107,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley - IEEE
- Sprache: Englisch
Understanding electromagnetic wave theory is pivotal in the design of antennas, microwave circuits, radars, and imaging systems. Researchers behind technology advances in these and other areas need to understand both the classical theory of electromagnetics as well as modern and emerging techniques of solving Maxwell's equations. To this end, the book provides a graduate-level treatment of selected analytical and computational methods.
The analytical methods include the separation of variables, perturbation theory, Green's functions, geometrical optics, the geometrical theory of diffraction, physical optics, and the physical theory of diffraction. The numerical techniques include mode matching, the method of moments, and the finite element method. The analytical methods provide physical insights that are valuable in the design process and the invention of new devices. The numerical methods are more capable of treating general and complex structures. Together, they form a basis for modern electromagnetic design.
The level of presentation allows the reader to immediately begin applying the methods to some problems of moderate complexity. It also provides explanations of the underlying theories so that their capabilities and limitations can be understood.
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 756
Veröffentlichungsjahr: 2016
Ähnliche
Table of Contents
Title Page
Copyright
Dedication
About the Author
Preface
Acknowledgements
Chapter 1: Background
1.1 Field Laws
1.2 Properties of Materials
1.3 Types of Currents
1.4 Capacitors, Inductors
1.5 Differential Form
1.6 Time-Harmonic Fields
1.7 Sufficient Conditions
1.8 Magnetic Currents, Duality
1.9 Poynting's Theorem
1.10 Lorentz Reciprocity Theorem
1.11 Friis and Radar Equations
1.12 Asymptotic Techniques
1.13 Further Reading
References
Problems
Chapter 2: Transverse Electromagnetic Waves
2.1 Introduction
2.2 Plane Waves
2.3 Oblique Plane Waves
2.4 Plane-Wave Reflection and Transmission
2.5 Multilayer Slab
2.6 Impedance Boundary Condition
2.7 Transmission Lines
2.8 Transverse Equivalent Network
2.9 Absorbers
2.10 Phase and Group Velocity
2.11 Further Reading
References
Problems
Chapter 3: Waveguides and Resonators
3.1 Separation of Variables
3.2 Rectangular Waveguide
3.3 Cylindrical Waves
3.4 Circular Waveguide
3.5 Waveguide Excitation
3.6 2D Waveguides
3.7 Transverse Resonance Method
3.8 Other Waveguide Types
3.9 Waveguide Discontinuities
3.10 Mode Matching
3.11 Waveguide Cavity
3.12 Perturbation Method
3.13 Further Reading
References
Problems
Chapter 4: Potentials, Concepts and Theorems
4.1 Vector Potentials and
4.2 Hertz Potentials
4.3 Vector Potentials and Boundary Conditions
4.4 Uniqueness Theorem
4.5 Radiation Condition
4.6 Image Theory
4.7 Physical Optics
4.8 Surface Equivalent
4.9 Love's Equivalent
4.10 Induction Equivalent
4.11 Volume Equivalent
4.12 Radiation by Planar Sources
4.13 2D Sources and Fields
4.14 Derivation of Vector Potential Integral
4.15 Solution Without Using Potentials
4.16 Further Reading
References
Problems
Chapter 5: Canonical Problems
5.1 Cylinder
5.2 Wedge
5.3 The Relation Between 2D and 3D Solutions
5.4 Spherical Waves
5.5 Method of Stationary Phase
5.6 Further Reading
References
Problems
Chapter 6: Method of Moments
6.1 Introduction
6.2 General Concepts
6.3 2D Conducting Strip
6.4 2D Thin Wire MoM
6.5 Periodic 2D Wire Array
6.6 3D Thin Wire MoM
6.7 EFIE and MFIE
6.8 Internal Resonances
6.9 PMCHWT Formulation
6.10 Basis Functions
6.11 Further Reading
References
Problems
Chapter 7: Finite Element Method
7.1 Introduction
7.2 Laplace's Equation
7.3 Piecewise-Planar Potential
7.4 Stored Energy
7.5 Connection of Elements
7.6 Energy Minimization
7.7 Natural Boundary Conditions
7.8 Capacitance and Inductance
7.9 Computer Program
7.10 Poisson's Equation
7.11 Scalar Wave Equation
7.12 Galerkin's Method
7.13 Vector Wave Equation
7.14 Other Element Types
7.15 Radiating Structures
7.16 Further Reading
References
Problems
Chapter 8: Uniform Theory of Diffraction
8.1 Fermat's Principle
8.2 2D Fields
8.3 Scattering and GTD
8.4 3D Fields
8.5 Curved Surface Reflection
8.6 Curved Wedge Face
8.7 Non-Metallic Wedge
8.8 Slope Diffraction
8.9 Double Diffraction
8.10 GTD Equivalent Edge Currents
8.11 Surface Ray Diffraction
8.12 Further Reading
References
Problems
Chapter 9: Physical Theory of Diffraction
9.1 PO and an Edge
9.2 Asymptotic Evaluation
9.3 Reflector Antenna
9.4 RCS of a Disc
9.5 PTD Equivalent Edge Currents
9.6 Further Reading
References
Problems
Chapter 10: Scalar and Dyadic Green's Functions
10.1 Impulse Response
10.2 Green's Function for A
10.3 2D Field Solutions Using Green's Functions
10.4 3D Dyadic Green's Functions
10.5 Some Dyadic Identities
10.6 Solution Using a Dyadic Green's Function
10.7 Symmetry Property of
10.8 Interpretation of the Radiation Integrals
10.9 Free Space Dyadic Green's Function
10.10 Dyadic Green's Function Singularity
10.11 Dielectric Rod
10.12 Further Reading
References
Problems
