In this section the canonical geometry used to assess the dispersion characteristics is presented. Field propagation is simulated within a notionally homogenous free space rectangular region of length L= 100m and width d=10m with open-circuited boundaries being placed on the top and bottom as shown in Figure 5.2. The electric field is polarised out of the plane. This configuration is chosen as it theoretically supports propagation of a dispersionless transversally uniform field along its axis which acts as a convenient reference. Both the input and output planes are terminated with the wave impedance of free space, where µo is magnetic
permeability and o is dielectric permittivity of the free space which theoretically
impedance-match the fundamental propagating field. The structure also supports infinity of higher order waveguide modes (Equation (5.14)) each of which has a transverse (i.e. with respect to x) field distribution that is orthogonal to that of all the others. Note that each of these higher order modes can be represented as interfering waves propagating a particular angle to axis. Theoretically, there should be no inter- mode coupling between these modes along the length of the uniform cross-section structure. However, the irregular nature of an unstructured mesh will actually cause non-physical coupling to occur, analogous to propagation through a medium with small-scale variations in its material parameters. A quantification of this mesh induced coupling provides a physically meaningful measure of the consequences of local mesh irregularities.
The bulk of the problem will be meshed by many different meshes and statistically processed results presented. In order to ensure consistent reference ports for the measurements, the input and output planes are always meshed with the same number of, N, equilateral triangles as shown in Figure 5.2.
The remainder of the problem space is meshed with a Delaunay mesh, the density of which is controlled by the number of fixed vertices at the input and output planes and by demanding a required triangle quality of the mesh generation software which leads to the insertion of additional vertices. Meshes are studied based on size and quality, and this can be obtained by sitting varieties of constraints on the command line to the
‘Triangle’ software in order to get a wide span of triangle size and quality. As mentioned in Section 3.5.1, two main switches are used in the ‘Triangle’ command line; ‘-a’ switch which is concerned with mesh size limitation, and ‘-q’ switch which relates to the mesh quality. For a mesh of certain average area (fixed ‘-a’ switch) different mesh quality can be obtained by setting different angle constrains following ‘-q’ switch. Then for a mesh of certain quality (fixed ‘-q’ switch) varieties of mesh sizes from coarse to fine can be obtained by setting upper limit for triangles area following the ‘-a’ switch. Each produced mesh has a unique distribution of triangle size or shape, so for the sake of comparison and classification each mesh is expressed by a average area Aav and average quality factor Q, in addition to the standard deviation
from the mean value.
To obtain a sequence of different meshes of the same statistical density and quality, the region is pre-seeded with a number of randomly placed vertices before meshing. Bear in mind that changing the number or the location of any of these vertices will produce different distribution of triangles, so it is almost impossible to reproduce the same mesh even by setting the same area and quality constraints.
Zo V1 Zo VN TLM nodes TLM nodes z x Open-circuit boundary Output plane Open-circuit boundary Input plane
L
o, µ
od
x=d x=0 x x Zo ZoFigure 5.2: Schematic diagram of parallel plate waveguide with the enforced triangularization of the input and output planes
In this study, 2D Delaunay meshes were generated using the ‘Triangle’ mesh generator provided by Shewchuk [6]. The triangulated mesh is then processed to obtain the
Voronoi mesh of transmission lines. Figure 5.3 shows an example of the Delaunay and Voronoi meshes of a rectangular space whilst keeping the input and output planes fixed.
Excitation side Observation side
d
L
z x Open-circuit boundary Open-circuit boundaryFigure 5.3: Delaunay (light) and Voronoi (dark) diagrams of a meshed rectangular space with fixed points along input and output planes.
In the simulations, the excitation to the model consists of voltage impulses on the N transmission line ports present in the input plane with a spatial distribution of
cos (5.24)
where i is the index of the input port, i=1,2, …N, Vo is the input signal amplitude, kx is
the transverse wavenumber given by kx=n/d, and x is the distance between adjacent
samples. Theoretically, after propagating along the structure length L and assuming the appropriate (frequency dependent for n0) source and termination impedances, the field at the output nodes has the form
cos (5.25)
where b is the phase constant. In the TLM model the discretisation will introduce numerical dispersion, this is first assessed by extracting the phase delay b which can be obtained using the 2D discrete Fourier transform over time and x of the voltages at all UTLM nodes in the input and output planes, which for the simplest case of n=0, reduces to evaluating
lim (5.26)
where
V
i(k
xi
x,
f)
denotes Fourier transformed voltage at the input or output plane, i.e.,(5.27)
where t is the TLM time step and T= mt denotes the total simulation time.
More generally, the same approach is used to evaluate the non-physical inter-modal coupling between different spatial modes. A particular mode of transverse order n is launched into the waveguide and the coupling to each mode of order m will be assessed by a 2D Fourier transform,
(5.28)
The 2D FFT will exhibit peaks at values of all modes that are present I the simulation.