Numerical Modeling of Electromagnetic Problems Using EM.Cube

Computational Electromagnetics

The electric field excited above a battleship illuminated by a plane wave source.

Mathematically speaking, all electromagnetic modeling problems require solving some form of Maxwell's equations in conjunction with certain initial and boundary conditions. Radiation and scattering problems are defined over an unbounded domain. Circuit and device problems are often formulated as shielded structures within finite domains. Aside from a few well-known canonical problems, there are no closed-form solutions available for most electromagnetic problems due to the complexity of their domains and boundaries. Numerical analysis, therefore, is the only way to solve such problems.

Click here for a brief review of Maxwell's Equations.

Using a numerical method to solve a certain electromagnetic modeling problem typically involves a recurring sequence of steps:

• Geometrical construction of the physical structure and material assignments
• Definition of the computational domain and boundary conditions
• Definition of excitation sources
• Definition of observables
• Geometrical reduction and mesh generation

The above steps reduce your original physical problem to a numerical problem, which must be solved using an appropriate numerical solver. Verifying and benchmarking different techniques in the same simulation environment helps you better strategize, formulate and validate a definitive solution.

A ubiquitous question surfaces very often in electromagnetic modeling: "Does one really need more than one simulation engine? A true challenge of electromagnetic modeling is the right choice of numerical technique for any given problem. Depending on the electrical length scales and physical nature of your problem, some modeling techniques may provide more accurate or computationally more efficient solutions than the others. Full-wave techniques provide the most accurate solution of Maxwell's equations in general. In the case of very large-scale problems, asymptotic methods sometimes offer the only practical solution. On the other hand, static or quasi-static methods may provide more stable solutions for extremely small-scale problems. Having access to multiple simulation engines in a unified modeling environment provides many advantages beyond getting the best solver for your particular problem. Some complex problems involve dissimilar length scales which cannot be compromised in favor of one or another. In such cases, a hybrid simulation using different techniques for different parts of the larger problem can lead to a reasonable solution.

An Overview of EM.Cube's Numerical Solvers

EM.Cube uses a number of computational electromagnetic (CEM) techniques to solve your modeling problems. All of these techniques are based on a fine discretization of your physical structure into a set of elementary cells or elements. A discretized form of Maxwell's equations or some variations of them are then solved numerically over these smaller cells. From the resulting numerical solution, the quantities of interest are derived and computed.

The numerical techniques used by EM.Cube are:

• Finite Different Time Domain (FDTD) method
• Shoot-and-Bounce-Rays (SBR) method
• Physical Optics (PO) method: Geometrical Optics - Physical Optics (GO-PO) method and Iterative Physical Optics (IPO) emthod
• Mixed Potential Integral Equation (MPIE) method for multilayer planar structures
• Wire Method of Moments (WMOM) based on Pocklington integral equation
• Surface Method of Moments (SMOM) with Adaptive Integration Equation (AIM) accelerator
• Finite Difference (FD) method solution of electrostatic and magnetostatic Laplace/Poisson equations

Module Name	Simulation Engine(s)	Solver Type	Modeling Accuracy	Frequency Range	Fundamental Solution
EM.Tempo	FDTD	Volumetric solver	Full-wave	Ultra-wideband time-domain	Electric and magnetic fields in the entire domain
EM.Terrano	SBR	Ray tracer	Asymptotic	High-frequency harmonic	Electric field ray tubes and power received at receiver locations
EM.Illumina	GO-PO & IPO	Surface solver	Asymptotic	High-frequency harmonic	Electric and magnetic currents on surfaces
EM.Ferma	FD	Volumetric solver	Static or quasi-static	DC or low-frequency	Electric or magnetic fields in the entire domain
EM.Picasso	MPIE	Planar solver	Full-wave	Arbitrary harmonic	Electric and magnetic currents on traces
EM.Libera	WMOM & SMOM	Surface & wire solvers	Full-wave	Arbitrary harmonic	Electric and magnetic currents on surfaces or wires

Module Name	Material Capability	Excitation/Sources	Lumped Devices
EM.Tempo	PEC, PMC, dielectric, anisotropic, dispersive, complex materials	Lumped and distributed sources, plane wave, Gaussian beam, arbitrary waveform	Passive and active, linear and nonlinear devices and circuits
EM.Terrano	Material surfaces, thin walls and material volumes	Transmitters, Hertzian dipoles	N/A
EM.Illumina	PEC, PMC, impedance surfaces	Hertzian dipole, plane wave, Huygens source	N/A
EM.Ferma	PEC, dielectric or magnetic materials	Charge, current and permanent magnet	N/A
EM.Picasso	PEC and slot traces, short vias, infinite substrate layers	Gap source, wave port, Hertzian dipole, plane wave, Huygens source	Simple passive RLC lumped elements
EM.Libera	PEC, homogeneous dielectric	Gap source, Hertzian dipole, plane wave, Huygens source	Simple passive RLC lumped elements

Module Name	Observables	Applications
EM.Tempo	Near-field, far-field, RCS, periodic R/T, S/Z/Y parameters, port current/voltage/power	General-purpose field simulator capable of handling complex geometrical and material variations
EM.Terrano	Far-field & received power	Radio wave propagation in very large scenes
EM.Illumina	Far-field & RCS	Scattering from very large surface structures & antenna-platform combinations
EM.Ferma	Electric or magnetic field & potential, voltage, current, energy, power	Small-scale devices and structures
EM.Picasso	Current distribution, far-field, periodic R/T, S/Z/Y parameters	Multilayer planar circuits, antennas & arrays, FSS, homogeneous substrates
EM.Libera	Current distribution, far-field, RCS, S/Z/Y parameters	Radiation and scattering problems involving metals and homogeneous dielectric materials

The Composition of Physical Structure

A structure made up of a PEC plate and different dielectric materials.