Chapter 11: Green's Functions Construction I
11.1 Sturm–Liouville Problem
11.2 Green's Second Identity
11.3 Hermitian Property
11.4 Particular Solution
11.5 Properties of the Green's Function
11.6 UT Method
11.7 Discrete and Continuous Spectra
11.8 Generalized Separation of Variables
11.9 Further Reading
References
Problems
Chapter 12: Green's Function Construction II
12.1 Sommerfeld Integrals
12.2 The Function
12.3 The Transformation
12.4 Saddle Point Method
12.5 SDP Branch Cuts
12.6 Grounded Dielectric Slab
12.7 Half Space
12.8 Circular Cylinder
12.9 Strip Grating on a Dielectric Slab
12.10 Further Reading
References
Problems
Appendix A: Constants and Formulas
A.1 Constants
A.2 Definitions
A.3 Trigonometry
A.4 The Impulse Function
Reference
Appendix B: Coordinates and Vector Calculus
B.1 Coordinate Transformations
B.2 Volume and Surface Elements
B.3 Vector Derivatives
B.4 Vector Identities
B.5 Integral Relations
Reference
Appendix C: Bessel's Differential Equation
C.1 Bessel Functions
C.2 Roots of
C.3 Integrals
C.4 Orthogonality
C.5 Recursion Relations
C.6 Gamma Function
C.7 Wronskians
C.8 Spherical Bessel Functions
References
Appendix D: Legendre's Differential Equation
D.1 Legendre Functions
D.2 Associated Legendre Functions
D.3 Orthogonality
D.4 Recursion Relations
D.5 Spherical Form
Reference
Appendix E: Complex Variables
E.1 Residue Calculus
E.2 Branch Cuts
References
Appendix F: Compilers and Programming
F.1 Getting Started
F.2 Fortran 90
F.3 More on the OS
F.4 Plotting
F.5 Further Reading
References
Appendix G: Numerical Methods
G.1 Numerical Integration
G.2 Root Finding
G.3 Matrix Equations
G.4 Matrix Eigenvalues
G.5 Bessel Functions
G.6 Legendre Polynomials
References
Appendix H: Software Provided
Index
End User License Agreement
Pages
xvii
xix
xx
xxi
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
485
486
487
488
489
Guide
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Background
Figure 1.1 Mathematical surfaces associated with the field laws. (a) Closed surface and volume, (b) open surface and contour , (c) boundary between regions 1 and 2.
Figure 1.2 (a) Contour bounds the disk . (b) Contour bounds the open surface .
Figure 1.3 (a) Wire loop in the plane with terminals 1–2. The closed path is tangential to the wire and crosses the terminal gap. (b) A toroidal inductor, showing the terminals1–2 and the directions of and .
Figure 1.4 A volume with material , , is bounded by a closed surface having an inward normal .
Figure 1.5 Percent error of versus number of terms, using the asymptotic formula (1.93).
Chapter 2: Transverse Electromagnetic Waves
Figure 2.1 Phasor diagram showing the loss tangent .
Figure 2.2 Time and space progression of a right-hand circularly polarized wave. (a) with increasing time; the electric field vector rotates counterclockwise. (b) as a function of ; the tip of the electric field vector traces out a helical path.
Figure 2.3 (a) Perpendicular polarization, is to the paper and (b) parallel polarization, is to the paper. is region 1 and is region 2.
Figure 2.4 and versus incidence angle for glass, .
Figure 2.5 Multilayer slab, with layers. In the th layer, the sum of right-travelling waves is and the sum of left-travelling waves is .
Figure 2.6 Impedance boundary, (a) thin sheet and (b) impedance boundary equivalent. Regions 1 and 2 are air.
Figure 2.7 (a) Air–dielectric interface and (b) impenetrable impedance boundary equivalent.
Figure 2.8 Illustration of the skin effect in a solid round wire of length and radius .
Figure 2.9 (a) Transmission line and LC-ladder equivalent. (b) One section of the ladder.
Figure 2.10 Voltage transfer function for an LC ladder with sections, and a transmission line (). (a) Magnitude, (b) phase.
Figure 2.11 LC ladder with losses.
Figure 2.15 Transmission line with matched generator and short-circuit load.
Figure 2.16 Voltage, current and on the transmission line.
Figure 2.17
Figure 2.18 Two types of absorbers. (a) Salisbury screen, and (b) Dallenbach layer.
Figure 2.19 Plot of Equation (2.179) at , with and .
Chapter 3: Waveguides and Resonators
Figure 3.1 (a) Rectangular metallic waveguide and (b) illustration of a plane wave propagating down the guide.
Figure 3.2 Electric field for the mode.
Figure 3.3 Interpretation of (a) phase velocity and (b) group velocity for the mode.
Figure 3.4 (a) Circular metallic waveguide and (b) electric field lines for the dominant mode.
Figure 3.5 Coax to waveguide transitions. (a) Loop coupling to . (b) Probe coupling to . (c) Probe coupling with dielectric sleeve.
Figure 3.6 Parallel-plate waveguide with (a) TEM mode, (b) TM mode and (c) TE mode.
Figure 3.7 Dielectric-slab waveguide on a ground plane, PEC or PMC.
Figure 3.8 Plot of and , showing the approximate solution .
Figure 3.9 Plot of and , showing the approximate solution .
Figure 3.10 Dielectric-slab waveguide with thickness .
Figure 3.11 Resonance on a transmission line. (a) One line, with characteristic impedance . (b) Two lines, impedance viewpoint. (c) Two lines, joined to form a resonator.
