**Preface.** **1 THE METHOD OF LINES.**

1.1 INTRODUCTION.

1.2 MOL: FUNDAMENTALS OF DISCRETISATION.

1.2.1 Qualitative description.

1.2.2 Quantitative description of the discretisation.

1.2.3 Numerical example.

**2 BASIC PRINCIPLES OF THE METHOD OF LINES.**

2.1 INTRODUCTION.

2.2 BASIC EQUATIONS.

2.2.1 Anisotropic material parameters.

2.2.2 Relations between transversal electric and magnetic fields – generalised transmission line (GTL) equations.

2.2.3 Relation to the analysis with vector potentials.

2.2.4 GTL equations for 2D structures.

2.2.5 Solution of the GTL equations.

2.2.6 Numerical examples.

2.3 EIGENMODES IN PLANAR WAVEGUIDE STRUCTURES WITH ANISOTROPIC LAYERS.

2.3.1 Introduction.

2.3.2 Analysis equations for eigenmodes in planar structures.

2.3.3 Examples of systemequations.

2.3.4 Impedance/admittance transformation in multilayered structures.

2.3.5 System equation in transformed domain.

2.3.6 System equation in spatial domain.

2.3.7 Matrix partition technique: two examples.

2.3.8 Numerical results.

2.4 ANALYSIS OF PLANAR CIRCUITS.

2.4.1 Discretisation of the transmission line equations.

2.4.2 Determination of the field components.

2.5 FIELD AND IMPEDANCE/ADMITTANCE TRANSFORMATION.

2.5.1 Introduction.

2.5.2 Impedance/admittance transformation in multilayered and multisectioned structures.

2.5.3 Impedance/admittance transformation with finite differences.

2.5.4 Stable field transformation through layers and sections.

**3 ANALYSIS OF RECTANGULAR WAVEGUIDE CIRCUITS.**

3.1 INTRODUCTION.

3.2 CONCATENATIONS OF WAVEGUIDE SECTIONS.

3.2.1 LSM and LSE modes in circular waveguide bends.

3.2.2 LSM and LSE modes in straight waveguides.

3.2.3 Impedance transformation at waveguide interfaces.

3.2.4 Numerical results for concatenations.

3.2.5 Numerical results for waveguide filters.

3.3 WAVEGUIDE JUNCTIONS.

3.3.1 E-plane junctions.

3.3.2 H-plane junctions.

3.3.3 Algorithm for generalised scattering parameters.

3.3.4 Special junctions: E-plane 3-port junction.

3.3.5 Matched E-plane bend.

3.3.6 Analysis of waveguide bend discontinuities.

3.3.7 Scattering parameters.

3.3.8 Numerical results.

3.4 ANALYSIS OF 3D WAVEGUIDE JUNCTIONS.

3.4.1 General description.

3.4.2 Basic equations.

3.4.3 Discretisation scheme for propagation between A and B.

3.4.4 Discontinuities.

3.4.5 Coupling to other ports.

3.4.6 Impedance/admittance transformation.

3.4.7 Numerical results.

**4 ANALYSIS OF WAVEGUIDE STRUCTURES IN CYLINDRICAL COORDINATES.**

4.1 INTRODUCTION.

4.2 GENERALISED TRANSMISSION LINE (GTL) EQUATIONS.

4.2.1 Material parameters in a cylindrical coordinate system.

4.2.2 GTL equations for *z*-direction.

4.2.3 GTL equations for *φ*-direction.

4.2.4 Analysis of circular (coaxial) waveguides with azimuthally-magnetised ferrites and azimuthallymagnetised solid plasma.

