Electromagnetic Modeling and Simulation E-Book

Levent Sevgi

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: IEEE/OUP Series on Electromagnetic Wave Theory (formerly IEEE only), Series Editor: Donald G. Dudley.
• Sprache: Englisch

Seitenzahl: 717

Veröffentlichungsjahr: 2014

Beschreibung

This unique book presents simple, easy-to-use, but effective short codes as well as virtual tools that can be used by electrical, electronic, communication, and computer engineers in a broad range of electrical engineering problems Electromagnetic modeling is essential to the design and modeling of antenna, radar, satellite, medical imaging, and other applications. In this book, author Levent Sevgi explains techniques for solving real-time complex physical problems using MATLAB-based short scripts and comprehensive virtual tools. Unique in coverage and tutorial approach, Electromagnetic Modeling and Simulation covers fundamental analytical and numerical models that are widely used in teaching, research, and engineering designs--including mode and ray summation approaches with the canonical 2D nonpenetrable parallel plate waveguide as well as FDTD, MoM, and SSPE scripts. The book also establishes an intelligent balance among the essentials of EM MODSIM: The Problem (the physics), The Theory and Models (mathematical background and analytical solutions), and The Simulations (code developing plus validation, verification, and calibration). Classroom tested in graduate-level and short courses, Electromagnetic Modeling and Simulation: * Clarifies concepts through numerous worked problems and quizzes provided throughout the book * Features valuable MATLAB-based, user-friendly, effective engineering and research virtual design tools * Includes sample scenarios and video clips recorded during characteristic simulations that visually impact learning--available on wiley.com * Provides readers with their first steps in EM MODSIM as well as tools for medium and high-level code developers and users Electromagnetic Modeling and Simulation thoroughly covers the physics, mathematical background, analytical solutions, and code development of electromagnetic modeling, making it an ideal resource for electrical engineers and researchers.

Leseprobe

Table of Contents

IEEE Press

Title page

Copyright page

Dedication

Preface

About the Author

Acknowledgments

CHAPTER 1: Introduction to MODSIM

1.1  Models and Modeling

1.2  Validation, Verification, and Calibration

1.3  Available Core Models

1.4  Model Selection Criteria

1.5  Graduate Level EM MODSIM Course

1.6  EM-MODSIM Lecture Flow

1.7  Two Level EM Guided Wave Lecture

1.8  Conclusions

References

CHAPTER 2: Engineers Speak with Numbers

2.1  Introduction

2.2  Measurement, Calculation, and Error Analysis

2.3  Significant Digits, Truncation, and Round-Off Errors

2.4  Error Propagation

2.5  Error and Confidence Level

2.6  Hypothesis Testing

2.7  Hypothetical Tests on Cell Phones

2.8  Conclusions

References

CHAPTER 3: Numerical Analysis in Electromagnetics

3.1  Taylor's Expansion and Numerical Differentiation

3.2  Numerical Integration

3.3  Nonlinear Equations and Root Search

3.4  Linear Systems of Equations

References

CHAPTER 4: Fourier Transform and Fourier Series

4.1  Introduction

4.2  Fourier Transform

4.3  Basic Discretization Requirements

4.4  Fourier Series Representation

4.5  Rectangular Pulse and Its Harmonics

4.6  Conclusions

References

CHAPTER 5: Stochastic Modeling in Electromagnetics

5.1  Introduction

5.2  Radar Signal Environment

5.3  Total Radar Signal

5.4  Decision Making and Detection

5.5  Conclusions

References

CHAPTER 6: Electromagnetic Theory: Basic Review

6.1  Maxwell Equations and Reduction

6.2  Waveguiding Structures

6.3  Radiation Problems and Vector Potentials

6.4  The Delta Dirac Function

6.5  Coordinate Systems and Basic Operators

6.6  The Point Source Representation

6.7  Field Representation of a Point/Line Source

6.8  Alternative Field Representations

6.9  Transverse Electric/Magnetic fields

6.10  The TE/TM Source Injection

6.11  Second-Order EM Differential Equations

6.12  EM Wave–Transmission Line Analogy

6.13  Time Dependence in Maxwell Equations

6.14  Physical Fundamentals

References

CHAPTER 7: Sturm–Liouville Equation: The Bridge between Eigenvalue and Green's Function Problems

7.1  Introduction

7.2  Guided Wave Scenarios

7.3  The Sturm–Liouville Equation

7.4  Conclusions

References

CHAPTER 8: The 2D Nonpenetrable Parallel Plate Waveguide

8.1  Introduction

8.2  Propagation Inside a 2D-PEC Parallel Plate Waveguide

8.3  Alternative Representation: Eigenray Solution

8.4  A 2D-PEC Parallel Plate Waveguide Simulator

8.5  Eigenvalue Extraction from Propagation Characteristics

8.6  Tilted Beam Excitation

8.7  Conclusions

References

CHAPTER 9: Wedge Waveguide with Nonpenetrable Boundaries

9.1  Introduction

9.2  Statement of the Problem: Physical Configuration and Ray-Asymptotic Guided Wave Schematizations

9.3  Source-Free Solutions

9.4  Test Problem: The 2D Line-Source-Excited Nonpenetrable Wedge Waveguide

9.5  The MATLAB Package “WedgeGUIDE”

9.6  Numerical Tests and Illustrations

9.7  Conclusions

Appendix 9A:  Formation of the Spectral IM integral in Section 9.3.3

References

CHAPTER 10: High Frequency Asymptotics: The 2D Wedge Diffraction Problem

10.1  Introduction

10.2  Plane Wave Illumination and HFA Models

10.3  HFA Models under Line Source (LS) Excitations

10.4  Basic MATLAB Scripts

10.5  The WedgeGUI Virtual Tool and Some Examples

10.6  Conclusions

References

CHAPTER 11: Antennas: Isotropic Radiators and Beam Forming/Beam Steering

11.1  Introduction

11.2  Arrays of Isotropic Radiators

11.3  The ARRAY Package

11.4  Beam Forming/Steering Examples

11.5  Conclusions

References

CHAPTER 12: Simple Propagation Models and Ray Solutions

12.1  Introduction

12.2  Ray-Tracing Approaches

12.3  A Ray-Shooting MATLAB Package

12.4  Characteristic Examples

12.5  Flat-Earth Problem and 2Ray Model

12.6  Knife-Edge Problem and 4Ray Model

12.7  Ray Plus Diffraction Models

12.8  Conclusions

References

CHAPTER 13: Method of Moments

13.1  Introduction

13.2  Approximating a Periodic Function by Other Functions: Fourier Series Representation

13.3  Introduction to the MoM

13.4  Simple Applications of MoM

13.5  MoM Applied to Radiation and Scattering Problems

13.6  MoM Applied to Wedge Diffraction Problem

13.7  MoM Applied to Wedge Waveguide Problem

13.8  Conclusions

References

CHAPTER 14: Finite-Difference Time-Domain Method

14.1  FDTD Representation of EM Plane Waves

14.2  Transmission Lines and Time-Domain Reflectometer

14.3  1D FDTD with Second-Order Differential Equations

14.4  Two-Dimensional (2D) FDTD Modeling

14.5  Canonical 2D Wedge Scattering Problem

14.6  Conclusions

References

CHAPTER 15: Parabolic Equation Method

15.1  Introduction

15.2  The Parabolic Equation (PE) Model

15.3  The Split-Step Parabolic Equation (SSPE) Propagation Tool

15.4  The Finite Element Method-Based PE Propagation Tool

15.5  Atmospheric Refractivity Effects

15.6  A 2D Surface Duct Scenario and Reference Solutions

15.7  LINPE Algorithm and Canonical Tests/Comparisons

15.8  The GrSSPE Package

15.9  The Single-Knife-Edge Problem

15.10  Accurate Source Modeling

15.11  Dielectric Slab Waveguide

15.12  Conclusions

References

CHAPTER 16: Parallel Plate Waveguide Problem

16.1  Introduction

16.2  Problem Postulation and Analytical Solutions: RevisitED

16.3  Numerical Models

16.4  Conclusions

References

APPENDIX A: Introduction to MATLAB

Vectors, Arrays, and Matrices

Basic Matrix Operations

Eigenvectors and Eigenvalues of a Matrix

Complex Numbers

Data Loading and Saving

Statistics

Presenting Output

Graphics Utilities

Sample CODE 3

Sample CODE 4

Sample CODE 5

Sample CODE 6

Sample CODE 7

Sample CODE 8

Sample CODE 9

APPENDIX B: Suggested References

EM Books

CEM Books

APPENDIX C: Suggested Tutorials and Feature Articles

IEEE AP Magazine

Testing Ourselves Column Tutorials

Electromagnetic Virtual Tool Tutorials

Index

IEEE PRESS SERIES ON ELECTROMAGNETIC WAVE THEORY

End User License Agreement

List of Tables

Table 1.1.  A Typical EM MODSIM Course Plan

Table 1.2.  Basic Time-Domain EM Virtual Tools

Table 1.3.  Basic Frequency-Domain EM Virtual Tools

Table 1.4.  Advanced-Level EM Virtual Tools

Table 2.1.  Some Confidence Levels and Corresponding Critical Values

Table 2.2.  Critical

Z

Values for the Hypothesis Testing

Table 2.3.  The 2 × 2 Table of the Case-Control Study

Table 2.4.  Data Table for the Reaction Time Test

Table 3.1.  Code for the Taylor's Expansion of Cosine Function

Table 3.2.  Numerical Solutions Using (3.9) and (3.17)

