≈ Z 0 αl cot βl + j Z0

dω αl + j l v ph

=

αl dω + jω p C l Z0 ωp

(2.112)

A parallel circuit according to Fig. 2.24a has input admittance as follows: 1 1 . + j ωC − Y = Rp ωL

(2.113)

In the vicinity of the resonant frequency, we can approximate ω2p LC = 1 and ω = ω p + dω and get: dω 1 + j2ω p C . (2.114) Y ≈ RP ωp Comparison of Eqs. (2.112) and (2.114) leads to Rp ≈

Z2 Z0 ≈ R 0l , αl 2

C l C= 2

(2.115)

With C, we can then calculate L as follows: L=

8 L I. π2

(2.116)

To help clarify the interpretation of Eqs. (2.115) and (2.116), Fig. 2.24b and c show the dual equivalent circuits corresponding to Fig. 2.23b and c for the λ/4 line that is short-circuited at the end. The quality factor Q of a parallel circuit is

92

M. Rudolph

Q = ωpC R p =

Rp ω p Z L C l β π ≈ = = . ωp L αl 2 2α αλ

(2.117)

The quality factor for the short-circuited λ/4 line is the same as that of the open λ/4 line if the quantity α is equal in both cases.

2.3.6 Reflection Coefficient, Transported Effective Power and Matching of Lossy Lines So far, we assumed that the reflection coefficient’s magnitude for a passive load cannot exceed 1, since the reflected power always needs to be lower than or equal to the transmitted power in absence of a power source on the load side. However, this statement tacitly assumed that the line’s characteristic impedance Z 0 is real and the surrounding circuit is passive. In the following discussion, we will now show that if we maintain our assumption of a passive circuit but remove the requirement for a real characteristic impedance, the magnitude of the reflection coefficient can indeed exceed a value of 1. We will then prove in the following section that this situation does not contradict the principle of conservation of energy. Moreover, all of the results obtained here for lines can also be applied analogously to two-port networks made of lumped components.

2.3.6.1

Reflection Coefficient for Lossy Lines

According to Eq. 2.67, the reflection coefficient is defined as a fractional-linear function Z /Z 0 − 1 (2.118) Γ = Z /Z 0 + 1 of the normalized impedance Z = Ae jψ = A(cos ψ + j sin ψ) Z0

(2.119)

For a real characteristic impedance Z 0 , the phase angle ψ of the normalized impedance agrees with the phase angle ϕ Z of the impedance Z , while for a complex characteristic impedance we have ψ resulting as the difference between ϕ Z and the phase angle ϕ Z 0 of the characteristic impedance. As we will demonstrate, this fact has a critical influence on the magnitude of the reflection coefficient. In the case of a lossless line and a passive load impedance, Z 0 is real and positive, ϕ Z 0 = 0, and for Z , the real part is zero or positive, R ≥ 0, −π/ ≤ ϕ Z ≤ π/2. As a result, the nominator in Eq. (2.118) is always smaller or equal in magnitude to the denominator, and |Γ | ≤ 1 holds.

2 Wave Propagation on Transmission Lines and Cables

93

For the magnitude of the reflection coefficient, we obtain |Γ |2 =

1 + A2 − 2 A cos ψ 1 + A2 − 2 A cos ψ = 1 + A2 − 2 A cos(π − ψ) 1 + A2 + 2 A cos ψ

(2.120)

We can see based on Eq. 2.120 that the magnitude of the reflection coefficient cannot be greater than 1 as long as cos(ψ) ≥ 0, i.e. it holds that −π/2 ≤ ψ ≤ π/2 for the phase angle of the normalized impedance. This condition is always fulfilled for a lossless line and a passive load. On the other hand, |Γ | becomes greater than 1 if ψ exceeds these limits. For a complex characteristic impedance, this is always the case if, for example, the line is terminated with a reactance. This result can also be derived graphically as shown in Fig. 2.25. The figure shows two vectors Z /Z 0 , and the resulting vectors of the nominator and denominator of Eq. (2.118). In Fig. 2.25a, the angle ψ is somewhere between 0 and π/2, resulting in |Γ | < 1. Figure 2.25b, shows that for ψ = 3π/4, Γ < 1 is observed. We would now like to investigate the maximum value that the magnitude of the reflection coefficient can assume for a lossy line. Obviously to begin, a peak value of ψ also leads to a peak value of |Γ |. The peak value of ψ arises when the line is terminated into a reactance j X and Z 0 shows a frequency dependence e.g. as seen in Fig. 2.9, providing ϕ Z 0 ≈ −π/4 between 3 and 100 Hz. ψ = φZ − φZ0 =

π π π + =3 . 2 4 4

(2.121)

For the angle ψ = 3π/4, we obtain the following expression from Eq. 2.120: |Γ |2ψ=3 π 4

√

√ 2 2A = =1+ √ √ 1 + A2 − 2 A 1 + A2 − 2 A 1 + A2 +

2A

(2.122)

The maximum for |Γ | is obtained by differentiating Eq. (2.122) and setting A = 1. |Γ |2max

√ ( 2 + 1)2 = √ , √ ( 2 − 1)( 2 + 1)

|Γ |2max = 1 +

√ 2.

(2.123)

Figure 2.25b illustrates this extreme case in the Z /Z 0 plane.

2.3.6.2

Transported Power, Reflection Matching, Impedance Matching

Given a lossless line, the results of the last section would violate the principle of conservation of energy in a passive circuit since more power could be reflected at the passive load then was supplied to it. However, no such contradiction arises for lossy lines in a passive circuit (and the presented results apply only to such lines). In order to demonstrate this, we will first calculate the complex power transported

94

M. Rudolph

a

jlm (Z/Z0)

b

Z/Z0-plane

jlm (Z/Z0)

2j

2j –1

j

+1 Z/Z0 =1·exp(j3p/4)

y

p–y

(Z/Z0)–1

Z/Z0

(Z/Z0)+1

–1

–1 (Z/Z0)–1

y –2

Z/Z0 -plane

j +1 (Z/Z0)+1 –3p/4

1

2 Re(Z/Z0)

–2

–1

1

2 Re(Z/Z0)

Fig. 2.25 Sketch highlighting the vectors Z /Z 0 ± 1. a Angle ψ towards the right half-plane results in |Γ | < 1. b Angle ψ towards the left half-plane results in |Γ | > 1

to the circuit using the characteristic admittance Y0 = G 0 + j B0 for the line under consideration. We have:

1 1 −γ z ∗ ∗ Vt e + Vr e+γ z Vt∗ e−γ z − Vr∗ e+γ z Y0∗ V (z)I ∗ (z) = 2 2 1 2 −2αz |Vt | e = − |Vr |2 e2αz + Vr Vt∗ e j2βz − Vt Vr∗ e− j2βz Y0∗ 2 1 2 −2αz ∗ |Vt | e = Y0 (1 − |Γ (z)|2 + Γ (z) − Γ ∗ (z) 2 1 2 −2αz ∗ = |Vt | e Y0 (1 − |Γ (z)|2 + 2 j {Γ (z)} 2

P(z) =

(2.124)

As the transported effective power, we are interested hereafter only in the real part from Eq. 2.124 for which we have the following: 1 B0 2 −2αz 2 G 0 1 − |Γ (z)| + 2 {Γ (z)} . P(z) = {P(z)} = |Vt | e 2 G0

(2.125)

For the effective power transport to the load, we must set z = l in Eq. 2.125. We will first consider Eq. 2.125 for the lossless case. Then, α and B0 are equal to zero and the magnitude of |Γ | cannot exceed a value of 1. Then, Eq. 2.125 states that the effective power transported to the location z can be calculated as the difference between the supplied power 1/2|Vt |2 · G 0 and the reflected power 1/2|Vt |2 · G 0 |Γ |2 . P(z) always remains greater than or equal to zero, which means the load acts as a power sink. In case of a negative P(z), the sink would act as a generator, but this is not possible since we are assuming a passive circuit. For a lossy line, α and B0 are not equal to zero and the magnitude of |Γ | can be greater than 1. For a passive circuit, the term 2B0 /G 0 (Γ (z)) which no longer disappears now also guarantees a positive power budget again in every case.

2 Wave Propagation on Transmission Lines and Cables

95

Based on the structure of Eq. 2.125, two distinct matching strategies can be derived. We speak of reflection matching if the reflection coefficient disappears, i.e. if we select Z = Z 0 . The effective power fed to the load is then equal to P(l) Z =Z 0 =

1 |Vt |2 e−2αl G 0 2

(2.126)

On the other hand, impedance matching is attained if we select Z = Z 0∗ . This requirement for impedance matching which is known per se can also be obtained based on Eq. 2.125 by seeking its extreme. Accordingly, we first write this equation in the following form: P(z) =

B0 1 (Γ (z) − Γ ∗ (z)) . |Vt |2 e−2αz G 0 1 − Γ (z)Γ ∗ (z) − j 2 G0

(2.127)

The necessary condition for an extreme of Eq. 2.127 is now ∂ P/∂Γ ∗ = 0, whence it also follows in the present form that ∂ P/∂Γ = 0. In other words, the following is required for impedance matching: − Γ (z) + j

B0 = 0, G0

(2.128)

The optimum reflection coefficient Γopt (z) is then Γopt (z) = j

B0 G0

(2.129)

This optimum reflection coefficient is associated with the optimum terminating impedance 1 + j GB00 1 + Γopt (z) 1 = Z 0 opt = Z 0 = Z 0∗ . (2.130) 1 − Γopt (z) G 0 + j B0 1 − j GB0 0 Under the load condition according to Eq. 2.130, the load is then fed the effective power 2 1 B0 2 −2αl G0 1 + P(I ) Z =Z 0∗ = |Vt | e . (2.131) 2 G0 Despite the finite reflection coefficient, in the case of impedance matching the effective power fed to the load is visibly greater compared to the case of reflection matching. For B0 = 0, i.e. a lossless line, reflection and impedance matching are identical. At this point, we should also mention a third matching variant to be covered in volume 2 of this book: noise matching.

96

M. Rudolph ΓL

a

Z0 Z0, b, a

ZL

Γ=0 Zin = Z0

b

Z0 Z0, b, a

Matching Network

ZL

Fig. 2.26 a Mismatched load b load reflection cancelled out by a matching network

2.4 Matching Techniques Reflection of the transmitted wave at the load can only be prevented if the load is matched to the line impedance, i.e. Z L = Z 0 . But commonly, it is to be expected that the input, or output, impedance of an electronic component like and amplifier is not matched to the line that will be connected to it. The solution is to include so-called matching networks into the design, which transform the load (or source) impedance into the line impedance, as depicted in Fig. 2.26. A matching network should not dissipate power, therefore it is commonly realized through reactive elements like capacitors, inductors, and short line sections. Matching of a load to a source is not unique to RF problems. Also in power electronics, the technique is used, commonly to reduce the reactive load to the power grid. In general, the design of a matching network aims at transforming the load impedance Z L = R L + j X L to a real value impedance Z 0 , which can be achieved for a certain frequency by adding an element in parallel and another in series to the load. If we connect a reactance j X s in series to the load, the input impedance becomes: Z Ls = Z L + j X s = R L + j (X L + X s )

(2.132)

which allows to change the reactive part of Z L , but not the resistive part. If we add an additional parallel element j B p , the input admittance reads:

2 Wave Propagation on Transmission Lines and Cables

1 Y Lsp = j B p ||(Z L + j X s ) = j B p + R L + j (X L + X s ) RL X L + Xs = 2 + j Bp − 2 R L + (X L + X s )2 R L + (X L + X s )2

97

(2.133) (2.134)

As we see, it is possible to alter the real part of the admittance (within certain bounds for R L and X L ) to match Y0 = 1/Z 0 by properly tuning X s , and than to cancel out the imaginary part through proper choice of B p . Although matching networks can be calculated and optimized mathematically based on formulas like this one, it not very intuitive. Instead, it is common to use a transmission-line chart, most probably the Smith chart, to derive the concept of a matching and to get an estimate of the element values, before performing the detailed design in a circuit simulation software.

2.4.1 Transmission-Line Charts So far we have seen that wave propagation on a transmission line is best understood in terms of transmitted and reflected waves, which are related to each other by the reflection coefficient. But at the line terminals, currents, voltages and impedances are often better to interpret. How to calculate one description from the other has been addressed in detail. This section will introduce transmission-line charts that graphically integrate both descriptions into a single graph. These are powerful tools, i.e. for the design of impedance-matching networks. Compared to analytical solution techniques, usage of line charts has the benefit of better clarity and speed. The graphical derivation therefore commonly is the first step to gain insight, before an exact solution is determined through numerical simulation. In our derivation of the line charts, we will assume the transformation properties of lossless lines.

2.4.1.1

Displaying Reflection Coefficients in the Impedance Plane—Buschbeck Rectangular Impedance Chart

One possibility to display reflection coefficients and impedances in one chart was proposed by Buschbeck. His chart is based on the cartesian plot of the complex impedance Z , onto which a grid is mapped denoting the corresponding reflection coefficients. The derivation of the chart first requires a few definitions concerning the standing waves and quarter-wavelength transformations: Since the minima and maxima of the standing wave are separated by a quarter wavelength, we can conclude that the values are related to each other according to the quarter-wavelength transformation according to Eq. (2.85): Rmax =

Z 02 . Rmin

(2.135)

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M. Rudolph

From this, we can derive the important relationship Rmin Rmax = Z 02 or Z0 =

Rmin Rmax .

(2.136)

(2.137)

There are further relationships we can discover between the impedance ratios and the matching coefficient m, which is the inverse of the V SW R defined in Eq. (2.92): m=

Vmin 1 − |Γ L | 1 = = VSWR Vmax 1 + |Γ L |

(2.138)

The power in the z -plane (where Vˆ = Vmin holds) must be equal to the power transported through the z -plane (where Vˆ = Vmax holds). In other words, we have 2 2 /Rmin = Vmax /Rmax and thus Vmin Vmin m= = Vmax

Rmin = Rmax

Rmax Rmin Z0 Rmin = = . 2 Rmax Rmax Z0

(2.139)

The transformation of a reflection coefficient along a lossless line only yields a phase shift, while its magnitude remains constant. We can therefore solve Eq. (2.138) for the reflection coefficient, considering |Γ L | = |Γin | |Γin | =

1−m 1+m

(2.140)

According to Eq. (2.84), Z can also be construed as the input impedance Z in of a lossless line which has characteristic impedance Z 0 and line length l and is terminated into impedance Z L . The quantities l and Z L are not determined at this point, and they will eventually establish the link to real and imaginary part of the reflection coefficient. Z 0 , in contrast, is the characteristic impedance the user of the chart intends to refer to. It thus connects impedance level and reflection coefficient. Changing Z 0 results in a differently normalized chart. From Sect. 2.3.4 on the current and voltage distribution on the line, it follows that the impedance Z in is real in the plane of a minimum or maximum of the current or voltage distribution. Equations (2.88)–(2.91) show, that when the voltage reaches its maximum Vmax , the current reaches its minimum Imin , since the current amplitudes are subtracted from each other while the voltage amplitudes are summed up. The input impedance observed at this point l = l B would be Z in = Rmax . The same holds for the case of the current maximum and voltage minimum at l = lk , where Z in = Rmin would be observed. This situation is expressed by the following equations:

2 Wave Propagation on Transmission Lines and Cables

Z in = Rin + j X in = Z 0 = Z0

ZL Z0

99

+ j tan(βl)

1 + j ZZL0 tan(βl) Rmin Z0

+ j tan(βlk )

1 + j RZmin tan(βlk ) 0

= Z0

Rmax Z0

+ j tan(βl B )

1 + j RZmax tan(βl B ) L

(2.141)

In mathematical terms, the individual expressions in the equation are all equivalent to one another. In order to attain independence from the particular value of the characteristic impedance Z 0 , we also normalize the impedance Z to Z 0 . Without limiting the generality, we thus base our derivation of the line chart on the following equation: Z R + jX m + j tan(βlk ) . = = Z0 Z0 1 + jm tan(βlk )

(2.142)

This equation expresses the fact that the pair of values m and l K /λ is equally well suited to characterizing an impedance Z as the pair of values R and X. Here, the matching coefficient m corresponds to the magnitude according to Eq. (2.140), while 4πlk /λ corresponds to the phase of the reflection coefficient. With the aid of Eq. (2.140), the complex plane of the reflection coefficient Γ can now be mapped onto the complex impedance (Z ) plane. The graphical representation of this mapping is the Buschbeck rectangular impedance chart. To represent it, we seek to map the coordinates m = const and l K /λ = const in the complex Z plane. The mapping will be restricted at this stage to passive impedances providing Γ ≤ 1, which means to resistances R ≥ 0. In order to calculate the line set m = const or l K /λ = const, the complex Eq. (2.142) is first split into real and imaginary parts. Real part: X lk R = m, (2.143) −m tan 2π Z0 Z0 λ Imaginary part:

X lk R lk = tan 2π . +m tan 2π Z0 Z0 λ λ

(2.144)

From Eq. (2.143), for example, we now calculate the value of tan(2πlk /λ) and apply it in Eq. (2.144). Or vice versa, we can also calculate m from one of the two equations and eliminate it in the other equation. In this manner, we obtain the following two equations:

2 2 1 2 1 X 1 1 R m+ −m − + = . Z0 2 m Z0 2 m

(2.145)

100

M. Rudolph

2 X 1 1 lK − + −

tan 2π Z0 2 λ tan 2π lλK 2 1 1 lK + =

tan 2π 2 λ tan 2π lλK

R Z0

(2.146)

Based on these equations, we can see that the lines m = const are represented in the Z plane by circles which have the center point coordinates Rm /Z 0 = 1/2(m + 1/m), X m /Z 0 = 0 and the radiuses ρ = 1/2(1/m − m). The lines lk /λ are likewise circles in the Z plane with the center point coordinates Rl =0 Z0

1 Xl 1 lk − = tan 2π

Z0 2 λ tan 2π lλk

(2.147)

and the radiuses ρ = 1/2 · (tan(2πlk /λ − 1/(tan(2πlk /λ)). We are thus now able to construct the coordinate network m = const, lk /λ = const in the Z plane. Since it is equivalent according to Eq. (2.141) whether we assume we have a termination of the line with Rmin spaced lk from the start of the line or a complex terminating impedance Z at the spacing l, we do not need to designate the line length with the k subscript. Figure 2.27a shows the plane of the reflection coefficient to be mapped while Fig. 2.27b shows the Z plane on which the reflection coefficient plane is to be mapped with the aid of Eq. (2.147). Figure 2.27b shows the result of this mapping: the Buschbeck rectangular impedance chart. Corresponding to the conformal mapping function given as Eq. (2.147), the polar (m, l/ λ) coordinates of the reflection coefficient plane also form an orthogonal network in the Z /Z L plane.

2.4.1.2

Displaying Impedance in the Reflection Coefficient Plane—Smith Chart

Instead of displaying a grid denoting the corresponding reflection coefficient on top of the impedance plane, we can display the relation the other way around: drawing a grid of equivalent impedances (or admittances) onto the complex reflection coefficient plane. This type of graph is called the Smith chart [3, 4]. The location of open and short circuit, reactive load and matched load were drawn in into the Γ plane already in Fig. 2.12. In order to get familiar with the Smith chart, it is helpful to learn where to locate these characteristic impedances. The Smith chart has a number of advantages over the Buschbeck diagram. First of all, passive impedances with R > 0 always correspond to reflection coefficients Γ < 1, so that all passive impedance values, i.e. the right half-plane of the complex impedance diagram, gets mapped to within the unity circle. Second, RF measurement heavily relies on the measurement of the reflection coefficient. Therefore, Γ takes precedence and Z is a quantity derived from it. Displaying such a measurement in the

2 Wave Propagation on Transmission Lines and Cables

a

101

b Γ plane

2/16

Z/Z0 plane

X / Z0

l/l

3/16 m = 0.2

2j

3/16

1/16

l /l m=0 m 0.4 0.5 0 8/16

2/16

1j

0.2

1/16 0.66 0.8 0.2

0.4

0.6

0.8

4/16 10 |Γ|

0

0.4

0.5 0.66 0.8

8/16

2

4/16 3 R/Z0

7/16 −1j

7/16

6/16

−2j

5/16

5/16 6/16

Fig. 2.27 a Polar diagram of the reflection coefficient; b Cartesian diagram of the normalized impedance (right half plane). Γ = −|Γ |e− j4πlλ in the range 0 ≤ |Γ | ≤ 1; m circles for constant matching coefficient, m = (1 − |Γ |)/(1 + |Γ |)

Γ plane reflects the relation between measured value and measurement uncertainty better. And finally, the Smith chart can be used as a tool to easily design for example matching circuits. Figure 2.14 already gave a hint to what will be possible, at the example of the transformation of a reflection coefficient over a quarter-wavelength line. In order to derive the Smith chart, we will make use of the equation relating the reflection coefficient Γ to load impedance Z = R + j X and characteristic line impedance Z 0 : Γ = Γ + jΓ =

R+ j X Z0 R+ j X Z0

−1 +1

=

|Z | jφ e Z0 |Z | jφ e Z0

−1 +1

.

(2.148)

We would now like to find the lines R/ Z 0 = const, X/Z 0 = const or |Z|/Z 0 = const, ϕ = const in the reflection coefficient plane. For this purpose, we will follow the same procedure we used for the Buschbeck line chart. If the impedance is characterized by real and imaginary parts, we obtain the following equations for the line R/Z 0 = const and X Z 0 = const: Γ −

R/Z 0 1 + R/Z 0

2

+ (Γ )2 =

1 1 + R/Z 0

2 ,

(2.149)

102

M. Rudolph

Γ l =0,125 l

0,1

0,15

j 0,2

0,05

l =0 l

–1 –0,8 –0,6 –0,4 –0,2

0

0,2

0,4

0,5

0,6

1

l =0,25 l

l =0,5 l

Γ

0,3

0,45 −j

0,4

0,35 l =0,375 l

Fig. 2.28 Reflection coefficient plane in polar coordinates

(Γ − 1) + Γ − 2

1 X/Z 0

2

=

1 X/Z 0

2 .

(2.150)

Figure 2.28 shows the reflection coefficient in polar coordinates. It is also indicated, by which angle the reflection coefficient would be turned if a line-length l/λ would be added (or removed). Plotting curves for R/Z 0 = const and X/Z 0 = const in this plane yields circles. The respective radius and origin are given by Eqs. (2.149) and (2.150). As an illustration, Fig. 2.29 shows lines of the constant real part in the Z plane and the images of these lines in the Γ plane. Figure 2.30 illustrates the relationship between the lines having a constant imaginary part in the Z plane and the Γ plane. A complete Smith chart is given in Fig. 2.31. If we prefer to characterize the impedance Z using the magnitude and phase according to Fig. 2.32, we form the lines |Z |/Z 0 = const and ϕ = const in the Γ plane. The following equations apply to these lines:

2 Wave Propagation on Transmission Lines and Cables

103

X / Z0

0

0,5 1

2

5

0

0,2

0,5

1

2

5

R / Z0

R / Z0

Γ plane

Z plane

Fig. 2.29 Conformal mapping of the lines R = const in the Γ plane X/Z0 5

1

0,5

2

2 0,2

1 0,5 0,2 0 −0,2 −0,5 −1

X Z0 X Z0

R / Z0

5

=0 = –0.2

-5 -2

−2 -0,5

-1

−5

Γ plane

Z plane

Fig. 2.30 Conformal mapping of the lines X = const in the Γ plane

(Γ )2 + (Γ + 1/tan φ)2 = 1/ sin2 φ Γ −

|Z | Z0 |Z | Z0

+1 −1

2 + (Γ ) = 2

2|Z | Z0 |Z | − Z0

(2.151)

2 1

.

(2.152)

104

M. Rudolph 0,4

0,3

Vmax

3.0

80°

4.0 5.0

20 50 ∞

10

V

min

0.15 70°

0.1 6 60

0.1 7 °

0. 18

5 0,

°

19 0.

50

2.0

°

0 13

°

110 0,7

7 0.0

0°

12

90°

100°

S

0.14

0,6

8 0.0

0.13

0.12

0.11

1.5

9 0.0

0. 06

2.0

Vmax

0.10

4 0,

0°

0.0 4

15

0,3

0.03

0.22

5.0

0,2

0,1 ° 210

20

10

5,0

3,0

4,0

2,0

1,5

0,9 1,0

0,8

0,6

0,7

0,5

0,4

0,2

0,1

2

00°

0.2 8

5,0 4,0

0°

33

0,3

0.2 9

0,2

6 0.4

0.2 7

0.48

10

0.49

j

X Z

0

0,3

X Z j

l/ λ

0

180°

Z

20 50

0.26

350°

0,1

340°

0.47

0.25

0°

190°

orcr

20 50

R

0.24

0.01

10

10°

170°

0.23

0.02

20°

160°

30°

4.0

0.21

3.

0

°

0 0.2

40

14 0°

1.5

1,0

m

0.0 5

1,0

0,5 0,6 0,7 0,8 0,9 1,0

Vmin

0,9

0,2

0,8

0,1

0

2,0

0,

0°

5

1,5

1,0

29 0.34

0°

0.35

280°

0.36

270°

0.37

0.38

0,9

0°

0,8

0,6

30

0,7

3 0.3

0° 22

23

0° 31

32 0.

44 0.

31 0.

5 0.4

0

3,

0°

32

0. 30

4

0,

250° 260° 0.39

0.4 0

0. 43

0°

24

0.4 2

0.4 1

Fig. 2.31 The Smith Chart, displaying normalized impedance in the reflection-coefficient plane

These two last equations are also circle equations. Figure 2.33 illustrates the relationship between the lines |Z |/Z 0 = const in the Z and the Γ plane and Fig. 2.34 illustrates this relationship for the lines ϕ = const. Figure 2.35 shows the complete chart, which is also commonly known as a Carter chart.

2.4.2 Narrow-Band Matching Techniques One of the key goals in RF circuit and system design is to avoid reflections on lines, i.e. to ensure that the power provided by a source is fully transferred to the

2 Wave Propagation on Transmission Lines and Cables Im(Z)

105

j–80° 60°

|Z| Z0

= const 40°

20° j

0,2

0,4

0,6

0,8

1,0

Re(Z)

–20°

–40°

–60° –80°

Fig. 2.32 Representation of the impedance Z /Z 0 by magnitude and phase in the impedance plane

load. Reflected power is not simply lost, if potentially can damage the source or disturb signal integrity. This section will introduce to the design of matching circuits consisting of reactive elements. The task of a matching network is to transform a load impedance Z L , corresponding to Γ L = 0 to Z 0 , corresponding to Γ = 0. In this section, matching using two elements will be introduced, and it will be shown how to design a matching circuit graphically using the Smith chart. At first, we focus on matching of one single frequency, therefore it is to be expected that the matching will be rather narrow-band. Figure 2.36 shows the benefit of using the Smith chart as a design tool. Two impedances are drawn, Z L = Z 0 · (0.6 − j) and Z L = Z 0 · (1.5 + j2.5). The arrows indicate how inserting a series element will change impedance and reflection coefficient. In case of lumped elements, it is clear that a capacitance in series reduces the imaginary part of Z L , while the inductance increases its imaginary part. In the Smith chart, an increasing imaginary part results in following the circle of constant real part in clockwise direction, eventually ending at the open circuit for L → ∞. Reducing the imaginary part through a series capacitance requires to follow the same circle counter-clockwise, eventually reaching the open circuit for C = 0. The grid

106

M. Rudolph

Im

|Z| |Z| Z0

0 0,1 0,2

0,5

Z0

= const

1,0

Re

0,1 0,2

0,5

= const

1,0

2,0

5,0 1,0

Fig. 2.33 Mapping of the circles |Z |/Z 0 = const in the Γ plane

of the Smith chart allows us to read the impedance of the capacitance or inductance required for a certain transformation of the load impedance. But it also allows to directly assess what impact the new element has on the reflection coefficient. Since the Smith chart plots impedances in the Γ plane, Γ can be determined by interpreting the location of the impedance in polar coordinates. In our example, we see that |Γ | is reduced by inserting an inductance to Z L and a capacitance to Z L . In addition, the Smith chart allows to easily determine how a series line with characteristic impedance Z 0 affects input impedance. Since a lossless line merely shifts the phase of Γ by φ = 4πl/λ in clockwise direction the new input impedance is found on a circle, and its value can readily be determined after the transformation. The Smith chart usually provides a scale encircling the graph shown here, indicating what angle corresponds to what fraction of l/λ. Achieving power match, which means to transform Γ → 0 and Z L → Z 0 , is not possible with one series component alone, with the exception of impedances for which Z L = Z 0 holds. Figure 2.37 shows, how adding a parallel capacitance or inductance alters the reflection coefficient. Since adding a reactive element is not changing the real part of Y L = 1/Z L , the transformation path follows a circle of constant Y L . In the parallel connection, adding a capacitance transforms the admittance value in clockwise direction, eventually reaching the short circuit for C → ∞.

2 Wave Propagation on Transmission Lines and Cables

107

Im 90°

80° 60°

j=

90° 80°

40°

60° 40° 20° 20° j

0°

0° Re

–20°

–20° –40° –60°

–40°

–80° j= –90°

–60° –90° –80°

Fig. 2.34 Mapping of the lines ϕ = const in the Γ plane

Adding a parallel impedance, on the other hand, transforms counter-clockwise, reaching the short circuit for L → 0. For sake of completeness, transformation through a line is also plotted in the admittance-type Smith chart, although the transformation is independent of the grid and resembles the transformation already addressed when discussing series connections. Comparing the different transformation paths for series and parallel connection, it becomes obvious that the proper combination of a series and a parallel element allows to match any impedance to Z 0 . This two-element matching is performed in two steps: 1. Start with a series (or parallel) element. Use it to transform the load impedance so that (1/Z L ) = Y0 holds (or in case of starting with a parallel element, that (1/Y L ) = Z 0 holds). 2. At this stage, only the imaginary part of the transformed admittance Y L (or impedance Z L ) needs to be resonated out. Add a parallel element providing Y = −Y L (or a series element providing Z = −Z L ). Figure 2.38 highlights the impedance and admittance values that can be matched through a single element, since either the real part of the impedance or of the admittance equals Z 0 . This transformation hence can be performed by the second element. It is the task of the fist element, to reach one of these two circles.

108

M. Rudolph

0

0,1

0,2

0,3

0,4

m

8

0 0,

°

110

°

120

70°

0,1 6

0, 17

60 °

80°

0°

0, 14 0 0° 6

0, 18

50 °

13

70°

40 °

15

0°

60°

40°

0,22

°

50°

0,23

160

20 50 ∞

10

20°

0,0 3

80°

90°

s

0,15

4,0 5,0

Vmax Vmin

° 30

20 50

10

4,0 5,0

3,0

0°

2,0

1,5

0,6 0,7 0,8 0,9 1,0

0,5

0,4

0,3

0,2

0,1

10°

190° 0,47

-40°

21

0°

300

0,3

3

23

° 290

0°

24

0°

37 0,38 0 ,3 9 0,36 0, 0,40 ,35 0,4 260° 4 0 280° 270° 0,3 1 25 °

44 0° 0, 22

0° 31

ϕ -90°

32 0,

45 0,

0°

-80°

0, 30

32

-70°

31 0,

6 0,4

0°

-60°

0,2 9

33

0°

-50°

0,2 8

° 200

-30°

340 °

0,48

350°

-20°

0,26 0,27

0,49

-10°

0,25

Z Z

10° orcr 0°

20°

0,24

170°

30°

180°

0,02

100°

0,13 0,14

90°

3,0

1 0,2

0 l/ λ0,01

0,12 0,11

2,0

0 0,2

0,0 4

9 0,0

0 0,1

1,5

19 0,

0, 05

07 0,

1,0

0,5 0,6 0,7 0,8 0,9 1,0

Vmin Vmax

0, 43

0, 42

Fig. 2.35 The Carter chart, displaying normalized impedance in magnitude and phase in the reflection-coefficient plane

Figures 2.39 and 2.40 show examples for possible matching networks and transformation paths that allow the matching of mainly capacitive load reflection coefficients. Capacitive impedances Z providing a negative reactance − j X are located on the lower half of the Smith chart. And in order to compensate the reactive part of the impedance, our matching network needs to be inductive. If R ≤ Z 0 holds for the real part of Z L = R − j X , it is possible to use a series inductance L in order to transform the impedance in a way so that (1/Z ) = 1/Z 0 = Y0 holds, as shown in Fig. 2.39. The group of arrows indicates the transformation path for any source impedance within the indicated area. At first, a series inductance is employed to reach a point on the circle where (1/Z ) = Y0 holds. If the conductance

2 Wave Propagation on Transmission Lines and Cables

109

C 2.5j Z'L Γ'L

ZL f' = 4πl / λ

Z'L + jwL

ZL – j / (wC)

Z'L – j / (wC) L 1.5

0.6

ΓLe–j4πl / λ 0

ZL

Γ'Le j4πl / λ

ZL + jwL ZL + jwL ZL – j / (wC) −0.3j ZL ΓL

f = 4πl / λ

−2.5j

ΓL

Z0, l / l

−1j

ΓLe–j4πl / λ

Fig. 2.36 Transformation of an impedance Z L by adding a series element

Y'L – j / (wL)

C

YL

L

YL

f' = 4πl / λ Y'L

YL + jwC

Γ'L

Y'L + jwC 0.1

0.3

0.6

1

1.5

2.4

13

4.4

ΓLe–j4πl / λ

Γ'Le j4πl / λ YL – j / (wL) YL ΓL

YL – j / (wL)

YL + jwC

Z 0, l / l

ΓL

f = 4πl / λ ΓLe–j4πl / λ

Fig. 2.37 Transformation of an admittance Y L by adding a parallel element

(1/Z L ) of the load larger than Y0 , it is possible to match with the two inductances as indicated in Fig. 2.39a. But it would also be possible to use a higher value of L and then transform back to Z 0 through a parallel capacitance C see Fig. 2.39b. The latter transformation path is also accessible for all load admittances with G L ≥ Y0 , and even those providing an inductive reactance.

110

M. Rudolph

Y L = 1 / Z0 add parallel C

add series C

add parallel L

add series L

ZL = Z0

Fig. 2.38 Sketch of the two circles where either the real part of the impedance or of the admittance equals Z 0 . It is possible to match these impedance and admittance values to Z 0 by means of a single L or C a

b Y = 1 / Z0 (2) add parallel C L

L Y = 1 / Z0

L

C

ZL

ZL

Z0

Z0 (2) add parallel L

(1) add series L

(1) add series L

Fig. 2.39 Examples of transformation paths for the matching of mainly capacitive loads providing Z L ≤ Z 0 . a Transformation path possible for Z L ≤ Z 0 , Y L ≥ Y0 . b Transformation path possible for loads providing either Z L ≤ Z 0 and Y L ≥ Y0 or Y L ≤ Y0

The example clearly showed that starting with a lumped series element does not allow to match load impedances featuring R L ≥ Z 0 . Since the load is still capacitive, we will again start with an inductance L, but this time in parallel connection to the load. Figure 2.40 sketches possible transformation paths. In contrast to the previous case, we now use an admittance to transform the load to an impedance providing R = Z 0 . The first transformation path therefore follows the constant-realpart lines on the admittance Smith chart, where the parallel inductance transforms in counter-clockwise direction. Otherwise, the matching is analog to the previous case.

2 Wave Propagation on Transmission Lines and Cables

a

111

b (2) add series C

L

C L

Z = Z0

ZL

Z0

L

Z = Z0

ZL

Z0

(2) add series L (1) add parallel L

(1) add parallel L

Fig. 2.40 Examples of transformation paths for the matching of mainly capacitive loads providing Y L ≥ Y0 . a Transformation path possible for Y L ≥ Y0 , Z L ≤ Z 0 . b Transformation path possible for loads providing either Y L ≥ Y0 and Z L ≤ Z 0 or Z L ≥ Z 0

Figure 2.40a shows transformation paths for loads providing G L ≤ Y0 , R L ≤ Z 0 , while the transformation paths indicated in Fig. 2.40b additionally covers all load impedances featuring R L ≥ Z 0 . These four types of transformation paths can be realized when starting with an inductance as the fist element, and it is evident that the region of inductive loads is only partly covered. In case of an inductive load, it might be wise to start matching relying on a capacitance instead. As the transformation path obtained is equivalent to the examples given above, we will leave it to the reader to deduce the matching network structures required to complete Figs. 2.39b and 2.40b into ying and yang symbols. The Smith chart is not only helping to visualize the transformation path in order to select an appropriate matching network topology. Proper charts also provide grid labels that allow to calculate the component values from the normalized reactance values X/Z 0 read from the chart. The graphical construction of a first-order estimate of a matching network should therefore be performed relying on the Smith chart, in order to get a sound starting point for the detailed design process using a numerical circuit simulator and more realistic component models. With increasing frequencies, it becomes more and more technologically challenging to realize impedance and capacitance components. On the other hand, the ratio between a certain line length and wave length increases with frequency, so that transformations through short low-loss lines becomes more and more feasible. At highest frequencies, matching through two transmission-line elements becomes a necessity since lumped elements are no longer available. As discussed starting with Sect. 2.3.3, a short open-circuit line shows an input impedance similar to that of a capacitance, while a short short-circuit line’s input impedance resembles that of an impedance. It is thereby quite straight forward to replace the parallel elements from the previous examples by parallel stub lines.

112

M. Rudolph

Instead of calculating the required line length using the equations derived above, we can also rely on the Smith chart to determine the line length. Figure 2.41 highlights how parallel stub lines can be designed. Figure 2.41a shows possible transformation paths for load admittances of Y L = Y0 . Admittance with different real parts would be transformed in a similar fashion. It would also be possible, of course, to transform Y L way into the upper part of the Smith chart using the short short-circuit stub, following the indicated circle of constant conductance. The same holds analogously for the open circuit stub and the capacitive (lower) half of the chart. The transformation achieved by parallel capacitances and inductances as shown in Fig. 2.40 thereby can be realized by the respective stub lines. Figure 2.41b highlights how the Smith chart can be used to determine the properties of the required capacitance or stub line. In this example, an admittance of Y L /Y0 = 1 + j1.5 is to be matched to Y L /Y0 = 1. Obviously, this can be achieved by connecting an inductance in parallel providing − j/(ωL · Y0 ) = − j1.5. In order to determine the respective line length, we use the Smith chart to determine the phase shift that would be caused by the inductance alone. Reactive loads such as an inductance or capacitance, as we have learned, provide a reflection coefficient of Γ = 1 together with a certain phase shift. We can determine the inductance properties therefore by mapping the difference in the imaginary part of the admittance caused by the inductance to the perimeter of the Smith chart, as shown by the dashed lines in the figure. This mapping to the perimeter equals the subtraction of the real part of the load impedance so that only the change in the imaginary part remains. The difference in Y before and after the transformation thus equals the admittance of the parallel inductance. The line length is now determined straight forward from the angle read from the perimeter of the Smith chart. In our example, the phase is shifted by 110◦ , or 4πl/λ = 0.6π , thus the line length required equals l/λ = 0.15. Smith charts designed to derive matching circuits commonly provide a scale around the chart that allows to directly read the angle in terms of relative line length l/λ. It is in principle also possible to replace lumped series elements through stub lines connected in parallel. If this is possible, however, depends very much on the geometry of the line. Series stub lines might be possible for twisted-pair and rectangular wave guide structures, but not for microstrip lines and coaxial cables. But it is always possible to transform a reflection coefficient by connecting a series line, shifting the reflection coefficient’s phase by 4πl/λ in clockwise direction. Figure 2.42 shows possible transformation paths, where a line is used to rotate the reflection coefficient’s phase until Y L /Y0 = 1 is reached. A parallel stub as the second element can then transform the admittance to Y L /Y0 = 1. Using lines, or combinations of lines and lumped elements, allows for many different approaches to match a load. It should also be remembered that a quarterwavelength line transforms a real impedance into another real impedance. When designing a matching circuit, one always can choose between different topologies. This gives some freedom to the designer. Some practical aspects are: • Long lines and high impedance elements add significant losses to the matching network, which attenuate the signal and add noise. The aspect of element losses

2 Wave Propagation on Transmission Lines and Cables

a

l/l

Z0

113

b −1j

−1.5j

−1.5j

ΓL

YL

C

−0.6j

−2.5j

−2.5j

−0.3j

-0.3j

YL + jwC

−6j

−6j

0.1

0.3

0.6

2.4

1.5

13

4.4

8

0.1

0.3

1

0.6

2.4

1.5

4.4

8

13

0

0

6j

6j

l/ Z 0,

l 0.3j 2.5j

2.5j

L

ΓL

YL

1.5j

4pl = 0.6p

0.6j

1.5j 1j

l

–j

YL – j / (wL)

wL·Y0

= –j1.5

Fig. 2.41 Sketch highlighting admittance transformation by means of parallel stub lines. a Transformation path of parallel lumped elements and equivalent stub lines at the example of a load impedance already providing Y L = Y0 . b Example of a transformation path by a parallel inductance or short-circuit stub line, and how to determine the respective element values

a

b l2 Z 0,

/l

l2 Z 0,

Y = 1 / Z0

/l

(2) add open-circuit stub line

ΓL

ΓL Y = 1 / Z0 Z0

(2) add short-circuit stub line

Z0

Z0, l1 / l

Z0, l1 / l

(1) insert line (1) insert line

Fig. 2.42 Examples of transformation paths for the matching of mainly capacitive loads using a line and a parallel stub. a Transformation path possible for short stub line. b Transformation path possible for open-circuit stub line

was not considered in this introduction, but it can’t be ignored when moving forward from the Smith-chart based design concept as described above to the circuit-simulator based design relying on accurate component models. • Matching networks are commonly designed at the input and output of RF circuits like amplifiers. It is common to reuse the components of the matching network for other purposes like:

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– Provide DC bias to a transistor. A parallel inductance to ground or a short-circuit stub can be connected to an RF short by a high capacitance C∞ in parallel to the DC supply voltage. – Provide DC decoupling through a series capacitance. – Optimizing the stability of an amplifier. Amplifiers are prone to oscillations, typically at frequencies below the target RF bandwidth the amplifier is designed for. At low frequencies, the gain of the transistor is high, and providing high reflection coefficients easily provokes oszillations. A certain damping is therefore advantageous. The low-pass or high-pass characteristic of the matching circuits can be exploited to connect frequency-selective damping to a transistor’s port. • Not at least consideration should be given to the practical question of element tolerances and production cost. Some theoretically possible circuits can’t be realized in good quality or with high yield due to the required element values. Other realizations might require higher effort in production or need more expensive parts. Examples for transistor matching circuits will be shown after discussing the bandwidth of the matching circuits.

2.4.3 Broadband Matching Techniques So far, matching circuits were discussed for a fixed frequency. One would determine for example a required reactance j X from which the inductance L = X/ω is derived. Any deviation in frequency ω hence leads to a certain mismatch. In reality, of course, also the impedance to be matched will vary with frequency. Therefore, the final broadband matching circuit design will be performed relying on circuit simulation software. But also in this case, it is required to start analytically with an approximate design concept as a basis for the numerical approach. The design of broadband matching networks is quite similar to designing a filter: the aim is that within a certain frequency band, the input impedance is matched and the electrical signal can pass without being reflected or attenuated. We can achieve it e.g. by increasing the number of segments from which the matching circuit (or filter) is composed. In filter theory, a number of approaches were developed to approximate the wanted frequency response scheme through the rational transfer function that can be realized with discrete elements. The same holds for matching circuits, including the derivation of the theoretical limit by Fano for typical load impedances. In practical cases, however, we might on one hand like to have a Smith-chart based quick estimation method for the network, before we design it considering the non-ideal component behavior using a numerical simulator. Which means to drop the classical exact mathematical approximation based on the assumption of ideal components. A common method to estimate the bandwidth of the matching goes through the quality factor Q as known from resonators. Our matching circuit, in fact, is a resonant circuit. As discussed in Chap. 1, the quality factor Q gives the ratio of the power that is

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stored in the reactances to the power that is dissipated by the resistance at resonance frequency, expressed by the ratio of the inductive part of the reactance X k to the resistance R: |X k | Q= (2.153) R It has also been shown that the 3-dB bandwidth Δω of the resonator depends on Q according to: ωr (2.154) Δω = Q with ωr denoting the resonance frequency. We will not apply these findings to a matching network. First of all, in order to increase the bandwidth of the matching, we need to reduce Q. It might be counter intuitive to aim at a low value for a quantity called quality factor, but it makes perfectly sense from the physics point of view. The goal of impedance matching is to deliver as much power to the load as possible. Maximizing reactive power stored in the matching network, on the other hand, only serves the purpose to translate the load impedance into Z 0 . An ideal matched load of Z L = Z 0 would not require any matching, leading to Q = 0 due to the missing reactive elements. At the same time, the match would not be limited to a specific frequency band. Second, if we assume that the load can be approximated close to the target center frequency ωr by either Z L = R L + j X L = R L + jωL L or 1/Z L = G L + j B L = G L + jωC L , it becomes obvious that the frequency dependence is already introduced by the load to be matched. If we manage to cancel the reactive part by means of a resonant circuit, its minimum Q min is given by Q min =

|X L (ωr )| RL

or

Q min =

GL |B L (ωr )|

(2.155)

where ωr denotes the center frequency. There is therefore a maximum bandwidth that can be achieved for a certain load. Third, since the maximum bandwidth is determined by the load impedance, it is impossible to increase it. But it is straight forward to reduce it by introducing high-value reactive elements to the matching network. The actual Q factor therefore needs to be analyzed every time a new element is added to the matching network. The approximate design method for a matching circuit providing a certain bandwidth would therefore follow the following steps: 1. Determine the Q factor corresponding to the desired bandwidth, load impedance and center frequency. 2. Determine which area of the Smith chart denotes impedances of lower Q factor. All the impedance transformation needs to stay inside this area. 3. Use one or more two-element transformations within the allowed area of the Smith chart to achieve matching at the center frequency.

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Constant- Q Circles in the Smith Chart

With the quality factor of a resonant circuit being defined as Q = |X |/R, contours for constant Q are easily drawn in the impedance plane. With the real part R drawn on the x axis, the imaginary part ±X is given by the two lines ±X = ±Q · R. In the Smith chart, the lines are transformed into circles that cut through the short circuit Γ = −1 and also through the open circuit Γ = 1. Considering the relation between reflection coefficient and impedance: Z = R + j X = Z0

1+Γ 1−Γ

(2.156)

we find the following equations for the real and imaginary parts: R = Z0

1 − Γ 2 − Γ 2 |1 − Γ |2

X = Z0

2Γ |1 − Γ |2

(2.157)

Determining Q = |X |/R yields: Q=

2Γ 1 − Γ 2 − Γ 2

(2.158)

which can be transformed into an equation expressing a circle: 1 1 2 2 1 + 2 = Γ + Γ ± Q Q

(2.159)

The center is given by: ±j and the radius is given by

1 Q

(2.160)

1 Q2

(2.161)

1+

Figure 2.43 shows the contour for Q = 1 as an example in the impedance plane and in the Smith chart.

2.4.4 Application Examples for the Smith Chart 2.4.4.1

Reflection Coefficient Along a Transmission Line

Problem 1: We are given an assumed lossless line with the characteristic impedance Z 0 = 50 , relative dielectric constant εr = 2.25 and length l = 12.66 cm. The line is terminated at f = 300 MHz with an impedance Z 2 = (30 − j50) . We would

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117

Fig. 2.43 Contours of constant Q a in the impedance plane, b in the Smith chart. Both cases show the example of the contour for Q = 1, and denote the areas of higher and lower Q

like to determine the reflection coefficient Γ2 at the end of the line, the reflection coefficient Γ1 at the start of the line and the input impedance Z 1 for the line. Solution: We plot the normalized impedance Z2 = 0.6 − j Z0

(2.162)

in the Smith chart in Fig. 2.44a (point [1]). Based on the length of the vector from the matching point R/Z 0 = 1 to point [1], we can read off the magnitude of the reflection coefficient Γ2 along with its phase on the outer division of the line chart. Γ2 = 0.57e− j4π(0.36−0.25) = 0.57e− j4π·0.11 = 0.57e− j79.4

◦

(2.163)

We now calculate l/λ for f = 300 MHz. λ= √

c = 66.66 cm εr f

l 12.66 = = 0.19 λ 66.66

(2.164)

Since the magnitude of the reflection coefficient does not vary on a lossless line, the geometric position of the reflection coefficient Γ1 and normalized impedance Z 1 /Z 0 we are seeking must lie on a circle with constant magnitude |Γ | = 0.57 for the reflection coefficient. According to Eq. (2.78), the phase of the reflection coefficient Γ2 varies corresponding to e− j4πl/λ . In the Smitch chart, this corresponds to a rotation of the vector for Γ2 through the arc ≡ l/λ = 0.19 in the clockwise

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direction (mathematically negative direction of rotation). In this manner, we form point [2] in the line chart and read off as follows: Γ1 = 0.57e j4π(0.25−0.05) = 0.57e j4π∗0.2 = 0.57e j144 Z1 = 0.3 + j0.3 ZL

◦

(2.165)

The absolute value of the input impedance is obtained by multiplying Z 1 /Z 0 by Z 0 = 50 . (2.166) Z 1 = 15(1 + j) .

2.4.4.2

Determine Admittance Value from Complex Impedance

Problem 2: We are given an impedance Z = (30 − j50) and would like to determine the admittance Y = 1/Z . Solution: We normalize Z to a suitable characteristic impedance. In this example, we will choose Z 0 = 50 and enter the point [1] in the line chart. According to Eq. (2.85), a line of length l = λ/4, produces a transformation corresponding to the reciprocal of the impedance. We imagine a λ/4 line connected in front of the impedance Z which transforms the normalized terminating impedance Z /Z 0 into the normalized input impedance Z 1 /Z 0 . Z 1 /Z 0 = Z 0 /Z = Z 0 Y.

(2.167)

In the Smith chart, this transformation corresponds to a rotation of the reflection coefficient vector from the point [1] through the arc π . We plot point [3] in the line chart and read off as follows: Z 0 /Z = Z 0 Y = 0.44 + j0.725.

(2.168)

We obtain the absolute value of the admittance we are seeking through denormalization, i.e. division by Z 0 = 50 or multiplication by 20 mS: Y = (8.8 + j14.5) mS.

(2.169)

Since transformation by means of a λ/4 line always corresponds to a rotation of the reflection coefficient vector through the arc π , we can also find point [3] by simply mirroring point [1] at the matching point.

2 Wave Propagation on Transmission Lines and Cables

2.4.4.3

119

Matching Using Line and Capacitance

Problem 3: We are given the same line as described in Problem 1, but now the line length l is to be defined. The circuit’s reflection coefficient is to be compensated by connecting a capacitive reactance in series. According to Fig. 2.44b, we are looking for the shortest length L to which the compensation is to be applied. Moreover, we would like to know the capacitance of the compensation capacitor. Solution: Using the circuit shown in Fig. 2.44b, we would like to compensate the reflection coefficient at the position l on the line. In other words, the total impedance of the circuit must assume the normalized value Z /Z 0 = 1 at the position l. Since only the imaginary part of an impedance can be compensated by connecting a reactance in series, we must first transform the terminating impedance Z 2 using a line section of length l such that the transformed, normalized real part already assumes the value {Z (l)/Z 0 } = 1. In our example, this is the case at point [4] at which the reflection coefficient circle and the line chart circle R/Z 0 = 1 intersect. From the Smith chart, we read off the value of l/λ which is necessary to reach the point [4]. l/λ = 0.313;

l = 0.313 · 66.66 cm = 20.8 cm.

(2.170)

By connecting a normalized series reactance of j X/Z 0 = − j1.4 = 1/( jωC Z 0 ).

(2.171)

we are able to move from point [4] to the matching point. At a frequency of f = 300 MHz, this reactance corresponds to a capacitance of C = 1/(1.4 ωZ 0 ) = 7.6 pF.

2.4.4.4

(2.172)

Narrowband and Broadband Matching

Problem 4: Find a matching circuit for a load consisting of a resistor R L = 100 in series with a capacitance C L = 8 pF to Z 0 = 50 providing (a) 1 GHz and (b) 2 GHz of bandwidth around 1 GHz. Solution: As suggested, we design two matching networks, using the Smith chart at a frequency of f 0 = 1 GHz. The load impedance then reads: Z L = (100 − j20) . The first step is to mark the normalized impedance Z L /Z 0 in the Smith Chart, see Fig. 2.45a and b. The bandwidth Δf is considered by constraining the quality factor Q of the matching circuit. Since Δ f = f 0 /Q, the matching circuit’s Q must not exceed Q = 1 for a bandwidth of 1 GHz, and Q = 0.5 for a bandwidth of 2 GHz. The second step is therefore to draw the respective Q circles by centering our compasses at ± j/Q,

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M. Rudolph 1,0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 Reflection coefficient |г|

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0.37

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Fig. 2.44 a Smith chart with application examples; b for problem 3: series compensation with C

reading the Smith chart as the cartesian coordinate representation of the reflection coefficient. The constant-Q circle cuts through Γ = ±1, see Fig. 2.45a and b. For the 1-GHz-bandwidth case (a), we see that it is possible to use parallel capacitance C1 = 1.3 pF which transforms Z L so that the real part of the impedance already matches Z 0 . The transformation path and the actual values for the reactive admittance would be read from the admittance grid printed on top of the impedance grid shown in Fig. 2.45a. The admittance grid and labels are omitted in the graph in order to make it easier readable. We observe that the capacitance transformed Z L to Z L = Z 0 − j Z 0 , which requires a series inductance L 1 = 8 nH to complete the matching to Z 0 . Figure 2.45c shows load and matching circuit. For the 2-GHz-bandwidth case (b), we can not use the two-element matching derived in part (a), since the condition Q = 0.5 restricts us to use a much smaller

2 Wave Propagation on Transmission Lines and Cables

a

121

b

ΓL

ΓL

Q=

Q=1

c

1 2

d ZL ΓL

Γin L'1

ZL ΓL

Γin RL CL

C'1

e

L1

L2

L3

C1

C2

C3

RL CL

f

Γ = –10 dB

Γin (dB)

0

ΓL broadband match

narrowband match

0.25–2.25 GHz

narrowband match

0.4–1.4 GHz

–10 –20

broadband match

–30 –40 1

2

3

4 f (GHz)

Fig. 2.45 Example for narrowband and broadband match. Smith chart showing transformation path and circle segments for maximum Q (a, b), c matching circuit corresponding to (a), d matching circuit corresponding to (b), frequency response for the load and the two matching circuits in the frequency range 0.5 . . . 2 GHz (e), frequency response drawn in dB (f)

area within the Smith chart, see Fig. 2.45b. Instead of using a two-element match, we therefore iterate towards the matching, by cascading two-element sections as shown in Fig. 2.45d. Determining the transformation paths and parameter values is otherwise identical to the solution of part (a). In the end, we obtain the following element values: C1 = 0.46 pF, L s = 6.5 nH, C2 = 0.9 pF, L 2 = 5.3 nH, C3 = 1.4 pF, and L 3 = 4.5 nH.

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The third step is to check the frequency response of our solution. At this stage, pen and paper are replaced by a computer with circuit simulator or other programming tool like python with the scikit-rf package as used to create the basis of these figures. Figure 2.45e shows the locus of the unmatched Γ L and the input reflection coefficients after 1-GHz (narrowband) and 2-GHz (broadband) matching for the frequency range 0.5 ≤ f ≤ 2 GHz. The locus turns clockwise with frequency, so that we know that f = 0.5 GHz corresponds to the lower end of the lines, while the upper ends correspond to f = 2 GHz. The matched curves cut through the center of the Smith chart for f = 1 GHz. As a guide, we drew a circle corresponding to |Γ | < −10 dB, which would be a boundary commonly considered to be a reasonably good match. As we see, the two-element match yields a frequency response where the impedance changes rapidly, covering quite a broad area of the Smith chart. The frequency response of the six-element match, on the other hand, stays within the circle, corresponding to the lower Q. Finally, Fig. 2.45f plots |Γin | in dB for both cases. We observe a negative peak in both curves at 1 GHz corresponding to the perfect match that we designed using the smith chart. For the two-element match, we obtain |Γin | < −10 dB in the frequency range of roughly 0.4 . . . 1.4 GHz, while the six-element matches provides matching for frequencies 0.25 . . . 2.25 GHz. Corresponding to the number of two-element sections, the frequency response of the narrowband match shows one minimum, while the three-section wideband match shows three minima. It is also observed that the response is not symmetric around f 0 = 1 GHz.

2.5 Scattering Parameters In this chapter so far, we focused on the transmission line that was terminated by a certain impedance. In the analysis of the wave phenomena on the line we derived the reflection coefficient that corresponds to the combination of load impedance and characteristic line impedance, and we learned that the voltage and current at the load terminal are defined by the superposition of transmitted and reflected wave voltage and current. It is now about time to shift the focus back to what we so far considered only by its property to be connected to the end of the line. In general, transmission lines will be used to connect all sorts of electronics, most of them being multiports. It makes therefore sense to extend the theory developed so far to linear multiports. Linear multiports are commonly described by a matrix connecting terminal currents and voltages. A four-port as depicted in Fig. 2.46a could for example be characterized through an Y matrix, defined as ⎞ ⎛ Y11 I1 ⎜ I2 ⎟ ⎜ Y21 ⎜ ⎟=⎜ ⎝ I3 ⎠ ⎝ Y31 I4 Y41 ⎛

Y12 Y22 Y32 Y42

Y13 Y23 Y33 Y43

⎞ ⎛ ⎞ Y14 V1 ⎜ V2 ⎟ Y24 ⎟ ⎟·⎜ ⎟ Y34 ⎠ ⎝ V3 ⎠ Y44 V4

(2.173)

2 Wave Propagation on Transmission Lines and Cables

a

123

b I3

I1 V1

V3

V2

V4

I4

I2

c

a1

a3

b1

b3

a2

a4

b2

b4

d Z0

I1 V1

I2

a1 b1

I3

Z0

I4

b2

b3

b4

Z0

Z0

Fig. 2.46 Current and voltage definitions for a four port (a), wave definitions (b). Measurement condition to determine Y11 , Y21 , Y31 and Y41 (c), measurement condition to determine S11 , S21 , S31 and S41 (d)

In order to determine an element of the matrix, e.g. Y11 , one would apply a voltage V1 and short-circuit all other ports, V2 = V3 = V4 = 0, see Fig. 2.46c. The first line of the matrix therefore simplifies to I1 = Y11 · V1

for V2 = V3 = V4 = 0

(2.174)

and Y11 can be determined form measurement of I1 . The relations between currents and voltages for a certain two-port can also be formulated in other ways, e.g. through Z parameters. As the different matrix representations describe the electrical properties of the multiport, it is possible to transform one representation into another representation. In the RF domain, it has to be considered that transmission lines will be connected to the ports, so that it becomes advantageous to describe the multiport in terms of incident and reflected waves instead of port voltages and currents. Figure 2.46b depicts the definition of incident waves labeled a1 . . . a4 and emitted, reflected or transmitted waves b1 . . . b4 . These waves are defined as normalized voltage waves as follows. At a port n, voltage Vn and current In are given by the superposition of waves Vn+ , In+ incident to the port and waves Vn− , In− emitted by the port: Vn = Vn+ + Vn− In = In+ − In−

(2.175) (2.176)

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where the characteristic impedance defines the ratio between current and voltage according to Z 0 = Vn+ /In+ = Vn− /(−In− ). If the focus should be on the waves instead of current and voltage, it is common to define normalized waves as follows:

V+ Z 0 In+ = √ n Z0 − V bn = Z 0 In− = √ n Z0

an =

incident wave at port n

(2.177)

emitted wave at port n

(2.178)

This definition leads to the known definition of the reflection coefficient, that is given by: V− bn (2.179) Γn = n+ = an Vn Port current and voltage are given by the normalized waves according to: Vn =

Z 0 (an + bn ) In = 1/ Z 0 (an − bn )

(2.180) (2.181)

√ The normalization by Z 0 has the advantage, that the incident power delivered towards port n is given by Pn+ = 1/2|an |2 , while the power delivered into the opposite direction is given by Pn− = 1/2|bn |2 . The power delivered into port n therefore can be expressed by:

in terms of normalized waves Pn,eff = 1/2 |an |2 − |bn |2

∗ Pn,eff = 1/2 · Vn · In in terms of current and voltage

(2.182) (2.183)

Instead of deriving the normalized waves from current and voltage, we can as well derive them from electromagnetic fields that constitute the waves towards and from the ports. This approach is advantageous e.g. in case of rectangular waveguides which are better understood in terms of an electromagnetic wave guided inside a metal structure than as in terms of current and voltage. The details like the field pattern will be discussed in detail in Chap. 5. ⎡ ⎤ !!

|an |2 1 ⎣ ∗ = Pna,eff = Eqa × Hqa d F⎦ 2 2 F ⎡ ⎤ ! ! 2

|bn | 1 ∗ = Pnb,eff = ⎣ Eqb × Hqb d F⎦ 2 2 F

(2.184)

(2.185)

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125

where Eq × Hq ∗ = S is the Pointing vector defining the power density delivered towards the multiport (index a) or into the opposite direction (index b), and F is the cross-sectional area of the wave port. The phase information is obtained from the phase of E q according to: |a| = |b| =

2Pa,eff ,

arc(a) = arc(E qa )

(2.186)

2Pb,eff ,

arc(b) = arc(E qb )

(2.187)

Based on these definitions, we can express the multiport in terms of an S- or Scattering-parameter matrix as follows: ⎞ ⎛ S11 b1 ⎜ b2 ⎟ ⎜ S21 ⎜ ⎟=⎜ ⎝ b3 ⎠ ⎝ S31 b4 S41 ⎛

S12 S22 S32 S42

S13 S23 S33 S43

⎞ ⎛ ⎞ S14 a1 ⎜ ⎟ S24 ⎟ ⎟ · ⎜ a2 ⎟ S34 ⎠ ⎝ a3 ⎠ S44 a4

(2.188)

In order to determine the S-parameters, it is required that only one port is excited by an incident wave an , while for all other ports am=n = 0 holds. The other ports are therefore to be terminated by Z 0 , as shown in Fig. 2.46d. In order to determine an element of the matrix, e.g. S11 , port 1 would be excited, while the other ports are terminated. b1 = S11 · a1

for a2 = a3 = a4 = 0

(2.189)

and S11 can be determined form measurement of b1 . The elements of the S-parameter matrix are called S-parameters. The elements on the diagonal, Snn , are given by: Snn

$ bn $$ = = Γn an $am=n =0

(2.190)

and therefore can be interpreted as the port reflection coefficient Γn under the condition that all ports are terminated by Z 0 . The other matrix elements Snm with m = n can be understood as a normalized power transfer coefficient: $ bm $$ (2.191) Smn = a $ n am=n =0

It is important to note that the interpretation of an S-parameter to represent a transfer or reflection coefficient is only valid in case of a proper termination. In other cases, input reflection coefficient and power transfer coefficients differ from the S-parameter values. The S-parameter matrix representation has a number of benefits:

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1. The coefficients of the S-parameter matrix are quantities that are easy to measure for an RF component. Measurement requires the conditions that all ports are matched by an impedance Z 0 , which is much easier to achieve at high frequencies than applying perfect short circuit or open circuit termination for Y-parameter or Z-parameter matrices, respectively. Terminating the ports by Z 0 in addition reduces the risk of oscillations or damage to the device that might not be safe for short-circuit or open-circuit operation. 2. The S-parameters are determined based on the normalized waves that can be determined through power measurement. This is always possible, even for transmission lines like rectangular waveguides where it becomes difficult to define and locate current and voltage. The characteristic impedance Z 0 required to derive S-parameters is designated hereafter as the normalization impedance. The S-parameter matrix is derived considering that the multiport to be described is connected to transmission lines of characteristic impedance Z 0 . The incident normalized waves an and outgoing normalized waves bn in this case correspond to real waves on the connected lines. But the S-matrix definition does not require that the respective lines are actually attached to the multiport, or that Z 0 matches the characteristic impedance of the connected lines. The mathematics even work when lumped elements or other multiports are connected to the terminals. Z 0 in the formulas above can basically be chosen like any other normalization parameter, and it is commonly set to 50 . The S-matrix therefore offers a higher level of abstraction, allowing us to understand and calculate multiports on the basis of travelling waves. Thus, normalized waves are primarily mathematical quantities that can be transformed into current and voltage at a lumped port, or renormalized to the actual characteristic impedance if required. It is also usual to normalize to Z 0 = 50 . This choice agrees with the characteristic impedance of coaxial cables and measurement equipment, but also allows for a comparison between S-parameters of different devices, e.g. in datasheets.

