Analysis of Multiconductor Transmission Lines

Clayton R. Paul, PhD, is Professor and the Sam Nunn Eminent Professor of Aerospace Systems Engineering in the Department of Electrical Engineering and Computer Engineering at Mercer University in Macon, Georgia. He is also Emeritus Professor of Electrical Engineering at the University of Kentucky. Dr. Paul is the author of numerous textbooks on EE subjects and technical papers, the majority of which are in his primary research area of EMC of electronic systems. He is a Fellow of the IEEE and an Honorary Life Member of the IEEE EMC Society.

• Preface xvii1 Introduction 11.1 Examples of Multiconductor Transmission-Line Structures 51.2 Properties of the TEM Mode of Propagation 81.3 The Transmission-Line Equations: A Preview 181.3.1 Unique Definition of Voltage and Current for the TEM Mode of Propagation 191.3.2 Defining the Per-Unit-Length Parameters 221.3.3 Obtaining the Transmission-Line Equations from the Transverse Electromagnetic Field Equations 281.3.4 Properties of the Per-Unit-Length Parameters 301.4 Classification of Transmission Lines 321.4.1 Uniform versus Nonuniform Lines 331.4.2 Homogeneous versus Inhomogeneous Surrounding Media 351.4.3 Lossless versus Lossy Lines 361.5 Restrictions on the Applicability of the Transmission-Line Equation Formulation 371.5.1 Higher Order Modes 381.5.1.1 The Infinite, Parallel-Plate Transmission Line 381.5.1.2 The Coaxial Transmission Line 431.5.1.3 Two-Wire Lines 441.5.2 Transmission-Line Currents versus Antenna Currents 451.6 The Time Domain versus the Frequency Domain 471.6.1 The Fourier Series and Transform 501.6.2 Spectra and Bandwidth of Digital Waveforms 521.6.3 Computing the Time-Domain Response of Transmission Lines Having Linear Terminations Using Fourier Methods and Superposition 56Problems 61References 692 The Transmission-Line Equations for Two-Conductor Lines 712.1 Derivation of the Transmission-Line Equations from the Integral Form of Maxwell’s Equations 712.2 Derivation of the Transmission-Line Equations from the Per-Unit-Length Equivalent Circuit 772.3 Properties of the Per-Unit-Length Parameters 782.4 Incorporating Frequency-Dependent Losses 792.4.1 Properties of the Frequency-Domain Per-Unit-Length Impedance ẑ(ω) and Admittance ŷ(ω) 81Problems 85References 883 The Transmission-Line Equations for Multiconductor Lines 893.1 Derivation of the Multiconductor Transmission-Line Equations from the Integral Form of Maxwell’s Equations 893.2 Derivation of the Multiconductor Transmission-Line Equations from the Per-Unit-Length Equivalent Circuit 993.3 Summary of the MTL Equations 1013.4 Incorporating Frequency-Dependent Losses 1023.5 Properties of the Per-Unit-Length Parameter Matrices L, C, G 103Problems 108References 1094 The Per-Unit-Length Parameters for Two-Conductor Lines 1104.1 Definitions of the Per-Unit-Length Parameters l, c,and g 1114.2 Lines Having Conductors of Circular, Cylindrical Cross Section (Wires) 1134.2.1 Fundamental Subproblems for Wires 1134.2.1.1 The Method of Images 1184.2.2 Per-Unit-Length Inductance and Capacitance for Wire-Type Lines 1194.2.3 Per-Unit-Length Conductance and Resistance for Wire-Type Lines 1304.3 Lines Having Conductors of Rectangular Cross Section (PCB Lands) 1444.3.1 Per-Unit-Length Inductance and Capacitance for PCB-Type Lines 1454.3.2 Per-Unit-Length Conductance and Resistance for PCB-Type Lines 148Problems 156References 1585 The