Table 3.3.  A Short MATLAB Script for 1D Poisson Equation

Table 3.4.  A Simple MATLAB Code for Numerical Integration

Table 3.5.  Numerical Integration Error Versus Number of Rectangles

Table 3.6.  A Simple MATLAB Code for Numerical Integration of Gauss Function

Table 3.7.  Fixed-Point Iteration MATLAB Code

Table 3.8.  Fixed-Point Root Search of

y

 = 

x

2

 − 4

x

 − 6 (

x

0

 = −1.0)

Table 3.9.  Newton–Raphson Root Search MATLAB Code

Table 3.10.  Newton–Raphson Root Search of

y

 = 

x

2

 − 4

x

 − 6 (

x

0

 = 5.0)

Table 3.11.  A Simple MATLAB Code for Gaussian Elimination Method

Table 3.12.  The Output of the Code Listed in Table 3.10

Table 4.1.  Some Functions and Their FTs

Table 4.2.  A MATLAB Module for DFT Calculations

Table 4.3.  A MATLAB Module for FFT Calculations

Table 4.4.  A MATLAB Module for Fourier Series Representations

Table 5.1.  System-Level Simulation Requirements

Table 5.2.  System-Level Simulation Challenges

Table 5.3.  Scripts for Random Number Generations

Table 5.4.  A MATLAB Code for Random Number Generation

Table 5.5.  A MATLAB Script for Signal Plus Noise Generation

Table 5.6.  MATLAB's AWGN Command

Table 5.7.  Critical

z

Values for the Hypothesis Testing

Table 5.8.  A MATLAB Code for the Calculation of CI or

α

Table 5.9.  A Short MATLAB Code for Plotting HOC

Table 5.10.  A MATLAB Code Calculating the Required Number of Samples

Table 5.11.  A Short MATLAB Code for the Generation of ROC

Table 5.12.  A MATLAB Code Which Plots ROC for I + Q CHANNEL

Table 5.13.  A Short MATLAB Code for the ROC Test

Table 6.1.  Duality Relations in Maxwell Equations

Table 6.2.  The

TE

and

TM

Sets of Equations for Figure 6.3.

Table 6.3.  The Two Sets of Equations for Fig. 6.4.

Table 6.4.  Sets of Equations for TE/TM Cases on the

xz

-Plane

Table 7.1.  Reduced SL Equations Under Different Parameter Sets

Table 8.1.  A MATLAB Mode Summation Module

Table 8.2.  A MATLAB Field Code Based on Eigenrays and Their Contributions

Table 8.3.  A MATLAB Module for the Calculation of Eigenray Paths

Table 8.4.  Eigenvalue Extraction MATLAB Code

Table 8.5.  A MATLAB Mode Summation Code with Tilted Beam Excitation

Table 9.1.  A MATLAB Code for NM Summation with HBC

Table 10.1.  A Sample MATLAB Code for the Exact Mode Summation (10.3)–(10.6).

Table 10.2.  A Sample MATLAB Code for the PO-PTD Formulations

Table 10.3.  Numerical Calculation of the Integrals in (10.8)

Table 10.4.  A Short MATLAB Code for the UTD Calculations

Table 10.5.  Numerical Calculation of Fresnel Integral

Table 10.6.  Numerical Calculation of PE Fields

Table 10.7.  A MATLAB Script for the Numerical Integration of (10.33b)

Table 10.8.  Sample Recorded Data for Angle versus Diffracted Fields

Table 11.1.  A MATLAB Module for Circular Array Investigations

Table 12.1.  A MATLAB Ray-Shooting Script

Table 12.2.  A Short MATLAB Code for the 2Ray Model

Table 12.3.  A Short MATLAB Code for 4Ray Model (Height Profile)

Table 12.4.  A Short MATLAB Code for 4Ray Model (Range Profile)

Table 13.1.  A Short MATLAB-MoM Code for the Plate Capacitor

Table 13.2.  A Short MoM Code for the Flat-Earth Problem

Table 13.3.  A Short MoM Code for 2D Scattering from a PEC Cylinder

Table 13.4.  A Short MATLAB Script for Segment Generation of a Square Cylinder

Table 13.5.  Script as in Table 12.4 but for Both Circular and Square Cylinders

Table 13.6.  A MoM Code for the Generation of Wedge-Diffracted Fields

Table 13.7.  A Wedge Waveguide MoM Code for HBC

Table 14.1.  A Simple 1D FDTD MATLAB Code

Table 14.2.  Primary Parameters of a TL

Table 14.3.  Analogous Parameters

Table 14.4.  The TRLine.m MATLAB Script

Table 14.5.  A MatLab Script for 1D FDTD with Wave Equation (Spatial Source Injection)

Table 14.6.  A MatLab Script for FDTD with 1D Wave Equation (Source Injection in Time from Left and Right Nodes)

Table 14.7.  A MatLab Script for FDTD with 1D Wave Equation (Source Injection in Time from a Given Node)

Table 14.8.  A Short MATLAB Code for the Time and Frequency Responses

Table 15.1.  A Simple FFT-Based SSPE Procedure

Table 15.2.  A Simple DST-Based SSPE Procedure

Table 15.3.  DST-Based SSPE versus 2Ray Model for Flat Earth with PEC Ground

Table 15.4.  A Short MATLAB Mode Sum Code for the Surface Duct Problem

Table 15.5.  A Short MATLAB Code for the Mode Sum and SSPE Methods

Table 15.6.  The Operational Parameters for Test 1

Table 15.7.  Computation Times and Memory Requirements for the Same Discretization Parameters

Table 15.8.  User-Supplied Parameters of the GrSSPE Package

Table 15.9.  MATLAB-Based SSPEvs4RAY.m Code

Table 15.10.  A MATLAB Code for SSPE, MoM, and 4RAY Model Comparisons under Tilted Beam-Type Excitations

Table 15.11.  User-Supplied DiSlab Input Parameters

Table 15.12.  A Matlab Script that Plots Even Mode Profiles

Table 16.1.  A MATLAB Mode Sum Code for Field versus Range/Height Calculations

Table 16.2.  A MATLAB Eigenray Code for the Calculation of Field versus Range/Height

Table 16.3.  A MATLAB Ray + Image Code for Field versus Range/Height Calculations

Table 16.4.  A MATLAB SSPE Code for Field versus Range/Height Calculations

Table 16.5.  A MATLAB MoM Code for Field versus Range/Height Calculations

Table 16.6.  A MATLAB MoM + IM Code for Field versus Range/Height Calculations

List of Illustrations

Figure 1.1.  Analytical-based modeling.

Figure 1.2.  Numerical-based modeling.

Figure 1.3.  Fundamental blocks of numerical MODSIM.

Figure 1.4.  A generic chart of EM-MODSIM.

Figure 2.1.  The normal distribution function.

Figure 2.2.  Critical value of a two-sided

Z

-test (95% CI,

α

 = 0.5).

Figure 3.1.  Cosine function and its Taylor's expansion at the origin.

Figure 3.2.  One-dimensional finite

z

-space.

Figure 3.3.  Numerical solution of Laplace equation for

Z

m

 = 1, KE = 100,

U

(0) = −2,

U

(1) = 1,

γ

 = 62, γ ≤ l

e

−3.

Figure 3.4.  Numerical solution of Poisson equation for

Z

m

 = 1, KE = 100,

U

(0) = −2,

U

(1) = 1,

γ

 = 146, γ ≤ l

e

−3. The dots show the charge distribution.

Figure 3.5.  Convergence of the Poisson equation for for

Z

m

 = 1, KE = 100,

U

(0) = −2,

U

(1) = 1,

γ

 = 256, γ ≤ l

e

−3. The dots show the charge distribution.

Figure 3.6.  The solution of 2D Poisson equation for 1 m

2

plate (100 × 100 space) for the boundary values of (left, right, top, bottom) (a) (1,1,−2,−2),

γ

 = 942, (b) (0,0,1,−2),

γ

 = 865, with similar Gaussian charge distributions inside (γ ≤ l

e

−3).

Figure 3.7.  The curve of a function and rectangular representation of the area.

Figure 3.8.  A three-loop electric circuit.

Figure 4.1.  Time variation of two sinusoids.

Figure 4.2.  Frequency variation of two sinusoids obtained with DFT.

Figure 4.3.  Frequency variation of two sinusoids obtained with FFT.

Figure 4.4.  Finite-length, finite-resolution DFT and FFT environment.

Figure 4.5.  The frequency spectrum for question 2a (±50 Hz cannot be observed since they exactly appear at ±

f

max

).