2.5.1 S-Matrix for Lossless Multiports The S-matrix is defined in a way that it relates incident to emitted normalized waves of a mulitport. Especially in the case of a lossless passive network, where the effective power delivered into the multiport is zero, the elements of the S-matrix are not independent of each other, but instead have to reflect that power delivered towards the multiport must either be reflected or emitted at other ports. We will derive the conditions that apply for a passive’s multiport S-matrix in the following. One can take advantage of these conditions for example in order to check if a multiport is or ideally can be lossless, or in order to support the estimation of the S-matrix if not all parameters are accessible to measurement.

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127

The effective power delivered into port n of a multiport was derived above in

Eq. (2.182) to be Pn,eff = 1/2 |an |2 − |bn |2 . The total power delivered to a mulitport can be given adding the effective powers delivered into the ports. Defining the vectors a and b according to: ⎛ ⎞ ⎛ ⎞ b1 a1 ⎜ b2 ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎜ ⎟ b=⎜ . ⎟ (2.192) a = ⎜ . ⎟, ⎝ .. ⎠ ⎝ .. ⎠ an bn The total effective power delivered to the mulitport thus reads: Peff =

1 T∗ a · a − bT ∗ · b 2

(2.193)

with a T ∗ denoting the transposed conjugate of vector a. Considering that b = S · a, we can write: 1 T∗ a · a − a T ∗ · ST ∗ · S · a 2 1 T∗

= a · E − ST ∗ · S · a 2

Peff =

(2.194) (2.195)

Where E denotes the unity matrix. For a lossless passive network, it holds that Peff = 0, which requires that E = ST ∗ · S −1

S

=S

T∗

(2.196) (2.197)

The inverse of the S-matrix describing a lossless passive mulitport therefore equals its transposed conjugate. Matrices of this type are called unitary. For a reactance two-port network that is commonly encountered in RF engineering (e.g. a filter with negligible losses), Eq. (2.197) leads to the relationships |S11 |2 + |S21 |2 = 1

(2.198)

|S22 | + |S12 | = 1

(2.199)

2

2

In case of transmission symmetry, we only require a single condition (since S21 = S12 ) (2.200) |S21 |2 = 1 − |S11 |2 In other words, for reactance two-port networks exhibiting transmission symmetry the magnitude of the transmission coefficient can be determined from the magnitude of the reflection coefficient.

128

M. Rudolph Zin Γin = S11

a Z0

Z1

Z0

Z3

Z2

Z0

Z0

ΓL = 0

ΓS = 0 ZS = Z0

ZL = Z0

I1 = I1+ – I1–

I2 = –I2+

b Z0

V1 = V1+ + V1–

Z1

Z3

Z2

V2 = V2+

Z0

Fig. 2.47 Direct calculation of the S-matrix of a two-port network. a Considering S11 , b considering S21

2.5.2 Deriving the S-Matrix of a Multiport There are basically two ways to determine an S-matrix for a known circuit. The first one is to directly determine the S-parameters, the second is to determine Y or Z parameters that are then transformed into the S-matrix by matrix manipulation. The direct approach is introduced here in order to deepen the understanding of the relation between the circuit network, defined based on voltages, currents and network elements on one side and the S-matrix, defined based on normalized waves and reflection- and transmission coefficients. In most practical cases, however, determining the Y- or Z-matrix is much more convenient and transformation into the S-matrix becomes part of the numerical circuit simulation. The condition for the derivation of the S-parameters is that all ports are matched by Z 0 . This situation is shown in Fig. 2.47 for the example of a T-type twoport that we will use as an example. Determining S11 and S22 is quite straight-forward, as depicted in Fig. 2.47a. The Sparameter S11 , for example, equals the input reflection coefficient when the other port is terminated into Z 0 . Therefore, we determine Z in and transform it into Γin = S11 :

2 Wave Propagation on Transmission Lines and Cables

129

Z 3 (Z 0 + Z 2 ) Z0 + Z2 + Z3 Z in − Z 0 = Γin = Z in + Z 0

Z in = Z 1 +

(2.201)

S11

(2.202)

In order to determine the transmission parameter S21 = b2 /a1 , we need to revert back to the waves travelling on the lines, as shown in Fig. 2.42b. a1 and b2 are normalized waves, but since we are interested into the ratio, the normalization cancels out and we work with the voltage waves: S21 = b2 /a1 = V2+ /V1+ . At the input port, the voltage V1 is given by the superposition of the forward and backward travelling waves: V1 = V1+ + V1− = V1+ (1 + Γin ) = V1+ (1 + S11 ) V1 V1+ = 1 + S11

(2.203) (2.204)

At the output, port 2, only the wave travelling towards the load exists, therefore we get: V2 = V2+

(2.205)

The transmission parameter S21 can therefore be determined from the ratio of the input and output voltages according to S21 =

V2+ V2 = (1 + S11 ) V1 V1+

(2.206)

The ratio V2 /V1 is easily derived: S21 = (1 + S11 )

Z2 Z 1 (Z 2 + Z 3 + Z 0 ) +1 · +1 Z 3 (Z 2 + Z 0 ) Z0

(2.207)

The other two S-parameters, S12 and S22 , are determined in the same way by exchanging the ports. It can be concluded that for very simple two-ports, deriving the S-matrix directly can be useful. But for more involved network topologies, determining Z- or Ymatrices might be much easier. The Z-matrix of the T-network, for example, reads: ZT =

Z3 Z1 + Z3 Z3 Z2 + Z3

(2.208)

Therefore, it is common to establish the link between network elements and multiport parameters through an analytical derivation of the Y- or Z-matrix, while the detailed calculations are left to a computer code. For example, one might derive the Y-matrix of a low-pass filter from the capacitances and inductances, but then leave

130

M. Rudolph

it to the computer to actually calculate its transmission coefficient S21 depending on element values as a function of frequency. The calculation would include the step of transforming the Y-matrix into the S-matrix. Also the reverse direction is common: determining device parameters, like a capacitance value, from an S-parameter measurement. We would analytically derive how the capacitance can be calculated once the Z-matrix is known, but leave it to the computer to transform the measured Sparameter values into Z-parameter values and then to apply the extraction algorithm that we derived. We would now like to calculate the S-matrix S for a multiport from a given Y- or Zmatrix. For this purpose, we must first choose a suitable normalization characteristic impedance Z 0 , which we write in the form of a matrix Z0 : ⎛

⎞ Z0 0 · · · 0 ⎜ 0 Z0 ⎟ ⎜ ⎟ Z0 = ⎜ . ⎟ .. ⎝ .. ⎠ . 0 Z0

(2.209)

The common choice is Z 0 = 50 , although it is possible to define different arbitrary values for the various ports if required in special cases. Next, we define normalized voltages V¯ and currents I¯ according to

V I

=

√ V 1/ Z0 √0 I Z0 0

(2.210)

where the quantities written in bold face represent vectors or matrices, respectively. Using these definitions, the relation between normalized waves quantities and current and voltage read in matrix notation: b = a V = I

V I 1 E E b a 2 −E E 1 2

E −E E E

(2.211) (2.212)

E is the identity matrix, e.g. for a four-port network ⎛

1 ⎜0 E=⎜ ⎝0 0

0 1 0 0

0 0 1 0

⎞ 0 0⎟ ⎟ 0⎠ 1

(2.213)

In order to calculate the S-matrix S in b=S·a

(2.214)

2 Wave Propagation on Transmission Lines and Cables

131

the following transformations are used: V = a + b = a + Sa = [E + S] a −1

a = (E + S)

(2.215)

V

(2.216)

I = a − b = a − Sa = (E − S) a

(2.217)

= (E − S) (E + S)−1 V

(2.218)

=Y·V

(2.219)

where the matrix Y stands for the normalized Y-matrix: Y = Z0 · Y · Z0

(2.220)

We can therefore calculate the Y-matrix from an S-matrix according to: Y = 1/ Z0 · (E − S) (E + S)−1 · 1/ Z0

(2.221)

Solving the equation for the S-matrix allows to calculate it from a known Y-matrix: −1

· E−Y S= E+Y

−1 = E−Y · E+Y

−1 =2· E+Y −E

(2.222) (2.223) (2.224)

Equivalent expressions are obtained for the transformation between Z-parameters and S-parameters: −1

· Z−E S= Z+E

−1 = Z−E · Z+E

−1 =E−2· Z+E where the matrix Z stands for the normalized Z-matrix: Z = 1/ Z0 · Z · 1/ Z0

(2.225) (2.226) (2.227)

(2.228)

2.5.3 Wave Chain Matrix If we introduce two port groups such that input ports have the quantities a1 , b1 and output ports the quantities a2 , b2 , we can write as follows:

132

M. Rudolph

b1 b2

=

S11 S12 S21 S22

a1 · a2

(2.229)

In case of port symmetry (number of input ports = number of output ports), we can define the wave chain matrix as follows: b1 T11 T12 a2 = · (2.230) a1 T21 T22 b2 T is the wave chain matrix or transmission matrix. The wave chain matrix allows for an easy calculation of the total transmission matrix Tt for cascaded multiports by matrix multiplication: (2.231) Tt = T1 · T2 · · · Tn The transmission matrix is in general not used unless the cascade of two-ports is to be determined. There are also different definitions of the matrix, e.g. interchanging the a and b vectors, or even reversing the direction of the a and b waves at the output to match with the input of the subsequent multiport. The relation between S-matrix and T-matrix is given by: T= S=

S12 − S11 S21 −1 S22 S11 S21 −1 −S21 −1 S22 S21 −1

T12 T22 −1 T11 − T12 T22 −1 T21 T22 −1 −T22 −1 T21

(2.232) (2.233)

2.5.4 Calculating Networks Based on S-Parameters RF systems consist of multiports, like filters or amplifiers, that are connected through transmission lines. Since perfect matching of all components is not possible in practice, we will have to deal with multiple reflections on the lines – reflections at load and source – mixed with lumped multiports. RF circuit and system design today is performed relying on a circuit or system simulator, that basically uses the chain matrix of the transmission line (see Sect. 2.3.1) to simulate everything like any other network, solving Kirchhoff’s equations for all nodes and branches. But in certain cases, it is required to derive everything analytically. One example is a measurement system, where we would like to remove the impact of cables, a low-noise amplifier and a filter from the measurement result by means of calibration. Using a circuit simulator is impractical – and the license costs might be prohibitive – so that we might want to determine attenuation and reflection coefficient of the structure by pen and paper. The result should be a compact formula that is easily coded into the calibration routine. One way of doing so is to use so-called Mason graphs [5, 6]. Figure 2.48 shows the Mason-graph representation of the S-matrix of a two-port. The normalized wave

2 Wave Propagation on Transmission Lines and Cables

133

S21

a1

b1

b2 S22

S11

a2

S12

Fig. 2.48 Expressing an S-matrix in terms of a Mason graph

a1 a1

S1

a2

S2

a3

S3

a4

S4

b S6

S5 a5

Fig. 2.49 Mason graph example showing series, parallel and loop connection of branches

quantities are drawn as the nodes, while the S-parameters are drawn as arrows denoting the direction of wave propagation. From the graph it is directly visible, that an incident wave a1 will add to the emitted waves b1 and b2 according to S11 · a1 and S21 · a1 , respectively. Other generic cases are shown in Fig. 2.49. If we are interested into the total transmission factor S41 = b/a1 , we have to consider a series of branches, branches in parallel and a feedback loop. The series connection yields a3 /a1 = S31 = S1 S2 , since a2 /a1 = S1 and a3 /a2 = S1 hold. Series branches therefore are multiplied. = a4 /a3 = S3 + S4 . For the For the parallel branch, S3 and S4 add up, so that S43 feedback loop, we find for the loop gain is Sl = S5 S6 , and we obtain b = a4 = a3 + Sl a4 . For the feedback loop, with the preceeding branch, we therefore obtain S43 /(1 − Sl ). We are now able to write down S41 = b/a1 = S1 · S2 · (S3 + S43 = S43 S4 )/(1 − S5 · S6 ).

2.5.4.1

Power Match Versus Termination into Z0

In the derivation of the reflection coefficient above, we considered a transmission line of a characteristic impedance Z 0 . There are actually waves on the line, and the maximum power is transferred for Γ L = 0, Z L = Z 0 . Power match in this case means the suppression of reflections on the line. If we consider the abstract concept of S-parameters, we use Z 0 as a normalization constant to calculate the normalized waves a and b. Of course, Z 0 is commonly set to 50 which is the target value of commercial cables’ characteristic impedance. But it is not required that there is any component within our system where electromagnetic waves actually are transmitted with this characteristic impedance.

134

M. Rudolph

A common application case is that a source is not perfectly matched, i.e. Γ S = 0, See Fig. 2.50. If we terminate it into a load providing an impedance of Z 0 , we obtain: b2 = a1

(2.234)

b1 = 0 1 1

PL = |b2 |2 − |b1 |2 = |a1 |2 2 2

(2.235) (2.236)

On the other hand, if the load is mismatched, we have to account for the feedback loop and get: a1 1 − ΓL ΓS a1 Γ L b1 = 1 − ΓL ΓS 1 1 − |Γ L |2 1

|a1 |2 PL = |b2 |2 − |b1 |2 = 2 2 |1 − Γ L Γ S |2 b2 =

(2.237) (2.238) (2.239)

In order to extract the maximum power from the source, it requires to select an appropriate Γ L = |Γ L |e jφL . From the denominator, we can determine the optimum phase, since |1 − Γ L Γ S | = |1 − |Γ L ||Γ S |e j (φL +φS ) | ≥ 1 − |Γ L ||Γ S |

(2.240)

The minimum is reached for φ L = −φ S . Determining the minimum for the term (1 − |Γ L |2 )/(1 − |Γ L ||Γ S |)2 yields the condition for the load reflection coefficient for maximum power transfer to be Γ L = Γ S∗

(2.241)

which equals the well-known condition Z L = Z S∗

(2.242)

A source delivers the maximum power to a matched load, i.e. a load of conjugatecomplex impedance. The impedance Z 0 , on the other hand, is merely a quantity we used for the normalization when calculating the S-parameters. Figure 2.50c and d shows the corresponding circuit representations of the two cases, where the source is either terminated by a load impedance Z 0 or by an other impedance value Z L , in order to support the finding that the maximum power is delivered to the load for Z L = Z S∗ , which is not necessarily equal to Z 0 . The maximum power available from a mismatched source is therefore given by:

2 Wave Propagation on Transmission Lines and Cables Fig. 2.50 Mason graph for a mismatched source terminated into a a load impedance Z 0 , Γ L = 0, and b a mismatched load. c Circuit-representation of a source terminated by Z 0 and d terminated by Z L

135

a

b

Source

Load 1

Source

1

a1

b2

1

a1

b2

ΓS

ΓS

b1

ΓL

b1 1

1

c

d

Source

Load

Source

V0

Load ZS

ZS

PS,av

Load 1

Z0

V0

2 1 |a1 |2 1 2 1 − |Γ S | = |a1 | = 2 |1 − |Γ S |2 |2 2 1 − |Γ S |2

ZL

(2.243)

At this point, it is required to revisit the definitions of a matched load and of the maximum available power. In a system where the source impedance and all transmission lines characteristic impedances all equal Z 0 , matching a port by Z 0 provides the maximum power. In all other cases, it still is common to use the terms, and it makes sense to do so. Speaking of a matched load refers to a load providing Γ L = 0. It establishes a constant reference value throughout the system. From a practical point of view, commercial components, cables, sources and measurement equipment is designed to provide an impedance level as close to 50 as possible. It therefore makes sense to consider an ideal Z 0 = 50 as matched, and also to give the maximum available power that a source provides in that case. Z 0 = 50 thereby serves as an interface definition in the commercial world. The maximum available power in this understanding refers to the power an optimized source delivers into a standard load. On the other hand, during the design phase, especially when it comes to the design of matching circuits, values of maximum available power to Z 0 and of maximum available power delivered to a conjugate match differ significantly, and one needs to be careful about the difference.

2.5.4.2

The Non-touching Loop Rule

Solving Mason graphs for complex structures is not feasible based on a branchby-branch analysis. Fortunately, Mason derived the so-called non-touching loop rule

136

M. Rudolph

a

b a1

a2

S1

a1

S2

a3

S3

a4

a1

S7 b

S8 a8

S10 S11

S9 S6

a7

S5

S12 S13 S4

a5

a3

a3

S3

a4

a4 S12

b

a8

S6

a7

S5

a6

a5

d a1

a2

a8

a7

S2

a1

S5

a6

S4

a2

a1 S14

b

S2

S14

a6

c a1

a2

S1

a1

S7 b

a5

S8 a8

S2

a3

S3

a4

S4

a5

S9

S14 a7

S5

a6

e a1

a1

S1

a2

S12 S13

S10 S11 b

a8

S6

a7

a4

a3

a6

S14 a5

Fig. 2.51 Mason graph to determine the input reflection coefficient for a network (a). Examples for (b) one of the direct paths, (c) a first-order loop, (d) a second-order loop, (e) a direct path and two non-touching first-order loops, at the same time a non-touching second-order loop

that allows for the derivation of reflection and transmission coefficients even in complicated cases. In order to introduce the non-touching loop rule, we need a few definitions that will be explained at the example of Fig. 2.51. As an example, we consider a cascade of three two-ports, which could be e.g. an amplifier, that is connected to load and source by cables. All the components provide mismatch to some extent, and the respective mason graph is shown in Fig. 2.51a. In order to determine, e.g. the input reflection coefficient Γin = b/a1 , we first need to determine the following parts: Direct paths P connecting the input, a1 with the output b. Figure 2.51b highlights one of the four direct paths that exist due to reflection. This path would read P1 = S1 · S2 · S12 · S5 · S6 . The network in the example provides 4 direct paths. First-order loops L(1) found anywhere in the system. Figure 2.51c shows one of them: L (1) 12 1 = S2 · S3 · S14 · S4 · S5 · S9 . The network in the example %provides first-order loops. We will denote the sum of all first-order loops by L (1) . Higher-order loops L(n) found anywhere in the system. A loop of order n is defined as the product of n first-order loops that are not sharing an node, therefore are nontouching. Figure 2.51d shows a second-order loop: L (2) 1 = (S7 · S8 ) · (S2 · S3 · S14 · S4 · S5 · S9 ). The network in the example provides 12 second-order loops, 7 thirdorder loops, one fourth-order loop. We will denote the sum of all nth-order % and loops by L (n) . Non-touching loops N are defined as the subset of loops of all order, that are not sharing a node with a certain direct path. Figure 2.51d shows an example of two (1) (1) = S11 · S12 and N22 = S13 · S14 and the second-order loop first-order loops N21

2 Wave Propagation on Transmission Lines and Cables

a

137

b a1

a1

b2

S21

ΓL

S11 S22 b1

S12

a1

S21

a1 ΓS

a2

b2 ΓL

S11 S22 b1

S12

a2

Fig. 2.52 Mason graph to determine the input reflection coefficient for twoport terminated by Γ L (a), and for a twoport connected to a mismatched source and load (b)

(2) N21 = (S11 · S12 ) · (S13 · S14 ) that are not touching the direct path P2 = S% 1 · S10 · Nm(n) . S6 . We will denote the sum of all nth-order loops not touching path m by

The non-touching loop rule reads: bi = aj

% n

% (1) % (2) % (3) Pn 1 − Nn + Nn − Nn ± · · · % % % 1 − L (1) + L (2) − L (3) ± · · ·

(2.244)

The non-touching loop rule allows to derive formulae for transmission and reflection coefficients quite straight-forward. • As mentioned when introducing the S-matrix, S11 of a two-port only equals the input reflection coefficient if the output is terminated into Z 0 . Consider the task to design an input and output matching network for a transistor in order to obtain a small-signal amplifier. How does the output reflection coefficient Γ L impact the input reflection coefficient? The mason graph is shown in Fig. 2.52a. For Γin = b1 /a2 , we have two direct paths P1 = S11 and P2 = S21 Γ L S12 , and one first-order loop L (1) = S22 Γ L , which is non-touching path P1 . By inserting these into the nontouching loop rule, we get: Γin =

b1 S11 (1 − S22 Γ L ) + S21 Γ L S12 S21 Γ L S12 = = S11 + a1 1 − S22 Γ L 1 − S22 Γ L

(2.245)

• Considering the mismatch at source and load, we end up with various definitions for linear amplifier gain. The transducer gain, for example, is defined as G tr =

PL power delivered to the load = available source power PS,av

(2.246)

The available source power was derived above in Eq. (2.243): PS,av

1 = 2

|a1 |2 1 − |Γ S |2

While the power delivered to the load is given by

(2.247)

138

M. Rudolph

PL =

1 2 |b2 | − |a2 |2 2

(2.248)

The system contains the following loops: L (1) 1 = Γ S S11

(2.249)

L (1) 2 L (1) 3 (2) L1

= Γ L S22

(2.250)

= Γ S S21 S12 Γ L

(2.251)

= (Γ S S11 )(Γ L S22 )

(2.252)

The direct path for a1 → b2 is given by S21 , and for a1 → a2 it is S21 Γ L . There are no nontouching loops, so that the relations are easily written down: b2 = a1 · a2 = a1 ·

1− 1−

S21 L (1) 1

+

L (1) 2

+ L (2) + L (1) 3 1

S21 Γ L

L (1) 1

(1) + L (2) + L (1) + L 2 3 1

(2.253)

(2.254)

Inserting everything into the equation for G tr , we arrive after a few steps the formula for the transducer gain: G tr =

1 − |Γ S |2 |S21 |2 1 − |Γ L |2 |(1 − S11 Γ S )(1 − S22 Γ L ) − S21 Γ L S12 Γ S |2

(2.255)

2.5.5 Example: FET and HBT Amplifier Matching To conclude this chapter, we would like to have a first look at matching of transistors in order to realize small-signal low-noise amplifiers. Figures 2.53 and 2.54 show example circuits based on the Avago ATF34143 HEMT and based on the Infineon BFP420F HBT, respectively. The corresponding S-parameters in the frequency range 0.5 . . . 4 GHz for the HEMT and HEMT-based amplifier are shown in Fig. 2.55. S11 of the HEMT alone follows the circle one would expect for the capacitive gate input impedance in series with extrinsic (metallization and contact) resistances and Rgs . S11 reaches into the inductive area of the Smith chart at the higher frequency end due to lead inductances within the SMD package. S22 also shows a circular frequency response, but extrapolating the locus towards lower frequencies is not pointing to the open circuit, but to a few 100 , which is the effect a limited output impedance has on the S-parameters. S21 is shown in polar coordinates, as it is a transmission coefficient

2 Wave Propagation on Transmission Lines and Cables

139

Vgg

Vdd C

C 900

10 pF 40 nH

5k 1 nH 15 pF

5

15 pF

2 nH ATF34143

Fig. 2.53 Example circuit for a low-noise amplifier based on the Avago ATF 34143 HEMT

that has no relation to an impedance. It shows a typical low-pass characteristic, seen in the phase shift and the decrease in amplitude. S12 , the feedback, is too low in amplitude to be visible in the graph. Matching and biasing the HEMT moves the reflection parameters closer to the origin, but not exactly to Γ = 0. Inspecting the circuit, Fig. 2.53, we see that in addition to the two inductances at the transistor’s gate and drain, serving as matching elements, a series and a parallel resistor are added at the gate branch. The 5 k resistor serves the purpose to provide the gate with bias voltage. The high-ohmic resistor is the easiest way to decouple the DC supply branch from the RF path, since no DC gate current flows. The series 5 resistance was inserted to prevent the amplifier from oscillating. At the drain side, an inductance of high value together with a large capacitance to ground is used to provide the DC bias. Input and output of the amplifier are DC decoupled through series capacitances at the ports. Designing a matching circuit is much more involved in this case than for the cases discussed so far, since the output reflection coefficient depends on the input matching

Fig. 2.54 Example circuit for a low-noise amplifier based on the Infineon BFP420F HBT

140

M. Rudolph

a

b HEMT Amplifier

S21

2 S22

HEMT

S11

4

6

8

10

Amplifier

Fig. 2.55 S-parameters of the Avago ATF 34143 HEMT for frequencies 0.5 . . . 4 GHz compared to the S-parameters of the circuit depicted in Fig. 2.53. Black lines: HEMT, grey lines: amplifier. a S11 , S22 , b S21

HBT

S21 20

HBT 10

S11 S22

Amplifier

Amplifier

Fig. 2.56 S-parameters of the Infineon BFP420F HBT for frequencies 0.5 . . . 4 GHz compared to the S-parameters of the circuit depicted in Fig. 2.54. Black lines: HBT, grey lines: amplifier. a S11 , S22 , b S21

circuit and vice versa, due to the transistor feedback. Also, it is required to ensure that the amplifier is stable and not oscillating. But even in this complicated case, using the Smith chart to analyze the circuit together with a numerical circuit simulator is advantageous over a pure trial-anderror approach. We will discuss it at the example of the output matching of the HBT circuit shown in Fig. 2.54. The S-parameters of the HBT and the corresponding amplifier are shown in Fig. 2.56. In contrast to the HEMT, S11 of a bipolar transistor is governed by the forward-biased base-emitter diode, providing usually some 100 at DC, following a semi-circle within the complex plane to cut the real axis at a low-ohmic value mainly determined by the base resistance. In the figure, we see the tail of the semicir-

2 Wave Propagation on Transmission Lines and Cables

141

b

a

S22 dB HBT feedback input match

0

HBT feedback input match

S22 of HBT

−5 −10 −15

HBT

HBT

feedback

feedback

−20 S22 of HBT −25 −30 0.5

2 1.5 Frequency (GHz)

1

2.5

3

d

c

S22 dB

0 parallel C −5

HBT feedback input match

−10 −15

HBT feedback input match

−20 −25 series L

parallel C

−30 series L −35 −40 0.5

1

f

e

S22 dB

2.5

3

HBT match db2 0

DC feed inductance

−5 DC feed inductance

1.5 2 Frequency (GHz)

−10 −15 parallel capacitance

−20 parallel capacitance

−25 −30 −35 −40 0.5

1

1.5 2 Frequency (GHz)

2.5

3

Fig. 2.57 Output matching of the circuit shown in Fig. 2.54. a and b S22 of HBT, HBT with feedback and HBT, feedback and input matching and DC supply circuit. c and d Impact of the parallel capacitance and series inductance on S22 . e and f Impact of DC feed inductance and parallel capacitance on S22

142

M. Rudolph

cle, and the extension of the curve into the inductive upper part of the Smith chart due to the lead inductances of the package. The output reflection coefficient S22 , on the other hand, is determined by the reversely biased base-collector junction and thereby starts at the open circuit for zero frequency. The shape of S21 and S12 resemble the HEMT case. A feedback through a resistor and a capacitance blocking DC is introduced in order to enhance bandwidth and stability, see Fig. 2.54. The feedback changes the output reflection coefficient, of course. But also the matching at the input of the HBT influences the output reflection coefficient. Figure 2.57a shows the change in the reflection coefficient for the frequency range of 0.5 . . . 2 GHz in the Smith chart. The same result is shown in dB in Fig. 2.57b. We see that the feedback already eases output matching by transforming S22 towards the center of the Smith chart. The input matching, unfortunately, has a rather big impact on S22 . In our case, we will take feedback and input matching for granted and focus on the analysis of the output match. The first element connected to the HBT’s collector is a capacitance to ground. Figure 2.57c shows that this parallel capacitance is used to transform the output impedance to the circle where its real part matches Z 0 at two frequencies. The series inductance then transforms the curve upwards so that it circles closer to the origin of the Smith Chart. Figure 2.57d shows the output reflection coefficients in dB. The transformation through the parallel capacitor is not changing much in terms of |S22 |, but it enables the transformation through the series inductance that yields the second minimum in |S22 |, which extends the matching bandwidth significantly. Finally, DC bias needs to be provided to the collector. This is done through the 40 nH choke inductor. As we see in Fig. 2.57e, the choke is not perfect but transforms S22 through its finite parallel inductance. The capacitance to ground is effectively connected in parallel to the inductance for RF signals, and adjusts S22 to provide even higher bandwidth.

References 1. 2. 3. 4. 5.

Lecher, E.: Eine Studie über elektrische Resonanzerscheinungen. Ann. Phys. 41, 850–870 (1890) Heaviside, O.: Electromagnetic Theory. 3 Volumes. London (1893–1912) Smith, P.H.: Transmission line calculator. Electronics 29–31 (1939) Smith, P.H.: An improved transmission line calculator. Electronics 130–133, 318–325 (1944) Mason, S.J.: Feedback theory—some properties of signal flow graphs. Proc. IRE 41(9), 1144– 1156 (1953). https://doi.org/10.1109/JRPROC.1953.274449 6. Mason, S.J.: Feedback theory—further properties of signal flow graphs. Proc. IRE 44(7), 920– 926 (1956)

Chapter 3

Impedance Transformers and Balanced-to-Unbalanced Transformers Holger Maune

Transformation elements are required to interconnect components and systems with different input and output impedances as well as cables with different characteristic impedances. Furthermore, the in- and output configuration of many system components as well as the cables, which typically have a coaxial implementation, are designed to be balanced or un-balanced with respect to ground. So, there are components that exhibit different balance properties with respect to ground and which must be interconnected in a reflection-free manner. In all of these applications, balanced-to-unbalanced transformers (“baluns”) are used which in many cases can also simultaneously transform the impedance (Fig. 3.1).

3.1 High-Frequency Transformers Overview The transformations illustrated in Fig. 3.2 are used in the field of RF engineering to enable matching of different impedances without reflections. Figure 3.2a shows the classic type of winding transformer with (or without) core, Fig. 3.2b the autotransformer, Fig. 3.2c a resonant transformation implemented as an two-port network in a -topology and Fig. 3.2d its dual implementation in T-topology. Figure 3.2e shows a homogeneous λ/4 line with characteristic impedance Z C and Fig. 3.2f an inhomogeneous line with a continuously varying characteristic impedance Z C (z). Transformers as shown in Fig. 3.2a, b are used up to about 100 MHz if the windings are realized as single-wire lines. On the other hand, if the windings are realized as two-wire or multiwire lines, components constructed based on this principle can currently be used up to about 10 GHz. Depending on the application area, the transition from transformers with lumped elements to transformers made from lines occurs H. Maune (B) Otto-V.-Guericke University Magdeburg, Magdeburg, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. L. Hartnagel et al. (eds.), Fundamentals of RF and Microwave Techniques and Technologies, https://doi.org/10.1007/978-3-030-94100-0_3

143

144

H. Maune

a

b

Z1 / Z2

Z1

Baluns

Z1

R2

R2 /2

R2 /2

Fig. 3.1 (Left) Impedance transformer from source Z1 to load R2 ; (right) balanced-to-unbalanced transformers (“baluns”) from asymmetric to symmetric transformation

n1

R2

1'

1

2'

g

2

1

2 I

R2

1'

R1+jX1

b

2

R1+jX1

1

R1+jX1

a

2I R2

2' I

1

C

L

R2

C

1'

e

d

2

L

2

2'

2'

f

2

1'

R2

C

1'

2'

1

L

1

R1+jX1

R1+jX1

c

2

Z C = R1.R2

R2

R1+jX1

R1+jX1

1 R1

Z (z) C

R2

R2

1' 1'

2' l- /4

z 2' l

Fig. 3.2 Conventional impedance transformation types. a Transformer; b auto-transformer; c -matching network (Collins filter); d T-matching network; e Homogeneous λ/4-line (stepped impedance) transformer; f Inhomogeneous line transformer; g transformer with bifilar windings and ferrite core

at frequencies starting at about 1 GHz. Construction involves line pairs, coaxial lines, waveguides or one of the various types of microstrips. Line transformers with constant characteristic impedance are realized as λ/4 transformers and often have adequate bandwidth as bandpass filters. For large bandwidths, it is possible to use inhomogeneous lines. Line transformers with a ferrite core as shown in Fig. 3.2g are broadband and space-saving. In all impedance transformations, the power loss should be as small as possible in relation to the transferred power. Then, the effective power P2 = I22 R2 absorbed by the real load resistance R2 is nearly equal to the effective power P1 = I12 R1 supplied on the source side, yielding the transformation ratio t of the effective resistances

3 Impedance Transformers and Balanced-to-Unbalanced …

R2 t= ≈ R1

I1 I2

2 =

145

1 , c2

(3.1)

with c the conversion ratio of the currents (or voltages) c=

I2 V1 = . I1 V2

(3.2)

3.1.1 Transformers for Impedance Transformation Figure 3.3 shows a classic transformer with two coupled coils. Based on the designations in this figure, we have the following: V1 = I1 (R L1 + jωL 1 ) + I2 jωM

(3.3)

V2 = I2 (R L2 + jωL 2 ) + I1 jωM.

Here, RL1/2 are the loss resistances, L 1/2 the coil inductances and M the mutual coupling inductance. Based on these equations, we can easily derive the equivalent circuit shown in Fig. 3.4. The conversion ratio of this transformer is with I2

I1

V1

L1

L2

RL1

RL2

V2

M

Fig. 3.3 Equivalent circuit of a transformer with two coupled coils

I1

V1

RL1

L1-M

L2-M

M

RL2

I2

V2

Fig. 3.4 Equivalent circuit without magnetic coupled elements of a transformer

R2

146

H. Maune

c=

N1 N2

(3.4)

dependent on the number of winding turns N 1 and N 2 . The transformer’s limit frequencies can be computed from the equivalent circuit in Fig. 3.4. At low frequencies, the inductivity of the primary inductance L 1 can no longer be neglected with respect to the transformed resistance R1 = c2 R2 . The resulting lower limit frequency f min is defined by R1 2

(3.5)

R1 4π L 1

(3.6)

ωmin L 1 = as f min =

At the upper limit frequency f max , the voltage drop across the leakage inductance can no longer be neglected with respect to the voltage drop across R1 . If we define ωmax σ L 1 = 2R1 with the leakage factor σ = 1 − M 2

(3.7)

L 1 L 2 it follows that

f max =

R1 , πσ L1

(3.8)

under the assumption that the inherent coupling capacitance is negligible at this frequency. From Eqs. (3.6) and (3.8), we thus obtain the frequency ratio 4(L 1 /L 2 ) f max 4 = = , f min σ (L 1 /L 2 ) − M 2

(3.9)

In transformers with a laminated, iron-powder or ferrite core with good magnetic continuity and windings realized as single-wire lines, we can obtain values for the leakage factor of σ ≤ 1%, i.e. frequency ratios of ≥400. By realizing the windings as two-wire (bifilar) or multiwire lines, we can obtain frequency ratios of ≥100,000. In air-core transformers, the σ values are in the range between about 2 and 25% resulting in a frequency ratio between 16 and 200.

3 Impedance Transformers and Balanced-to-Unbalanced …

147

3.1.2 Resonance Transformers Consisting of Lumped Elements Two-port networks can transform impedance values exactly at the center frequency f c and approximately in a frequency range (the “bandwidth”) that is dependent on the circuit. Based on the impedance transformer in Fig. 3.5 the conditions for a real-valued source at port 1 can be defined as Re{Y S } = G 1 = 1/R1 = t/R2 Im{Y S } = 0.

(3.10)

We can exactly determine the two reactances for continuous wave (CW) operation at the center frequency f c with B0 = ± t/(1 − t)G 2

∧

B0e = ∓ t(1 − t)G 2 .

(3.11)

However, the circuits are usable only for certain conversion ratios within a defined bandwidth. Networks with more than two reactances can also be used for impedance transformation purposes. The additional circuit elements provide degrees of freedom that make it possible to influence the bandwidth by means of the dimensioning or use a more favorable selection of component values in the practical implementation. Consisting of three reactances, the Collins filter shown in Fig. 3.6 is commonly used. Due to its structured it is also referred to as -configuration. The input admittance of this circuit is Y I = jωC1 +

1 jωL +

1 G 2 +jωC2

.

(3.12)

It thus follows [20] that

jB0 e

Ys

jB0

G2

Fig. 3.5 Impedance transformer in L-configuration with two reactances G2 and jB0 or G2 and jB0 e

148

H. Maune L

R1+jX1

1

2

C2

C1

1'

R2

2'

Fig. 3.6 Impedance transformer in -configuration with three reactances (Collins filter)

G2 Re{Y I } = 2 2 1 − ω LC2 + (ωLG 2 )2 ωC2 1 − ω2 LC2 − wLG 22 Im{Y I } = ωC1 + . 2 1 − ω2 LC2 + (ωLG 2 )2

(3.13)

Using the substitution x = C1 /C2 , Eqs. (3.10) are fulfilled at the center frequency f c for ωc C1 =

1−t x G2 x 2 /t − 1

1−t G 2 = ωc C1 /x −1 (1 − t) x 2 /t − 1 R2 . ωc L = |x − t|

ωc C2 =

x 2 /t

(3.14)

If center frequency ωc , transformation ratio t and load impedance R2 are specified, this leaves only the capacitance ratio x to be selected. According to Eq. (3.14), the elements are real-valued for t t. C2

(3.15)

√ For the case √ in which x → ∞, i.e. C2 → 0, we obtain ωc C1 = (1 − t)/t · G 2 and ωc L = (1 − t)/t · R2 . These formulae are consistent with Eq. (3.11) which were derived for reactance transformers consisting of two elements. Additionally, we also obtain real-valued elements for

3 Impedance Transformers and Balanced-to-Unbalanced …

t >1

∧

x=

149

√ C1 < t. C2

(3.16)

The frequency range in which the magnitude || of the input reflection coefficient = S11 is less than a specified maximum value max is designated as the bandwidth B of the impedance matching network. Figure 3.7a shows the relative bandwidth B/ f c as a function of x = C1 /C2 for max = 10% versus different transformation ratios t. The parameter is t > 1. For t < 1, we replace t with 1/t and obtain the curves mirrored on the lines C1 /C2 = 1 since swapping the terminal designations inverts the resistance conversion ratio while the bandwidth does not change. Figure 3.7a shows the achievable relative bandwidth versus transformation ratios t for different maximum reflection max . The T circuit shown in Fig. 3.8 is electrically equivalent to the Collins filter. If the Collins filter is constructed using equal capacitances x = C1 /C2 = 1, the dimensioning formulae in Eq. (3.14) are greatly simplified to ωc L =

1 = X K = R1 R2 . ωc C

(3.17)

From Fig. 3.7a, however, we can see that 1 < t < 4 optimal bandwidths are not achieved in practice. Here unequal capacitances with x = C1 /C2 ≤ 0.5 should be chosen instead. Figure 3.7b shows the relative bandwidth versus transformation ration t for x = 0 and √ different reflection coefficients. Since half of the power is reflected for max = 1/2 = 0.707, much smaller values of max are intended for RF transformation networks. Thus, connecting two ports with different impedances 100 % 60

100 % 60

t = 1,5

m

40

= 70,7%

40 2

20

40

20 4

20 10 8

10

6

B/fm

B/fm

10 8

20

4

10

6 5

4 3 100

2

2

1 0,8

1 0,8

1%

0,6 0,1

0,2

0,4 0,6 0,81,0 C1/C2

2

4

6

8 10

0,6 0,1

0,2

0,4 0,6 0,8 1,0

2

4

6

8 10

t

Fig. 3.7 Left: Relative bandwidth B/ f c of the Collins filter as a function of x = C1 /C2 for different transformation ratios t = R2 /R1 for a maximum reflection of max = 10%. Right: Optimal relative bandwidth as a function of the transformation ratio t for different maximum reflections for C 1 = 0

150

H. Maune 1

L1

L2

2

C

R2

1'

2'

Fig. 3.8 Impedance transformer in T-configuration with three reactances

via a matching network will always limit the bandwidth and achievable reflection limit max (see also Bode Fano). Perfect matching can only be achieved at a single frequency point. The larger the bandwidth gets the higher the reflection will be. By connecting several reactance transformers with individually smaller conversion ratios in series, we can increase the bandwidth or reduce the reflection coefficient max at the cost of higher insertion loss.

3.1.3 Line Transformers Consisting of Homogeneous, Low-Loss Lines 3.1.3.1

Single-Stage Transformers with λ/4 Line

As shown earlier, a transmission line of characteristic impedance Z C length l and load Z 2 has an input impedance Z 1 of Z1 = ZC

Z2 ZC

+ j tan

2πl λ j ZZC2 tan 2πl λ

.

(3.18)

2πl l=λ/4 π −→ tan = ∞ λ 2

(3.19)

1+

For the special case of a λ/4 line tan follows directly Z1 =

Z C2 Z2

(3.20)

3 Impedance Transformers and Balanced-to-Unbalanced …

151

i.e. the λ/4 line produces a transformation that is proportional to the reciprocal of the impedance. Accordingly, a load capacitance appears as an inductance at the input and an inductive load appears as a capacitance at the input. Given an ohmic terminating impedance Z 2 = R2 , Eq. (3.20) indicate that the input impedance Z 1 = R1 is real valued as well. For matching, the characteristic impedance of the line is determined by ZC =

R1 R2

(3.21)

One drawback of the λ/4 transformer is that it only can match a real load impedance Im{Z 2 } = 0. Complex load impedances can always be transformed to real valued ones by compensating elements, at the expense of narrow bandwidth. Furthermore, the electrical length of the line is λ/4 only at the design frequency. This length changes with frequency, invalidating Eq. (3.20) and thus resulting in mismatch and reflections. For the input reflection coefficient 1 = S11 referred to Z 0 = R1 corresponding to the circuit in Fig. 3.9, we obtain the following 1 =

Z 1 − R1 . Z 1 + R1

(3.22)

By combining with (3.18) we get t −1

1 = √ t + 1 + j2 t tan π2

√ t − 1/ t = √ √ . 2 p + t + 1/ t √

f fc

(3.23)

where the substitution p = j tan{π f /(2 f c )} is known as the Richards transformation [1] and reflects the periodic dependence on the frequency. The center frequency f c is defined as the frequency at which we have the electrical length l = λ/4. We thus obtain the relationship 4l f = . fc λ R1

~

(3.24)

/4

m

Zc

R2

Fig. 3.9 Single-stage transformers with λ/4 line of characteristic impedance Z C between source with impedance R1 and load impedance R2

152

H. Maune

For this value p = ∞ the input reflection becomes ( f = f c ) = 0 as intended. For values l/λ = 0, 0.5, 1, . . . corresponding to f / f c = 0, 2, 4, . . . we get p = 0 and thus maximum reflection. According to (3.23) this maximum value is 0 =

t −1 . t +1

(3.25)

Combining this again with (3.23) we obtain 1 =

0

1 + j 1 − 02 tan

π f 2 fc

.

(3.26)

Figure 3.10 shows ( f / f c ) as a function of frequency according to Eq. (3.26). Due to the symmetry of the tangents around π/2 the reflection coefficient is symmetrical around f c with

π fc + δ f tan 2 fc

π fc − δ f = − tan . 2 fc

(3.27)

Due to the periodicity of the tangent, it is periodic with a period of f c . If we now allow the magnitude of to attain a maximum value max at the limits of the frequency range to be transferred, we obtain a lower limit frequency fl and an upper limit frequency f u . We define the relative bandwidth or fractional bandwidth as f u − fl fl B . = =2 1− fc fc fc

(3.28)

within which the λ/4 transformer can be operated. The bandwidth increases as we allow the maximum reflection max to increase and/or as the impedance transformation t approaches 1 and max thus decreases. The lower frequency fl can be calculated 1.0 %

ZL/Z0 = 10, 0.1 0.75 ZL/Z0 = 4, 0.25

50 | |

f = 16% fm

0.5

0.25 10

ZL/Z0 = 2, 0.5 0

0,23

0,25

0,27

0,5

f = l/ 4fm

0

1 f/f0

Fig. 3.10 Reflection coefficient as function of frequency for different transformation ratios t

2

3 Impedance Transformers and Balanced-to-Unbalanced …

b

a Z1 = R1

Zt = R1.R2

Z2 = R 2

Z1 = R 1

Zt = R1.R2

153

c Z2 = R2 Z1 = R1

Zt

Z2 = R2

C

d Z1 = R1 0

0

4

4C

e Zt

Z1 = R1

–

Zt

Z2 = R2

Z2 = R2

–

3/4

Fig. 3.11 Examples of λ/4-line transformers. a and b Show coaxial realizations for R2 > R1 while in (a) the size of the inner conductor is changed (b) utilizes a dielectric filling to obtain different line impedance. In c and d a two-wire line transformers is shown for R2 < R1 with (c) changing line diameter or (d) different line distances. In e a planar microstrip line transformer with different line width is shown

by (3.26) to |0 | −1 fl 2 −1 max = tan . fc π 1 − |0 |2

(3.29)

Inserting into (3.28) results in

B 4 −1 1 t(1 − max ) − (max + 1) . = 2 − tan √ fc π 2 max t

(3.30)

Figure 3.10 shows the reflection for different transformation ratios t. For a transformation from R2 = 600 to R1 = 150 and a maximum reflection max = 0.1 a fractional bandwidth of 16% can be achieved. Figure 3.11 shows different line of λ/4-line transformers in coaxial and two-wire topology.

3.1.3.2

Multi-section λ/4-Line Transformers

If the required properties for the transformation ratio t, bandwidth B and permissible input reflection coefficient max cannot be fulfilled using a single-stage λ/4 transformer, we can connect N λ/4-line sections with different characteristic impedances ranging from Z c1 to Z cN in a chain (see Fig. 3.12). The length l of each line sections is chosen to l = λc /4. A selection of unequal lengths results in an increase in the period of r 1 = r 1 (f /f c ). In general, the transfer properties are not improved. In the following we assume equal length and monotonically increasing or decreasing characteristic

154

H. Maune R1

~

/4

/4

m

m

Zc

m

Zc

1

/4

Zc

2

N-1

/4

m

Zc

R2 N

Fig. 3.12 Multi-section transformer consisting of N λ/4-line sections with different impedances Z c1 to Z cN

impedance across the transformer. Thus, Eqs. (3.23) and (3.24) hold also for the case in which N > 1. There are various possibilities for the selection of the N characteristic impedances. If the rule for forming characteristic impedances Z c1 to Z cN is given, it is relatively easy to calculate the input reflection coefficient |1 ( f / f c )| based on this information. The disadvantage of such “analysis” is that the maximum permissible input reflection coefficient max and the fractional bandwidth B/ f c cannot be predetermined, as is the objective of a “synthesis”. Four typical possibilities to determine the characteristic impedances Z c1 to Z cN are discussed in the next sections.

Binomial/Geometrical Multi-section Transformers The binominal multi-section transformer is based on the line impedances of 2N

Z c(n+1) = Z cn

⎛

⎛

⎞

⎝

⎞

N⎠ n

N⎠ Z c(n+1) t n ←→ ln = Z cn 2N ⎝

ln t for n = 0 . . . N

(3.31)

with Z c0 = R1 and Z c(N +1) = R2 . For geometrical multi-section transformers the impedances are chosen according to √ Z c(n+1) 1 = ln t for n = 1 . . . (N − 1) Z c(n+1) = Z cn n t ←→ ln Z cn n

(3.32)

In general, in case of an odd value of N, the characteristic impedance of the middle stage is always Z m = Z c( N 2+1 ) =

R1 R2 .

(3.33)

Moreover, the product of the characteristic impedances which are equally spaced from the center of the transformer is always Z cn Z c(N +1−n) = R1 R2 = R12 t for n = 1 . . . N .

(3.34)

3 Impedance Transformers and Balanced-to-Unbalanced … R1

155

/4

/4

R2 = tR1 ZC

ZC

~

2

1

Fig. 3.13 Multi-section transformer consisting of two λ/4-line sections with different impedances Z c1 and Z c2

From now on, for the characteristic impedances normalized to R1 we will write Z cn /R1 = wn

(3.35)

wn w N +1−n = t for n = 1 . . . N .

(3.36)

Inserting into (3.34) leads to

In the following we will investigate transformers with different number of stages starting with a two-section transformer according to Fig. 3.13. (a)

Two-section transformer

The symmetry condition implies w1 w2 = t. If we now apply Eq. (3.18) twice to calculate the input impedance Z in of the transformer terminated with R2 = t R1 we obtain √ √ √ √ p 2 w12 / t − t/w12 + t − 1/ t Z in − R1 1 = = 2 2 √ √ √ √ √ √ Z in + R1 p w1 / t + t/w12 + 2 p w1 / t + t/w1 + t + 1/ t (3.37) with p = j tan{π f /(2 f c )}. The reflection coefficient is thus dependent only on t and the selectable ratio w1 = Z c1 /R1 . According to (3.31) and (3.32), we now can calculate the characteristic impedances for both types (binomial and geometrical) to √ √ √ 4 Z c1 = R1 4 t and Z c2 = R1 t 3 = R2 / 4 t,

(3.38)

which results in w1 =

√ 4 t.

Inserting (3.39) in (3.37) results in the reflection coefficient

(3.39)

156

H. Maune

a

b

ln(Zt /R1) lnt 1

60 %

ln(R2/R1) ln(Zt2/R1)

3/4 1/2

40

1

t=R2/R1 ln(Zt1/R1)

20

1/4

0

0,5

1

z/l

0

0,5

1,0

1,5

2,0

f/fm

Fig. 3.14 Two-stage transformer with total length λ/2 with a characteristic impedances Z c1 and Z c2 normalized to R1 and b Reflection coefficient 1 versus frequency when terminated with R2 = 4R1

√ t − 1/ t √ √ √ t + 1/ 4 t + t + 1/ t √

1 =

2 p2 + 2 p

√ 4

(3.40)

For f / f c = 0 and 2, we have p = 0 and 1 = (t − 1)/(t + 1) agrees with 0 according to Eq. (3.25), while for f / f c = 1, in contrast to Eq. (3.23), 1 has a double zero in the band center due to the power p2 in the denominator. As consequence, the fractional bandwidth for max = 0.1 climbs from 16% to nearly 48% (see Fig. 3.14). (b)

Three-section transformer

The symmetry conditions for the three-section transformers are Z c1 = Z c3 = R1 R2 = t R12 and Z c2 = Z m =

√ R1 R2 = R1 t.

(3.41)

If we now apply Eq. (3.18) three times to calculate the input reflection 1 of the transformer terminated with R2 = t R1 we obtain √ √ √ √ p 2 2(w1 − 1/w1 ) − t/w12 − w12 / t + t − 1/ t 1 = √ √ √ √ √ √ 2 p 3 + p 2 2(w1 + 1/w1 ) + t/w12 + w12 / t + 2 p t/w1 + 1 + w1 / t + t + 1/ t

(3.42) with p = j tan{π f /2 f c }. In the band center, we have | p| = tan{π/2} = ∞, i.e. 1 ∝ p 2 / p 3 has a single zero at mid-band if the factor of p 2 in the numerator does not disappear. In case of binomial impedance distribution, the characteristic impedances of the three transformation stages have the following values referred to R1 : w1 =

√ Z c1 √ Z2 Z3 √ 8 = 8 t, w2 = = t and w3 = = t 7. R1 R1 R1

(3.43)

3 Impedance Transformers and Balanced-to-Unbalanced …

157

With this, Eq. (3.42) thus simplifies to 13-binom =

√ √ √ √ √ √ p 2 2 8 t − 1/ 8 t − 4 t − 1/ 4 t + t − 1/ t √ √ √ √ √ √ √ √ 8 3 8 2 p 3 + p 2 2 8 t + 1/ 8 t + 4 t + 1/ 4 t + 2 p t + 1 + 1/ t 3 + t + 1/ t

(3.44) The single zero from Eq. (3.40) is preserved but the factor of p2 in the numerator is extremely small and negative (for t > 1). The denominator always remains positive. In case of geometrical impedance distribution, the characteristic impedances of the three transformation stages have the following values referred to R1 : w1 =

√ Z c1 √ Z2 Z3 √ 6 = 6 t, w2 = = t and w3 = = t 5. R1 R1 R1

(3.45)

With this, Eq. (3.42) thus simplifies to 3-geo 1

√ √ √ √ 6 t − 1/ 6 t + t − 1/ t √ √ = √ √ √ √ 2 p 3 + 3 p 2 6 t + 1/ 6 t + 2 p 3 t + 1 + 1/ 3 t + t + 1/ t p2

(3.46)

In contrast to binomial impedance distribution, the factor of p2 in the numerator is positive and relatively large. Accordingly, there are two additional zeroes lying symmetrically about the zero in the band center whose position follows from the disappearance of the numerator where √ √ √

√ t − 1/ t t − 1/ t 2 π f =√ p = −√ √ → tan √ . 6 6 2 fc t − 1/ 6 t t − 1/ 6 t 2

(3.47)

Resulting in the zeros at ⎫ ⎧ √ √ ⎬ ⎨ f t − 1/ t 2 = tan−1 ± √ √ . 6 ⎩ fc π t − 1/ 6 t ⎭

(3.48)

A comparison of the binomial and geometric impedance generators is shown in Fig. 3.15. The introduction of two additional zeros is clearly visible in Fig. 3.15b. For further comparison√let’s have a closer look at a impedance transformation ratio t = R2 /R1 = 16, so 4 t = 2. For the binomial transformer, the numerator gets −0.0858 p 2 + 3.75, resulting in very flat behavior of 13-binom around f c . In contrast, the nominator in geometric transformers is +0.9574 p 2 + 3.75 which results in a 3-geo around f c . Due to the two additional zero apart from much steeper behavior of 1 f = f c the fractional bandwidth B/ f c is increased for the latter one while keeping the maximum reflection max at the same level. While the reflection of the binomial transformer is maximally flat, the response of geometric transformers exhibits ripples up to the maximum reflection max . Table 3.1 compares the bandwidths for binomial

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a

b

ln(Zt /R1) lnt

60 %

ln(R2 /R1)

f/fm = 76%

1 40

5/6 ln(Zt3/R1)

f/fm = 50%

ln(Zt2/R1) 1

1/2

20 ln(Zt1/R1)

4,4

1/6 0

1/3

1/2

2/3

1 z/l

0

0,5

1,0

1,5

2

f/fm

Fig. 3.15 Three-stage transformer with total length 3λ/4 with a the characteristic impedances Z c1 … Z c3 normalized to R1 for the binomial (dashed line) and geometric (solid line) construction rules and b Reflection coefficient 1 versus frequency for both cases when terminated with R2 = 4R1 Table 3.1 Relative bandwidth of three-section λ/4 transformers t= √ 2 2 4 8

R2 R1

Binomial transformer w1 = √ 16 2 √ 8 2 √ 4 2 √ 8 3 2

Z C1 R1

Geometrical transformer

max (%)

B/ f c (%)

1.1

52.1

2.2

51.8

4.4

50.0

6.4

47.1

w1 = √ 12 2 √ 6 2 √ 3 2 √ 2

Z C1 R1

max (%)

B/ f c (%)

1.1

78.1

2.2

77.8

4.4

76.0

6.4

73.3

√ and geometrical transformers for four transformation ratios t between 2 and 8. The maximum max is chosen from the geometrical transformer and used also for evaluation of the binomial transformer. With this, the bandwidths for geometrical graduation are roughly 1.5 times the bandwidths for binomial graduation. (c)

Four- and five-section transformers

Here again, the characteristic impedances are generated symmetrically around the center according to Eqs. (3.31)–(3.34). Like all transformers with an even number of stages, the reflection coefficient 1 for the four-section arrangement has a double zero for f = f c , while the five-stage binomial transformers have a single zero in the center of the band. Figure 3.16a shows the reflection coefficient for different number of stages. The geometrical transformer with five sections has five distinct zeroes with (in each case) two maxima of different magnitudes for the reflection coefficient. Figure 3.16b shows, that the bandwidth for a given max can be smaller for higher number of stages as one of the ripples can violate max . Thus, the bandwidth B of geometric transformers exhibits unsteady behavior as function of N for given max . (d)

Synthesis techniques for N-stage transformers

The last example with maxima of different magnitudes shows that geometrical transformers are not optimal for larger values of N. Typically, we seek the following: (I)

3 Impedance Transformers and Balanced-to-Unbalanced …

a

159

b 60 %

60 %

40

40

1

1

f/fm n = 4 f/fm n = 5 20

20 10

0

0,5

1,0

1,5

2,0

0

0,5

1,0

f/fm

1,5

2,0

f/fm

Fig. 3.16 Reflection coefficient 1 for a binomial and b geometrical transformers for different number of stages N

The maxima of 1 should have the same magnitude and (II) their magnitude in the passband max should be specifiable with, say, 1% or 3% or 10% accuracy. In this case, the reflection coefficient must be specified using Chebyshev functions and the necessary characteristic impedances are determined on this basis, see the following section. If we require an extremely small reflection coefficient near the band center (“maximally flat” curve, Butterworth approximation [2, 3]), then an N-stage transformer must have an N-fold zero in the band center. The example of the three-stage transformer showed that although the numerator factor of p2 in Eq. (3.42) does not disappear for binomial transformers of the characteristic impedances, it is indeed very small (for t = 4, this factor is ≈ –0.0067). This means the frequency dependency of the reflection coefficient 1 and the characteristic impedances for a “maximally flat” curve come very close to the values for binomial graduation with N ≥ 3. For four stages and t = 4, for example, the relative bandwidth for a maximally flat curve with max = 10% is about 82.7% versus 82.1% for binomial transformers, while a two-stage binomial transformer precisely match the values for a “maximally flat” behavior.

Chebyshev Multi-section Transformers (a)

Two-section transformer

√ If, in contrast to Eq. (3.38), we choose the ratio w1 = Z c1 /R1 = 4 t, then the double zero in the band center of (3.37) disappears. For f = f c equal to p → ∞, 1 obtains an auxiliary maximum max which corresponds to the ratio of the factors of p2 in the √ numerator and denominator. Thus, following multiplication with t/w12 , we obtain max

1 − t/w14 = 1 + t/w14

↔

Z c1 √ w1 = = 4t R1

4

1 + max . 1 − max

(3.49)

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H. Maune 1.0

n=0

n=1 0.5

Tn (x)

n=4

n=2

0.0 n=3 n=5

–0.5

–1.0 –1.0

–0.5

0.5

0.0 x

1.0

Fig. 3.17 Plot of the amplitude of Chebyshev polynomials of the first type |TN {x}| for degree n = 0, 1, 2, 3, 4, 5 in the interval –1 < x < 1

In this simple case involving a two-stage transformation, 1 is easy to determine using Eq. (3.37) as soon as the value of max is chosen. By fixing max , the ratio Z c1 /R1 = R2 /Z c2 is determined simultaneously. (b)

Three-section transformer

For three stages, calculation of w1 = Z c1 /R1 as a function of the magnitude max of the two auxiliary maxima that occur symmetrically with respect to the band center according to Eq. (3.42) becomes challenging, while the calculation is extremely difficult for more than three stages. However, it is possible to express the square magnitude of the input reflection coefficient |1 |2 in a clear manner with the aid of the Chebyshev function of the 1st type, Nth order TN {x} for N stages [2, 3]. We thus obtain Tn |1 |2 = 2 1−max 2 max

1− p12 1− p2

+ Tn

!2

1− p12 1− p2

!2 .

(3.50)

with p = j tan{π f /2 f c } and p1 = j tan{π f 1 /2 f c }. The Chebyshev function TN {x} has N single zeroes in the range 0 ≤ |x| < 1. Like the two boundary values for x = ±1, the intermediate N − 1 maxima all have the same magnitude of 1. For |x| > 1, |TN {x}| increases monotonically to ∞ for increasing |x|, see Fig. 3.17. In Eq. (3.50), p1 = j tan{π f 1 /2 f c } is tied to the lower limit frequency f 1 . For f = f 1 , we have p = p1 and TN {1} = 1 and thus

3 Impedance Transformers and Balanced-to-Unbalanced …

161

60 %

40 f/fm n = 5 1

f/fm n = 4 f/fm n = 2

20

0.5

0

1.0

1.5

2.0

f/fm

Fig. 3.18 Input reflection 1 of a Chebyshev transformer with different number of sections N

|1 ( f = f 1 )|2 =

1 2 1−max 2 max

+1

2 = max .

(3.51)

For f 1 / f c , there exists a simple relationship with max and 0 = (t − 1)/(t + 1): For p = 0, we have 1 = 0 and thus according to Eq. (3.50) TN

1 − p12 = TN

⎧ ⎨

1

⎫ ⎬

= ⎩ cos π f 1 ⎭ 2 fc

2 1 − max 2 max

2 1 − max t −1 1 = √ = . 2 A 2 t 2 max 1 − 0 0

(3.52)

With the quantity 1/A which follows directly from the transformation ratio t and the maximum reflection max , we can explicitly specify the relative bandwidth B for any number of stages N. Namely, we have 1 B 4 " " ## . = 2 − arccos fc π cosh N1 arcosh A1

(3.53)

As example, the reflection coefficient for max = 0.1 and t = 4 is shown in Fig. 3.18. A manual procedure for finding the characteristic impedances as a function of max and the number of stages N for the Chebyshev transformer is given in [3]. Table 3.2 shows the characteristic impedances normalized to R1 for n = 2, 3, 4 and 5 for t = 4 for all previously discussed algorithms. Based on the small differences in the numerical values, we can see that the difference between binomial and “maximally flat” behavior is only a question of manufacturing tolerances for real-world transformers. It is therefore not worth the effort to present the synthesis of transformers with “maximally flat” graduation since they are nearly identical to transformers with binomial graduation.

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Table 3.2 Characteristic impedances for λ/4-line transformers for a transformation ratio of t = 4 with N = 2–5 stages for different algorithms Number of stages N

Normalized characteristic impedance

Binomial

Geometric

Chebyshev with max = 0.1

Maximally flat

w1

1.414

1.414

1.487

1.414

w2

2.828

2.828

2.690

2.828

w1

1.189

1.260

1.323

1.191

w2

2.000

2.000

2.000

2.000

w3

3.364

3.175

3.025

3.359

w1

1.091

1.189

1.252

1.092

w2

1.542

1.682

1.683

1.544

w3

2.594

2.378

2.376

2.590

w4

3.668

3.364

3.196

3.663

w1

1.044

1.149

1.214

1.045

w2

1.297

1.516

1.519

1.300

w3

2.000

2.000

2.000

2.000

w4

3.084

2.639

2.633

3.077

w5

3.830

3.482

3.295

3.827

wn = 2 3

4

5

Algorithm

Z cn R1

Given the same number of stages N and the same maximum reflection coefficient max at the band limits, the bandwidth of the Chebyshev transformers is about 50– 55% larger than that of the transformers whose reflection rises constantly from the band center. However, if the group delay in the passband has not to fluctuate by more than ~20%, the full bandwidth of Chebyshev transformers cannot be utilized. Compensated λ/4 Transformers In order to improve the transfer properties of a λ/4-line transformer, there is—in addition to a cascade connection with N line sections—also the possibility to connect further M line sections in parallel or in series. Known as stubs or compensation lines, such low-loss line sections have a nearly purely imaginary input impedance because of their open-circuit or short-circuit termination. As a result, if properly connected they can compensate the reactance component of the input impedance of an N-stage λ/4-line transformer which occurs in case of deviation from the center frequency in the vicinity of this center frequency. In order to obtain a curve for the input reflection coefficient that is symmetrical about f = f c , these compensation lines are made to have the same length l = λc /4 as the cascaded line sections. The two simplest arrangements (m = 1, n = 1) for a compensated λ/4 transformer are shown in Fig. 3.19; the stub can be arranged on the generator side as well as on the load side.

