Impedance Boundary Condition
The Impedance Boundary Condition provides a boundary condition that is useful at exterior boundaries, where the electromagnetic wave can penetrate only a short distance outside the boundary. This is usually the case when the material outside is a good conductor. The boundary condition approximates this penetration to avoid the need to include another domain in the model. The material properties that appear in the equation are those for the conductive material excluded from the model.
In the general case, penetration of electromagnetic waves through a conductive medium 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.
The tangential magnetic field and the surface current at the boundary are relates as
where n is an outer normal. In frequency domain, the surface current can be related to the tangential electric field using the following admittance form:
where the surface admittance is computed as
where ω = 2πf, and
is the wave number (complex-valued).
The skin depth can be computed as
For a good conductor, the expression simplifies into
For time domain and eigenfrequency computations, the surface admittance is 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.
Thus
where the skin depth is computed at a certain reference frequency ωc. The current is computed then as
where the auxiliary electric fields satisfy the equations
Impedance Boundary Condition
In this section, the following material properties can be defined for the domain outside the boundary, which this boundary condition approximates. The default use the values From material. 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)
Time Domain and Eigenfrequency
Use this section to configure and compute approximation of the surface admittance, which is needed for time domain and eigenfrequency computations.
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 the 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 at the Time Domain and Eigenfrequency section; this approximation computation will be done automatically at the start of a time-dependent or an eigenfrequency solver.