Figure 3.12
Figure 3.13
Figure 3.14 (a) Ridge waveguide, electric field lines and parallel-plate model. (b) Transverse network.
Figure 3.15 Ridge waveguide with . Normalized cutoff wavelength versus ridge width for several gap sizes .
Figure 3.16 (a) Finline geometry. (b) Transverse network.
Figure 3.17 Finline with . Normalized cutoff wavelength versus gap size .
Figure 3.18 (a) Reflection by a thin metal iris at . (b) Equivalent transmission line and shunt susceptance .
Figure 3.19 Possible reflection coefficients for thin metal irises; (a) shunt inductance and (b) shunt capacitance.
Figure 3.20
Figure 3.21 Waveguide step discontinuities in the (a) H plane and (b) E plane.
Figure 3.22 Rectangular waveguide with an H-plane step at .
Figure 3.23 H-plane step, mode-matching solution for the normalized impedance looking right at . The left waveguide is WR90, and .
Figure 3.24 Rectangular waveguide with inductive iris at ; .
Figure 3.25 Inductive iris normalized susceptance . Mode-matching and variational solutions, for a WR90 waveguide and . The number of modes is .
Figure 3.26 Rectangular cavity resonator, at for the mode.
Figure 3.27 Dielectric resonator with microstrip excitation.
Figure 3.28 (a) Empty metal cavity resonant at with fields . (b) Metal cavity resonant at with material and fields .
Figure 3.29
Figure 3.30 (a) Empty metal cavity resonant at with fields . (b) Metal cavity resonant at with a small dent, and fields .
Figure 3.31 Waveguide, (a) air-dielectric interface, (b) quarter-wave transformer.
Figure 3.32 Waveguide attenuator.
Figure 3.33 Cavity with (a) dielectric slab on the bottom, (b) square metallic post at on the side wall and (c) hemispherical metallic bump at bottom centre.
Figure 3.34 Corrugated conductor.
Chapter 4: Potentials, Concepts and Theorems
Figure 4.1 Coordinates for the vector potential integral in Equation (4.13).
Figure 4.2 Far-field approximation for the vector potential integral.
Figure 4.3 Rectangular metallic waveguide with cross section , partially filled with a dielectric, assuming (a) and (b) .
Figure 4.4 (a) Rectangular metallic waveguide with cross section , partially filled with a dielectric. (b) Transmission line equivalent.
Figure 4.5 Uniqueness theorem. (a) and generate , everywhere. (b) and generate , everywhere.
Figure 4.6 (a) Vertical electric dipole above a perfect electric conductor and (b) equivalent in free space. (c) Horizontal electric dipole above a perfect electric conductor and (d) equivalent in free space.
Figure 4.7 (a) Electric dipoles and magnetic dipoles above a perfect electric conductor and (b) equivalent in free space. (c) Electric and magnetic dipoles above a perfect magnetic conductor and (d) equivalent in free space.
Figure 4.8 A curved perfect electric conductor. (a) Source produces the field . (b) Source produces the field .
Figure 4.9 An incident field illuminates a perfect electric conductor. (a) Infinite, flat conductor. (b) Finite, smoothly curved conductor.
Figure 4.10 (a) Original problem with the scatterer, source and fields . The ambient medium is free space. (b) Equivalent problem, having the same within (outside ) and zero fields outside (inside ). (c) Equivalent problem in free space.
Figure 4.11
Figure 4.12 (a) Original problem with the scatterer, source and fields . (b) Equivalent problem in free space, having the same outside and zero fields inside .
Figure 4.13
Figure 4.14 (a) Original problem with the scatterer, source and fields . (b) Induction equivalent, with , outside and , inside . (c) Original source and fields . (d) Induction equivalent.
Figure 4.15 (a) Body in the presence of impressed sources . (b) Impressed sources produce the incident field in free space. (c) Free-space equivalent: volume equivalent currents inside plus the impressed sources produce the original field .
Figure 4.16 Coordinates showing the -directed and -directed electric current ribbons and .
Chapter 5: Canonical Problems
Figure 5.1 Plane wave incident on a perfectly conducting circular cylinder.
Figure 5.2 Perfectly conducting cylinder, normalized echo width versus , for the backscatter case. (a) Small and (b) arbitrary .
Figure 5.3 Line source illuminating a perfectly conducting circular cylinder.
Figure 5.4 Slot with a -directed electric field in the wall of a perfectly conducting circular cylinder.
Figure 5.5 Line source illuminating a perfectly conducting wedge with faces at and . The exterior wedge angle is .
Figure 5.6 (a) Half plane. A plane wave is broadside incident from above, has a strength of and a wavelength of . (b) Surface currents and geometrical optics (GO) approximation.
Figure 5.7
Figure 5.8 Normalized radar cross section versus for a perfectly conducting sphere.
Figure 5.9 Behaviour of the integrand, when , and . The stationary phase point is at . The areas cancel when the oscillation is rapid, away from the stationary phase point.
Figure 5.10 Perfectly conducting circular cylinder. (a) Line source produces . (b) Line dipole produces .
Chapter 6: Method of Moments
Figure 6.1 MoM and exact solutions of the Fredholm integral equation (6.16), with pulses.
Figure 6.2 2D perfectly conducting strip of width and plane wave incidence.
Figure 6.3 (a) Metal strip with top and bottom surface currents. (b) Same currents, on a surface equivalent in free space. (c) Strip thickness . (d) using pulse bases. The match points are at the dots •.
Figure 6.4
Figure 6.5 (a) Perfectly conducting thin wire with radius . (b) Array of thin wires, with spacing .
Figure 6.6
Figure 6.7 on wire surface and field point on the axis.
Figure 6.8 (a) on wire surface; (b) on axis; (c) equivalent configuration with .