3 Impedance Transformers and Balanced-to-Unbalanced …

163

a l R1

Zp Zt

~

R2

l

l

b Zs

R1 ~

R2

Zt l

Fig. 3.19 Sketch of the two simplest circuits for a compensated λ/4 transformer, a Parallel and b Series compensation

Two of the possibilities (parallel connection of a line that is open at the end and series connection of a line that is short-circuited at the end) are disregarded since in these cases an input reflection coefficient with a magnitude of 1 results for f = f c . In contrast to the uncompensated λ/4 transformer with 1 ( f / f m ∈ {0, 2, 4, . . .}) = 0 < 1

(3.54)

we have the following if the λ/4 transformer has at least one stub: 1 ( f / f m ∈ {0, 2, 4, . . .}) = 1.

(3.55)

This is because at these frequencies the stubs result in a short-circuit (Fig. 3.19a) or a discontinuity (Fig. 3.19b). The parallel compensated circuit is discussed in more detail hereafter. Using Eq. (3.1), we obtain the following expression for the input reflection coefficient 1 referred to R1 : 2 Z p 2 Rt2 − t + p RZ1t t − 1 − ZZ pt − ZZ pt t . (3.56) 1 = 12 p 2 ZRt2 + t + p RZ1t t + 1 + ZZ pt + ZZ pt t 1

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H. Maune

The two characteristic impedances Z t and Z p are determined based on the requirement that the function 1 = 1 ( f / f c ) in the range 0 < ( f / f c ) < 2 should have the maximum possible number of two real zeroes. Given p = j tan(π f /(2 f c )) it follows that in the numerator of Eq. (3.56), the factor of the imaginary frequency function must be t −1−

Zt = 0. Zp

(3.57)

Zt . t −1

(3.58)

We thus obtain the relationship Zp =

Since Z t and Z p must be positive, it follows from Eq. (3.58) that we must have t > 1; this means that the compensation must be implemented by means of a single stub connected in parallel at the low-impedance end of the λ/4 transformer. If the input reflection coefficient in the band center ( f / f c = 1, | p| = ∞) is to disappear, then it follows from Eq. (3.56) that the following relationship must be fulfilled: √ Z t = R1 t.

(3.59)

If Eq. (3.58) is also fulfilled simultaneously, then we obtain a double zero for f / f c = 1; see Fig. 3.20. For t = 4 and max = 10%, a relative bandwidth f / f c ≈ 33% follows in this case. On the other hand, if the input reflection coefficient is to have the maximum permissible value max in the band center, we then find with Eqs. (3.56) and (3.58) that 1 − max . (3.60) Z t = R1 t 1 + max For t = 4 and max = 10%, the relative bandwidth we obtain here is f / f c ≈ 43%. If we perform the corresponding calculation for the series compensated line shown in Fig. 3.19b, we discover that the series compensation must be implemented using a single stub at the high-impedance end of the λ/4 transformer. For t = 4 and max = 10%, we again obtain the curve shown in Fig. 3.20 for the input reflection coefficient. Figure 3.21 shows examples of single-stage, compensated λ/4 transformers in coaxial and balanced arrangements. Compensation of a λ/4 transformer with a stub is also possible with multistage transformers (n > 1). Moreover, more than one stub (m > 1) can be used for

3 Impedance Transformers and Balanced-to-Unbalanced …

165

100 %

75 67

1

50

25 10

0

0.5

1.0

1.5

2.0

f/fC

Fig. 3.20 Input reflection coefficient as a function of frequency for a single-stage λ/4 transformer with a compensation line with t = 4 and max = 10% with double zero at f = f c (–––) and with maximum allowed mismatch at f = f c (– – –) leading to a bandwidth of f / f c ≈ 33% and

f / f c ≈ 43%, respectively

a

c

2

4

R2

1

2'

1'

R1 = Z1

Z2 = R2

Zt Zs

L =

d

Zp

Zt 4 2

1 R1

b L

1

R2

Zt 1'

=

2' Zs

m 4

Zp

2 Zt

Z1 1' L =

m 4

R2 2'

Fig. 3.21 Examples of single-stage, compensated λ/4 transformers for t > 1. a Compensation with short-circuited stub, coaxial version; b Compensation with short-circuited stub, balanced version; c Compensation with open-circuited stub, coaxial version; d Compensation with open-circuited stub, balanced version

166

H. Maune

4

l

Fig. 3.22 λ/4 transformer with double compensation

-2 R1-Z1

R2

Zt -2'

compensation. Thus, several structures arise from which the optimal structure can be chosen for a given set of parameters ( f / f c , max , t, total length, etc.). Calculation of the λ/4 transformer for m > 1 and n > 1 involves specifying the behavior of the magnitude of the input reflection coefficient as a function of frequency (maximally flat or Chebyshev curve); in this manner, we can obtain according to Riblet [3] the characteristic impedances of the cascaded line sections by elimination from the impedance function for p = 1; the characteristic impedances of the stubs are obtained by eliminating the poles for p = 0 from the impedance function or admittance function since in the p-plane lumped elements (L or C) ensue formally from the stubs. Only in the case of the double-compensated, single-stage λ/4 transformer (m = 2, n = 1) as shown in Fig. 3.22 with a maximally flat curve for the input reflection coefficient we can specify simple relationships for the characteristic impedances: √ Z t = R1 t, Z p = Z t

2 t −1 , and Z s = Z t t −1 2

(3.61)

For t = 4 and max = 10%, we obtain a relative bandwidth of f / f c ≈ 68% for this transformer.

3.1.4 Line Transformation with Inhomogeneous Low-Loss Lines The total length ltot of an N-section λ/4-line transformer is ltot = N λc /4. For the center frequency f c we obtain the following expression:

(3.62)

3 Impedance Transformers and Balanced-to-Unbalanced …

fc =

167

c0 N . 4 ltot

(3.63)

If we now increase the number of sections N while holding the total length ltot constant, then the center frequency f c will grow with the number of stages N according to Eq. (3.63). As N increases, the difference between the characteristic impedances of two successive stages also becomes smaller. If the lower limit frequency of the transformer converges to a finite limit for N → ∞, then the stepped-impedance line with periodic bandpass behavior becomes the inhomogeneous line with highpass behavior since its “center frequency” according to Eq. (3.63) and its “upper limit frequency” go to infinity.

3.1.4.1

Mathematical Description of the Inhomogeneous Line

In order to describe the inhomogeneous line, we start from the unit cell description of a transmission line, see Fig. 3.23. In contrast to the derivation of the telegrapher’s equation, the distributed quantities R , L , G and C are now dependent on the position along the length of the line. However, their relative change should be small on a path that is comparable to the conductor spacing. Since when used as a transformer the inhomogeneous line is at most a few wavelengths long, we can neglect the losses, as was the case with the λ/4-line transformer. Thus, for a line element of length δz, we obtain ∂ V (z) = jωL (z)I (z) ∂z ∂ I (z) = jωC (z)V (z). − ∂z −

(3.64)

For a homogeneous line, we have L C = const = 1/υ 2 in case of small losses. If we specify 1/υ 2 , L and C dependent on each other, resulting in L (z) = L 0 · fct(z)

(3.65)

C (z) = C0 /fct(z). R'

Fig. 3.23 Unit cell of a transmission line

i1

1

2

dz

L' 2

G'dz

dz

L' 2

dz

C'dz

R' 2

dz

i2

2

168

H. Maune

The function fct(z) is always positive in its range. For the characteristic impedance of the line, we obtain Z (z) =

L (z) = C (z)

L0 1 . · fct(z) = υ L (z) = C0 υC (z)

(3.66)

Differentiating Eqs. (3.64) and substituting I and dI/dz, we obtain ∂ 2 V (z) 1 ∂ L (z) ∂ V (z) + ω2 L (z)C (z)V (z) = 0 − ∂z 2 L (z) ∂z ∂z ∂ 2 I (z) 1 ∂C (z) ∂ I (z) − + ω2 L (z)C (z)I (z) = 0. ∂z 2 C (z) ∂z ∂z

(3.67)

If we now replace L and C according to Eq. (3.65) with the characteristic impedance Z(z) and the factor β02 = ω2 L C , we obtain ∂ V (z) ∂ 2 V (z) ∂ ln{Z (z)} + β02 (z)V (z) = 0 − 2 ∂z ∂z ∂z ∂ I (z) ∂ 2 I (z) ∂ ln{Z (z)} + β02 (z)I (z) = 0. + ∂z 2 ∂z ∂z

(3.68)

These equations differ from the differential equations for the homogeneous line in the second element which reflects the spatial dependency of the line constant. We have 1 ∂ Z (z) 1 ∂fct(z) ∂ ln{Z (z)} = = . ∂z Z (z) ∂z fct(z) ∂z

(3.69)

For the current and voltage, the differential equations are now also different so that their solutions must also differ. Since we wish to use the inhomogeneous line as a transformer, we are interested primarily in the reflection coefficient . Instead of calculating it based on a detour via the current and voltage, we would like to obtain it here directly from a differential equation for which should have only the spatial dependency of the characteristic impedance and frequency [4]. The reflection coefficient is defined as =

V /I − Z . V /I + Z

(3.70)

From this, we can compute the ratio V /I. V 1+ =Z . I 1−

(3.71)

3 Impedance Transformers and Balanced-to-Unbalanced …

169

We now differentiate V /I: ∂ VI 1 ∂V V ∂I = − 2 . ∂z I ∂z I ∂z

(3.72)

∂ V /∂z and ∂ I /∂z are given by Eqs. (3.64) while the ratio V /I is given by Eq. (3.71). Applying these equations, we now obtain 1+ 2 ∂ 1+ Z = −jωL + jωC Z . ∂z 1 − 1−

(3.73)

Replacing L and C according to Eq. (3.66) by Z(z) and differentiating by z, we obtain ∂ ∂ − jβ0 + 1 − 2 ln Z = 0. ∂z ∂z

(3.74)

For the reflection coefficient, we obtained a nonlinear differential equation of the 1st order. The solution contains only one integration constant. This also makes sense in physical terms since the reflection coefficient on the line is determined solely by the terminating impedance. With the definition of the “reflection function” as P(z) =

1 ∂ ln Z . 2 ∂z

(3.75)

we obtain 2

∂ − j2β0 + 1 − 2 P(z) = 0. ∂z

(3.76)

A solution to this Riccati equation can be found only in special cases. Accordingly, we will seek an approximate solution. For small reflection coefficients (which is all we are interested in), we can neglect 2 with respect to 1. We can thus solve the differential equation as follows ⎛ (z) = ej2β0 z ⎝C1 −

$z

⎞ P(z)e−j2β0 z ∂z ⎠.

(3.77)

0

We calculate the integration constant C 1 from the boundary condition at the end of the line z = l. We assume that the inhomogeneous line is terminated there in a reflection-free manner |(z = l)| = 0. C 1 is then:

170

H. Maune

$l C1 =

P(z)e−j2β0 z ∂z.

(3.78)

0

At the input to the inhomogeneous line (z = 0), we then calculate the input reflection coefficient 1 as $l 1 = (z = 0) =

P(z)e−j2β0 z ∂z.

(3.79)

0

Assuming the behavior of the characteristic impedance along the line is known, this relationship makes it possible to calculate the reflection coefficient as a function of frequency. On the other hand, the reflection coefficient is often given as a function of frequency and we are interested in the characteristic impedance of the inhomogeneous line. In Eq. (3.79), P(z) is thus unknown. To solve this, we continue the inhomogeneous line at both ends with a homogeneous line with characteristic impedances Z(0) and Z(l), respectively. The function P(z) is then equal to zero for z < 0 and z > l. We can then write the integral as $+∞ 1 = (β0 ) =

P(z)e−j2β0 z ∂z.

(3.80)

−∞

This is a Fourier integral which can be inversed easily $+∞ P(z) = (β0 )ej2β0 z ∂β0 .

(3.81)

−∞

In the context of the approximation (theory of small reflections) we applied above, we are able to calculate the reflection coefficient from the characteristic impedance or the reflection function P(z) as a function of frequency or the necessary characteristic impedance behavior along the line from the specified reflection coefficient |(β0 )| [4]. For each of the two calculation possibilities described by Eqs. (3.79) or (3.81), we provide an example hereafter which was also discussed in the calculation of a transformer with N λ/4-line sections. The calculation techniques which yield zeroes for this transformer in the input reflection coefficient in the band center only (binomial or maximally flat transformer) cannot be applied to the inhomogeneous line since as N grows, not only the center frequency f c but also the lower limit frequency f 1 goes to infinity.

3 Impedance Transformers and Balanced-to-Unbalanced …

3.1.4.2

171

Exponential Inhomogeneous Line as Transformer

Extending the geometrical multi-section transformers, Eq. (3.32), with N = ltot /l and n = zl + 1 z z Zn Z (z) 1 lim ln ln t = = ln = lim + ln t N →∞ N →∞ l tot R1 R1 2N ltot

(3.82)

Substituting μ = ln t/ltot we finally obtain Z (z) = R1 eμz

(3.83)

Due to the exponential dependency of the characteristic impedance Z(z) on the spatial coordinate z, we refer to this inhomogeneous line as an “exponential line” or “exponential taper”. If we plug this characteristic impedance relationship into the differential Eqs. (3.68), we obtain ∂ V (z) ∂ 2 V (z) + β02 (z)V (z) = 0 −μ 2 ∂z ∂z ∂ 2 I (z) ∂ I (z) + β02 (z)I (z) = 0. +μ 2 ∂z ∂z

(3.84)

Which only contains constant coefficients. Thus, the input reflection coefficient of the exponential line can be calculated exactly, we finally obtain 1 =

sin βltot sin ϕ . cos βltot cos ϕ + j sin βltot

(3.85)

with % β=

β02 −

2π μ μ2 ω , β0 = = , and sin ϕ = 4 υ λ 2β0

(3.86)

The behavior of the input reflection coefficient versus frequency is very similar to the case of geometrical stepped transformer. Likewise for the exponential line, we obtain maxima having magnitudes that decrease with increasing frequency; see Fig. 3.24. Moreover, since max is determined for a given t like in the case of geometrical-stepped transformer, this line with exponential impedance characteristic does not deliver optimal results.

172

H. Maune 60 %

1

40

20

0

0.5

1.0

1.5

2.0

lges/λ

Fig. 3.24 Input reflection coefficient 1 for a nine-section geometrical-stepped transformer (solid line) in comparison to an inhomogeneous line with exponential impedance characteristic (dashed line) for a transformation ratio of t = 4

3.1.4.3

Chebyshev Inhomogeneous Line as Transformer

If we realize the limiting process n → ∞ in Eq. (3.50) while holding the total length ltot = nl constant, then we obtain the exact relationship for the input reflection coefficient of this inhomogeneous line 2 cos (β0 ltot )2 − (β1ltot )2 |1 |2 = 2 2 1−max 2 2 + cos l − l (β ) ) (β 0 tot 1 tot 2

(3.87)

max

where β1ltot

2 1 − ltot 0 max = 2π = cosh−1 λ1 max 1 − 02

(3.88)

Here, ltot /λ1 is the total length of the line normalized to the wavelength λ1 of the limit frequency. In order to be able to, at least approximately, calculate the characteristic impedance behavior of this inhomogeneous line with the aid of Eq. (3.81), we will again make use of the approximation || 1. We then obtain the relationship [5] 1 with

& & & 2 & 2 & ≈ max &cos (β0 ltot ) − β1ltot &&

(3.89)

3 Impedance Transformers and Balanced-to-Unbalanced …

β1 ltot = cosh−1

0 max

and 0 =

173

1 ln t 2

(3.90)

According to Klopfenstein [5], integration of Eq. (3.81) provides in terms of the behavior of the characteristic impedance, assuming we introduce the coordinate transformation z = z/ltot − 1/2 the following relationship

⎫ ⎧ ⎧ 2z $ 2 ⎪ ⎪ ⎪ β I l 1 − y 1 ⎪ ⎪ 2 ⎪ 1 tot ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ β l dy ⎨ ⎬ ⎪ tot 1 & & ⎪ 2 β l 1 − y ⎪ ⎪ tot 0 + max for &z & ≤ 21 1 ⎨ 0 Z z ⎪ ⎪ ⎪ ⎪ ⎪ ln = ⎪ + H z − 1 − H −z − 1 ⎪ ⎪ ⎩ ⎭ ⎪ R1 ⎪ ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ln t for z > 21 ⎪ ⎩ 0 for z < − 21 (3.91) Here, I 1 is the modified Bessel function of the 1st type and 1st order. H is the Heaviside step function. H (x) =

0 for x < 0 1 for x ≥ 0

(3.92)

One notable property of this line is that it exhibits a step in the characteristic impedance curve at the start and end. Figure 3.25 illustrates this with a numerical example in comparison to the continuous behavior of an exponential tapered transformation line. The error which results from the assumption that || 1 can be seen in Fig. 3.26. There, the solutions with the precise behavior according to Eq. (3.87) compared to the approximation according to Eq. (3.89).

174

H. Maune

ln(Z/R1)/lnt

1.0

0.5

0 0

0.5

z/lges

1.0

–0.5

0

z'

0.5

Fig. 3.25 Comparison of the characteristic impedance Z (z) for inhomogeneous lines with exponential (dashed line) and Chebyshev gradient (solid line) (t = 10, max = 3%)

3.1.4.4

Compensated Inhomogeneous Lines

As was the case with stepped λ/4 transformers, we can also improve the transmission properties of inhomogeneous lines by adding reactances; in particular, we can reduce the total length of an inhomogeneous line in this manner. In order to preserve the high-pass nature of the inhomogeneous line, these reactances are realized here using lumped elements. The stub that is connected in parallel and short-circuited at the end corresponds to a parallel coil, while the stub that is connected in series and open at the end corresponds to a series capacitor. For single- or double-compensated inhomogeneous lines, we thus obtain the circuits shown in Fig. 3.27. Figure 3.28 illustrates the dependency of the total length L ≡ ltot on the transformation ratio t for zero to two compensation elements. Given the same compensation type, the superiority of the inhomogeneous Chebyshev line (T) is also apparent from its steady behavior compared to the exponential line (E).

3 Impedance Transformers and Balanced-to-Unbalanced …

175

a 120 %

1

80

40

0

0.2

0.4

0.6

0.8

1.0

lges/λ

b

6 % 5

4

1

3

2

1

0 0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

lges/λ

Fig. 3.26 Input reflection coefficient of a Chebyshev designed with the exact Eq. (3.87) (––––) and approximated Eq. (3.89) (– – –) behavior. (– · – · –) shows the solution of the differential Eq. (3.74) with the coordinate transformation Eq. (3.91). a Shows the overview for all reflection coefficient, while b shows a zoomed view for |1 | < 6%

3.1.4.5

Cosine-Squared Inhomogeneous Lines Transformers.

Using equation (3.79), it is possible to calculate the reflection coefficients for various inhomogeneous lines. Bolinder [4] provides a wide range of examples. It turns out that the exponential line does not possess the most favorable transformation properties. Another example is the cosine-squared transmission line. We set the reflection function z R2 1 1 (3.93) ln cos2 π − P(z) = ltot R1 ltot 2

176

H. Maune

a

R1

~

L

b

R2

In

C

R1

~

R2

In

c

C

R1

~

L

R2

In

Fig. 3.27 Different prototypes of compensated inhomogeneous lines 1.0

E0

E1

0.8 T0

0.6

L/

1

T1 E2

0.4

T2

0.2

0

1 1.2 1.4 1.6 1.8 2

4 t

6

8 10 12 14 16 18 20

Fig. 3.28 Comparison of electrical length L/λ of the inhomogeneous line with Chebyshev behavior (T) with the exponential line (E) for max = 10%; The index 0, 1, and 2 stand for lines without compensation, with single compensation, and with double compensation, respectively (The bend in the curves at t = 2 is due to the change in scale.)

3 Impedance Transformers and Balanced-to-Unbalanced …

a

177

c

Ln

%

Z R1

Ln

R2 R1

60 40

Exponential line

0 0

0,5

cos2 line Expon. line

20

cos2 line 1,0 z/l

0

0,5

1,0

1,5

l/

b

(

Z R1 dz

d ln

) cos2 [ (z/l -0,5)]

0

0,5

1,0 z/l

Fig. 3.29 Comparison of the cosine-squared transmission line with the exponential line for the same transformation ratio t = 4. a Curve of the characteristic impedances along the line; b Curve of the reflection function; c Curve of the reflection coefficient over l/λ

The characteristic impedance Z(z) then

Z (z) =

⎞⎫

⎧ ⎛ ⎨ z ⎬ sin 2π ltotz − 21 R2 ⎝ ⎠ R1 R2 exp ln 1+ 2π z ⎩ ltot ⎭ R1 l

(3.94)

tot

For the reflection coefficient, we obtain the following according to Eq. (3.79): () R |1 | =

2

R1 R2 R1

* +& #& " sin 2πlλtot && π 1 R2 && 1 R2 −K t − ln e λ + ln & & 2 R1 & π 2 − 2πltot 2 2πlλtot & + 1 2 R1 λ (3.95) −1

In Fig. 3.29, the reflection coefficient is plotted versus frequency along with that of the exponential line. Here, the first maximum of the cosine-squared transmission line after the first zero is equal to 2.6% of the initial value for ω = 0 versus 21.7% for the exponential line. However, this favorable behavior comes at the cost of a greater total length for the transformer since the first zero does not occur until l = λ (vs. l = λ/2 for the exponential line).

3.1.5 Transformers in Microstrip Technology The trend towards miniaturization in microwave engineering has favored replacement where feasible of previously conventional transformers based on coaxial and

178

H. Maune

b

W

a /4

m

W1

m

m

/4

/4

W2

Fig. 3.30 Structures of λ/4-transformers based on microstrip technology a series connection of line sections b parallel connection of line sections

waveguide technology with planar waveguide technologies. Typical benefits of these devices include ease of integration, compact dimensions, low weight, dependability and the ability to realize even complex conductor structures in a cost-effective and reproducible manner. On the other hand, typical disadvantages are higher losses accompanied with lower power handling compared to coaxial and waveguide technology and limitation to planar conductor geometries. This section will discuss the properties of microstrips that are relevant especially when building transformers with these line components. See Sects. 4.7, 4.8 and 4.9 for a discussion of conventional design models for these devices, calculation of their static parameters (and especially the realizable characteristic impedances), wave propagation, dispersion, losses and connection at the line ends.

3.1.5.1

Transformers in Microstrip Technology

λ/4-Line Stepped-Impedance In Sect. 1.3.2, cascaded stepped-impedance λ/4-line sections are used to construct stepped-impedance transformers. Additional stubs are added in some cases. These are either connected in parallel and short-circuited at the end or connected in series and left open at the end. Series connection and parallel connection of line sections using microstrip technology does not pose any inherent difficulties. Figure 3.30 provides schematic depictions of the geometry of conductor structures of this sort. On the other hand, it is not possible to realize a series connection of line sections as is feasible, say, using coaxial technology (see Fig. 3.21c, d) due to the microstrip’s two-dimensional structure in this technology. The possibility to arrange two microstrips over or adjacent to one another over a base plate does not lead to a system consisting of two independent lines. Instead, a coupled three-conductor system arises having transfer properties that differ fundamentally from those of their

3 Impedance Transformers and Balanced-to-Unbalanced …

a

b

l 2 R1

~

Z1

~

4

a

l R1

R2

3

b

1

179

R2

V2 = V4 = 0

c

ZP1

ZP2

l

l

l

R1

üp1

Zt2

l

R2 üs:1

Fig. 3.32 Realization of a series stub using microstrip technology with coupled lines a implementation as a microstrip; b Equivalent circuit with uncoupled lines; c Transformation according to Kuroda (see Sect. 4.14.2)

a

b

Fig. 3.33 Cascade connection of a four-stage λ /4 transformer and an inhomogeneous line in microstrip environment

3 Impedance Transformers and Balanced-to-Unbalanced …

181

line are arranged in a cascade configuration. The spatial arrangement of the input and output in a component of this sort constructed using microstrip technology can be chosen within wide limits based on the shaping of the conductor. Some examples of applications of this technique are broadband matching of a coplanar line to a microstrip line on a superconductive substrate [4] and broadband impedance matching of a laser diode (3–10 ) to a 50 system [6].

3.2 Matching Between Balanced and Unbalanced Lines In the previous sections, we examined the processes that occur on lines without considering their structure and the influence of their environment. We will now consider three different types of lines: the single line over ground, the double line (line pair) over ground and the coaxial cable (Fig. 3.34). In the single line over ground, the operating state is clearly determined. The current flows in the conductor to the load while the ground carries the return current (I1 = −I0 ). The voltage V is between the conductor and ground. The characteristic impedance follows from the as Z c = 1/υC10 . capacitance per unit length C10 In contrast, the state is ambiguous for the line pair shown in Fig. 3.34b. We can operate the line such that conductors 1 and 2 have opposing voltages versus ground (V1 = −V2 ) and conductor 2 carries the return current of conductor 1 (I2 = −I1 ∧ I0 = 0). This is known as the balanced operating state (differential or odd mode). The other possibility is for conductors 1 and 2 to have the same voltage over ground (V1 = V2 ) and the currents I2 = I1 flow in the same direction (common or even mode) while the return current flows through the ground I0 = −2I1 . This unbalanced operating state (Fig. 3.34c) basically does not differ from the operating mode illustrated in Fig. 3.34a. The characteristic impedance for the even-mode Z ce is Z ce =

1 υ C10 + C20

(3.96)

The balanced operating state (odd-mode) clearly has three prerequisites: The line pair must be fed by the generator in a balanced manner, the balance may not be

a

b 1

c

V 1

2

1

V=0

d

2

1 C12

C12 V

C10 0

V1 0

C10

C20

V2= –V1 0

V1

C10

C20

V2

2a 2b

C20

Fig. 3.34 Different line types over ground. a single line over ground, b balanced line pair over ground, c unbalanced line pair over ground and d coaxial cable over ground

182 Fig. 3.35 Load impedances for a line pair over a conductive plane

H. Maune

a

Z1

I1

Z2

3

1

Z30

V1

b

I2

0

2

I1

V2

V1

1

2

l2 Z10 0

Z20

V2 0

disrupted by the terminating impedance and the line pair must be constructed to be = C20 . The characteristic impedance for balanced over ground. This means that C10 balanced mode is then obtained as follows: Z co =

1 υ C11 +

C10 2

(3.97)

In the general case, the load impedance of a line pair over conductive ground represents a T or topology (Fig. 3.35). According to the definition of balanced mode, we must have the currents I2 = −I1 and the voltages V1 = −V2 . This is the case only if point 3 is at ground potential V3 = 0, i.e. if Z 1 = Z 2 . Since no current will then flow via Z 30 . For the balanced wave we obtain the load impedance. Z lo = Z 1 + Z 2 = 2Z 1 .

(3.98)

Z 30 does not enter into Z l and can thus be of any arbitrary value in differential mode without disrupting the matching. In case of unbalanced operation, we have V1 = V2 . We obtain the load impedance for the common-mode wave for Z 1 = Z 2 as Z le =

Z1 + Z 30 2

(3.99)

By choosing a suitable value for Z 30 , we can ensure reflection-free termination for the common-mode wave. Analogously, in Fig. 3.2/2b we must have Z 30 > 100Z 1 not to disrupt the balance of this arrangement. Then, the terminating impedance for the differential-mode wave is Z lo =

2Z 10 Z 12 2Z 10 + Z 12

(3.100)

Z 10 2

(3.101)

and for the common-mode wave Z le =

3 Impedance Transformers and Balanced-to-Unbalanced …

a

183

b 5A

a

E 2A

1A

4A

5A

5A

+ 2A

3A

b 3A

Sheath current Ground current

4A 1A Differential Common mode mode

2A 2A Ground current

Ground

Fig. 3.36 Transition from a coaxial cable to a balanced line pair without balun

Figure 3.34d shows a cross-sectional view of a coaxial cable over ground. This three-conductor system is unbalanced to ground: Although the inner conductor 1 has a capacitance C12 versus the inner skin 2a of the cable shield, it has zero or only a negligible capacitance to ground. In other words, the partial capacitance C10 is missing compared to Fig. 3.34b. The characteristic impedance of the coaxial cable for f ≥10 kHz (see Sect. 2.1.4), is real and is equal to Z inner =

1 υC12

(3.102)

Analogous to Fig. 3.34a, the outer skin 2b of the cable shield forms an independent conductor system along with ground. Its characteristic impedance is Z outer =

1 υC20

(3.103)

In Fig. 3.36, a balanced line is connected directly to an unbalanced line over ground. The current on the inner conductor of, say, 5 A enters undisturbed into one wire of the line pair. In the coaxial cable, the current on the inner skin of the cable shield must be equal and opposite to the current on the inner conductor. However, at the junction from the coaxial cable to the line pair, a current branch forms. The current flowing on the inner skin of the cable shield consists of the current on the second wire of the line pair and the ground current. This is regardless of whether we locate the short-circuit between the cable shield and ground (directly at point b or, as shown in the figure, at a point E that is apart from the cable end). In our numerical example, it is assumed that the return current is distributed in the form of 3 A on the other wire and common-phase with 2 A on ground. Figure 3.36b shows that we can imagine this situation as the superimposition of a differential-mode current of 4 A with a common-mode current of 1 A. This distribution of the currents is dependent on the length of the line and the different terminating impedances which are encountered by the common-mode and differential-mode waves at the end of the line. Accordingly, the current and impedance relationships are undefined when we

184

H. Maune Z1'

Z2' 2a

1b ' Z30

R1

Z1

1a 2

1b

V1

Z1

Z2

2b

S 2b

Z2

V2

0

1b 2b

2

Z30

2b Zb

Z30

0

0

Fig. 3.37 Connection of a unbalanced and a balanced line

connect a balanced line to an unbalanced line as shown in Fig. 3.36. Accordingly, undesired reflections can occur even if the balanced line is properly terminated with its characteristic impedance for the differential-mode wave. If such a line is used to feed a balanced antenna, for example, the antenna’s radiation pattern can also be distorted by the radiation produced by the feeder line’s common-mode wave. For a more systematic approach, we investigate the circuits in Fig. 3.37. If we directly connect Pins 1a to 2a and 1b to 2b, the differential-mode current that flows into the balanced line at 2a does not flow out at Pin 2b with the same magnitude since part of it can flow via Z 30 and Z b0 . In order to eliminate this partial current, there are the following possibilities: 1.

2.

3.

4.

We can eliminate the direct connection between terminal pair 1 and terminal pair 2 and connect a balun as transformer (Fig. 3.38a). Then, Z b0 can be of low-impedance or even be short-circuit between cable shield and ground. Even if it were possible to keep Z 30 > 100Z 1 or Z 30 > 100Z 2 (unfeasible in a larger frequency range), the shunt current via Z 30 would be negligible but we would not necessarily have V2 = −V1 which is the condition for differential mode. In order to fulfill this condition, Z b0 must be as large as possible (e.g. |Z b0 | > 10 k ). This is possible on a frequency-selective basis using sleeve baluns (Fig. 3.38b). It is better to include a circuit to form a balanced bridge. For Z 1 = Z 2 the balancing unit (balun) must add an impedance Z a0 as an image of Z b0 on the inner conductor of the coaxial cable at 1a (Fig. 3.38c). In Fig. 3.38d, an interesting possibility involves connecting terminal 1a directly to 2a and moreover connecting 2b via an inner λ/2 bypass line to 1a (see Figs. 3.2/20). Then, the current in the coaxial cable is twice as large as the current at the start of the balanced line. Moreover, the voltage between 1a and 1b is half as large as the voltage between 2a and 2b such that a 1:4 transformation (R = 4Z) is associated with this balancing mechanism. For example, this transformation offers a practical way to simultaneously match balanced lines with a characteristic impedance of 240 to a coaxial cable with Z = 60 .

3 Impedance Transformers and Balanced-to-Unbalanced …

a

185 R = 4Z

d 1a

2a

2a

2b U

U

I

II

2Z l

l 1b

2b

I/2 1a

b

I/2

Zsleve

Isleve

1b

Z

I

2Z

U

Isleve

Z

2b

1a

R-Z 1b

2a

Vsleve

S

/2

Vsleve /4

c

/4

/4

Z1

Z2 = Z1

1a 1b

2a

2b

Symmetric line

Fig. 3.38 Balun circuits a transformer b sleeve balun, c balanced bridge d λ/2 bypass

3.2.1 Balancing Transformer If we wish to connect from an unbalanced line such as a coaxial cable to a balanced line (or vice versa), the preceding discussion states that we must insert an intermediate element to prevent a common-mode wave from arising. In the long-, medium- and shortwave ranges where the geometrical dimensions are small with respect to wavelength, we can use the classic winding transformer shown in Fig. 3.39a. Without a static shield between the primary and secondary coils, parasitic capacitances between the windings must be considered. Even if the capacitances between 1a–2a and 1b–2b are identical, the capacitive currents will have different magnitudes due to different voltages. If we provide an electrostatic shield connected to ground between the two windings, the same voltage will be present on the ground capacitances of the

186

H. Maune

a

b 1a

2a

1a

2a

1b

2b

1b

2b

c

d 1a

I2

2a

A

2a 1a

I1

I1

M

1b 1b

B

I1–I2 I2

2b

2b

Fig. 3.39 Balancing transformer a standard transformer, b standard transformer with static shield, c auto-transformer with unbalance connection d auto-transformer with unbalance connection

balanced winding (Fig. 3.39b). The windings are thus loaded in the same manner by the parasitic capacitances to ground so the balance is retained. Instead of a transformer with two separate windings, an auto-transformer is commonly used at higher frequencies. An arrangement that is suitable for providing balance in air coils is shown in Fig. 3.39d. The shield of the coaxial cable is connected at point M which is at ground potential from the perspective of the balanced side. The inner conductor is fed in a coaxial manner within on half of the coil tube, comes out in an isolated manner after n windings (point A) and is connected to point B which is the mirror image via the center M of A.

3.2.2 Baluns Consisting of Line Elements 3.2.2.1

Sleeve Baluns

As seen in Fig. 3.36, the differential-mode wave arises at a transition from a coaxial cable to a line pair due to the current branch at point b. We can force balance by making the current flowing on the outer skin of the cable shield at point b equal to zero. The line formed by the outer skin of the cable shield and ground with characteristic impedance Z a is short-circuited at its end at point E. The input impedance of this line is Z 1 = jZ 2 tan

2πl λ

(3.104)

3 Impedance Transformers and Balanced-to-Unbalanced …

a

187

b

m

/4

m

/4

Zsleve

Isleve

Isleve

Z

2b

1a

R=Z 1b

Usleve

2a S

Usleve /4

mast / feedline

Fig. 3.40 Sleeve balun a Theory, b technical realization at a dipole antenna

if we neglect the losses. The current disappears if the input impedance of this stub line exhibits very large values, i.e. for a λ/4-long short-circuit stub line”). In the technical realization, we do not connect the coaxial cable at a distance λ/4 from the end to ground; instead, we wrap the end of the cable with a tube of length λ/4. This tube is connected to the shield the far-end end and forms a short-circuited λ/4 line (“sleeve balun”). Such an arrangement is sketched in Fig. 3.40a together with the current and voltage behavior. If the operating frequency is not equal to the resonant frequency of the sleeve balun, the impedance drops off between 2a and S and a sheath current flows via the outer skin of the balun and the adjacent cable shield. The balance is disrupted in this manner. Accordingly, the sleeve balun is useful only in a relatively narrow band around λ/4 (narrowband balancing). Figure 3.2/7 shows an example with a clever design. There is an open dipole antenna with the lower half of the dipole formed simultaneously with the sleeve balun formed by the coaxial feed cable within the stand pipe and a folded end of the stand pipe.

3.2.2.2

Collinear Baluns and Balancing Loops

Another way of providing balance involves loading the inner conductor additionally with an impedance that is just as large as that resulting on the outer conductor due to the sheath current. Such an arrangement is known as a collinear balun and is sketched in Fig. 3.41. The left side contains the feeder coaxial cable on which a sheath current forms. The inner conductor is connected to the image arranged on the right side. If

188

H. Maune

a

b /4

l< /4

/4

Z1

Z2 = Z1 1a

1a 1b

2a

2b

1b

2a

2b

Symmetric line

Symmetric line

Fig. 3.41 Collinear baluns a with symmetric image load and b with additional capacitive loading

the characteristic impedances Z 1 = Z 2 as well as the lengths of the short-circuited lines are equal, the desired balance is obtained and is independent of frequency. The input impedance of the line pair at the connecting point 2a–2b has a reactance in parallel of magnitude X |2a−2b = 2Z 1 tan(2πl/λ). For l = λ/4, its influence disappears. As the frequency diverges from the resonant frequency, it causes a mismatch which limits the usable bandwidth. The total length can be reduced by providing capacitive loading as shown in Fig. 3.41b. In both examples shown in the figure, the fields are fully shielded by the housing with respect to the environment which also prevents radiation by the λ/4 resonators. In principle, this shielding is not necessary for the balance. While in this case we have considered the image opposite to the cable end, in Fig. 3.42 it has been turned by 180°, arranged parallel to the cable and connected at point E to the cable shield. This arrangement is known as a “balancing loop”. Here again, the balance is independent of frequency. We can also imagine how the balancing loop works in another way: If we reduce the number of windings in Fig. 3.39d until there is only a single winding left for the secondary and primary side, we obtain Fig. 3.43. This circuit is known as “EMI loop” developed by “Electrical & Musical Ind. Ltd.”.

VUsymm inner conductor

outer conductor

a

E image line

b

symmetric line

Vsymm l = /4

Fig. 3.42 Collinear baluns with image line parallel to unbalanced coaxial cable (balancing loop)

3 Impedance Transformers and Balanced-to-Unbalanced …

189 l = /4

Fig. 3.43 EMI loop Z Z

M 2a

1a

R=Z

2b 1b

b

a Z 1a

2a 1b

R=Z 2b movable short /4

Fig. 3.44 a Balancing loop with adjustable short-circuit and b Balancing loop as magnetic dipole antenna

The load impedance of the coaxial cable is the parallel connection of the input impedance of the balanced line pair and the short-circuited line which forms the balancing loop. If we assume the line pair is terminated in reflection-free manner, we obtain a real terminating impedance for the coaxial cable if the length of the balancing loop is equal to λ/4 at the operating frequency. If the frequency needs to be changed more often, a sliding short-circuit bridge as shown in Fig. 3.44a can be used in place of the fixed short-circuit. Figure 3.44b shows an interesting application of the balancing loop. This is a magnetic dipole antenna. Here, the folded dipole itself forms the balun. In the “balancing half-shell” in Fig. 3.45, the outer sheath of concentric line 1 is enclosed by a half-shell 2 of length l = λ/4 which is connected by a segment 3 to the inner conductor and a segment 4 to the outer conductor of the concentric line. This arrangement is especially practical in cases where a small Z p is needed to compensate reactance components of λ/4 transformers (Sect. 3.1.3.3).

3.2.2.3

Split Tube Baluns

The split tube balun shown in Fig. 3.46 represents a special type of balancing arrangement. The image of the cable shield for the conductor that is connected to the inner

190

H. Maune

1

3

1

2

4

2 l = /4

Fig. 3.45 Balancing half-shell C12

C13 A

1 2

3

C23 3

2

2

R2 3

1

1

B

V1

A–B

V3

V2

l– /4

Fig. 3.46 Split tube balun a cross section alon line b cross section and c cross section with capacitances

conductor of the coaxial cable is not arranged next to the cable sheath; instead, it is part of the cable shield itself. From a concentric coaxial line, the outer shield is cut open at a length l = λ/4 to form two symmetrical half-shells. The inner conductor is connected at the end with one half-shell. The split tube balun differs from the balancing loop in that the current and voltage on the inner conductor are also coupled with the half-shells. It represents a three-conductor system (or a four-conductor system over a conductive plane). The transformation properties can be characterized as follows: At the end of the split tube balun, the load resistance R2 is in parallel to a reactance X 2 = Z s tan ßl. The latter arises due to the line which the two half-shells form. Z co is the characteristic impedance of the two half-shells and l is the length of the slot. Z co can be calculated from the partial capacitances shown in Fig. 3.46c: Z co =

1 υ C12 +

For the parallel circuit, we obtain

C13 2

=

1 υ C12 +

C23 2

(3.105)

3 Impedance Transformers and Balanced-to-Unbalanced …

Z2 =

jR2 Z co tan{βl} R2 + jZ co tan{βl}

191

(3.106)

It now turns out that for a continuous inner conductor with an unchanged diameter d, R2 = 4Z 0 must be connected as the load resistance so that for l = λ/4, Z 0 appears as the input impedance at the input to the split tube balun at z = 0 and thus matching to the characteristic impedance Z c of the unsplit coaxial line is possible. For a continuous inner conductor, we have Z ce = Z c . At the input according to Eq. (3.106), we then have Z 2 /4. For l = λ/4, the input impedance of the slotline with characteristic impedance Z s which is short-circuited √ at z = 0 is high that we have Z 2 /4 ≡ R2 /4. In other words, we must choose Z ce = Z 0 R2 /4. The magnitude of Z ce follows from the equation for the common-mode wave that propagates between the inner conductor and the two parallel half-shells as shown in Fig. 3.46c: Z ce

1 1 1 60 Dt = = = = √ ln υ(C13 + C23 ) 2υC13 2υC23 εr dt

(3.107)

Here, d t and Dt are the inner diameter and outer diameter within the slotted transformation path, respectively.

3.2.2.4

λ/2 Bypass Line

Balanced operation of a line pair requires the currents as well as the voltages on both lines to be out of phase with respect to one another. We can produce an outof-phase current and an out-of-phase voltage very easily using a line of length l = λ/2. We connect one conductor of the line pair directly to the inner conductor of the coaxial cable and the second conductor via a λ/2 bypass line to the inner conductor (Fig. 3.47a). Since from the perspective of the incoming feeder cable with characteristic impedance Z c the two lines I and II appear to be in parallel, each of them must have the characteristic impedance 2Z c for reflection-free connection to the feeder cable. From the perspective of the terminating resistance R, the characteristic impedances of lines I and II appear to be in series. Thus, for reflection-free termination we must have R = 2•2Z = 4Z. Along with the balancing effect, the λ/2 bypass line also produces a 1:4 transformation. Mismatches do not change the balance since the path difference is always λ/2. The balance condition that one path must always be longer than the other by λ/2 limits the bandwidth of this arrangement. If we wish to use a λ/2 bypass line in different frequency ranges, we can employ a trombone-type design or a design like the “phase transformer”, shown in Fig. 3.47b, where the connection point and hence the length difference can be adjusted.

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2b

R = 4Z 2a

2b V

V II

I

2Z 2Z l I I/2

1a

I/2 1b

Z I

2Z

V

/2

Z 1b

/4

1a

/2

/2

Fig. 3.47 a λ/2 bypass line balun and b adjustable phase transformer

3.2.3 Broadband Line Transformers for Transformation and Balancing Made of Lines and Ferrite Components We will now take another look at the junction between a coaxial cable and a balanced line pair. According to the discussion there, the desired pure differential-mode operation on the balanced line pair can be achieved using measures on the coaxial cable as well as on the balanced line pair. Besides the possibilities considered in previous sections, such balance can also be achieved using ferrite components. If we enclose the cable sheath with a material having the best possible magnetic properties as shown schematically in Fig. 3.48a, this increase in inductance will boost the value of |Z b0 | as desired and thus reduce the cable sheath current. If we enclose the line pair with a ferrite core of this sort or wind the line on a bar or ferrite ring as shown schematically in Fig. 3.48b, c, then the differential-mode current

3 Impedance Transformers and Balanced-to-Unbalanced …

a

193

Ferrite tube Ri

Z1

Z30

Z1

Coaxial cable ZL

b

c Ferrite core Rl

Z1

l1 l2

Serrite ring core Ri

Z1

Z30

Z1 Symmetric (double) line ZL

Z30 l0

l0

Z1 Symmetric (double) line ZL

Fig. 3.48 Balancing with ferrite components a suppression of the sheath current on the coaxial cable and b/c suppression of the common-mode current on the balanced line pair

will be practically unaffected while the common-mode current will be subject to a large inductive impedance in comparison to the characteristic impedance, thereby reducing this undesired current. The different impact on the differential-mode and common-mode currents by the magnetic material occurs for the following reasons: 1.

2.

For currents that are equal but opposite (I 1 = –I 2 , differential mode), no magnetic flux is developed in the magnetic material (I 0 = 0, Fig. 3.48b) and the mutual inductance between two successive windings (Fig. 3.48c) is negligible since the magnetic field is concentrated primarily in the region between the two conductors, For currents flowing in the same direction (I 1 = I 2 , common mode), magnetic flux is developed in the magnetic material causing an increase in the inductance such that the common magnetic field induces voltages in the adjacent windings, resulting in increase of the mutual inductance.

Besides balancing effects, these line transformers also allow transformation. Since the current via the common ground connection can be suppressed in the circuits shown in Fig. 3.48, it is possible to build transformers with these circuits as the basic element. The simplest arrangement (a 1:1 transformer with polarity reversal) is shown for a balanced line pair in Fig. 3.49. It is now possible to realize transformers with a conversion ratio |c| = 1 by connecting several of these basic elements to the inputs in series and to the outputs

a

b R1

R1 V1

V2

R2 = R 1

V1

V2

R2 = R1

ZL ZL

Fig. 3.49 Examples of 1:1-transformers with polarity reversal; for matching we have Z L = R1 = R2 a Transformer with ferrite core; b Transformer with ferrite ring

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in parallel (|c| > 1, t < 1) or to the inputs in parallel and to the outputs in series (|c| < 1, t > 1). Examples for a transformer with c = 2 (t = 1/4) are shown in Fig. 3.50. Using the circuit, a transformation is realized along with balancing of the junction. Transformers built using this principle exhibit an integral transformation ratio. If we extend this principle such that series circuits as well as parallel circuits consisting of these basic elements are allowed at the input and output, then it is also possible to realize transformation ratios with fractional values [7, 8]. Based on a simple example, we would now like to investigate the limit frequencies of baluns and transformers made of line sections and ferrite components. We will somewhat rearrange the circuit in Fig. 3.49a for this purpose. As was mentioned above, the differential-mode current through the ferrite core is practically unaffected while the common-mode current that flows via the common ground connection is subject to an inductive impedance. We thus obtain the equivalent circuit shown in Fig. 3.51. In terms of the lower limit frequency, the line transformer is no different from a normal winding transformer, as we can see by comparing Fig. 3.51 with Fig. 3.4. We have the following for the lower limit frequency: ωmin L C =

a

(3.13)

b

ZL

R1

R1 R1 → f min = 2 4π L C

R2 = R1/4 ZLgg

ZLu

ZL

ZL

Fig. 3.50 Line transformer with c = 2 (t = 1/4). a Operated unbalanced to ground on both sides, Z L = R1 /2 = 2R2 ; b With transition from balanced to unbalanced line, Z L = Z Lgg /2 = 2Z Lu

R1 V1

V1

R1

Lc

Fig. 3.51 Equivalent circuit for the transformer in 3.49a at low frequencies; L C = Inductance of short-circuit loop via the common ground connection

3 Impedance Transformers and Balanced-to-Unbalanced …

195

However, there is a significant difference in the upper limit frequency. The leakage inductances present in the winding transformer do not occur at all in the coaxial line transformer shown in Fig. 3.48a. Moreover, in the line transformer shown in Fig. 3.48b, c, they can be made smaller by orders of magnitude compared to the winding transformer by using a spatially compact arrangement of the two conductors. The winding capacitances which also influence the upper limit frequency for the winding transformer enter only into the line’s characteristic impedance for the line transformer and have no direct influence on the upper limit frequency. In fact, the upper limit frequency of the line transformer is determined essentially by the wave propagation on the line, i.e. by the upper limit frequency of the line itself as well as by any reflections that occur. In the interest of the most broadband transmission possible, usage of a suitable line is critical along with the following points: 1. 2. 3.

The characteristic impedance Z L of the line that is used must be matched to the generator’s internal impedance and the load impedance When using multiple line sections as shown, say, in Fig. 3.50, the propagation times must be precisely equal. The discontinuities that are unavoidable when connecting multiple lines may only exhibit slight deviations from the calculated characteristic impedance behavior.

In actual practice, this approach can be used to create matching and balancing circuits over a very wide frequency range. An individual component can exhibit a bandwidth of several hundred MHz. Values can be realized for the lower limit frequency of approx. 1 kHz and for the upper limit frequency of approx. 10 GHz. Such components are used, for example, when connecting antennas to feed lines, in broadband amplifier technology, pulsed technologies such as radar and especially in components needed to meet miniaturization requirements in microwave engineering. Numerous examples of these components as well as technical realizations, theoretical discussions and experimental measurement results can be found in [7, 9].

References 1. Richards, P.I.: Resistor-transmission-line circuits. Proc. IRE 36(2), 217–220 (1948). https://doi. org/10.1109/JRPROC.1948.233274 2. Collin, R.E.: Theory and design of wide-band multisection quarter-wave transformers. Proc. IRE 43(2), 179–185 (1955). https://doi.org/10.1109/JRPROC.1955.278076 3. Riblet, H.J.: General synthesis of quarter-wave impedance transformers. IRE Trans. Microw. Theor. Tech. 5(1), 36–43 (1957). https://doi.org/10.1109/TMTT.1957.1125088 4. Bolinder, F.: Fourier transforms in the theory of inhomogeneous transmission lines. Proc. Inst. Radio Eng. 38(11), 1354–1354 (1950) 5. Klopfenstein, R.W.: A transmission line taper of improved design. Proc. IRE 44(1), 31–35 (1956). https://doi.org/10.1109/JRPROC.1956.274847 6. Carvalho, M.C.R., Margulis, W.: Transmission line transformer. Electron. Lett. 27(2), 138–139 (1991). https://doi.org/10.1049/el:19910090

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7. MacDonald, M.: Design broad-band passive components with ferrites. Microw. RF 32(10), 81–000 (1993) 8. Myer, D.: Synthesis of equal delay transmission line transformer networks. Microw. J. 35(3), 106–111 (1992) 9. Ruthroff, C.L.: Some broad-band transformers. Proc. IRE 47(8), 1337–1342 (1959). https://doi. org/10.1109/JRPROC.1959.287200

Chapter 4

Properties of Coaxial Cables and Transmission Lines, Directional Couplers and RF Filters Matthias Rudolph

Abstract This chapter addresses properties of technically important RF and microstrip lines, such as coaxial, microstrip, coplanar waveguide and striplines. It is discussed in detail, how the geometrical dimensions and material properties translate into line parameters as characteristic impedance or propagation constants, and the respective formulas are given. The second part of the chapter is devoted to passive microwave circuits that can be realized on the basis of lines: couplers, dividers, and line-based filters. Special emphasis is laid on the comprehensive treatment of surface-acoustic wave filters in the final section of the chapter.

4.1 Properties of Coaxial Cables and Transmission Lines This section addresses properties and dimensions of coaxial cables and transmission lines. Coaxial cables are the most common types of transmission lines for flexible connections over longer distances, while microstrip lines are typically used on circuit boards and in integrated circuits. At highest frequencies and at highest powers, when line losses need to be at the absolute minimum achievable, rectangular waveguides offer significant advantages. The discussion of transmission line properties requires the analysis of the electromagnetic fields of the propagating waves. This is rather straight forward in case of coaxial cables which will be discussed in detail. The field vectors, as we will see, are oriented orthogonaly to each other and to the direction of propagation. These socalled TEM (transversal electromagnetic) waves are non-dispersive, which means that line impedance is constant over frequency, and phase and group velocity are constant and equal to the velocity of light, depending on the dielectric. On striplines, in contrast, propagating waves are only approximately of the TEM type. They do have a field component parallel to the propagation direction, which is not very strong. These waves are often called quasi-TEM waves, which means M. Rudolph (B) Brandenburg University of Technology, Cottbus, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. L. Hartnagel et al. (eds.), Fundamentals of RF and Microwave Techniques and Technologies, https://doi.org/10.1007/978-3-030-94100-0_4

197

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M. Rudolph

that although properties change over frequency, there is a broad range of frequencies where it is safe to ignore the dispersion in practical circuit design. Rectangular waveguides and free-space waves require more involved reasoning based on Maxwell’s equations, which requires a chapter on it’s own: Chap. 5. Some properties of traveling waves that we use in the analysis of coaxial lines and striplines, such as the wave impedance, will be properly derived in Chap. 5.

4.1.1 Concept of the Wave Impedance Analogous to the familiar concept of the characteristic impedance of a line from transmission line theory, it is found that the ratio of the electric and magnetic fields of an electromagnetic wave can be understood as the “wave impedance”. For a pure traveling wave, the wave impedance1 Z F is the ratio of the transverse field components E and H: ZF =

E . H

On a transmission line, or in free space when observing wave propagation in one spacial direction, the same value of ZF is obtained for the forward wave with transverse components E p and H p and for the reflected wave with E r and H r . ZF =

Ep Er =− . Hp Hr

The wave impedance for a transverse electromagnetic (TEM) wave (see Chap. 5) is determined solely by the material constants of the medium and (to a small extent) of the conductors and is thus spatially-independent in the case of a homogeneous medium. The ratio E p /H p has a constant value at high frequencies: Ep = ZF = Hp

μ0 μr = Z F0 · 0 r

μr . r

(4.1)

Here, we have Z F0 =

μ0 = 120 π = 377 0

(4.2)

which is the wave impedance of free space (vacuum). The wave impedance of air is very close to Z 0 since μr = 1 and εr ≈ 1.0006. For lines and cables, we often have μr = 1 but εr > 1 and thus 1

In general, Z F is complex. However, the imaginary part can be neglected as an approximation for low-loss lines.

4 Properties of Coaxial Cables and Transmission Lines …

199

Z F0 Z F = √ < 377 . r For a given current and voltage definition, the characteristic impedance of a line which is determined based on the voltage and current differs from the wave impedance only by a numerical factor that is dependent on the geometry of the line’s cross section.

4.1.2 Characteristic Impedance of a Line and Capacitance Per Unit Length √ Given v = 1/ L C and Z 0 = CL (see Chap. 2), we obtain a direct relationship between the line characteristic impedance Z 0 and the capacitance per unit length C : Z0 · v = Or, given the phase velocity ν =

ω β

L 1 1 ·√ = C C L C =

Z0 = For c =

√1 μ0 ε0

√c εr

(4.3)

, we have

√ r 1 · . C c

(4.4)

≈ 3 · 1010 cm/s, we obtain √ r 1 s · Z0 = · 10 cm C 3 · 10

and since s/ = 1 F (or 1/1012 s/ = 1 pF), we obtain the adapted quantity equation √ pF/cm Z 0 = 33.3 r · C

(4.5)

In other words, determination of the characteristic impedance Z 0 of a line does not require insight into the inductance per unit length L and capacitance per unit length C . Instead, it suffices to determine only C , e.g. by performing a capacitance measurement on a section of the line that is open at the end. Z 0 and C are then inversely proportional according to Eq. (4.5).

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4.1.3 Characteristic Impedance of a Line and Inductance Per Unit Length Analogous to our the previous section, we obtain the following by √ observations in √ dividing Z 0 = L /C by ν = 1/ L C : Z0 = L v

(4.6)

Given that c v = √ , c ≈ 3 · 1010 cm/s and 1 s = 1 H r we thus obtain Z0 L 30 cm L = √ · 3 · 1010 =√ · r s r nH/cm

(4.7)

The characteristic impedance and inductance per unit length L are thus proportional. Equations (4.5) and (4.7) are both valid for any arbitrary dimensions of a cable or line pair. The inductance per unit length L can be determined by measuring the inductance of a section of the cable with the end short-circuited at a measurement frequency at which the wavelength is greater than the length of the cable section by a factor of at least 30 (l ≤ λ/30).

4.1.4 Power Transfer and Power Density The power P that is transferred in a cable is equal to the following in case of matching (Z 0 = Z L ): 2 V = I 2 Z0. P=V I = Z0

(4.8)

and Here, V I are rms values. This transferred power must be strictly distinguished from the power dissipation I 2 R dz that arises in the conductors and the thermal 2 loss V G dz produced in the dielectric. In the space permeated by the field, we have ¨ ¨ P= SdA = S Z dbda. (4.9)

4 Properties of Coaxial Cables and Transmission Lines …

201

· H is the temporal average of the power density (temporal average Here, Sz = E of the Poynting vector S) and dA = dbda is the surface permeated vertically by Sz . is the effective value of the electric transverse field strength and H is the E effective value of the magnetic transverse field strength. Since da = d U and H db = d E I We thus obtain ¨ P=

H db da = E

¨

dV I,

i.e. P=V I

(4.10)

4.1.5 Voltage Loading, Line Attenuation and Heat Limitation in High Power Cables In the presence of electromagnetic fields, the highest load on the dielectric occurs there where the electric field strength exhibits a maximum. In a coaxial cable, this is the case on the surface of the inner conductor. In other words, if we apply voltage V to a cable with capacitance per unit length C , the linear charge density Q = C V will arise on the inner and outer conductors. Based on the requirement that the dielectric flux that passes through a potential surface must be equal to the charge that d A = Q), we obtain the following for the coaxial cable: it encloses ( D ε0 εr E2πr z = Q z

(4.11)

and E=

CV Q 1 1 · = · 2π 0 r r 2π 0 r r

(4.12)

If the inner conductor has diameter d, then the maximum field strength E max =

2V CV 1 · = π 0 r d d ln D/d

(4.13)

will occur on its surface. Since this field strength E max may not exceed a specified maximum value taking into account the electric strength of the dielectric material, the

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permissible voltage V max is limited for a given cable according to Eq. (4.13). Thus, assuming a voltage that exhibits sinusoidal behavior versus time, the maximum power the cable can transport is given by Pmax =

2 Vmax , 2Z 0

(4.14)

if the cable is terminated into its characteristic impedance. In actual practice, such power can be transported only if the heat loss that occurs does not thermally overload the cable. o into a line with attenuation coefficient α at its If we feed the power P0 = Io · V start (z = 0), the power 0 e−αz = P0 e−2αz P(z) = I0 e−αz V

(4.15)

will pass through the line cross-section at position z. The power loss pL per unit of length that must be transferred to the environment in the form of heat is then pv = −

dP = 2α P0 e−2αz = 2α P. dz

(4.16)

For a low-loss line, the attenuation coefficient α of a line is α≈

1 1 R + G Z 0 = α R + αG Z 0 ≈ (Z 0 ) 2 Z0 2

(4.17)

as we can conclude from Eqs. (2.22) and (2.28). The component α R = 21 ZR0 is associated with the current attenuation (resistance attenuation) and the component αG = 21 G Z 0 is associated with the voltage attenuation (conductance attenuation). For the latter, we obtain in conjunction with Eq. (4.4) and the relationship for the loss angle δ G of the dielectric tan δG =

G ωC

the following relationship: √ r 1 1 1 G Z 0 = ωC tan δG · 2 2 C c √ r αG = π f tan δG . c Substituting the speed of light c by its value, we obtain

(4.18) (4.19)

4 Properties of Coaxial Cables and Transmission Lines …

αG ≈ 1.05 f

√

εr tan δG

203

10−2 m MHz

(4.20)

Although tanδ G is generally dependent on frequency, it is nearly constant in the frequency range of interest for most cable insulator materials used in engineering applications. The component associated with the resistance attenuation α R = 21 ZR0 is determined by the resistance of the forward and return conductors. If the skin effect occurs on both conductors, the current will flow only on the surface of the conductor that is facing the field; the effective conductor cross-section is equal to the circumference times the skin depth δ. For example, if a coaxial cable has an outer conductor with inner diameter D and inner conductor with diameter d and both have resistivity ρ, then the total resistance per unit length is R =

1 1 + π dδ π Dδ

= R

1 π

1 1 . + d D

(4.21)

R is the surface resistivity. At high frequencies, it is equal to ρ = πρμf δ 1 1 1 1 + αR = · · πρμf . 2Z 0 π d D R =

(4.22) (4.23)

δ G must From Eq. (4.19), we can see that α G ~ f . The prerequisite here is that tan√ be independent of frequency. According to Eq. (4.23), we then have α R ~ f . If a suitable dielectric is used, we can neglect α G with respect to α R . The specific power loss pL is then equal to pv ≈ 2α R P ∼

fP

√ [cf. Eq. (4.16)]. For pL to not exceed a certain maximum value, the product P f must be held constant. The power that can be transported thus falls off with the root of the frequency (Fig. 4.1). While it is limited by the voltage at low frequencies, the limitation at higher frequencies is tied to the maximum permissible heating. The maximum heat output that can be dissipated depends on the diameter of the cable and the thermal conductivity of its insulation. A typical value for the power dissipation for the example of an excess temperature on the inner conductor of 40 °C is about 20–100 W/m.

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voltage limit

thermal limit

Fig. 4.1 Maximum power transfer for a cable as a function of frequency (schematic). The corner frequency happens to lie at 1 MHz

4.1.6 Optimal Coaxial Cables If we have a coaxial cable and wish to increase its electric strength, decrease its attenuation or increase the power it can transfer, we could simply increase the crosssection. For reasons of economy, however, we generally attempt to obtain the best possible properties for the given outer diameter. The diameter of coaxial cables is also limited to a fraction of the wavelength, which is a severe restriction in the mmwave range and beyond. For a given dielectric, our only choice is therefore to choose a suitable value for the inner conductor diameter.

4.1.6.1

Characteristic Impedance of a Coaxial Cable

In order to proceed in this manner, we will first compute the characteristic impedance Z 0 of a coaxial cable having an outer conductor with inside diameter D and an inner conductor with diameter d (Fig. 4.2). We can obtain the capacitance per unit length C from Eq. (4.12) by computing the cable voltage V from E = E r .

D/2 V = d/2

Q Er dr = 2π 0 r C =

D/2 d/2

Q dr D = · ln r 2π 0 r d

2π 0 r Q = V ln Dd

(4.24)

4 Properties of Coaxial Cables and Transmission Lines …

205

Fig. 4.2 Cross-sectional dimensions of a coaxial cable

Applying Eq. (4.4) with c = coaxial cable:

√1 μ0 ε0

, we obtain the characteristic impedance of a

Z0 = Since

μ0 0

ln Dd μ0 · √ 0 2π r

(4.25)

= 120 π , we obtain 60 D Z 0 = √ ln r d

(4.26)

Figure 4.3 shows Z 0 as a function of D/d.

4.1.6.2

Cables with Minimum Attenuation

For transporting signals over long distances, we prefer cables with the smallest possible attenuation. We are thus interested in the value of the characteristic

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Fig. 4.3 Characteristic impedance and attenuation α R of a coaxial cable as a function of the diameter ratio D/d. In the marked tolerance zone, a single line represents a deviation of 1% from the optimum value

impedance of the line Z 0 for which the attenuation coefficient α = α R + α G is minimized for a given inside diameter D of the outer conductor.2 From Eq. (4.19), we can see that α G is independent of the line cross-section and is thus a constant for the purpose of the following observations. All that remains to investigate is α R = 21 ZR0 . Using Eqs. (4.21) and (4.26), we obtain αR =

√ R εr 1 + Dd π D120 ln Dd

substituting the field impedance ZF :

2

If the ratio D/d is constant, then the resistance attenuation decreases proportional to 1/D. However, since the cost of the cable increases roughly proportional to D, we must minimize D.

