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Waveguide Input Port Boundary Condition

The waveguide input port boundary condition models the electromagnetic field that goes into the computational domain from an attached waveguide. The waveguide input boundary condition assumes that the injected radiation is a well-established waveguide mode. The complex amplitude \(\tilde{\mathbf{E}}^\text{in}\) of the input electric field is

\[ \tilde{\mathbf{E}}^\text{in} = \tilde{\mathbf{e}}(\mathbf{r})\, e^{-j\beta \mathbf{r} \cdot \hat{\mathbf{n}}}, \]

where \(\tilde{\mathbf{e}}(\mathbf{r})\) describes the mode profile, defined in the plane of the port, \(\beta\) is the mode propagation constant, \(\mathbf{r}=(x,y,z)\) is the radius-vector and \(\hat{\mathbf{n}}\) is the unit vector normal to the boundary. mufem computes the mode profile \(\tilde{\mathbf{e}}\) and the propagation constant \(\beta\) from a two-dimensional eigenvalue problem on the port surface. The solver finds hybrid modes [1], not only TE and TM modes. Hybrid modes occur in waveguides with an inhomogeneous or anisotropic filling. Examples are a partially loaded waveguide, a microstrip line, and a coplanar waveguide.

The port eigenvalue problem uses the material data of the model:

  • The effective permittivity is \(\varepsilon_\text{eff} = \varepsilon - j\sigma/\omega\), where \(\varepsilon\) is the electric permittivity (F/m), \(\sigma\) is the electric conductivity (S/m), and \(\omega\) is the angular frequency (rad/s). For a lossy filling, \(\beta\) (rad/m) is complex.
  • mufem uses the permittivity \(\varepsilon\), the conductivity \(\sigma\), and the magnetic reluctivity \(\nu = \mu^{-1}\) (m/H) as full \(3\times3\) tensors. Thus, the port supports anisotropic materials, such as a sapphire substrate.

mufem sorts the modes by their propagation constant. The mode with the largest \(\beta\) (the fundamental mode) has the index 0. mufem scales each mode to a time-averaged power of 1 W through the port:

\[ \frac{1}{2} \mathrm{Re} \int_{\Gamma} \left(\tilde{\mathbf{e}} \times \tilde{\mathbf{h}}^*\right) \cdot \hat{\mathbf{n}} \, d\Gamma = 1~\text{W}, \]

where \(\Gamma\) is the port surface, \(\tilde{\mathbf{h}}\) is the magnetic field of the mode (A/m), and \(^*\) is the complex conjugate.

Boundaries of the port surface

The edges of the port surface get one of two conditions in the eigenvalue problem:

  • Wall (PEC): An edge that touches a boundary with a Perfect Electric Conductor, Surface Impedance, or other waveguide port condition gets a zero tangential electric field.
  • Open (PMC): All other edges get a perfect magnetic conductor condition. This applies, for example, at an edge that touches an Absorbing boundary or a symmetry plane.

Conductors that go through the port surface are also walls. Examples are the inner conductor of a coaxial cable and the signal trace and ground planes of a coplanar waveguide. Thus, you do not need a closed metal wall around the port. Apply the PEC or surface impedance condition to these conductors.

Note

If two modes have almost the same propagation constant (a relative difference less than 1%), mufem writes a warning to the log. For such degenerate modes, the eigensolver can select any combination of the modes. Thus, mode-specific results can change between runs. An example is the pair of quasi-TEM modes of a coplanar waveguide.

With these assumptions, the waveguide input port boundary condition gives the equation below for the amplitude \(\tilde{\mathbf{E}}\) of the electric field at the port boundary:

\[ \hat{\mathbf{n}} \times \left(\frac{1}{\mu} \nabla \times \tilde{\mathbf{E}}\right) = j\frac{\beta}{\mu} \hat{\mathbf{n}} \times \left( \tilde{\mathbf{E}} \times \hat{\mathbf{n}} \right) -j\frac{2\beta}{\mu} \hat{\mathbf{n}} \times \left( \tilde{\mathbf{E}}^\text{in} \times \hat{\mathbf{n}} \right), \]

where \(\mu = \mu_0 \mu_r\) is the magnetic permeability and \(\mu_0\) is the vacuum permeability. For an anisotropic material, mufem replaces \(1/\mu\) with the reluctivity tensor \(\nu = \mu^{-1}\).

To create a WaveguideInputPortCondition, give a name, the boundary Marker, and the mode index (0-based) to launch from the input port:

condition = WaveguideInputPortCondition(
    name = "My Waveguide Input Port Condition",
    marker = my_marker,
    mode_index = 0,
)

After the solve, you can read the complex propagation constant \(\beta\) (rad/m) of the launched mode:

beta = condition.get_propagation_constant()

Typical industrial applications are the feed of a horn antenna from a rectangular waveguide, coaxial connectors, and coplanar waveguides on microwave and quantum circuit chips.

References

[1] J.-F. Lee, D.-K. Sun, and Z. J. Cendes, "Full-wave analysis of dielectric waveguides using tangential vector finite elements", IEEE Transactions on Microwave Theory and Techniques, vol. 39, no. 8, pp. 1262-1271, 1991

Usage Examples