Figure 4.6.  The frequency spectrum for question 2b (exp(−206

πt

) enters the spectrum from the right and is 53 Hz away from the right boundary; note the amplitude differences).

Figure 4.7.  The frequency spectrum for question 2c (exp(

j

104

πt

) enters the spectrum from the left and is 2 Hz away from the left boundary).

Figure 4.8.  The function and its Fourier series approximation with five terms.

Figure 4.9.  The function and its Fourier series approximation with 10 terms.

Figure 4.10.  The function and its Fourier series approximation with 25 terms.

Figure 4.11.  The function and its 25-term Fourier series approximation for 4 ≤ 

x

 ≤ 8.

Figure 4.12.  A rectangular pulse and its FS representations.

Figure 5.1.  A typical multisensor system. HF, high-frequency; MW, microwave; PR, profiling; SAR, synthetic aperture radar; OCC, operation control center.

Figure 5.2.  Normally distributed random array and its histogram.

Figure 5.3.  An array of Gaussian random numbers and its histogram (

n

 = 1000).

Figure 5.4.  Histograms of Gaussian random numbers (L)

n

 = 5000, (R)

n

 = 100,000.

Figure 5.5.  An array of Rayleigh random numbers and its histogram (

n

 = 5000).

Figure 5.6.  Synthetically generated sinusoidal signal + noise (SNR = 10 dB); solid: uniform noise; dashed: Gaussian noise.

Figure 5.7.  Signal + noise (SNR = 0 dB). Dashed: single generation; solid: oversampled and averaged.

Figure 5.8.  A typical HF radar ocean wave spectra [5].

Figure 5.9.  Signal + noise + clutter vs. time (I channel).

Figure 5.10.  A typical HF radar spectrum (SNR = 13 dB, CNR = 30 dB, noise = −30 dB).

Figure 5.11.  Another spectrum with the same parameters given in Fig. 5.10.

Figure 5.12.  HOC prepared via MATLAB code listed in Table 5.3.

Figure 5.13.  A computer generated white noise, sinusoidal signal, and both.

Figure 5.14.  Detection probability and false alarm rates for white noise receivers.

Figure 5.15.  ROC prepared using MATLAB code in Table 5.11 (solid: single channel; dashed: I + Q channel).

Figure 5.16.  CFAR detection tests performed using the code in Table 5.13.

Figure 6.1.  Rectangular, cylindrical, and spherical coordinates.

Figure 6.2.  Delta Dirac function (left) of a point source and (right) an impulse in time.

Figure 6.3.  Transverse field components in 2D on the

yz

-plane, (a) TM

yz

case, and (b) TE

yz

case.

Figure 6.4.  Transverse field components on the

xy

-plane: (a) TM

xy

(= TE

z

) case and (b) TE

xy

(= TM

z

) case.

Figure 6.5.  Transverse field components in 2D on the

xz

-plane, TM

xz

case, (a) on the

xy

interface and (b) on the

yz

-interface.

Figure 6.6.  Field excitation on

yz

-plane for the TM

yz

case: (a)

H

x

is excited,

H

x

is observed, (b)

H

x

is excited, Poynting vector is observed, (c)

H

x

is excited,

E

y

is observed, (d)

H

x

is excited,

E

z

is observed (produced by packages presented in Refs. 7 and 8).

Figure 6.7.  Field excitation on

xy

-plane for the TM

yz

case: (a)

E

z

is excited,

E

z

is observed; (b)

E

z

is excited,

H

x

is observed, (c)

E

y

is excited,

E

z

is observed, (d) and

E

y

is excited,

H

x

is observed (produced by the virtual tools presented in [Refs. 7 and 8).

Figure 6.8.  Field components on the

xz

-plane and the duality between TE and TM cases.

Figure 6.9.  Typical scattering integral contours.

Figure 7.1.  Canonical waveguiding problems (a) 2D parallel plate waveguide with nonpenetrable boundaries, (b) wedge waveguide with penetrable and/or nonpenetrable boundaries, (c) 2D dielectric (optical) film, and (d) surface/elevated ducting and antiducting on flat earth.

Figure 7.2.  The transverse and longitudinal decomposition strategy for GWT.

Figure 7.3.  Interconnections for source-free and source-driven problems characterized by the SL equation.

Figure 7.4.  Singularities on complex

λ

-plane and the integration contour.

Figure 8.1.  Physical configuration: 2D, homogeneously filled, parallel plate waveguide with PEC boundaries.

Figure 8.2.  Spatial and spectral coordinates (transverse and longitudinal wave numbers); local plane wave (i.e., ray) trajectories: generic ray and

n

-indexed eigenray; spectral angles

W

and

W

n

(measured with respect to the

x

-direction).

Figure 8.3.  Trajectories for eigenrays, where

n

and

i

tag the number of reflections and the eigenray species, respectively. Relations between

n

and

i

are as follows:

i

 = 1:

n

reflections at both boundaries (− − for

x

>

 = 

x

s

,

x

<

 = 

x

p

and + + for

x

>

 = 

x

p

,

x

<

 = 

x

s

);

i

 = 2:

n

reflections at upper and (

n

 + 1) reflections at lower boundaries (− +);

i

 = 3: (

n

 + 1) reflections at upper and

n

reflections at lower boundaries (+ −);

i

 = 4: (

n

 + 1) reflections at both boundaries (+ + for

x

>

 = 

x

s

,

x

<

 = 

x

p

and − − for

x

>

 = 

x

p

,

x

<

 = 

x

s

). This

i

-ordering corresponds to upward (+) or downward (–) departure directions at the source (first symbol inside the parenthesis), and the corresponding arrival directions at the observer (second symbol inside the parenthesis).

Figure 8.4.  Schematizations of eigenrays and eigenmode ray congruences, with inclusion of the analytic generating function for eigenray and eigenmode expansions. (a) Eigenrays and (b) eigenmode ray congruences.

Figure 8.5.  As in Fig. 8.4, but for hybrid ray-mode partitioning. (a) Partitioning in physical domain and (b) hybrid ray-mode schematization.

Figure 8.6.  Vertical profiles of the first, third, and sixth modes.

Figure 8.7.  Fields inside the waveguide. Left to right: Vertical extend of the fifth mode, source profile and its mode sum equivalent, and mode fields at two ranges.

Figure 8.8.  The same as in Fig. 8.7 but for another source profile.

Figure 8.9.  Eigenray paths obtained from image method.

Figure 8.10.  Multisector partitioning as in (8.51) showing groupings of eigenrays and modes. These intervals can be filled either with rays or modes depending on the problem conditions; for example, a few LOM and HOM plus a few eigenrays may be used in an efficient hybrid form to account for the field distribution.

Figure 8.11.  Front panel of the RAYMODE package. The user supplies plate height, source/observer heights, horizontal distance, frequency, and number of reflections (

n

) for the ray solution. The package calculates the modal (reference) solution and searches for the eigenrays for the specified geometrical parameters.

Figure 8.12.  The front panel of the HYBRID package. Field versus height and/or range can be calculated via (i) mode summation, (ii) ray summation, and (iii) hybrid ray-mode formulation.

Figure 8.13.  Eigenray paths calculated via RAYMODE for the following parameters:

a

 = 1 m, distance = 5.6 m, source height = 0.3 m, observer height = 0.7 m,

f

 = 2387 MHz (

ka

 = 50),

n

 = 6. Top:

n

 = 0, middle:

n

 = 1, bottom:

n

 = 6.

Figure 8.14.  Field versus. height calculated via RAYMODE. The parameters are

a

 = 1 m, distance = 5.6 m, source height = 0.3 m,

f

 = 2387 MHz (

ka

 = 50). Left:

n

 = 0, middle:

n

 = 1, right:

n

 = 6.

Figure 8.15.  Field versus range calculated via the HYBRID package.

Figure 8.16.  Field versus height calculated via the HYBRID package.

Figure 8.17.  Eigenrays and wave field versus height obtained via RAYMODE package. All parameters are the same as in Fig. 8.15, except the range is more than doubled. The number of reflections

n

 = 10.

Figure 8.18.  Same as in Fig. 8.14, but with

n

 = 19.

Figure 8.19.  The flowchart of the MATLAB package.

Figure 8.20.  (a) Windowed longitudinal correlation function and (b) the frequency spectrum of the correlation function (eigenvalues are also listed).

Figure 8.21.  3D color map of field versus range–height variations inside a parallel plate PEC waveguide. (Top) Waveguide height = 1 m,

f

 = 3 GHz, beam width = 3°, antenna height = 0.5 m, beam tilt = −45°; (bottom) waveguide height = 8 m,

f

 = 1 GHz, beam width = 7°, antenna height = 4 m, beam tilt = −30°.

Figure 9.1.  Geometry of waveguiding regions comprising fixed boundaries or interfaces, interior guiding inhomogeneous medium profiles, or both, in original (

x

,

y

) and better matched, locally separable (

u

,

v

) coordinates. Also shown are portions of modal ray congruences pertaining to modes confined by (1) physical fixed boundaries and (2) caustics. If the boundaries are penetrable, fields are refracted into the exterior.