4 Properties of Coaxial Cables and Transmission Lines …

√ R r 1 + Dd αR = · ZF D ln Dd

207

with Z F =

μ0 0

(4.27)

Figure 4.3 shows the function def

fα = α R

1 + Dd ZF D √ = R εr ln Dd

versus D/d. If we define x = D/d as a single variable, then fα = (1 + x)/ln x = u(x)/v(x) exhibits a minimum if d/dx(u/v) = 0 or u/v = u (x)/v (x) or 1 + x(α) 1 = 1 ln x(α) x

(α)

As a solution to this transcendental equation, we obtain x(α) =

D d

(α)

D = 3.6 or ln d

(α)

= 1.28

such that the cable with minimal attenuation has the characteristic impedance 77 Z 0(α) = √ . r

(4.28)

If the outer and inner conductors are made of different materials (e.g. outer conductor = aluminum, inner conductor = copper), then the surface resistivity values are different. This is true also if the outer conductor is made of copper braiding instead of copper tubing. In this case, the optimal dimensions will change. In the latter case, for example, we obtain the attenuation-optimized characteristic impedance as follows: 95 Z 0(α) ≈ √ . r

4.1.6.3

Cables with Maximum Electric Strength

Section 4.1.5 already mentioned the voltage loading of a cable. Now we will investigate the characteristic impedance a cable must have in order to minimize the field strength for a given inside diameter D of the outer conductor and a given operating voltage V on the inner conductor. According to Eq. (4.13), the field strength on the inner conductor is

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M. Rudolph

E max =

CV 1 · . π 0 r d

Replacing C here according to Eq. (4.24), we obtain E max =

2V Dd def 2V = fE . D ln Dd D

(4.29)

If we again compute the extreme values as in Sect. 4.1.6.2, we obtain the following for minimal field strength on the inner conductor:

D ln d

(E)

= 1 or

D d

(E)

= 2.718, respectively

Thus, according to Eq. (4.26) we have 60 Z 0(E) = √ . r

(4.30)

Figure 4.4 shows the function

2

·D ·

Fig. 4.4 Field strength on the inner conductor and possible power transfer for a coaxial cable. In the marked tolerance zone, a single line represents a deviation of 1% from the optimum value

4 Properties of Coaxial Cables and Transmission Lines …

fE =

D d ln Dd

=

209

D E max 2V

versus D/d.

4.1.6.4

Cables with Optimal Power Transfer

Now we will determine the characteristic impedance for a cable in which the possible power transfer P is maximized for a given inside diameter D of the outer conductor and a given field strength E max on the inner conductor. Here, the cable must be terminated into its characteristic impedance and the voltage v must exhibit the temporal function v = V sin ωt. The transferred power is then P=

V2 . 2Z 0

(4.31)

If the dielectric may be loaded maximally with field strength E max , then the maximum permissible cable voltage according to Eq. (4.29) is Vmax =

E max D ln Dd · D 2 d

(4.32)

In combination with Eq. (4.26), we obtain √ 2 2 E max D 2 r ln Dd Vmax P= · 2 . = 2Z 0 240 2 Dd

(4.33)

Performing the extreme value calculation described in Sect. 4.1.6.2, we find the cable with the best power transfer as follows: ln

D d

= P

1 or 2

D d

=

√ e = 1.65, respectively

P

Thus, the characteristic impedance of the cable with the best power transfer is 30 Z 0(P) = √ . r

(4.34)

Figure 4.4 plots the function f P = 100P ·

ln Dd 240 = 100 √ 2 . 2 D2 E max r 2 D d

versus D/d.

(4.35)

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M. Rudolph

Table 4.1 Optimal cables and lines Coaxial cable

Line pair Open

Shielded

d d D D

d D

Da √

√

D/d

Z 0 · εr

D/d

Z 0 · εr

D/d

D/Da

√ Z 0 · εr

Minimum attenuation

3.6

77

2.276

175.6

2.47

0.428

147

Maximum electric strength

2.72

60

2.932

208.6

2.69

0.48975

142

Maximum power

1.65

30

2.146

167.7

1.85

0.4935

94.7

Compromise values

2.3

50

2.42

185

2.50

0.490

133

Typical coaxial cables used in practice have characteristic impedances of 50 . If the space between the inner and outer conductors is filled entirely √ with Teflon (εr = 2.05) or polystyrene (εr = 2.5), then for a 50 cable Z 0 εr is equal to 72.5 or 79 , i.e. a cable of this sort is nearly optimal in terms of its attenuation. The optimal properties of open and shielded balanced line pairs can be derived in the same manner as for coaxial cables. Table 4.1 lists diameter ratios and characteristic impedances for minimum attenuation, maximum electric strength and maximum power transfer. The derivations of these quantities can be found in the 1st and 2nd editions of this “Lehrbuch der Hochfrequenztechnik” (in German).

4.2 Striplines 4.2.1 Overview of Different Designs and Applications Striplines in their various embodiments represent the main type of transmission line used in VHF and microwave systems as well as in high-speed digital circuits except

4 Properties of Coaxial Cables and Transmission Lines …

211

in cases where special requirements such as low attenuation or high power transfer necessitate usage of coaxial lines or waveguides. Table 4.2 compares the properties of striplines, coaxial lines and waveguides. Here, we use the term stripline as a general designation for all waveguides in which at least one conductor is in the form of a strip and the other conductors are at least planar (disregarding the panels of the enclosure). Striplines are situated on a layer known as the substrate. Figure 4.5 shows the most important types of striplines. All of the types represent lines with defined characteristic impedances and propagation Table 4.2 Comparison of properties of important types of lines (according to [1]) Property

Stripline

Coaxial line

Waveguide

Line attenuation

High

Medium

Low

Resonator quality factor

Low

Medium

High

Power transfer

Low

Medium

High

Decoupling of adjacent circuit elements

Low

Very high

Very high

Bandwidth

Large

Large

Small

Miniaturization

Outstanding Unsatisfactory Unsatisfactory

Volume, weight

Low

Production: passive circuits

Very simple Simple

Simple

Simple

Possible

High

High

Integration of • Semiconductor elements

Possible

• Lumped passive elements

Very good

Possible

Possible

• Ferrite components (e.g. in circulator)

Very good

Moderate

Good

• Dielectric components (e.g. as dielectric oscillator)

Very good

Moderate

Good

a

b

c

i e

g

d

j

k

f

h

Fig. 4.5 a–k Cross-section of striplines, coplanar waveguides, slotlines and finlines. a Stripline (triplateline); b suspended substrate line; c microstrip; d double-band line; e coplanar waveguide; f coplanar strip; g unbalanced coplanar waveguide; h slotline; i–k finlines: i unilateral, j bilateral, k antipodal

212

M. Rudolph

delays. However, unlike coaxial lines and waveguides, they can be produced using the same simple photoetching, thin-film or thick-film processes that are applied to produce printed circuit boards and hybrid circuits for lower frequencies. The stripline types shown in Fig. 4.5 make it easy to produce integrated microwave circuits. Filters, directional couplers, transformers, circulators and similar passive devices are constructed using suitably dimensioned striplines on an insulated substrate as the base and supplemented with lumped passive components and semiconductors to form so-called hybrid integrated microwave circuits. Alternatively, semiconductor substrates can be used on which the active components and lumped passive components are integrated on-chip as so-called monolithic integrated microwave circuits (MMICs). In digital circuit engineering, striplines are basically used to connect logic circuits on printed circuit boards with the proper characteristic impedance and to connect different printed circuit boards. Due to the many lines that are required, it is common to arrange multiple layers of lines over one another on multilayer boards. Despite using the same production technologies, the stripline types shown in Fig. 4.5 have different application areas. Stripline (triplateline) as shown in Fig. 4.5a is used for modules consisting largely of line components such as filters, couplers and tees. This line type was commonly used in the early days of integrated microwave circuits and is characterized by low dispersion and suppression of radiation losses due to its shielded design. However, hybrid integration of lumped elements is difficult due to the complete dielectric filling. The suspended substrate line (Brenner line [2, 3]) shown in Fig. 4.5b can be used with identical dimensions to achieve higher quality factors and higher characteristic impedances than the stripline in Fig. 4.5a or the microstrip in Fig. 4.5c. However, the suspended substrate line is more difficult to manufacture especially if ground connections are required. The microstrip in Fig. 4.5c is an open stripline that is unbalanced to ground. This is the main type of line in integrated microwave circuits. Open lines, parallel branches and components in series with the line are very easy to implement. The double-band line (Fig. 4.5d) is created by mirroring a microstrip on the ground plane; it is normally used only in conjunction with other line types. Coplanar waveguides as shown in Fig. 4.5e–g allow production of lines on substrates that are metalized on one side. They favorably complement the properties of microstrips, e.g. they allow easy implementation of short-circuits, high-impedance lines (Z 0 > 100 ) and components in parallel with the line. For the striplines shown in Fig. 4.5a–g, the characteristic impedance and line propagation time exhibit low frequency dependency compared to the lines in Fig. 4.5i–k, which means these lines are also suitable for distortion-free transmission of signals with DC components, as generally required in digital circuits, for example. In contrast, the finlines in Fig. 4.5i–k transport waves for which the energy of the wave is concentrated in the slot only at sufficiently high frequencies. At frequencies f → 0, the characteristic impedance goes to zero and the line propagation time changes noticeably such that low-frequency broadband signals cannot be transported distortion-free via finlines. Accordingly, slotlines are used customarily in the microwave range in conjunction with microstrip or coplanar waveguides, allowing

4 Properties of Coaxial Cables and Transmission Lines …

213

very easy implementation of short-circuits, series branches and components diagonal to the line. There is an extensive body of literature on the different types of striplines as well as the related fields, line constants and applications [1, 3, 4].

4.2.2 Field Types in Striplines Fields in striplines, and as a consequence, the line properties, are much more difficult to calculate than the fileds of coaxial cables. The easiest case is the stripline shown in Fig. 4.5a, that could be understood approximately as a coaxial line of rectangular cross section and a flat center conductor. All other types are unsymmetric and two types of dielectric are to be considered: substrate and air. Slotlines and finlines are the most involved structures in this respect as they guide an electromagnetic wave in the gap between grounded metal plates. These types of lines are therefore not able to carry DC, and exhibit a so-called cut-off frequency, which denotes the lowest frequency where a wave propagation is possible. This example of a slotline illustrates that a true understanding of these types of transmission lines requires concepts that are derived from the analysis of the electromagnetic fields. These concepts will be derived and discussed in detail at the example of rectangular waveguides in Chap. 5. It will be derived that waves that have field components parallel to the propagation direction are dispersive, i.e. that their properties including characteristic impedance, phase and group velocity change over frequency. Finally, there is an infinite number of possible solutions when solving Maxwell’s equations, which correspond to different forms of fields and are called modes.3 Dispersion on striplines is, however, not dominant if the physical dimensions are chosen properly for the envisioned frequency range. Therefore striplines are already introduced now, and the reader is referred to Chap. 5 for a detailed fieldoriented discussion of electromagnetic wave propagation. Different field types are propagated on the different stripline types shown in Fig. 4.5. In the following discussion, we will assume that the substrate material is isotropic, homogeneous and purely dielectric (μr = 1) such that it can be fully characterized in electrical terms by its relative permittivity εr . The stripline has a homogeneous dielectric and thus transports Lecher waves as the fundamental wave (see Sect. 5.2) as is the case, for example, with a coaxial line. (At high frequencies, Lecher waves correspond to transverse electromagnetic (TEM) waves but they take into account the conductor losses.) In the lossless case, the stripline thus transports TEM waves. The suspended substrate line, microstrip, double-band line and various coplanar waveguides in Fig. 4.5b–g have a layered dielectric in the field-permeated space and transport quasi-TEM waves as their fundamental wave, i.e. waves for which the longitudinal components of the electric and magnetic field strengths are 3

The EH 0 and EH 1 modes that will be mentioned in the following, refer to a field type where E and H fields do have components in the transmission direction.

214

M. Rudolph

still present even in the lossless case but are negligible at sufficiently low frequencies with respect to the corresponding transverse components. Finlines (Fig. 4.5i–k) also have a layered dielectric in the field-permeated space in addition to strip conductors connected to one another via the surrounding shielding, and thus transport—instead of the quasi-TEM wave that cannot exist on these lines— quasi-H waves as the lowest field type, i.e. waves for which the electric longitudinal field strength is negligible but the magnetic longitudinal field strength is not. All of the discussed field types are uniquely associated with a complex propagation coefficient γ = α + jβ = α + jω/vph

(4.36)

For lines with a layered dielectric, we introduce as a useful auxiliary quantity the effective relative permittivity εr eff defined by 2 εr eff = c0 /vph = (λ0 /λ)2

(4.37)

(c0 speed of light, vph phase velocity, λ0 free-space wavelength, λ wavelength on line). This definition of the effective relative permittivity implies that the wave on the line with a layered dielectric propagates with the same phase velocity as a TEM wave in a homogeneous dielectric with relative permittivity εr eff . εr eff is less than εr of the substrate material since the field is located not only in the substrate but also partially in air (εr = 1). The phase velocity is thus √ vph = c0 / εr eff ,

(4.38)

β = ω/vph = ω εr eff /c0 ,

(4.39)

and the phase constant is

√ such that the wavelength on the line is reduced by the factor εr e f f with respect to the free-space wavelength. In quasi-TEM wave lines, the phase velocity vph and thus εr eff are dependent on the line dimensions and to a slight extent also on the frequency. This frequency dependency is known as dispersion. By solving the wave equation, e.g. for the case of the microstrip [5–7], it is revealed in agreement with measurement results [8, √ 9] that for increasing frequency, the phase velocity vph tends towards c0 / εr and thus—because of (4.37)—εr eff also tends towards εr of the substrate material since the field is increasingly concentrated in the substrate at higher frequencies. If we take into account this dispersion, i.e. this frequency dependency of the phase velocity, in the development of a line circuit, we can unambiguously operate striplines to the lower limit frequency of the next higher field type.

4 Properties of Coaxial Cables and Transmission Lines …

215

Radiation

Fig. 4.6 Effective relative permittivity for the fundamental wave (quasi-TEM wave) and higher field types of a microstrip (according to [3])

As an example of this, Fig. 4.6 illustrates the frequency dependency of εr eff for the fundamental wave (HE0 or quasi-TEM wave) and the first two higher field types HE 1 wave, HE 2 wave on a microstrip [3]. Apart from the discrete field types, a continuous radiation spectrum also exists with the microstrip as an open line which is excited by discontinuities and causes energy loss due to radiation. The quasi-H waves of the slotlines exhibit a basically similar dependency of the phase velocity (and thus the effective relative permittivity) on the frequency. However, the frequency dependency is more pronounced than is the case for quasiTEM waves.

4.2.3 Quasi-static Line Constants While the complex propagation constants and the fields of the fundamental waves of striplines are clearly defined at any arbitrary frequency, this is true of the remaining line constants only in the quasi-static case, i.e. at frequencies that are so low that the

216

M. Rudolph

longitudinal components of the fields are negligible with respect to the corresponding transverse components. In this case, line wave approximations for the field provide good approximate values for the line constants. Given the static effective relative permittivity (see Fig. 4.6) εr eff,stat = εr eff ( f = 0) and the characteristic impedance Z 01 of a line with the same conductor arrangement but without a substrate, i.e. with εr = 1, we obtain the following TEM wave approximations for the remaining line constants which are valid in the quasi-static case: characteristic impedance √ Z 0 = Z 01 / r eff,stat , r eff,stat = (Z 01 /Z 0 )2 ,

(4.40)

inductance per unit length L = Z 01 /c0 = Z 0 /vph,stat ,

(4.41)

capacitance per unit length C = r eff,stat /(c0 Z 01 ) = 1/ vph,stat · Z 0 ,

(4.42)

At frequencies in the lower microwave range, this quasi-static analysis of striplines which transport quasi-TEM waves as their field type generally provides sufficient accuracy. At higher frequencies and for lines such as slotline and finline which transport quasi-H waves, however, it is necessary to solve the wave equation. An analysis of this sort provides, in addition to the frequency dependency of the phase velocity and thus of εr eff , also the frequency dependency of the characteristic impedance which can be defined from the voltage and current, voltage and power or current and power. (In contrast to the static case, these three definitions provide slightly different characteristic impedance values; see, e.g. [3].) Since striplines are typically used in high to extremely high frequency ranges, we will limit our focus hereafter to the case of small losses and nearly total current suppression in the conductors due to the skin effect, i.e. all conductors are at least three skin depths δ thick—a condition which almost always holds in practice. In this manner, we can calculate satisfactory approximate values for the characteristic impedance, inductance per unit length and capacitance per unit length from the fields in the lossless case along with the resistance per unit length and the power loss using the power-loss method based on the assumption of the surface resistivity R for all conductors R =

π f μρ

(4.43)

4 Properties of Coaxial Cables and Transmission Lines …

217

where f is the frequency, μ = μ0 μr the permeability and ρ the resistivity of the conductors.

4.2.4 Stripline (Triplateline) In the case of stripline (also known as triplate line), the strip conductor (width w, thickness t) is arranged as shown in Fig. 4.7 in the center between two ground planes in a homogeneous dielectric with relative permittivity εr . For practical reasons, we typically choose the inside breadth b to be large with respect to the inside height h and the strip width w, thereby making it unimportant whether the side panels shown in Fig. 4.7 are present or not. For a relative line breadth b/h > 2 + w/h, the characteristic impedance changes remain under 2% compared to a line with infinite breadth (b/h → ∞), allowing us to restrict our focus hereafter to this case (b > 2 h + w). For extremely high-frequency applications, soft organic materials such as tetrafluoroethylene (Teflon) (εr = 2.05) or polyethylene (εr = 2.32) are typically used as the dielectric so the space between the conductors can be completely filled. In the lossless case, the stripline transports a TEM wave with a frequency√ independent phase velocity vph = vph.stat = c0 / εr and a frequency-independent √ characteristic impedance Z 0 = Z0 stat = Z 01 / εr (see above); static analysis is therefore adequate to characterize the fundamental wave. For line breadths b → ∞, the characteristic impedance Z 0 can be calculated exactly (even for conductor thicknesses t = 0) by means of conformal mapping using elliptical integrals. In the special case of strip conductor thickness t = 0, we obtain the following for the characteristic impedance according to Cohn [10]: ZF K (k) , with Z F = Z0 = √ · 4 r K (k )

μ0 ≈ 120 π 0

(4.44)

where k = 1/ cosh

πw , 2h

(4.45)

w h/2

t

h/2 b Fig. 4.7 Cross-section of stripline (triplateline)

218

M. Rudolph

k =

1 − k 2 = tanh

πw , 2h

(4.46)

K(k) is the complete elliptical integral of the first kind of modulus k [11]. Simple approximations for Eq. (4.44) have been given by Hilberg [12]. For b h and t/h ≤ 0.25, the following holds approximately with errors < 1.2% for w/(h − t) ≥ 0.35 [13]: Z0 =

√ 94.25 · (1 − t/ h)/ r w h

+

2 π

ln

2−t/ h 1−t/ h

−

t πh

ln

(4.47)

t (2−t/ h) h(1−t/ h)2

and for w/(h − t) ≤ 0.35:

Z0 60 t 8h = √ · ln / 1+ (1 + ln 4π w/t) + 0.51(t/w)2 r πw πw

(4.48)

√ The characteristic impedances Z 01 = Z 0 · εr are illustrated in Fig. 4.8. Based on the assumptions above (low losses, high-frequency case), the conductor attenuation coefficient α ρ can be calculated using Wheeler’s incremental inductance rule [14] from the relationships for the characteristic impedance, Eqs. (4.47) and (4.48), and the surface resistivity R (Eq. (4.43)) of the strip conductor and shielding. For broad strip conductors, i.e. w/(h − t) ≥ 0.35, we obtain [3, 13] √ 2.02 × 10−6 · r · Z0 · f /GHz · ρ/ρcu αρ = dB/cm h/cm 2wh h(h + t) 2h − t h + , + ln × h−t (h − t)2 π(h − t)2 t

(4.49)

and for narrow strip conductors, i.e. w/(h − t) ≤ 0.35 √ 0.0114 · f /GHz · ρ/ρcu αρ = dB/cm Z 0 / · h/cm 0.65 1.65 t + 0.5 ln 4πw + 0.1947 wt − 0.0767 wt 2h 0.5 + 2πw π t · × 1+ 1.65 w 1 + t 1 + ln 4πw + 0.236 t πw

t

w

(4.50) Here, ρ is the resistivity of the conductor and shielding material and ρcu = 17.2n · m is the resistivity of copper. Figure 4.9 illustrates the reference conductor attenuation coefficient αe∗ = √

αe · h/cm . f /GHz · εr · /cu

4 Properties of Coaxial Cables and Transmission Lines …

219

Fig. 4.8 Characteristic impedance of stripline (according to [3]). Parameter: Relative conductor thickness t/h

The conductor attenuation coefficient α ρ follows from the reference conductor attenuation coefficient αρ∗ as √ α = α∗ ·

f /GHz · εr · h/cm

√ /cu

;

√ For the given assumptions, it grows proportional to f . If the dielectric also exhibits losses characterized by its loss factor tan δε , the total attenuation coefficient α of the line ensues from the conductor attenuation coefficient α ρ and loss factor tan δ ε as follows: α = α +

β tan δε . 2

(4.51)

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M. Rudolph

Fig. 4.9 Reduced conductor attenuation coefficient αρ∗ for stripline [3]

For conventional substrate materials, tan δε is constant in the microwave range and is on the order of magnitude of 10–3 . Thus, the dielectric losses increase proportional to frequency in accordance with Eq. (4.51), but the conductor losses generally predominate. We can obtain higher characteristic impedances and lower attenuations compared to the stripline in Fig. 4.7 if we do not embed the strip conductor in a full dielectric; instead, we arrange it as shown in Fig. 4.10 on a thin substrate roughly in the center of the line. The suspended substrate line (also Brenner line) [2, 3, 15–18] that is created in this manner makes it possible to even realize lines having characteristic impedances of over 100 in a cost-effective manner. Due to the substrate’s low share of the volume permeated by the field, a low value of εr eff is obtained compared to the value of εr for the substrate.

4 Properties of Coaxial Cables and Transmission Lines …

s

221

w

s

h1

h

h2 b Fig. 4.10 Cross-section of suspended substrate line

w er = 1.0

t

er = 9.7

h t

Fig. 4.11 Cross-section of microstrip. Typical dimensions for Z 0 = 50 : w = 610 μm, h = 635 μm, t = 5 μm, substrate Al2 O3 (99.5%), εr = 9.7

4.2.5 Microstrip The microstrip in Fig. 4.11 is an unbalanced, open stripline. It is created from the suspended substrate line in Fig. 4.10 by taking away the upper and side panels and eliminating the air space under the substrate (h2 → h/2, b > 4 h + w).4 For typical substrates with a high value of εr such as Al2 O3 ceramic, the bulk of the electric field lines and thus also the bulk of the transported power flows in the substrate under the strip conductor. With this line type, there is a risk of occurrence of transverse leaky modes (see Sect. 5.4.3.3).

4.2.5.1

Quasi-static Line Constants for Microstrip

As mentioned above, the microstrip transports as its fundamental wave a quasi-TEM wave that contains all six field components. At low frequencies, definable for the microstrip based on the requirement that √ w, h < λ0 / 40 εr ,

4

(4.52)

In practical terms, the side and upper panels have no influence on the line properties if the following conditions are fulfilled: normalized line breadth b/h > 4 + w/h and normalized cover height h1 /h > 4 [19]. In this case, the characteristic impedance changes remain below 2% compared to the microstrip without panels and cover.

222

M. Rudolph

i.e. the lateral dimensions are small with respect to quarter wavelength of line, the longitudinal components of the fields are negligible and line wave approximations for the field thus provide good estimates of the line constants. (For the line shown in Fig. 4.11, this requirement (4.52) is met up to frequencies of f = 3.8 GHz.) At significantly higher frequencies, the longitudinal components of the electric and magnetic field strengths can no longer be neglected such that solutions of the wave equation are required for more precise analysis. Due to the layered dielectric, even in the static case there is no exact solution in closed form for the line constants for the microstrip. However, there are many techniques that provide solutions such as modified conformal mapping [20–22], the relaxation method [23], the variation method [24], the method of Green’s functions [25, 26], the subsurface method [27], the method of moments [28] and the method of straight lines [29]. For computer-aided design and optimization, solutions in closed form are preferred. The results of Wheeler [20, 21] are widely used in this context. For wide lines (w/h > 1) with very thin, low-loss conductive tracks (t/w 0.1), he found the following: For the characteristic impedance: Z0 =

w 2h

√ 188.5/ r w r +1 + 0.441 + 2π + 0.94 + 1.451 + ln 2h r

0.082(r −1) r2

(4.53)

For narrow lines (w/h < 1) according to [20]: Z0 60 1 w 2 1 r − 1 0.2416 8h = + 0.4516 + · ln − w 32 h 2 r + 1 r r +1

(4.54)

2

The errors in these characteristic impedance formulae have maximum values of approx. 2%. The other line constants can be computed subsequently with Eqs. (4.39) to (4.42). More exact analytical expressions for the characteristic impedance Z 1 = Z 0 (εr = 1) without the dielectric and effective relative permittivity εr eff have been derived by Hammerstad and Jensen [30]. By approximating the exact results of Magnus and Oberhettinger [31] as well as other authors that can be derived with conformal mapping for the case without the substrate (εr = 1), they determined the following for the characteristic impedance Z 01 with an infinitely thin strip conductor: ⎡ ⎤ 2 f Z 01 (w/ h) 2h 1 ⎦ = 60 ln⎣ + 1+ w/ h w

(4.55)

The adaptation function is f 1 (w/ h) = 6 + (2π − 6)e−(30.666h/w)

0.7528

(4.56)

4 Properties of Coaxial Cables and Transmission Lines …

223

The inaccuracy of this approximation for Z 01 is 1), a part of the field that is no longer negligible is located in the air space below the substrate, resulting in lower effective relative permittivity and higher characteristic impedance compared to thick substrates. Figures 4.23 and 4.24 show families of curves for the characteristic impedance and εr eff that were computed for Al2 O3 ceramic (εr = 9.8) as the substrate material. For substrates with a different value of εr , Eqs. (4.78) and (4.79) can be used to provide an approximate conversion. We can obtain formal approximate solutions for the static effective relative permittivity based on [44] from Eq. (4.80) where 600 Ω 550

h/d = 0

500

0.05 0.1 0.2

450

0.4 400

0.8...1 ∞

Z0

350 300 250 200 150 100 50 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

0.95

1.00

s/d

Fig. 4.23 Characteristic impedance Z 0 of coplanar strips as a function of the gap width s referred to d = s + 2w. εr = 9.8. Conductor thickness t → 0 (in part according to [3])

4 Properties of Coaxial Cables and Transmission Lines …

237

6.0 h/d = 5.5

1

0.8

5.0 0.6 4.5 0.4

er eff,stat

4.0

0.3 0.2

3.5 0.15 3.0 0.1 2.5 2.0 h/d = 0.05 1.5 1.0 0

0.1

0.2

0.3

0.4 0.5 s/d

0.6

0.7

0.8

0.9

1.0

Fig. 4.24 Static effective relative permittivity εr eff.stat of coplanar strips for t = 0, εr = 9.8 (according to [3])

tanh[π s/(4h)] , tanh[π d/(4h)] k1 = 1 − k12 ,

k1 =

(4.90) (4.91)

k follows from Eq. (4.86) and k from Eq. (4.87), resulting in the characteristic impedance of the coplanar strip according to Eq. (4.85). Like with the coplanar waveguide, the conductor path thickness t can be taken into account by replacing the conductor path width w by the effective conductor path width weff = w + w and the slot width s by seff = s − w [4]. However, this correction is generally not required for typical metallization thicknesses. The formulae for the characteristic impedance Z 0 and effective relative permittivity εr eff.stat provide highly accurate approximations up to the frequency limit estimated by Eq. (4.84). At higher frequencies, εr eff exhibits a slight increase while the characteristic impedance calculated from the power and longitudinal current falls off slightly [47, 48].

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M. Rudolph

The resistance attenuation of coplanar strips is smaller for high-impedance lines (Z 0 > 100 ) compared to microstrips with similar dimensions; however, it is significantly higher for low-impedance lines (Z 0 < 50 ). Like with all striplines, the conductor losses generally dominate the dielectric losses. The conductor attenuation coefficient can be calculated like for the coplanar waveguide, yielding [3, 45] 8.68R αρ = dB 4Z 0 d K 2 (k ) · 1 − (s/d)2

4π s(1 − s/d) 4π d(1 − s/d) 2d π + ln + 2 π + ln × s t (1 + s/d) t (1 + s/d)

(4.92)

Here, R is the surface resistivity according to Eq. (4.43), Z 0 is the charactercomplete elliptical integral of the first kind of istic impedance, and K(k ) is the the complementary modulus k = 1 − (s/d)2 . As before, the prerequisite is total current suppression (skin effect) in the conductors. The conductor attenuation coef√ ficient increases according to Eq. (4.92) proportional to f . At higher frequencies, additional losses arise as before due to cross-currents in the conductors. Like for the coplanar waveguide, the conductor attenuation coefficient can be deduced from Fig. 4.21. The dielectric losses follow in turn from Eq. (4.67) or (4.68) and increase proportional to f . For the coplanar strip, the effective relative permittivity follows from Eq. (4.80) in conjunction with Eqs. (4.86), (4.87), (4.90) and (4.91) (or (4.72) in the case of thick substrates). The total attenuation coefficient follows from Eq. (4.69).

4.2.8 Slotlines The slotline consists of a dielectric substrate with metallization on one side which is interrupted by a slot (Fig. 4.5h). We can imagine a slotline as a microstrip (Fig. 4.1c) in which the unmetallized regions of the substrate surface have been metalized and vice versa. Originally proposed in 1968 [49], the slotline never became as important as the microstrip. Reasons include the strong dispersion of the phase coefficient and characteristic impedance, the relatively large transverse field elongation and the junction required for coaxial test equipment. Nevertheless, the slotline does have applications in conjunction with the microstrip in balanced mixers, PIN diode attenuators and directional couplers. The field pattern of the fundamental type on the slotline (see Fig. 4.25) differs substantially from the quasi-TEM type on the microstrip. While the electric field primarily contains transverse components, there exists a magnetic z component on the same order of magnitude as the transverse H field. In a simplified representation, the field pattern resembles, assuming the x and y coordinates are swapped, that of the H 10 wave in the rectangular waveguide. Due to the circular polarization of the

4 Properties of Coaxial Cables and Transmission Lines …

239

y H H E x E

er

l/2 z

H E l/2

S

Fig. 4.25 Qualitative behavior of the electric and magnetic field lines of the fundamental type on a slotline

magnetic field, non-reciprocal components (unidirectional lines, circulators) can also be realized using a premagnetized ferromagnetic substrate material. Like the microstrip, the transmission properties of the slotline can be characterized using the effective relative permittivity εr eff = (ß/k 0 )2 = (c/vph )2 and the characteristic impedance Z 0 = V 2 /(2P). The voltage V is defined as the integral over the electric field strength along the shortest path between the slot edges. P is the transported power. Among the different computational techniques [19, 50] used to determine εr eff and Z 0 , we should mention the transverse resonance technique applied in [49]. Here, a standing wave on the line is assumed. At the nodes where the electric transverse field strength tends towards zero, electrically conductive panels can be arranged in the transverse plane without disrupting the fields. Two such panels arranged parallel to the line with sufficient spacing from the slot result in a rectangular waveguide which is provided with a dielectrically loaded slot cover plate. This arrangement can be calculated by developing the fields based on waveguide wave types. Figure 4.26 shows the effective relative permittivity εr eff as a function of h/λ0 (or of frequency for a typical substrate thickness h). εr eff grows as the frequency increases and the slot width s decreases. As a coarse approximation for s h, we can assume that εr eff ≈ (εr + 1)/2 according to [51]. Figure 4.27 shows the characteristic

240

M. Rudolph 5.0 0.06 0.04 s/h=0.02

4.5

0.1 0.2 0.4

4.0

0.6 0.8 1

er eff

3.5 1.4 1.6 2 1.8

1.2

3.0

2.5

t=0

s

h

2.0

er = 9.7

1.5 0

0

0.005 0.010 0.015

2

4

6

0.020 0.025 0.030 0.035 h/l

8

10

12

14

0.040

16 GHz18

f

Fig. 4.26 Effective relative permittivity εr eff (for the fundamental type on a slotline) as a function of substrate height h referred to λ0 . Frequency scale for h = 0.635 mm (25 mil). Parameter: Slot width s referred to h (according to [3])

impedance Z 0 of the slotline according to the voltage–power definition as a function of the same variables. Since the field is concentrated in the slot region as the frequency increases, assuming the power is constant the voltage grows over the slot such that the characteristic impedance rises. Further data for additional values of εr can be found in [52].

4.3 Coupled TEM-Wave Lines 4.3.1 Line Differential Equations Multiconductor arrangements occur on the one hand in telecommunications and data communications systems where multiple lines are commonly routed close to

4 Properties of Coaxial Cables and Transmission Lines …

241

200 s/h = 2.0 1.8 1.6 1.4 1.2 1.0

160 140 120

Z0

100 80 60 40

0.2 0.1 0.06 0.02

20

0

0.005

0.01

0.8 0.6 0.4 0.02

0.03

0.04

h /l0

Fig. 4.27 Characteristic impedance Z 0 of the slotline as a function of h/λ0 (or frequency) for h = 0.635 mm. The parameter is s/h. Substrate material εr = 9.7 (in part according to [3])

one another along some path. Here, mutual influences between lines (e.g. “crosstalk” [53]) cause the most significant disruption. Mitigation of these effects is the subject of the field of electromagnetic compatibility. On the other hand, multiconductor systems are important components in RF engineering. Here, the mutual coupling properties of lines are exploited, e.g. as directional couplers, filters, phase shifters, power dividers and balanced-to-unbalanced transformers. We will introduce coupled line properties in this section, laying the basis for the discussion of directional couplers. In our approach to the line differential equations for coupled TEM wave lines, we will assume we have two coupled lines and the applicable equivalent circuit (Fig. 4.28). Analogous to the procedure we applied with the single line, we obtain the following differential equations for two coupled lines (steady state, complex calculation) [54]: d VI = jω L 11 I I + L 12 I I I dz d VI I = jω L 12 I I + L 22 I I I − dz −

d II = jω c10 VI − c12 VI + c12 + c12 VI I (VI − VI I ) = jω c10 dz d II I VI I − c12 VI I + c12 + c12 VI − = jω c20 (VI I − VI ) = jω c20 dz

(4.93)

−

(4.94)

242

M. Rudolph

a

z

l

l1

II(z)

V1

1

l2

VI(z)

l3

2

Line I

l4

III(z)

V3

3

Line II

VII(z)

4

z

z/I = 0

b 2

z

I I(z

)

L' 11

z

+ I I(z)

1

I I(z)

VI(z) + VI(z)

4

C'10 z z C' 12

L' 22 ) I II(z

V4

z/I = 1

L'12 z

VII(z)

V2

z

)+ I II(z

) I II(z

VII(z) + VII(z)

3

C'20 z VII(z)

Fig. 4.28 a, b Two coupled lines. a Basic concept; b equivalent circuit for a section of length z

The line differential equations for n coupled lines can be derived in an analogous manner. In matrix form, we then have −

d V = jωL I dz

(4.95a)

−

d I = jωC V dz

(4.95b)

with the voltage vector V and the current vector I

4 Properties of Coaxial Cables and Transmission Lines …

⎛

⎞ VI ⎜ VI I ⎟ ⎜ ⎟ V = ⎜ . ⎟, ⎝ .. ⎠

243

⎛

⎞ II ⎜ II I ⎟ ⎜ ⎟ I = ⎜ . ⎟, ⎝ .. ⎠

Vn

(4.96)

In

as well as the matrix L of the inductance per unit length coefficients ⎞ L 11 L 12 . . . L 1n ⎜ L L ... L ⎟ 21 22 2n ⎟ L = ⎜ ⎠ ⎝ ... L n1 L n2 . . . L nn ⎛

(4.97)

and the matrix C of the capacitance per unit length coefficients ⎞ C 11 C 12 . . . C 1n ⎜ C C . . . C ⎟ 21 22 2n ⎟ C = ⎜ ⎠ ⎝ ... C n1 C n2 . . . C nn ⎛ ⎞ c10 + c12 + c13 + · · · −c −c1n 12 . . . ⎜ −c ⎟ c20 +c21 +c23 + · · · . . . −c2n 21 ⎟, =⎜ ⎝ ⎠ ··· −cn2 . . . cn0 +cn1 +cn2 + · · · −cn1 (4.98) ⎛

where c ik are the partial capacitance per unit length values in the equivalent circuit (cf. Fig. 4.28b). We have L ik = L ki and C ik = C ki (or c ik = c ki ).

4.3.2 Even- and Odd-Mode Excitation In addition to the model for coupled lines based on couplings associated with capacitance per unit length and inductance per unit length coefficients, there exists a model based on even- and odd-mode excitation that leads to the same computational results [55, 56]. The following definition of the even- and odd-mode voltages V even and V odd as well as the even- and odd-mode currents I even and I odd forms the basis for the treatment of three-conductor systems in [3, 55–58] (Fig. 4.29): Even-mode system Veven =

1 (V1 + V2 ) 2

244

a

M. Rudolph l2

l1

C'20

C'10

V2

V1

2C'12 2C'12

f

S

b

e

c

d S1

l1

S2

l1

l2

l2

l1 + l 2

Fig. 4.29 a–f Three-conductor system. a Schematic representation with directional arrows; b stripline; c fields in odd-mode operation; d fields in even-mode operation; e microstrip; f partial capacitances per unit length and symmetry plane S; S 1 “Electrical wall” (χ = ∞), S 2 “Magnetic wall” (μ = ∞)

Ieven =

1 (I1 + I2 ) 2

Odd-mode system 1 (V1 − V2 ) 2 1 = (I1 − I2 ) 2

Vodd = Iodd

(4.99)

Based on these relationships and the differential equations (4.95), we obtain the following differential equations for even- and odd-mode operation in the case of electrically equivalent lines (C 11 = C 22 = C; L 11 = L 22 = L , decoupling conditions):

4 Properties of Coaxial Cables and Transmission Lines …

245

d Veven d Vodd = jω L + L 12 Ieven , − = jω L − L 12 Iodd dz dz d Ieven d Iodd Vodd − = jω C + C12 Veven , − = jω C − C12 dz dz

−

(4.100)

Unlike Eqs. (4.95), these two systems of differential equations are decoupled from one another. The problem of analyzing doubly coupled lines is thus reduced to separate consideration of two operating cases. Analogous to the single line, it is conventional in publications [3, 55–58] to define an even-mode characteristic impedance Z 0e (“even”) and an odd-mode characteristic impedance Z 0o (“odd”) in order to characterize the three-conductor system: Z 0e =

L + L 12 , C + C12

Z 0o =

L − L 12 C − C12

(4.101)

For striplines and microstrips5 (Fig. 4.29b, e), these characteristic impedances are specified as a function of the geometric dimensions in [59].

4.3.3 Chain Matrix The chain matrix of the coupled line will be derived in the following. The derivation will also establish a number of important properties of the coupled line. It will be shown that the magnetic and electric coupling factors are equal, and we will see that the even and odd-mode characteristic impedances can be derived from the characteristic impedance of the single line and the coupling factor. The derivation starts by differentiating the line differential Eq. (4.95a) and plugging in Eq. (4.95b). We obtain −

d2V = ω2 L C V . dz 2

(4.102)

For TEM waves, the propagation coefficient γ for all n conductors is equal to ω γ = jβ = j , v

(4.103)

v = phase velocity. Assuming a forward TEM wave along the n conductors, we obtain for each component of the voltage vector 5

No TEM waves are propagated on microstrips (cf. e.g. [5]). However, in the lower GHz range we can apply the TEM wave approach as an approximation in case of conventional dimensions and substrates [23]. For an improved approximation, we can take into the account the different phase velocities of the even- and odd-mode wave in coupled microstrips [19, 23].

246

M. Rudolph ω

Vi (z) = Vi e− j v z

(4.104)

From Eq. (4.102), it then follows that 0=−

ω2 V + ω2 L C V , v2

or multiplied by V T from the left 1 T V V = V T L C V , v2

(4.105)

These expressions are identical if the following holds: 1 E = L C v2

(4.106)

V T is the transposed vector of V and E is the identity matrix. From Eq. (4.102), it follows in conjunction with (4.106) that ω2 d2 V + V =0 dz 2 v2

(4.107)

or expressed component-by-component (v = 1, …, n): ω2 d2 Vv + 2 Vv = 0 2 dz v

(4.108)

The solution to this homogeneous differential equation of the 2nd order is analogous to the single line: V = V p e−γ z + V r e+γ z

(4.109)

where γ =j

ω = jβ. v

For the current, it holds correspondingly that I = vC V p e−γ z + V r e+γ z

(4.110)

V p , V r are the vectors of the forward and reverse voltage waves. With Eqs. (4.100) and (4.109), we can develop the chain matrix of the n coupled lines analogous to the single line. We obtain the following result: Vin = cosh(γ I )Vout + sinh(γ I )vL Iout

(4.111a)

4 Properties of Coaxial Cables and Transmission Lines …

247

Iin = sinh(γ I )vC Vout + cosh(γ I )Iout

(4.111b)

where the indices in designate the input quantities and out the output quantities. L and C are related via Eq. (4.106) and γ is given according to Eq. (4.103). For an arrangement consisting of two coupled lines (Fig. 4.28a), it is conventional to define characteristic impedances Z 01 =

L 11 , C11

Z 02 =

L 22 , C22

(4.112)

and coupling factors L 12 L 12 , , k L2 = L 11 L 22 C C12 = 12 , kC2 = − , C11 C22

k L1 =

(inductive coupling factors).

kC1

(capacitive coupling factors)

(4.113)

The minus sign for k C2 was introduced because −C 12 = c 12 is the positive partial capacitance per unit length that is actually measurable; cf. Eq. (4.98). From Eq. (4.106), it follows that L 12 C12 = − = k L1 = kC2 = k1 L 11 C22 L 12 C12 = − = k L2 = kC1 = k2 L 22 C11

(4.114)

i.e. the inductive and capacitive coupling factors defined according to Eq. (4.113) are equal to one another. Moreover, we can demonstrate with Eq. (4.106) that the following relationship holds: k1 Z 01 = k2 Z 02

(4.115)

With Eqs. (4.106), (4.111), (4.112) and (4.114), we obtain the following for the chain matrix A of an arrangement consisting of two coupled lines (Fig. 4.28a): ⎞ ⎛ ⎞ V2 V1 ⎜ V4 ⎟ ⎜ V3 ⎟ ⎜ ⎟ = A⎜ ⎟ ⎝ I2 ⎠ ⎝ I1 ⎠ I3 I4 ⎛

(4.116)

248

M. Rudolph

⎡

cos βl 0 j (sin βl) Zχ01 jk1 (sin βl) Zχ01 ⎢ 0 cos βl jk2 (sin βl) Zχ02 j (sin βl) Zχ02 ⎢ A=⎢ k2 1 cos βl 0 ⎣ j (sin βl) Z 01 χ − j (sin βl) Z 01 χ k1 1 − j (sin βl) Z 02 χ j (sin βl) Z 02 χ 0 cos βl

⎤ ⎥ ⎥ ⎥ (4.117) ⎦

√ with χ = 1 − k1 k2 . Instead of the characteristic impedances in Eq. (4.112) and coupling factors in Eq. (4.114), we can introduce even- and odd-mode characteristic impedances Z 0e and Z 0o as described in Sect. 4.3.2. For equivalent lines (k 1 = k 2 = k, Z 01 = Z 02 = Z 0 ), the following relationships hold with Eqs. (4.101), (4.106), (4.112) and (4.114):

1+k , 1−k 1−k , = Z0 · 1+k

Z 0e = Z 0 · Z 0o

(4.118)

and Z0 =

Z 0e · Z 0o , k =

Z 0e − Z 0o Z 0e + Z 0o

(4.119)

4.4 S-Matrix for Matched Couplers and Power Dividers Up to this point, the reader might be under the impression that the wave propagation on transmission lines mainly causes trouble like reflections and attenuation that have to be mitigated through careful circuit design. But this section will address waveguide structures that exploit the nature of traveling waves in order to realize coupliner and divider or combiner structures. It is often desired to access the waves traveling on a waveguide. For example, one might like to measure forward and/or backward waves, or to superimpose an additional wave that travels only into one direction on the waveguide. Figure 4.30 shows such a coupler. The requirements to such a structure would be, in the ideal case: 1. 2. 3. 4. 5. 6. 7. 8.

It has four ports. The device is lossless. The device is reciprocal. Two ports are part of the main line. Port 1 and 2 are directly coupled. The third port couples into the forward wave of the main line. An incident wave at port 1 is coupled to port 3, but port 3 is decoupled from an incident wave at port 2. The fourth port couples into the backward wave of the main line. All four ports are matched.

4 Properties of Coaxial Cables and Transmission Lines …

249

Fig. 4.30 Schematic of an ideal directional coupler

In general, a phase shift is observed between the wave leaving port 2 to the coupled wave at port 3, which also might be exploited, e.g. to generate differential signals. Another functionality that is often desired is the splitting of a wave into two waves on different waveguides. A splitter should also be matched at all ports, be lossless, and it should suppress the transmission of an incident wave at one of the two output ports to the other ports. The device can also act as a combiner that merges the waves from two waveguides into one wave traveling on a third waveguide. An application is distributed power amplification, where the output power of a number n of power amplifiers is combined to obtain an n times higher output power compared to a single power amplifier. This operation has to be distinguished from a simple parallel connection of the n power amplifiers. A parallel connection would reduce the port impedance of the circuit to 1/n the value of a single amplifier, which makes power matching difficult. The individual power amplifiers would also directly interact with each other through the interconnected ports. A divider and combiner solution, in contrast, maintains the characteristic impedance Z0 through splitting and combination stages, and decouples the individual amplifiers. Before discussing different types of couplers and combiners, the desired Smatrices will be investigated in order to establish if the desired structures are theoretically possible at all.

4.4.1 Conditions for Non-dissipative Combiners and Dividers and the Even-Mode—Odd-Mode Analysis 4.4.1.1

Conditions for Non-dissipative Combiners and Dividers

The requirements for an ideal coupler as listed above can be expressed in terms of an S-matrix. We define port 1 and 4 to belong to the through path, with a transmission coefficient S14 = S41 . Port 3 should be coupled to an incident wave at port 1 through S31 , and port 4 is coupled to port 2 through S42 . No transmission should be observed from port 1 to port 4 and from port 2 to port 3 (S41 = S32 = 0), which are denoted as isolated ports with respect to each other. All ports are matched, i.e. S11 = S22 = S33 = S44 = 0. Finally, we consider the coupler to be reciprocal, which means Sij =

250

M. Rudolph

Sji , and symmetric, which means that also port 2 and 3 form a through path, with an incident wave at port 2 only coupled to port 3 and 4, and an incident wave at port 3 is coupled only to port 2 and 1. The respective S-matrix reads: ⎛

S11 ⎜ S21 S=⎜ ⎝ S31 S41

S12 S22 S32 S42

S13 S23 S33 S43

⎞ ⎛ 0 S14 ⎜ ⎟ S24 ⎟ ⎜ 0 = S34 ⎠ ⎝ S13 S44 S14

0 0 S14 S13

S13 S14 0 0

⎞ S14 S13 ⎟ ⎟ 0 ⎠ 0

(4.120)

So far, the requirements of matching at all ports and of coupling an incoming wave only to two specific ports are defined. It is equally important that the coupler is passive and lossless. As derived above, the S-matrix of a lossless multiport is a unitary matrix, so that S−1 = S+ holds. In the case of the coupler, the condition can be written as ⎛

0 ⎜ 0 S · S+ = E = ⎜ ⎝ S13 S14

0 0 S14 S13

S13 S14 0 0

⎞ ⎛ 0 0 S14 ⎜ 0 0 S13 ⎟ ⎟·⎜ ∗ ∗ 0 ⎠ ⎝ S13 S14 ∗ ∗ 0 S13 S14

∗ S13 ∗ S14 0 0

⎛ ⎞ ∗ ⎞ 1000 S14 ∗ ⎟ ⎜ ⎟ S13 ⎟ = ⎜ 0 1 0 0 ⎟ (4.121) 0 ⎠ ⎝0 0 1 0⎠ 0

0001

Which leads to the following equations defining magnitude and phase of the S-parameters: |S13 |2 + |S14 |2 = 1

(4.122)

∗ Re S14 S13 = 0

(4.123)

The first condition refers to the absolute power of the transmitted waves and simply states that for a matched lossless coupler, no power is dissipated but√ transmitted to the output ports. If we set, e.g. |S14 | = k < 1, it follows that |S13 | = 1 − k 2 . From the second condition, if follows that the phase difference between the vectors S14 and S13 is ±90◦ so that the product becomes purely imaginary. The generic S-matrix of a symmetric, matched lossless coupler now reads: ⎛

0 0 ⎜ 0 0 √ S=⎜ ⎝ 1 − k2 jk √ 1 − k2 jk

√

⎞ 1 − k 2 √ jk 1 − k2 ⎟ jk ⎟ ⎠ 0 0 0 0

(4.124)

Other phase relations between the transmission factors are possible if the symmetry condition is dropped, as we will see for the rat-race line coupler below and in case of the magic Tee in Chap. 5.

4 Properties of Coaxial Cables and Transmission Lines …

251

So far, we were able to prove that a matched lossless coupler is theoretically possible, and established also the phase relation at the output ports independent of the actual realization. Depending on the application, couplers providing different values of k are designed. For measurement purposes, for example one would like to observe the traveling waves without too much of attenuation, thus setting k ≈ 1, and a ratio of 20…40 dB between transmitted and coupled wave. On the other hand, the coupler can also √ be used as a power splitter, if transmitted and coupled powers are equal, and k = 2. For this purpose, port 1 would be defined as the input port, and port 3 and 4 as the output port. Port 2 is terminated. Ports 3 and 4 would be decoupled, so that a reflection on one of the ports is not affecting the output power at the other port. We now might ask whether it is also possible to directly realize a three-port network that is matched on all sides, lossless and exhibits transmission symmetry. Such a device is called power divider, splitter, or combiner. We will illustrate the three required properties using an S-matrix. Matching on all sides: S ii = 0 ⎛

⎞ 0 S12 S13 S = ⎝ S21 0 S23 ⎠, S31 S32 0

(4.125)

S+ S = E,

(4.126)

|S21 |2 + |S31 |2 = 1,

(4.127)

|S12 |2 + |S32 |2 = 1,

(4.128)

|S13 |2 + |S23 |2 = 1

(4.129)

Lossless (Sect. 2.5.1):

i.e.

∗ S32 S31

= 0.

(4.130)

Transmission symmetry: Sik = Ski .

(4.131)

Let us assume that, e.g. S 23 = S 32 = 0; it thus follows from Eqs. (4.130) and (4.131) that S 31 = S 13 = 0. However, this contradicts Eqs. (4.127), (4.128) and (4.131). Based on these considerations, we can state the following:

252

M. Rudolph

Unlike a lossless four-port network that exhibits transmission symmetry, it is not possible to obtain matching on all sides with a lossless three-port network that exhibits transmission symmetry. Since we cannot achieve these three properties at the same time, we can choose to abandon one of them. Two of the possibilities have interesting technical applications: 1.

The three-port network with transmission asymmetry. Here, the S-matrix according to Eq. (4.125) holds where, e.g. |S 13 | = |S 21 | = |S 32 | = 1 and S 12 = S 23 = S 31 = 0, i.e. give up the requirement of reciprocity Sik = Ski . The S-matrix reads ⎛ ⎞ 001 S = ⎝1 0 0⎠ (4.132) 010

2.

The matrix is obviously unitarian and the network is lossless. This so-called three-port circulator (cf. Sect. 5.8.2.1) transfers power entering port 1 only to port 2, power entering port 2 only to port 3 and from port 3 only to port 1. While it can not be used to comine or split signals, it can be used to separate forward and backward traveling waves at a port. In a transceiver, it can be used to separate transmitter (at port 1) from receiver (at port 3) sharing a matched antenna (at port 2). The lossy three-port network, known as power divider (Fig. 4.46). The S-matrix for the ideal Wilkinson power divider is as follows [60–62]: ⎛ ⎞ 011 j ⎝ S = −√ 1 0 0⎠ 2 100

(4.133)

This S-matrix is obviously not unitarian, S−1 = S+ , and therefore lossy. But it performs the following tasks perfectly: 1. 2. 3.

It is matched at all ports It splits incident power from port 1 without losses equally to port 2 and 3 It combines even-mode waves incident at port 2 and 3 without losses and transmits the whole power to port 1. In order to understand the concept of even-mode excitation, consider two combiners connected back-to-back. Half of the incident power is transmitted to each of the two output ports of the first divider. Since the structure is reciprocal, the powers are transmitted to the output of the second divider (acting as a combiner) without losses. In S-parameters, if S denotes the S-matrix of the divider and S stands for the combiner: S11 = S12 · S21 + S13 · S31 = −1 and therefore |S11 |2 = 1.

4 Properties of Coaxial Cables and Transmission Lines …

4.

253

Any other arbitrary excitation on port 2 or port 3 is not transmitted to the other output port. Half of the power is transmitted to the input port according to |S12 |2 = |S13 |2 = 0.5. The other half of the incident wave is dissipated inside the structure. These losses are commonly advantageous. For example if the divider-combiner configuration is used to operate power amplifiers in parallel, the only desired mode of operation is that all amplifiers work synchronized and in phase. Any other asynchronous signal results from unwanted oscillations or from a damaged amplifier. The lossy divider thus helps to suppress oscillations and reduces the impact of a damage in a single device on the whole amplifier circuit.

A realization of this concept is the Wilkinson power divider that will be discussed in the following [60].

4.4.1.2

Even-Mode—Odd-Mode Analysis

Calculating the S-matrix of a four-port device can be very tedious. But if the fourport happens to be symmetrical as in the case of couplers, we can decompose the problem into two much simpler ones through the so-called even-mode—odd-mode analysis. The principle has been used already in order to understand coupled lines and will be extended to generic symmetrical four-ports described by an S-matrix in the following. If a symmetrical four-port is excited symmetrically at two ports, identical voltages will be observed at the line of symmetry so that no current crosses it, which corresponds to a virtual open. In case of an excitation by asymmetrical signals that are 180° out of phase, on the other hand, the line of symmetry must be a virtual ground. The four-port is in both cases split into two two-ports that do not interact, which simplifies the calculation significantly. Figure 4.31 illustrates the basic principle. In terms of the S-matrix, we obtain the following relations for the outgoing waves at the four ports when, for example, port 1 is excited and all other ports are matched, so that a2 = a3 = a4 = 0 holds: b1 b2 b3 b4

= = = =

S11 · a1 S21 · a1 four port S31 · a1 S41 · a1

(4.134)

Now, if port 1 is excited in even mode by a1,e = a1 /2 and port 2 is excited by the same signal a2,e = a1 /2, we obtain:

254

M. Rudolph

a

b a2 = a2,e – a2,o = 0

a4 = 0

a2 = 0 2

a4 = 0

2

4

1

3

4 symmetry 3

1 a1

a3 = 0

a1 = a1,e + a1,o

c

a2,e = a1,e a2,o = a1,o a1,e = a1,o = a1/2

a3 = 0

d

even mode

odd mode

a2,e

–a2,o

a4 = 0

2

a4 = 0

2

4

4

virtual open at symmetry 1

3

a3 = 0

a1,e

virtual ground at symmetry 1

3

a1,o

a3 = 0

Fig. 4.31 Schematic explaining the even-mode—odd-mode analysis of a symmetrical fourport in the S-parameter domain. a Excitation of the fourport at port 1 b interpretation of the excitation as the superposition of even and odd excitation, and subdividing the fourport at the symmetry c evenmode analysis, assuming open circuit and d odd-mode analysis, assuming short-circuit termination at the symmetry plane

b1,e b2,e b3,e b4,e

= = = =

S11 · a1,e + S12 · a2,e S21 · a1,e + S22 · a2,e S31 · a1,e + S32 · a2,e S41 · a1,e + S42 · a2,e

= (S11 + S12 ) · a1 /2 = (S21 + S22 ) · a1 /2 even mode = (S31 + S32 ) · a1 /2 = (S41 + S42 ) · a1 /2

(4.135)

If port 1 is excited in odd mode by a1,o = a1 /2 and port 2 is excited by with 180° phase shift by −a2,o = −a1 /2, we obtain: b1,o b2,o b3,o b4,o

= = = =

S11 · a1,o − S12 · a2,o S21 · a1,o − S22 · a2,o S31 · a1,o − S32 · a2,o S41 · a1,o − S42 · a2,0

= (S11 − S12 ) · a1 /2 = (S21 − S22 ) · a1 /2 odd mode = (S31 − S32 ) · a1 /2 = (S41 − S42 ) · a1 /2

(4.136)

Superimposing even and odd case yields the original four-port response, Eq. (4.134). Symmetry of the four port means, in case of our port numbering, port 1 is symmetrical to port 2, and port 3 is symmetrical to port 4. Equations (4.134)–(4.136) can be simplified since S12 = S21 , S11 = S22 , and S31 = S42 , S32 = S41 . Considering the virtual open at the line of symmetry for the even case, we can therefore solve the equations for the two-port with ports 1 and 3, and in a second step solve it for the odd mode case. For port 1 excitation, one obtains reflection coefficients 1,e , 1,o and transmission coefficients T31,e , T31,o which are defined as:

4 Properties of Coaxial Cables and Transmission Lines …

b1,e a1,e b1,o a1,o b3,e a3,e b3,o a3,o

255

= 1,e = S11 + S21 = 1,o = S11 − S21 = T31,e = S31 + S41 = T31,o = S31 − S41

(4.137)

Thus, once the reflection and transmission factors in even and odd mode are determined, it is easy to obtain the four-ports S-matrix by solving (4.137) for the respective parameters. The remaining missing parameters are determined analogously. This approach can be applied to all symmetrical four-ports. It will be used in the following in the analysis of the Wilkinson power divider and of the line coupler.

4.5 Ring Couplers (180° and 90° Hybrid) The ring coupler in Fig. 4.32 provides an example of the top conductor configuration for a realization using microstripline technology. This device has applications as matched power dividers (3 dB couplers), 180° or 90° phase shifters or in sum and difference circuits. The operation principle bases on the phase difference of the waves that reach a certain port. An incident wave from one port travels clockwise and counter-clockwise around the ring. The other ports are located at points where the two waves superimpose constructively (transmit and coupled port) or destructively (decoupled port). The S-matrix of the 180° hybrid ring coupler (“rat-race coupler”) is as follows for the frequency corresponding to the line lengths in Fig. 4.32a [64, 65]: a

1

b

3

Z0

l/4

3

Z0

Z0 l/4 Z0

l/4

l/4 Z0

Z0

2

Z0

Z0

4

2

Z0

2

4 Z0

Z0

2

2

Z0

Z0

Z0

3l/4 1

Fig. 4.32 a, b Ring coupler a 180° Hybrid; b 90° Hybrid

2

256

M. Rudolph

⎛

0 0 −j ⎜ 0 0 S= √ ⎜ ⎝ 2 1 1 1 −1

⎞ 1 1 1 −1 ⎟ ⎟. 0 0⎠ 0 0

(4.138)

Based on the S-matrix in Eq. (4.138), we can see that power division occurs for a wave supplied on one port, all ports are matched, and ports 2 and 1 as well as 3 and 4 are decoupled from one another. Moreover, the four-port network exhibits transmission symmetry (S ik = S ki ). The relative phase shift between port 2 and 4 is 180° (“180° hybrid”) compared to that between port 2 and port 3. The S-matrix (4.138) of the 180° hybrid corresponds to that of the magic tee (cf. Sect. 5.7.1). The rat-race coupler has the following interesting features: • Defining port 2 as the input port yields in-phase signals at ports 4 and 3, while port 1 is isolated. An equivalent behavior is found for input at port 3. • Defining port 1 as the input yields a differential signal at ports 1 and 4, i.e. 180° phase shift between the signals. • If two signals are injected at ports 4 and 3, we obtain the difference of the signals at port 2 and the sum of the two signals at port 1. The S-matrix for the 90° hybrid (Fig. 4.32b) is [64, 65]: ⎛

0 1 ⎜ 0 S = −√ ⎜ 2⎝1 j

0 0 j 1

1 j 0 0

⎞ j 1⎟ ⎟. 0⎠ 0

(4.139)

Here, the relative phase shift is 90° (“90° hybrid”). The 90° hybrid can be built to be very broadband with a multistage implementation (“branch-line coupler”) (cf. e.g. [66]). This structure is suitable to realize a coupler with a coupling ratio of 3 dB (3 dB coupler). It is often used as a power splitter or combiner. The fourth unused port is in this case terminated with Z0 , and is in the ideal case not receiving any power as long as the combined waves are in proper phase relation and of identical magnitude.

4.6 Directional Couplers 4.6.1 S-Matrix for Termination with the Characteristic Impedance of the Line In this section, we will determine the S-matrix of an arrangement consisting of two coupled lines (Fig. 4.33) that are provided with connecting lines having characteristic

4 Properties of Coaxial Cables and Transmission Lines …

257

impedances Z 01 , Z 02 (i.e. the normalization impedance, cf. Sect. 2.5, is equal to the characteristic impedance of the respective coupled line). According to Sect. 2.5.3, the wave chain matrix T is first determined from the chain matrix A, Eq. (4.117). With Eqs. (2.233), (2.234) and (4.117), we obtain ⎛

T11 ⎜ 0 T=⎜ ⎝ 0 T∗14

0 T11 T∗14 0

0 T14 T∗11 0

⎞ T14 0 ⎟ ⎟, 0 ⎠ T∗11

(4.140)

with the coefficients sin 2π λl l T11 = cos 2π − j λ 1 − Z 01 k 2

Z 02 1

Z 01 l Z 02 T14 = jk1 sin 2π λ 1 − Z 01 k 2 Z 02

(4.141)

1

Here, l is the length of the coupling path; the coupling factor k 1 is defined in Eq. (4.114) and k 2 is expressed by Eq. (4.115). The S-matrix for the arrangement of doubly coupled lines as shown in Fig. 4.33 is obtained with Eqs. (2.233), (4.140) and (4.141) as ⎛

0 ⎜ S12 S=⎜ ⎝ S13 0

S12 0 0 S24

S13 0 0 S34

⎞ 0 S24 ⎟ ⎟ S34 ⎠ 0

01

02

Fig. 4.33 Four-port network consisting of two coupled lines with connecting lines

(4.142)

258

M. Rudolph

where ∗ 1 T14 T14 1 = ∗ |T11 |2 − |T14 |2 = ∗ , ∗ T11 T11 T11 ∗ T14 =− ∗ T11

S12 = S34 = T11 − S13 =

T14 ∗ , S24 T11

(4.143)

and |S 13 | = |S 24 |. With Eqs. (4.141) and (4.143), we have S11 =

1 l sin 2π l cos 2π λ + j ( Z 01λ )2 1− Z

S13 = S24 =

jk1 sin 2π λl

(4.144)

k 02 1

Z 01 Z 02 Z

1− Z 01 k12 02

sin 2π l cos 2π λl + j ( Z 01λ )2 1− Z

(4.145)

k 02 1

Based on the interpretation of the S-matrix, Eq. (4.142), we can conclude as follows (cf. Sect. 4.4.1.1): The main diagonal elements S ii , i.e. the input reflection coefficients, are equal to zero; the coefficients S 14 = S 41 , S 23 = S 32 , i.e. the transmission coefficients between ports 1 and 4 or 2 and 3 as well as the inverse in each case, are likewise equal to zero. The four-port network configured with connecting lines having Z 01 , Z 02 and made of coupled lines (Fig. 4.33) is thus matched on all sides, and the ports 1 and 4, as well as 2 and 3, are decoupled from one another. An arrangement of this sort is known as a directional coupler. Taking the feeding of port 1 as an example, the effective transmission coefficients are illustrated in Fig. 4.34 using arrows.

