Transition Boundary Condition
The Transition Boundary Condition is used on interior boundaries to model a conductive layer of a medium that should be geometrically thin but does not have to be electrically thin.
The theory is based on the assumption that the wave propagates in the normal direction in the thin layer. Thus, the wave could be incident in the normal direction or the wave could be refracted to propagate in a direction close to the normal direction. The latter condition is fulfilled for a good conductor. In addition, the thickness of the layer should also be less than the radius of curvature for the boundary, and the material properties in the thin layer are assumed to be isotropic.
In the general case, transmission of electromagnetic waves through a layer of conductive material is a complex process that can be significantly frequency dependent. Thus, modeling of such phenomenon in time domain needs to be based on the frequency domain representation.
In frequency domain, it is described by a relation between the electric field discontinuity and the induced surface current density:
where indices 1 and 2 refer to the different sides of the layer, and the surface and transfer admittances are written as
where the surface and transfer impedances are found as
where ω = 2πf, and
is the wave number (complex-valued).
The above relations can be simplified depending of the ratio of the layer thickness ds and the skin depth δ(ω) that can be calculated as
For a good conductor, the expression simplifies into
For time domain and eigenfrequency computations, the surface and transfer admittances are approximated using the partial fraction fit. The frequency-dependent relations for the surface currents are replaced by a number of distributed ODE solved in time domain for auxiliary variables that represent time-dependent contributions to the tangential electric fields.
Transition Boundary Condition
In the Transition Boundary Condition section, you can select the boundary Type
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Electrically very thin layer
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Electrically thin layer (default)
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Electrically thick layer
The Electrically thick layer option decouples the two domains adjacent to the boundary. This setting is suitable, for example, when the thickness exceeds several times the skin depth. The condition becomes effectively equivalent to two Impedance Boundary Condition which are applied independently on two sides of the layer. This option can be also considered as a high-frequency case, since the skin depth becomes smaller as the frequency increases.
If the thickness is significantly smaller than the skin depth, select Electrically very thin layer. The tangential electric field is assumed to be continuous over the boundary in this case. This option can be also considered as a low-frequency case, as the skin depth becomes larger as the frequency decreases.
Enter the following properties for the layer material, which this boundary condition approximates:
The defaults use the values From material, taking the properties from the material specified for the boundary. For User defined, enter different values or expressions.
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Relative permittivity, εrb (dimensionless)
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Relative permeability, μrb (dimensionless)
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Electric conductivity, σb (SI unit: S/m)
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Surface thickness, ds (SI unit: m)
However, notice that only isotropic (scalar) material parameters are supported for this boundary condition.
For time domain and eigenfrequency computations, the approximation for the layer admittance is precomputed for Electrically very thin layer case to be valid when
where δ is the skin depth.
For Electrically thin layer, it is precomputed to cover the range:
For Electrically thick layer the approximation needs to be computed using the Time Domain and Eigenfrequency section.
Time Domain and Eigenfrequency
This section becomes available only in case of Electrically thick layer. Use this section to configure and compute approximation of the surface admittance, which is needed for time domain and eigenfrequency computations. For other layer types, computations using time dependent or eigenfrequency study will work automatically, and they do not require any further inputs.
For Electrically thick layer, you need to specify the range of validity of the approximation because this layer type can represent a high-frequency case. For Frequency range, you can specify either Bandwidth and center (default) or Minimum and maximum.
Enter either
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Center frequency, fc (SI unit: Hz)
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Bandwidth (decades), nd (dimensionless)
or alternatively
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Minimum frequency, fmin (SI unit: Hz)
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Maximum frequency, fmax (SI unit: Hz)
In both cases, you can specify the approximation Accuracy. Note that both higher accuracy and wider frequency range can require using more auxiliary variables together with the corresponding ODEs to be solved.
The approximation computation needs to be done as a preprocessing step. Use the Compute approximation () action button available in the upper-right corner of Time Domain and Eigenfrequency section. The approximation computation is quick, and it will also compute and show the skin depth value at the center frequency. Once the computation has been performed, you can preview it using the Preview plot () action button that will become active at this section. You can also check the Show approximation data checkbox to inspect the actual computed values.
It is advised to compute and inspect the approximation quality manually prior to any lengthy calculations. However, if you do not compute the approximation, COMSOL Multiphysics will attempt to compute it using all information available in the Time Domain and Eigenfrequency section; this approximation computation will be done automatically at the start of a time-dependent or an eigenfrequency solver.