Figure 9.2.  Physical configuration and coordinate designation for homogeneously filled waveguide with weakly wedge-tapered PEC boundaries (weakly nonseparable in rectangular coordinates: [

u

,

v

] → [

x

,

y

]. Strictly separable in cylindrical coordinates: [

u

,

v

] → [

ρ

,

φ

]). Note that the “height” coordinate

y

increases in the downward direction.

Figure 9.3.  Wedge waveguide and modal caustic formation, showing track of incident-reflected modal ray tube.

Figure 9.4.  Transverse mode fields at different ranges.

Figure 9.5.  (a) Longitudinal variation of the dominant

m

 = 1 mode showing the propagation–cutoff transition: surfer plot. (b) Longitudinal variation of the dominant

m

 = 1 mode showing the propagation–cutoff transition: contour plot.

Figure 9.6.  (a) As in Fig. 9.5a, but for

m

 = 2. (b) As in Fig. 9.5b, but for

m

 = 2.

Figure 9.7.  Dominant mode versus range ( 

f

 = 15 MHz,

α

 = 3°,

φ

s

 = 1.5°,

φ

o

 = 1.5°). Solid: NM, dashed: AM (modal cutoff [caustic] range is 192 m).

Figure 9.8.  Integration path

C

β

and singularities in the complex-

β

plane: Branch cuts. Poles associated with propagating, evanescent, and nonphysical modes.

Figure 9.9.  Integration paths

C

θ

and in the complex angular spectrum

θ

-plane

β

 = 

k

sin

θ

. Relevant local steepest descent paths (LSDPs) through the SP(s). Permissible paths should terminate in the shaded valley regions.

Figure 9.10.  Integration path in the complex

ξ

-plane.

Figure 9.11.  The run-time window of the WedgeGUIDE MATLAB package.

Figure 9.12.  The flowchart of WedgeGUIDE algorithm.

Figure 9.13.  Test 1; upslope propagation. Source:

x

 = 1000 m,

y

 = 40 m, observation along

y

 = 20 m,

f

 = 15 MHz,

α

 = 3°. Number of propagating modes

m

 = 5. Modal cutoff ranges:

x

c

1

 = 191 m,

x

c

2

 = 382 m,

x

c

3

 = 572 m,

x

c

4

 = 763 m,

x

c

5

 = 954 m.

Figure 9.14.  (a) Wave field versus range for the line source (

x

s

 = 1000 m) in terms of NM (9.27) and IM summations (9.41). The range extends from the source to 100 m. The height of the source is

y

 = 15 m. The observation track is parallel to the top

x

-axis at a height

y

 = 2 m (

f

 = 15 MHz,

α

 = 3°). The wave field versus height profile at

x

 = 300 m is plotted as well. (b) Same as Fig. 9.14a with NM, AM, and IM (9.49) summations.

Figure 9.15.  Wave field versus range and height for NM, AM, and IM formulations at longer ranges. The parameters are

f

 = 15 MHz,

α

 = 3°,

x

s

 = 2500 m,

y

s

 = 65 m,

x

o

 = 2000 m,

y

o

 = 52 m (

x

 = 2400 m).

Figure 9.16.  Wave field versus range and height for NM, AM, and IM formulations with the following parameters:

f

 = 100 MHz,

α

 = 3°,

x

s

 = 1000 m,

y

s

 = 26 m,

x

o

 = 970 m,

y

o

 = 25 m. The number of propagating modes is 34 (

x

 = 970 m).

Figure 9.17.  Wave field versus range and height computed from NM, AM, and IM formulations with the following parameters:

f

 = 15 MHz,

α

 = 15°,

x

s

 = 250 m,

y

s

 = 33 m,

x

o

 = 20 m,

y

o

 = 3 m. The number of propagating modes is 6. The wave field height profile is taken at

x

 = 140 m. The AM/IM spikes near modal caustics are explained in the text.

Figure 9.18.  Wave field versus range and height computed from NM, AM, and IM formulations. The parameters are

f

 = 15 MHz,

α

 = 15°,

x

s

 = 1000 m,

y

s

 = 135 m,

x

o

 = 800 m,

y

o

 = 110 m. The number of propagating modes is 26. The height profile is taken at

x

 = 900 m (TM Pol/SBC).

Figure 9A.1.  Branch cuts and SDPs in the complex-

β

plane for propagating modes.

Figure 9A.2.  Branch cuts and SDP in the complex-

β

plane for evanescent-cutoff modes.

Figure 10.1.  Geometry of the wedge scattering problem. SSI, single side illumination; DSI, double side illumination.

Figure 10.2.  Exact versus GO solutions for the total fields. (Top)

φ

0

 = 60°,

r

 = 50 m,

kr

 = 31.4159,

N

 = 200; (bottom)

φ

0

 = 45°,

r

 = 500 m,

kr

 = 314.1593,

N

 = 800 (

α

 = 350°,

f

 = 30 MHz).

Figure 10.3.  Front panel of WedgeGUI.

Figure 10.4.  The basic flowchart of WedgeGUI.

Figure 10.5.  WedgeGUI: Diffracted fields versus angle for HBC (

α

 = 350°,

φ

0

 = 45°,

f

 = 30 MHz,

r

 = 50 m,

kr

 = 31.4159).

Figure 10.6.  WedgeGUI: Total fields vs. angle for SBC (

α

 = 330°,

φ

0

 = 70°,

f

 = 30 MHz,

r

 = 50 m,

kr

 = 31.4159).

Figure 10.7.  WedgeGUI: Diffracted fields versus angle for HBC (

α

 = 250°,

φ

0

 = 150°,

f

 = 30 MHz,

r

 = 50 m,

kr

 = 31.4159).

Figure 10.8.  Diffracted fields versus angle for (left) SBC and (right) HBC computed with exact and UTD models (

α

 = 240°,

φ

0

 = 110°,

f

 = 30 MHz,

r

 = 50 m,

kr

 = 31.4159).

Figure 10.9.  Total fields versus angle for (left) SBC and (right) HBC computed with exact and UTD models (

α

 = 240°,

φ

0

 = 110°,

f

 = 30 MHz,

r

 = 50 m,

kr

 = 31.4159).

Figure 10.10.  Diffraction coefficients versus (top) frequency and (bottom) range for the PW illumination and SBC (

α

 = 350°,

φ

0

 = 60°,

f

 = 30 MHz).

Figure 11.1.  A vertical wire antenna and its 3D radiation pattern at resonance.

Figure 11.2.  An isotropic radiator located at (

x

1

,

y

1

,

z

1

). The

i

th radiator's location vector and the observation vector are and , respectively.

Figure 11.3.  Geometrical configurations of planar arrays of isotropic radiators (a) arbitrary, (b) linear, (c) planar, and (d) circular.

Figure 11.4.  Front panel of ARRAY package and radiator locating.

Figure 11.5.  Horizontal radiation pattern of arbitrarily located seven-element array. The beam angles are selected to be

θ

 = 0° and

φ

 = 245°.

Figure 11.6.  Radiation pattern of a three-element linear array on the horizontal plane (

f

 = 300 MHz,

d

 = 0.5 m [

d

 = 

λ

/2]; beam angles are

θ

0

 = 90° and

φ

0

 = 0°).

Figure 11.7.  The 3D radiation pattern of the same three-element array (

f

 = 300 MHz,

d

 = 0.5 m [= 

λ

/2]; beam angles are

θ

0

 = 90° and

φ

0

 = 0°).

Figure 11.8.  A 21-element circular array (left) its horizontal plane (right) 3D patterns (

f

 = 300 MHz,

r

 = 0.5 m [

λ

/2]; beam angles are

θ

0

 = 90° and

φ

0

 = 245°).

Figure 11.9.  A 10 × 10 planar array and its (left) horizontal plane, (right) 3D patterns (

f

 = 300 MHz,

dx

 = 

dy

 = 0.25 m; beam angles are

θ

0

 = 45° and

φ

0

 = 45°).

Figure 11.10.  A 3 × 10 planar array and its vertical radiation for

θ

0

 = 

ϕ

0

 = 0°.

Figure 11.11.  A 3 × 10 planar array and its vertical radiation for

θ

0

 = 90°,

ϕ

0

 = 30°.

Figure 11.12.  A 3 × 10 planar array and its horizontal radiation for

θ

0

 = 90°,

ϕ

0

 = 60°.

Figure 11.13.  A 2 × 5 planar array and its horizontal radiation patterns without line phasing (

dx

 = 0.25 m,

dy

 = 0.40 m,

f

 = 300 MHz) (FBR is zero).

Figure 11.14.  A 2 × 5 planar array and its horizontal radiation patterns with 90° line phasing (

dx

 = 0.25 m,

dy

 = 0.40 m,

f

 = 300 MHz) (more than 30 dB FBR is achieved).

Figure 12.1.  A 2D projection of an urban propagation environment with buildings along the propagation path and GO/GTD components: (1) direct ray between transmitter and receiver, (2) reflected rays (from the top of buildings as well as from the bottom surface), and (3) tip-diffracted rays.

