2.5 Boundary Conditions
The solution of the finite element equations can only proceed once the boundary con- ditions have been correctly applied. The walls of the cavity and feed waveguide, shown in Figure 2.3, are assumed to be perfect conductors. Thus on the walls we have,
E×n=0. (2.45)
This condition can be applied by setting the values of the electric field corresponding to edges that lie on the boundary to zero [Silvester & Ferrari, 1990]. For boundary value problems, such as the one described in this paper, it is also necessary to specify a source term. It is assumed that only a TE10 mode exists in the waveguide, which makes it necessary to model a sufficient length of waveguide, as shown in Figure 2.3, to ensure that disturbances caused by the aperture are not present at the point where the excitation is applied. For frequency domain problems it is simply necessary to specify the tangential components of the field at the terminating plane in the waveguide [Webb, 1981]. This can be carried out since the distribution of the transverse field across the waveguide is known;
Ey = E0sin
µπx a
¶
,
Ex = 0, (2.46)
wherea is the width of the waveguide. E0 is normally chosen to be 1 and the fields can then be rescaled after the normalised distribution has been found (see§4.3).
The amplitude of the field at the terminating plane is the vectorial sum of both forward and reflected waves. In the time domain the amplitude of the reflected wave will vary as the system moves from a transient to a steady state condition. Consequently the amplitude of the field at the terminating plane cannot be determined in advance.
It is therefore not possible to prescribe an inhomogeneous Dirichlet boundary on this plane in the time domain. To overcome this problem the waveguide is excited by a current sheet some distance from the terminating plane and the mesh terminated by an absorbing boundary, Figure 2.3. This technique has previously been used with the time domain finite difference method [Iskander, 1993]. In practice this approach has a good
Figure 2.3: Multimode cavity with waveguide feed and absorbing boundary plane for time domain calculation.
physical basis when the applicator is supplied via an iso-circulator which has the effect of absorbing the majority of reflected energy.
2.5.1 Absorbing boundary conditions (ABCs)
Having determined that an ABC is required to terminate the waveguide feed in the time domain its implementation needs to be considered. Development of high quality ABCs has seen intensive effort for open problems, such as finding the scattering cross section of objects. Engquist & Majda [1977] proposed a method for generating good approximations to an ABC. This was extended by Mur [1981] explicitly for use with the finite difference method. Mur’s ABC has become a standard for finite difference calculations, however, other methods which appear greatly superior to Mur have been suggested such as Liao’s ABC [Liao et al., 1984; Chew, 1990] and the Berenger ABC [Katz et al., 1994]. These methods, while offering extremely good approximations to a perfect absorber are designed for open problems and are of a complexity that is unnecessary for this application. Second order Engquist-Majda and Mur ABCs lead to unsymmetric matrices while Liao’s method requires the field at several time steps to be stored for points near to the boundary plane. For a simple waveguide we have a
2.5 Boundary Conditions 43
x y z
Figure 2.4: Rectangular waveguide.
considerable advantage: the field distribution is known a priori. This enables a simple but effective ABC to be derived that can be implemented with very little effort and that does not increase the computational effort during the time stepping. The ABC employed here simply involves terminating the waveguide with its characteristic impedance, which can be done by evaluation of the surface integral term of equation (2.22) [Dibben &
Metaxas, 1994b]
In a rectangular waveguide, as shown in Figure 2.4, the transverse components of the field for the TE modes are related by [Collin, 1992]:
Ex = Z(ω)Hy,
Ey = −Z(ω)Hx, (2.47)
whereZ(ω) is the characteristic impedance of the waveguide for a given mode. For the TE10 mode
Z(ω) =Z0λg
λ0, (2.48)
where λg and λ0 are the waveguide and free space wavelengths respectively and Z0 is the impedance of free space. Equation (2.47) enables us to evaluate the surface integral of equation (2.22) over the terminating plane of the waveguide, where the cross product may be written as,
n×∂H
∂t = −∂Hx
∂t bj
0.000 0.005 0.010 0.015 0.020
2.3 2.35 2.4 2.45 2.5 2.55 2.6
Frequency, GHz Reflection Coefficient, ρ
Figure 2.5: Magnitude of the reflection coefficient,ρ, for a waveguide terminated by the simple ABC,Z(ω) chosen at ω = 2.45 GHz.
= 1
Z(ω)
∂Ey
∂t bj, (2.49)
since for the TE10 mode, Hy =Ex = 0. The surface integral term therefore becomes µ0
Z
Γψ·
Ã
n× ∂H
∂t
!
dΓ → µ0
Z(ω)
Z
Γψ· ∂Ey
∂t dΓ. (2.50)
This integral can now be discretised in the same fashion as before, using equations (2.23). Only ∂Ey/∂t is involved here, so the terms can be added directly into the matrix [Tσ] during the matrix assembly process. The connectivity remains unchanged so the sparsity of the matrix is not reduced and the matrix remains symmetric. This ABC can be seen as equivalent to adding a layer of lossy material at the terminating plane.
Z(ω) is frequency dependent, therefore this boundary will only be strictly valid at a single frequency, for which Z(ω) is specified. However, in practice we are interested in a narrow spectrum of frequencies and this method appears to provide a satisfactory approximation to an absorbing boundary when a mean value ofZ(ω) is chosen. Figure