Preface ...................................................... xiii
1. THE METHOD OF LINES ......................................... 1
1.1. INTRODUCTION .......................................... 1
1.2. MOL: FUNDAMENTALS OF DISCRETISATION ................... 5
1.2.1. Qualitative description ....................... 5
1.2.2. Quantitative description of the
discretisation ................................ 7
1.2.3. Numerical example ............................ 11
2. BASIC PRINCIPLES OF THE METHOD OF LINES .................... 15
2.1. INTRODUCTION ......................................... 15
2.2. BASIC EQUATIONS ...................................... 16
2.2.1. Anisotropic material parameters .............. 16
2.2.2. Relations between transversal electric
and magnetic fields - generalised
transmission line (GTL) equations ............ 19
2.2.3. Relation to the analysis with vector
potentials ................................... 21
2.2.4. GTL equations for 2D structures .............. 22
2.2.5. Solution of the GTL equations ................ 23
2.2.6. Numerical examples ........................... 25
2.3. EIGENMODES IN PLANAR WAVEGUIDE STRUCTURES WITH
ANISOTROPIC LAYERS ................................... 26
2.3.1. Introduction ................................. 26
2.3.2. Analysis equations for eigenmodes in planar
structures ................................... 30
2.3.3. Examples of system equations ................. 33
2.3.4. Impedance/admittance transformation in
multilayered structures ...................... 35
2.3.5. System equation in transformed domain ........ 36
2.3.6. System equation in spatial domain ............ 38
2.3.7. Matrix partition technique: two examples ..... 40
2.3.8. Numerical results ............................ 43
2.4. ANALYSIS OF PLANAR CIRCUITS .......................... 45
2.4.1. Discretisation of the transmission line
equations .................................... 45
2.4.2. Determination of the field components ........ 52
2.5. FIELD AND IMPEDANCE/ADMITTANCE TRANSFORMATION ........ 52
2.5.1. Introduction ................................. 52
2.5.2. Impedance/admittance transformation in
multilayered and multisectioned structures ... 53
2.5.3. Impedance/admittance transformation with
finite differences ........................... 61
2.5.4. Stable field transformation through layers
and sections ................................. 66
3. ANALYSIS OF RECTANGULAR WAVEGUIDE CIRCUITS ................. 73
3.1. INTRODUCTION ......................................... 73
3.2. CONCATENATIONS OF WAVEGUIDE SECTIONS ................. 75
3.2.1. LSM and LSE modes in circular waveguide
bends ........................................ 76
3.2.2. LSM and LSE modes in straight waveguides ..... 80
3.2.3. Impedance transformation at waveguide
interfaces ................................... 82
3.2.4. Numerical results for concatenations ......... 84
3.2.5. Numerical results for waveguide filters ...... 87
3.3. WAVEGUIDE JUNCTIONS .................................. 90
3.3.1. E-plane junctions ............................ 93
3.3.2. H-plane junctions ............................ 96
3.3.3. Algorithm for generalised scattering
parameters ................................... 98
3.3.4. Special junctions: E-plane 3-port junction ... 99
3.3.5. Matched E-plane bend ........................ 100
3.3.6. Analysis of waveguide bend
discontinuities ............................. 103
3.3.7. Scattering parameters ....................... 110
3.3.8. Numerical results ........................... 110
3.4. ANALYSIS OF 3D WAVEGUIDE JUNCTIONS .................. 115
3.4.1. General description ......................... 116
3.4.2. Basic equations ............................. 117
3.4.3. Discretisation scheme for propagation
between A and В ............................. 118
3.4.4. Discontinuities ............................. 121
3.4.5. Coupling to other ports ..................... 122
3.4.6. Impedance/admittance transformation ......... 125
3.4.7. Numerical results ........................... 126
4. ANALYSIS OF WAVEGUIDE STRUCTURES IN CYLINDRICAL