Figure 12.2.  2D propagation environment with a PEC obstacle along the path. The medium is assumed multiplanar layers with constant refractive indexes. The height (Δ

x

) of each layer is user selected and constant. The ray path segments (Δ

s

) and range segments (Δ

z

) are variable and determined by the Snell's law. (1) Consecutive applications of Snell's law and calculations of Δ

s

and Δ

z

. (2) Reflection from the PEC bottom surface. (3) Reflection from the top of the obstacle. (4) Ray end at the left of the PEC obstacle (the package may be improved also to handle ray reflection at walls of the obstacles). (5) Maximum chosen height.

Figure 12.3.  Flowchart of SNELL (input parameters: maximum range/height, layer height, source location, first/last ray angles measured from surface normal, ray increment, trilinear refractivity parameters, and obstacle heights and locations).

Figure 12.4.  Ray paths through trilinear atmosphere.

Figure 12.5.  Front panel of the SNELL package. The left plot is reserved for vertical refractivity profile

n

 = 

n

(

x

). The right plot is for ray paths.

Figure 12.6.  Ray shooting through an environment with the homogeneous atmosphere (i.e.,

n

 = 

n

0

 = 1.0) and with two obstacles where rays are straight lines (first obstacle: height = 10 m, base length = 200 m, range = 300 m; second obstacle: height = 100 m, base length = 200 m, range = 1800 m).

Figure 12.7.  Ray shooting through an atmosphere with a typical trilinear refractivity profile and with no obstacles. Ray reflection, refraction, bending, and caustics are clearly observed.

Figure 12.8.  Ray shooting through an atmosphere with a typical bilinear refractivity (elevated ducting) profile and with no obstacles.

Figure 12.9.  Ray shooting with the same trilinear refractivity profile in Fig. 12.6, but with different source locations (no obstacles, source height = 100 m, maximum height = 300 m, layer height = 0.2 m, maximum range = 2500 m, first ray angle = 45°, last ray angle = 120°, ray increment = 0.375°, the slope of refractivity = 0.001, and the heights of ducting and antiducting are 80 and 150 m, respectively).

Figure 12.10.  Ray shooting with the same refractivity in Fig. 12.6, with two obstacles (source height = 50 m, maximum height = 300 m, layer height = 0.2 m, maximum range = 2500 m, first ray angle = 45°, last ray angle = 120°, and ray increment = 0.375°).

Figure 12.11.  Ray shooting with a trilinear refractivity profile and two obstacles (source height = 50 m, maximum height = 300 m, layer height = 0.2 m, maximum range = 2500 m, first ray angle = 45°, last ray angle = 120°, ray increment = 0.375°, the slope of refractivity = 0.001, and the heights of ducting and antiducting are 150 and 200 m, respectively).

Figure 12.12.  Propagation over PEC flat-earth scenario.

Figure 12.13.  Field versus range showing interaction of direct and ground-reflected rays.

Figure 12.14.  Field versus height showing interaction of direct and ground-reflected rays.

Figure 12.15.  Received power versus range.

Figure 12.16.  Single knife-edge problem and possible four rays.

Figure 12.17.  PF versus range/height for horizontal polarization over PEC ground with a 75-m-high knife edge at 15-km range (source height = 20 m,

f

 = 3 GHz).

Figure 12.18.  PF versus (left) height and (right) range computed with the 4Ray model.

Figure 12.19.  Two knife edges and possible ray (either reflected or diffracted) paths at three receiver locations between source and receiver which contribute the total EM field at the receiver.

Figure 12.20.  3D PF maps for double knife edges above flat earth at 3 and 5 km ranges illuminated by a line source at 25 m height (Hor Pol, GO + UTD,

f

 = 30 MHz).

Figure 13.1.  Normalized capacitance of the parallel plate capacitor versus plate distance, simulated with MoM (plates are identical and square with width = length = 2

a

).

Figure 13.2.  Periodic rectangular function with

P

 = 2

π

and its FS representation.

Figure 13.3.  A smooth Gauss function and its FS approximations with (top) first term and (bottom) 14 terms.

Figure 13.4.  A sine-modulated smooth Gauss function and its FS approximations with (top) 20 terms and (bottom) 64 terms.

Figure 13.5.  A function and its FS representation in 0 ≤ 

x

 ≤ 20. The FS expansion is used for 0 ≤ 

x

 ≤ 10 and perfect agreement is obtained with the first 32 terms. The function and its FS are totally different for 10 ≤ 

x

 ≤ 20.

Figure 13.6.  Residue function versus

x

.

Figure 13.7.  Exact and computed functions.

Figure 13.8.  Parallel plate capacitor having identical PEC plates with length = width = 2

a

, the patch area is Δ

s

 = 4

b

2

, and the number of patches on both plates is

N

.

Figure 13.9.  PF versus range: A comparison of MoM–2Ray models; solid: 2Ray, dashed: MoM.

Figure 13.10.  A typical transmit antenna system for HF radars and/or communication systems.

Figure 13.11.  Vertical (a) and horizontal radiation patterns of the transmit antenna shown in Fig. 13.10 (

ε

g

 = 15,

σ

g

 = 0.005 S/m).

Figure 13.12.  A typical terrain profile (with range in kilometers and height in meters) and a MoM versus SSPE comparison via signal strength (given as PF in decibels as a function of range in kilometers for a TM type of problem (

f

 = 10 MHz, transmitter height of 400 m, receiver height above ground of 300 m).

Figure 13.13.  A propagation scenario with two smooth, 150-m high, 4-km wide and 200-m high, 4-km wide Gaussian-shaped hills, and MoM versus SSPE comparisons: signal strength in dBV/m as a function of range in kilometers for the TE type of problem (

f

 = 10 MHz, maximum range of 30 km, transmitter height of 400 m). (Top) The results for a receiver height above ground of 400 m and (bottom) for a receiver height above terrain of 10 m.

Figure 13.14.  Geometry of the 2D scattering problem.

Figure 13.15.  Polar plot showing MoM versus analytic results (

a

 = 0.25 m).

Figure 13.16.  Polar plot showing MoM versus analytic results (

a

 = 1 m).

Figure 13.17.  RCS versus angle of PEC cylinders (

a

 = 1 m,

φ

inc

 = 0°).

Figure 13.18.  Typical discrete MoM models of air targets.

Figure 13.19.  (Top) Discrete FDTD and MoM models of 15-m-long F-16 aircraft. (Bottom) Horizontal and vertical RCS variations computed with both MoM and FDTD methods (bistatic case).

Figure 13.20.  Discrete MoM models of two F-16 flying together.

Figure 13.21.  RCS versus frequency of single F-16 compared with side-by-side two F-16 15 m apart (monostatic case).

Figure 13.22.  Wedge geometry and MoM modeling of scattered fields for the line source illumination. MoM-computed scattered fields contain reflected and diffracted fields (

θ

 = 2

π

 − 

α

).

Figure 13.23.  Infinite-plane geometry and MoM modeling of scattered fields for the line source illumination. MoM-computed scattered fields contain only reflected fields.

Figure 13.24.  Wedge scattering for TM/SBC, exact, UTD, and MoM solutions:

α

 = 300°,

r

 = 50 m,

r

0

 = 100 m,

φ

0

 = 90°, and

f

 = 30 MHz. (Left) Total fields versus Angle; (right) diffracted fields versus angle.

Figure 13.25.  Wedge scattering for TM/SBC, exact, UTD, and MoM solutions:

α

 = 270°,

r

 = 50 m,

r

0

 = 100 m,

φ

0

 = 60°, and

f

 = 30 MHz. (Left) Total fields versus Angle; (right) diffracted fields versus angle.

Figure 13.26.  Wedge scattering for TE/HBC, exact solution versus MoM:

α

 = 3000°,

r

 = 50 m,

r

0

 = 100 m,

φ

0

 = 50°, and

f

 = 30 MHz. (Left) Total fields versus Angle; (right) diffracted fields versus angle.

Figure 13.27.  Wedge scattering for TE/HBC, exact, UTD, and MoM solutions:

α

 = 240°,

r

 = 50 m,

r

0

 = 100 m,

φ

0

 = 30°, and

f

 = 30 MHz. (Left) Total fields versus Angle; (right) diffracted fields versus angle.

Figure 13.28.  Wave field versus range computed with NM and MoM models. The parameters are

f

 = 15 MHz,

α

 = 15°,

x

s

 = 1000 m,

y

s

 = 15 m,

x

o

 = 150 m,

y

o

 = 2 m (TM Pol/SBC).

Figure 13.29.  Wave field versus range computed with NM and MoM models. The parameters are

f

 = 15 MHz,

α

 = 15°,

x

s

 = 1000 m,

y

s

 = 15 m,

x

o

 = 150 m,

y

o

 = 2 m (TE Pol/HBC).

Figure 13.30.  Wave field versus height computed with NM and MoM models. The parameters are

f

 = 15 MHz,

α

 = 15°,

x

s

 = 1000 m,

y

s

 = 15 m,

x

o

 = 150 m,

y

o

 = 2 m. (Left) TM Pol/SBC; (right) TE Pol/HBC.

