Hierarchical Electromagnetics
EmCAD – Cloud-native EM Simulator for Circuit-Level Modeling
Dispersive Materials
General Framework
EmCAD treats dispersive media using an innovative circuit-based representation. Unlike traditional simulators that embed frequency-dependent material models into Maxwell’s equations via convolution or auxiliary differential equations, EmCAD works at the circuit level.
The tetrahedral mesh of the domain generates a lumped equivalent circuit with capacitive and inductive elements. In the form used for model reduction, inductive behavior is captured using capacitors paired with impedance inverters. As a result, the equivalent circuit consists entirely of “direct” capacitors (representing electric fields) and “inverted” capacitors (representing magnetic fields).
Material Parameter Localization
In the circuit structure, the values of the direct capacitive elements are proportional to the permittivity of the material, while the inverted ones are proportional to its permeability. All dependencies of the electromagnetic response on the material properties are localized in these two sets of coefficients.
Dispersive behavior is introduced by replacing each direct capacitor with a small subcircuit that implements a frequency-dependent permittivity. All substitutions occur before the model order reduction step, ensuring that the final reduced circuit faithfully represents the dispersive behavior of the original domain.
EmCAD supports a wide class of dispersive models, including Debye, Lorentz, Drude, and generalized multi-pole models.
EmCAD also allows modeling of magnetic permeability dispersion. This is implemented in a fully symmetric way: all rational function models used for electric permittivity (such as Drude, Lorentz, or Debye) can be applied to the magnetic permeability. While these models were historically introduced for permittivity, they can equally characterize the frequency dependence of permeability, enabling accurate modeling of magnetically dispersive media.
Nonlinear Materials Based on D(E) Relationship
Unlike general nonlinear devices (e.g., varactors), the class of nonlinearities addressed here stems from the constitutive relation between electric displacement D and electric field E. In the linear case, this relation reduces to a constant permittivity. In dispersive media, it becomes frequency-dependent, and is realized through a rational admittance replacing the original capacitance.
For nonlinear materials such as Kerr media, the D(E) relation remains local and functional but becomes nonlinear. EmCAD can represent this nonlinearity exactly by replacing the direct capacitances with nonlinear admittance elements that directly implement the nonlinear D(E) behavior. This yields a nonlinear equivalent circuit that, apart from the usual discretization errors, faithfully models the nonlinear electromagnetic response of the structure.