Figure 6.9 (a) Dipole antenna with delta-gap feed. (b) External equivalent, with surface currents on the wire and gap. (c) Internal equivalent, with surface currents on the gap.
Figure 6.10 Monopole on a ground plane and magnetic frill excitation in the coaxial aperture.
Figure 6.11 (a) Perfect electric conductor surrounded by surface . (b) Same currents and fields in free space.
Figure 6.12 (a) Dielectric body and source . (b) External equivalent, everywhere. Fields are zero on . (c) Internal equivalent, everywhere. Fields are zero on .
Figure 6.13 Different types of subdomain basis functions for wires: (a) pulse, (b) triangle, (c) piecewise sinusoidal. (d) One piecewise sinusoid.
Figure 6.14 (a) One RWG basis function covers a domain with two triangles. (b) Three triangles showing current across edge 3–4. (c) Three triangles showing current across edge 2–4.
Chapter 7: Finite Element Method
Figure 7.1 (a) Cross-sectional view of a TEM transmission line; region and bounding surfaces (contours) . (b) Mesh representation.
Figure 7.2 (a) Disconnected elements and (b) connected elements.
Figure 7.3 Some for (a) disconnected elements and (b) connected elements.
Figure 7.4 First quadrant of the transmission line. (a) Electric field lines. (b) Mesh showing fixed nodes • and free nodes .
Figure 7.5 (a) Connected elements with permittivities and . (b) Contour along the junction.
Figure 7.6 (a) Connected elements with permittivities and . (b) Definition of , and outward normal for . (c) Contour bounds . (d) Limit as .
Figure 7.7 Shielded microstrip line.
Figure 7.8
Figure 7.9 An air region and -directed current density , surrounded by a high-permeability material.
Figure 7.10 Rectangular waveguide with dimensions . There are 64 elements and 45 nodes.
Figure 7.11
Figure 7.12 (a) First-order element. (b) Second order in potential. (c) Second order in geometry and second order in potential.
Figure 7.13 (a) First-order tetrahedron. (b) Second-order tetrahedron. (c) Second-order isoparametric tetrahedron.
Figure 7.14 Edge-based element, showing (a) , (b) and (c) .
Figure 7.15 (a) Metal cylinder with magnetic line source. is bounded by and an absorbing boundary is at . (b) Relation between , , and .
Figure 7.16 (a) Some elements at the boundary . (b) Element (1) with node values . (c) Linear interpolation between nodes 1 and 2.
Figure 7.17 (a) Ideal parallel-plate capacitor with no fringing and (b) mesh with fixed • and free nodes.
Figure 7.18 Stripline with centre conductor surrounded by ground.
Chapter 8: Uniform Theory of Diffraction
Figure 8.1 Fermat's principle. (a) Direct path, reflection and refraction. (b) Edge-diffracted ray. (c) Creeping wave.
Figure 8.2 Electric line source above a perfect electric conductor.
Figure 8.3 (a) GO incident and reflected rays for a line source near a conducting wedge. (b) Incident shadow boundary (ISB) and reflected shadow boundary (RSB). The ISB occurs at and the RSB occurs at . (c) Geometry used for the diffraction coefficient.
Figure 8.4
Figure 8.5
Figure 8.6
Figure 8.7 (a) 3D edge-fixed coordinates, (b) cross-sectional view. The source point is at and the field point is at . (c) Keller cone of edge-diffracted rays; .
Figure 8.8 (a) Perfectly conducting rectangular plate and magnetic current . (b) GO and diffracted fields.
Figure 8.9 UTD radiation patterns for a slot and a monopole on a rectangular plate, of width .
Figure 8.10 Astigmatic ray tube. Caustic 1–2 is associated with and ; caustic 3–4 is associated with and . (Colour version at www.wiley.com/go/paknys9981.)
Figure 8.11 Point source and reflected field. (a) Reflected rays showing ray tube cross sections and . (b) Unit vectors used in the dyadic reflection coefficient .
Figure 8.12 Edge-diffracted ray tube. Caustic 3–4 is on the diffracting edge. Both the source and the caustic 1–2 are at the same distance from . (Colour version at www.wiley.com/go/paknys9981.)
Figure 8.13 (a) Perfectly conducting circular disc and monopole . (b) Diffracted ray caustics.
Figure 8.14 Radiation pattern for a short monopole on a circular disc, radius . Comparison of GO, UTD and MoM.
Figure 8.15 Line source and curved surface reflected field. (a) The reflected ray caustic distance is measured between the image and . (b) The reflecting surface radius of curvature at is ; the law of reflection requires .
Figure 8.16 Parameters for slope diffraction by a dipole.
Figure 8.17 TE double diffraction by a strip.
Figure 8.18 (a) Edge diffraction and (b) equivalent current .
Figure 8.19 Diffraction by a circular cylinder. (a) Incident and reflected rays in lit region; creeping waves in shadow region. (b) Multiple encirclement by a creeping wave.
Figure 8.20 Scattering by a circular cylinder. Incident and reflected rays reach in the lit region; creeping wave reaches in the shadow region.
Figure 8.21 Pekeris functions and .
Figure 8.22 (a) Radiation by a source on a circular cylinder. Direct ray reaches in the lit region; creeping wave reaches in the shadow region. Source types: (b) magnetic line source , (c) magnetic line dipole and (d) electric line dipole .
Figure 8.23 Fock radiation functions and . (a) Magnitude and (b) phase.
Figure 8.24 Coupling problem, source on a circular cylinder. Creeping wave reaches point in the shadow region.
Figure 8.25 Fock coupling functions and .