4.2.5 GTL equations for *r*-direction.

4.3 DISCRETISATION OF THE FIELDS AND SOLUTIONS.

4.3.1 Equations for propagation in *z*-direction.

4.3.2 Equations for propagation in *φ*-direction.

4.3.3 Solution of the wave equations in *z*- and *φ*-direction.

4.3.4 Equations for propagation in *r* -direction.

4.4 SOLUTION IN RADIAL DIRECTION.

4.4.1 Discretisation in *z* -direction – circular dielectric resonators.

4.4.2 Discretisation in *z* -direction – propagation in *φ*-direction.

4.4.3 Discretisation in *φ*-direction – eigenmodes in circular multilayered waveguides.

4.4.4 Eigenmodes of circular waveguides with magnetised ferrite or plasma – discretisation in *r* -direction.

4.4.5 Waveguide bends – discretisation in *r* -direction.

4.4.6 Uniaxial anisotropic fibres with circular and noncircular cross-section – discretisation in *φ*-direction.

4.5 DISCONTINUITIES IN CIRCULAR WAVEGUIDES – ONE-DIMENSIONAL DISCRETISATION IN RADIAL DIRECTION.

4.5.1 Introduction.

4.5.2 Basic equations for rotational symmetry.

4.5.3 Solution of the equations for rotational symmetry.

4.5.4 Admittance and impedance transformation.

4.5.5 Open ending circular waveguide.

4.5.6 Numerical results for discontinuities in circular waveguides.

4.5.7 Numerical results for coaxial line discontinuities and coaxial filter devices.

4.5.8 Non-rotational modes in circular waveguides.

4.5.9 Numerical results and discussion.

4.6 ANALYSIS OF GENERAL AXIALLY SYMMETRIC ANTENNAS WITH COAXIAL FEED LINES.

4.6.1 Introduction.

4.6.2 Theory.

4.6.3 Regions with crossed lines.

4.6.4 Two special cases.

4.6.5 Port relations of section D.

4.6.6 Numerical results.

4.6.7 Further structures and remarks.

4.7 DEVICES IN CYLINDRICAL COORDINATES –TWO-DIMENSIONAL DISCRETISATION.

4.7.1 Discretisation in *r*- and *φ*-direction.

4.7.2 Numerical results.

4.7.3 Discretisation in *r*- and *z*-direction.

4.7.4 Discretisation in *φ*- and *z* -direction.

4.7.5 GTL equations for *r*-direction.

**5 ANALYSIS OF PERIODIC STRUCTURES.**

5.1 INTRODUCTION.

5.2 PRINCIPLE BEHAVIOUR OF PERIODIC STRUCTURES.

5.3 GENERAL THEORY OF PERIODIC STRUCTURES.

5.3.1 Port relations for general two ports.

5.3.2 Floquetmodes for symmetric periods.

5.3.3 Concatenation of *N* symmetric periods.

5.3.4 Floquet modes for unsymmetric periods.

5.3.5 Some further general relations in periodic structures.

5.4 NUMERICAL RESULTS FOR PERIODIC STRUCTURES IN ONE DIRECTION.

5.5 ANALYSIS OF PHOTONIC CRYSTALS.

5.5.1 Determination of band diagrams.

5.5.2 Waveguide circuits in photonic crystals.

5.5.3 Numerical results for photonic crystal circuits.

**6 ANALYSIS OF COMPLEX STRUCTURES.**

6.1 LAYERS OF VARIABLE THICKNESS.

6.1.1 Introduction.

6.1.2 Matching conditions at curved interfaces.

6.2 MICROSTRIP SHARP BEND.

6.3 IMPEDANCE TRANSFORMATION AT DISCONTINUITIES.

6.3.1 Impedance transformation at concatenated junctions.

6.4 ANALYSIS OF PLANAR WAVEGUIDE JUNCTIONS.

6.4.1 Main diagonal submatrices.

6.4.2 Off-diagonal submatrices – coupling to perpendicular ports.

6.5 NUMERICAL RESULTS.

6.5.1 Discontinuities in microstrips.

6.5.2 Waveguide junctions.

**7 PRECISE RESOLUTION WITH AN ENHANCED AND GENERALISED LINE ALGORITHM.**

7.1 INTRODUCTION.

7.2 CROSSED DISCRETISATION LINES AND CARTESIAN COORDINATES.

7.2.1 Theoretical background.

7.2.2 Lines in vertical direction.

7.2.3 Lines in horizontal direction.

7.3 SPECIAL STRUCTURES IN CARTESIAN COORDINATES.

7.3.1 Groove guide.

7.3.2 Coplanar waveguide.

7.4 CROSSED DISCRETISATION LINES AND CYLINDRICAL COORDINATES.

7.4.1 Principle of analysis.

7.4.2 General formulas for eigenmode calculation.

7.4.3 Discretisation lines in radial direction.