Per-Unit-Length Parameters for Multiconductor Lines 1605.1 Definitions of the Per-Unit-Length Parameter Matrices L, C, and G 1615.1.1 The Generalized capacitance Matrix c 1675.2 Multiconductor Lines Having Conductors of Circular, Cylindrical Cross Section (Wires) 1715.2.1 Wide-Separation Approximations for Wires in Homogeneous Media 1715.2.1.1 n + 1 Wires 1735.2.1.2 n Wires Above an Infinite, Perfectly Conducting Plane 1735.2.1.3 n Wires Within a Perfectly Conducting Cylindrical Shield 1745.2.2 Numerical Methods for the General Case 1765.2.2.1 Applications to Inhomogeneous Dielectric Media 1815.2.3 Computed Results: Ribbon Cables 1875.3 Multiconductor Lines Having Conductors of Rectangular Cross Section 1895.3.1 Method of Moments (MoM) Techniques 1905.3.1.1 Applications to Printed Circuit Boards 1995.3.1.2 Applications to Coupled Microstrip Lines 2115.3.1.3 Applications to Coupled Striplines 2195.4 Finite Difference Techniques 2235.5 Finite-Element Techniques 229Problems 237References 2396 Frequency-Domain Analysis of Two-Conductor Lines 2406.1 The Transmission-Line Equations in the Frequency Domain 2416.2 The General Solution for Lossless Lines 2426.2.1 The Reflection Coefficient and Input Impedance 2446.2.2 Solutions for the Terminal Voltages and Currents 2476.2.3 The SPICE (PSPICE) Solution for Lossless Lines 2506.2.4 Voltage and Current as a Function of Position on the Line 2526.2.5 Matching and VSWR 2556.2.6 Power Flow on a Lossless Line 2566.3 The General Solution for Lossy Lines 2586.3.1 The Low-Loss Approximation 2606.4 Lumped-Circuit Approximate Models of the Line 2656.5 Alternative Two-Port Representations of the Line 2696.5.1 The Chain Parameters 2706.5.2 Approximating Abruptly Nonuniform Lines with the Chain-Parameter Matrix 2736.5.3 The Z and Y Parameters 275Problems 2787 Frequency-Domain Analysis of Multiconductor Lines 2827.1 The MTL Transmission-Line Equations in the Frequency Domain 2827.2 The General Solution for An (n + 1)-Conductor Line 2847.2.1 Decoupling the MTL Equations by Similarity Transformations 2847.2.2 Solution for Line Categories 2917.2.2.1 Perfect Conductors in Lossy, Homogeneous Media 2927.2.2.2 Lossy Conductors in Lossy, Homogeneous Media 2937.2.2.3 Perfect Conductors in Lossless, Inhomogeneous Media 2967.2.2.4 The General Case: Lossy Conductors in Lossy, Inhomogeneous Media 2987.2.2.5 Cyclic-Symmetric Structures 2987.3 Incorporating the Terminal Conditions 3057.3.1 The Generalized Thevenin Equivalent 3057.3.2 The Generalized Norton Equivalent 3087.3.3 Mixed Representations 3107.4 Lumped-Circuit Approximate Characterizations 3127.5 Alternative 2n-Port Characterizations 3147.5.1 Analogy of the Frequency-Domain MTL Equations to State-Variable Equations 3147.5.2 Characterizing the Line as a 2n-Port with the Chain-Parameter Matrix 3167.5.3 Properties of the Chain-Parameter Matrix 3187.5.4 Approximating Nonuniform Lines with the Chain-Parameter Matrix 3227.5.5 The Impedance and Admittance Parameter Matrix Characterizations 3237.6 Power Flow and the Reflection Coefficient Matrix 3277.7 Computed and Experimental Results 3327.7.1 Ribbon Cables 3327.7.2 Printed Circuit Boards 335Problems 338References 3428 Time-Domain Analysis of Two-Conductor Lines 3438.1 The Solution for Lossless Lines 3448.1.1 Wave Tracing and the Reflection Coefficients 3468.1.2 Series Solutions and the Difference Operator 3568.1.3 The Method of Characteristics and a Two-Port Model of the Line 3618.1.4 The SPICE (PSPICE) Solution for Lossless Lines 3658.1.5 The Laplace Transform Solution 3688.1.5.1 Lines with Capacitive and Inductive Loads 3708.1.6 Lumped-Circuit Approximate Models of the Line 3738.1.6.1 When is the Line Electrically Short in the Time Domain? 