Figure 8.26 (a, c) Magnetic line source and 2D radiation; (b, d) magnetic dipole and 3D radiation. The cylinders are -directed and the radius is .
Figure 8.27 Second-order TE diffractions on a strip.
Figure 8.28 Parallel plate waveguide of width in an infinite ground plane.
Figure 8.29 (a) Pyramidal horn. (b) 2D E-plane model and (c) 2D H-plane model.
Figure 8.30 Some geometries with curved surface reflection. (a) 2D concave reflector. (b) 2D semicircular cylinder. (c) 3D cylinder and hemispherical endcaps.
Figure 8.31 Diffraction at the junction of a PEC half plane and a thin dielectric slab.
Figure 8.32 UTD rays for a circular cylinder. (a) Scattering, (b) radiation and (c) coupling.
Figure 8.33 Coupling formulation for an electric dipole. (a) Source produces . (b) Source produces .
Chapter 9: Physical Theory of Diffraction
Figure 9.1 (a) TE plane wave incident on a perfectly conducting half plane. (b) Exact surface current and GO approximation; and .
Figure 9.2 Magnetic line source and surface current on a half plane.
Figure 9.3 Integrand of (9.7) in arbitrary units, with , the source at , and the field point at .
Figure 9.4 (a) 2D parabolic reflector antenna and magnetic line source of strength ; the surface current is . (b) Unit normal and tangent at the surface point .
Figure 9.5 (a) 2D parabolic reflector, showing edge diffraction points and . (b) Local geometry showing tangent plane and diffraction angles.
Figure 9.6 Comparison of PO, PTD and MoM radiation patterns for the parabolic reflector. and .
Figure 9.7 Perfectly conducting circular disc and -polarized incident plane wave.
Figure 9.8 Circular disc, (a) edge-diffracted rays and (b) edge-diffracted ray caustics.
Figure 9.9 Circular disc, monostatic radar cross section , and comparison of PO, PTD and MoM. , disc radius and .
Figure 9.10 Double-diffraction backscatter from a perfectly conducting circular disc, broadside incidence.
Chapter 10: Scalar and Dyadic Green's Functions
Figure 10.1 2D surface and bounding curve in the plane.
Figure 10.2 An -directed magnetic current at with the field point at .
Figure 10.3 Field point is (a) outside and (b) inside the source region . (c) Exclusion of from integration.
Chapter 11: Green's Functions Construction I
Figure 11.1 (a) Branch cut in the complex plane and counterclockwise integration path . (b) Integration path in the complex plane.
Figure 11.2 plane showing the poles of (11.51) at and the closed integration path.
Figure 11.3
Figure 11.4
Figure 11.5
Figure 11.6 (a) Cylindrically tipped metallic wedge with faces at and . The cylinder radius is . (b) Singularities of and in the plane.
Chapter 12: Green's Function Construction II
Figure 12.1 (a) Branch points for and for . (b) With the plane maps to the lower plane.
Figure 12.2 Analytical properties of the function in the plane.
Figure 12.3 One possible choice of branch cuts for in the plane. The branch cuts define a rule that ensures the single valuedness of the function for all points along the integration path .
Figure 12.4 Proper and improper regions on the top sheet of the plane.
Figure 12.5 Sommerfeld's choice of branch cuts are along the dashed lines where .
Figure 12.6 The non-unique mapping .
Figure 12.7 Some examples of how -plane integration paths map into the plane.
Figure 12.8 Example of how a -plane integration path maps into the plane when part of the path passes onto the bottom sheet of the plane.
Figure 12.9 Functions or showing a saddle point at .
Figure 12.10 Integration path in the plane.
Figure 12.11 -Plane and -plane SDP topologies. The shaded regions represent descending directions away from the saddle point that lead to valleys.
Figure 12.12 plane, showing the angle at the saddle point.
Figure 12.13
Figure 12.14 Plane, showing SDP and (a) pole not crossed, ; (b) pole is crossed, .
Figure 12.15 The correspondence between the plane and plane. What's proper on the -plane top sheet is improper on the bottom sheet and vice versa.
Figure 12.16 Grounded dielectric slab excited by an electric line source; image at .
Figure 12.17 (a) Integration path on top sheet of the plane and surface wave poles. (b) Bottom sheet of the plane, showing leaky wave poles. (c) The top and bottom sheets are mapped into the plane.
Figure 12.18 Deformation of the integration path to the steepest descent path , passing through the saddle point .
Figure 12.19 Leaky wave pole trajectories in the plane and plane, with increasing frequency . Top figure: the poles tend to as . Bottom figure: as is increased, the leaky wave pole will move upwards from point and pass , evolving into a surface wave pole.
Figure 12.20 Existence region in space, for a leaky wave.
Figure 12.21 (a) Half space with air in region 1 and material in region 2 . The incident and reflected angles are . (b) Image at and image coordinates .
Figure 12.22 has a branch point at and has branch points at .
Figure 12.23 Location of the Zenneck wave poles, on the top sheet of the plane. is real, but the branch cuts are shown slightly displaced from the -plane axes for clarity. The branch cuts map to the plane as shown.
Figure 12.24 SDP in the plane and plane, when . The pole is only present for the polarization.
Figure 12.25 Half space with (a) magnetic dipole . (b) Reciprocal problem with electric dipole . (c) Conventional coordinates for vertical dipole in 3D.
Figure 12.26
Figure 12.27 Attenuation function versus numerical distance .
Figure 12.28 (a) Circular cylinder with radius , source at and field point at . (b) Singularities of and in the plane.
Figure 12.29 (a) Angular domain for . (b) Singularities of and in the plane. (c) Interpretation of for the encirclements.