7.4.4 Discretisation lines in azimuthal direction.

7.4.5 Coupling to neighbouring ports.

7.4.6 Steps of the analysis procedure.

7.5 NUMERICAL RESULTS.

**8 WAVEGUIDE STRUCTURES WITH MATERIALS OF GENERAL ANISOTROPY IN ARBITRARY ORTHOGONAL COORDINATE SYSTEMS.**

8.1 GENERALISED TRANSMISSION LINE EQUATIONS.

8.1.1 Material properties.

8.1.2 Maxwell’s equations in matrix notation.

8.1.3 Generalised transmission line equations in Cartesian coordinates for general anisotropic material.

8.1.4 Generalised transmission line equations for general anisotropic material in arbitrary orthogonal coordinates.

8.1.5 Boundary conditions.

8.1.6 Interpolationmatrices.

8.2 DISCRETISATION.

8.2.1 Two-dimensional discretisation.

8.2.2 One-dimensional discretisation.

8.3 SOLUTION OF THE DIFFERENTIAL EQUATIONS.

8.3.1 General solution.

8.3.2 Field relation between interfaces A and B.

8.4 ANALYSIS OF WAVEGUIDE JUNCTIONS AND SHARP BENDS WITH GENERAL ANISOTROPIC MATERIAL BY USING ORTHOGONAL PROPAGATING WAVES.

8.4.1 Introduction.

8.4.2 Theory.

8.4.3 Main diagonal submatrices.

8.4.4 Off-diagonal submatrices – coupling to other ports.

8.4.5 Steps of the analysis procedure.

8.5 NUMERICAL RESULTS.

8.6 ANALYSIS OF WAVEGUIDE STRUCTURES IN SPHERICAL COORDINATES.

8.6.1 Introduction.

8.6.2 Generalised transmission line equations in spherical coordinates.

8.6.3 Analysis of special devices – conformal antennas.

8.6.4 Analysis of special devices – conical horn antennas.

8.6.5 Numerical results.

8.7 ELLIPTICAL COORDINATES.

8.7.1 GTL equations for *z*-direction.

8.7.2 GTL equations for *ξ*-direction.

8.7.3 GTL equations for *η*-direction.

8.7.4 Hollow waveguides with elliptic cross-section.

**9 SUMMARY AND PROSPECT FOR THE FUTURE.**

**A. DISCRETISATION SCHEMES AND DIFFERENCE OPERATORS.**

A.1 DETERMINATION OF THE EIGENVALUES AND EIGENVECTORS OF *P.*

A.1.1 Calculation of the matrices *δ.*

A.1.2 Derivation of the eigenvalues of the Neumann problem from those of the Dirichlet problem.

A.1.3 The component of *εr* at an abrupt transition.

A.1.4 Eigenvalues and eigenvectors for periodic boundary conditions.

A.1.5 Discretisation for non-ideal places of the boundaries.

A.2 ABSORBING BOUNDARY CONDITIONS (ABCs).

A.2.1 Introduction^{1}.

A.2.2 Factorisation of the Helmholtz equation.

A.2.3 Pad´e approximation.

A.2.4 Polynomial approximations.

A.2.5 Construction of the difference operator for ABCs.

A.2.6 Special boundary conditions (SBCs).

A.2.7 Numerical results.

A.2.8 ABCs for cylindrical coordinates.

A.2.9 Periodic boundary conditions.

A.3 HIGHER-ORDER DIFFERENCE OPERATORS [11].

A.3.1 Introduction.

A.3.2 Theory.

A.3.3 Numerical results.

A.4 NON-EQUIDISTANT DISCRETISATION.

A.4.1 Introduction.

A.4.2 Theory.

A.4.3 Interpolation.

A.4.4 Numerical results.

A.5 REFLECTIONS IN DISCRETISATION GRIDS.

A.5.1 Introduction.

A.5.2 Dispersion relations.

A.5.3 Reflections at discretisation transitions.

A.6 FIELD EXTRAPOLATION FOR NEUMANN BOUNDARY CONDITIONS.

A.7 ABOUT THE NATURE OF THE METHOD OF LINES.

A.7.1 Introduction.

A.7.2 Relation between shielded structures and periodic ones.

A.7.3 Method of Lines and discrete Fourier transformation.

A.7.4 Discussion.

A.8 RELATION BETWEEN THE MODE MATCHING METHOD (MMM) AND THE METHOD OF LINES (MoL) FOR INHOMOGENEOUSMEDIA.

A.9 RECIPROCITYAND ITS CONSEQUENCES.

**B TRANSMISSION LINE EQUATIONS.**

B.1 TRANSMISSION LINE EQUATIONS IN FIELD VECTOR NOTATION.

B.2 DERIVATION OF THE MULTICONDUCTOR TRANSMISSION LINE EQUATIONS.

**C SCATTERING PARAMETERS.**

**D EQUIVALENT CIRCUITS FOR DISCONTINUITIES.**

**E APPROXIMATE METALLIC LOSS CALCULATION IN CONFORMAL STRUCTURES.**

**Index.**