3748.1.7 The Time-Domain to Frequency-Domain (TDFD) Transformation Method 3758.1.8 The Finite-Difference, Time-Domain (FDTD) Method 3798.1.8.1 The Magic Time Step 3858.1.9 Matching for Signal Integrity 3928.1.9.1 When is Matching Required? 3988.1.9.2 Effects of Line Discontinuities 3998.2 Incorporation of Losses 4068.2.1 Representing Frequency-Dependent Losses 4088.2.1.1 Representing Losses in the Medium 4088.2.1.2 Representing Losses in the Conductors and Skin Effect 4108.2.1.3 Convolution with Frequency-Dependent Losses 4158.2.2 The Time-Domain to Frequency-Domain (TDFD) Transformation Method 4218.2.3 The Finite-Difference, Time-Domain (FDTD) Method 4238.2.3.1 Including Frequency-Independent Losses 4238.2.3.2 Including Frequency-Dependent Losses 4278.2.3.3 Prony’s Method for Representing a Function 4318.2.3.4 Recursive Convolution 4348.2.3.5 An Example: A High-Loss Line 4398.2.3.6 A Correction for the FDTD Errors 4438.2.4 Lumped-Circuit Approximate Characterizations 4478.2.5 The Use of Macromodels in Modeling the Line 4508.2.6 Representing Frequency-Dependent Functions in the Time Domain Using Pade Methods 453Problems 461References 4679 Time-Domain Analysis of Multiconductor Lines 4709.1 The Solution for Lossless Lines 4709.1.1 The Recursive Solution for MTLs 4719.1.2 Decoupling the MTL Equations 4769.1.2.1 Lossless Lines in Homogeneous Media 4789.1.2.2 Lossless Lines in Inhomogeneous Media 4799.1.2.3 Incorporating the Terminal Conditions via the SPICE Program 4829.1.3 Lumped-Circuit Approximate Characterizations 4879.1.4 The Time-Domain to Frequency-Domain (TDFD) Transformation Method 4889.1.5 The Finite-Difference, Time-Domain (FDTD) Method 4889.1.5.1 Including Dynamic and/or Nonlinear Terminations in the FDTD Analysis 4909.2 Incorporation of Losses 4969.2.1 The Time-Domain to Frequency-Domain (TDFD) Method 4989.2.2 Lumped-Circuit Approximate Characterizations 4989.2.3 The Finite-Difference, Time-Domain (FDTD) Method 4999.2.4 Representation of the Lossy MTL with the Generalized Method of Characteristics 5019.2.5 Model Order Reduction (MOR) Methods 5129.2.5.1 Pade Approximation of the Matrix Exponential 5129.2.5.2 Asymptotic Waveform Evaluation (AWE) 5159.2.5.3 Complex Frequency Hopping (CFH) 5189.2.5.4 Vector Fitting 5189.3 Computed and Experimental Results 5249.3.1 Ribbon Cables 5269.3.2 Printed Circuit Boards 530Problems 537References 54110 Literal (Symbolic) Solutions for Three-Conductor Lines 54410.1 The Literal Frequency-Domain Solution for a Homogeneous Medium 54810.1.1 Inductive and Capacitive Coupling 55410.1.2 Common-Impedance Coupling 55610.2 The Literal Time-Domain Solution for a Homogeneous Medium 55810.2.1 Explicit Solution 56010.2.2 Weakly Coupled Lines 56210.2.3 Inductive and Capacitive Coupling 56410.2.4 Common-Impedance Coupling 56710.3 Computed and Experimental Results 56710.3.1 A Three-Wire Ribbon Cable 56810.3.2 A Three-Conductor Printed Circuit Board 569Problems 575References 57611 Incident Field Excitation of Two-Conductor Lines 57811.1 Derivation of the Transmission-Line Equations for Incident Field Excitation 57811.1.1 Equivalence of Source Representations 58511.2 The