Lossy Boundary Conditions

Modeling Imperfect Conductors

In many practical microwave and photonic applications, boundaries cannot be idealized as perfect electric conductors (PEC). Surface losses due to finite conductivity must be accounted for — especially at high frequencies or when dealing with unconventional materials such as resistive sheets or thin metallic coatings.

Equivalent Circuit Representation via Inverter Mapping

In EmCAD, lossy boundaries are modeled by introducing effective impedances, but these are not directly applied to the tangential electric fields of the surface triangles. Instead, the treatment proceeds through an intermediate step: a network of impedance inverters is used to interface the electric fields with scalar circuit elements that embed the correct frequency-dependent impedance behavior.

This transformation ensures a physically consistent and topologically compatible representation within the lumped-element model. The inverter network serves to project the boundary interaction onto a circuital form, where each transformed impedance captures the intended response while being compatible with the global nodal formulation.

Surface Impedance Models: Causal Huray and Gradient-Based

EmCAD supports two physically motivated models for characterizing conductive surface losses. The first is the Causal Huray Model, which accounts for microscopic current crowding effects due to surface texture. It yields a frequency-dependent surface impedance that respects causality.

More recently, EmCAD has adopted the Gradient Model by Gold, which is now often preferred due to its simplicity and physical clarity. It depends only on two input parameters: bulk conductivity and surface roughness. Both models produce an impedance function of frequency that characterizes surface losses across a wide range.

From Impedance Function to Circuit Realization

Once the surface impedance function is specified (for either model), it is sampled across a set of frequencies and approximated using a rational fit via Vector Fitting. The resulting rational function is then realized as a compact equivalent circuit made of lumped elements. This circuit is inserted into the inverter-mapped formulation, thereby implementing the lossy boundary condition in a passive, causal, and circuit-compatible manner.

Integration with Model Order Reduction

Importantly, the substitution of lossy boundary impedances with rational circuit realizations occurs before model order reduction (MOR). This ensures that frequency-dependent dissipation effects are accurately preserved during the compression process. Since the added elements affect only a small portion of the overall circuit (i.e., near boundary tetrahedra), they contribute minimally to the computational cost of the reduced model.

As a result, EmCAD produces compact yet accurate reduced-order models that include realistic boundary loss mechanisms, enabling the simulation of high-frequency structures with imperfect conductors.

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