Fig. 4.34 Directional coupler consisting of two coupled TEM wave lines

4 Properties of Coaxial Cables and Transmission Lines …

259

4.7 TEM Wave Directional Couplers 4.7.1 Definitions and Illustration of the Directional Effect Directional couplers allow separate measurement of the forward and reverse waves on a line and can also be used as a power divider, an attenuator, a phase shifter and for low-reflection output coupling of signals. For output coupling of signals in telecommunications systems, a small bandwidth is often sufficient, while other applications such as RF test and measurement typically require a large bandwidth. For a directional coupler, the S-matrix representation, Eq. (4.142), ⎞ ⎛ 0 b1 ⎜ b2 ⎟ ⎜ S12 ⎜ ⎟=⎜ ⎝ b3 ⎠ ⎝ S13 b4 0 ⎛

S12 0 0 S24

S13 0 0 S12

⎞⎛ ⎞ a1 0 ⎜ a2 ⎟ S24 ⎟ ⎟⎜ ⎟ S12 ⎠⎝ a3 ⎠ 0 a4

(4.146)

is typically used to define the following quantities. In accordance with Fig. 4.34, the signal is supplied on port 1 while ports 2–4 are terminated with the normalization impedances (Sect. 4.11.2). Input reflection coefficient 1 =

b1 a1

(4.147)

(equal to zero regardless of frequency in the ideal6 directional coupler): Reverse transfer ratio AB12 = S12 ,

(4.148)

Coupling transmission coefficient AB13 = S13 ,

(4.149)

Directivity factor AR =

b3 S13 = b4 S14

(4.150)

(infinitely large regardless of frequency for an ideal directional coupler, for S 13 = 0). The associated values are often indicated in decibels, e.g. −20 log (AB12 ) = ac (coupling attenuation), 20 log (AR ) = aD (directional attenuation). 6

Practically speaking, such values cannot be attained due to losses, manufacturing tolerances, higher wave types, etc. Typical values are roughly |r 1E | ≤ 5%, |AR | ≥ 10 (or ≥20 dB).

260

M. Rudolph

Ik z<

0)

x(t )

CI

R1

C12

R2

···

Rn

CO

y(t )

Fig. 4.61 Structure of a linear coupled tunable filter with tunable resonators R1 . . . Rn , but fixed input- (C I ), output- (C O ) and inter-resonator coupling (C12 . . . C(n−1)(n−2) )

288

M. Rudolph

C1n

x(t )

CI

R1

C12

R2

···

Rn

CO

y(t )

C2n

Fig. 4.62 Structure of a linear coupled tunable filter with tunable resonators R1 . . . Rn , and tunable input- (C I ) and output- (C O ) as well as inter-resonator coupling (Cmn )

The main drawback of this kind of realization is the fixed coupling between the individual resonators. This results in a specific fractional bandwidth, which results in changing absolute bandwidth B while tuning the center frequency f c . For universality, a filter independently tunable in center frequency f c and bandwidth B is highly desired, compare class (c) in Fig. 4.60. Such a realization requires the implementation of reconfigurable coupling coefficients. Moreover, for the introducing of additional nulls into the transmission coefficient, cross coupling between resonators is highly desired, compare Sect. 1.3. Figure 4.62 shows the schematic of such a tunable filter with tunable cross coupling. In coupling matrix representation, compare Sect. 1.3 such filters can be described by the coupling matrix C: ⎤ R1 C12 · · · C1N ⎥ ⎢ C12 R2 ⎥ ⎢ C=⎢ . ⎥ .. ⎦ ⎣ .. . ⎡

C1N

(4.167)

RN

with the resonators Rn on the main diagonal and the inter-resonator coupling Cmn on the first side diagonals. While the tuning of cavity resonators either by means of geometric changes or by tuning of the cavity’s material is straight-forward, the implementation of tunable coupling structures is more challenging. One example used for non-planar filters is iris coupling with a variable iris [92, 93]. This implementation is efficient but increases system complexity. Also, system reliability suffers from mechanically moving parts. As alternative the variable couplings can be emulated by non-resonating nodes—resonators operated out of resonance—[94–96] as shown in Fig. 4.63. A similar concept is followed in planar structures where components of the J- and K-inverters of the coupling structures are absorbed in the resonators itself. Figure 4.64 shows the example of a simple third-order filter with the introduction of K-inverters and the absorption of its components into adjacent resonators. This leads to additional boundaries for the coupling elements as the negative capacitor value −C K must be compensated by the series resonant circuit capacitors

4 Properties of Coaxial Cables and Transmission Lines …

···

R1

R2*

C12

Resonating node

289

C23

R3

Non-resonating node

···

Resonating node

Fig. 4.63 Realization example of a tunable coupling of two resonators R1 , R3 by a non-resonating node R2∗ with fixed coupling C12 , C23

C1 . . . C3 , resulting in: C K < C1 , C3

∧

2C K < C2

⇒

CK

εr2 μr2 . According to the law of refraction (5.3.44b), already at an angle of incidence ϑ < π/2 we have the angle of refraction ϑ t = π/2. The angle ϑ at which this occurs is known as the critical angle ϑ total of total reflection. For it, the following holds: sin ϑtotal =

εr2 μr2 . εr1 μr1

(5.3.51)

At the critical angle of incidence, we have cos ϑ t = 0 and it follows from Eq. (5.3.45) that E r1 = E f1 The incident wave is totally reflected. In medium 2, a homogeneous plane wave is propagated in the y direction according to Eqs. (5.3.42) and (5.3.43); for the amplitude of its electric field strength, the following holds according to Eq. (5.3.46): E f2 = 2E f1 We now ascertain that Eq. (5.3.44a) is satisfied even for the critical angle of incidence: k1 sin ϑtotal = k2 , √ ω εr1 μr1

εr2 μr2 √ = ω εr2 μr2 . εr1 μr1

The relationships become more complex for ϑ > ϑ total . Here too, there is total reflection; the wave in medium 2 can thus again exhibit only a y propagation direction.

376

H. Arthaber

If it were a homogeneous plane wave, its wavenumber would have to be k 2 . According to Eq. (5.3.44), the y component of the incident wave has the wavenumber k 1 sin ϑ. According to Eq. (5.3.51), we now have the following for ϑ > ϑ total : sin ϑ >

εr2 μr2 εr1 μr1

or k1 sin ϑ > k2 . Thus, the boundary conditions at the separating plane for ϑ > ϑ total with a homogeneous plane wave in medium 2 can no longer be satisfied independent of y. Instead, in medium 2 a wave must propagate in the y direction with a phase constant β 2 > k 2 . In Sect. 5.3.2, we encountered plane waves with this characteristic and called them “inhomogeneous waves”. According to Eq. (5.3.27a), they have the phase constant β22 = k22 + α22 . With this phase constant, we can always satisfy the condition of equal y components of the phase velocities: k1 sin ϑ =

!

k22 + α22 .

By solving this equation for α 2 , we now also obtain a conditional equation for α 2 : α2 =

! k12 sin2 ϑ − k22 .

(5.3.52)

In summary, we can state that for ϑ > ϑ total , an inhomogeneous plane TE wave is propagated in medium 2 in the y direction. In medium 1, the incident wave is totally reflected. The amplitudes of the TE wave decrease according to e−α2 z . If ϑ is constant, α 2 grows linearly with frequency while the intensity of the wave in medium 2 decreases accordingly at higher frequencies. Applying the same approach to the case in Fig. 5.9b, we conclude that a TM wave is propagated in medium 2.

5.4 Dielectric Waveguides Electromagnetic waves can be guided by certain material (physical) structures along an axis, with the energy transported by the wave being concentrated in proximity to the axis. Structures that are capable of transporting waves are known as waveguides. In Chaps. 2 and 4, we already investigated the two-wire line and the coaxial line as such. Additional types of waveguides can be created by exploiting the reflection

5 Field-Based Description of Propagation on Waveguides

377

of plane waves on metallic walls or their total reflection at the boundary surface between two dielectric materials. In dielectric waveguides, the total reflection of plane waves on a medium with lower optical density is exploited in order to transport an electromagnetic wave in a medium with higher optical density. Dielectric waveguides have obvious applications in the transmission of light waves since the waves follow even a curve in the waveguide due to the total reflection at the boundary surface to the air. Theoretical investigations of such waveguides began around the start of the twentieth century [12].

5.4.1 Dielectric Slab Waveguides The simplest mathematical treatment involves an infinitely extended dielectric slab of thickness a. We will use a coordinate system as shown in Fig. 5.10 and assume that μr1 εr1 > μr2 εr2 . In order to characterize the wave propagation along a slab of this sort, it suffices to allow two plane waves to interfere with one another. We can imagine Medium 2

Medium 1

mr , er

mr , er 2 2

1

Medium 2

mr2, er2

1

y

J 3 J

x 2

z

1

a

Fig. 5.10 Dielectric slab waveguide, total reflection in the slab waveguide, inhomogeneous plane wave in the external space. The exponential functions in the external space characterize the amplitude decay of the inhomogeneous wave

378

H. Arthaber

that each wave arises due to total reflection of the other at the boundary plane to the medium with lower optical density. For the angle of incidence and reflection ϑ of the two waves, we must therefore have ϑ ≥ ϑ total . We will first consider only the processes at the boundary plane at z = 0 (Fig. 5.10). We assume that a homogeneous plane wave impinges onto it which is characterized by Eqs. (5.3.33) and (5.3.34). The polarization direction of its electric field strength is parallel to the boundary plane. In medium 1, this results in a reflected wave characterized by Eqs. (5.3.35) and (5.3.36). In accordance with our discussion in Sect. 5.3.3, in medium 2 for z ≥ 0 we should expect an inhomogeneous plane wave that propagates in the positive y direction (ϑ t = π/2) and has an amplitude that decreases exponentially in the positive z direction. For the propagation constant γ2 of this wave, we must therefore have γ 2 = α2 ez + jβ2 ey .

(5.4.1)

Using Eqs. (5.3.26) and (5.3.23), we then obtain the field components of the refracted wave: Ef2 = E f2 ex e−(jβ2 y+α2 z) , Hf2 = −

E f2 β2 ez + jα2 ey e−(jβ2 y+α2 z) . ωμ0 μr2

(5.4.2) (5.4.3)

In order to satisfy the boundary conditions for the tangential components of E and H at z = 0, the following relationships must hold between the amplitudes of the incident, reflected and refracted waves:

E f1 + E r1 e−jk1 y sin ϑ = E f2 e−jβ2 y ,

E f1 − E r1

cos ϑ Z W1

e−jk1 y sin ϑ =

α2 E f e−jβ2 y . jωμ0 μr2 2

Both equations must be satisfied independently of y such that we obtain β2 = k1 sin ϑ.

(5.4.4)

However, according to Eq. (5.3.52) α 2 is also determined with β 2 . α2 =

!

! k12 sin2 ϑ − k22 = k1 sin2 ϑ − sin2 ϑtotal .

(5.4.5)

Finally, after a simple calculation we obtain the following for the reflection coefficient r and the transmission factor t at the boundary plane at z = 0:

5 Field-Based Description of Propagation on Waveguides

r=

E r1 = E f1

Ef t= 2 = E f1

jωμo μr2 α2 ωωμ0 μt2 α2

− +

Z W1 cos ϑ Z W1 cos ϑ

jωμ0 μr2 α2 jωμ0 μr2 Z + cosWϑ1 α2

2

379

= ejψ ,

(5.4.6)

= ejψ + 1.

(5.4.7)

It is worth noting that in case of total reflection, the reflected wave experiences a phase shift by the phase angle Ψ corresponding to Eq. (5.4.6). α2 Z W1 , cos ϑ ωμ0 μr2 Z W1 π α2 0 ≤ arctan ≤ . cos ϑ ωμ0 μr2 2

ψ = 2 arctan

(5.4.8)

At the boundary plane at z = –a, we encounter the same conditions; the wave impinging onto it is equal to the wave reflected at the boundary plane at z = 0. During reflection back and forth between points 1, 2 and 3 in Fig. 5.10, the phase of the wave at point 3 can differ from that at point 2 only by integer multiples of 2π since otherwise the interfering waves would not be plane waves. For the phase on the path from point 1 to point 3, we must therefore have the following: e"−jk1 a(tan ϑ#$sin ϑ+cos ϑ)% 1→2

ej2ψ "#

%$e"−jk1 a(tan ϑ#$sin ϑ+cos ϑ)% = ejv2π e−jk1 a(2 tan ϑ sin ϑ) ,

2→3 total reflection (2,3) ej2(ψ−k1 a cos,ϑ) = ejv2π , ψ − k1 a cos ϑ = vπ, v = 0, ±1, ±2, . . .

(5.4.9)

For conciseness, we introduce η=

α2 Z W1 cos ϑ ωμ0 μr2

and rewrite Eq. (5.4.9) in the form arctan η −

π k1 a cos ϑ = v . 2 2

(5.4.9a)

For v = 1, 2, 3, …, we thus have arctan η =

k1 a π cos ϑ + p , 2 2

p = 1, 2, 3, . . .

(5.4.9b)

380

H. Arthaber

and for v = 0, –1, –2, –3, … arctan η =

π k1 a cos ϑ − n , 2 2

n = 0, 1, 2, . . .

(5.4.9c)

Due to the limitation of arctan η to values between 0 and π/2 [see Eq. (5.4.8)] and because (k 1 a/2) cos ϑ is always positive, only Eq. (5.4.9c) is relevant as an equation of condition. By forming the tangent of this equation, we finally obtain nπ k1 a cos ϑ − 2 2 k1 a cos ϑ for n = 0, 2, 4, . . . = tan 2 k1 a = − cot cos ϑ for n = 1, 3, 5, . . . 2

η = tan

(5.4.10a) (5.4.10b)

Equations (5.4.10) should be interpreted as conditional equations for u = (k 1 a/2) cos ϑ. Before we discuss their solution, we will transform η so this quantity also appears explicitly as a function of u. We first introduce in the form of β P the phase constant of the wave that propagates along the slab in the y direction. This follows from Eq. (5.3.44) βP − k1 sin ϑ

(5.4.11)

and we can now see that β P is determined if (k 1 a/2) cos ϑ is known.

βP a 2

2

+ u2 =

k1 a 2

2 .

(5.4.12)

Equation (5.4.12) is a circle equation. Of the circle with radius (k1 a/2) = only the range in the first quadrant is relevant because β P > 0 and u > 0. For three different angular frequencies, this range of Eq. (5.4.12) is depicted graphically in Fig. 5.11. Taking into account Eq. (5.4.5), we can now write the following for η: √ 1 ωa μ0 μr1 ε0 εr1 , 2

!μ

0 μr1

! √ μ0 μr1 ε0 εr1 βP2 − k22

ε0 εr1 Z W1 α2 = cos ϑ ω μ0 μr2 k1 cos ϑμ0 μr2 ! 2 2 μr k12 − k22 − k12 cos2 ϑ μr βP − k2 = 1 = 1 μr2 k1 cos ϑ μr2 k12 cos2 ϑ

η=

5 Field-Based Description of Propagation on Waveguides

381

ba/2 w3 w w2 w1

u

Fig. 5.11 Relationship between phase constant β P of the dielectric slab waveguide and u = (k 1 a/2) cos ϑ for three different angular frequencies ω

=

μr1 μr2

k12 − k22 k12 cos2 ϑ

−1=

μr1 μr2

a 2 2 k1 − k22 2 − 1. u2

(5.4.13)

Since η is a real function, according to Eq. (5.4.13) the expression k 1 cos ϑ must satisfy the following inequality: 0 ≤ k1 cos ϑ ≤

!

k12 − k22

(5.4.14)

Based on this condition and Eq. (5.4.12) for β P , it follows that the phase constant of a dielectric slab waveguide must always lie between the limit values k 1 and k 2 . k2 ≤ βP ≤ k1 .

(5.4.15)

We can now attempt to solve Eq. (5.4.10). Since it is transcendent in the quantity of interest u, only a numerical or graphical solution is possible in which we represent the left and right sides of Eq. (5.4.10) in a common coordinate system as a function of u and find solutions as points of intersection between the individual functions. Figure 5.12 shows solutions for even and odd n at three different angular frequencies ω. For small values of u, the function η has practically a hyperbolic shape and is defined in the real domain only for η = ∞ to 0. This condition determines the end points of the function η on the abscissas in Fig. 5.12. According to Eq. (5.4.13), we have the following for these points: k12 cos2 ϑ = k13 − k22 , sin ϑ =

k2 = sin ϑtotal . k1

(5.4.16)

In physical terms, the end points can be interpreted as characterizing the transition from total reflection to refraction. If

382

H. Arthaber

a

b tan (u) h (u)

–cot (u) h (u)

2

n=0

n=1

4

3

5

2p

3p u

w w1

w2

w3 w

p

w1

2p

3p

u

w3

w2

p

Fig. 5.12 Graphical solution of Eq. (5.4.10). a For n = 0, 2, 4, …; b for n = 1, 3, 5, … Parameter: angular frequency ω

k1 cos ϑ is greater than k12 − k22 (otherwise stated: ϑ < ϑ total ), the slab waveguide is no longer able to transport a wave and radiation occurs due to the refracted wave. Based on Fig. 5.12, we will now have a closer look at the wave propagation process on a dielectric slab waveguide. We first realize that at a sufficiently high angular frequency ω, there is more than one solution in u and thus for β P which belong to the same ω. Each β P is associated with its own mode which we designate by the order n of the tan or cot branch on which the associated solution intersection lies. In other words, we must distinguish between the different βPn . For example, if we consider solutions for ω = ω3 , we discover based on Fig. 5.12 that with βP0 to βP5 a total of six modes can simultaneously exist on the dielectric waveguide. If we reduce the angular frequency of operation, then the η curve and thus also its end point on the abscissa are shifted towards smaller values of u. In this manner, the solution intersections migrate closer and closer to the zeroes of the tan function (or the cot function). In the example in Fig. 5.12, the solution intersection for βP5 is the first to reach a zero of this sort. For this mode, the corresponding ω = ωc5 is then the one for which total reflection is just still possible. If we have ω < ωc5 , then the mode with βP5 can no longer exist. In general, the zeroes of tan(u) and –cot(u) thus define lower cutoff frequencies f cn = (1/2π)ωcn for corresponding modes. Expressed in the variables u and with the aid of Eq. (5.4.16), the condition for the cutoff frequency of the nth mode is ! k1 a a π cos ϑ = k12 − k22 =n . u cn = 2 2 2 cn cn

5 Field-Based Description of Propagation on Waveguides

383

We can obtain the actual cutoff frequency by solving this equation for f cn . f cn =

c n , n = 0, 1, 2, 3, . . . , √ 2a μr1 εr1 − μr2 εr2

c= √

(5.4.17)

1 , c = Speed of light in a vacuum. μ0 ε0

According to Eq. (5.4.17), for a dielectric slab waveguide there is a mode with the lower cutoff frequency f c0 = 0. This is also clear from Fig. 5.12a in which for any ω always at least one intersection is found with the tan branch of order n = 0. For f = f c , α 2 is equal to zero according to Eq. (5.4.5) because of Eq. (5.4.16). The relevant mode is still transported by the waveguide but the transported energy is no longer concentrated in its proximity. In the waveguide’s external space, a wave is propagated with an amplitude that no longer diminishes in the z direction. In the following discussion, we will provide insight into the basic shape of the dispersion curves βPn = f (ω). Due to Eq. (5.4.15), these curves must lie between the two limit lines β P = k 1 and β P = k 2 . From Eqs. (5.4.12) and (5.4.5) in the forms βP2 = k12 − α2 =

!

2 u a

2 ,

βP2 − k22

and based on Fig. 5.12, it follows moreover that ω → ∞, u = const, βP → k1 , α2 → ∞, ! a 2 2 ω = ωc , u = 2 k1 − k2 , βP = k2 , α2 = 0. Finally, we can also make an assertion about the slope dβ P /dω of the dispersion curves. We have k2 dβP = 1 − dω ωβP

2 u du 2 . a βP dω

(5.4.18)

In order to evaluate this equation, we must calculate du/dω. We take the derivative of Eq. (5.4.13) with respect to ω, dη = dω

d (tan u) dω d (− cot u) dω

=

1 du cos2 u dω 1 du sin2 u dω

μr = 1 μr2

a 2 2 k1 − k22 u du 2 ! 2 − a ω dω k 2 −k 2 u 3 ( 2 ) (u 21 2 ) − 1

384

H. Arthaber

introduce the term

tsc

u du = dω ω

−1 cos2 u = 2 −1 and solve for (du/dω). sin u μr1 μr2

a 2 2 k1 − k22 tsc 2

! 2 a k 2 −k 2 u 3 ( 2 ) (u 21 2 ) − 1 +

μr1 μr2

a 2 2 k1 − k22 tsc 2

.

(5.4.19)

From Eq. (5.4.19), it first follows that du/dω < u/ω. If we now replace du/dω in Eq. (5.4.18) by u/ω, we obtain the inequality dβ P /dω > β P /ω, which implies that the slope of all dispersion curves must always be greater than β P /ω (which is equivalent to saying that the group velocity of each mode must always be less than its phase velocity). For ω → ∞, we have β P → k 1 , du/dω → 0 and according to Eq. (5.4.18) it thus follows that dβP k1 (5.4.20) = . dω ω→∞ ω Thus, all dispersion curves asymptotically approach the limit line β P = k 1 for ω → ∞. In the vicinity of the cutoff frequency ω = ωc , we have β P → k 2 , ! du u 1a → = k 2 − k22 dω ω ω2 1 and it thus follows from Eq. (5.4.18) that

dβP dω

ω→ωe

k12 1 2 2 a = − ωk2 ωk2 a 2

2

k2 k12 − k22 = . ω

(5.4.21)

All dispersion curves begin at ω = ωcn on the limit line β P = k 2 , and start with its slope. Based on this discussion of curves, we provide a qualitative representation in Fig. 5.13 of the dispersion diagram for TE waves on a dielectric slab waveguide. Taking into account Eqs. (5.4.6) and (5.4.7) along with Eq. (5.4.9), we obtain the complete set of equations characterizing the field of the H waves along a dielectric slab waveguide. In the range –a ≤ z ≤ 0, we have ψ −jk1 y sin ϑ e , E1 = 2E f1 ex cos k1 z cos ϑ + 2 2 ψ H1 = − E f1 jey cos ϑ sin k1 z cos ϑ + Z W1 2 ψ e−jk1 y sin ϑ +ez sin ϑ cos k1 z cos ϑ + 2 For the range z ≥ 0, it follows that

(5.4.22)

5 Field-Based Description of Propagation on Waveguides

385

bp

n

k1 = w m0 mr e0 er 1

1

6

5 4 3 2 1 n=0

k2 = w m0 mr e0 er 2

wc

1

wc

2

wc

wc

3

4

wc

5

wc

2

6

w

Fig. 5.13 Dispersion diagram of the TE modes of a dielectric slab waveguide ωc1 c π√ a

=

μr1 εr1 −μr2 μr2

E2 = 2E f1 ex cos H2 = −

ψ −(α2 z+jkt y sin ϑ) e , 2

2E f1 ψ jey α2 + ez k1 sin ϑ e−(α2 z+jk1 y sin ϑ) cos ωμ0 μr2 2

(5.4.23)

and finally we obtain for the range z ≤ –a E2 = ±2E f1 ex cos H2 = ±

ψ α2 (z+a)−jk1 y sin ϑ e , 2

2E f1 ψ jey α2 + ez k1 sin ϑ eα2 (z+a)−jk1 y sin ϑ . cos ωμ0 μr2 2

(5.4.24)

In Eqs. (5.4.24), the positive sign applies to even values of v and the negative sign to odd values [see Eq. (5.4.9)]. Separate treatment of the E waves on a dielectric slab waveguide is unnecessary since our discussion of H waves applies analogously. In particular, Figs. 5.11, 5.12, 5.13 and Eq. (5.4.17) hold with no changes. Figure 5.14 shows a qualitative field pattern for the wave with v = 0 (commonly known as the fundamental mode). The closed lines in the y, z plane are electric field lines of the E 10 mode. If we swap E r1 with μr1 , then we have H 10 modes with the E and H lines swapped. As we have seen in conjunction with Fig. 5.12, for the dielectric waveguide at a specific frequency there is in each case only a finite number of modes that can exist, which are also known as modes. Accordingly, these modes do not suffice to solve the excitation problem in the dielectric waveguide. Instead, we must also consider radiation fields that are invariably co-excited.

386

H. Arthaber lP

y

z

Fig. 5.14 Qualitative field pattern for the E 10 mode (fundamental mode, v = 0) on a dielectric slab waveguide, λP = 2π/β P

5.4.2 Cylindrical Dielectric Waveguides Inspired by the work of Sommerfeld [13], Hondros [14] investigated in greater detail the conditions under which electromagnetic waves can be transported by a single metallic wire with finite conductivity. He discovered that further auxiliary modes are possible in addition to the so-called main mode calculated by Sommerfeld. Whereas the propagation constant of Sommerfeld’s main mode is determined primarily by the properties of the dielectric (air) surrounding the wire, the propagation constants of Hondros’ auxiliary modes primarily depend on the properties of the wire material. The auxiliary modes on the Sommerfeld wire are thus so highly attenuated that in practical terms they are no longer observable. We can imagine this difference between the main and auxiliary modes as follows: For the main mode, the field within the wire is pushed to the wire surface, while the field of the auxiliary modes is concentrated inside the wire and thus very highly attenuated (attenuation values of 105 dB/cm). In terms of the field concentration, the auxiliary modes on the Sommerfeld wire correspond to the waves along a dielectric slab waveguide. D. Hondros and P. Debye then also replaced the metallic wire with a dielectric wire. Since the thermal losses that occur in such a wire are low, they were justified in expecting that it would be capable of transporting waves of observable intensity. In a paper published in 1910, they provided theoretical proof [12], while Zahn [15] and Schriever [16] experimentally

5 Field-Based Description of Propagation on Waveguides

387

corroborated these theoretical insights. This was followed by a number of further publications. For a good summary, see [17, p. 481 ff.]. Transmission of waves along cylindrical dielectric tubes was investigated by Mallach [18, 19] and Unger [20, 21]. Compared to a dielectric solid wire, tubes have the advantage of more favorable transmission figures in a wider frequency range. Even today, wave propagation on dielectric waveguides in general as well as on dielectric wires and tubes in particular remains an area of intensive research. Dielectric rods and tubes made of rigid material have found practical applications in the construction of corresponding antennas. Using flexible materials, flexible waveguides can be constructed for centimeter, millimeter and light waves. In many respects, there is a close relationship between wave propagation on a dielectric slab and on a dielectric wire. In both cases, the wave transport can be explained based on successive total reflection at the boundary surface between the media with electrically higher and lower densities. In the slab waveguide, application of a single plane wave is sufficient. Due to the cylindrical boundary surface, however, infinitely many plane subwaves must be superimposed for the cylindrical dielectric waveguide with propagation directions that form a cone with an aperture angle 2ϑ with respect to the axis of the waveguide. For a discussion on how to proceed in principle, see, e.g. [2, p. 364]. As shown in Fig. 5.15, we choose as the axial direction of the wire the z direction of a cylindrical coordinate system ρ, ϕ, z. In the integration of the subwaves over real azimuth direction angles, we are lead to Bessel functions J n for the ρ dependency of the resulting electromagnetic field. Bessel functions of order n = 0, 1 and 2 are illustrated in Fig. 5.16. They are appropriate as radial solution functions for the inside of the wire. In the external space, the radial field dependency must be characterized by a function that exhibits an exponential decrease for large values of ρ as is the case with the slab waveguide. We obtain this function by integrating the subwaves with respect to complex azimuth direction angles. They are known as modified Bessel functions K n of the second kind. They are shown in Fig. 5.17 for orders of n = 0 and 1. The complete solution for the radial dependency consists of linear combinations of the given functions. It is, e.g. according to [22], as follows: x

x er = 1, k2 2

er > 1, k1 1

z mr = mr = 1 1

2

y 2a

Fig. 5.15 Cylindrical dielectric waveguide with cylindrical coordinate system ρ, ϕ, z

388

H. Arthaber 1.0 J0(r ) 0.8 J1(r ) 0.6

2 pr

J2(r )

0.4

J0(r )

J1(r )

J2(r )

0.2 0 –0.2 2 pr

–0.4 –0.6

0

2

4

6

8

10

12

r

Fig. 5.16 Bessel functions (of the first kind) versus argument r. Here, we have r = k c ρ. J 0 (r) = Bessel function of the first kind, 0th order, J 1 (r) = Bessel function of the & first kind, 1st order, J 2 (r) = Bessel function of the first kind, 2nd order, with argument r. k c = γ 2 + k 2 ρ = radius 5

4

3

K1 (r ) 2

1

K0 (r )

0

0.5

1.0

1.5

2.0

2.5

3.0

r

Fig. 5.17 Modified Bessel functions of the second kind K 0 (r) and K 1 (r) with argument r

5 Field-Based Description of Propagation on Waveguides

389

⎫ E z = AnJn , ⎪ ⎪ ⎪ ⎪ βD nωμ0 E ρ = − j h An Jn + h 2 ρ Bn Jn , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ nβD ωμ0 E ϕ = − h 2 ρ An Jn + j h Bn Jn , ⎬ e−j(βD z+nϕ) ⎪ Hz = B for ρ ≤ a, ⎪ n Jn , ⎪ ⎪ nωε0 εr1 βD ⎪ Hρ = An Jn − j h Bn Jn , ⎪ 2 ⎪ ⎪ h ρ ⎪ ⎪ ωε0 εr1 H =− j A J + nβD B J ⎭ ϕ

n n

h

h2 ρ

n n

⎫ Ez = C ⎪ ⎪ n K n , ⎪ ⎪ 0 ⎪ E ρ = j βαD2 Cn K n + nωμ , D K ⎪ 2 n n ⎪ α ρ 2 ⎪ ⎪ ⎪ nβD ωμ0 E ϕ = α2 ρ Cn K n − j α2 Dn K n , ⎬ e−j(βD z+nϕ) 2 ⎪ for ρ ≥ a. Hz = D ⎪ n Kn, ⎪ βD ⎪ ⎪ 0 ⎪ Hρ = −j nωε C K + j D K 2 n n n ⎪ n α α ρ 2 ⎪ 2 ⎪ ⎪ nβD ωε0 D K , ⎭ H = j C K + ϕ

α2

n

n

α22 ρ

n

(5.4.25)

(5.4.26)

n

In Eqs. (5.4.25) and (5.4.26), which we obtain in this form as solutions of the wave equation in cylindrical coordinates, we introduced the term h = k 1 cos ϑ. The argument of the Bessel functions J n is hρ and that of the modified Bessel functions K n is α 2 ρ. Bessel functions with the prime mark indicate derivatives with respect to the argument. An , Bn , C n and Dn are integration constants. Between the phase constants β D of the dielectric wire and k 1 , k 2 as well as α 2 , we have the relationship βD2 = k22 + α22 = k12 − k12 cos2 ϑ = k12 sin2 ϑ. This equation is identical to Eqs. (5.4.11) and (5.4.12) for the slab waveguide. Now we must determine the conditional equation for β D . For this purpose, we develop the continuity conditions of the tangential field strengths at the boundary surface for ρ = a with Eqs. (5.4.25) and (5.4.26) and thus obtain a homogeneous system of equations for the constants An , Bn , C n , Dn . This has a non-zero solution only if its determinant disappears. Instead of Eq. (5.4.10), we thus have for the dielectric wire the conditional equation Jn 0 − nβD J h2 a n ωε0 εr1 −j h Jn or further realized

=0 2 nβD nβD ωε0 − h 2 a Jn −j α2 K n − α2 a K n 0 −K n 0 0 −K n Jn nβD ωμ0 0 j ωμ J − K j Kn 2 n n h α2 α a 2

390

H. Arthaber

εrt Jn 1 K n + ha Jn α2 a K n

1 Jn 1 K n + ha Jn α2 a K n

=

nβD k2

2

2 1 1 + . (ha)2 (α2 a)2 (5.4.27)

Only graphical or numerical solutions are possible for Eq. (5.4.27). For the special case of n = 0, we wish to discuss a graphical solution in qualitative terms. This special case is characterized in that it leads to rotationally symmetrical fields and pure E or H fields are possible only for it. Here, Eq. (5.4.27) splits into two equations. Given dJ 0 (x)/dx = –J 1 (x), dK 0 (x)/dx = –K 1 (x), they are −

εr1 J1 1 K1 = ha J0 α2 a K 0

E-modes,

(5.4.28a)

−

1 J1 1 K1 = ha J0 α2 a K 0

H -modes.

(5.4.28b)

Except for the factor εr1 , the two equations are identical such that it suffices to consider Eq. (5.4.28b) for a well-founded discussion. Figure 5.18a shows the curve of the function K 1 /(α 2 a! K0 ) and Fig. 5.18b the curve of the function –J 1 /(haJ 0 ). Using the relationship α2 =

k12 − k22 − h 2 , we can plot the function K 1 /(α2 a K 0 ) with

a

b

1 K1(a2a) • a2a K0(a2a)

–

a2a

1 •

J1(ha)

ha J0(ha)

K1(a2(h)a) 1 • a2(h)a K0(a2(h)a) (ha)max = a k12– k22 ha

Fig. 5.18 a The function K 1 /(α2 a K 0 ) of the argument α 2 a. b The function –J 1 /(haJ 0 ) of the argument ha and solution intersections with the function (l/α 2 a) K 1 /K 0 of the argument α2 a = ! k12 − k22 − h 2 a

5 Field-Based Description of Propagation on Waveguides a

E01 – mode

b

H01 – mode

391 c

HE11 – mode

e1 = 81 e0 e2 = e0

e1 = 81e0 e2 = e0

l/ 2 l = 14.4a < lc = 23.6a

l/ 2

l/ 2

l = 3.25a < lc

Fig. 5.19 Excitation and qualitative field patterns on the cylindrical dielectric waveguide. a Field pattern for the E 01 mode, b field pattern for the H 01 mode, c field pattern for the HE 11 mode

! (ha)max = a k12 − k22 in the latter representation. We can see that under the assumed conditions, we obtain two solution intersections, corresponding to the existence of two rotationally symmetrical E and H modes. Similar to the slab waveguide, for the dielectric wire there is only a finite number of modes that can exist at a certain frequency. Unlike the slab waveguide, there are no E or H waves on a dielectric wire with a cutoff frequency of zero. This is a characteristic difference between the two waveguides. For n > 0, E and H waves can no longer propagate separately on a cylindrical dielectric waveguide. They always occur together and are then designated as hybrid modes or EH and HE modes for short. The first letter indicates the dominant mode. Among hybrid modes, the HE 11 mode, which is also commonly known as the dipole type, is especially important because it does not have a lower cutoff frequency. It is the fundamental mode of the dielectric wire. For the indexing of hybrid modes, note that the first index corresponds to the number n (azimuthal distribution) and the second index indicates which solution intersection is implied. Figure 5.19 shows qualitative field patterns for E 01 , H 01 and HE 11 modes along with an indication of how these modes can be basically excited. When using the dielectric wire for communications, however, the modes are usually excited by a metallic circular waveguide with a dielectric rod or tube inserted into its cone-shaped end (Fig. 5.20). The aperture diameter of the cone must be approximately equal to twice the limit radius ρ 0 of the dielectric wire mode. The limit radius is defined as the radius of a cylinder that coaxially encloses the volume in which roughly 90% of the total energy is transported.

5.4.3 Optical Fibers As an optical fiber (or optical waveguide), the cylindrical dielectric waveguide covered in Sect. 5.4.2 plays a crucial role in optical communications. In the arrangement shown in Fig. 5.15, the total reflection at the boundary surface from the dielectric with higher optical density to the air with lower optical density is disrupted at each contact. In optical fibers used in communications, the fiber core is thus enclosed by

392

H. Arthaber

Fig. 5.20 Excitation with a conical horn

a fiber cladding with a somewhat lower refractive index, which itself is adjacent to the outside or has a coating for mechanical reinforcement.

5.4.3.1

Structure of Optical Fibers and Their Refractive Index Profile

The breakthrough in usage of the optical fiber for optical transmission of signals was not achieved until the 1970s when it became feasible to manufacture quartz-based optical fibers with extremely low attenuation and low dispersion for the wavelength range around 1 μm. Their main advantage over metallic waveguides is their low attenuation and large transmission bandwidth. Light at a frequency of >100 THz is used as the carrier signal while the modulation is in the MHz to GHz frequency range. The optical properties of an optical fiber are determined by the refractive index profile n(r, ϕ, z) which is defined with the aid of relative permittivity εr (r, ϕ, z): n(r, ϕ, z) =

&

εr (r, ϕ, z).

(5.4.29)

For the permeability of materials for optical fibers, we have: μ = μ0 . The refractive index n is thus the reciprocal factor by which the phase velocity vph of light in the dielectric is less than in a vacuum: vph = c/n (c is the speed of light in a vacuum). In general, the refractive index profile is independent of the coordinates ϕ and z. An important class of refractive index profiles is represented by the following relationship according to Gloge and Marcatili [23]: n(r ) =

! g for r < a, n 0 1 − 2 ar √ n 0 1 − 2 = n a for r ≥ a.

(5.4.30)

5 Field-Based Description of Propagation on Waveguides

a

393

b Coating

r

r Cladding Core

a n (r)

–a

a –a

nan0

n (r) nan0

Fig. 5.21 Refractive index profile and longitudinal section of a step index fiber, b gradient index fiber

Here, a is the core radius and (r/a)g characterizes the index in the core with the profile parameter g. Moreover, n0 is the refractive index on the fiber axis and na is the refractive index of the cladding. should be interpreted as a relative refractive index difference: =

n0 − na n 20 − n 2a ≈ . 2 n0 2n 0

(5.4.31)

The refractive indexes of the core and cladding are very close to one another (typical value for quartz glass n ≈ 1.5) and the refractive index difference is thus on the order of 1%; the above approximation holds for weakly guiding fibers with 1. Figure 5.21 shows the refractive index profile for the cases of the step index fiber (g → ∞) and the gradient index fiber (g = 2).

5.4.3.2

Geometrical Optics and Wave Optics

The theory of wave propagation in an inhomogeneous medium follows from Maxwell’s equations as demonstrated in Sect. 5.4.2 for the cylindrical dielectric waveguide. Solution of the wave equations derived from Maxwell’s equations leads to the field distributions and propagation constants for the modes (or eigenmodes, wave types) that can be propagated. Optical fibers on which many modes can be propagated are known as multimode fibers. In case of a sufficiently small core diameter 2a, only the fundamental mode can be propagated and the fiber is said to be a monomode fiber. For multimode fibers, the way in which light is transported in the fiber core can be characterized very clearly based on the total reflection of beams at the boundary surface from the medium with higher optical density to the medium with lower optical density. We can calculate the beam paths using geometrical optics. For multimode fibers, frequently only the propagation speed of the individual modes or the connected mode dispersion is relevant and the amplitude distribution of the fields plays no role. In such cases, it is adequate to characterize the propagation effects with the aid of geometrical optics.

394

H. Arthaber

Although wave theory holds for general refractive index profiles, the beam pattern from geometrical optics can only be applied to such structures with good accuracy if their dimension is large with respect to the light wavelength λ, i.e. optical fibers where 2a λ. A typical value for the diameter of the fiber core in multimode fibers is 2a = 50 μm at a wavelength λ ≈ 1 μm. The beam pattern is physically apparent but it represents a rough approximation; it does not suffice if we wish to understand all of the properties of optical fibers. Accordingly, the effect that not any arbitrary number of discrete light beams but rather only a finite number thereof at angles γ i < γ total (critical angle of total reflection) can be propagated in an optical fiber can be explained only by interference due to the wave nature of light. Summaries of the theory of optical fibers can be found in [24–35]. Reference [26, Chap. 6] compares solutions obtained using geometrical optics with various approximate solutions based on wave theory.

5.4.3.3

Characterization of Multimode Optical Fibers Based on Geometrical Optics

In the limit case where the wavelength λ of the applied electromagnetic radiation is small with respect to the dimensions of the optical components that are used, geometrical optics can be employed to characterize the wave propagation. When applied to optical fibers, this means, for example, that light beams that are totally reflected at the core/cladding boundary surface in a step index fiber do not penetrate into the cladding, which is contrary to the results from Sect. 5.4.2. In geometrical optics, it is assumed that light propagates approximately by means of laterally restricted light beams. These light beams are represented by local plane waves. In an inhomogeneous dielectric material, the following two wave equations for the electric and magnetic fields can be derived from Maxwell’s equations in case of harmonic excitation [30]: grad ε E , ε

(5.4.32)

grad ε × curlH. ε

(5.4.33)

E = −k 2 n 2 E − grad H = −k 2 n 2 H −

In a weakly inhomogeneous dielectric material,10 the term (grad ε/ε) can be neglected such that the same wave equation is obtained for the electric and magnetic fields: E = −k 2 n 2 (x, y, z)E, H = −k 2 n 2 (x, y, z)H. 10

(5.4.34)

For a gradient index fiber, the dielectric is only weakly inhomogeneous due to the small refractive index difference and the continuous curve of the refractive index. The same also applies to a real step index fiber since the refractive index curve is actually smoothed as opposed to having an exact step form.

5 Field-Based Description of Propagation on Waveguides

395

In Cartesian coordinates, these wave equations reduce to the scalar wave equation which holds for each component of the fields: A = −k 2 n 2 (x, y, z)A.

(5.4.35)

The simplest solution of the wave equation in the homogeneous medium is the harmonic homogeneous plane wave (see Sect. 5.3.1) for which each component is characterized by a constant complex amplitude. In the weakly inhomogeneous medium, waves no longer propagate in straight lines but rather are bent; however, plane waves are present locally. Thus, in the weakly inhomogeneous medium, we apply a wave with a complex amplitude that varies slowly: ˆ A(x, y, z) = A(x, y, z) · e−jk S(x,y,z) .

(5.4.36)

Here, S(x, y, z) is known as the normalized phase, optical path length or eikonal. Wavefronts are surfaces with constant phase such that the following applies to them: S(x, y, z) = const. Using the approximation from geometrical optics λ → 0 (or k → ∞), application of Eq. (5.4.36) in the scalar wave Eq. (5.4.35) leads to what is known as the eikonal equation: (grad S)2 = n 2 .

(5.4.37)

This is the basic equation of geometrical optics since it ties the refractive index distribution to the behavior of the wavefronts. However, it is often more practical to consider the behavior of the light beams instead of the wavefronts. Figure 5.22 illustrates the relationship between beams and wavefronts. Here, s is the path and

S

S + dS

n0 grad n

n0 + Dn n0 + 2Dn

dr (s)

s=0

n0 + 3Dn n0 + 4Dn n0 + 5Dn

s

r(s)

r(s + ds)

0 Fig. 5.22 Wavefronts and beams in an inhomogeneous dielectric material

396

H. Arthaber

a

b

c

d

Fig. 5.23 Beams and their projection on the fiber cross-section in a step index fiber. a Meridional rays; b skew rays and in a gradient index fiber; c meridional rays, d skew rays

r(s) is the position vector along the observed beam. dr is the differential change in the position vector on the path ds. Using the eikonal equation, we can derive an equation to calculate the behavior of the beam r(s): dr d n = grad n. ds ds

(5.4.38)

This is known as the beam differential equation of geometrical optics. It says that beams in an inhomogeneous dielectric material are curved in the direction of increasing refractive index (grad n). In relation to beam propagation in optical fibers, we must differentiate two beam types: meridional rays cross the fiber axis and travel in a plane. On the other hand, skew rays propagate in a helical pattern and have an external and an internal cylindrical boundary surface (caustic). For a ray that is transported in a step index fiber, a polygonal line arises due to the total reflection at the core/cladding boundary surface as shown in Fig. 5.23. One of the most important properties of an optical fiber is the dispersion. This is defined as the spreading of a short optical pulse on the path through the optical fiber. In a multimode fiber, the mode dispersion generally dominates. It arises due to different phase constants for the individual modes that can propagate. Using geometrical optics, we can calculate a good approximation of the mode dispersion in a multimode fiber. From the perspective of geometrical optics, mode dispersion represents multipath propagation. The difference between the longest and shortest optical path length (S max and S min ) through the fiber core determines the mode dispersion. The optical path length S is determining by integrating with respect to the refractive index along the path of the considered light beam: S=

n(s)ds.

(5.4.39)

5 Field-Based Description of Propagation on Waveguides

397

The temporal spreading due to mode dispersion is thus T = Tmax − Tmin =

1 (Smax − Smin ). c

(5.4.40)

For the sake of simplicity, we will only consider meridional rays in the following discussion. In a step index fiber, the path traveled through the fiber core has a zigzag shape. The optical path length is determined by the angle γ between the considered beam and the fiber axis. From the perspective of geometrical optics, the angle γ can assume a continuum of values between zero and the critical angle γ total = arccos(na /n0 ) right at the limit of total reflection. The temporal spreading is thus 1 n 20 L n0 L T = − n0 L ≈ · . c na c

(5.4.41)

It is proportional to the length L of the optical fiber and the relative refractive index difference. For realistic numerical values (n0 = 1.48; na = 1.46; = 1.38%; L = 5 km), a relatively large value is obtained for the temporal spreading (T = 340 ns) such that multimode step index fibers can be used only for short distances and slow data rates. The temporal spreading can be reduced by using a profile with a continuous change in the refractive index (gradient index fiber) instead of the step index refractive index profile. Here, the refractive index profile should have a curve such that a beam propagates faster the further it is located from the fiber axis. The refractive index profile must be optimized such that all transported beams have on average as close as possible to the same propagation velocity in the z direction. Using a quadratic Gloge and Marcatili index profile of Eq. (5.4.30) with g = 2, approximate propagation time equalization can be attained such that the mode dispersion is considerably reduced. The path of a meridional ray is obtained by solving the beam differential equation in cylindrical coordinates (r, ϕ, z) where we set (∂/∂ϕ) = 0. The radial component of the beam differential equation and the component in the propagation direction are as follows: dr dn d n = , (5.4.42) er : ds ds dr dz d ez : n = 0. (5.4.43) ds ds Integration of Eq. (5.4.43) leads to the beam invariant: n

dz = n(r ) cos γ (r ) = n γ = const. ds

(5.4.44)

Here, γ is the angle between the light beam and the fiber axis. The beam invariant nγ is determined by the entrance location and entrance angle into the fiber core and is

398

H. Arthaber

constant for every beam. Using the quadratic index profile (g = 2), we can transform Eq. (5.4.42) into the following oscillating differential equation: 2n 20 d2 r + r = 0. dz 2 a 2 n 2γ

(5.4.45)

Solution of this equation leads to harmonic oscillations characterizing how the beam travels through the fiber core: z r (z) = rˆ cos 2π + ϕz . zp

(5.4.46)

The spatial period zp of the oscillations is dependent on the amplitude rˆ since the beam invariant nγ which can be expressed with rˆ is contained in the oscillating differential Eq. (5.4.45): zp =

2 2 rˆ πa 1 − 2 . a

(5.4.47)

Since the spatial period is not equally large for all amplitudes rˆ , the optical path lengths of different beams through the fiber do not exactly coincide. The optical path length for long fibers (L zp ) is '

( 1 2 rˆ 4 S ≈ n0 L 1 + . 2 a

(5.4.48)

This equation shows that the beam that passes straight through (ˆr = 0) has the shortest optical path length. The longest optical path length is associated with a beam that is right at the limit of transport in the fiber core (ˆr = a). The temporal spreading is thus T ≈

n0 L 1 2 · . c 2

(5.4.49)

If we use the same numerical values like for the step index fiber, we obtain significantly lower temporal spreading due to mode dispersion compared to the step index fiber: T = 2.35 ns. However, this does not represent the minimum possible temporal spreading. The optimum index profile deviates only slightly from the quadratic profile; it has the parameter g = 2 − 2. The minimum temporal spreading is obtained with this refractive index profile: T ≈

n0 L 1 2 · . c 8

(5.4.50)

5 Field-Based Description of Propagation on Waveguides na n0

g (r )

399

2a

Jmax

Fig. 5.24 Calculation of the numerical aperture for a gradient index fiber, acceptance angle ϑ max and acceptance cone

By means of a continuous transition in the refractive index from the core to the cladding, the mode dispersion in a gradient index fiber can be reduced by two to three powers of ten compared to a step index fiber. It can be demonstrated that with a Gloge and Marcatili index profile, the meridional rays and skew rays have the same temporal spreading [28]. Up to now no refractive index profile has been discovered to exactly equalize the propagation times of meridional rays and skew rays. In order for a beam to be transported in a fiber, the angle ϑ between the beam and fiber axis (Fig. 5.24) must not exceed a certain value. At the other end of the fiber, light is likewise emitted only below this angle. For the sake of simplicity, we will only consider meridional rays. Applying Snell’s law to the fiber end face gives the following: n(r ) · sin γ (r ) = 1 · sin ϑ.

(5.4.51)

A beam that strikes the cladding (i.e. a beam at the limit of transport in the fiber core) has the beam invariant: n(r ) · cos γ (r ) = n a · cos 0.

(5.4.52)

Applying this equation with the law of refraction at the fiber end face and solving for the sine of the maximum permissible angle of incidence ϑ max (known as the numerical aperture AN ), we obtain: AN = sin ϑmax =

!

n 2 (r ) − n 2a .

(5.4.53)

The maximum permissible angle of incidence ϑ max is known as the acceptance angle. For a step index fiber, the numerical aperture is independent of the radius r. For a gradient index fiber, the numerical aperture continuously decreases from the !

maximum value at the center AN = n 20 − n 2a to zero at the core/cladding boundary. If we also consider skew rays, we obtain acceptance cones with a half aperture angle that is equal to the acceptance angle ϑ max (r). The numerical aperture governs the coupling efficiency between the optical signal source and the optical fiber. When coupling a large-scale planar Lambertian radiator, e.g., LED planar emitter, a gradient index fiber with g = 2 accepts exactly half as

400

H. Arthaber y

kn(r )

y

kj ktr

g

kr j

b

x

J

z x 2a

2a

Fig. 5.25 Local wave vector kn and its components k ϕ , k r and β

much power as a step index fiber. The lower coupling efficiency is the cost of the significantly lower mode dispersion. In our discussion so far on the propagation of beams in multimode fibers, a mode continuum was assumed. However, wave optics shows that actually only a finite number of modes can be propagated because the wavelength is not infinitely small with respect to the core diameter of a multimode fiber. We can consider this fact using transverse resonance conditions (self-consistency conditions) analogous to Eq. (5.4.9). Here, the wave vector kn of a beam is broken down into the azimuthal component k ϕ , radial component k r and axial component β (see Fig. 5.25). Everywhere in the optical fiber, the magnitude k n of the wave vector is equal to the product of the local refractive index n(r) and the wavenumber k = 2π/λ where λ is the wavelength of light in free space: r kn (r ) = n(r )k = n 0 k 1 − 2 a

g

for r < a,

(5.4.54)

In this √ manner, k n always lies between the extreme values k 0 = n0 k and k a = na k = k 0 1 − 2. The axial component of the wave vector of a mode is the propagation coefficient β whose value is location-independent. Unlike β, the azimuthal component k ϕ and the radial component k r are not constant along the propagation direction. The transverse resonance condition requires a beam to constructively interfere with the original after one revolution about the fiber axis. We thus obtain the following for the azimuthal component k ϕ of the wave vector:

5 Field-Based Description of Propagation on Waveguides

401

2π kϕ · r dϕ = kϕ · 2πr = v · 2π

⇒

kϕ =

v . r

(5.4.55)

0

Here, v is the azimuthal mode number (v = 0, 1, 2, …). It determines that 2v intensity maxima lie on the circumference (r = const). In conjunction with k ϕ according to Eq. (5.4.55), we can express the radial component as follows: kr =

kn2 (r ) − β 2 −

v r

2

.

(5.4.56)

For modes that are transported in the fiber with low losses, all components of the wave vector k r , k ϕ and β must be real. From Eq. (5.4.56), it is immediately clear that the expression under the root must be positive. At the boundary with geometrical optics, this implies that light beams exist in the range v

2

r

< n 2 (r )k 2 − β 2

(5.4.57)

> n 2 (r )k 2 − β 2

(5.4.58)

while there is shade in the range v r

2

For different types of modes, Fig. 5.26 illustrates the right and left sides of the inequalities (5.4.57) and (5.4.58) for a gradient index fiber. The intersections of the functions determine the radii r 1 to r 3 which demark the limits between light and shade. These cylindrical boundary surfaces are known as caustics. The shaded ranges are hatched in Fig. 5.26. Case a represents meridional modes (meridional rays) with v = 0 which have only an outer caustic for r = r 2 . Beams can propagate only in the range |r| < r 2 . For generally guided modes (skew rays—case b), we obtain the two caustics r = r 1 and r = r 2 between which k r is real and light beams can propagate. For the special case in which the two caustics r = r 1 and r = r 2 converge, i.e. r 1 = r 2 = r H (dashed lines in Fig. 5.26b), we obtain helix modes (helix rays). They propagate in a helical pattern and the spacing of the beams from the fiber axis is constant. For step index fibers, the outer caustic is determined by the core/cladding boundary surface: r 2 = a. Case c (β < k a ) reveals a class of modes that are guided but which also simultaneously radiate power. These more or less highly lossy and thus attenuated modes are known as leaky modes (leaky rays). The main difference between guided modes and leaky modes is that a third caustic arises for r = r 3 . The radiation of a beam transported in the range r 1 < |r| < r 2 cannot be explained with the aid of geometrical optics. Wave optics is required to demonstrate that in the shade range r 2 < |r| < r 3 , a fast decaying (evanescent) field is present such that power can also reach the range

402

H. Arthaber

a

b

(n/r ) 2

kn2 – b 2

kn2 – b 2

(0/r )2

–a

0

– r2

r

a

r2

– rH

–a

– r2 – r1

c

rH

0 r1

r

a r2

d

kn2 – b 2

(n/r ) 2

(n/r)2 kn2 – b 2

– r3 – a – r2

– r1

0

r1

r2 a

r3

r

–a

– r1

0

r1

a

r

Fig. 5.26 Differentiation of different types of modes for a gradient index fiber by analyzing inequalities (5.4.57) and (5.4.58). a Meridional modes; b generally guided modes; c leaky modes and d radiation modes

|r| > r 3 and be radiated. This phenomenon can be interpreted as a tunneling effect in which photons pass through the forbidden shade range region. If β 2 is too small (case d), the light beams can exit the core into the cladding; they lead to the continuum of radiated waves (refracted beams) which are also known as radiation modes (or unbound modes). These beams can be partially transported in the cladding; in this case, they are known as cladding modes. For ϕ = const, the transverse resonance condition requires a beam to constructively interfere with the original beam after a reflection on the inner caustic and a reflection on the outer caustic. For the radial component of the wave vector, we thus obtain: r2 kr dr +

φ1 + φ2 = (μ − 1)π. 2

(5.4.59)

r1

Here, φ 1 and φ 2 are the phase shifts which occur for the total reflections at the caustics. μ is the radial mode number (μ = 1, 2, 3, …); it determines the number

5 Field-Based Description of Propagation on Waveguides

a

403

b

m/V

m/v

b = Constant Guided modes

b = Constant

Radiation modes

Leaky modes

Guided modes

Leaky modes

Radiation modes

1/p 1/4

1/4

1/8

1/8

0

0.25

0.50

0.75

1.00

1.25

n/v

0

0.25

0.50

0.75

1.00

1.25 n/v

Fig. 5.27 Differentiation of different types of modes by the mode numbers v and μ relative to the fiber parameter V for a fiber with a quadratic power profile, b step index profile

of radial intensity maxima. For a given v, the propagation constant β is determined with Eq. (5.4.59). The mode numbers v = 0, 1, 2, … and μ = 1, 2, 3, … define what is known as the LPvμ modes which have fields with linear polarization due to the beam approach. We will consider this topic in greater depth in the next section. Figure 5.27 illustrates the different types of modes in the v/μ plane of the mode numbers. The parameter V introduced in Fig. 5.27 is known as the fiber parameter or normalized optical frequency since it is proportional to the optical angular frequency ω: ! √ ω n 20 − n a2 . (5.4.60) V = ak0 2 = a c The fiber parameter V is independent of the shape of the refractive index profile and has a value of about 50 in typical multimode fibers. The area of the individual regions in Fig. 5.27 multiplied by 4V 2 is equal to the number of modes contained in them. Here, the factor 4 can only be justified in terms of wave optics since each LPvμ mode (v = 0) can occur in two mutually orthogonal polarizations and in two different orientations (see Sect. 5.4.3.4). The total number of modes transported in the fiber is proportional to the square of fiber parameter V; for a waveguide with a power profile, we obtain [34]: M=

V2 g · . g+2 2

(5.4.61)

Thus, a gradient index fiber with g = 2 can transport only half as many modes as a fiber with a step index profile (g → ∞).

5.4.3.4

Characterization of Optical Fibers Based on Wave Optics

Geometrical optics cannot be applied in the case of optical fibers in which only a small number of modes can propagate (i.e. monomode fibers) since the diameter of

404

H. Arthaber

the fiber core is no longer large with respect to the wavelength of the transported signal. Accordingly, we generally require a solution to the vectorial wave equation. The result is summarized in Sect. 5.4.2 for a step index fiber in Eqs. (5.4.25) and (5.4.26). In optical communications technology, weakly guiding fibers are generally used in which the relative refractive index difference is less than one percent. In weakly guiding fibers ( 1), the guided waves in the core and cladding are almost purely transverse (consistently polarized) such that the field distributions can be determined as a very good approximation by solving the scalar wave Eq. (5.4.35). Analytical solutions to the scalar wave equation are known for the step index profile and for the infinitely extended parabolic index profile [34]. We will consider only the step index profile in the following discussion. Contrary to Sect. 5.4.2, we will use the standard nomenclature from the field of optical communications here. The scalar wave equation in cylindrical coordinates is given by the following expression: 1 ∂2 A ∂2 A ∂2 A 1 ∂ A + + 2 + k 2 n 2 A = 0. + ∂r 2 r ∂r r 2 ∂ϕ 2 ∂z

(5.4.62)

This partial differential equation can be solved using the separation approach: A(r, ϕ, z) = Ar (r ) · Aϕ (ϕ) · Az (z)

(5.4.63)

Applying this approach to the wave equation, we obtain three ordinary differential equations: ) v 1 ∂ Ar ∂ 2 Ar + k 2n2 − β 2 − + 2 ∂r r ∂r r

2

* Ar = 0,

(5.4.64)

∂ 2 Aϕ + v 2 Aϕ = 0, ∂ϕ 2

(5.4.65)

∂ 2 Az + β 2 Az = 0. ∂z 2

(5.4.66)

Equations (5.4.65) and (5.4.66) are oscillating differential equations with solutions that result in a sinusoidal or cosinusoidal dependence of the field amplitude on the angle ϕ and unattenuated wave propagation in the positive or negative z direction: Aϕ (ϕ) = c1 cos(vϕ) + c2 sin(vϕ),

(5.4.67)

Az (z) = e∓jβz .

(5.4.68)

Equation (5.4.64) is a Bessel differential equation with cylinder functions as its solution. In the fiber core in which the transverse component of the wave vector is

5 Field-Based Description of Propagation on Waveguides

405

real: ktr2 = kr2 + kϕ2 = k 2 n 20 − β 2 > 0,

(5.4.69)

we obtain Bessel functions, and in the cladding in which the transverse component is imaginary (ktr2 < 0), we obtain modified Bessel functions of the second kind (mod. Hankel functions). This result for the radial field distribution is the same as that for the Cartesian components E z and H z of the vectorial solution according to Eqs. (5.4.25) and (5.4.26). The complete solution for guided waves that propagate in the positive z direction is thus: Jv u ar [c1 cos(vϕ) + c2 sin(vϕ)]e−jβz for r ≤ a A(r, ϕ, z) K v w ar [c3 cos(vϕ) + c4 sin(vϕ)]e−jβz for r ≥ a

(5.4.70)

In the fiber core, we use the term ! u = aktrk = a k 2 n 20 − β 2

(5.4.71)

! w = −jaktrm = a β 2 − k 2 n 2a .

(5.4.72)

and in the cladding the term

The ratio of the coefficients c1 /c3 = c2 /c4 is obtained by matching the tangential fields at the core/cladding boundary surface. This matching leads to the characteristic equation with which we can determine the parameters u and w for the individual modes: u

K v+1 (w) Jv+1 (u) =w . Jv (u) K v (w)

(5.4.73)

The parameters u and w both depend on the propagation constant β, but it can be shown that the sum of u2 and w2 is independent of β and equal to the square of the normalized optical frequency: u 2 + w2 = V 2 .

(5.4.74)

Solution of the equation system (5.4.73) and (5.4.74) leads to value pairs u, w which we can use to calculate the propagation constant β according to Eq. (5.4.71) or (5.4.72) and the ratio c1 /c3 = c2 /c4 with the aid of the following relationship: c1 K v (w) . = c3 Jv (u)

(5.4.75)

406

H. Arthaber 8 LP01 LP11 LP21

7 6

LP02 LP12LP41LP03 LP51 LP61 LP13 LP22 LP32

LP31

V=6

LP42

5

w

4 3 2

LP71

V=2

1 0

1

2

3

4

5

6

7

8

9

10

11

u

Fig. 5.28 Graphical solution of the characteristic equation for V = 2 and V = 6

The equation system (5.4.73) and (5.4.74) can be solved numerically or graphically with the aid of Fig. 5.28. Each intersection between the arc V = const. and Eq. (5.4.73) corresponds to a mode LPvμ that can be propagated. The scalar solution of the scalar wave equation leads to linearly polarized LPvμ modes which are transverse electromagnetic (TEM) waves. Each LPvμ mode is associated with a mode quartet since each mode defined by the mode numbers v and u (with the exception of modes with v = 0) can exist in two mutually orthogonal polarizations (parallel ex or ey ) and two different orientations. The two orientations correspond to an angle dependency of the fields which can be proportional to cos ϕ or sin ϕ. In the case where v = 0, only two modes can be distinguished which have orthogonal polarizations. Figure 5.29 illustrates the distributions of the intensity (power density) of the lowest LPvμ modes. In weakly guiding optical fibers, the linearly polarized LPvμ modes can be represented as a linear combination of degenerated hybrid modes (see Sect. 5.4.2) which follow from the exact solution of the vectorial wave equation [26, 32–35]. As long as the normalized optical frequency V is less than the cutoff frequency V g = 2.405 of the first higher order mode LP11 , the fiber is monomode since only the fundamental mode LP01 (HE11 hybrid mode) can be propagated. At the cutoff frequency of the LP11 mode, about 1/5 of the power of the fundamental mode is transported in the cladding and a correspondingly larger share at lower normalized frequencies. It therefore does not suffice to manufacture only the core of the monomode fiber from extremely low-attenuation material; a cladding layer of suitable thickness must also exhibit low attenuation. In a monomode fiber with absolute cylindrical symmetry, two mutually independent LP01 modes can always be propagated that exhibit identical properties except for the polarization. Due to unavoidable asymmetries that are also caused by external

5 Field-Based Description of Propagation on Waveguides

407

LP03

LP02

LP12

LP22

LP01

LP11

LP21

LP31

LP41

LP51

Fig. 5.29 Qualitative depiction of the intensity distribution of the lowest-order LPvμ modes

influences such as pressure and temperature, coupling occurs between these modes such that the state of polarization at the end of a monomode fiber is undetermined and fluctuates over time. To avoid variation of the state of polarization along a monomode fiber, we can use polarization-maintaining fibers [33, 35, 36]. In such fibers, birefringence (different propagation constants for the two orthogonal polarizations) is exploited to obtain negligible coupling of the modes. Birefringence is produced with asymmetry in the fiber core (e.g. elliptical) and especially with mechanical tensions in the fiber core. Mechanical tensions are created by asymmetrically introducing materials with divergent thermal expansion coefficients. The mechanical tensions are established during cooling in the manufacturing process. The main benefit of an ideally circular monomode fiber is that no mode dispersion occurs. Nevertheless, dispersion still arises in a monomode fiber since the group delay of all guided modes (i.e. including that of the fundamental mode) is dependent on the optical frequency. Due to the finite spectral width of the optical signal source, a short optical pulse becomes spread since the waveguide has a different group delay for the individual spectral components of the transmit signal. This effect is known as waveguide dispersion. The group delay’s dependence on the frequency can be explained using Fig. 5.28 based on the characteristic equation: If the normalized frequency V is changed, the intersections with the curves are shifted in accordance with Eq. (5.4.73). The shifted coordinates of the intersections u, w lead to a different propagation constant, thereby also generally resulting in a different group delay. The waveguide dispersion is illustrated in Fig. 5.30: There, the effective group index n geff of the lowest modes is shown as a function of the normalized frequency V. The effective group index is defined as the refractive index of a homogeneous medium

408

H. Arthaber n0 (1+Δ) LP51

LP32 LP42 LP61

LP41 LP31

n0 LP22

ng

eff

LP21

LP01

LP11

LP12

LP02

LP03

LP13

n0 (1-Δ) 0

2

4

6

8

10

V

Fig. 5.30 Effective group index n geff of the lowest modes of a step index fiber as a function of normalized optical frequency V

in which a plane wave propagates at the same velocity as the observed mode in the waveguide. The group delay coefficient tg of a mode through a waveguide is thus obtained as follows: tg =

n geff . c

(5.4.76)

The frequency dependency can be interpreted as follows: At low normalized frequencies (close to the cutoff frequency V g of the observed mode), the field penetrates deep into the cladding and the refractive index of the cladding na determines the propagation velocity. At high normalized frequencies, the field of each mode is concentrated in the core such that the refractive index of the core determines the propagation constant. For each individual mode, we have lim n geff = n a ,

v→Vg

lim n geff = n 0 .

v→∞

(5.4.77)

The mode dispersion of a multimode fiber is also recognizable in Fig. 5.30: At high optical frequencies, the effective group indices of the different modes are distributed over the interval n0 ≤ n geff ≤ n0 (1 + ). This is exactly the range predicted by geometrical optics.