Figure 14.1.  A plane wave propagating along

x

-direction with the speed of light.

Figure 14.2.  1D FDTD and discretization in space (along

z

-axis).

Figure 14.3.  Leap-frog FDTD discretization.

Figure 14.4.  FDTD and Gaussian plane wave formation (

k

s

 = 35,

T

 = 25Δ

t

).

Figure 14.5.  FDTD simulations and locating a Gaussian pulse in space.

Figure 14.6.  FDTD space, ±

z

propagating waves, and termination types.

Figure 14.7.  Front panel of FDTD 1D.

Figure 14.8.  Finite region and plane wave propagation (at 27th time step).

Figure 14.9.  Finite region and plane wave propagation (at 65th time step).

Figure 14.10.  Finite region and plane wave propagation (at 117th time step).

Figure 14.11.  Finite region and plane wave propagation (at 167th time step).

Figure 14.12.  Signal versus time at node 70 for the source injection at node 30.

Figure 14.13.  Resonance frequencies of the 1D 100-m-long resonator.

Figure 14.14.  Resonance frequencies of the 1D 100-m-long resonator after 1000Δ

t

.

Figure 14.15.  Second scenario and traveling pulses at 119th time step.

Figure 14.16.  Signal versus time at the 25th node plotted after the simulation ends.

Figure 14.17.  Frequency spectrum of the second scenario after 1000Δ

t

.

Figure 14.18.  (Top) Rectangular pulse observed at 95 m of a 100-m-long region, source being at 30 m, (bottom) the frequency spectrum and resonances.

Figure 14.19.  First mode (resonance) of the structure.

Figure 14.20.  Second mode (resonance) of the structure.

Figure 14.21.  A sketch of a two-wire TL.

Figure 14.22.  The two-wire TL and its discretized models.

Figure 14.23.  A full-symmetric-discrete FDTD model of the two-wire TL.

Figure 14.24.  Leap-frog FDTD scheme of the two-wire TL model.

Figure 14.25.  Discretization at the source end via Norton equivalent.

Figure 14.26.  (Top) Parallel

RC

load and (bottom) serial

RL

load.

Figure 14.27.  (Top) Voltage along the TL and (bottom) signal versus time at a chosen observation point.

Figure 14.28.  Front panel of the MATLAB-based TDRMeter package.

Figure 14.29.  Signal versus time plot of a 50 Ω, 0.5 m lossless SC TL excited by a Gaussian pulse generator with 50 Ω-internal resistor with

G

f

 = 4 S/m at 0.3 m.

Figure 14.30.  The scenario of the echoes observed in Fig. 14.29.

Figure 14.31.  Signal versus time for the resistive termination showing the incident rectangular pulse and the echo (reflected pulse) along a 50 Ω, 0.5 m TL (pulse length = 400 ps,

R

s

 = 50 Ω,

R

L

 = 350 Ω).

Figure 14.32.  Signal versus time at midpoint of a 50 Ω, 0.5 m TL under (a) serial

RL

termination (

R

L

 = 50 Ω,

L

L

 = 250 nH) and (b) parallel

RC

termination (

R

L

 = 50 Ω,

C

L

 = 5 pF). Matched termination is used at the source (

R

s

 = 50 Ω). Pulse length = 400 ps.

Figure 14.33.  Voltage reflection coefficient versus frequency of Fig 14.32b obtained with the FFT procedure. Solid: TD simulation result; dashed: analytical exact solution.

Figure 14.34.  (a) Signal versus time at midpoint of a 50 Ω, 0.5 m TL. Load: parallel

RLC

(

R

L

 = 50 Ω,

C

L

 = 5 pF,

L

L

 = 10 nH). Source:

R

s

 = 50 Ω. Pulse length = 400 ps. (b) Voltage reflection coefficient versus frequency obtained with the FFT procedure. Solid: TD simulation result; dashed: analytical exact solution.

Figure 14.35.  The sketch of the Laplace procedure; prediction of the inductive load from the time signature of the echo by measuring the time constant

τ

.

Figure 14.36.  Signal versus time (step response) at midpoint of a 50 Ω, 0.5 m TL under serial

RL

termination (

R

L

 = 50 Ω,

L

L

 = 27 nH). Matched termination is used at the source (

R

s

 = 50 Ω). The value of the inductor calculated from the plot is 26.2 nH.

Figure 14.37.  The sketch of the Laplace procedure: Prediction of the capacitive load from the time signature of the echo by measuring the time constant

τ

.

Figure 14.38.  Signal versus time (step response) at midpoint of a 50 Ω, 0.5 m TL under parallel

RC

termination (

R

L

 = 50 Ω,

C

L

 = 50 pF). Matched termination is used at the source (

R

s

 = 50 Ω). The value of the capacitor calculated from the plot is 49.4 pF.

Figure 14.39.  Signal versus time at midpoint of a 50 Ω, 0.5 m TL for a step voltage source under serial resonance termination (

Z

L

 = 50 Ω,

L

L

 = 5 nH,

C

L

 = 5 pF,

R

s

 = 50 Ω).

Figure 14.40.  Signal versus time at midpoint of a 50 Ω, 0.5 m TL for a step voltage source under parallel resonance termination (

Z

L

 =50 Ω,

L

L

 = 5 nH,

C

L

 = 5 pF,

R

s

 = 50 Ω).

Figure 14.41.  The last scenario and time variation of the echoes.

Figure 14.42.  Signal versus time at midpoint of the TL for an unknown fault.

Figure 14.43.  The scenario of time the signal observed in Fig. 14.42.

Figure 14.44.  Signal versus time of the second fault scenario.

Figure 14.45.  The scenario of time the signal observed in Fig. 14.44.

Figure 14.46.  Step response closes the end of a 1 m long lossless OC TL (

Z

0

 = 50 Ω,

R

s

 = 10 Ω). The total simulation step is 3000.

Figure 14.47.  Step response close the end of a 1-m-long lossless OC TL (

Z

0

 = 50 Ω,

R

s

 = 300 Ω). The total simulation step is 3000.

Figure 14.48.  Step response close the end of a 1-m-long lossless SC TL (

Z

0

 = 50 Ω,

R

s

 = 10 Ω). The total simulation step is 3000.

Figure 14.49.  Reduced 2D FDTD components on

xz

-plane.

Figure 14.50.  The front panel of the MGL2D virtual tool.

Figure 14.51.  Rural propagation modeling: User-specified 1-km-long terrain profile and 18 MHz CW signal scattering.

Figure 14.52.  Outdoor propagation modeling along a vertically projected user-specified street after 1000 time steps.

Figure 14.53.  Outdoor propagation modeling along a vertically projected user-specified street after 11,000 time steps.

Figure 14.54.  Outdoor propagation modeling along a horizontally projected user-specified street after 2100 time steps.

Figure 14.55.  Outdoor propagation modeling along a horizontally projected user-specified street after 8600 time steps.

Figure 14.56.  Indoor propagation modeling: Top view of a typical two-room apartment with various penetrable and nonpenetrable objects. The source is located inside left room.

Figure 14.57.  Indoor propagation modeling: Late time response of a CW signal and the coverage.

Figure 14.58.  Indoor propagation modeling: Same as in Fig. 14.56 but with a different color scale.

Figure 14.59.  Propagation through a user-specified tunnel and a CW source location.

Figure 14.60.  Propagation through a user-specified tunnel: Late time response.

Figure 14.61.  Two rectangular resonators and pulse scattering inside.

Figure 14.62.  Pulse scattering inside the resonator recorded at two different observation points.

Figure 14.63.  Fourier transform of the recorded signals shown in Fig. 14.61. The observed resonance frequencies determined by the sizes of the resonator.

Figure 14.64.  A horn-type element excited with a CW source inside.

Figure 14.65.  A horn-type element and a kind of parabolic reflector in the front.

Figure 14.66.  A double-layered parallel plate waveguide with coupling apertures in the middle.

Figure 14.67.  A parallel plate waveguide with asymmetrically located periodic triangular barriers.

Figure 14.68.  User-drawn three-part closed resonator.

Figure 14.69.  The 2D PEC wedge problem, line source (LS) illumination, and three characteristic regions separated by RSB and ISB.

Figure 14.70.  (Left) Total and (right) diffracted fields around the PEC wedge. Exact by series versus UTD solution (HBC,

α

 = 240°,

f 

= 30 MHz,

r

 = 50 m,

kr

 = 31.4).

Figure 14.71.  (Left) Total and (right) diffracted fields around the PEC wedge. Exact by series versus UTD solution (SBC,

α

 = 350°,

f

 = 30 MHz,

r

 = 50 m,

kr

 = 31.4).

Figure 14.72.  2D FDTD cells on the

xy

-plane and locations of the field components.

Figure 14.73.  Physical and nonphysical case for the TE

z

problem (thick arrows show incident and reflected waves; thin arrows show tip-diffracted waves).

Figure 14.74.  Dey–Mittra conformal FDTD scenarios.