Figure 12.30 Integration paths in the plane and plane.
Figure 12.31 Circular cylinder, and rays. (a) Shadow region, (b) lit region.
Figure 12.32 Plane wave incident on a metal strip grating, on a grounded dielectric slab.
Figure 12.33 Grounded dielectric slab with no strips. (a) due to magnetic line source, (b) due to magnetic line source and (c) due to electric line dipole . (d) Source and field points at .
Figure 12.34 TE incident and reflected plane waves , in the presence of a grounded dielectric slab.
Figure 12.35 (a) A mode having a radiating space harmonic at . (b) An incident plane wave excites a mode. (c) By symmetry this is the same as (b); .
Figure 12.36 TE strip grating; the parameters are , , and and the frequency is . (a) Phase of the reflection coefficient . (b) The field amplitude at is .
Appendix B: Coordinates and Vector Calculus
Figure B.1 Rectangular, cylindrical and spherical coordinates and basis vectors.
Figure B.2 Differential volumes and surfaces.
Figure B.3 3D mathematical surfaces associated with the integral formulas. (a) Closed surface and volume . (b) Open surface and bounding contour .
Figure B.4 2D mathematical surfaces associated with the (a) divergence theorem and (b) Stokes theorem.
Appendix C: Bessel's Differential Equation
Figure C.1 Bessel functions of the first kind, .
Figure C.2 Bessel functions of the second kind, .
Appendix E: Complex Variables
Figure E.1 Some branch cuts in the complex plane. (a) Cut for . (b) Cut for . (c) Cut for .
List of Tables
Chapter 1: Background
Table 1.1 Duality principle
Chapter 2: Transverse Electromagnetic Waves
Table 2.1 Transmission-line parameters
Table 2.2 Types of velocities
Chapter 3: Waveguides and Resonators
Table 3.1 Summary of TE and TM grounded slab fields for
Table 3.2 Inductive shunt susceptances
Table 3.3 Capacitive shunt susceptances
Chapter 5: Canonical Problems
Table 5.1 TM cylindrical field transforms
Table 5.2 TE cylindrical field transforms
Chapter 6: Method of Moments
Table 6.1 Propagation constants for the space harmonics
Chapter 8: Uniform Theory of Diffraction
Table 8.1 Edge diffraction parameters, rectangular plate
Chapter 9: Physical Theory of Diffraction
Table 9.1 Edge diffraction parameters
Chapter 12: Green's Function Construction II
Table 12.1 Calculation of and Where and
Appendix B: Coordinates and Vector Calculus
Table B.1 Unit vector dot products
Appendix C: Bessel's Differential Equation
Table C.1 First few roots of and
Appendix F: Compilers and Programming
Table F.1 Some Linux commands. It is assumed that gfortran has been installed
Table F.2 Some Windows terminal commands
Table F.3 Input/output unit numbers
Appendix H: Software Provided
Table H.1 Software list
APPLIED FREQUENCY-DOMAIN ELECTROMAGNETICS
Robert Paknys
Concordia University, Montreal, Canada
This edition first published 2016. © 2016 John Wiley & Sons Ltd
Registered office
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.
The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book.
Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought.
Library of Congress Cataloging-in-Publication Data
Names: Paknys, Robert, author.
Title: Applied frequency-domain electromagnetics / Robert Paknys.
Description: Hoboken, New Jersey : John Wiley & Sons, 2016. | Includes bibliographical references and index.
Identifiers: LCCN 2016003736 (print) | LCCN 2016009152 (ebook) | ISBN 9781118940563 (cloth) | ISBN 9781118940556 (pdf) | ISBN 9781118940549 (epub)
Subjects: LCSH: Electromagnetic waves. | Electromagnetism–Mathematics.
Classification: LCC QC670 .P35 2016 (print) | LCC QC670 (ebook) | DDC
537–dc23
LC record available at http://lccn.loc.gov/2016003736
A catalogue record for this book is available from the British Library.
To Marilyn and Michael
About the Author
Robert Paknys was born in Montreal, Canada. He received the BEng degree from McGill University in 1979, and the MSc and PhD degrees from Ohio State University in 1982 and 1985, respectively, all in electrical engineering.
He was an assistant professor at Clarkson University during 1985-1987 and an engineer at MPB Technologies during 1987-1989. He joined Concordia University in 1989 as a faculty member in electrical and computer engineering, and is a professor. He has served as a consultant for the government and industry.
He was a visiting professor at the University of Auckland in 1996, the University of Houston in 2004 and the Ecole Polytechnique de Montreal in 2010.
Professor Paknys is a registered professional engineer, a member of CNC-URSI Commission B, a senior member of the IEEE, and a past associate editor for the IEEE Transactions on Antennas and Propagation.
Preface
The technologies related to electromagnetic waves go back to Hertz, Marconi and the radar systems of World War II. The knowledge gained during those eras propelled the subsequent development of microwave and satellite communications and the ubiquitous wireless technology of today. Understanding electromagnetic scattering is pivotal in the applications of radar target identification, underground geophysical probing as well as security applications such as airport scanners and seeing through walls. Computational electromagnetic modelling is a key element in the design of commercial and military aircraft, and navy ships, where the placement of dozens of collocated antennas must be carefully considered, so that intersystem interference can be mitigated.