5 Field-Based Description of Propagation on Waveguides

5.4.3.5

409

Attenuation

The attenuation of an optical fiber is described by Beer-Lambert law: P(z) = P(0)e−α D z ,

(5.4.78)

i.e. the decrease in light power P takes place exponentially with fiber length z. The attenuation coefficient α D is composed of the material-dependent components of absorption and scatter along with the radiation component. Today, low-attenuation optical fibers are constructed on the basis of extremely pure quartz (SiO2 ) and doping materials such as Ge, B, P or F. Absorption losses arise due to impurities and the self-absorption of the glass material that is used. Selfabsorption arises in the ultraviolet (UV) range due to atomic electron transitions, but the absorption maximum lies at λ = 140 nm so we can neglect electron transitions in the infrared (IR) range. In the infrared range, absorption occurs due to excitation of molecular vibrations in the quartz or the doping materials. Pure quartz glass has absorption resonances at λ = 9, 12.5, 21 and 36.4 μm. Attenuation maxima occur at these resonances as well as their harmonics and mixing products; the absorption resonance with the shortest relevant wavelength is at λ = 3 μm. Superposition of the slopes of all of the absorption resonance spectra determines the attenuation of optical fibers in the wavelength range (λ ≥ 1550 nm). Scattering losses are caused primarily by light scattering at microscopic refractive index fluctuations in the amorphous glass material. Since the correlation length of these refractive index fluctuations in the glass network is very small with respect to the wavelength and the refractive index fluctuations are small with respect to the average refractive index, Rayleigh scattering occurs as a result of refractive index fluctuations. The attenuation contribution due to Rayleigh scattering is proportional to 1/λ4 , thereby causing higher attenuation towards the UV range [34]: αD =

2 4π3 2 · n i − n 2 Vi . 4 3λ " #$ %

(5.4.79)

VS

Here, ni is the refractive index of a single microscopic scattering body, n 2 is the quadratic average of the refractive index and V i is the volume of the scattering body. The average value V s that appears in Eq. (5.4.79) is known as the scattering volume. For liquids, the scattering volume is proportional to the absolute temperature due to molecular thermal agitation. For amorphous materials such as quartz glass, the solidification temperature is critical. In addition to microinhomogeneities in pure quartz glass, microscopic mixing fluctuations occur in doped quartz glass that is mixed from multiple materials which also cause Rayleigh scattering. The share of the Rayleigh scattering in the reverse direction which is transported by the optical fiber can be detected at the start of the fiber. This effect is exploited by optical time-domain reflectometers which transmit a short light pulse into the fiber

410

H. Arthaber 10 4 IRabsorption

a D (dB/km)

2 1 0.8 0.6 0.4 0.2 0.1 800

Rayleigh scattering 1000

1200 l (nm)

1400

160 0

1800

Fig. 5.31 Attenuation of typical optical fibers (monomode fibers and gradient index fibers)

and then measure the power of the backscattered light. In this manner, the attenuation coefficient can be determined from one end of the fiber based on the fiber length. The actual fiber length can also be determined if the effective refractive index is known. Scattering due to faults such as bubbles, inclusions and diameter fluctuations with dimensions in the range of the light wavelength or greater are known as Mie scattering. The associated losses only decrease proportional to 1/λm with the wavelength; the parameter m is in the range 0 < m < 4 depending on the size of the fault. In modern optical fibers, Mie scattering is generally negligible. Radiation occurs especially at bends in the fiber because a part of the guided modes in the straight fiber is transformed into highly attenuated leaky modes. This is especially critical in monomode fibers. Coating of the fiber generally helps to prevent microbends due to external deformations with a small period length. The basic limits on the attenuation of quartz glass are related to Rayleigh scattering in the range of short wavelengths and the absorption in the excitation of molecular resonances in the range of larger wavelengths. Figure 5.31 illustrates the range of the attenuation curve for a typical monomode fiber (this also corresponds to the attenuation range of a multimode gradient index fiber). With modern fibers, the attainable attenuation is close to the theoretical limit over a wide wavelength range. The minimum attenuation is attained for λ ≈ 1550 nm and is equal to α D ≈ 0.2 dB/km. Deviations from the theoretical curve occur primarily due to impurities. The attenuation due to impurities is proportional to the concentration thereof. For productionrelated reasons, a low level of water contamination occurs which leads to disruptive absorption resonances due to the harmonics at λ = 1.39 μm and λ = 0.93 μm resulting from OH vibration of the hydroxyl group at λ = 2.78 μm. Thus, contamination by OH ions with a weight proportion of 10–6 results in additional attenuation of approx. 48 dB/km at λ = 1.39 μm. Nowadays, contamination due to metal ions (e.g. Fe, Cr, Ni) is no longer a problem with quartz glass fibers since it is technically feasible to maintain concentrations εr μr 2a/m, both quantities are imaginary and α h = β h /j is real. In this case, no wave propagation is possible on the line, but there still exists a field that decays exponentially in the y direction corresponding to α h . This mode is known as the evanescent mode. The transition from the propagation mode to the evanescent mode √ occurs at λ0 = εr μr 2a/m. This wavelength is known as the cutoff wavelength λc : λc =

√ εr μr 2a/m.

(5.6.3)

The cutoff wavelength λc corresponds to a cutoff frequency c m . fc = √ εr μr 2a

(5.6.4)

A quantity k c is often introduced as well. In the treatment of higher modes by solving the wave equation, it appears as a separation constant that is dependent only on the geometry of the line. k c is linked to λc via the following equation: kc =

2π √ mπ . εr μr = λc a

(5.6.5)

We can consider E waves in a parallel-plate line as a dual case. They arise if we apply a polarization corresponding to Fig. 5.9b for the homogeneous plane wave. All of the equations given so far in this section also apply with no changes to E waves since β E = β h . Differences arise only in the wave impedances Z WH and Z WE . From Eqs. (5.3.39) and (5.3.40), it follows that Z WH = ZW /sin ϑ and for E waves Z WE = ZW sin ϑ and thus

5 Field-Based Description of Propagation on Waveguides

+ Z WH = Z H

1−

+ Z WE = Z H

1−

λ0 λc λ0 λc

419

2 ,

(5.6.6a)

.

(5.6.6b)

2

We use the integer m as an index to characterize the individual modes. It simultaneously indicates the number of sine or cosine half-cycles in the transverse field distribution along the path a. In the direction that is transverse and perpendicular to a, the field components are spatially in dependent. This is expressed by a second index 0. H m0 and E m0 fields with m = 1, 2, 3, etc. can therefore exist on a parallel-plate line. Compared to the dielectric waveguide, we find two characteristic differences: 1. 2.

For a finite a, there is no TE or TM mode with cutoff frequency f c = 0. (Conversely, the TEM wave with f c = 0 can naturally propagate.) For every frequency f , all of the modes can always exist (in accordance with the ratio f cm0 /f ) as evanescent or propagation mode.

5.6.2 The Rectangular Waveguide If we arrange two further conductive walls with a mutual spacing b ≤ a in a parallelplate line perpendicular to its walls and parallel to the propagation direction, we obtain a rectangular waveguide as shown in Fig. 5.35b. The two new plates have no influence on an H m0 field because the electric field lines are always perpendicular with respect to them and the magnetic field lines are parallel to them everywhere. The H 10 mode in the rectangular waveguide is the wave with the lowest cutoff frequency and is thus designated as the fundamental mode. However, an E m0 field cannot exist since without a dependence of the field components along the path b, a tangential electric field strength would occur at the new metal plates and would be short-circuited. Figure 5.36 shows the field pattern of the H 10 mode in the rectangular waveguide. We can imagine that H 0n fields in the rectangular waveguide arise through successive reflection of a homogeneous plane wave along the “parallel-plate line” with the spacing b between the two plates. E 0n fields are not possible for the same reason as E m0 fields. The interference principle can also be applied to H mn and E mn fields in the rectangular waveguide. Here, we imagine that a homogeneous plane wave with suitable polarization is incident on a metal wall as seen in Fig. 5.9 such that its plane of incidence no longer coincides with the y, z plane. Due to reflection of this wave at a second metal wall in the y, z plane, an interference pattern arises with nodal planes of the electric field strength along the z axis as well as along the x axis. We can again arrange metal walls in these planes and in the planes with spacings a and b without disrupting the resultant field pattern. This interference pattern also arises

420

H. Arthaber

x

z

a

y b

y

z

a

x

b

Fig. 5.36 Fields of the H 10 mode in the rectangular waveguide. a Electric field lines –––––, magnetic field lines ---- in three intersecting planes; b wall surface currents on the inner sides of the surfaces as a continuation of the displacement lines

through superposition of four homogeneous plane waves with a suitable inclination of the propagation directions and the appropriate polarization. The E 11 mode in the rectangular waveguide is the E wave with the lowest cutoff frequency. If we formulate the equations given for the parallel-plate line with λc , we can apply them with no changes to the rectangular waveguide too. However, the individual modes differ in terms of their k c or λc . In the general case, we now have the following

5 Field-Based Description of Propagation on Waveguides

421

lc

l c /2

b/n

b/n

l c /2 a/m

Fig. 5.37 Construction of the cutoff wavelength λc for a rectangular waveguide

in place of Eq. (5.3.5): kc2mn =

mπ a

2

+

nπ b

2

.

(5.6.7)

Further indexing of k c is not required since it is identical for the E and H fields. From Eqs. (5.3.7) and (5.3.5), we obtain the following for λc : λcmn =

√

εr μr &

2ab (mb)2 + (na)2

(5.6.8)

Using this equation with the geometric mean theorem, we can also determine λc graphically as illustrated in Fig. 5.37. Figure 5.38 shows a normalized representation of α and β as well as vph = ω/β and vgr = dω/dβ as a function of f / f c . The normalized representation of Z WH and Z WE as a function of f / f c appears in Fig. 5.39. Based on superposition of parallel-plate line waves or by solving the wave equation, we find the following for the field components of the H mn fields in a rectangular waveguide:

422

H. Arthaber 7

7 6

6

b 2p m e lc r r

4

4

3

3 b 2p m e lc r r

2

2

a 2p m e lc r r

1 0

uph u

ugr u

5

a 2p m e r r lc

5

,

uph u

1 ugr u

b =0 0

1

2

a =0 3 f/fc

4

5

6

7

0

Fig. 5.38 Phase constant β –––, attenuation constant α phase velocity vph and group velocity vgr in a normalized representation. Valid for all H and E waves in a homogeneous, lossless waveguide with arbitrary cross-sectional geometry. Differentiation into individual cases only with √ numerical value of f c or λc . We have v = c/ μr εr

4

Z WE Z WH , ZW ZW

3

2 Z WH 1 Z WE 0

1

2

3 f/fc

4

5

6

7

Fig. 5.39 Wave impedance Z WH and Z WE for a homogeneous lossless waveguide with arbitrary cross-sectional geometry in a normalized representation. Differentiation into individual cases only √ with λc or f c . We have Z W = Z 0 μr εr

5 Field-Based Description of Propagation on Waveguides

423

⎫ nπy −j 2π mπx z ⎪ cos e λh , ⎪ ⎪ a b ⎪ ⎪ ⎪ ⎪ π k μr mπ mπx nπy −j 2π ⎪ λh z+ 2 ⎪ ,⎪ E y = Hz0 2 Z 0 sin cos e ⎪ ⎪ kc εr a a b ⎪ ⎪ ⎪ ⎪ π mπx nπy −j 2π 2π mπ ⎪ ⎪ λh z− 2 ⎬ sin cos e , Hx = Hz0 2 λh k c a a b ⎪ mπx nπy −j 2π k μr nπ z− π , ⎪ ⎪ cos sin e λh 2 ⎪ E x = Hz0 2 Z 0 ⎪ ⎪ ⎪ kc εr b a b ⎪ ⎪ ⎪ ⎪ π mπx nπy −j 2π 2π nπ ⎪ λh z− 2 ⎪ cos sin e Hy = Hz0 , ⎪ ⎪ 2 ⎪ λh k c b a b ⎪ ⎪ ⎭ E y = −Z WH Hx , E x = Z WH Hy . Hz = Hz0 cos

(5.6.9)

For the E mn fields, we have nπy −j 2π mπx z sin e λh , a b π 2π mπ mπx nπy −j 2π λh z+ 2 E x = E z0 , cos sin e 2 λh k c a a b mπx nπy −j 2π k 1 mπ z+ π cos sin e λh 2 Hy = E z0 2 ! μ kc Z 0 r a a a

E z = E z0 sin

εr

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ,⎪ ⎪ ⎪ ⎬

⎪ π ⎪ mπx nπy −j 2π 2π nπ ⎪ λh z+ 2 ⎪ sin cos e , E y = E z0 ⎪ ⎪ λh kc2 b a b ⎪ ⎪ ⎪ 2π π ⎪ mπx nπy −j λ z− 2 ⎪ k 1 nπ h ⎪ sin cos e ,⎪ Hx = E z0 2 ! ⎪ ⎪ k c Z 0 μr b a b ⎪ ⎪ ⎪ εr ⎪ ⎪ ⎭ E x = Z WE Hy , E y = −Z WE Hx .

(5.6.10)

Figure 5.40 shows the field patterns for the H 11 and E 11 modes. In Eqs. (5.3.9) and (5.3.10) as well as in Figs. 5.36 and 5.40, we introduced the more customary orientation of the coordinate system with the z direction as the direction of propagation.

5.6.3 The Circular Waveguide The theoretical treatment of the circular waveguide is exactly analogous to that of the rectangular waveguide. The results for the waveguide wavelength λh , wave impedance Z WH and Z WE as well as the phase and group velocity are thus the same for rectangular and circular waveguides. Hence, it is only necessary to specify the mathematical representation of the E and H waves and determine their cutoff frequencies or cutoff wavelengths.

424

H. Arthaber

z

y

x

y

H11-field x

z

y

y

H11-field Fig. 5.40 Electric (–––) and magnetic (----) field lines in the rectangular waveguide for the H 11 mode; b E 11 mode. In the cross-sectional representations (right), the top left quarter of the a H 11 field corresponds to the bottom right quarter of the E 11 field if (––––) and (– – –) are swapped

The only conceivable solution to the wave equation in cylindrical coordinates which can hold for a circular waveguide is the Bessel function J m . In terms of the indexing of the field strengths with integers m and n, we have the following: For H waves, in the cross-section we have m diameters on which the axial magnetic field strength disappears, and n concentric circles about the waveguide axis with a diameter that does not disappear on which the electric field components that are tangential to these circles disappear. For E waves, in the cross-section we have m diameters and n concentric circles about the waveguide axis with a diameter that does not disappear on which the axial electric field strength disappears. For both modes, m simultaneously designates the number of periods on the circumference and the order of the Bessel function which represents the axial field components. For H fields, we have ⎫ −j 2π z Hz,m,n = Hz0 Jm (kc ρ) cos mϕe λh , ⎪ ⎪ ⎪ ! π ⎪ −j 2π ⎪ μr km λh z− 2 ,⎪ E ρ,m,n = Hz0 k 2 ρ Z 0 εr Jm (kc ρ) sin mϕe ⎪ ⎪ c ⎪ ⎪ 2π π ⎪ −j λ z− 2 ⎬ 2πm h Hρ,m,n = Hz0 λh k 2 ρ Jm (kc ρ) sin mϕe , c ! ⎪ −j 2π z− π ⎪ E ρ,m,n = Hz0 kkc Z 0 μεrr Jm (kc ρ) cos mϕe λh 2 , ⎪ ⎪ ⎪ ⎪ ⎪ 2π π ⎪ −j λ z− 2 2π ⎪ h Hρ.m,n = Hz0 λh kc Jm (kc ρ) cos mϕe , ⎪ ⎪ ⎭ E ρ = Z WH Hρ , E ϕ = −Z WH Hρ .

(5.6.11)

5 Field-Based Description of Propagation on Waveguides

425

At the waveguide wall, the field strengths E ϕ and H ρ must disappear. Here, it is obviously necessary that for ρ = D/2 the derivative of the Bessel function (Jm ) disappears. Just like the actual Bessel function (see Fig. 5.16), its derivative also has a discrete sequence of infinitely many zeros (roots). We designate these zeroes as xmn where m indicates the order of the corresponding Bessel function and n the number of the zero. Here, we do not count a zero for k c ρ = 0. The necessary boundary conditions are thus satisfied if kc

2π D D = = xmn 2 λc 2

The relationship for the cutoff wavelength of H mn modes is therefore λcHmn =

πD xmn

(5.6.12)

Table 5.3 lists the initial values of xmn . With the given explanation of the indices m and n it is now no longer difficult to develop the field pattern of an H wave in the rectangular waveguide Fig. 5.41a illustrates this principle based on the example of an H 32 mode. Corresponding to the periodicity of cos m ϕ = cos3 ϕ, three nodal diameters are initially plotted for which H z , H ρ and E ϕ disappear. We can determine the diameters of the n = 2 nodal circles on which H ρ and E ϕ disappear from the curve of the function Jm (x) = J3 (x) shown in the bottom half of the figure as twice the spacing between its first and second zero from the origin. The outermost nodal circle (in our example, the second), is simultaneously the waveguide wall. In the resultant sectors, the electric field lines of H waves, entirely run in a transverse plane, are shown in principle. Figure 5.41b shows the field pattern of an H 01 mode drawn in detail based on the described principles and Fig. 5.41c shows the detailed field pattern of an H 11 mode for a longitudinal section and a cross-section. For drawing the field patterns, the component representation of the waveguide field can be utilized. For E waves, we have

Table 5.3 The first three zeroes of the Bessel functions J0 (x), Jx (x) and J3 (x). A zero for x = 0 is not counted. x = k c ρ n

m 0

1 2 3

1

2

3

H 01

H 11

H 21

H 31

3.832

1.841

3.054

4.201

H 02

H 12

H 22

H 32

7.016

5.331

6.706

8.015

H 03

H 13

H 23

H 33

10.173

8.536

9.969

11.346

426

H. Arthaber D

a

1

2

φ 1

3

J3(x)

0.5

2

J3′ (x) J3(x)

Hz32 ~ J3(x) cos 3ϕ Hρ32 ~ J3′ (x) cos 3ϕ

Eφ32 ~ J3′ (x) cos 3ϕ Hφ32 ~ J3(x) sin 3ϕ Eρ32 ~ J3(x) sin 3ϕ

5

10

X≡kc⋅ρ

J3′ (x) X′32=8.01 b φ

D c φ

Fig. 5.41 a Nodal circles (– - – - – - –), electric field lines (–––) of an H 32 mode in the circular waveguide. Behavior of J 3 (x), J3 (x) and dependency of the field components on ρ and ϕ; b circular electric field lines (––––) and (ellipse-like) magnetic field lines (– – –) of the H 01 mode in the circular waveguide; c electric (––––) and magnetic (– – –) field lines of the H 11 mode in the circular waveguide

5 Field-Based Description of Propagation on Waveguides

427

⎫ −j 2π z ⎪ E z = E z0 Jm (kc ρ) cos mϕe λh , ⎪ ⎪ ⎪ ⎪ ⎪ 2π π 2π ⎪ −j λ z+ 2 ⎪ h ⎪ Jm (kc ρ) cos mϕe , E ρ = E z0 ⎪ ⎪ λh k c ⎪ ⎪ ⎪ 2π π ⎪ k 1 −j λ z+ 2 ⎪ h ! Jm (kc ρ) cos mϕe Hϕ = E z0 , ⎪ ⎪ ⎪ ⎪ k c Z 0 μr ⎬ ε r

2π m −j 2π z+ π Jm (kc ρ) sin mϕe λh 2 , 2 λh k c ρ 1 km −j 2π z− π ! Jm (kc ρ) sin mϕe λh 2 Hρ = E z0 2 k c ρ Z 0 μr

E ϕ = E z0

εr

E ρ = Z WE Hϕ ,

E ϕ = −Z WE Hρ .

(5.6.13)

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ,⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

The missing relationship for the cutoff wavelength again follows from the requirement that the field strengths E ϕ and H ρ must disappear for ρ = D/2. Designating the zeroes of J m as x mn , this condition is satisfied if kc

2π D D = = xmn 2 λc 2

and we obtain the following for the cutoff wavelength of an E mn mode in the circular waveguide: λcEmn =

πD xmn

(5.6.14)

Table 5.4 lists some of the zeroes. In order to graphically depict the field pattern of E waves, we again exploit our explanation of the indices m and n from Sect. 5.6.3. Unlike the case of H waves, we now find the diameters of the nodal circles on which E z , E ϕ and H ρ disappear as twice the spacing between the zeroes of J m (x) = J 3 (x). Figure 5.42a now plots Table 5.4 Initial three zeroes of the Bessel functions J 0 (x), J 1 (x) and J 2 (x) and J 3 (x). A zero for x = 0 is not counted. x = k c ρ n

m 0

1 2 3

1

2

3

E 01

E 11

E 21

E 31

2.405

3.832

5.136

6.380

E 02

E 12

E 22

E 32

5.520

7.016

8.417

9.761

E 03

E 13

E 23

E 33

8.654

10.173

11.620

13.015

428

H. Arthaber D

a –

+

+ +

– +

–

2

1

φ

+

– 1

3

+

–

2

J3(x) J3′ (x) 0.5 J3(x) Ez32 ~ J3(x) cos 3ϕ Eρ32 ~ J3′ (x) cos 3ϕ Hφ32 ~ J3′ (x) cos 3ϕ Eφ32 ~ J3(x) sin 3ϕ Hρ32 ~ J3(x) sin 3ϕ

5

10

X≡kc⋅ρ

J3′ (x)

X32=9.76

b

φ

D

φ

c

D

Fig. 5.42 a Nodal circles and magnetic field lines of an E 32 mode in the circular waveguide. Behavior of J 3 (x), J3 (x) and dependency of the field components on ρ and ϕ; b electric (––––) and magnetic (– – –) field lines of the E 01 mode in the circular waveguide; c electric (––––) and magnetic (– – –) field lines of the E 11 mode in the circular waveguide

5 Field-Based Description of Propagation on Waveguides

429

the nodal circles on which E z , E ϕ and H ρ disappear and the nodal diameters for the example of an E 32 mode. The outermost nodal circle is again simultaneously the waveguide wall. Also plotted is the curve of the magnetic field lines which for E waves run entirely in a transverse plane. As is clear from comparing the component equations, the magnetic field lines of the E waves have the same form as the electric field lines of the H waves (see Fig. 5.41). Figure 5.42b and c shows detailed field patterns for the E 01 mode and E 11 mode. Also note that the distribution of E z over the waveguide cross-section corresponds to the mechanical buckling of a circular membrane that is clamped on its circumference and suitably excited. The alternating + and – signs in Fig. 5.17a are intended to recall this fact. Unlike the rectangular waveguide, E 0n fields are possible in the circular waveguide. However, no E m0 or H m0 fields can exist since at least one nodal circle must always occur in the form of the waveguide wall.

5.6.4 Generalized telegrapher’s Equations. Waveguide Equivalent Circuits and Attenuation of Waveguide Waves If we introduce into Maxwell’s equations the field components in the direction of the transverse coordinates and the field components in the direction of the z coordinate, we obtain the following relationships which are equivalent to the original equations: curlz Et = −jωμez Hz , t = −jωμ(ez × Ht ), gradt E z − ∂E ∂t curlz Ht = jωεez E z , t = j ω ε(ez × Et ), gradt Hz − ∂H ∂t

(a) (b) (c) (d)

(5.6.15)

In these equations, the index t is intended to express that among the vector components indexed in this manner, only the transverse components are implied and the differential operator gradt only contains derivatives with respect to the transverse coordinates. For the transverse field components, we use: Et (x1 , x2 , z) = V (z)t E (x1 , x2 ), Ht (x1 , x2 , z) = I (z)t H (x1 , x2 ). Here, V (z) and I(z) are scalar functions that characterize the z dependency of the wave propagation process and the vectors t E and t H characterize the direction and spatial dependency of the transverse field components on the transverse coordinates x 1 and x 2 (e.g. x, y or ρ, ϕ). We will first consider the case of an E field where H z = 0 for which Eq. (5.6.15d) assumes the following form: −

dI (z) 1 t H = ez × t E jωεV (z) dz

430

H. Arthaber

Since on the right side of this equation there is only a function of the transverse coordinates, it cannot be a function of z. The product of the z-dependent functions dI(z)/dz and −1/(jωεV (z)) must therefore be equal to a constant which we will call 1/K E . We thus have the following two functions: dI (z) jωε =− V (z) = −YE V (z), dz KE

(5.6.16a)

t H = K E (ez × t E )

(5.6.16b)

By plugging Eqs. (5.6.16b) and (5.6.15c) into Eq. (5.6.15b) and rearranging, we obtain the following relationship: gradt (divt t E ) = t E

dV dz

+ jωμK E I(z) 1 K I(z) jωε E

.

On the left side of this equation, we again have only a function of the transverse coordinates; accordingly, the second factor on its right side must be equal to a constant √ which we will call −kc2 . In conjunction with the wavenumber k = ω με, it thus follows that KE 2 dV (z) = −j k − kc2 I (z) = −Z E I (z), dz ωε

(5.6.17a)

gradt (divt t E ) + kc2 t E = 0.

(5.6.17b)

The equation pair (5.6.16a) and (5.6.17a) is known as the generalized telegrapher’s equations for an E-wave waveguide. Fully analogously, we obtain the following for H fields with E z = 0: dV (z) = −jωμK H I (z) = −Z H I (z), dz

(5.6.18a)

t E = −K H (ez × t H ),

(5.6.19a)

j 2 dI (z) =− k − kc2 V (z) = −YH V (z), dz ωμK H gradt (divt t H ) + kc2 t H = 0.

(5.6.18b) (5.6.19b)

The generalized telegrapher’s equations for waveguides correspond to the waveguide equivalent circuits in Fig. 5.43a for H fields and in Fig. 5.43c for E fields. Contrasted to the equivalent circuits for TEM wave lines, the distinction is that for H fields a parallel inductance is added and for E fields a series capacitance is added.

a

kc2

m0 m r

•

Dz

KH

e0er Dz

KE

Dz =

e e DCs = 20 r • 1 Kc KE Dz

C'p • KH

e 0 er • Dz

R's • Dz

d

R's • Dz

b

L'si • Dz

L'si • Dz

C'p • Dz

L's • Dz

DRp

DLp

C'p • Dz

DL'pi

L's • Dz

e L's• Dz

DRp

DLp

DL'pi

DCp

Fig. 5.43 Equivalent circuits for rectangular and circular waveguides. a H fields, c E fields without, b, d with consideration of wall losses; L s , L p , C p and C s are frequency-independent. Due to the skin effect, Rs , L si , Rp and L pi are frequency-dependent, e waveguide equivalent circuit for H 0n modes in the circular waveguide

C'p • Dz =

L's • Dz = m0 mr DzKE

c

DLp =

L's • Dz = m0 mr • DzKH

5 Field-Based Description of Propagation on Waveguides 431

432

H. Arthaber

By means of the parallel inductance, the magnetic longitudinal field strength H z is taken into account for H fields while the series capacitance takes into account the electric longitudinal field strength E z for E fields. Up to now, the constants K E , K H and k c are undetermined. For E fields with H z = 0, it follows from Eq. (5.6.15a) that the transverse electric field strength is irrotational and thus can be represented as a gradient of a scalar function ϕ. The same holds for the magnetic transverse field strength in the case of an H field with E z = 0 [see Eq. (5.6.15c)]. We will call the associated scalar function Ψ . We thus have t E = –gradt ϕ for E fields and t H = –grad Ψ for H fields. If we now plug t E or t H into Eq. (5.6.17b) or (5.6.19b), respectively, we obtain the following since divt gradt = t : t ϕ + kc2 ϕ = 0, t ψ + kc2 ψ = 0. The constants k c are thus determined if these equations are solved taking into account the boundary conditions. They turn out to be separation parameter which are defined for rectangular waveguides by Eq. (5.6.7) and for circular waveguides by /D for H fields but by k c = 2x mn /D for E fields. More precise determinakc = 2xmn tion of the constants K E and K H involves the power transported by the waveguide. It is obtained by integrating the Poynting vector formed from the transverse field components with respect to the waveguide cross-sectional plane A. P=

1 Re 2

1 Etp × H∗tp · dA = ReVp Ip∗ (t E × t H ) · dA 2 A

A

For E fields, we set t H according to Eq. (5.6.16b) and for H fields, we eliminate t E corresponding to Eq. (5.6.19a). For the power transported by E waves, we then obtain 1 1 ∗ ∗ K E · t E · t E dA = ReVp Ip K E |t E |2 dA P = ReVp Ip 2 2 A

A

and the corresponding relationship for H waves is P=

1 ReVp Ip∗ 2

K H t H · t H dA =

1 ReVp Ip∗ K H 2

A

|t H |2 dA. A

In order to calculate the power like in the transmission line theory with Vp Ip∗ , the following conditions must therefore be satisfied:

1 Re 2

|t E |2 dA = 1,

KE A

(5.6.20a)

5 Field-Based Description of Propagation on Waveguides

433

|t H |2 dA = 1.

KH

(5.6.20b)

A

Regarding the p indices in the above equations, note that a wave is implied that propagates in the positive z direction. Even with the Eqs. (5.6.20), K E and K H are still undetermined. In order to determine them, we will proceed as described by Zinke [52]: For E fields with wall surface currents that are only oriented axially, we define I p as the integral over the magnitude of the displacement currents prevailing in the waveguide cross-section: Ip = ωε

E zp dA.

A

I p thus has the dimension of current and t H the dimension of reciprocal length. V p must then have the dimension of voltage and t E also has the dimension of reciprocal length. K E and K H can thus only be pure numerical factors. O. Zinke demonstrated in [52] that the electric longitudinal field energy stored per length unit in the waveguide field by E zp is equal to the energy stored in the series distributed capacitor C s : 1 ε 2

A

2 2 I 2 ωε A E zp dA E zp dA = 1 p = 1 . 2 ω2 Cs2 2 ω2 ε

We thus obtain the equation of condition for the constant K E : 2 E zp dA K E = A 2 . kc2 A E zp dA

(5.6.21)

For the K E01 mode in the circular waveguide, we have K E01 = 1/4π = 0.0796. For H waves, the current I p is calculated with the aid of the magnetic transverse field strength H tw oriented tangentially to the waveguide wall. Ip =

|Ht |w dst . St

In this equation, the transverse integration path S t is to be arranged such that it comprises all of the equally directed axial wall surface currents. If we now set the magnetic cross field energy stored per length unit in the waveguide field equal to the energy stored in the series distributed inductance L’s, we obtain the equation of condition for the constant K H :

434

H. Arthaber

2 Htp dA K H = A . 2 |H | ds t w t St

(5.6.22)

For illustration purposes, we will calculate the constant K H10 for the H 10 mode in a rectangular waveguide. For H tp = H x max sin π x/a [see Eq. (5.6.9)], we have b a 0

2 Htp dxdy = H 2 x max

0

sin2 (π x/a)dxdy = Hx2max ab/2, 0

0

a |Ht |w dst = Hx max

St

b a

sin(π x/a)dx = 0

2a Hx max . π

For the constant K H10 , it thus follows that K H10 =

Hx2 max ab/2 π2 b = Hx2 max (2a/π )2 8 a

Rauskolb [53] gives the constants for further modes in the rectangular and circular waveguides while Lorek [54] provides results for the ridge waveguide. Using the quantities of “equivalent voltage” V p and “equivalent current" I p determined in this manner, we can also specify a characteristic impedance Z L = V p /I p for a waveguide. It is equal to the wave impedance of the mode under consideration multiplied by the relevant constant K E or K H . These constants are pure numerical factors that depend only on the mode and line geometry: Z LH = Z WH K H ,

Z LE = Z WE K E .

The characteristic impedance of a waveguide governs the reflection coefficient that occurs, for example, if waveguides for the same mode but with different crosssectional dimensions must be interconnected. This is avoided if we interpose a λ/4 transformer, for example. Using the same formulae from Chap. 3, we can calculate its characteristic impedance and then apply the formulae in this section to determine its cross-sectional dimensions. Up to now, we have assumed that the metal walls of our waveguides exhibit ideal conductivity. Accordingly, the attenuation constant due to thermal losses of the waveguide is equal to zero for f > f c . For f < f c , the phase constant β is equal to zero and α is a pure return attenuation (Fig. 5.38). Practically speaking, however, we must always assume finite wall conductivity, which leads to a finite value of α for the waveguides even for f ≥ f c . Rigorous solutions of the wave propagation problem in lossy waveguides are then impossible if E and H waves are coupled via the wall surface currents in a waveguide with finite wall losses. An exception exists only for circularly symmetrical modes (m = 0) in the circular waveguide since for the E 0n

5 Field-Based Description of Propagation on Waveguides

435

modes the wall surface currents only flow axially and for the H 0n modes they only flow circularly and are thus not coupled. However, the rigorous solutions obtained for these special cases are rather unwieldy such that we must typically make do with more or less comprehensive approximations. The simplest but also the least comprehensive approximation method for calculation of α R is what is known as the power loss method. If P(z) and P(z + z) are the transported powers in two cross-sectional planes of the waveguide separated by z, we obtain the following for the distributed power dissipation P v per length unit: Pv z = P(z) − P(z + z) = P(z) − P(z) −

dP(z) z dz

i.e. Pv = −

dP(z) . dz

Given that P(z) = P0 e−2αz it follows that α=

1 Pv . 2 P(z)

(5.6.23)

The approximation provided by the power loss method consists in that Pv and P(z) are calculated with the aid of the field solution in the lossless waveguide (undisturbed wave type). P(z) is obtained by integrating the Poynting vector over the waveguide cross-section and Pv is calculated based on the wall surface current coefficient (n × Htan ) and the sheet resistance R of the waveguide walls. For the fundamental mode H 10 in the rectangular waveguide and for the H 11 mode and the H 01 mode in the circular waveguide, we obtain the following using the power loss method: For H 10 in the rectangular waveguide

α10

R 1 + 2 ab ffc = bZ W 1 −

For H 11 in the circular waveguide

fc f

2

2

,

(5.6.24)

436

H. Arthaber

) 2R α11 =

fc f

2

* +

D ZW 1 −

1 2 x11 −1

fc f

2

,

(5.6.25)

For H 01 in the circular waveguide α01 =

2R

2

fc f

DZW 1 −

fc f

2

,

(5.6.26)

For f = f c , infinitely high attenuation is always obtained according to these equations. This is true regardless of the specific mode that is chosen. In reality, the waveguide attenuation remains finite and the power loss method for calculation of the waveguide attenuation fails for f ≤ f c . This is because the transported power P(z) is calculated to be equal to zero for the lossless waveguide and f ≤ f c . However, the power loss method also always fails for f ≥ f c if in order to satisfy the boundary conditions in the lossy waveguide, E and H modes with the same f c must be applied (degenerate case) and they have wall surface current components in the same direction. For rectangular waveguides, this involves all E and H fields with the same index. For Eqs. (5.6.24) to (5.6.26), this case does not occur such that their implications are usable for f > 1.01 f c . If we have f f c , we can see that waveguide and the H 11√wave in the attenuation of the H 10 wave in the rectangular √ the circular waveguide increases with R = π fμ0 μr /κ proportional to f . The frequency dependency of α 01 in the circular waveguide exhibits special behavior: It decreases monotonically with f and proportional to l/(f )3/2 if f f c . Due to the low attenuation of optical fibers, wave propagation in circular waveguides with H 01 modes is no longer relevant for future long-distance telecommunications. Below and at the cutoff frequency of the H 01 mode, additional modes are also possible: E 11 , H 11 , E 01 , H 21 . Especially due to the fact that the H 01 mode has the same cutoff frequency as the E 11 mode, the requirement for mode conversion-free operation of the waveguide is therefore a task that can be managed, for example, by means of dielectric coating of the waveguide wall. A more extensive approximation is possible when calculating α by applying the waveguide equivalent circuits. Zinke [52] extended them by adding the wall sheet impedances for the longitudinal and transverse wall surface currents. Rauskolb [53] additionally considers the corresponding inner inductances (see Fig. 5.43b and d). The propagation constant γ = α + jβ is calculated using methodology from transmission line theory. Figure 5.44 shows the result for a rectangular waveguide. Figure 5.45a has an excerpt from this presentation. By way of comparison, Fig. 5.45b shows α 0 and β 0 for the lossless waveguide and α 1 based on the power loss method. In contrast, the attenuation now remains finite even for f = f c . The equivalent circuits must be extended with couplings in case of degeneracy with wall surface current

5 Field-Based Description of Propagation on Waveguides

437

10

10 b

1

1

10–1

10–1

10–2

10–2 b

10–3

10–3

10–4 10–5 10 Hz

a

100

1

10 kHz

100

1

10 MHz 100

1

10

b ( cm–1 )

aH10 (Np·cm–1)

a

10–4 10–5 100

f (GHz)

Fig. 5.44 Frequency response of attenuation constant α and phase constant β for a real waveguide (calculated according to Fig. 5.38) over a very wide frequency range. H 10 mode

components in the same direction. This is not the case for the attenuation curves in Fig. 5.46. Using the waveguide equivalent circuits, we can provide a very clear explanation of the abnormal behavior of the attenuation of H 0n modes in the circular waveguide. Since for these modes the magnetic field at the wall only has a longitudinal component, there are only circumferential currents in the wall which do not produce any longitudinal voltage drop. In Fig. 5.43b, Rs and L si are thus equal to zero. The equivalent circuit for H 0n modes in the circular waveguide is thus shown separately in Fig. 5.43e. As the frequency increases, according to this equivalent circuit the series connection of Rp and jω(L pi + L p ) is capacitively bridged such that the attenuation decreases with 1/f 3/2 . A very extensive method for calculation of the waveguide attenuation is based on methodology from perturbation theory. Details are given by Collin [22, p. 182] as well as in [55–57]. Perturbation theory can deliver usable results for the waveguide attenuation even in cases where the other methods we have described fail. Figure 5.47 illustrates the attenuations of the H 11 and E 11 modes for a rectangular waveguide. For comparison purposes, the attenuation values that were calculated using the power loss method as described in [22, p. 182] are also plotted. Clearly, αH11 turns out to be much greater and αE11 significantly less than the values calculated using the power loss method.

438

a

H. Arthaber 10

10 b

1

1

aH

10–1

10 –2

10–2

10 –3

10–3

10 –4

10–4

a

b

10 –5

10–5

10

b

b cm–1)

10 –1

)

–1 10 (Np·cm )

a

10

a1 b0

1

1

10–1

10–2

10–2

10–3

10–3

10–4

10–4

b cm–1)

10

aH

10–1

)

(Np·cm–1)

a0

a1 10–5

10–5

0.5·10–5 0 0.3 0.4

0.5·10–5

b0 0.6

0.8 1

a0 2 fc 3 f (GHz)

4

6

8

10

0 20

30

Fig. 5.45 a Section from Fig. 5.44; b by way of comparison, α 0 and β 0 according to Fig. 5.38 for the lossless waveguide with H 10 field. Curve for α 1 calculated from the wall surface currents according to Eq. (5.6.24)

5.6.5 Coaxial Line with Higher Modes In addition to Lecher waves, higher modes can also occur on a coaxial line under certain conditions. In our exploration of such modes, we will limit our consideration to a lossless coaxial line with air as the dielectric. We will ignore the influence of supporting spacers like spiral strands.

5 Field-Based Description of Propagation on Waveguides

a

2.0

b

439

3.0

1.5 2.0 α/αcoaxial

α/αcoaxial

1.0 H10

0.7

H11

0.5 0.04

0.020

(

H11

m

Np·cm

3/2

(

D

0.005 0.004

f

m

Np·cm

0.02

( α·D/2·√D/2

0.010

b=a/2

a=D

0.007

D

0.03

3/2

H10

E11

0.15

E21

E01

0.01 10 1

0.003

15 20

30 40 50 70 100 GHz·cm

1.5

3

2

4

5

7

10

300 20

30

f(for D = 10cm) (GHz) H01

0.002

0.001

E11 E21

0.68 0.56

0.7

0.4 0.233

(

E01

1.0

0.5 0.3

α·D/2·√D/2

1.5

10

1

15 20

1.5

2

30 40 50 70 100 GHz·cm 3

4

5

7

300

10

20

30

f(for D = 10cm) (GHz)

Fig. 5.46 a Attenuation of the H 11 and H 01 modes in the circular waveguide in comparison to attenuation of the H 10 mode in the rectangular waveguide. Also plotted are the values referred to σ coaxial of a Lecher wave [52]. b Attenuation of the E 01 , E 11 and E 21 modes in the circular waveguide. Also plotted are the values referred to α coaxial of a Lecher wave [52]

Because the electromagnetic field is present solely in the dielectric due to the assumed infinitely good conductivity of the inner and outer conductors, i.e. the regions 0 ≤ ρ ≤ d/2 and ρ ≥ D/2 are entirely field-free, we apply the following solutions for the electric and magnetic longitudinal field strength, respectively: E z = C1 Jm (r ) + C2 Nm (r );

Hz = C3 Jm (r ) + C4 Nm (r ).

(5.6.27)

On the conductor surfaces ρ = d/2 and ρ = D/2, the infinite conductivity requires the following: Ez = 0 ∂ Hz =0 ∂ρ

for

ρ=

d and 2

D . 2

& & Using the terms d2 γ 2 + k 2 = rd and D2 γ 2 + k 2 = rD , we obtain the following conditional equations for γ from these boundary conditions: Nm (rd ) Nm (rD ) = , Jm (rd ) Jm (rD )

(5.6.28)

440

H. Arthaber 6 •10–4

a (Np/cm)

4

H11

3 E11 2

H11 1 E21 0

2

4

6

8

10

f / fc

Fig. 5.47 Attenuation of the H 11 and E 11 modes in the rectangular waveguide for a = 2b = 2.54 cm. Conductor material = Copper Perturbation method; power loss method

Nm (rd ) N (rD ) = m , Jm (rd ) Jm (rD )

(5.6.29)

Equation (5.6.28) determines γ for modes in the coaxial cable with an electric longitudinal field strength and Eq. (5.6.29) for modes with a magnetic longitudinal field strength (E and H waves). A rigorous, closed solution of these transcendent equations is not possible. However, the following expressions for the cutoff wavelengths of waveguide types in the coaxial cable provide a relatively straightforward approximation [2]: λcE01 ≈ D − d, λcH11 ≈ π

D+d . 2

(5.6.30)

The H 11 mode in the coaxial cable is the mode with the largest cutoff wavelength. Waveguide modes in coaxial lines are practically unused. They only play a negative role in the form of undesired interfering waves. Accordingly, we must make certain to preserve a well-defined Lecher wave during experimentation with a coaxial measuring line, for example. According to Eq. (5.6.30), for example, well-defined operation of a coaxial line with d = 6 mm and D = 16 mm is possible only up to

5 Field-Based Description of Propagation on Waveguides 0.6

441

N0(r ) N (r ) 1 N2(r )

0.4

2 pr

0.2 0 –0.2 2 − pr

–0.4 –0.6 –0.8 –1.0 –1.2 –1.4 0

2

4

6 r

8

10

12

Fig. 5.48 Neumann functions versus argument r. Here, we have r = k c ρ. N 0 (r) = Neumann function 0th order, N 1 (r) = Neumann function 1st order, N 2 (r) = Neumann function 2nd order, with argument r

about λ = 3.5 cm. Figure 5.48 illustrates the behavior of the Neumann functions N 0 (r), N 1 (r) and N 2 (r).

5.7 Components Used in Waveguide Technology In the preceding sections, we treated the waveguide exclusively as a tubular element which is used as a transmission line between a transmitter and a receiver. For test and measurement purposes as well as for construction of complete transmission systems, however, various waveguide components are required additionally and we intend to discuss certain key components here. For an in-depth treatment of waveguide components, see [17].

5.7.1 Junctions with Rectangular Waveguides We can differentiate between parallel and series junctions. For a parallel junction, it is characteristic that at the junction point the currents (and magnetic fields) of the junction lines divide according to their loads into I 3 and I 4 while the voltage and the electric fields are equal for both. For the main line, the loads transformed to the junction point appear to be connected in parallel (Fig. 5.49a).

442

H. Arthaber

a

b (3)

(3) I3

I1

I2 I4

(1)

U3 (2)

U1

U2 U4

(4) (4)

c

(2)

d (3)

(3)

(1)

(4)

e

(4)

f

(2)

(1) (3)

(4)

(3)

(4)

Fig. 5.49 Junctions for the H 10 mode. a Parallel junction; b series junction; c H junction; d E junction; e splitting of E when supplying port (1); f splitting of E when supplying port (2)

For the series junction, the voltage V 2 is divided out of phase with respect to the junction line into V 3 and V 4 . This is illustrated in Fig. 5.49b. A waveguide junction in the plane of the H field lines of an H 10 mode exhibits the same behavior as a parallel junction (Fig. 5.49c and e), while a junction in the plane of the E field lines behaves like a series junction (see Fig. 5.49d and f). By combining an E junction with an H junction, we obtain the four-port network in Fig. 5.50. Its properties are best characterized by the scattering matrix S. Here, we assume that only the H 10 mode can propagate in the individual arms of the waveguide. Based on the labeling of the ports in Fig. 5.50 and by exploiting the symmetry, we can state the following for the elements of S: In

5 Field-Based Description of Propagation on Waveguides (3)

443

(2)

(4)

(1)

Fig. 5.50 E–H junction for the H 10 mode, commonly called “magic tee” for 3 dB split ratios

⎛

S11 ⎜ S21 S=⎜ ⎝ S31 S41

S12 S22 S32 S42

S13 S23 S33 S43

⎞ S14 S24 ⎟ ⎟ S34 ⎠ S44

we have S13 = S14 but S23 = –S24 . Since the four-port network exhibits transmission symmetry, we can also state in general that Sμv = Svμ . Finally, we see that ports (1) and (2) are decoupled from one another. If a signal is supplied to port (1), only an E wave could be excited in arm (2) and if a signal is supplied to port (2), only an H 20 mode could be excited in arm (1) (see Fig. 5.49f). However, since only the H 10 mode should be propagated, we have S12 = 0 and S21 = 0. For the S-matrix of an E–H junction, we can thus write ⎛

SE,H

S11 0 ⎜ 0 S22 =⎜ ⎝ S13 S23 S13 −S23

S13 S23 S33 S34

⎞ S13 −S23 ⎟ ⎟. S34 ⎠ S44

(5.7.1)

If the two ports (1) and (2) are configured with sliding short-circuits (reactance lines, stub lines), we obtain an “E–H tuner”. This device makes it possible to connect independent, arbitrary reactances in series or parallel into the main line (3)–(4). The E–H tuner is thus useful in a wide range of matching tasks. The E–H junction can also be used to measure an unknown impedance like a bridge. A calibrated, variable impedance is connected to port (3) and the unknown impedance to port (4). A signal is supplied to port (1) and port (2) is terminated with an indicator in a reflection-free manner (a2 = 0). For the outgoing wave quantity b2 on port (2), we then have b2 = S23 (a3 − a4 )

444

H. Arthaber

where a3 = r3 b3 = r3 S13 a1 , a4 = r4 b4 = r4 S13 a1 . The indicator display will go to zero only if the two reflection coefficients r 3 and r 4 , i.e. the impedances connected to ports (3) and (4), are equal. We can express the power consumed in a multiport, i.e., the losses, as the difference between the power supplied and consumed on all ports. Based on the wave quantities a and b in matrix form, we obtain Pl : Pl =

1 † a · a − b† · b 2

where b† = a† · S† and b = S · a leading to11 Pl =

1 † a 1 − S† · S a. 2

If the multiport is lossless, i.e. Pl = 0, it follows that12 S† · S = I

(5.7.2)

Equation (5.7.2) says that the S-matrix of a lossless (but otherwise arbitrary) multiport must be unitary. We make use of this fact in the following discussion. Let us imagine that for an E–H junction we set elements S11 and S22 equal to zero using appropriate matching elements (which is always possible independently since ports (1) and (2) are decoupled). According to Eq. (5.7.2), we then obtain the following for the individual rows of (5.7.1): 2|S13 |2 = 1, 2|S23 |2 = 1, |S13 |2 + |S23 |2 + |S33 |2 + |S34 |2 = 1, |S13 |2 + |S23 |2 + |S34 |2 + |S44 |2 = 1. Plugging the first two equations into the latter two and adding them, we obtain

11 12

M† is the Hermitian conjugate of M. I is the identity matrix.

5 Field-Based Description of Propagation on Waveguides

445

|S33 |2 + 2|S34 |2 + |S44 |2 = 0. This equation can be satisfied only if S33 = S34 = S44 = 0. Based on our initial assumptions for S11 and S22 , it thus necessarily follows that the input reflection coefficients on ports (3) and (4) must also disappear and these two ports must be decoupled from one √ another. From the first two equations, it follows that S13 = √ ejϕ / 2 and S23 = ejϕ / 2. By shifting the port planes to port (1) or (2), we can obtain ϕ = Ψ . The S-matrix for our E–H junction thus assumes the following form: ⎛

SMT

⎞ 0 01 1 1 ⎜ 0 0 1 −1 ⎟ ⎟. =√ ⎜ 2⎝1 1 0 0⎠ 1 −1 0 0

(5.7.3)

We have thus created a four-port network that allows matched operation on all ports (such operation is basically possible with a lossless three-port network that exhibits transmission symmetry). A four-port network of this sort in which all of the elements in the main diagonal of its S-matrix are equal to zero is generally known as a directional coupler. Here, we will pay special attention to the 3 dB directional coupler because the power supplied to one port is divided into half among the other two ports if they are matched. We can also say that the coupling attenuation is equal to 3 dB. The coupling attenuation is defined as the ratio of power flowing into the main line to the power flowing out of the secondary line. If we supply a signal to port (1) and consider port (4) as the output of the secondary line, we obtain the following for the coupling attenuation aK : aK = 20 log

|a| = 3 dB. |b4 |

In literature, a 3 dB coupler of this sort made of waveguide junctions is commonly referred to as a “magic tee”. It is used, for example, in balanced mixers in extremely high frequency applications and also as a duplexer.13 However, it can also function as a sum and difference generator or a power splitter in redundancy circuits. If signals at the same frequency are fed to ports (1) and (2) corresponding to a1 and a2 , we obtain the sum and difference signal on ports (3) and (4), respectively: 1 1 b3 = √ (a1 + a2 ), b4 = √ (a1 − a2 ). 2 2 Let us now assume that we wish to redundantly supply each of two loads with one half of the power from a generator. By redundant, we mean that if the generator fails, a reserve generator will take its place. One possible implementation involves 13

A duplexer is an element that allows to route transmit and receive signals over a single path.

446

H. Arthaber

RF switches. However, such switches are not necessary if one generator is connected to port (1), the second to port (2) and the two loads to ports (3) and (4). One generator remains switched off and is activated only if the other fails. In test and measurement applications, 3 dB couplers are typically not the best choice because it is uneconomical to expend half of the total power to simply display a process on the main line. Instead, directional couplers with a significantly higher value of aK are used.

5.7.2 Metallic Irises and Posts in Waveguides Reactance circuits are necessary in order to perform matching tasks as well as to construct filters. We have already seen one such circuit in the form of the E–H tuner. However, it requires a large amount of space. We can commonly do without its tuning capabilities if all we need is reactances with fixed values. In waveguide technology, reactances are commonly realized as metallic discontinuities in the form of irises and posts (Fig. 5.51). By introducing discontinuities of this sort, a reflection coefficient is produced such that higher modes than the H 10 mode are excited. If they cannot be propagated, energy is stored in the vicinity of the discontinuity. Since the iris with its assumed ideal conductivity does not consume any power of its own, we can alternatively imagine that the reflection coefficient is caused by a reactance. Rigorous treatment of waveguide discontinuities is very difficult if not impossible. For information about the calculation and representation of quantitative results, see [17, 58]. Equivalent circuits for irises and posts provide some insight. On the iris, the electric tangential field strength and the magnetic normal field strength must disappear. Let a b

a

b

c

d

Fig. 5.51 Metallic irises and post for H 10 modes in the rectangular waveguide. a Asymmetrical and symmetrical capacitive iris; b asymmetrical and symmetrical inductive iris; c Iris as parallel resonant circuit corresponding to a and b; d post as T-type highpass element

5 Field-Based Description of Propagation on Waveguides

447

b a

a

b

Fig. 5.52 a Influence on the electric field — by an iris as shown in Fig. 5.51a; b influence on the magnetic field – – – by an iris as shown in Fig. 5.51b

us first consider the iris types shown in Fig. 5.51a. Figure 5.52 shows how the electric field is influenced by the iris in a longitudinal section through the waveguide. We can see that a longitudinal electric field strength necessarily arises which entails, in addition to the H 10 mode that can be propagated, the excitation of E mn fields that cannot be propagated. Below its cutoff frequency, 1/(ωC s ) dominates ωL s in the equivalent circuits for the E mn fields such that electric energy is stored by the E mn fields. Since the addition of H mn fields is not necessary, an iris as shown in Fig. 5.51a acts like a parallel capacitance for the H 10 mode. Figure 5.52b shows how the magnetic field is altered by an iris according to Fig. 5.51b. A field pattern of this type can be generated if the H 10 mode that can be propagated has H m0 fields that cannot be propagated superimposed on it. Superposition of E mn fields is not necessary. Irises as shown in Fig. 5.51b thus act like parallel inductances for the H 10 mode. The combination of a capacitive and an inductive iris can be realized with the window-shaped design in Fig. 5.51c which behaves like a parallel resonant circuit in accordance with our discussion so far. A post as shown in Fig. 5.51d has an effect on the H 10 mode like that of a T-type highpass element [58]. An iris/post combination is used, for example, as a matching circuit for the magic tee. It makes it possible to visually distinguish a simple E–H junction from a 3 dB coupler.

5.7.3 Waveguide Loaded with Inhomogeneous Dielectric Material Waveguides can be partially loaded with a dielectric material such as shown in Fig. 5.54. This gives them characteristics that turn them into phase shifters or attenuators, for example. Gyromagnetic material can be introduced to create microwave isolators and circulators (see Sect. 5.8.2). In a rectangular waveguide that is loaded with an inhomogeneous material, E and H fields can generally no longer exist separately from one another. The only exception to this rule is provided by the H 0n or H m0 fields if their electric field strength oriented parallel to the boundary surface

448

H. Arthaber

of the loading material. However, a combination of E and H fields with the same cutoff frequency is possible and leads to what is known as longitudinal section waves. We distinguish between E and H longitudinal section waves (LSE and LSH fields). Figure 5.53 shows how longitudinal section waves arise through superposition of E and H waves. Their name reflects the fact that E or H only have components in the plane of a longitudinal section through the waveguide. The possibility of their existence was first verified by Buchholz [59]. We would now like to analyze the effect of dielectric loading in a rectangular waveguide as shown in Fig. 5.54 with the aid of a perturbation calculation. Here, all quantities with the index zero refer to the waveguide without loading and all quantities without an index refer to the waveguide with loading. In the field equations, we apply E0 (or E) and H0 (or H) with the propagation constant exp(jωt – γ 0 z) (or exp(jωt – γ z)). By combining the resultant equations, we obtain the following relationship [22] ∗ ∗ A (εr − 1)E0 · EdA . (5.7.4) γ + γ0 = jωε0 ∗ ∗ A E0 × H + E × H0 · dA In this (still exact) equation, we must integrate over the waveguide cross-section A and the cross-section A of the loading. We now assume that the waveguide field of the H 10 mode is practically undisturbed by the loading. This is true especially if s a is satisfied. We then have E ≈ E0 = ey E max sin π x 1 /a and H ≈ H0 = –E0 /Z WH . 2 sin2 (π x1 /a)sb and For the numerator in Eq. (5.7.4), we thus obtain (εr − l)E max 2 for the denominator, we obtain E max ab/Z WH . The following approximation is then obtained: a

Hz

– Hy

Hz

Ez

Ez LSE

+

– Ex

Ex Hy

= Ex

Hy – Hy

b Hz

Hz

Ez – Hy Ex

+

– Ex

Ez LSH

=

Hy

Fig. 5.53 Longitudinal section waves in the rectangular waveguide. a LSE field; b LSH field

5 Field-Based Description of Propagation on Waveguides

449

Z

X b

S X1 a

y

Fig. 5.54 Waveguide with inhomogeneous loading

γ + γ0∗ ≈ j

s sin2 π x1 /a ω εr − 1 & . c a 1 − ( f c / f )2

(5.7.5)

For εr = ε −jε , we will first assume that ε = 0. In this case, we have γ = jβ and = −jβ0 such that we have a phase shifter. If ε = 0, then the propagation constants are complex which leads to an attenuator for ε > ε . The magnitude of the phase shift or the attenuation is dependent on the one hand on the design length of the loading and on the other hand also on its spacing x 1 from the waveguide wall. By modifying this spacing, we can also modify the phase shift or attenuation. However, such simple variable phase shifters or attenuators are not suitable for precision measurements. Moreover, loaded waveguides are also used in polarization rotators and converters. Such components make it possible to modify the polarization of a wave or to convert a linearly polarized wave into a circularly polarized wave and vice versa. Such rotators and converters can also be used to build precision phase shifters and attenuators. Uher et al. [60] discuss applications of waveguide loading for realization of multistage impedance transformers.

γ0∗

5.7.4 Cavity Resonators Just like with TEM wave lines, superposition of two waves of the same amplitude but opposite propagation direction also leads with waveguides to standing waves with nodes and antinodes of the electric and magnetic field strength. We can arrange conductive walls in the nodal planes for E without disrupting the field. Since the nodes are spaced successively by λh /2, the mutual spacing c of the walls must generally

450

H. Arthaber

equal an integer multiple of λh /2: c = qλh /2.

(5.7.6)

Waveguides with both ends short-circuited are known as cavity resonators. They correspond to the λ/2 line resonator that is short-circuited on both sides. Figure 5.55 shows a rectangular resonator and a cylindrical resonator. By plugging Eq. (5.6.2b) for the waveguide wavelength into Eq. (5.7.6), we obtain the relationship for its resonant wavelength λr that is critical for dimensioning a resonator. We have

1 λr

2

1a = 4 c

2

+

1 λc

2 .

(5.7.7)

In this form, Eq. (5.7.7) is applicable to all cylindrical cavity resonators. Specialization for the individual design types requires application of the corresponding cutoff wavelengths. According to Eq. (5.6.8), we obtain the following for the rectangular resonator:

2 λr

2 =

m a

2

+

n

2

b

+

q

2

c

.

(5.7.8)

This equation applies equally to E and H waves since λc follows from the same relationship for both modes. However, in cylindrical resonators we must distinguish between the two modes. Applying Eqs. (5.6.12) and (5.6.14), we obtain the following for H waves

1 λr

2

=

H

xmn πD

2 +

1q 4 c

2

(5.7.9)

a b c

a

c

D

b

Fig. 5.55 Cavity resonators. a Rectangular resonator; b cylindrical resonator

5 Field-Based Description of Propagation on Waveguides

451

and for E waves

1 λr

2 = E

x

mn

πD

2

+

1q 4 c

2

.