COORDINATES ............................................... 131
4.1. INTRODUCTION ........................................ 131
4.2. GENERALISED TRANSMISSION LINE (GTL) EQUATIONS ....... 132
4.2.1. Material parameters in a cylindrical
coordinate system ........................... 132
4.2.2. GTL equations for z-direction ............... 133
4.2.3. GTL equations for ø-direction ............... 137
4.2.4. Analysis of circular (coaxial) waveguides
with azimuthally-magnetised ferrites and
azimuthally-magnetised solid plasma ......... 140
4.2.5. GTL equations for r-direction ............... 144
4.3. DISCRETISATION OF THE FIELDS AND SOLUTIONS........... 150
4.3.1. Equations for propagation in z-direction .... 150
4.3.2. Equations for propagation in ø-direction .... 153
4.3.3. Solution of the wave equations in z- and
ø-direction ................................. 155
4.3.4. Equations for propagation in r-direction .... 155
4.4. SOLUTION IN RADIAL DIRECTION ........................ 155
4.4.1. Discretisation in z-direction - circular
dielectric resonators ....................... 155
4.4.2. Discretisation in z-direction - propagation
in indirection .............................. 162
4.4.3. Discretisation in ø-direction - eigenmodes
in circular multilayered waveguides ......... 171
4.4.4. Eigenmodes of circular waveguides with
magnetised ferrite or plasma -
discretisation in r-direction ............... 186
4.4.5. Waveguide bends - discretisation
in r-direction .............................. 202
4.4.6. Uniaxial anisotropic fibres with circular
and noncircular cross-section -
discretisation in ø-direction ............... 208
4.5. DISCONTINUITIES IN CIRCULAR WAVEGUIDES - ONE-
DIMENSIONAL DISCRETISATION IN RADIAL DIRECTION ...... 216
4.5.1. Introduction ................................ 216
4.5.2. Basic equations for rotational symmetry ..... 217
4.5.3. Solution of the equations for rotational
symmetry .................................... 218
4.5.4. Admittance and impedance transformation ..... 219
4.5.5. Open ending circular waveguide .............. 220
4.5.6. Numerical results for discontinuities in
circular waveguides ......................... 223
4.5.7. Numerical results for coaxial line
discontinuities and coaxial filter
devices ..................................... 223
4.5.8. Non-rotational modes in circular
waveguides .................................. 225
4.5.9. Numerical results and discussion ............ 228
4.6. ANALYSIS OF GENERAL AXIALLY SYMMETRIC ANTENNAS
WITH COAXIAL FEED LINES ............................. 229
4.6.1. Introduction ................................ 229
4.6.2. Theory ...................................... 230
4.6.3. Regions with crossed lines .................. 239
4.6.4. Two special cases ........................... 244
4.6.5. Port relations of section D ................. 247
4.6.6. Numerical results ........................... 248
4.6.7. Further structures and remarks .............. 249
4.7. DEVICES IN CYLINDRICAL COORDINATES -
TWO-DIMENSIONAL DISCRETISATION ...................... 250
4.7.1. Discretisation in r- and ø-direction ........ 250
4.7.2. Numerical results ........................... 253
4.7.3. Discretisation in r- and ø-direction ........ 253
4.7.4. Discretisation in ф- and ø-direction ........ 254
4.7.5. GTL equations for r-direction ............... 255
5. ANALYSIS OF PERIODIC STRUCTURES ........................... 267
5.1. INTRODUCTION ........................................ 267
5.2. PRINCIPLE BEHAVIOUR OF PERIODIC STRUCTURES .......... 269
5.3. GENERAL THEORY OF PERIODIC STRUCTURES ............... 274
5.3.1. Port relations for general two ports ........ 274
5.3.2. Floquet modes for symmetric periods ......... 274
5.3.3. Concatenation of N symmetric periods ........ 280
5.3.4. Floquet modes for unsymmetric periods ....... 281
5.3.5. Some further general relations in periodic
structures .................................. 283
5.4. NUMERICAL RESULTS FOR PERIODIC STRUCTURES IN ONE
DIRECTION ........................................... 286
5.5. ANALYSIS OF PHOTONIC CRYSTALS ....................... 291