Frequency-Domain Solution 58611.2.1 Solution of the Transmission-Line Equations 58611.2.2 Simplified Forms of the Excitations 59211.2.3 Incorporating the Line Terminations 59411.2.4 Uniform Plane-Wave Excitation of the Line 59811.2.4.1 Special Cases 60211.2.4.2 One Conductor Above a Ground Plane 60611.2.5 Comparison with Predictions of Method of Moments Codes 61011.3 The Time-Domain Solution 61111.3.1 The Laplace Transform Solution 61111.3.2 Uniform Plane-Wave Excitation of the Line 62011.3.3 A SPICE Equivalent Circuit 62511.3.4 The Time-Domain to Frequency-Domain (TDFD) Transformation 62811.3.5 The Finite-Difference, Time-Domain (FDTD) Solution Method 62811.3.6 Computed Results 635Problems 638References 63912 Incident Field Excitation of Multiconductor Lines 64112.1 Derivation of the MTL Equations for Incident Field Excitation 64212.1.1 Equivalence of Source Representations 64812.2 Frequency-Domain Solutions 65012.2.1 Solution of the MTL Equations 65112.2.2 Simplified Forms of the Excitations 65312.2.3 Incorporating the Line Terminations 65512.2.3.1 Lossless Lines in Homogeneous Media 65812.2.4 Lumped-Circuit Approximate Characterizations 66012.2.5 Uniform Plane-Wave Excitation of the Line 66012.3 The Time-Domain Solution 66712.3.1 Decoupling the MTL Equations 66812.3.2 A SPICE Equivalent Circuit 67412.3.3 Lumped-Circuit Approximate Characterizations 68112.3.4 The Time-Domain to Frequency-Domain (TDFD) Transformation 68112.3.5 The Finite-Difference, Time-Domain (FDTD) Solution Method 68212.4 Computed Results 686Problems 691References 69213 Transmission-Line Networks 69313.1 Representation of Lossless Lines with the SPICE Model 69613.2 Representation with Lumped-Circuit Approximate Models 69913.3 Representation via the Admittance or Impedance 2n-Port Parameters 69913.4 Representation with the BLT Equations 71213.5 Direct Time-Domain Solutions in Terms of Traveling Waves 72113.6 A Summary of Methods for Analyzing Multiconductor Transmission Lines 726Problems 727References 728Publications by the Author Concerning Transmission Lines 729Appendix A. Description of Computer Software 736A.1 Programs for the Calculation of the Per-Unit-Length Parameters 738A.1.1 Wide-Separation Approximations for Wires: WIDESEP.FOR 738A.1.2 Ribbon Cables: RIBBON.FOR 740A.1.3 Printed Circuit Boards: PCB.FOR 743A.1.4 Coupled Microstrip Structures: MSTRP.FOR 745A.1.5 Coupled Stripline Structures: STRPLINE.FOR 746A.2 Frequency-Domain Analysis 747A.2.1 General: MTL.FOR 747A.3 Time-Domain Analysis 748A.3.1 Time-Domain to Frequency-Domain Transformation: TIMEFREQ.FOR 748A.3.2 Branin’s Method Extended to Multiconductor Lines: BRANIN.FOR 748A.3.3 Finite Difference-Time Domain Method: FINDIF.FOR 749A.3.4 Finite-Difference-Time-Domain Method: FDTDLOSS.FOR 749A.4 SPICE/PSPICE Subcircuit Generation Programs 749A.4.1 General Solution, Lossless Lines: SPICEMTL.FOR 750A.4.2 Lumped-Pi Circuit, Lossless Lines: SPICELPI.FOR 750A.4.3 Inductive-Capacitive Coupling Model: SPICELC.FOR 751A.5 Incident Field Excitation 752A.5.1 Frequency-Domain Program: INCIDENT.FOR 752A.5.2 SPICE/PSPICE Subcircuit Model: SPICEINC.FOR 753A.5.3 Finite-Difference, Time-Domain (FDTD) Model: FDTDINC.FOR 754References 755Appendix B. A SPICE (PSPICE) Tutorial 756B.1 Creating the SPICE or PSPICE Program 757B.2 Circuit Description 758B.3 Execution Statements 763B.4 Output Statements 765B.5 Examples 767B.6 The Subcircuit Model 769References 771Index 773