(5.7.10)

Unlike the line resonator in which the resonant wavelength is dependent only on the electric length, in the cavity resonator λr is determined by the volume or cross-section. It can thus be tuned in principle based on volume deformation. Many reflex klystrons take advantage of this possibility for coarse tuning. However, in most cases the resonator volume is altered with a sliding short circuit. In the construction of the tuning short, careful attention must be paid to ensure that the contact resistance between the sliding element and the waveguide wall is extremely low. Non-contact tuners [61] are thus preferred in many applications. In order to calculate the quality factor of a cavity resonator, we proceed just like we did for the line resonator. However, we must consider the fact that further resonances of undesired modes will arise in addition to the resonance of the desired mode. In order to easily determine which resonances are possible for a given resonator, we plot Eqs. (5.7.8) or (5.7.9) and (5.7.10) for different values of m, n and q in what is known as a mode or mode map (Fig. 5.56). We can see that overlaps occur. Therefore, we generally choose an operating range for the resonator in which unambiguous operation is possible. In order to designate a specific resonator field, we use m, n and q as indices, thereby distinguishing between H mnq and E mnq resonators. Figure 5.56 shows the unambiguous ranges for the H 101 rectangular resonator and the cylindrical a

b

Fig. 5.56 Mode or mode maps for cavity resonators. a Rectangular resonator with a = 2b. For E waves, we must have m and n = 0. The indices of the sloping lines are thus all associated with H waves; b cylindrical resonator

452

H. Arthaber

H 111 resonator. One peculiarity occurs in E wave resonators in relation to the index q. To illustrate this, we write in conjunction with the terms Hzf = −

Hz0 −j 2π z e λh , 2j

Hzr =

Hz0 j 2π z e λh 2j

E zr =

E z0 j 2π z e λh 2

or E zf =

E z0 −j 2π z e λh , 2

for the resonator fields the following short forms: H waves 2π z , Hz = Hz0 sin λh 2π E x1 = E x10 sin z , λh 2π Hx2 = Hx20 cos z , λh 2π E x2 = E x20 sin z , λh 2π Hx1 = Hx10 cos z , λh E waves

2π E z = E z0 cos z , λh 2π E x1 = E x10 sin z , λh 2π Hx2 = Hx20 cos z , λh 2π E x2 = E x20 sin z , λh 2π Hx1 = Hx10 cos z . λh Regardless of the specific resonator design, x 1 and x 2 denote transverse coordinates, i.e. x, y in the rectangular resonator and ρ, ϕ in the cylindrical resonator. Based

5 Field-Based Description of Propagation on Waveguides

453

on the above equations in conjunction with Eqs. (5.6.9), (5.6.10) as well as (5.6.11) and (5.6.12), we can see that resonator operation with λh = ∞ is non-existent for H waves since in this case all of the field components disappear with H z . In contrast, this operating case is possible for E wave resonators. The transverse electric field components disappear but not E z , Hx2 and Hx1 . For λh = ∞ and q = 0, the design length c of the resonator remains undetermined according to Eq. (5.7.6) and we can thus choose any arbitrary value. It no longer has any influence on the resonant wavelength although it does influence the resonator’s quality factor. With E wave resonators, we have, for example, an E 110 rectangular resonator and a cylindrical E 010 resonator. Figure 5.57 shows field patterns for both resonator types. A work by Schmidt [62] examines the possibility of tuning cavity resonators with dielectric loading as well as the application of resonators to measure material constants at high frequencies. Due to the increased difficulty associated with construction of rectangular resonators, cylindrical resonators have greater practical relevance. For an H 011 resonator, it is possible to attain a computational unloaded quality factor of 25,000 at 10 GHz [63]. In the real world, this quality factor is degraded by the surface roughness of the conductor material. With careful surface treatment, the calculated quality factor is reduced by about 20%. With superconducting resonators, unloaded quality factors on the order of 105 can be attained. a

b

c

a

b c

D

Fig. 5.57 Field patterns in the cavity resonator with λr independent of c. a E 110 rectangular resonator; b E 010 cylindrical resonator

454

H. Arthaber

5.7.5 Waveguide and Dielectric Resonator Based Filters 5.7.5.1

Rectangular Waveguide Bandpass Filter

Bandpass filters with a narrow passband are built for frequencies in the GHz range using waveguide resonators with a high quality factor in order to minimize the losses in the passband. If we wish to insert the bandpass filter into a circuit consisting of H 10 rectangular waveguides, a design with directly coupled H 101 resonators with a length of λh0 /2 as shown in Fig. 5.58 is practical. The waveguide broadside should be chosen such that only the H 10 mode can be propagated in the frequency range of interest. Dimensioning formulae [64–66] are derived corresponding to Fig. 5.58a from a bandpass filter with inverter coupling. When selecting the reference lowpass filter that satisfies the requirements for the passband and stopband, the dispersion of the waveguide wave must be taken into account. According to Fig. 5.58b, the series resonant circuits from the equivalent circuit are replaced with λh0 /2-waveguide resonators and the impedance inverters with parallel inductances. They can be realized using irises or posts in the waveguide as seen in Fig. 5.58c and d. Although the longitudinal reactances of the iris equivalent circuit are negligible for very thin irises, they must be considered in the case of posts (see Fig. 5.58e, f). The resonator length must be corrected with the resulting intrinsic length of the impedance inverter. Moreover, the resonator length is reduced to some extent in order to allow tuning for an ideal response using a screw inserted at the

a Z0

b K 01

K 12

K 23

c

K34

Z0

Z0

l h /2

f

0

jX Z 0

Z0

Z0

Z0

d ~lh /2 0

b

Irises

Posts

a

e jXa

f jX

=

jXa jXb

Fig. 5.58 H 10 rectangular waveguide. Bandpass with inductive irises. a Equivalent circuit with impedance inverters; b equivalent circuit with waveguide resonators and inductive irises; c construction of the bandpass with irises; d construction of the bandpass with posts; e equivalent circuits of an inductive iris

5 Field-Based Description of Propagation on Waveguides

455

maximum of the electric field strength. Mechanical tolerances can be compensated in this manner. Since the relative passband is only between 0.5 and 2% in many application areas, the filters must be manufactured using a material with a low thermal expansion coefficient, e.g. from drawn Invar waveguides or from Invar plates in order to minimize the center frequency offset as a function of temperature. This is especially important if higher continuous power levels are to be transmitted since the power dissipation due to the passband attenuation leads to heating of the bandpass filter. The energy stored in the filter resonators is very high in case of a small bandwidth. The pulse power that can be transferred is thus limited by the maximum permissible electric field strength in the resonators. The losses in the passband range decrease if we use H 111 or H 011 circular waveguide resonators instead of H 101 rectangular resonators. The H 011 resonance has an especially high unloaded quality factor, but it is difficult to avoid impairment of the filter stopband attenuation due to adjacent oscillation modes. Since coaxial lines are commonly used in the high frequency equipment, bandpass filters built from cavity resonators mostly use coaxial connectors. The H 101 or E 110 rectangular resonators or the E 010 circular resonators are configured as shown in Fig. 5.59. The inner conductors of the coaxial connectors are coupled to the electric field of the outermost resonators. Coupling between the resonators is implemented using irises. Using radially inserted dielectric tuning posts made of quartz or aluminum oxide ceramic, the filter can be tuned to different center frequencies f 0 . The posts influence the input and intermediate coupling such that the absolute passband remains approximately constant in the tuning range. A different design is shown in Fig. 5.60a. Here, the resonators are arranged over one another to allow selection of the coupling between resonators 2 and 3 of either capacitive by means Tuning posts

2

D/L

2...2.5

a

a/b

a

b (L)

a

b

D

c

Electric field strength vector

Fig. 5.59 Bandpass filter with waveguide resonators and coaxial connectors. a Longitudinal crosssection through the bandpass; b cross-section with rectangular resonators a × a × b; c cross-section with circular resonators D × L

456

a

H. Arthaber

b

aB (dB)

c

(MHz)

Fig. 5.60 Fourth order bandpass filter with or without coupling between resonators 1 and 4. a Cross-sections of the resonator arrangement; b equivalent circuit; c attenuation of a filter of this sort

of a central coupling hole or inductive by means of an opening arranged at the edge of the resonator. In case of demanding requirements for the selectivity, transfer curves with attenuation poles (Cauer or elliptic-function filters) are advantageous. They can be created by introducing additional couplings that bridge multiple filter circuits [67, 68]. With the fourth order bandpass filter in Fig. 5.60a, this can be easily realized with a coupling 1–4 which must have an opposite sign to that of the inner intermediate coupling 2–3 (Fig. 5.60b). An attenuation pole is obtained on both sides of the passband as seen in Fig. 5.60c. With the sixth order bandpass filter in Fig. 5.61a, we can obtain a Cauer characteristic with two attenuation poles above as well as below the passband by introducing two couplings. The coupling 1–6 must have the same sign as the inner intermediate coupling 3–4 while the coupling 2–5 must have the opposite sign. In case of a

5 Field-Based Description of Propagation on Waveguides

a 1

2

3

6

5

4

1

2

457

b

3

1

2

3

4

5

6

c aB

f

Fig. 5.61 Sixth order Cauer bandpass filter with rectangular resonators. a Arrangement of the resonators; b equivalent circuit; c attenuation curve

different choice for the sign of the couplings, the filter’s transfer function contains an all-pass component which can be used to level out the group delay in the passband [69].

5.7.5.2

Dual-Mode Waveguide Resonators

If we select a square-shaped H 101 resonator or an H 111 circular resonator instead of a rectangular resonator with the side ratio a/b ≈ 2, two orthogonal modes can exist in one resonator which can be coupled to one another by means of a defined asymmetry, e.g. a screw inserted at an angle of 45° with respect to the electric field strength vectors as shown in Fig. 5.62 [70]. Since both orthogonal modes are used for transmission, the number of spatial resonators and thus the volume of the filter are cut in half. This is especially advantageous in satellite applications where it is critical to minimize the volume and weight. Figure 5.63 illustrates the design of a sixth order bandpass filter with three dual-mode H 111 circular resonator cavities. The coupling between resonant circuits 2 and 3 as well as between 4 and 5 is implemented by means of inductive irises, while the coupling between resonant circuits 1 and 2, 3 and 4 and 5 and 6 is implemented using coupling screws. As connecting lines, rectangular waveguides can be coupled via irises in the end faces of resonator cavities 1 and 3 to resonant circuits 1 and 6 or coaxial lines in the center of resonators 1 and 3 opposite tuning

458

H. Arthaber 1

Coupling screws

1

a

45

E1

E1 2

a

2 E2

E2 Electric field strength vectors a 1

1

b

E1

45

E1

2

D

2 E2

E2

c

1

2

1

2

Fig. 5.62 Resonators with coupled orthogonal modes. a Coupling in the square H 101 resonator; b coupling in the H 111 circular resonator; c equivalent circuits: coupling sign reversal when coupling screw position is changed

elements 1 and 6. As shown in Fig. 5.62, the coupling changes sign if the position of the coupling screw is turned by 90° relative to the E vectors of the modes. Based on the example of a fourth order bandpass filter with square H 101 resonators (Figs. 5.64 and 5.65), we can see that it is possible to obtain either a steeper attenuation curve or a leveling of the delay in the passband. Figure 5.66 shows a sixth order Cauer bandpass filter with dual-mode H 101 resonators [69, 71].

5 Field-Based Description of Propagation on Waveguides

459

a 6 6 5–6

5 Tuning elements

3

5

Resonator 3

3 3–4

Iris 4−5

4 Coupling screw

4

2

1–2

2

Resonator 2

Iris 2−3 1

Resonator 1

1 Electric field strength vectors

b

1

2

3

Resonator 1

4

Resonator 2

5

6

Resonator 3

Fig. 5.63 Sixth order dual H 111 mode bandpass filter. a Design; b equivalent circuit 1–2

3–4

Coupling screws

c

a

a

t

aB 1

3

4

a

2

lh0

2–3

1–4

2

b

1

Stopband transition not steepened 2

3

f

Delay flattened

f

4

Fig. 5.64 Fourth order dual H 101 mode bandpass filter with coupling 1–4. a Design; b equivalent circuit; c attenuation curve: not steepened; delay curve: flattened

460

H. Arthaber

a

1–2

c

Coupling screws 3–4

1

2

3

2–3

aB

4

1–4

b

Stopband transition steepened 1

2

3

f

Delay not flattened

f

4

Fig. 5.65 Fourth order dual H 101 mode bandpass filter with coupling 1–4. a Design: modified position of coupling screw 3–4 compared to Fig. 5.64; b equivalent circuit; c attenuation curve with attenuation poles, delay curve not flattened

2–3 1–2

1–4

I 3

1–6

3–4

2

4

II

c 5–6

4–5

aB (dB)

a

III

1

5

6

40

Output

Input

Electric field strength vectors

b

30

Measured

25

calculated (Q = 8700)

20

10 1

2

Resonator

3

I

4

II

5

6

III

f0 = 4017.9 MHz 0 3960

3980

4000

4020

4040

4080

f (MHz)

Fig. 5.66 Sixth order Cauer bandpass filter with dual H 101 mode technology, a design; b equivalent circuit; c attenuation of filter

5.7.5.3

Filters with Dielectric Resonators

Cylindrical disks made of insulating material with high relative permittivity εr and simultaneously low loss factor tan δ and a small temperature coefficient can be used as dielectric resonators. They have smaller dimensions than cavity resonators with a somewhat lower quality factor, but a significantly higher quality factor compared to stripline resonators. They can thus help to reduce the space required for filter circuits. Like for the metallic cavity resonators, we classify E and H modes. For a length to diameter ratio L/D ≈ 0.4, the H 10q resonance occurs as the magnetic

5 Field-Based Description of Propagation on Waveguides

461

fundamental mode. Then, the separation to the next modes H 11q and E01q is also maximized and can be further improved to some extent with an axial drill hole in the resonator [72]. Figure 5.67 shows the field distribution for the H 01q resonance. The electric field lines are concentric circles enclosed by the magnetic field lines. The field disperses into the external space. Some 70% of the field energy is stored within the resonator for εr = (35–90) [73]. The connecting lines are coupled to the magnetic Hz

a

Ej

b

L

H

Resonator

c

D

Ej

Fig. 5.67 Dielectric resonator (cylinder resonator). a Behavior of magnetic and electric field strength for the magnetic fundamental mode; b magnetic field lines; c electric field lines

462

H. Arthaber

stray field as shown in Fig. 5.68. In order to avoid radiation losses, the resonators must be installed in a metal enclosure. As a result, wall surface current losses arise in addition to the dielectric losses in the resonator which diminish the resonator quality factor. The resonant frequency is shifted upwards. The minimum required spacing from the enclosure is about L/2. The most common resonator material is barium zirconate titanate (BZT) with εr ≈ 38 and tan δ ≤ 2 · 10–4 up to 10 GHz. Based on the composition, the temperature coefficient of the resonant frequency can be adjusted within a range from –25 < TKf < +60 · 10–6 /K to about 1 · 10–6 /K. The influence of the enclosure on the TKf can also be compensated in this manner.

Open circuit

a

/4 la

b

c

Fig. 5.68 Options for inductive coupling to the fundamental oscillation of a dielectric resonator. a Open-circuited inner conductors of a coaxial connector; b coupling loops with coaxial connector; c coupling to microstrip

5 Field-Based Description of Propagation on Waveguides

a

463

C

A

L

Resonator

Holder (Quarz)

b

D

B

Dimensions for f0 = 6.8 GHz and D = 8.5 mm

A = 8.1 mm

L = 3.5 mm

B = 20 mm

r

= 37.8 C = 74 mm

Fig. 5.69 Fourth order bandpass filter with dielectric resonators. a Cross-sectional view; b top view and main dimensions

Figure 5.69 shows construction details for a bandpass filter with four dielectric resonators. At a passband frequency of 6.8 GHz, they have diameter D = 8.5 mm and length L = 3.5 mm; they are fastened to the metal enclosure using quartz disks. The intermediate coupling is determined by the axial spacing. The coupling to the connecting lines is realized at the current maximum before their open-circuited ends. Metal screws are used for fine adjustment of the frequency. For a 3 dB bandwidth of 49 MHz, the passband attenuation is 0.85 dB which corresponds to a quality factor of about 4,100. Further applications of dielectric resonators in microwave circuits are found in [74, 75] along with detailed bibliographical material.

5.7.6 Waveguide Directional Couplers 5.7.6.1

Aperture Couplers

One very popular type of waveguide directional coupler is the aperture coupler (Fig. 5.70). Here, two identical rectangular waveguides are coupled by means of multiple holes in the wide or narrow common waveguide wall [76–81]. The coupler principle is best illustrated with the two-hole coupler (Fig. 5.70). Assume that a wave

464

H. Arthaber

a

b 4

Bre–jbd

Bf

Br

B f e–jbd e–jbd

1

3 2

d Z1

Z2

Fig. 5.70 Waveguide aperture coupler. a H-plane coupler as example; b basic principle of directivity in the two-hole coupler

arriving at port 1 with (normalized) amplitude 1 induces in the secondary line a field with amplitude Bf in the forward direction and Br in the reverse direction. The total amplitude of the wave occurring in the forward direction in the secondary line at plane z2 is equal to 2Bf e−jβ d . In contrast, the total wave in the reverse direction at plane z1 is Br (l + e−2jβ d ). Since the path lengths in the forward direction are identical in both lines, the two wave components are in phase and their amplitudes are added. In contrast, the wave components flowing in the reverse direction are out of phase and thus subtract if 2βd = nπ (n = 1, 3, 5, …). Consequently, we see that a value equal to a quarter of the waveguide wavelength λg (d = λg /4) leads to cancellation of the wave components flowing in the reverse direction (forward-wave coupler in contrast to the reverse-wave coupler or backward-wave directional coupler, TEM wave coupler). The coupling attenuation is aK = −20 log 2|Bf |,

(5.7.11)

and the directional attenuation is |Bf | 2|Bf | = 20 log |Br || cos βd| |Br | 1 + e−2jβd Bf 1 = 20 log + 20 log . B cos βd

aR = 20 log

(5.7.12)

r

Multi-hole arrangements can be used to create directional attenuation characteristics, e.g. with Chebyshev behavior; see also [76–81].

5.7.6.2

Branch-Guide Couplers

Branch-guide couplers are also widely used and have been extensively studied [76, 78, 80–90]. They are well suited for measurement applications (20 dB or 10 dB coupler) as well as power splitter applications (3 dB coupler). Moreover, they are easy to build and are relatively broadband devices. This coupler is typically analyzed using network theory [80–91]. However, rigorous field theoretical methodology has also been applied to design optimal E-plane directional couplers [89, 90]. Figure 5.71

5 Field-Based Description of Propagation on Waveguides

465

a

(dB)

b

(GHz)

Fig. 5.71 E-plane branch-guide coupler [90]. a Basic coupling principle; b S-parameters S11 to S14 versus frequency f

shows a possible design for a 3 dB coupler [90]. A slot height of about λg /4 is optimal; however, it is often more practical to use a (mechanically simpler) coupling plate (approx. 200 μm thick) since the coupler slots can be produced with an etching process. Besides E-plane branch-guide couplers, H-plane branch-guide couplers are also used. Although this coupler type does not offer the same good directional coupler characteristics as the E type, it is preferred in high-power applications because the E type tends towards field breakdowns in the y direction due to inhomogeneities [76].

466

5.7.6.3

H. Arthaber

Further Waveguide Directional Couplers

Figure 5.72 shows further waveguide directional coupler types that are also encountered in practice [79].

5.8 Wave Propagation in Gyromagnetic Media (Directional Components, Ferrites and Yttrium Iron Garnet Garnets) 5.8.1 Basic Principles Ferrites are metal oxide compounds with the chemical formula MeOFe2 O3 (Me = bivalent metal). In contrast to ferromagnetic materials (Fe, Ni, Co), they have ceramic properties. Due to their high resistivity (up to 1012 cm), they can be used without eddy current losses in the microwave range. Alongside these ferrites, there exist ferrimagnetic materials with other crystal structures such as barium ferrites and yttrium iron garnets (YIG). Nowadays, all materials with ferrimagnetic properties are said to be ferrites [92]. Usage of pre-magnetized ferrites in microwave applications to realize components that exhibit transmission asymmetry (non-reciprocity) is based on the electron spin

a

b Mz

lL/4 Mz

Output

Input

c

T slot in common broadside

Fig. 5.72 Further waveguide directional couplers [79]. a Directional coupler with inverted phase as described by Schwinger; b directional coupler with crossed waveguides as described by Moreno; c T-slot directional coupler as described by Riblet

5 Field-Based Description of Propagation on Waveguides

467

precession. Due to the pre-magnetization, the permeability of these ferrite materials is no longer independent of the direction of the fields, i.e. it is a tensor that we will designate hereafter with ||μ||. ||μ|| gives the relationship between the induction B and the magnetic field strength H: B = μ · H or in matrix form: ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ μ11 μ12 μ13 Hx μ11 Hx + μ12 Hy + μ13 Hz Bx ⎝ By ⎠ = ⎝ μ21 μ22 μ23 ⎠⎝ Hy ⎠ = ⎝ μ21 Hx + μ22 Hy + μ23 Hz ⎠. Bz μ31 μ32 μ33 Hz μ31 Hx + μ32 Hy + μ33 Hz Bx is thus dependent on H x , H y and H z , i.e. an induction in the x direction can be produced by a magnetic field in the y direction. Based on the model of the electron spinning about its own axis, the direction-dependent permeability can be calculated.

5.8.1.1

Direction-Dependent Permeability μ

Due to its charge we can attribute a magnetic dipole moment jB (Bohr magneton [93]) to an electron spinning about its own axis and due to its mass, we can attribute an angular momentum D (spin magnetic moment) to it. We have: j B = −Γ · D, Γ = μ0

e . me

(5.8.1)

Here, G is the gyromagnetic ratio.14 For μ0 = magnetic field constant = 1.257 · 10–8 s/cm, e = electron charge = 1.6·10–19 As, me = electron mass = 9.1 · 10–35 Ws3 /cm2 , it follows that G = 22.1 MHz cm/A. Due to mass inertia, the direction of D is constant. Under the influence of the resultant field strength H r inside the ferrite, the vector of the magnetic moment and thus the angular momentum vector experiences a mechanical moment of force M which is perpendicular to jB and H r : M = j B × H r . Since D and M are related by the equation M = dD/dt, we obtain the equation of motion for jB [95]: d jB = −Γ · j B × H. dt

(5.8.2)

If H r = H 0 is independent of time (constant magnetic field), the vector of jB travels with the angular velocity ω0 = +G · |H 0 | on the surface of a cone with H 0 In literature, γ is often used instead of G. In order to avoid confusion with the propagation constant, the designation G was introduced in [94].

14

468

H. Arthaber H0

Fig. 5.73 Precession of the magnetic moment around the direction of the constant magnetic field

djB/dt

jB

D

as the axis (Fig. 5.73). This precession motion experiences damping which is taken into account in Eq. (5.8.6). The magnetic field strength H r inside the ferrite consists of the externally applied magnetic field H a and the anisotropy fields H n of the magnetic domains. Without an external field, the directions of the anisotropy fields are randomly distributed such that the sum over all H n is equal to zero. If the ferrite is biased, all H n are turned in the direction of the applied field strength. The direction of H r increasingly matches that of H a as H a approaches saturation. According to Kittel [96], in case of saturation the magnetic field strength H r can be calculated as a function of the geometric dimension, the applied field and the saturation magnetization. We will assume hereafter that the ferrite is saturated and H r is known. In order to obtain the direction-dependent permeability ||μ||, we must discover the relationship between jB and Br since B = ||μ|| · H · Br consists of the induction of free space μ0 H r and the sum of all of the magnetic moments in the observed volume V: . j Bn /V. B r = μ0 H r + n

/

n j Bn /V is the magnetic polarization Bi = Br − μ0 H r . Applying Eq. (5.8.2), we obtain the relationship between Br and H r :

d (B r − μ0 H r ) = −Γ (B r − μ0 H r ) × H r = −Γ B r × H r . dt Assuming that

(5.8.3)

5 Field-Based Description of Propagation on Waveguides

1.

2.

469

The induction and the magnetic field strength are composed of temporally independent (constant) components along with relatively small alternating components and The constant components only have components in the z direction which are so large that the ferrite is saturated. Br and H r then take the following form: B r = B + B 0 = ex · Bx + ey · By + ez (Bz + B0 ), H r = H + H 0 = ex · Hx + ey · Hy + ez (Hz + H0 ).

Applying the approaches in Eq. (5.8.3) and neglecting the products of alternating quantities, we obtain the following in complex notation [97]: jω(Bx − μ0 · Hx ) + Γ By · H0 − Hy B0 = 0, jω By − μ0 · Hy + Γ (Hx · B0 − Bx H0 ) = 0, jω(Bz − μ0 · Hz ) = 0 such that B = || μ || · H: ⎞ ⎛ ⎞⎛ ⎞ μ1 jμ2 0 Hx Bx ⎝ By ⎠ = ⎝ −jμ2 μ1 0 ⎠⎝ Hy ⎠ Bz Hz 0 0 μ0 ⎛

(5.8.4)

with the components15 ω0 ωm ωωm , μ2 = μ0 2 μ1 = μ0 1 + 2 , ω0 − ω2 ω0 − ω2 Γ Γ Bis . ω0 = Γ H0 , ωm = (B0 − μ0 H0 ) = μ0 μ0

(5.8.5)

Bis is the magnetic saturation polarization and ω0 /2π is the gyromagnetic resonance frequency. If the frequency ω/2π of the excitation magnetic field strength is equal to ω0 /2π, then μ1 , μ2 and thus Bx , By are infinitely large according to Eq. (5.8.5). In reality, however, they only attain a finite maximum value because the precession motion is damped, and a damping constant is missing from the denominator of Eq. (5.8.5) which we can introduce as follows: The denominator function N = ω02 − ω2 can be converted into the form N = p 2 + ω02 = ( p − p1 )( p − p2 ) with p = jω where p1,2 = ±jω0 represent poles of μ1 and μ2 . p1,2 lie on the jω axis of the complex p plane. In order to determine the damping, the poles are shifted by the magnitude aω0 into the domain of negative real parts: 15

In literature, μ1 is commonly referred to as μ and μ2 as –K or –κ.

470

H. Arthaber p1,2 = ±jω0 − aω0 .

a is the damping factor introduced by Landau and Lifschitz. As we will demonstrate later, a can be determined from resonance width measurements. With the poles p1,2 , the denominator function becomes N = p − p1 p − p2 . We thus obtain the following for p = jω: N = ω02 1 + a 2 − ω2 + 2 jaω0 ω = (ω0 + jaω)2 − ω2 1 − a 2 + a 2 ω02 and for |a| 1: N = −ω2 + (ω0 + jωa)2 . Comparison of this function with the one for a = 0 shows that the damping is taken into account if we replace ω0 with ω0 + jωa. We thus obtain the following like in [95]: μ1 = μ0 1 + μ2 = μ0

5.8.1.2

(ω0 + jωa)ωm , (ω0 + jωa)2 − ω2

ωωm (ω0 + jωa)2 − ω2

(5.8.6a) (5.8.6b)

Wave Propagation in Pre-magnetized Ferrites

In order to illustrate the direction-dependent permeability, it is practical to consider the behavior of an electromagnetic wave in an infinitely extended medium with the characteristics described above. From Maxwell’s equations, we obtain the following after eliminating E: curl curl H − ω2 εμH = 0.

(5.8.7)

Assuming a solution of the form e j (ωt−k·r) (k = ex kx + ey ky + ez kz and r = ex · x + ey · y + ez · z) in Cartesian coordinates for H = ex Hx + ey Hy + ez Hz and initially setting k x = k y = 0 and jk z = γ , we obtain the following three component equations from Eq. (5.8.7): 2 ω εμ1 + γ 2 Hx + jω2 εμ2 Hy + 0 = 0, −jω2 εμ2 Hx + ω2 εμ1 + γ 2 Hy + 0 = 0, 0 + 0 + ω2 εμ0 Hz = 0.

5 Field-Based Description of Propagation on Waveguides

471

For nontrivial solutions, the coefficient determinant must disappear such that we obtain √ y± = α± + jβ± = jω εμ±

(5.8.8)

where μ± = μ0 μ± − jμ± = μ1 ± μ2 = μ0 1 +

ωm . ω0 ∓ ω + jωa

(5.8.9)

The corresponding solutions for H are as follows: H± ∼ ex ∓ je y ejωt−γ± z . In other words, two modes with different propagation constants can be propagated. The components H x and H y have the same magnitude but are phase-shifted by ±π/2, i.e. they are circularly polarized. The direction of H+ (H– ) rotates around the z axis in the mathematically positive (negative) sense. Figure 5.74 illustrates the rotation direction of the polarization and the effect of the different propagation constants: A positive and a negative circulating wave are propagated in the medium. The real parts of H± are represented at a fixed time point. At position z1 , the vectors H+ and H– trail those at position z = 0 by the spatial angles ϕ ± = β ± z. Assuming we neglect the different damping, the direction of the total field strength H = H+ + H– forms at position z = z1 an angle ϕ = 21 (ϕ+ − ϕ− ) = 21 (β+ − β− )z 1 with the x axis. In other words, if a linearly polarized wave that can be decomposed into two circularly polarized waves with the same amplitude and frequency but different directions of rotation encounters a ferrite that is pre-magnetized in the propagation direction of the wave (Fig. 5.75), the wave experiences a polarization rotation when passing through the ferrite (Faraday effect). The magnitude and direction of the rotation are dependent on μ+ and μ– and thus also on ω and H 0 . Figure 5.76 shows the relative quantities μ+ , μ− , μ+ , and μ− as a function of the constant field for a fixed frequency ω/2π. While the effective permeability μ– of the wave rotating in opposition to the precession only changes slightly in the entire range, μ+ passes through a resonance point at H 0 = ω/G, i.e. ω = ω0 , at which the damping component μ+ becomes very large. Based on the measurable line width H 0 (see Fig. 5.76b), we obtain in conjunction with Eq. (5.8.9) the damping factor a: a=

Γ H0 1. 2ω

In case of low pre-magnetization, μ+ and μ− increase again (low field losses). The polarizations of the magnetic domains no longer have a uniform direction. The transitions between the domains (Bloch walls) realize spatial oscillations at the frequency of the alternating field and can thus absorb energy. In order to avoid such losses,

472

a

H. Arthaber z

z

b

H−(z = z1)

z = z1

z = z1

wt wt y

H+(z = z1)

y

z=0

z=0

j – = b –·z1

wt wt H+(z = 0)

H−(z = 0)

j + = b+·z1

x

x

Fig. 5.74 Propagation of a wave with a right-hand circular polarization b left hand circular polarization y

y m+(H0) ± m–(H0)

H

x

j H

x

H0 j

E

E z=0

S

z = z1

S

z

Fig. 5.75 Faraday effect: rotation of the polarization of a linearly polarized wave in a premagnetized ferrite

5 Field-Based Description of Propagation on Waveguides

473

a μ′ 1 + (wm / 2a · w)

μ′+

μ′− wG

Saturated

Unsaturated

μ′r

H0

1 – (wm / 2a · w)

b

μ″ (μ″+ )max

μ″+ Saturated

Unsaturated

wm /a · w

H0

1/2 (μ″+ )max

μ″r Low field losses wG

μ″– H0

Fig. 5.76 a μ+ and μ− ; b μ+ and μ− as a function of the constant field. Here, H 0 points in the positive z direction

we must strive to produce ferrimagnetic materials with the lowest possible crystal anisotropy and low saturation magnetization (e.g. substituted yttrium iron garnets). We will now investigate the case in which we assume that the wave in the x direction propagates perpendicular to the pre-magnetization, i.e. k y = k z = 0, jk x = γ . If the vector of the magnetic field H lies in parallel to the pre-magnetization H 0 , we obtain the following for γ as discussed at the start of this section: √ γ = γ|| = jω μ0 ε. If H is perpendicular to H 0 , we obtain γ = γ⊥ = jω ε

μ21 − μ22 . μ1

474

H. Arthaber

The effective permeability and thus the propagation constant are dependent on the polarization direction of the wave. A medium that possesses such properties is said to be birefringent.

5.8.2 Application in Nonreciprocal Components By exploiting the direction-dependent characteristics of pre-magnetized ferrites, we can create diverse components for the microwave range (and in some cases down to several tens of MHz) including circulators, isolators (nonreciprocal attenuators or unidirectional lines), controllable attenuators and phase shifters, modulators, microwave switches, gyrators (see [60]) and absorbers. The circulator and the isolator are the most important components in this group.

5.8.2.1

Circulators (Waveguide Circulators)

A circulator is a nonreciprocal component with three or more ports. In general, the S-matrix16 of the three-port circulator is as follows: ⎛

⎞ S11 S12 S13 S = ⎝ S21 S22 S23 ⎠. S31 S32 S33 In case of rotational symmetry in the three-port circulator, S assumes a simpler form with only three factors: ⎞ S1 S2 S3 S = ⎝ S3 S1 S2 ⎠. S2 S3 S1 ⎛

Here, S1 (= S11 = S22 = S33 ) are the reflection coefficients which ideally should disappear. Moreover, for a circulator rotation direction as shown in Fig. 5.77, S2 (= S12 = S23 = S31 ) are the transmission coefficients in the reverse direction which should be as small as possible and S3 (= S13 = S21 = S32 ) are the transmission coefficients in the forward direction which should ideally have a magnitude of 1. For an ideal circulator, the S-matrix for the rotation direction l → 2 → 3 is thus ⎛

⎞ 0 0 ejψ S = ⎝ ejψ 0 0 ⎠ 0 ejψ 0 16

The scattering matrix S associates the power waves a flowing into a multiport to the outgoing power waves b based on the relationship b = Sa.

5 Field-Based Description of Propagation on Waveguides

475 R3 3

R1

V0

~ ~

1

2

R2

Fig. 5.77 Circuit symbol for a circulator with connected ports

and for the rotation direction 1 → 3 → 2, it is ⎛

⎞ 0 ejψ 0 S = ⎝ 0 0 ejψ ⎠. ejψ 0 0 Ψ is an arbitrary phase. If port 2 is matched, the power supplied to port 1 (Fig. 5.77) is completely dissipated in resistor R2 . Port 3 is “isolated". If port 2 is unmatched, part of the power is reflected and if port 3 is matched, it is dissipated in resistor R3 . It can be demonstrated that a lossless three-port network that is matched on all sides must be nonreciprocal and exhibit ideal circulator behavior [98]. Ferrite circulators exploit the difference in the phase constants β = β + − β − of waves with positive and negative circular polarization. Using Eq. (5.8.8), we can calculate17 that β is proportional to ω2 for ω ω0 , approximately equal to zero for ω = ω0 and nearly frequency-independent for ω ω0 [99]. Therefore, it is not possible to create circulators for ω = ω0 . Most microwave circulators operate in the range ω ω0 . Ferrites for circulators in the frequency range from about 4 to 100 GHz have saturation magnetizations H is = 1/μ0 Bis between 300 and 500 kA/m and line widths H of the resonance in the range 5–70 kA/m. The relative permittivity εr is equal to about 9–18 and the loss factor is 10–4 to 10–3 . Three-port circulators basically consist of a resonator in which three waveguides terminate which are spatially offset by 120°. Within the resonator, there is a ferrite cylinder that is pre-magnetized perpendicular to the plane of the three waveguides. It typically has a circular or a triangular cross-section. We will discuss its operation based on a waveguide circulator as shown in Fig. 5.78. In the initially unmagnetized 17

Although Eq. (5.8.8) holds only for infinitely extended space, it can be used to approximately characterize the electric conditions in the circulator.

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a

b 1

H0 = 0

3

2

1

H0 > 0

3

2

Fig. 5.78 H-plane waveguide circulator. a Without pre-magnetization; b with magnetic field H 0

ferrite, an H 10 mode fed into port 1 generates a resonance field that is symmetrical with respect to the direction of the supplying waveguide. This field can be decomposed into two circularly polarized rotating fields with the same amplitude but opposite directions of rotation: H = H + + H – . The resonant frequency of the fundamental mode which is customarily used is determined, except for the cylinder dimensions, from the material constants of the ferrite without pre-magnetization. With pre-magnetization applied, different permeability values apply for the two rotating fields: B+ = μ+ · H + , B− = μ– · H – . A resonant frequency arises that results approximately from the arithmetic mean of the phase coefficients β + and β − . Like when there is no pre-magnetization, a linearly polarized oscillation again arises from the superposition. If port 2 is matched, the magnetic field strength has on average a polarization direction that is perpendicular to the plane of port 3 such that no H 10 mode can be excited here; port 3 is decoupled. Circulators with this design are also known as H-plane circulators since the three waveguides lie in the H plane of the excitation waves. Most circulators are constructed based on this principle. However, for high power levels and for the case in which the waveguide arms must be rotated by 90° about the respective propagation axes due to spatial constraints, E-plane circulators have been developed [100, 101]; see Fig. 5.79. Whereas in H-plane circulators the magnetic H x component and the electric E y component are the main field components of the H 10 mode that excite the ferrite resonator, in the E-plane circulator this role is played by the H z component and the highly reduced E y component on the edge of the waveguide. The maximum tolerable field strengths and the associated power levels are thus higher. For usage in microwave integrated circuits (MICs), circulators can also be realized with microstrip or stripline technology [102] (Fig. 5.80a). Here, an opening in the substrate (typically Al2 O3 ceramic) accommodates the ferrite element (drop-in circulators). The functioning is basically identical to that of H-plane circulators. If a permeable material such as ferrite or yttrium iron garnet is used as the substrate

5 Field-Based Description of Propagation on Waveguides

477

a

H

Ey

H0

b

ferrite

3

1 Hz

Ey

2

Fig. 5.79 E-plane waveguide circulator. a View of port 2; b field patterns with decoupled port 3

a

b 2 Microstrip line 2

3

1

Ferrite

Ground conductor

Microstrip line Ferrite cylinder

3 1

H0 Copper-cladded insulator (e.g. printed circuit board)

Fig. 5.80 Circulator designs with planar conductors. a With microstrip line; b with stripline

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for the circuit, pre-magnetization of the otherwise isotropic substrate suffices at the junction between the three transmission lines. For coaxial line systems, circulators are commonly realized using stripline technology (Fig. 5.80). The junction between the three inner conductors is located between two ferrite disks. Since the dimensions of the resonators increase with wavelength, circulators are constructed at low frequencies down to several tens of MHz as lumped-element circulators in which the resonator is replaced by a balanced three-arm transformer connected in a star or delta configuration. The pre-magnetized ferrite is located in the field of the transformer. If port 2 is matched, the vector of the linearly polarized induction lies in the plane of the coil associated with port 3 such that no voltage can be induced in it [103]. Lumped-element circulators generally require matching networks that limit the bandwidth. Circulators are used in radar applications [104] to isolate the transmitter and receiver on a common antenna and in directional radio applications [105] to decouple the channel filters. Circulators are also used in the operation of reflection amplifiers (reactance, IMPATT diode amplifiers [106]) or for reduction of load reflections. By reversing the pre-magnetization, a microwave switch can be realized. For lower frequencies down to about 10 MHz, circulators can also be realized with the aid of active components [107, 108]. However, such low-frequency circulators have other application areas too. For example, they can be used to create floating inductors in the form of integrated circuits.

5.8.2.2

Unidirectional Lines (Microwave Isolators)

Unidirectional lines (microwave isolators) are two-port networks that are matched on both sides with nearly lossless transmission in the forward direction (attenuation 20 dB). They are used to decouple components or equipment units along a transmission path and especially to reduce the reflection coefficient by attenuating the reflected wave. Ferrite-based isolators built using waveguide technology are realized with narrowband characteristics by exploiting the Faraday effect. The resonant unidirectional line exploits the different attenuation components μ+ and μ− of the positive and negative circularly polarized waves in the vicinity of the gyromagnetic resonant frequency. The ferrite which is pre-magnetized perpendicular to the alternating magnetic field is located in the waveguide at a position where the magnetic field strength exhibits circular polarization. For an H 10 mode in the rectangular waveguide, these positions are located between the center and the side walls of the waveguide (Fig. 5.81). The pre-magnetization direction is chosen such that the rotation direction of the polarization of a wave incident on port 1 does not coincide with the spin precession. As a result, the wave passes through the ferrite region nearly unattenuated. If the propagation direction is reversed, the rotation direction of the polarization also changes in terms of the pre-magnetization such that the spin precession is highly excited and can absorb energy. A wave fed into port 2 is thus nearly completely absorbed in the ferrite.

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479

x

H0

H0

0

H (y)

Hz (y)

0

1/4

a

y

1

y/a

Hy (y)

1/2

3/4

Fig. 5.81 Basic design of resonance isolators using waveguide technology

Unidirectional lines based on the Faraday effect consist of a section of ferrite-filled circular waveguide operated in H 11 mode with transitions to rectangular waveguides offset at a 45° angle (Fig. 5.82). In the transition regions, damping sheets are arranged parallel to the broadside of the waveguide to suppress undesired modes. The polarization of an H 10 mode (E 1 ) that is fed into port 1 is spatially rotated by 45° in the ferrite region such that the wave can exit the unidirectional line on port 2 nearly unattenuated. Since the rotation direction is constant with respect to the pre-magnetization direction and does not depend on the propagation direction of the wave, a wave (E 2 ) that is fed into port 2 is rotated by the same angle (45°). The wave is attenuated because its polarization is now no longer perpendicular to the damping sheet. In coaxial lines with a homogeneous dielectric, there is no location with circular polarization as long as only the fundamental mode is propagated [109].

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2

Ferrite Damping sheet

1

45°

1

E1

2

E1

E1 E1

E2

E2 E2

Fig. 5.82 Unidirectional line based on the Faraday principle. See text for discussion

References 1. 2. 3. 4. 5. 6. 7.

8.

9. 10. 11. 12. 13. 14. 15. 16. 17.

Maxwell, J.C.: A Treatise on Electricity and Magnetism. London (1873) Stratton, J.A.: Electromagnetic Theory. New York, London (1941) Lagally, M.: Vorlesungen über Vektorrechnung. Akad. Verlagsgesellschaft, Leipzig (1956) Mahr, H.: Ein Beitrag zur Theorie der im Grundwellentyp angeregten Koaxialleitung. Der Fernmeldeingenieur 23. H. 5–7 oder Bad Windsheim: Heidecker (1969) Mie, G.: Elektrische Wellen an zwei parallelen Drähten. Ann. Phys. 2, 201–249 (1900) Zinke, O.: Anwendung Maxwellscher und Kirchhoffscher Gleichungen auf homogene LecherLeitungen beliebiger Leitergeometrie. NTZ 24, 369–374 (1971) Lorenz, R.W.: Über Lecher-Wellen, Leitungs-Wellen und TEM-Wellen auf verlustbehafteten Mehrleitersystemen und die Bedeutung der Diffusionsgleichung zur Ermittlung der Leitungsbeläge. Frequenz 25, 208–215 (1971) Lorenz, R.W.: Berechnung der frequenzabhängigen Leitungsbelagsmatrizen von Systemen verlustbehafteter Leiter mit rundem Querschnitt. Frequenz 25, 227–234 (1971). Frequenzabhängigkeit von Induktivität und Verlustwiderstand der Doppelleitung bei Gegentakt- und Gleichtakterreung. Frequenz 26, 1–7 (1972) Piefke, G.: Zusammenfassung der Maxwellschen und Londonschen Theorie mit Anwendung auf supraleitende Bandleitungen. AEÜ 17, 249–253 (1963) Kaden, H.: Über den Verlustwiderstand von Hochfrequenzleitern. Archiv für Elektrotechnik 28, 818–825 (1934) Wheeler, H.A.: Formulas for the skin effect. Proc. IRE 30, 412–424 (1942) Hondros, D., Debye, P.: Elektromagnetische Wellen an dielektrischen Drähten. Ann. Phys. 32, 465 (1910) Sommerfeld, A.: Ann. d. Phys 67, 233 (1899) Hondros, D.: Ann. Phys. 30, 905 (1909) Zahn, H.: Über den Nachweis elektromagnetischer Wellen an dielektrischen Drähten. Ann. Phys. 49, 907 (1916) Schriever, O.: Elektromagnetische Wellen an dielektrischen Drähten. Ann. Phys. 63, 645 (1920) Harvey, A.F.: Microwave Engineering. Academic, London, New York (1963)

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18. Mallach, P.: Dielektrische Richtstrahler. FTZ 2, 33 u (1949). FTZ 3, 325 (1950) 19. Mallach, P.: Untersuchungen an dielektrischen Wellenleitern in Stab- und Rohrform. FTZ 8, 8 (1955) 20. Unger, H.G.: Dielektrische Rohre als Wellenleiter. AEÜ 8, 241–252 (1954) 21. Unger, H.G.: Übertragungswerte von Wellen an dielektrischen Rohren. FTZ 8, 438–443 (1955) 22. Collin, R.: Field Theory of Guided Waves, p. 481. McGraw-Hill (1960) 23. Gloge, D., Marcatili, E.A.J.: Multimode theory of graded-core fibers. Bell Syst. Tech. J. 52, 1563–1578 (1973) 24. Marcuse, D.: Theory of Dielectric Optic Waveguides. Academic, New York (1974) 25. Unger, H.G.: Planar Optical Waveguides and Fibres. Clarendon Press, Oxford (1977) 26. Okoshi, T.: Optical Fibers. Academic, New York (1982) 27. Kersten, R.T.: Einführung in die optische Nachrichtentechnik. Springer, Berlin, Heidelberg, New York, Tokyo (1983) 28. Snyder, A.D., Love, J.D.: Optical Waveguide Theory. Chapman & Hall, London (1983) 29. Gowar, J.: Optical Communication Systems. Prentice-Hall, London (1984) 30. Heinlein, W.: Grundlagen der faseroptischen Übertragungstechnik. Teubner, Stuttgart (1985) 31. Geckeler, S.: Lichtwellenleiter für die optische Nachrichtenübertragung. Springer, Berlin, Heidelberg, New York, Tokyo (1985) 32. Börner, M., Trommer, G.: Lichtwellenleiter. Teubner, Stuttgart (1989) 33. Grimm, E., Nowak, W.: Lichtwellenleitertechnik. Hüthig, Heidelberg (1989) 34. Unger, H.G.: Optische Nachrichtentechnik, Teil I: Optische Wellenleiter. Hüthig, Heidelberg (1990) 35. Grau, G., Freude, W.: Optische Nachrichtentechnik. Springer, Berlin, Heidelberg, New York, Tokyo (1991) 36. Unger, H.G.: Optische Nachrichtentechnik, Teil II: Komponenten, Systeme, Meßtechnik. Hüthig, Heidelberg (1992) 37. Miller, S.E., Kaminow, I.P.: Optical Fiber Telecommunications II. Academic, Boston (1986) 38. Gloge, D.: Optical power flow in multimode fibers. Bell Syst. Tech. J. 51, 1767–1783 (1972) 39. Gloge, D.: Impulse response of clad optical multimode fibers. Bell Syst. Tech. J. 52, 801–816 (1973) 40. Agrawal, G.P.: Nonlinear Fiber Optics. Academic, Boston (1989) 41. Dianov, E.M., Mamyshev, P.V., Prokhorov, A.M., Serkin, V.N.: Non-linear Effects in Optical Fibers. Harwood Academic Publishers, Chur (1989) 42. Zenneck, J.: Über die Fortpflanzung elektromagnetischer Wellen längs einer ebenen Leiterfläche und ihre Beziehung zur drahtlosen Telegraphie. Ann. Phys. 23, 846–866 (1907) 43. Harms, F.: Elektromagnetische Wellen an einem Draht mit isolierender zylindrischer Hülle. Ann. Phys. 23, 44–60 (1907) 44. Goubau, G.: Single-conductor surface-wave transmission lines. Proc. IRE 39, 619 (1951) 45. Kaden, H.: Eine allgemeine Theorie des Wendelleiters. AEÜ 5, 534–538 (1951) 46. Kaden, H.: Fortschritte in der Theorie der Drahtwellen. AEÜ 5, 399–414 (1951) 47. Zinke, O.: Kabel und Funkweg im Mikrowellenbereich. NTZ 10, 425–430 (1957) 48. Thomson, J.J.: Recent Researches in Electricity and Magnetism. London (1893). Lodge, O.J.: The work of Hertz. Proc. Roy. Inst. 14, 321–340 (1894) 49. Rayleigh, L.: On the passage of electric waves through tubes or vibrations of dielectric cylinders. Philos. Mag. 43, 125–132 (1897) 50. Becker, A.: Interferenzröhren für elektrische Wellen. Ann. Phys. 8, 22–62 (1902) 51. Vlcek, A.: Zurückführung elektrodynamischer Felder auf statische Lösungen. AEÜ 18, 706– 717 (1964) 52. Zinke, O.: Runde und rechteckige Hohlleiter und ihre Ersatzschaltungen. Arch. Elektrotechn. 41, 364–384 (1955) 53. Rauskolb, R.: Fortpflanzungskonstante und Feldwellenwiderstand von Hohlleitern in der Umgebung der Grenzfrequenz und bei tiefen Frequenzen. AEÜ 16, 427–435 (1962)

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54. Lorek, W.: Eine Erweiterung der quasistationären Hohlleitertheorie auf Steghohlleiter und ihre Anwendung bei der parametrischen Steuerung des H10 —Wellenfeldes. Darmstädter Dissertation, 1971 55. Papadopoulos, U.M.: Propagation of electromagnetic waves in cylindrical waveguides with imperfectly conducting walls. Q. J. Mech. Appl. Math. 7, 325–334 (1954) 56. Karbowiak, A.E.: Theory of imperfect waveguides. The effect of wallimpedance. J. Inst. Electr. Eng. 102(Part B), 698–708 (1955) 57. Bayer, H., Schaffeld, W.: Über das Verhalten elektromagnetischer Wellen in kreiszylindrischen Hohlleitern im Bereich der Grenzfrequenz unter Berücksichtigung der endlichen Wandleitfähigkeit. AEÜ 10, 89–97 (1956) 58. Marcuvitz, N.: Waveguide Handbook. MIT Rad. Lab. Series, vol. 10. McGraw-Hill (1951) 59. Buchholz, H.: Elektr. Nachrichtentechnik 16, 73 (1939) 60. Uher, J., Arndt, F., Bornemann, J.: Field theory of design of ferrite-loaded waveguide nonreciprocal phase shifters with multisection ferrite or dielectric step impedance transformers. IEEE Trans. Microwave Theory Tech. MTT-35, 552–560 (1987) 61. Deutsch, J., Zinke, O.: Kontaktlose Kolben für Mikrowellenmeßgeräte. FTZ 7, 419–424 (1954) 62. Schmidt, S.: Untersuchungen an einem kreiszylindrischen Hohlraumresonator mit axial geschichtetem Dielektrikum. AEÜ 16, 436–448 (1962) 63. Otto, W.: Berechnung von Hohlraumresonatoren höchster Güte. Nachrichtentechnik 10, 205– 209, 267–272, 365–372 (1960) 64. Matthaei, G.L., Young, L., Jones, E.M.T.: Microwave Filters, Impedance Matching Networks and Coupling Structures. McGraw-Hill, New York (1964) 65. Young, L.: Direct-coupled cavity filters for wide and narrow bandwidths. IEEE Trans. MTT-II, 162–178 (1963) 66. Levy, R.: Theory of direct-coupled-cacity filters. IEEE Trans. MTT-15, 340–348 (1967) 67. Levy, R.: Filters with single transmission zeros at real or imaginary frequencies. IEEE Trans. MTT-24, 172–181 (1976) 68. Williams, A.E.: A four-cavity elliptic waveguide filter. IEEE Trans. MTT-18, 1109–1114 (1970) 69. Pfitzenmaier, G.: Synthesis and realisation of narrow-band canonical microwave band-pass filters exhibiting linear phase and transmission zeros. IEEE Trans. MTT-30, 1300–1311 (1982) 70. Butterweck, H.J.: Mikrowellenbandfilter unter Verwendung mehrerer Eigenschwingungen in einem Hohlraum. Nachr.-Tech. Fachber. 23, 46–52 (1961) 71. Pfitzenmaier, G.: An exact solution for a six-cavity dualmode elliptic bandpass filter. In: IEEE MTT-S International Microwave Symposium Digest, San Diego, June 1977, pp. 400–403 72. Pöbl, K., Wolfram, F.: Dielektrische Resonatoren, neue Bauelemente der Mikrowellentechnik. Siemens-Components 20, 14–18 (1982) 73. Cohn, S.B.: Microwave bandpass filters containing dielectric resonators. IEEE Trans. MTT16, 218–227 (1968) 74. Plourde, J.K., Ren, C.-L.: Application of dielectric resonators in microwave components. IEEE Trans. Microwave Theory Techn. MTT-29, 754–770 (1981) 75. Fiedziuszko, S.J.: Microwave dielectric resonators. Microw. J. 29, 189–200 (1986) 76. Levy, R.: Directional couplers. In: Young, L. (ed.) Advances in Microwaves, vol. 1. Academic, New York (1966) 77. Levy, R.: Improved single and multiaperture waveguide coupling theory including explanation of mutual interactions. IEEE Trans. Microwave Theory Tech. MTT-28, 331–338 (1980) 78. Matthaei, G.L., Young, L., Jones, E.M.T.: Microwave Filters, Impedance-Matching Networks and Coupling Structures. McGraw-Hill, New York (1964) 79. Collin, R.E.: Grundlagen der Mikrowellentechnik. Verlag Technik, Berlin (1973) 80. Young, L.: Branch guide directional couplers. Proc. Nat. Electron. Conf. 12, 723–732 (1956) 81. Reed, J., Wheeler, G.J.: A method of analysis of symmetrical fourport networks. IRE Trans. Microwave Theory Tech. MTT-4, 246–252 (1956)

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82. Reed, J.: The multiple branch waveguide coupler. IRE Trans. Microwave Theory Tech. MTT6, 398–403 (1958) 83. Patterson, K.G.: A method for accurate design of a broadband multibranch waveguide coupler. IRE Trans. Microwave Theory Tech. MTT-7, 466–473 (1959) 84. Young, L.: Synchronous branch-guide directional couplers for low and high power applications. IRE Trans. Microwave Theory Tech. MTT-IO, 459–475 (1962) 85. Levy, R., Lind, L.F.: Synthesis of symmetrical branch-guide directional couplers. IEEE Trans. Microwave Theory Tech. MTT-16, 80–89 (1968) 86. Levy, R.: Analysis of practical branch-guide directional couplers. IEEE Trans. Microwave Theory Tech. MTT-17, 289–290 (1969) 87. Levy, R.: Zolotarev branch-guide couplers. IEEE Trans. Microwave Theory Tech. MTT-21, 95–99 (1973) 88. Kühn, E.: Improved design and resulting performance of multiple branch-waveguide directional couplers. Arch. EI. Übertragung 28, 206–214 (1974) 89. Bräckelmann, W., Hess, H.: Die Berechnung von Filtern und 3-dB-Kopplern für die Hm0 Wellen im Rechteckhohlleiter. Arch. EI. Übertragung 22, 109–116 (1968) 90. Arndt, F., et al.: Field theory analysis and numerical synthesis of symmetrical multiplebranch waveguide couplers. Frequenz 36, 262–266 (1982) 91. Lutzke, D.: Lichtwellenleiter-Technik. Pflaum-Verlag, München (1986) 92. Wolff, J.: Felder und Wellen in gyrotropen Mikrowellenstrukturen. Habilitationsschrift TH Aachen (1970) 93. Westphal, W.H.: Physik. 25./26. Aufl. Springer, Berlin, Göttingen, Heidelberg, p. 616 (1963) 94. Deutsch, J.: Ferrite und ihre Anwendungen bei Mikrowellen. 1. Teil: NTZ 11, 473–481 (1958); 2. Teil: NTZ 11, 503–507 (1958) 95. Helszajn, J.: Principles of Microwaves Ferrite Engineering. Wiley (1969) 96. Kittel, C.: On the theory of ferromagnetic resonance absorption. Phys. Rev. 73, 155–161 (1948) 97. Polder, D.: On the theory of ferromagnetic resonance. Philos. Mag. 40, 99–115 (1949) 98. Penfield, P.: A classification of lossless three-ports. Transact. IRE CT-9, 215–223 (1962) 99. Motz, H., Wrede, H.W.: Ferrite für Resonanz-Richtungsisolatoren und Zirkulatoren. Telefunken-Zeitung 38, 187–195 (1965) 100. Wright, W., McGowan, J.: High-power Y-junction E-plane circulator. IEEE Trans. MTT-16, 557–559 (1968) 101. Solbach, K.: E-plane circulators 30 through 150 GHz for integrated mm-wave circuits. In: Proceedings of the 13th European Microwave Conference, 1983, pp. 163–167 102. Bosma, H.: On the principle of stripline circulation. Proc. IEEE (B) (Suppl. 21), 137–146 (1961) 103. Bex, H., Schwarz, E.: Wirkungsweise konzentrierter Zirkulatoren. Frequenz 24, 288–293 (1970) 104. Meinel, H., Plattner, A.: Radartechnik mit Millimeterwellen. Wiss. Ber. AEG-Telefunken 54, 164–171 (1981) 105. Fox, A.G., Miller, S.E., Weiss, M.T.: Behaviour and applications of ferrites in the microwave region. Bell Syst. Tech. J. 34, 95–97 (1955) 106. Holpp, W.: Hohlleiterzirkulatoren für den Millimeterwellen-Bereich. Wiss. Ber. AEGTelefunken 54, 212–218 (1981) 107. Tanaka, S., Shimomura, N., Ohtake, K.: Active circulators—the realisation of circulators using transistors. Proc. IEEE 53, 260–267 (1965) 108. Rembold, B.: Ein 3-Tor-Zirkulator mit aktiven Bauelementen. NTZ 24, 121–125 (1971) 109. Rehwald, W., Vöge, K.H.: Untersuchungen an einer koaxialen Ferrit-Richtungsleitung. Frequenz 16, 367–375 (1962)

Chapter 6

Antennas Jan Hesselbarth

6.1 Introduction In radio-frequency (RF) circuits, the signal energy is transported along waveguides in the form of a guided electromagnetic wave. Most waveguides rely on currents and charges on metallic structures, and electric and magnetic fields are related to them. On the other hand, the plane wave is a well-known solution of Maxwell’s equations. In a plane wave, electric and magnetic fields interact, and there are no currents nor charges required for the existence of the wave. Even though a plane wave cannot exist physically, as it has infinite extend and carries infinite power, electromagnetic waves propagating in free space can be approximated by plane waves in many cases. An electromagnetic wave propagating in free space shall be named space wave. The device transforming a guided wave into a space wave, and vice versa, is called antenna (or: aerial). This is in line with the definition by the IEEE, namely, an antenna (aerial) being a “means for radiating or receiving radio waves” [1]. From an engineering perspective, a slightly widened approach can be advantageous, using terms and concepts of antenna engineering also for, e.g., circuitry of integrated optics (that is, no currents nor charges) combined with optical radiators (that is, frequencies much higher than typical radio waves). For the sake of simplicity, antennas are treated in the following as linear, timeinvariant and reciprocal components. The transmit antenna presents a load for the transmitter, which can be described by the load’s impedance. The impedance has real and imaginary parts and varies over frequency. In many engineering applications, a good impedance match is required over a decent bandwidth. This leads to various specific techniques in antenna design as discussed in the following. For a receiver circuit, the antenna represents a source with a complex source impedance, which again varies over frequency. On the “free space side of the antenna”, however, J. Hesselbarth (B) University of Stuttgart, Stuttgart, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. L. Hartnagel et al. (eds.), Fundamentals of RF and Microwave Techniques and Technologies, https://doi.org/10.1007/978-3-030-94100-0_6

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things are more complex. The radiated energy (from a transmitting antenna) can be directed in different directions of space. The so-called radiation pattern describes the intensity of the radiated wave as a function of direction. Since “direction” has two variables (often denoted as θ and ϕ in spherical coordinates), a corresponding graphical representation will be three-dimensional. In addition, the antenna radiates in two orthogonal polarizations, and the radiation pattern is frequency-dependent. As a result, many parameters are derived from the radiation pattern, mostly aiming to compress or reduce the amount of information and to bring key antenna characteristics into a simple graph or few key numbers. Such parameters like directivity, beam-width, side lobe level etc. are discussed in the following. Fortunately, because of the above restriction to linear and reciprocal antennas, the parameters describing an antenna remain the same no matter if it is used in transmit or in receive application. The design of practically relevant antennas is now a more than 100 years old art. New frequency bands, new technologies and new applications drive the development of new antennas, but many “historic” antenna designs are still powerful if adapted, scaled or otherwise optimized. In that sense, the use of electromagnetic field simulation software helps a lot in optimization and shortens development time, but an understanding of the basic principles of the many classes of antennas developed over time is crucial for making the right choice when designing an antenna for a specific task. In the following, a mathematical treatment of the radiation process will first result in a demonstration of key aspects of antennas. Then, typical parameters to describe antenna characteristics are introduced. In the remaining sections, typical classes of antennas are described, discussed with their salient features, advantages and problems.

6.2 The Hertzian Dipole Assuming that an antenna must carry some current (conduction current and/or displacement current), the most simple radiating structure to consider is likely a very short straight current element in free space (note that a single point in space, though seemingly even simpler, cannot carry a current which needs a direction). This structure is called Hertzian dipole.1 The current path is so short that it is assumed constant over the length. Due to the current, there will be charge of one polarity at one end of the dipole, and charge of opposite polarity at the other end. Obviously, charges and currents create electric and magnetic field in the surrounding space, governed by Maxwell’s equations. The calculation of the electromagnetic fields surrounding the Hertzian dipole is a useful exercise. It allows to find typical parameters describing 1

As a professor in Karlsruhe, Germany, Heinrich Hertz (February 22, 1857, to January 1, 1894) demonstrated in 1886 that analogous to light waves, electromagnetic waves can be refracted and reflected and are propagated like light waves. In his experiments, he employed short dipole antennas and small loop antennas to transmit and receive the waves.

6 Antennas

487 z E

θ

H

R

J

y

ϕ x

Fig. 6.1 Hertzian dipole, z-oriented in the origin of both cartesian and spherical coordinate systems, and point of observation at R

these fields and the combination of the radiation of many Hertzian dipoles (both Maxwell’s equations and antenna are linear, so the principle of superposition applies) leads to good approximations of many practically relevant antennas. It is possible to calculate the fields, including the radiated fields, of a current distribution by direct integration of Maxwell’s equations. However, this integration can be a mathematically very complex task. Therefore, so-called vector potentials are introduced as an intermediate step. Finding them involves an integration, and the resulting electric and magnetic fields follow then from rather straightforward differentiation. Figure 6.1 shows the simplified scenario of a short current element of length L, placed in the origin of both cartesian and spherical coordinate systems. The current is of constant of magnitude along L and is oriented in direction of z-axis, J = Jz . In the time-harmonic case (frequency ω), Maxwell’s equations read ∇ × H = jωε E + J ∇ × E = − jωμ H with ∇ · B = 0 (no magnetic charges) and B = μ H , where ∇ denotes the Nabla operator. Because of ∇ · (∇ × V ) ≡ 0 holds for any vector field V , the magnetic vector potential A can be defined such that B = ∇ × A = 0. If the curl of a vector field V is zero, Substitution gives ∇ × ( E + jω A) ∇ × V = 0, then this vector field can be expressed as the gradient of a potential field,

488

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Φ, as V = ∇Φ, because of the identity ∇ × (∇Φ) ≡ 0. Thus, the electric scalar potential Φe is introduced as E + jω A = −∇Φe Here, the minus sign can be chosen according to the definition of Φe . Backsubstitution in to Maxwell’s equations results in − → ∇ × ∇ × A = jωεμ − jω A − ∇Φe + μ J − ∇ × ∇ × A, allows to The Laplace operator, , defined as A = ∇(∇ · A) re-write this in the form − → A + ω2 εμ A = −μ J + ∇ ∇ · A + jωεμΦe This expression simplifies by relating the newly introduced parameters, the magnetic vector potential A and the electric scalar potential Φe , according to ∇ · A = − jωεμΦ e (this condition is called Lorentz gauge), leading to the inhomogeneous wave equation for the vector potential A + ω2 εμ A = −μ J A particular solution for this inhomogeneous wave equation in A is μ A = 4π

˚

e J

− jk R

R

dV

V

√ where k denotes the wave number, k = ω με, and R is the distance between the current element and the point of observation as shown in Fig. 6.1. Once A is known, the electric and magnetic fields are obtained straightforwardly by differentiation from 1 H = ∇ × A μ E = − jω A −

j ∇(∇ · A) ωμε

or, for source-free regions, directly from Maxwell’s equations

6 Antennas

489

1 ∇ × H E = jωε The current distribution with the single current element as shown in Fig. 6.1 can be described using the Dirac delta function, δ, as J(x, y, z) =

0 · ex + 0 · ey + I0 δ(x)δ(y) · ez for − L/2 ≤ z ≤ L/2 0 elsewhere

is given by In this case, the solution of the magnetic vector potential, A, μ A(x, y, z) = 4π

+L/2 +∞ +∞

ez I0 δ(x)δ(y) −L/2 −∞ −∞

I0 μL − jkr e− jkr d xd ydz = ez e r 4πr

which can be transformed into spherical coordinates as θ, ϕ) = I0 μL e− jkr (er cos θ − eθ sin θ ) A(r, 4πr The magnetic and electric fields are now obtained by applying the curl operation as described in the above equations, resulting in (in spherical coordinates!) on A, Hr ≡ 0 Hθ ≡ 0

1 k I0 L sin θ 1+ e− jkr 4πr jkr 1 I0 L cos θ 1 + e− jkr Er = Z 0 2πr 2 jkr 1 1 k I0 L sin θ e− jkr 1+ − Eθ = j Z 0 4πr jkr (kr )2 Eϕ ≡ 0

Hϕ = j

where Z0 =

μ0 ε0

denotes the field impedance of free space, a value of approximately 377 . Thus, the particular scenario depicted in Fig. 6.1 (a spherical coordinate system with the source in the origin and with the source oriented in the direction θ = 0) leads to a solution with only three non-zero vectorial field components. Further inspection of

490

J. Hesselbarth

these three components suggests that far away from the source current (where r is large), the terms in the round brackets approach “one” in all three cases. When r is large, and with the exception of the dipole axis, E r 1 K by the Rayleigh-Jeans approximation. 2kTH H≈ (8.152) λ2 The radiation of a black body is unpolarized and of a random nature. This application involves an inhomogeneously tempered black body. If, as shown in Fig. 8.18a, the antenna is assumed to be disposed inside the black body, this means that TH is a function of θ and φ. According to Fig. 8.18b, it is assumed that a surface element is considered on the surface of the black body, the projection of which onto a surface perpendicular to the r -direction is designated as d A H . The power H d A H is isotropically emitted from this surface element for each bandwidth unit. At the distance r , this therefore gives a spectral noise power density per surface unit of: d SA = H

d AH = H d H 4πr 2

(8.153)

With d A H /(4πr 2 ) = d H , the solid-angle element d H is introduced. This is the solid angle at which the vertically oriented surface element d A H is to be viewed from the observation location. The antenna at the origin of the coordinate system in Fig. 8.18a is impedance matched, i.e. Z = Z ∗A , if Z A means the impedance in the antenna feed point. The available power provided to the load, d Nv is then absorbed z

a

b TH (f, q)

dAH q

antenna

dAH r

dWH

y matched load

f

point of observation

x

Fig. 8.18 a Antenna inside a black body. TH is a function of θ, φ; b surface element d A H and solid-angle element d H

8 Interference and Noise

789

for the frequency interval f and the heat radiation contribution radiated from the direction θ , φ. 1 k · Aw (θ, φ) · d S A (θ, φ) · δ f = 2 TH (θ, φ) · Aw (θ, φ) · δ f · d H 2 λ (8.154) where Aw (θ, φ) means the active area of the antenna in the θ, φ direction. The reason for the factor 1/2 is that Eq. (8.152) applies to the total radiation in all polarization directions, but the antenna processes only a preferred polarization (co-polarization component). In order to determine the total noise power Nv , it is required to integrate d Nv over the entire antenna environment. ⎧ ⎫ ⎨1 % ⎬ A (θ, φ) · T (θ, φ) · d Nv = k f (8.155) w H H ⎩ λ2 ⎭ d Nv =

4π

If Nv is expressed in terms of an equivalent antenna temperature T A according to Nv = kT A f , the following is found: 1 TA = 2 λ

% Aw (θ, φ) · TH (θ, φ) · d H

(8.156)

4π

In order to express T A in terms of the antenna gain G(θ, φ), Aw (θ, φ) can be substituted via the relationship Aw (θ, φ) = (λ2 /4π )G(θ, φ): TA =

1 4π

% G(θ, φ) · TH (θ, φ) · d H

(8.157)

4π

The antenna noise temperature is therefore to be understood as the mean value of the noise temperature of the antenna environment weighted with the antenna active area or the antenna gain. It naturally depends on the spatial orientation of the antenna. Two special cases are considered. In the first case, a constant distribution of TH , changing only slightly with θ, φ and an antenna with very high gain in the primary transmitting direction are assumed. If the antenna is aligned in direction θ0 , φ0 referred to a space-fixed coordinate system according to Fig. 8.18a, the antenna noise temperature T A (θ0 , φ0 ) is measured. The following then applies: T A (θ0 , φ0 ) =

1 4π

% G(θ, φ) · TH (θ, φ) · d H 4π

1 · TH (θ0 , φ0 ) = 4π

(8.158)

% G(θ, φ) · d H = TH (θ0 , φ0 ) 4π

4π

(8.159)

790

M. Rudolph

a 106 solar noise

noisy quiet TH /K

105

104

1

10

100

f / GHz

b

c 1000

1000 cosmic noise (milky way)

atmosphere

Φ horizon 100 0o

TH /K

TH /K

100 max.