Researchers behind these and other advances in technology need to understand both the classical theory of electromagnetics and modern techniques for solving Maxwell's equations. To this end, this book provides a graduate-level treatment of selected topics. Chapters 1 and 2 present background material on Maxwell's equations, plane waves and rigorous and approximate boundary conditions. Chapter 3 develops solutions for rectangular, cylindrical and dielectric waveguides and resonators. In Chapter 4, some crucial theorems, principles and potential theory are explained in detail. Chapter 5 presents the solutions to some canonical problems that have an exact solution, such as the cylinder, wedge and sphere. Chapter 6 describes the method of moments. Chapter 7 covers the finite element method. Chapter 8 is about the uniform geometrical theory of diffraction, and Chapter 9 covers physical optics and the physical theory of diffraction. Chapters 10–12 are about Green's functions and their applications.
Analytical methods provide physical insights that are valuable in the design process and the invention of new devices. The separation of variables method is applied to waveguides, cylinders, wedges and other canonical shapes. Asymptotic methods address the evaluation of integrals, as well as diffraction theory. Green's function concepts are presented in the two-dimensional (2D) scalar and three-dimensional (3D) dyadic forms, and their interpretation is given in relation to the surface equivalence principle.
Numerical methods are indispensable as they allow us to solve highly arbitrary and realistic problems that the purely analytical techniques cannot. The method of moments and the finite element method are described in dedicated chapters. The level of presentation allows the reader to immediately begin applying the methods to some problems of moderate complexity. It also provides an explanation of the underlying theory so that its capabilities and limitations can be understood. This has value as it helps one make informed decisions when using modern CAD tools.
Often, in the preliminary stages of research, it is very useful to investigate field behaviour by using 2D problems. This way, it is often possible to greatly simplify the problem while still retaining the essential characteristics of the fields. It is also a good way to learn the subject, as it minimizes the mathematical complexity and makes the field solutions easier to physically interpret. The book emphasizes a 2D approach, however, where appropriate, 3D is also used.
The book is aimed at graduate students and engineers in industry and R&D labs. The minimum assumed background is an undergraduate course in waves and transmission lines. The first three chapters aim to put all readers on an equal footing – thereby readers with diverse backgrounds and levels of familiarity are accommodated. The coverage is intended to assist research students who are beginning to explore the current engineering literature, as well as more experienced researchers who need to learn about new topics.
The way people look for relevant literature has changed dramatically in the past 20 years. For this reason, no attempt has been made to compile a comprehensive list of references, which in any case would be prone to rapid obsolescence. Rather, each chapter contains a small list of references that should help readers proceed and find the key books and papers that address their specific interests. Many fine works have been omitted, and should any authors feel slighted, I offer my apologies in advance.
The topics are not necessarily arranged by the subject category, but in the order that they are most easily learned and applied. Some topics are revisited at a gradually increasing depth. For instance, waveguides are in Chapters 3 and 4, and the surface equivalence principle is in Chapters 4 and 10.
The homework problems have been developed with an intention to provide motivation and opportunities for practice, as well as revealing new concepts. There are problems for review purposes, for analytical development and for programming.
For both analytical and numerical techniques, it is a rewarding step to generate numerical results. The computer-oriented homework problems allow the reader to apply numerical techniques. Some of the problems involve minor modifications of existing programs instead of coding from scratch. Therefore, larger amounts of material and more ambitious problems can be covered in a given time. Many other problems involve little or no computer work, so instructors can choose to opt out of the computation-oriented format or else solve some of the problems with their own code.
The supporting code is written in Fortran 90, which is widely used in computational science and high-performance computing. Well-tested subroutines are provided for special functions, diffraction coefficients, root finding, numerical integration and matrix manipulations. The Netlib repository is extensively used. In this book, the object-oriented capability of Fortran 90 has been used to develop easy-to-use interfaces that hide the complexity of large subroutines.
Computing and plotting can be done with public-domain software that is available under Linux, Windows and Mac OS X. It is assumed that the reader has some prior experience with a programming or scripting language, but not necessarily Fortran 90. Appendix F summarizes the essentials, so that the reader can begin computational work with little difficulty.
R. Paknys Montreal September 2015
Acknowledgements
I would like to thank a few colleagues for their help, encouragement and friendship over the years: Dr Amy R. Pinchuk, Infield Scientific Inc.; Professors Chris Trueman, Concordia University; Ayhan Altintas, Bilkent University; Michael J. Neve, University of Auckland; David R. Jackson and Donald R. Wilton, University of Houston; Jean-Jacques Laurin, Ecole Polytechnique de Montreal; and Derek McNamara, University of Ottawa. Particular appreciation goes to Professor Jackson for sharing his knowledge of Riemann surfaces, leaky waves and periodic structures.
Going back to my early years, I would like to acknowledge some inspiring professors. Their imparted knowledge and wisdom have served me well to this day. At McGill: Professors G. L. d'Ombrain, G. W. Farnell, E. L. Adler, C. W. Bradley, J. E. Turner, R. Vermes, and G. Bach; at Ohio State: Professors C. H. (Buck) Walter, R. G. Kouyoumjian, W. D. Burnside, P. H. Pathak, N. Wang, R. J. Marhefka, J. H. Richmond, E. H. Newman, B. A. Munk, and (visiting professor) R. E. Collin. A few, regrettably, are deceased – nevertheless their ideas live on.
I would like to thank our students. Their helpful feedback led to many improvements in this book. Most importantly, without them, there would have been no course and no book.
Finally I would like to thank a few people who helped transform my ideas into a book; in particular, Anna Smart and Sandra Grayson of the Wiley editorial staff, and Lincy Priya, the project manager at SPi Global.
Chapter 1Background
This chapter provides a review of Maxwell's equations in integral and differential forms. The capacitor and inductor are used to demonstrate and interpret the integral forms. The Poynting theorem, Lorentz reciprocity theorem, Friis transmission formula and radar range equation are also described. Some of the properties of high-frequency asymptotic techniques are reviewed.