Your Physical Structure in EM.Cube consists of a number of CAD objects you draw in the project workspace. In all EM.Cube modules, you use CubeCAD's common drawing tools and/or its import capability to construct your geometrical structure. In order to perform an electromagnetic simulation, you need to assign material properties to all of your CAD objects in the project workspace. The drawn CAD objects are organized together based on their common properties under one or more object groups or nodes in the Navigation Tree. The grouping of CAD objects is essentially based on their material composition and their associated boundary conditions.

Among EM.Cube's computational modules, EM.Tempo is the most comprehensive in view of material variety offering. EM.Ferma and EM.Libera offer PEC and dielectric material groups similar to EM.Tempo. EM.Illumina offers PEC, PMC and impedance surfaces. EM.Terrano groups objects into material block types that are characterized by their ray interaction properties. EM.Picasso groups objects based on their trace location (i.e. Z-coordinate) in the substrate layer hierarchy.

From an electromagnetic analysis point of view, materials are categorized by the constitutive relations or boundary conditions that relate electric and magnetic fields. EM.Cube offers a large variety of material types listed in the table below:

Material Type	Supporting Module(s)
Perfect Electric Conductor (PEC)	EM.Tempo, EM.Illumina, EM.Ferma, EM.Picasso, EM.Libera
Perfect Magnetic Conductor (PMC)	EM.Tempo, EM.Illumina, EM.Picasso
Dielectric	EM.Tempo, EM.Ferma, EM.Picasso, EM.Libera, EM.Terrano
Impedance Surface	EM.Illumina
Conductive Sheet	EM.Picasso
Anisotropic Material	EM.Tempo
Dispersive Material	EM.Tempo
Inhomogeneous Material	EM.Tempo
Thin Wire	EM.Tempo, EM.Libera

Constitutive Parameters of a Material Medium

In general, an isotropic material medium is macroscopically characterized by four constitutive parameters:

• Permittivity (ε) having units of F/m
• Permeability (μ) having units of H/m
• Electric conductivity (σ) having units of S/m
• Magnetic conductivity (σ_m) having units of Ω/m

The permittivity and permeability of a material are typically related to the permittivity and permeability of the free space as follows:

[math] \epsilon = \epsilon_r \epsilon_0 [/math]

[math] \mu = \mu_r \mu_0, \quad \quad [/math]

where ε₀ = 8.854e-12 F/m, μ_r = 1.257e-6 H/m, and ε_r and μ_r are called relative permittivity and permeability of the material, respectively.

The constitutive parameters relate the field quantities in the material medium:

[math] \mathbf{D} = \epsilon \mathbf{E}, \quad \quad \mathbf{J} = \sigma \mathbf{E} [/math]

[math] \mathbf{B} = \epsilon \mathbf{H}, \quad \quad \mathbf{M} = \sigma_m \mathbf{H} [/math]

where E and H are the electric and magnetic fields, respectively, D is the electric flux density, also known as the electric displacement vector, B is the magnetic flux density, also known as the magnetic induction vector, and J and M are the electric and magnetic current densities, respectively.

The electric conductivity and magnetic conductivity parameters represent the material losses. In frequency-domain simulations under a time-harmonic (e^jωt) field assumption, it is often convenient to define a complex relative permittivity and a complex relative permeability in the following manner:

[math] \epsilon_r = \epsilon^{\prime}_r -j\epsilon^{\prime\prime}_r = \epsilon^{\prime}_r -j\frac{\sigma}{\omega \epsilon_0} [/math]

[math] \mu_r = \mu^{\prime}_r -j\mu^{\prime\prime}_r = \mu^{\prime}_r - j\frac{\sigma_m}{\omega \mu_0}[/math]

where ω = 2πf, and f is the operational frequency. It is also customary to define electric and magnetic loss tangents as follows:

[math] \tan \delta = \epsilon^{\prime\prime}_r / \epsilon^{\prime}_r [/math]

[math] \tan \delta_m = \mu^{\prime\prime}_r / \mu^{\prime}_r [/math]

Three special media frequently encountered in electromagnetic problems are:

• Vacuum or Free Space: ε_r = μ_r = 1 and σ = σ_m = 0
• Perfect Electric Conductor (PEC): ε_r = μ_r = 1, σ = ∞, σ_m = 0
• Perfect Magnetic Conductor (PMC): ε_r = μ_r = 1, σ = 0, σ_m = ∞