10

10 min.

10o 30o 5o 50o

1

10 f / GHz

cosmic 1

100

10 f / GHz

absorption by water, oxygen 100

Fig. 8.19 Noise temperatures: a solar noise; b cosmic background noise; c atmospheric noise; the bold solid line applies to the sum of the thermal noise and atmospheric noise at = 50◦

In this way, the direction-dependent background radiation of the antenna environment can be measured. In Fig. 8.19b, c, the cosmic and atmospheric noise temperatures TH are represented as a function of frequency. For the second case, a virtually discrete noise source of the temperature TH is assumed in a narrow solid-angle area H . The other background radiation can be ignored. The equivalent solid angle ω A = 4π/G of the antenna is greater than H . If the antenna is aligned with the noise source, it follows: T A = TH

H A

(8.160)

This relation allows for a calculation of TH , if H of the source and A of the antenna are known and T A is measured. A practical example of a discrete, cosmic noise source is the sun, wherein approx. 6.5 × 10−5 sr (steradiant) can be assumed for H . The frequency dependence of the solar noise temperature is shown in Fig. 8.19a

8 Interference and Noise

791

for quiet and noisy sun. The antenna noise temperature plays an essential role in the dimensioning of a radio path only if it is of the same order of magnitude as or exceeds the effective noise temperature of the receiver. If T A is known, it can be decided, on the other hand, whether it is also worth using a specific low-noise amplifier.

References 1. Papoulis, A., Pillai, S.U.: Probability, Random Variables and Stochastic Processes, 4th ed. McGraw Hill, New York (2002) 2. Schottky, W.: Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern. Ann. Phys. 57, 541–567 (1918) 3. van der Ziel, A.: Thermal noise at high frequencies. J. Appl. Phys. 21, 399–401 (1950) 4. Johnson, I.B.: The Schottky effect in low-frequency circuits. Phys. Rev. 26, 71–85 (1925) 5. Hooge, F.N.: 1/ f noise sources. IEEE Trans. Electron Devices 41(11), 1926–1935 (1994) 6. Haus, H., Adler, R.: Canonical form of linear noisy networks. IRE Trans. Circuit Theor. 5(3), 161–167 (1958) 7. Rothe, H., Dahlke, W.: Theory of noisy fourpoles. Proc. IRE 44(6), 811–818 (1956) 8. Hillbrand, H., Russer, P.: An efficient method for computer aided noise analysis of linear amplifier networks. IEEE Trans. Circuits Syst. 23(4), 235–238 (1976) 9. Hillbrand, H., Russer, P.: Correction to “An efficient method for computer aided noise analysis of linear amplifier networks”. IEEE Trans. Circuits Syst. 23(11), 691 (1976) 10. Pucel, R.A., Struble, W., Hallgren, R., Rohde, U.L.: A general noise de-embedding procedure for packaged twoport linear active devices. IEEE Trans. Microwave Theor. Tech. 40(11), 2013– 2024 (1992) 11. Pucel, R.A., Haus, H.A., Statz, H.: Signal and noise properties of gallium arsenide microwave field-effect transistors. Adv. Electron. Electron Phys. 38, 195–265 (1975) 12. Pospieszalski, M.W.: Modeling of noise parameters of MESFETs and MODFETs and their frequency and temperature dependence. IEEE Trans. Microwave Theor. Tech. 37(9), 1340– 1350 (1989) 13. Heymann, P., Rudolph, M., Prinzler, H., Doerner, R., Klapproth, L., Bock, G.: Experimental evaluation of microwave field-effect-transistor noise models. IEEE Trans. Microwave Theor. Tech. 47(2), 156–163 (1999) 14. Rudolph, M., Doerner, R., Klapproth, L., Heymann, P.: An HBT noise model valid up to transit frequency. IEEE Electron Device Lett. 20(1), 24–26 (1999) 15. Fukui, H.: Optimal noise figure of microwave GaAs MESFETs. IEEE Trans. Electron Devices 26(7), 1032–1037 (1979) 16. Van Der Ziel, A.: Noise in junction transistors. Proc. IRE 46(6), 1019–1038 (1958) 17. Rudolph, M., Heymann, P.: Comparative study of shot-noise models for HBTs. In: Microwave Integrated Circuit Conference (EuMIC), pp. 191–194 (2007) 18. Rudolph, M., Korndorfer, F., Heymann, P., Heinrich, W.: Compact large-signal shot-noise model for HBTs. IEEE Trans. Microwave Theor. Tech. 56(1), 7–14 (2008) 19. Pucel, R.A., Rohde, U.L.: An exact expression for the noise resistance Rn for the Hawkins bipolar noise model. IEEE Microwave Guided Wave Lett. 3(2), 35–37 (1993)

Chapter 9

Amplifiers Rüdiger Quay

Abstract This chapter describes one of the main application of RF-and microwave concepts, i.e., the concept of amplification and related procedures.

Acronyms AlGaN BiCMOS BT CMCD CMOS CS DE DG DSP FET GaAs GaN HBT HPA IC InP LINC LNA LP MAG MDS MMIC

Aluminum gallium nitride Bipolar complementary metal oxide Bipolar transistor Current-mode class-D (amplifier) Complementary metal oxide semiconductors Common-source Drain efficiency Dual-gate Digital-signal processing Field-effect transistor Gallium arsenide Gallium nitride Hetero-bipolar transistor High-power amplifier Integrated circuit Indium phosphide LInear amplification with Nonlinear Components Low-noise amplifier Loadpull Maximum available gain Minimum detectable signal Microwave monolithically integrated circuit

R. Quay (B) Fraunhofer-Institute for Applied Solid State Physics IAF, Freiburg im Breisgau, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. L. Hartnagel et al. (eds.), Fundamentals of RF and Microwave Techniques and Technologies, https://doi.org/10.1007/978-3-030-94100-0_9

793

794

MOSFET MSG PA DPD PAE PCB PUF SiC SiGe TRX VCO VGA

R. Quay

Metal-oxide semiconductor field-effect transistor Maximum stable gain Power amplifier Digital predistorsion Power-added efficiency Printed circuit board Power utilization factor Silicon carbide Silicon germanium Transmit-receive Voltage controlled oscillator Variable gain amplifier

9.1 Amplifier Characteristics in Complex Functions Amplifiers are active two-port or multi-port networks which amplify input signals using external energy sources. Electronic amplifiers are divided up into small-signal amplifiers (direct-current amplifiers, low-power, low-frequency amplifiers, broadband amplifiers, narrowband high-frequency amplifiers) and large-signal amplifiers (high-power, low-frequency amplifiers, transmitter amplifiers). Important characteristics of amplifiers are the input and output level (saturation level), input and output impedance and transmission factors as a function of frequency. Further aspects to be considered in the design of amplifiers are signal-to-noise ratio and distortions, efficiency and power consumption, stability to withstand temperature fluctuations, aging of components and changes in the terminal impedance. Figure 9.1 gives the schematic of a very simple transmit and receive (TRX) function or module, where the importance of amplifiers becomes visible. RF- and microwave amplifiers are needed due to the radiation laws with a reduction of the received signal by 1/r4 in distance in r for active sensing, due to power dissipation, and due to the undesired coupling of waves. In the transmit path high-gain driver amplifiers (DRA) and high-power amplifiers (PA) are needed. In the receive path low-noise amplifiers (LNA) add the most decisive contributions to the signal distortion and thus require particular attention. The pre-amplifier (Pre-Amp) and post amplifiers (post amplifier) are typically amplifier with variable gain (VGA). To distinguish a tuned amplifier, that amplifies the highfrequency signals at a specific frequency f for a bandwidth Δf, can be characterized by several figures of merit. These include: • Amplification or gain, • Bandwidth, • Stability,

9 Amplifiers

795

Driver Amplifier

Pre-Amp

DSP

Attenuator

Phase shifter

High Power PA Amplifier

antenna

Post Amplifier

Low Noise Amplifier

Fig. 9.1 Schematic of a TRX function or module, DSP (digital signal processing) Fig. 9.2 Principal schematic of an active device with input and output currents and voltages

Uin Iin

Uout Iout

• Noise (both amplitude as well as phase noise), • Efficiency in its various forms, and • Linearity, again expressed in its various forms. These aspects, all relevant to the functionality of the module depicted in Fig. 9.1, are discussed in the following. Additional aspects may be of strong importance, such as ruggedness and susceptibility to damages from thermal or electrical exposure, however, will not be covered in the course of this chapter (Fig. 9.2).

9.1.1 Amplification and Gain Amplification is defined either as voltage and current amplification: Uout ( f ) Uin ( f ) Iout ( f ) Ai = Iin ( f )

Av =

(9.1) (9.2)

Both are frequency dependent. The combination of both is called power amplification, and will be discussed in the following.

796

R. Quay

9.1.2 RF-Device Configurations Any active device can be used in different principal configurations. These include: 1. common source/common emitter; 2. common gate/common base; 3. common drain/common collector. The situations are explained in Fig. 9.3. For the use in RF-amplifiers the three principal configurations are found useful in different applications, with respect to both voltage and current amplification, their impedance transformation, and isolation. For principal analysis we start with the configurations with their properties for frequency f → 0. The common-source configuration makes use of both the high current-gain and of the voltage gain.

FET

D

D

D Vout out Vout out

Vout out

Vinin

Vinin G

G

G

S

S

S

Vinin

Bipolar C C

C Vout out

Vout out

B

Vinin

B

Vinin

E

G E

Vout out

Vinin

Fig. 9.3 Definition of the RF-device configurations given for an N-channel Schottky field-effect transistor (FET) and an p-based bipolar transistor (BT)

9 Amplifiers

797

AV =

Uout gm · R D ≈ Vin 1 + gm · R S Iout AI = ≈∞ Iin

(9.3) (9.4)

The voltage gain is thus determined by the loading resistances R S and R D for gm 1. Common gate devices have a unity short circuit current gain and a voltage gain as given by the approximation in Eq. 9.5. They are useful to reduce feedback (see below). Uout gm · R L ≈ Vin 1 + R S · gm Iout AI = ≈1 Iin

AV =

(9.5) (9.6)

The common-drain/common-collector is typically called a source/emitter follower. In this configuration, again with the approximation for low frequencies, a unity voltage gain and a very high current gain are achieved. Uout gm · R S ≈1 = Vin gm · R S + 1 Iout AI = =∞ Iin

AV =

(9.7) (9.8)

In addition, the impedance transformation of the device configuration has to be considered. In order to describe this behavior we switch to the powerful tool of RF-parameters.

9.1.3 RF-Parameter Description of Small-Signal Amplifiers 9.1.3.1

Basic Parameters and Signal-Flow

Any basic two-port (or later multiport) configuration can be described by the four fundamental parameter systems z, y, h, and p. Figure 9.4 gives the small-signal equivalent circuit diagrams of active two ports in the four descriptions: (a) z-parameter, (b) y-parameter, (c) h-parameter, and (d) p-parameter. The two-port equations can be written: U1 = z 11 · I1 + z 12 · I2 U2 = z 21 · I1 + z 22 · I2 I1 = y11 · U1 + y12 · U2 I2 = y21 · U1 + y22 · U2

(9.9) (9.10) (9.11) (9.12)

798

R. Quay

a

I1

(z)

U1

z11

z12 I2

z22

~

b

I2

U z21 I1 2

~

I1

(y)

I2

U1

~

U2

~

y11

y22 y21U1

y12 U2

c

d I1

(h)

U1

h12 U2

~

U2

~

p22

I1

I2

h11

(p)

U1

~

~

p11

h22

I2

U2 p21 U1

p12 I2

h21 I1

Fig. 9.4 Principal schematic of an active device in a z, b y, c h, and d p-parameter configuration Fig. 9.5 S-parameters and ai /b j wave description of the amplifier

ZS a1

Transistor

a2 ZL

b1 (S)

b2

U1 = h 11 · I1 + h 12 · U2 I2 = h 21 · I1 + h 22 · U2

(9.13) (9.14)

I1 = p11 · U1 + p12 · I2 U2 = p21 · U1 + p22 · I2

(9.15) (9.16)

In addition, Fig. 9.5 gives the description with both the S-parameters and the ai and b j waves with the relations for a multiport with n-ports: bk =

n

Sk j a j with k = 1, 2, .., n

(9.17)

j=1

with Sk j =

bj |a =0 for m= k ak m

(9.18)

The waves bk are results of the incoming wave quantities a j and thus represent the dependent variable. The coefficients of the linear equation systems are the Sparameters Si j .

9 Amplifiers

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Example The actual parameters of a two-port with n= 2 are described as: b1 |a =0 a1 2 b2 = |a1 =0 a2 b1 = |a1 =0 a2 b2 = |a2 =0 a1

S11 =

(9.19)

S22

(9.20)

S12 S21

(9.21) (9.22)

where S11 is the input reflection coefficient, S22 is the output reflection coefficient, S12 is the reverse isolation, and S21 is the forward transmission or gain. These have to mentioned with respect to a reference impedance, typically 50 , 75 , or even 10 for lower-impedance systems.

9.1.3.2

Bipolar Amplifier Example

As an example the description of an active bipolar transistor with S-parameters is given in the following. The top figure gives the equivalent circuit with input and output capacitances, feedback capacitance, and parasitic resistances. The lower figure of Fig. 9.6 gives the measured S-parameters of an InP bipolar transistor for the frequency range between 0.25 and 110 GHz. As seen in the equivalent circuit in the top, the input circuit (emitter-base) consist of the emitter-junction capacitance represented by two parallel capacitances, seen in the S11 for f→ 0, which also has a parallel real contribution. At the output, we observe a finite DC-input resistance represented by S22 for f→ 0, again in with a real contribution in parallel. The feedback, expressed by the S12 in the polar-chart, is significant, indicated by the radius of the right top polar chart, serving for the S12 Further, we observe a high absolute gain S21 for a given area or transistor width. This is indicated by the radius of the top polar chart in the graph, where the radius is given as another number. The current amplification of bipolar transistors is rather high, especially per area used, in comparison to FET, which is why the radii are typically higher for BTs.

9.1.3.3

Field Effect Transistor Example

As an example for field-effect small-signal characteristics Fig. 9.7 gives the measured S-parameters of a GaN field-effect transistor between 0.25 and 110 GHz and a simplified small-signal equivalent circuit for the modelling. Contrary to the bipolar example, we observe the purely capacitive input behavior of the FET for S11 for f→ 0, when is expressed by the input capacitance in the equivalent circuit in Fig. 9.7.

800

R. Quay

Base

Collector Cj,BC

RB

RC gm

CD,BE

gEC Cout

Cj,BE gEB

RE

Emitter

radius = 30

radius = 0.2 S12

S21

radius = 1 S11 S22 radius = 1

Fig. 9.6 Equivalent circuit and measured S-parameters of an InP-bipolar transistor with an emitter width of 1 µm and a length of 4 µm and simplified small-signal equivalent circuit (hybrid pi-model)

Further we observe a finite output resistance for f→ 0, while also the output-side S22 is capacitive represented by the output capacitance Cds . The reverse isolation or feedback, expressed by the S12 , is also important, given by the scale of the top right polar chart. The forward transmission S21 is lower in absolute numbers than in the bipolar example, again indicated by the radius of the top left polar chart. Further, as the reader may note, the measurement data for S11 and S22 turns from the capacitive region of the Smith-Polar diagram into the inductive region for higher frequencies. This is not reflected in the small-signal equivalent circuit model given. It can be reflected by adding series parasitic inductive elements at source, gate, and drain.

9 Amplifiers

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Drain RD

Cgd

Cds RG

gm

gds

Gate

Cgs Rs

Source

radius = 0.2 S21 radius = 10

S12 S22 radius = 1

S11 radius = 1

Fig. 9.7 Measured S-parameters of a GaN FET with a gate length of lg = 100 nm and a gate width of 6×45 µm and a simplified equivalent-circuit model for FETs

9.1.3.4

Idealized Basic Circuits

The principal variants for (unilateral) amplifiers derived from the parameters are depicted in Fig. 9.8. In this case, no feedback from the output to the input is considered. In this simplification we can give idealized equivalent-circuit description of amplifiers. Figure 9.8a gives the ideal transimpedance amplifier. Figure 9.8b gives the ideal transconductance amplifier with y21 = γT = gm . The current amplification factor in Fig. 9.8c, ki = A I represents the factor α or β of a transistor. The ideal

802 Fig. 9.8 Idealized principal circuits, a transimpedance amplifier, b transconductance amplifier, c ideal current source, d ideal voltage amplifier

R. Quay I1

a 0

0

zT

0

(z)=

~

zT I 1

I2

b 0

0

(y)=

U1

~

0

1

1 U1

c

I1

0

I2

0

~

(h)= 0

Ki

K i I1

d 0

0 U1

(p)= KU

KU U1

~

U2

0

voltage amplifier AU in Fig. 9.8d without infinite input impedance, lossless output and a real voltage amplification also has a infinite power amplification. To get closer to real amplifiers including the feedback effect we will discuss this effect in the next section.

9.2 RF-Feedback 9.2.1 Basic Principles RF-feedback is used with the feedback of a small portion of the output signal to the input. The delay in the feedback path must be chosen carefully to control stability. Once you supply the output current or voltage back to the input of an active

9 Amplifiers

803

a

b I1a

I1

I2a (Za)

U1a

U2

(Zb)

U2a

U2 I1b U1b

U2b

c

I2

U1

I2b

I1b

I2a (ya)

U1a

U2a

U1

U1b

I1a

I1

I2b (yb)

U2b

d I1a U1a

I2a (ha)

U2a

U2

U1b

I2 U2a

U2

U1 I1b

I2b (hb)

I2a (pa)

U1a

U1 I1b

I1a

I1

I2

U2b

U1b

I2b (pb)

U2b

Fig. 9.9 The four types of feedback: a serial current feedback, b parallel voltage feedback, c serial voltage feedback, d parallel current feedback Fig. 9.10 Control circuit with feedback loop

Xin(p)

Xw(p)

H(p)

Xout(p)

Xfeed(p) G(p)

two-port this is called feedback. A two-port can be described through the four sets of parameters given above. The resulting four types of feedback are given in Fig. 9.9. Figure 9.10 gives the general case of feedback in a control loop. Let us assume: X out ( p) X w ( p) X f eed ( p) G( p) = X out ( p) p = σ + iω

H ( p) =

(9.23) (9.24) (9.25)

804

R. Quay

The feedback circuit then can be described as: F( p) =

X out ( p) H ( p) = X in ( p) 1 + G( p) · H ( p)

(9.26)

This general case will now be discussed with some examples.

9.2.1.1

Negative Feedback

If an output parameter of an active two-port network (e.g. amplifier) is fed back to its input, a negative feedback is obtained. In an active electrical two-port network, one or both input parameters (input voltage, input current) can be modified by one or both output parameters (output voltage, output current). With restriction to the case where only one input parameter is influenced in each case by one output parameter, the four negative feedback types shown in Fig. 9.9 are obtained according to the four possible combinations. A two-port network can be described by various forms of two-port network equations, which interconnect input and output parameters. For the four coupling circuits, the form of the two-port network equations can be selected in each case in such a way that the equations of the resulting two-port network are derived from the addition of the two-port network equations of the active two-port network and the feedback two-port network (see Fig. 9.9). For the real, frequency-independent transmission factors H(p) = A0 and G(p) = K, the following is obtained: Ak =

A0 1 + K A0

(9.27)

Negative feedback occurs if |1 + K A0 | > 1. Positive feedback is designated accordingly by the condition |1 + KA0 | < 1. Negative feedback is used for the purposes discussed in the following paragraph. This gives, e.g. as the relationship between the relative changes in the transmission factors: ΔA0 1 ΔAk = Ak 1 + K A0 A0

(9.28)

Accordingly, the following relationship applies approximately to the distortion factors: 1 k0 (9.29) k≈ 1 + K A0 To investigate the stability of feedback networks, the poles of F(p), i.e. the zero values of the characteristic equation 1 + H ( p)G( p) = 0

(9.30)

9 Amplifiers

805

must be determined from the poles and zero values of the transmission factors H(p) and G(p). There are various methods for performing the stability calculation, e.g. according to Hurwitz, Cremer-Leonhard, Nyquist or the root locus method, which are described in detail in the literature [1–3]

9.2.1.2

Discussion of Special Cases

As a special case, going to real quantities without frequency dependence, we get: H ( p) = A0 G( p) = K Ak =

A0 1 + K · A0

(9.31) (9.32) (9.33)

Thus for |1 + K · A0 < 1| we get positive feedback and for |1 + K A0 > 1| we get negative feedback. These relations are very well known from a DC-perspective at low frequencies. The frequency dependence will render the situation more complex, thus we will start with some examples.

9.2.2 Basic Applications The applications and advantages of feedback are multiple: 1. Feedback can be used to stabilize a transistor against aging, temperature, and changes of the operation voltage. 2. Feedback enables to flatten the gain characteristics of an amplifier, i.e., to counter the frequency characteristics of the MAG/MSG for wideband amplifiers with frequency-selective feedback. 3. Further, the output signal can be subtracted from the input signal for the frequency band of interest in order to raise the stability (later will we call this k-factor above unity) and ensure unconditional stability. 4. Series feedback is often used by adding in inductor in the source path of an active device. This has a stabilizing effect, but at the same time moves the optimum noise match (see below) closer to the conjugate match, as explained below [5]. 5. Parallel feedback is used from the output of the device (collector/drain) to the input (base/gate). This path has both DC- and RF-requirements. 6. Feedback is suitable to reduce distortion in amplifiers, as explained below.

806

9.2.2.1

R. Quay

Example: Stabilization Against Temperature Variation

The microwave gain parameters, maximum stable gain (MSG), and maximum available gain (MAG) of individual RF-devices are strongly temperature-dependent: M AG/M SG = M AG/M SG(T ).

(9.34)

The gain can both decrease with increasing temperature (parts of the FET world) and increase with increasing temperature (e.g. bipolar world). One of the most common examples of feedback is to compensate this (increasing) temperature dependency using temperature-dependent feedback of the signal. Figure 9.11 gives a common example for an emitter- and base-based feedback to be used for electrical feedback and for temperature control. A voltage divider network consisting of R B1 and R B2 provides a voltage divider for the internal base resistance. In some cases,R B2 can also be replaced by a diode, which automatically compensates the temperature effect [4]. The resistor R E in the emitter path is used in series with the device emitter lead to provide voltage feedback. It must be carefully bypassed for RF-signals, and this impact shall be considered also for stability (see below). Further, carefully chosen R B1 , R B2 , and R E with their resistance increase with temperature helps to construct stable bipolar feedback.

Fig. 9.11 Emitter and base feedback as a means of temperature control

VCC

RB1

RB2 (T)

RE

9 Amplifiers

807

Table 9.1 Proposed feedback network for a silicon bipolar VC E = 2 V R B1 = 889

VCC = 2.7 V R B2 = 2169

IC = 5 mA R E = 2169

H F E = 80

Numerical Example for Temperature Compensation To establish an emitter-based feedback loop, as shown in Fig. 9.11, the following values are proposed for the Silicon BT, (Keysight HBFP-0405) (Table 9.1).

9.2.2.2

Example: Gain Flattening

A frequency-dependent feedback can be used to flatten the gain over the targeted bandwidth. As seen in this chapter, the main gain parameters are strongly frequencydependent. Specifically for broadband amplifiers this poses the problem, as the gain is thus not inherently flat over the targeted bandwidth, especially for wideband circuits. Frequency-dependent feedback enables a flat gain characteristics. The following simplified example is given to illustrate the mechanism. Let us assume an amplifier with a high bandwidth up to the upper band edge fup . Figure 9.12 gives the example. Numerical Example As shown in Fig. 9.12 frequency-dependent feedback is used to flatten the gain. This is achieved by an increasing (to lower frequencies) negative feedback, which is minimum for the upper band-edge (to make the most of the technology), and increases for lower frequencies. The following simplified example for a broadband amplifier shall illustrate the findings: We use a real approach: Ak =

A0 1 + K · A0

(9.35)

If A0 is 100 (20 dB) at fup , we will use K(fup )= 0. Further we demand the gain to be flat (20 dB) down to fup /10, i.e. over a decade in frequency (This is not an easy task to design, however, is good in the understanding). For lower frequencies than fup , we

Xin(p)

Xw(p)

Xout(p) H(p)

gain fup

Xfeed(p) G(p) frequency

Fig. 9.12 Feedback as a means of gain flattening

808

R. Quay

will thusconstruct a K-function as a function of frequency, which yields Ak (fup /10)= 100. Thus, we find for a transistor, which is to be fully in the MSG-region of its gain characteristics: (with a gain slope of −10 dB/dec). We get: Ak = 100 (fup ), which yields K(fup /10)= 9/100. This example is of course simplified, as we have to take the phase into account, as well over the broad bandwidth, and especially whave to accound for the losses, and the stability considerations.

9.2.2.3

Example: RF-Stabilization

Amplifier stability is a critical criterion. Design of feedback is an essential tool to achieve the stability over bandwidth. RF-Feedback for stabilization can used be in order to compensate the built-in internal feedback of a circuit, which is one source for instability. This built-in feedback is always there, as any real active RF device is nonunilateral, i.e., the increased output signal due to the forward amplification will result in a non-neglegible impact on the input signal. This is due to the intrinsic feedback capacitance (Cgd for FETs, and C j,bc in bipolar transistors). The compensation of this feedback can be achieved over frequency. This, however, is dangerous, as at lower frequencies, the amplifier is only conditionally stable, i.e. is susceptible to the matching conditions achieved at the input and output of the transistor over frequency. For further explanation, see the sections on neutralization and the stability factor below. 9.2.2.4

Example: Series Feedback Inductive Noise Matching

Series feedback can help to overcome another problem of matching: For noise matching the requirement to be met is, on the one side to achieve optimum noise matching for minimum noise, while on the other side, to achieve maximum gain, i.e. conjugate complex matching at the input of the transistor. These two conditions are typically not met simultaneously, as the required matching conditions differ for plain transistors. The following inductive-series feedback method can be used to help in this situation. Figure 9.13 gives the image of the small-signal modeling of a source feedback. The series inductor induces a purely reactive feedback, which simplifies the matching, and in resonance, actually, does not add additional noise from the matching networks. This is done, as the Q-factor of any passive technology is not ideal. Numerical Example The so called noise degeneration introduces an additional reactive component in the source path. The trick of source degeneration is based on the following analysis of the input impedance of a device Zin . We do this for a FET, however, without limiting the generality: The input impedance reads: Z in = Z LG + Z (C gs ) + Z L S + gm · Z (C gs ) · Z L S

(9.36)

9 Amplifiers

809

Drain

Cgd

LG

RG

gm

gds

Gate

Cgs RS

Source

LS

Fig. 9.13 Inductive series feedback as a means to unify both noise as well as conjugate complex matching requirements

The last term in Eq. 9.36 is due to feedback. The current generator gm is assumed to resonate at the frequency fT . The trick of the source dgeneration is to use both inductor LG and L S and to choose the input impedance we see from the outside world, which is to be matched while maintaining the noise performance. The real part of the input impedance Zin amounts to: gm · f T ≈ fT · L S (9.37) Re(Z in ) = C gs With the application of series feedback in the transistor path, we can move the optimum noise matching towards the matching obtained for the conjugate complex situation. Ideally no noise will be added, while the matching of the low-noise situation is greatly improved. Example: Distortion Reduction As a last example, feedback is a very useful tool for the reduction of non-linearity of amplifiers. In RF-feedback, a portion of the RF-output signal from the amplifier, is fed back and thus subtracted from the RF-input signal. The gain is again reduced to:

810

R. Quay

Ak =

A0 1 + K · A0

(9.38)

This is a reduction of the gain, however, in the same way, the distortion is reduced to by the very same factor.

9.2.2.5

Neutralization of Transistor Amplifiers

The internal (mostly capacitive) feedback (according to y12 of the conductance matrix) of an amplifier quadripole may result in instability. This effect can be prevented by counteracting the unwanted feedback through neutralization. Narrowband small-signal transistor amplifiers can be neutralized into the short-wave range [6]. A frequently used circuit is shown in Fig. 9.14. A current, which, in terms of amount, is as high as the current through Ccb , but is phase-shifted through 180◦ , is fed via the capacitor C N to the base. The same result is obtained if the internal collector base capacitance is matched with the operating frequency by means of a parallel-connected inductance. In broadband transistor small-signal amplifiers, complex neutralization measures are often avoided by allowing for a greater number of transistor stages with lower amplification; in transistor power amplifiers, an exact neutralization is not possible due to the modulation-dependent feedback capacitance.

9.2.2.6

RC-Coupled Amplifiers

In AC- and RF-voltage amplifiers, the DC-current path is normally separated from the AC-current path using coupling networks. In the case of RC-coupled amplifiers, these coupling networks consist of resistors and coupling capacitors. The emitter

UB

CN

C cb

Fig. 9.14 Narrowband transistor amplifier stage with the neutralization capacitor C N

9 Amplifiers

811

U R1 Ri

RC1

C1

UO ~

R2

C2

CE1

RE1

R3

RC2

C3

R4

Ua

CE2

RE2

Fig. 9.15 Two-stage RC amplifier with DC voltage negative feedback Ri

C1

C2

C3

lB1 UO

~

R12

IB2 ~

rBE1 b1 IB1

rCE1

RC1

R34

rBE2

~

rCE2

RC2

RL Ua

b2 IB2

Fig. 9.16 Small-signal equivalent circuit of the RC amplifier for low frequencies

circuit is used in most cases. A typical two-stage amplifier is shown in Fig. 9.15. To minimize the DC voltage drift, the DC voltages of the two transistors are negatively coupled via R E1 and R E2 . Conversely, for AC voltages, the full voltage transmission factor is effective if C E1 and C E2 are selected as so high that their impedances can be regarded as short circuits at the operating frequency. Along with the advantage of drift decoupling, the circuit shown in Fig. 9.15 offers the facility to set the operating point of each amplifier stage separately at the optimum setting in terms of noise, input resistance, output resistance, and transmission factor. Figure 9.16 shows the small-signal equivalent circuit for low frequencies. Where R12 = R1 R2 /(R1 + R2 ) and R34 = R3 R4 /(R3 + R4 ), the transmission factors become Aγ =

Ai =

Az =

jωC1 IB1 1 = γBE1 R12 γBE1 U0 1+ R 1 + jωC + 1 i R12 R12 + γBE1

jωC2 IB2 β1 RC1 ≈− γBE2 R34 γBE2 IB1 1+ R34 1 + jωC2 RC1 + R34 + γBE2 jωC3 Ua with rCE RC . ≈ −β2 R L RC2 IB2 1 + jωC3 (RC2 + R L )

(9.1/21)

(9.1/21)

812

R. Quay CBC1

Ri

UO

~

CB1

Ra

CBC2

~

RC1

CC1

Rb

~

RC2

CC2

RL

Ua

Fig. 9.17 Small-signal equivalent circuit of the RC amplifier for high frequencies

The lower edge-frequencies: 1 C · (R C1 · (Ri + C2 · (Ri + 3 C2 + R L ) (9.39) can thus be read off, the highest of which determines the lower limit frequency of the entire circuit. In circuits with RF-transistors, the influence of switching capacitances in relation to the transistor capacitances cannot be ignored. If both influences are combined, a simplified high-frequency equivalent circuit according to Fig. 9.17 is obtained. The parallel connection of the divider resistances R1 , R2 and the baseemitter resistance r B E1 is combined in Ra , and the parallel connection of R34 and r B E2 is combined accordingly in Rb . ω1 =

9.2.2.7

1

R12 r B E1 ) R12 +r B E1

, ω2 =

1

R34 r B E2 ) R34 +r B E2

, ω3 =

Transformer-Coupled Amplifiers

The coupling of amplifier stages by means of a transformer enables a relatively broadband impedance transformation with simultaneous separation of DC voltages. The disadvantages lie in the costs, weight, and volume of the transformer. Due to the unavoidable winding capacitances and leakage inductances, the upper frequency limit is of the order of magnitude of 100 MHz, maybe even GHz. The electrical equivalent circuit of a transformer with the winding resistances R1w and R2w is shown in Fig. 9.18. Provided that Rh ω Lh (small (iron) losses) and Lh Lσ 1 , Lσ 2 (small leakage), the equivalent circuit can be simplified (Fig. 9.19). Here, R = R1w + R2w /u¨ 2 and Lσ = Lσ 1 + Lσ 2 /u¨ 2 . The wired transformer is loaded at the input with the internal resistance Ri1 of the prestage and at the output with the terminal impedance Z2 (Fig. 9.20), which can be converted on the primary side (Z2 = Z2 /u¨ 2 ). Here, 1/Z2 = 1/R2 + ωC2 is divided into the effective conductance 1/R2 and the susceptance ωC2 . (C2 comprises the input capacitance of the following stage and the winding capacitance, 1/R2 the effective conductance of the following stage.)

9 Amplifiers

813

L R1w

Ls1

R2w

2

ü2

Rh

ü2

1:ü

U2

Lh

U2

Fig. 9.18 Equivalent circuit of the technical transformer using an ideal transformer where N1 : N2 = U2 : U2 =1:u¨

L

R

1:ü

U2

Lh

U2

Fig. 9.19 Simplified equivalent circuit of the technical transformer R

su 1

R i1

L

1:ü

Lh

R2

U2

ü2 C2·ü 2

su 1

Fig. 9.20 Equivalent circuit of the transformer loaded on both sides

U2

814

R. Quay

su1

1:ü

R2

R i1

ü2

U2

U2

su1 Fig. 9.21 Equivalent circuit of the transformer for the center-frequency range

(a) Transformation of center frequencies Here, the equivalent circuit (Fig. 9.20) can be even further simplified to provide the circuit in Fig. 9.21, since, with appropriate dimensioning, the following applies: ωL h Ri1 ||R2 /u¨ 2 ; Ri1 ||R2 /u¨ 2 R, ωL σ ; ωC2 u¨ 2 u¨ 2 /R2 .

(9.40)

A voltage-transmission factor ABm = U2 /U1 ≈ −S Ri1 R2 /(u¨ Ri1 + R2 ).

(9.41)

then follows from Fig. 9.21. In some cases, e.g. if the transformer output is loaded with field effect transistors, this equation is further simplified due to the resulting high input resistance R2 to give the following: ABm = U2 /U1 ≈ −S Ri1 .

(9.42)

(b) Transformation of high frequencies The influences of the transformer leakage inductance Lσ and the input capacitance C2 u¨ 2 of the second stage become noticeable at high frequencies. The associated equivalent circuit is shown in Fig. 9.22. The voltage transmission factor is derived from this as follows:

AB =

U2 −S Ri1 = U1 1 + Ri1 u¨ 2 /R2 − ω2 L σ C2 u¨ 2 + jωC2 Ri1 u¨ 2 + jωL σ u¨ 2 /R2 (9.1/22a)

With ABm , according to Eq. 9.41, the following applies:

9 Amplifiers

815

L

su 1

R i1

1:ü

R2

U2

C 2◊ü 2

ü2

U2

su 1 Fig. 9.22 Equivalent circuit of the transformer for high frequencies

A B jϕ = e A Bm ABm AB

=

(1 + Ri1 u¨ 2 /R2 )e jϕ 2 ) + ω4 L 2 C 2 u¨ 4 (1 + Ri1 u¨ 2 /R2 )2 + ω2 (L 2σ u¨ 4 /R22 − 2L σ C2 u¨ 2 + C22 u¨ 4 Ri1 σ 2

(9.1/22b) It is evident from Fig. 9.22 that Lσ and C2 u¨ 2 form a series resonant circuit, which is damped by Ri1 and R2 /u¨ 2 . Depending on the damping of the circuit by the effective resistances, a resonance rise of the voltage transmission factor is obtained, which can be used to increase the upper limit frequency of the amplifier (see Fig. 9.23). Without Lσ , the upper limit frequency would be: f0 =

Ri1 + R2 /u¨ 2 2π C2 Ri1 R2

(9.43)

according to Sect. 9.2.2.7. This 3-dB limit frequency is obtained by equating the real part and imaginary part of the denominator. Without the influence of Lσ , according to curve 1, where f > f0 , the decrease would be 1/ω, i.e. 20 dB/frequency decade (or 6 dB/octave). Since Lσ is always present, the curves 2–4 in Fig. 9.23 decrease according to Sect. 9.2.2.7 1/ω2 by 40 dB/frequency decade. With little leakage (curve 2), AB decreases at a constantly slow rate. The phase angle ψ increases relatively slowly. By increasing Lσ , the 3-dB limit frequency can be shifted slightly to higher frequencies (curve 3). Where R2 ≡R2 /u¨ 2 > Ri1 , this gives a resonance rise AB

ABm max

=

at the angular frequency:

Ri1 u¨ 2 /R2 + 2

ωr =

R2 /u¨ 2 Ri1

1 − Ri1 u¨ 2 /R2 L σ C2 u¨ 2

(9.44)

(9.45)

816

R. Quay 1 0.8 0.6 0.4

0.707

1

A B A Ba

0.2

0 –3 –6 –10

2

0.1 0.08 0.06 0.04

3

–20

–30 4

0.02

dB

0.01 180*

–40 4

135*

3 2

90*

1 f

45* 0 –45* fo

fu –90* 101

2

4 6 8102 2

4 6 8103

2

4 6 8104

2

4 6 8105

2

4 Hz 106

f

Fig. 9.23 Voltage transmission factor of a transformer between 10 Hz and 1 MHz; top, amount; bottom, phase; L H = 1 H; u¨ 2 C2 = 1.6 nF; Ri1 = 1 k; R2 = R2 /u¨ 2 = 2 k; 1 Lσ = 0; 2 Lσ = 0.1 mH; 3 Lσ = 0.45 mH; 4 Lσ = 3.2 mH

This is associated with a lowering of the 3-dB limit frequency (curve 4 in Fig. 9.23). For pulse transformers, the curve 1 with its relatively small phase rise should be aimed for, so that Lσ can be minimized in these transformers. (c) Transformation of low frequencies The behavior at low frequencies is determined above all by the shunt by the main inductance Lh of the transformer. The equivalent circuit applicable to low frequencies is shown in Fig. 9.24. The voltage transmission factor becomes: The lower 3-dB limit frequency in Fig. 9.23 is derived from the equality of the real and imaginary parts in the denominator of AB as fu = R p /(2π Lh ). Where Ri1 = 1 k and R2 ≡ R2 /u¨ 2 = 2 k, R p = 2/3 k. Where Lh = 1 H, fu = 106 Hz follows. Here, φ = -45◦ .

AB = U2 /U1 = −S R P

Rp with R p = Ri1 R2 /(u¨ 2 Ri1 + R2 ). 1+ jωL h (9.1/23)

9 Amplifiers

817

9.2.3 Selective Amplifiers If signals with a very narrow spectrum or only a single frequency are to be amplified, low-bandwidth selective amplifiers are preferred to broadband amplifiers. They provide a better signal-to-noise ratio and above all noise signals lying outside the frequency band to be transmitted are suppressed.

9.2.3.1

Single-Circuit Amplifiers

The voltage transmission factor A B of a single-circuit amplifier according to Fig. 9.25 can be calculated from its equivalent circuit shown in Fig. 9.26. Assuming feedbackfree transistors (Ccb → 0), the following voltage transmission factor is obtained: AB =

U2 1 = −S Z P = −S U1 YP

su 1

R i1

(9.46)

1:ü

R2

Lh

U2

ü2

U2

su 1 Fig. 9.24 Equivalent circuit of the transformer for low frequencies

+ L

U1

Rp

C1

R1

U2

R2

Fig. 9.25 Single-circuit amplifier: a circuit with coupling to the next stage

818

R. Quay

~

Ca

Ri

Cs

Rp

L

C

R1

R2

Re

Ce

U2

S◊U i

Fig. 9.26 Single-circuit amplifier: b equivalent circuit

The following applies here: YP =

1 1 + RP Ri

= G P + iωC +

(9.47) 1 iωL

(9.48)

The internal resistance Ri of the first transistor and the input resistance of the second transistor, including its base voltage divider resistances thus lie parallel to the resonant resistance RP of the resonating circuit. They additionally clamp the circuit. The effective resonating circuit capacitance C is made up of the capacitance C’ of the resonating circuit capacitor, the switching capacitance C S , the output capacitance Ca of the first amplifier stage and the input capacitance Ce of the following stage. The following applies: YP = YP RP = 1 + i V (9.49) GP so that (Eq. 9.46) can be written in the form AB =

A Bm U2 S RP =− = =− U1 1 + iV 1 + iV

(9.50)

The following applies: V = Qv = ωr C R P

ω ωr

−

RP ω ωr

ωr

= − ω ωr L ωr ω

(9.51)

With this scaling, the two parameters A Bm and V are sufficient to describe the amplifier characteristics instead of the 5 parameters S, R p , L, C and f. If the voltage transmission factor AB = |AB|e jω scaled to A Bm is represented in the Gaussian plane, a circle is obtained (see Fig. 9.27). The more V deviates from zero, the smaller the amount of A B becomes and the more its phase φ deviates from π . The following is obtained from Eq. 9.51: A Bm |A B | = √ (9.52) 1 + V2

9 Amplifiers Fig. 9.27 Locus of the scaled voltage transmission factor A B /A Bm in the single-circuit amplifier

819

Im

V

( AA ( B

Bm

1.0

V=Vc1=+1

0.5

0.5

V=

0.5

AB (V) ABm V 0 –1

–0.5

Re

( AA ( B

Bn

–0.5

V= –

0.5

–0.5 V=Vc2=–1

–1.0

and φ = π − arctan V

(9.53)

The bandwidth of the amplifier is defined by the frequencies fc1 and fc2 , at which √ A B =A Bm / 2 (3 dB drop). This is the case where V = Vc1 = 1 and V = Vc2 = −1. In the single-circuit amplifier, the phase of A B deviates by −45◦ or +45◦ from 180◦ at the limit frequencies defined in this way (in multi-circuit amplifiers, this no longer applies!). The graphical representation of the function 1 A B /A Bm = √ 1 + V2

(9.54)

produces the curve shown in Fig. 9.28. With the two limit frequencies fc1 and fc2 (Fig. 9.28), and with reference to Chap. 1, the quality: Q=

fr fr = Δf c f c1 − f c2

(9.55)

820

R. Quay

AB ABm 1.0 0.8

00V

0.6 0.4 0.2

–5

–4

–3

–2 f'

–1

0

1

fc2

fr

fc1

2

3

4

f'

5

V f

fc Fig. 9.28 Amount of the scaled voltage transmission factor depending on the frequency or the scaled detuning V

is defined. For the product bandwidth Δf c · maximum amplification A Bm , with A Bm = S R p and Q = ωr C R p , this provides the fundamental relationship Δf c A Bm =

fr S R p fr S A Bm = = Q 2π fr C R p 2πC

(9.56)

independent from the resistance R p ! Therefore, if, for example, the greatest possible amplification A Bm is to be achieved for a predefined bandwidth, a transistor with high transconductance S must be selected and the effective resonant circuit capacitance C must be minimized. If the resonant circuit capacitance is left out (C = 0), C is then made up of only the switching capacitance Cs , the output capacitance Ca of the transistor and the input capacitance Ce of the following stage. Thus, in the most favorable case, the value Cmin = Ca + Ce + Cs is obtained for C. However, since Cmin is not sufficiently constant (new matching required following operating point modification and transistor exchange), a compromise has to be found between high amplification and constancy of the resonant frequency.

9.2.3.2

Multi-stage Selective Amplifier

Since the effective quality of a resonant circuit cannot be set at arbitrarily high values, the selection of a single circuit does not generally meet the imposed requirements. In practice, a selective amplifier therefore comprises several of the stages shown in

9 Amplifiers

821

Fig. 9.25, each of which, as an external resistance, has a resonant circuit tuned to the same frequency fr . The total transmission factor A B is then AB =

−A Bmn −A Bm1 −A Bm2 −A Bm3 ... 1 + i Q1v 1 + i Q2v 1 + i Q3v 1 + i Qn v

(9.57)

If stages with the same maximum amplification A B St and the same resonant circuit quality Q are used, the following complex total transmission factor is obtained in the case of n stages: AnB St A B = (−1)n (9.58) 1 + i QV n The amount thereof is |A B | =

AnB St 1+

V 2 n/2

=

A Bm 1 + V 2 n/2

.

(9.59)

The maximum amplification is therefore A Bm = AnB St . If the bandwidth is again defined by the limit frequencies fc1 and fc2 , at which 1 1 A B = √ A2B St = √ A Bm 2 2

(9.60)

(3 dB drop), the following applies at the band limits:

or:

2 = (1 + VC2 )n

(9.61)

√ n VC1,2 = ± 2−1

(9.62)

According to Eq. 9.55, the following then applies: fr Δf c = Q

√ 0.87 fr n 2−1≈ √ Q n

(9.63)

The higher the quality Q and the number of stages n, the smaller the bandwidth Δf c becomes. Figure 9.58 shows the maximum voltage transmission factor A Bm =AnB St and the bandwidth Δf c dependent on n for the discussed case of the constant stage amplification A B St independent from n (Fig. 9.29).

822

R. Quay

10

1.0 InABm

4

0.4

Q

0.6

Vcf 0.2

2 0

fr

6

. fC

0.8

InABSt

Vcf =

InABm InABSt

8

0

2

4

6

8

10

0

n Fig. 9.29 Maximum amplification and bandwidth depending on the number of stages

9.3 Gain and Matching 9.3.1 Power Gain and Impedance Matching Every active elements has the ability to amplify at all frequencies below the cutoff frequencies. A measure is to be found to judge on the impact of the matching networks to stability. These conditions shall be derived in the following. Figure 9.30 gives the terms for the analysis of the active two-port with respect to stability. Two active two-ports are wired with the complex impedances Z S and the load Z L . In due course bG is the power wave, which is given from the generator to the a complex impedance Z. bG is composed from b S (which the generator can give to Z0 ) and from a second contribution r S ·aG , which arises from an arbitrary Z=Z0 . The following equations hold [7] which are deduced from Fig. 9.30:

ZS a1

bG bS rS aG

r1 b1

Two-port (S)

Fig. 9.30 Active two-port investigated for stability analysis

b2 r2 a2

aL rL

ZL bL

9 Amplifiers

823

bG = b S + r S · aG

(9.64)

a L = b2 a1 = b G a2 = b L

(9.65) (9.66) (9.67)

aG = b1 bL = r L · aL

(9.68) (9.69)

b1 = r1 · a1 b2 = r2 · a2

(9.70) (9.71)

The reflection coefficients can be rewritten: S12 · S21 · r L (1 − S22 · r L ) S12 · S21 · r S r2 = S22 + (1 − S11 ) · r S r1 = S11 +

(9.72) (9.73)

The related powers can be written: 1 1 |u L · i L∗ | = |a L |2 2 2 1 2 = (1 − |r L | )|a L |2 2 1 |b S |2 PV = . 2 (1 − |r S |2 ) P2 =

(9.74) (9.75) (9.76)

P2 : power effective in load, PV : power available at source The transducer gain GT is defined as the ratio of the power transmitted to the load and the available power at power PV and thus: a 2 L G T = (1 − |r S |2 ) (1 − |r L |2 ) bS 1 − |r L |2 2 . |S | · GT = 21 |1 − r S ·11 |2 |1 − r L · r22 |

(9.77) (9.78)

In this description the transducer gain thus only depends on the S-parameters Si j and the reflection coefficients. In this case GT is maximized, if we obtain power matching with the conditions: r S = r1∗ r L = r2∗

(9.79) (9.80)

824

R. Quay

These two conditions are called the conjugate-complex matching conditions. The transmission power amplification is the maximum, i.e. GT = Gmax , if power matching is set simultaneously on port 1 and port 2. If we insert Eqs. 9.79 and 9.80 into Eqs. 9.72 and 9.73, we obtain: ∗ ∗ · S21 · r2 S12 ∗ (1 − S22 · r2 ) (S ∗ − r ∗ ) r2 = S11 + 11 ∗ 1∗ Δ · r1 · S22 with : Δ = S11 · S22 − S12 S21 . ∗ + r1∗ = S11

(9.81) (9.82) (9.83)

From Eqs. 9.81 to 9.82 we obtain a second order equation with a solution which maximizes the transducer gain. The maximum transducer gain GT,max can further be rewritten:

S21 · k ± k2 − 1 . (9.84) G T,max ( f ) = S12 The GT,max obtained is independent from the external biasing. In this case the Rollet stability factor is defined in Eq. 9.85 [8]. 1 − |S11 |2 − |S22 |2 + |S11 S22 − S12 S21 | 2 · |S12 ||S21 | 2 | (1 + |Δ|2 ) − |S11 |2 − |S22 = 2|S12 ||S21 |

k( f ) =

(9.85) (9.86)

GT can further be factorized as: (1 − |r S |2 )(1 − |r1 |2 ) 1 − |r2 |2 · |1 − r L · S22 |2 (1 − |r1 |2 ) |1 − r S · r1 |2 2 = |S21 | · G S · G L

G T = |S21 |2 ·

(9.87) (9.88)

From the measured S-parameters, both G L and G S can be calculated. Further, the curves with G L = const and G S = const are circles. In order to achieve a maximum transmission power amplification, a radiofrequency transistor must be connected to the input and to the output with matching networks, in order to satisfy the conditions of Eqs. 9.79 and 9.80. The matching networks then consist of distributed or, for lower frequencies, concentrated elements with minimal loss. Due to the frequency dependence of the scattering parameters, power matching with matching networks, e.g. comprising an L and a C, or with distributed transmission lines, is possible in a narrow frequency range only. Stability investigations and definition of the elements of the matching networks can be performed in parallel, e.g. graphically in the complex reflection factor plane with the Smith diagram, or using electronics design automation tools. From the scattering

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parameters given for a transistor, G L can be calculated according to Eq. 9.87 and G S by 9.88. The loci G L = const and G S = const produce circles in the complex reflection factor plane, as shown below. r L and r S can be determined for power matching according to Eqs. 9.79 and 9.80. It can be shown that a two-port network is not unconditionally stable if |r L | ≤1 and |r S | ≤ 1. With |r L |= 1 and |r S |= 1, stability circles are obtained, which can similarly be plotted in the complex reflection factor plane.

9.3.2 Small-Signal Amplifier with Field Effect Transistors The basic principles of field effect transistors (Si-JFET, Si-MOSFET, MESFET and heterojunction-FET) have been dicussed in the semiconductor chapter. At microwave frequencies, amplifiers are advantageously designed with scattering parameters. Some of the concepts and resources required for this purpose are discussed below. Any type of FET can be used in the examples.

50j 100j

25j

250j

10j

10

25

50

100

250

–250j

–10j

–100j

–25j –50j

Fig. 9.31 Circles of constant gain in the input plane of G S in the Smith-Polar Chart

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50j 25j

100j

250j

10j

10

25

50

100

250

-250j

-10j

-25j

-100j -50j

Fig. 9.32 Circles of constant gain in the output plane of G L in the Smith-Polar Chart

ZS

GS

rS

S11

G0

S22 r L

GL

ZL

Fig. 9.33 Transducer gain in a two-port network with two matching networks

9.3.2.1

Scattering Parameters of a FET

Terminal voltages and terminal currents are used to define the Y and Z parameters. These can easily be measured, provided that the current and voltage on the feed lines to the quadripole undergo little local change. At microwave frequencies, the wave characteristics of current and voltage are noticeable, so that it is advantageous in this case to use scattering parameters. The small-signal characteristics of any FETs are therefore normally described by scattering parameters. Example Figure 9.34 indicates the characteristics of the loci of the scattering parameters for a GaN HEMT with a gate length of 100 nm in the source circuit, U DS = 15 V, I D = 100 mA/mm; In addition, Sect. 9.3.2.1 sets out the numerical values of the scattering parameters in polar coordinates. According to the essentially capacitive input and output impedance of the GaN-HEMT, the reflection factor S11 has negative phase up to around 30 GHz and S22 up to 60 GHz. The amount of S11 decreases from 0.99 at

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10 MHz (broadband matching very difficult) to 0.65 at 15 GHz. The amount of S22 similarly decreases with increasing frequency, from 0.9 at 1 GHz to 0.6 at 15 GHz. The forward transmission S21 decreases steadily with the frequency in terms of amount from 8 at 10 MHz to 1 at 60 GHz, whereas the backward transmission increases from 0.0001 at 10 MHz to a maximum of approx. 0.09 at 120 GHz. This results in an amplification decrease with increasing frequency. At low frequencies, S21 has a phase of around 180◦ and S12 a phase of approx. 90◦ . The phases decrease steadily with increasing frequency in each case to approx. 8◦ at 15 GHz. The scattering parameters can be converted into the Z, Y, H and P parameters customary at low frequencies.

S22 S11

Radius= 1

S21

Radius= 10

S12

Radius= 1

Fig. 9.34 Scattering parameters of a GaN-HEMT for the 1–110 GHz frequency range. Characteristic impedance Z0 = 50 . Reference planes; contacting points of measurement probes on the 50 connection pads: a input reflection S11 with scaled input resistance Z1 and output-reflection S22 with scaled output resistance Z2 ; b forward transmission S21 scale (x10) and backward transmission, S12 x1

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Table 9.2 Scattering parameters for a GaN-HEMT as a function of the frequency f S11 S21 S12 S22 Amount Phase Amount Phase Amount Phase Amount 0.0100 0.0150 0.020 0.025 0.030 0.035 0.040 0.050 0.100 0.500 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 15.00 20.00 25.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 100.00 110.00 115.00 120.00

0.9990 0.9989 0.998 0.998 0.997 0.998 0.997 0.995 0.998 0.991 0.983 0.971 0.958 0.933 0.917 0.902 0.882 0.863 0.849 0.840 0.805 0.786 0.781 0.737 0.762 0.751 0.751 0.754 0.756 0.747 0.745 0.734 0.737 0.740

359.90 359.88 359.79 359.73 359.65 359.57 359.48 359.31 358.25 352.93 345.72 331.37 317.42 304.95 293.23 282.88 273.43 264.98 256.86 250.21 223.19 205.89 191.45 181.68 167.31 152.38 139.45 127.19 114.26 100.91 87.83 74.37 67.70 59.38

8.2565 8.2067 8.228 8.224 8.253 8.286 8.304 8.328 8.419 8.575 8.536 8.312 7.981 7.534 7.125 6.677 6.215 5.817 5.429 5.067 3.753 2.921 2.381 1.974 1.547 1.243 1.040 0.883 0.780 0.705 0.645 0.587 0.565 0.544

179.46 180.293 180.54 180.60 180.60 180.61 180.684 180.601 180.056 176.381 171.363 162.182 152.368 144.126 136.416 129.475 123.168 117.218 111.871 107.149 86.992 70.971 57.785 44.884 28.593 8.417 349.910 332.350 314.455 297.049 279.375 261.812 253.832 245.793

0.00014 0.00022 0.00029 0.00038 0.00046 0.00061 0.00070 0.00088 0.00156 0.0080 0.01592 0.03107 0.04469 0.05635 0.06655 0.07485 0.08121 0.08677 0.09101 0.09460 0.10411 0.10686 0.10825 0.10716 0.10961 0.10711 0.10261 0.09801 0.09581 0.0966 0.0970 0.0969 0.0983 0.0988

93.09 101.56 86.427 98.050 88.172 88.178 91.757 87.762 90.179 85.771 81.604 72.669 64.437 56.938 50.039 44.058 38.569 33.498 28.844 24.906 8.507 356.62 347.191 338.023 325.762 311.106 299.159 287.388 277.79 266.83 256.51 244.53 240.63 235.48

0.6300 0.6274 0.6262 0.6242 0.625 0.623 0.623 0.621 0.617 0.609 0.602 0.586 0.570 0.545 0.521 0.499 0.475 0.452 0.433 0.421 0.373 0.363 0.357 0.355 0.391 0.422 0.463 0.504 0.529 0.551 0.588 0.613 0.6129 0.6306

Phase 0.33 0.249 0.16670 359.99 359.87 359.75 359.68 359.43 358.72 354.77 349.94 340.18 330.20 321.26 313.18 306.10 299.19 293.47 287.38 282.50 261.41 246.45 234.23 228.30 213.72 199.16 186.14 170.20 156.10 142.34 128.28 112.67 105.27 97.17

Table Scattering parameters for an unmatched GaN-HEMT as a function of the frequency (Table 9.2).

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9.3.3 Signal Flow Diagrams The characteristics of a small-signal amplifier or more generally a linear network, can be described by means of a linear equation system. In the case of a microwave network, scattering parameters are often used to link the variables. The cause of the returning wave variables b j lies in the ingoing wave variables ak . The a j are therefore independent variables and the bk are dependent variables. The scattering parameters Sk j are the coefficients of the equation system. To analyze the network, the equation system must be solved according to the relevant variables. This can be done, for example, by applying the matrix calculation. Instead of the algebraic determination of the network characteristics, a graphical solution by means of signal flow diagrams [9, 10] is also possible, wherein the cause-effect relationship is taken into account. The graphical solution is physically descriptive and often requires less time than the algebraic solution. The relationship between the equation system and the signal flow diagram is established by the definitions described in Fig. 9.35: 1. The wave variables a j and bk are described in the signal flow diagram by nodes. a j and bk are referred to as node signals. 2. The S-parameters are represented by directed branches. 3. These directed branches connect the nodes and therefore describe the signal flow. This takes place from the independent node a j (source) to the dependent node bk (sink). 4. The node signal bk of a sink is derived from the sum of all incoming node signals Sk j a j . Fig. 9.36 shows signal flow diagrams for basic circuits, from which more complex circuits can be constructed. Figure 9.36a describes a two-port network, Fig. 9.36b shows a load resistance (b L = r L a L ) and Fig. 9.36c shows a generator (bG = b S + r S aG ). r L is the reflection factor of the load, r S that of the de-activated source (b S = 0). b S is the power wave that can be delivered by the generator to a matched consumer (aG = 0). For a given signal flow diagram, a transmission factor U¨ k j between two nodes a j and bk can be defined as bk U¨ = ] (9.89) aj U¨ k j can be determined from the structure of the signal flow diagram. The following rules must be observed here: 1. A path is a continuous sequence of similarly oriented branches which connect the node a j (source) to the node bk (sink). No node may be touched more than once. The path transmission factor P is derived from the product of the branch transmission factors along the path. 2. A loop is a self-contained path. No node may be touched more than once. The loop transmission factor L is derived from the product of the branch transmission factors along the loop.

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aj

Source

Branch

Drain

aj

Sjk

bk

bk

Node with a parting power wave

bk

Node with an incident power wave

Node with incident power waves

Fig. 9.35 Definition of signal flow diagrams

a

a1

a1

S21

b2

b2

S

b1

a2

S11

b1

b

S22

S12

a2

aL aL rL

ZL bL bL

c bS

1

ZS bS =

uS ZS

uS rS

b6 = a6

b6

ZD +

ZD

bs = 0

Fig. 9.36 Signal flow diagrams for: a two-port network; b load resistance; c generator

rS

a6

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bs =us/2 Z0

a1

S21

b2

1 S11

b1

S22

rL = (ZL – Z0)/(ZL+Z0)

a2

S12

Fig. 9.37 Signal flow diagram for a two-port network provided with a source (Z S = Z0 ) and load

3. The following applies: U¨ k j =

n

Pν Δν Δ

ν=1

(9.90)

where: Σ =1− Σν = 1 −

ν

L(1) + L(1) +

L(2) + L(2) +

ν

L(3) + · · ·

(9.91)

L(3) + · · ·

(9.92)

In the denominator,the symbols have the following meanings: • Σ L(1): sum of all loop transmission factors occurring in the mitsignal flow diagram (1st order loops). • Σ L(2): sum of all possible products of the loop transmission factors of two nontouching loops (2nd order loops). • Σ L(3): sum of all possible products of the loop transmission factors of three non-touching loops (3rd order loops). In the numerator, the symbols have the following meanings: • P1 , P2 , …the path transmission factors of the n possible paths between a j and bk . • Σ (1) L(1): sum of the loop transmission factors of all 1st order loops occurring in the signal flow diagram which do not touch the path P1 . • Σ (1) (L(2): sum of the loop transmission factors of all 2nd order loops which do not touch the path P1 . • Σ (2) L(1): sum of the loop transmission factors of all 1st order loops which do not touch the path P2 . Example: Fig. 9.37 shows the signal flow diagram for a two-port network which is provided with a source with the internal resistance Z S = Z0 and any given load Z L . The required transmission factor is U¨ 11 = b1 /a1 . This corresponds to the input reflection factor r1 with any given load reflection factor r L . The following is derived from Eq. 9.90:

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P1 = S11 P2 = S21r L S12 Δ = 1 − S22 r L Δ1 = 1 − S22 r L Δ2 = 1. P1 Δ1 + P2 Δ2 S12 S21r L U¨ 11 = r1 = = S11 + Δ 1 − S22 r L

(9.1/79)

from which the following is obtained: The output reflection factor r2 for any given source reflection factor r S is obtained due to the circuit symmetry from Sect. 9.3.3 by replacing the input variables wit