5.5.1. Determination of band diagrams .............. 291
5.5.2. Waveguide circuits in photonic crystals ..... 297
5.5.3. Numerical results for photonic crystal
circuits .................................... 299
6. ANALYSIS OF COMPLEX STRUCTURES ............................ 311
6.1. LAYERS OF VARIABLE THICKNESS ........................ 311
6.1.1. Introduction ................................ 311
6.1.2. Matching conditions at curved interfaces .... 312
6.2. MICROSTRIP SHARP BEND ............................... 315
6.3. IMPEDANCE TRANSFORMATION AT DISCONTINUITIES ......... 318
6.3.1. Impedance transformation at concatenated
junctions ................................... 318
6.4. ANALYSIS OF PLANAR WAVEGUIDE JUNCTIONS .............. 320
6.4.1. Main diagonal submatrices ................... 322
6.4.2. Off-diagonal submatrices - coupling to
perpendicular ports ......................... 323
6.5. NUMERICAL RESULTS ................................... 327
6.5.1. Discontinuities in microstrips .............. 328
6.5.2. Waveguide junctions ......................... 333
7. PRECISE RESOLUTION WITH AN ENHANCED AND GENERALISED
LINE ALGORITHM ............................................ 345
7.1. INTRODUCTION ........................................ 345
7.2. CROSSED DISCRETISATION LINES AND CARTESIAN
COORDINATES ......................................... 346
7.2.1. Theoretical background ...................... 346
7.2.2. Lines in vertical direction ................. 351
7.2.3. Lines in horizontal direction ............... 357
7.3. SPECIAL STRUCTURES IN CARTESIAN COORDINATES ......... 361
7.3.1. Groove guide ................................ 361
7.3.2. Coplanar waveguide .......................... 363
7.4. CROSSED DISCRETISATION LINES AND CYLINDRICAL
COORDINATES ......................................... 366
7.4.1. Principle of analysis ....................... 366
7.4.2. General formulas for eigenmode
calculation ................................. 366
7.4.3. Discretisation lines in radial direction .... 367
7.4.4. Discretisation lines in azimuthal
direction ................................... 368
7.4.5. Coupling to neighbouring ports .............. 369
7.4.6. Steps of the analysis procedure ............. 373
7.5. NUMERICAL RESULTS ................................... 373
8. WAVEGUIDE STRUCTURES WITH MATERIALS OF GENERAL
ANISOTROPY IN ARBITRARY ORTHOGONAL COORDINATE SYSTEMS ..... 377
8.1. GENERALISED TRANSMISSION LINE EQUATIONS ............. 377
8.1.1. Material properties ......................... 377
8.1.2. Maxwell's equations in matrix notation ...... 377
8.1.3. Generalised transmission line equations in
Cartesian coordinates for general
anisotropic material ........................ 379
8.1.4. Generalised transmission line equations
for general anisotropic material in
arbitrary orthogonal coordinates ............ 381
8.1.5. Boundary conditions ......................... 383
8.1.6. Interpolation matrices ...................... 384
8.2. DISCRETISATION ...................................... 385
8.2.1. Two-dimensional discretisation .............. 385
8.2.2. One-dimensional discretisation .............. 386
8.3. SOLUTION OF THE DIFFERENTIAL EQUATIONS .............. 388
8.3.1. General solution ............................ 388
8.3.2. Field relation between interfaces A and В ... 389
8.4. ANALYSIS OF WAVEGUIDE JUNCTIONS AND SHARP BENDS
WITH GENERAL ANISOTROPIC MATERIAL BY USING
ORTHOGONAL PROPAGATING WAVES ........................ 389
8.4.1. Introduction ................................ 389
8.4.2. Theory ...................................... 389
8.4.3. Main diagonal submatrices ................... 391
8.4.4. Off-diagonal submatrices - coupling to
other ports ................................. 393
8.4.5. Steps of the analysis procedure ............. 398
8.5. NUMERICAL RESULTS ................................... 398
8.6. ANALYSIS OF WAVEGUIDE STRUCTURES IN SPHERICAL
COORDINATES ......................................... 399
8.6.1. Introduction ................................ 399
8.6.2. Generalised transmission line equations