Figure 14.75.  Exact solution versus FDTD for (a) staircase approximation (b) Dey–Mittra model.

Figure 14.76.  The front panel of WedgeFDTD package.

Figure 14.77.  The front panel of WedgeFDTD package showing reflected and tip-diffracted waves.

Figure 14.78.  Total fields versus angle computed with both FDTD and exact integral models (

α

 = 300°,

f

 = 20 MHz,

r

 = 70 m,

φ

0

 = 120°).

Figure 14.79.  Diffracted fields versus angle solution computed with the FDTD, UTD, and exact integral models (

α

 = 300°,

f

 = 20 MHz,

r

 = 70 m,

φ

0

 = 120°).

Figure 14.80.  Total fields versus angle solution computed with FDTD, UTD, and exact integral models (

α

 = 330°,

f

 = 30 MHz,

r

 = 80 m,

φ

0

 = 45°).

Figure 14.81.  Diffracted fields versus angle solution computed with, FDTD, UTD, and exact integral models (

α

 = 330°,

f

 = 30 MHz,

r

 = 80 m,

φ

0

 = 45°).

Figure 14.82.  (Left) Total fields and (right) diffracted fields around PEC wedge as a function of angles at 30 MHz using exact series, UTD, MoM, and FDTD solutions:

α

 = 300°,

r

 = 50 m,

r

0

 = 100 m,

φ

0

 = 45° (TM/SBC case).

Figure 14.83.  (Left) Total fields and (right) diffracted fields around PEC wedge as a function of angles at 30 MHz using exact series, UTD, MoM, and FDTD solutions:

α

 = 300°,

r

 = 50 m,

r

0

 = 100 m,

φ

0

 = 45° (TE/HBC case).

Figure 15.1.  The 2D projected propagation region,

x

and

z

are the height and range coordinates, respectively. The region is

N

x

-segments vertically and

N

z

-segments longitudinally. An antenna pattern is located vertically at the initial range and height. SSPE and FEMPE are longitudinally marching methods and calculate the vertical field profiles at the next range step from the previous one. Both sweep the region longitudinally up to the desired range step by step.

Figure 15.2.  Vertical field above PEC flat earth at (left) 5 km and (right) 10 km.

Figure 15.3.  Surface and elevated duct formation because of vertical refractivity variations; (left and middle) surface duct and (right) elevated duct.

Figure 15.4.  (L–R) First mode, initial vertical source, and fields at two ranges.

Figure 15.5.  (L–R) Thirteenth mode, initial vertical source, and fields at two ranges.

Figure 15.6.  (L–R) Twentieth mode, initial vertical source, and fields at two ranges.

Figure 15.7.  Field versus range–height calculated with the mode sum and SSPE models.

Figure 15.8.  Flowchart of the MATLAB-based LINPE code which uses analytical exact representation, SSPE and FEMPE.

Figure 15.9.  Maps of E-field strength versus range–height for a given Gaussian antenna pattern obtained from the analytical representations (top) without a tilt and (bottom) antenna 1° uptilted (

f

 = 300 MHz,

a

0

 = 1.2 × 10

−6

,

N

x

 = 256,

N

z

 = 1000).

Figure 15.10.  Maps of E-field strength versus range–height for a given nontilted Gaussian antenna pattern obtained via (top) SSPE and (bottom) FEMPE (

f

 = 300 MHz,

a

0

 = 1.2 × 10

−6

,

N

x

 = 256,

N

z

 = 1000).

Figure 15.11.  Maps of E-field strength versus range–height variations for a given 1° uptilted Gaussian antenna pattern obtained via (top) SSPE and (bottom) FEMPE (

f

 = 300 MHz,

a

0

 = 1.2 × 10

−6

,

N

x

 = 256,

N

z

 = 1000).

Figure 15.12.  E-field versus height at three ranges. Solid: analytic; dashed: SSPE; dash dotted: FEMPE. (Left) The source profile and its mode sum representation, (middle) E-field at 25 km, and (right) E-field at 50 km (

f

 = 300 MHz,

a

0

 = 1.2 × 10

−6

,

N

x

 = 256,

N

z

 = 1000, without antenna tilt).

Figure 15.13.  E-field versus range at two heights. Solid: analytic; dashed: SSPE; dash dotted: FEMPE. (Top) 100 m and (bottom) 50 m above the ground (

f

 = 300 MHz,

a

0

 = 1.2 × 10

−6

,

N

x

 = 256,

N

z

 = 1000, without antenna tilt).

Figure 15.14.  E-field versus height at three ranges. Solid: analytic; dashed: SSPE; dash dotted: FEMPE. (Left) The source and its mode sum representation, (middle) E-field at 15 km, and (right) E-field at 75 km (

f

 = 300 MHz,

a

0

 = 1.2 × 10

−6

,

N

x

 = 256,

N

z

 = 1000, with 1° upward antenna tilt).

Figure 15.15.  E-field versus range at two heights. Solid: analytic; dashed: SSPE; dash dotted: FEMPE. (Top) 100 m and (bottom) 50 m above the ground (

f

 = 300 MHz,

a

0

 = 1.2 × 10

−6

,

N

x

 = 256,

N

z

 = 1000, with 1° upward antenna tilt).

Figure 15.16.  The 3D color plots of E-field versus range–height variations for a given 1° upward tilted Gaussian antenna pattern obtained via (top) SSPE and (bottom) FEMPE (

f

 = 300 MHz,

a

0

 = 3.6 × 10

−6

,

N

x

 = 256,

N

z

 = 1000).

Figure 15.17.  E-field versus height at three ranges. Solid: analytic; dashed: SSPE; dash dotted: FEMPE: (Left) The source and its mode sum representation, (middle) E-field at 50 km, and (right) E-field at 100 km (

f

 = 300 MHz,

a

0

 = 3.6 × 10

−6

,

N

x

 = 256,

N

z

 = 1000, with 1° upward antenna tilt).

Figure 15.18.  E-field versus range at two heights. Solid: analytic; dashed: SSPE; dash dotted: FEMPE. (Top) 100 m and (bottom) 50 m above the ground (

f

 = 300 MHz,

a

0

 = 3.6 × 10

−6

,

N

x

 = 256,

N

z

 = 1000, with 1° upward antenna tilt).

Figure 15.19.  The same as in Fig. 15.12 but with SSPE:

N

x

 = 256,

N

z

 = 1000, FEMPE:

N

x

 = 512,

N

z

 = 2000.

Figure 15.20.  The same as in Fig. 15.13 but with SSPE:

N

x

 = 256,

N

z

 = 1000; FEMPE:

N

x

 = 512,

N

z

 = 2000.

Figure 15.21.  Maps of E-field strength versus range–height over a user-specified nonflat terrain profile. (Top) SSPE and (bottom) FEMPE results (the parameters are taken from fig. 13 in Ref. 4,

f

 = 100 MHz,

N

x

 = 256,

N

z

 = 1000).

Figure 15.22.  E-field versus height at three ranges. Solid: SSPE; dashed: FEMPE. (Left) The source and its mode sum representation, (middle) E-field at 30 km, and (right) E-field at 60 km (

f

 = 100 MHz,

N

x

 = 256,

N

z

 = 1000).

Figure 15.23.  E-field versus range at two heights. Solid: SSPE; dashed: FEMPE. (Top) 900 m and (bottom) 750 m above the ground (

f

 = 100 MHz,

N

x

 = 256,

N

z

 = 1000).

Figure 15.24.  Maps of E-field versus range–height variations over a user-specified nonflat terrain profile. (Top) SSPE and (bottom) FEMPE results (the parameters are taken from fig. 17 in Ref. 4,

f

 = 100 MHz,

N

x

 = 256,

N

z

 = 1000).

Figure 15.25.  E-field versus height at three ranges. Solid: SSPE; dashed: FEMPE. (Left) The source and its mode sum representation, (middle) E-field at 25 km, and (right) E-field at 50 km (

f

 = 100 MHz,

N

x

 = 256,

N

z

 = 1000).

Figure 15.26.  E-field versus range at two heights. Solid: SSPE; dashed: FEMPE. (Top) 1200 m and (bottom) 250 m above the ground (

f

 = 100 MHz,

N

x

 = 256,

N

z

 = 1000).

Figure 15.27.  SSPE and analytic propagators with tilted waves at 200 and 400 m with −0.5° and 0.5° tilts, respectively.

Figure 15.28.  Vertical field profiles at two different ranges (

dM

/

dx

 = −600M/km).

Figure 15.29.  Front panel of the GrSSPE MATLAB package. The user locates terrain points on the screen by just left clicking the mouse.

Figure 15.30.  Terrain profile is generated from the marked points by using cubic-spline curve fitting technique.

Figure 15.31.  (Left) Flowchart of the main SSPE routine and (right) the relation between the transverse spatial and wave number domains.

Figure 15.32.  Propagation through standard atmosphere over flat earth: frequency of 100 MHz, maximum range of 20 km, maximum height of 1000 m, transmitter height of 800 m, antenna is tilted 1.4° downward, and beamwidth is 1°. Linearly increasing refractivity with a slope of 117 M/km corresponds to the standard atmosphere (including earth's curvature).