1.1 Field Laws
Maxwell's equations in integral form are
We will use the MKS system of units. The Volt, Ampere, Coulomb, Weber, and Tesla are abbreviated as V, A, C, Wb, and T. The electric field is in ; the magnetic field is in ; the electric flux density is in , and the magnetic flux density is in (equivalent to ). The electric current density is in ; charge is in , and charge density is in .
The surface and volume integrals are associated with the mathematical surfaces shown in Figure 1.1. The first equation is Faraday's law. The second one is credited to Ampère and Maxwell, and the third one is Gauss's law. The fourth equation is called Gauss's law for magnetism. The group of four equations is usually referred to as Maxwell's equations.
Figure 1.1 Mathematical surfaces associated with the field laws. (a) Closed surface and volume, (b) open surface and contour C, (c) boundary between regions 1 and 2.
If a region has fields , , and a charge is moving through those fields with a velocity u, the charge will experience a force, in accordance with the Lorentz force law
Charge cannot be created or destroyed. Any increase or decrease of charge occurs because there is a current. This is stated mathematically as the continuity equation. In integral form, the outflux of current across a closed surface S equals the time rate of decrease of the charge that is inside S
The point-form equivalent is
By integrating both sides of (1.7) over a volume V and applying the divergence theorem to the left-hand side, the integral form (1.6) is obtained.
From the electric field, the voltage is
An electric field having is said to be irrotational. This occurs in electrostatics and in the transverse cross section of a transmission line. In these cases, the line integral becomes path independent, and hence, the voltage is uniquely defined by the endpoints a and b.
From the magnetic field, the current is
where is in the direction of the right-hand thumb and C is a closed contour in the direction of the fingers. This relationship is strictly true for steady (DC) currents. It is still true in the AC case if there is no component perpendicular to the surface bounded by C.
1.2 Properties of Materials
The electrical properties of materials are governed by their physical makeup. In this book, the physics and chemistry of these topics will not be covered, and the reader is referred to the references at the end of the chapter. It will be adequate for our purposes to describe the mathematical models that account for the presence of materials.
In free space, we have the constitutive relations
Interestingly, in any system of units, the value of or can be arbitrarily chosen. However, must equal the speed of light. In the MKS system, in (1.13) is chosen as an exact value. Then, in (1.12) is determined.
In dielectric materials,
The term is what we have in free space. If a dielectric is present, the applied electric field will push its atomic charges, positive towards one direction and negative in the opposite direction, forming dipoles. These dipoles contribute an additionalelectric flux density , the polarization in . Generally, the relation between and can be complicated, that is, non-linear. In the special case of linear materials, is linearly proportional to the applied field. More precisely, where the constant of proportionality is called the electric susceptibility. In this case,
Therefore, in linear materials, we can use the simple relation where the permittivity is .
In magnetic materials,
The term is what we have in free space. If a magnetic material is present, the applied magnetic field will reorient the material's electronic orbits (which act as current loops) and contribute an additional magnetic flux density , the magnetization in . In non-linear materials, the relation between and can be complicated. In the special case of linear materials, where the constant of proportionality is called the magnetic susceptibility. In this case,
Therefore, in linear materials, we can use the simple relation where the permeability is .
In a good conductor, when an electric field is applied, the charges move immediately. Dipoles (as in a dielectric) do not have a chance to form. Therefore, and consequently . In non-magnetic materials, . Such approximations are good for non-magnetic conductors such as aluminium or copper.
1.3 Types of Currents
The convection current is associated with charges that are moving with a velocity u
Such a ‘stream’ of charged particles occurs, for example, in a vacuum tube, a cathode ray tube or a scanning electron microscope.
Inside a conductor, an electric field will push on the charges and cause a conduction current
The conductivity is in S/m (Siemens/m, or equvalently, mho/m). The main difference between a convection current and a conduction current is that the latter type occurs in an electrically neutral material. For example, in a wire, for every charge that enters at one end, a charge leaves at the other end. Therefore there is no net charge and .
An impressed current is independent of the field around it, but the field around it depends on the impressed current. An example of an impressed current is a dipole antenna. An induced current comes from the interaction of a field with any surrounding media and/or boundaries. As an example, if a dipole antenna illuminates a metal body, it will cause surface currents to flow on the body; these are induced currents. The purpose of induced currents is that they adjust themselves in just the right way so that their field, when added to the impressed field, will give a total field that satisfies the boundary conditions, that is, on the metal. Inside dielectrics there are volume-equivalent induced currents; these are discussed in Chapter 4.
1.4 Capacitors, Inductors
To gain a better understanding of Maxwell's equations in the integral form, this section demonstrates their application to the fields inside capacitors and inductors.
First, the Ampere-Maxwell equation
will be applied to a capacitor, in Figure 1.2. The capacitor supports an electric field in the region . With in case (a), the current density pierces . Because is zero outside the capacitor, will be zero on , so that (1.20) becomes
With in case (b), the current density is zero on and so that
The right-hand side of (1.21) is the total current . Since C is the same in both cases, the left-hand side of (1.21) and (1.22) are equal. This leads to
or
Recognizing the capacitance , we see that
Figure 1.2 (a) Contour C bounds the disk . (b) Contour C bounds the open surface .
If is right at the surface of the plate, then , and we can say that from which we obtain the well-known result . Equating this with (1.23) implies that , which gives us the capacitance .
Next, we apply Faraday's law to a wire loop and a toroidal inductor. Figure 1.3(a) shows a wire loop in the plane. The integration path C is tangent to the wire and crosses the gap at the terminals 1–2. Because on the wire, the line integral is zero everywhere except at the gap, and
The direction of C implies that . Let us denote the flux through the loop as