in spherical coordinates .................... 400
8.6.3. Analysis of special devices - conformal
antennas .................................... 408
8.6.4. Analysis of special devices - conical horn
antennas .................................... 413
8.6.5. Numerical results ........................... 419
8.7. ELLIPTICAL COORDINATES .............................. 420
8.7.1. GTL equations for z-direction ............... 421
8.7.2. GTL equations for ξ-direction ............... 422
8.7.3. GTL equations for η-direction ............... 423
8.7.4. Hollow waveguides with elliptic
cross-section ............................... 424
9. SUMMARY AND PROSPECT FOR THE FUTURE ....................... 429
A. DISCRETISATION SCHEMES AND DIFFERENCE OPERATORS ........... 433
A.1. DETERMINATION OF THE EIGENVALUES AND EIGENVECTORS
OF P ................................................ 433
A.1.1. Calculation of the matrices 5 ............... 436
A.1.2. Derivation of the eigenvalues of the
Neumann problem from those of the
Dirichlet problem ........................... 438
A.1.3. The component of er at an abrupt
transition .................................. 439
A.1.4. Eigenvalues and eigenvectors for periodic
boundary conditions ......................... 441
A.1.5. Discretisation for non-ideal places of the
boundaries .................................. 442
A.2. ABSORBING BOUNDARY CONDITIONS (ABCs) ................ 444
A.2.1. Introduction1 ............................... 444
A.2.2. Factorisation of the Helmholtz equation ..... 445
A.2.3. Pade approximation .......................... 446
A.2.4. Polynomial approximations ................... 447
A.2.5. Construction of the difference operator
for ABCs .................................... 449
A.2.6. Special boundary conditions (SBCs) .......... 450
A.2.7. Numerical results ........................... 450
A.2.8. ABCs for cylindrical coordinates ............ 453
A.2.9. Periodic boundary conditions ................ 455
A.3. HIGHER-ORDER DIFFERENCE OPERATORS [11] .............. 456
A.3.1. Introduction3 ............................... 456
A.3.2. Theory ...................................... 457
A.3.3. Numerical results ........................... 459
A.4. NON-EQUIDISTANT DISCRETISATION ...................... 460
A.4.1. Introduction ................................ 460
A.4.2. Theory ...................................... 460
A.4.3. Interpolation ............................... 464
A.4.4. Numerical results ........................... 466
A.5. REFLECTIONS IN DISCRETISATION GRIDS ................. 468
A.5.1. Introduction ................................ 468
A.5.2. Dispersion relations ........................ 468
A.5.3. Reflections at discretisation transitions ... 471
A.6. FIELD EXTRAPOLATION FOR NEUMANN BOUNDARY
CONDITIONS .......................................... 475
A.7. ABOUT THE NATURE OF THE METHOD OF LINES ............. 476
A.7.1. Introduction ................................ 476
A.7.2. Relation between shielded structures and
periodic ones ............................... 477
A.7.3. Method of Lines and discrete Fourier
transformation .............................. 478
A.7.4. Discussion .................................. 479
A.8. RELATION BETWEEN THE MODE MATCHING METHOD (МММ)
AND THE METHOD OF LINES (MoL) FOR INHOMOGENEOUS
MEDIA ............................................... 480
A.9. RECIPROCITY AND ITS CONSEQUENCES .................... 483
В. TRANSMISSION LINE EQUATIONS ............................... 491
B.1. TRANSMISSION LINE EQUATIONS IN FIELD VECTOR
NOTATION ............................................ 491
B.2. DERIVATION OF THE MULTICONDUCTOR TRANSMISSION LINE
EQUATIONS ........................................... 492
С. SCATTERING PARAMETERS ..................................... 497
D. EQUIVALENT CIRCUITS FOR DISCONTINUITIES ................... 499
E. APPROXIMATE METALLIC LOSS CALCULATION IN CONFORMAL
STRUCTURES ................................................ 501
Index ......................................................... 503
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