Figure 15.33.  Propagation over a user-plotted terrain, through bilinear atmosphere (including earth's curvature).

Figure 15.34.  Propagation through bilinear atmosphere over irregular terrain: frequency is 100 MHz, maximum range of 60 km, maximum height of 1500 m, transmitter height of 750 m, antenna tilt 2° downward, and beamwidth is 1.5°. Atmospheric refractivity increases up to 700 m with a slope of 2000 M/km, and then decreases with the same slope.

Figure 15.35.  PF versus range/height over PEC flat earth 75-m-high knife edge at 15 km (Hor pol, source height = 20 m,

f 

= 3 GHz).

Figure 15.36.  PF versus height at four ranges: 12, 15.1, 16, and 18 km (

f 

= 3 GHz; solid: 4Ray; dashed: SSPE; Δ

x 

= 0.1 m; Δ

z 

= 50 m).

Figure 15.37.  (a) Vertical field distributions of some sources and (b) their wave number domain behaviors.

Figure 15.38.  Excitation coefficients of various Gaussian beams. (Left) Untilted, (solid)

θ

bw

 = 20°, (dashed)

θ

bw

 = 40°, (dashed dotted)

θ

bw

 = 60°; (right) tilted, (solid)

θ

bw

 = 20°, (dashed) untilted, (dashed dotted)

θ

tilt

 = 10°.

Figure 15.39.  Field versus range–height for a given tilted Gaussian source: (a) SSPE, (b) 2Ray, (c) MoM (one-way propagation, DBC,

f 

= 30 MHz, source height = 200 m,

θ

bw

 = 30°,

θ

tilt

 = −15°, Δ

z

 = 1 m, Δ

x

 = 5 m).

Figure 15.40.  Field versus range for a given tilted Gaussian source (DBC): (Top) Untilted source and (bottom) tilted source (at 200 m height,

θ

bw

 = 30°,

f 

= 30 MHz,

θ

tilt

 = −15°).

Figure 15.41.  PF versus range–height for a given tilted Gaussian source: (a) SSPE, (b) FEMPE, and (c) 4Ray (one-way propagation, DBC,

f 

= 300 MHz, with 50-m-high knife edge at 800 m range, source height = 60 m,

θ

bw

 = 15°,

θ

tilt

 = −5°, Δ

z

 = 1 m, Δ

x

 = 0.2 m).

Figure 15.42.  PF versus range at 60-m height for the scenario given in Fig. 14.43 (Δ

z

 = 1 m, Δ

x

 = 0.2 m).

Figure 15.43.  The canonical 2D dielectric slab waveguide.

Figure 15.44.  The first four modes of the dielectric slab waveguide.

Figure 15.45.  Front panel of the DiSLAB package.

Figure 15.46.  DiSLAB screen after run for typical values.

Figure 15.47.  Full DiSLAB screen (slab width = 4 m,

f

 = 1 GHz,

n

0

 = 1.01,

n

0

 = 1.00).

Figure 15.48.  Full DiSLAB screen for example 2: Slab width = 4 m,

f

 = 1 GHz,

n

0

 = 1.05 inside, and

n

0

 = 1.00 outside. Nine beta values of the propagating modes are 21.9790, 21.9426, 21.8818, 21.7969, 21.6878, 21.5549, 21.3989, 21.2219, and 21.0308.

Figure 15.49.  Full DiSLAB screen for example 2: Slab width = 4 m,

f

 = 1 GHz,

n

0

 = 1.05 inside, and

n

0

 = 1.00 outside. Five beta values of the propagating modes are 21.9487, 21.8219, 21.6128, 21.3286, and 21.0026.

Figure 16.1.  Parallel plate waveguide and source–observer locations.

Figure 16.2.  Magnetic field versus range with Green's function: TM

z

case,

a

 = 1 m,

z

′ = 0,

x

′ = 0.3 m,

x

 = 0.7 m,

k

0

a

 = 50. (Solid) Only propagating modes (15 modes) and (dashed) 100 modes.

Figure 16.3.  Magnetic field versus height with Green's function: TM

z

case,

a

 = 1 m,

z

′ = 0,

x

′ = 0.3 m,

k

0

a

 = 50, (a) at

z

 = 2

λ

and (b) at

z

 = 20

λ

. (Solid) Only propagating modes (15 modes) and (dashed) 100 modes.

Figure 16.4.  Eigenrays with (a)

n

 = 0 and (b)

n

 = 1 inside a 1-m-wide 300-m-long parallel plate waveguide; source height = 0.3 m, receiver height = 0.7 m.

Figure 16.5.  E-field versus range with (solid) Green's function (100 modes), (dashed) eigenray representation (

n

 = 50), and (dashed dotted) ray + image method (20 images): TE

z

case,

a

 = 1 m,

z

′ = 0,

x

′ = 0.3 m,

x

 = 0.5 m,

k

0

a

 = 50.

Figure 16.6.  E-field versus range/height with a tilted Gaussian beam. (a) Narrow SSPE, (b) wide SSPE, and (c) Green's function (31 modes): TE

z

case,

a

 = 1 m,

z

′ = 0,

x

′ = 0.4 m,

k

0

a

 = 50,

θ

bw

 = 30°,

θ

elv

 = −10°.

Figure 16.7.  E-field versus height for a tilted Gaussian beam: TE

z

case,

a

 = 1 m,

z

′ = 0,

x

′ = 0.4 m,

k

0

a

 = 50,

θ

bw

 = 30°,

θ

elv

 = −10° (a) at

z

 = 2

λ

and (b)

z

 = 20

λ

. (Solid) Green's function (31 modes), (dashed) narrow SSPE, and (dashed dotted) wide SSPE.

Figure 16.8.  E-field versus range/height at three different time instants for a tilted Gaussian beam computed with FDTD: TE

z

case,

a

 = 1 m,

z

′ = 0,

x

′ = 0.5 m,

k

0

a

 = 50,

θ

bw

 = 40°,

θ

elv

 = −20°,

t

a

 = 3.42 × 10

−9

 s,

t

b

 = 6.84 × 10

−9

 s,

t

c

 = 1.03 × 10

−9

 s.

Figure 16.9.  E-field versus height for a tilted Gaussian beam: TE

z

case,

a

 = 1 m,

z

′ = 0,

x

′ = 0.5 m,

k

0

a

 = 50,

θ

bw

 = 40°,

θ

elv

 = −20° (a) at

z

 = 2

λ

and (b)

z

 = 20

λ

. (Solid) Green's function (42 modes), (dashed) SSPE, and (dashed dotted) FDTD

dx

 = 

dz

 = 0.5 m,

dt

 = 1.14 × 10

−11

 s.

Figure 16.10.  Parallel plate waveguide and source/observer for the MoM (

ρ

m

: source segment;

ρ

n

: observer segment).

Figure 16.11.  Propagation factor (PF) versus range (TM

z

case): (solid) mode sum and (dashed) Mi-MoM,

a

 = 100 m,

z

′ = 0,

x

′ = 50 m,

x

 = 5 m,

k

0

a

 = 209.5.

Figure 16.12.  Field versus height (TE

z

case): (solid) mode sum and (dashed) multi-MoM,

a

 = 1 m,

z

′ = 0,

x

′ = 0.4 m,

k

0

a

 = 50.

Figure 16.13.  Field versus range/height (TE

z

case): (top) mode sum with 42 modes and (bottom) Mi-MoM with 40 iterations,

a

 = 1 m,

z

′ = 0,

x

′ = 0.3 m,

k

0

a

 = 50,

dz

 = 

dx

 = 0.01 m,

θ

bw

 = 45°, no tilt.

Figure 16.14.  Field versus height at

x

 = 0.2 m: (top) TE

z

case, (bottom) TM

z

case, (solid) mode sum with 282 modes, and (dashed) Mi-MoM with 50 iterations,

a

 = 1 m,

z

′ = 0,

x

′ = 0.4 m,

k

0

a

 = 200,

dz

 = 

dx

 = 0.0025 m,

θ

bw

 = 80°, no tilt.

Figure 16.15.  Field versus height at

x

 = 0.2 m: (top) TE

z

case, (bottom) TM

z

case, (solid) mode sum with 298 modes, and (dashed) multi-MoM with 50 iterations,

a

 = 1 m,

z

′ = 0,

x

′ = 0.4 m,

k

0

a

 = 200,

dz

 = 

dx

 = 0.0025 m,

θ

bw

 = 45°,

θ

tilt

 = −20°.

Figure 16.16.  (Top) field versus range/height produced with SSPE, (bottom) field versus range at

x

 = 0.4 m, both for TE

z

case, (solid) SSPE, and (dashed) Mi-MoM with 50 iterations,

a

 = 1 m,

z

′ = 0,

x

′ = 0.4 m,

k

0

a

 = 200,

dz

 = 

dx

 = 0.0025 m,

θ

bw

 = 80°, no tilt.

Guide

Cover

Table of Contents

Start Reading

Preface

CHAPTER 1

Index

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