Numerical Modeling of Electromagnetic Problems Using EM.Cube

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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

The table below compares EM.Cube's computational modules and its simulation engines with regards to modeling accuracy, frequency limitations and the type of numerical solution they offer:

Module Name	Simulation Engine(s)	Modeling Accuracy	Solver Type	Frequency Range	Fundamental Solution	Applications
EM.Tempo	FDTD	Full-wave	Volumetric solver	Ultra-wideband time-domain	Electric and magnetic fields in the entire domain	General-purpose field simulator capable of handling complex geometrical and material variations
EM.Terrano	SBR	Asymptotic	Ray Tracer	High-frequency harmonic	Electric field ray tubes & power received at receiver locations	Radio wave propagation in very large scenes
EM.Illumina	GO-PO & IPO	Asymptotic	Surface solver	High-frequency harmonic	Electric and magnetic currents on surfaces	Scattering from very large surface structures & antenna-platform combinations
EM.Ferma	FD	Static or quasi-static	Volumetric solver	DC or low-frequency	Electric or magnetic fields in the entire domain	Small-scale devices and structures
EM.Picasso	MPIE	Full-wave	Planar solver	Arbitrary harmonic	Electric and magnetic currents on traces	Multilayer planar circuits, antennas & arrays, FSS, homogeneous substrates
EM.Libera	WMOM & SMOM	Full-wave	Wire & surface solvers	Arbitrary harmonic	Electric and magnetic currents on surfaces or wires	Radiation and scattering problems involving metals and homogeneous dielectric materials

The table below compares EM.Cube's computational modules with regards to their computational domain type, boundary conditions, mesh type and solver type:

Module Name	Domain Type	Domain Boundary Conditions
EM.Tempo	Finite box	PEC, PMC, PML
EM.Terrano	Open-boundary free space with optional half-space ground	Radiation BC
EM.Illumina	Open-boundary free space	Radiation BC
EM.Ferma	Finite box	Dirichlet & Neuman
EM.Picasso	Open-boundary with multilayer background medium	Radiation BC
EM.Libera	Open-boundary free space	Radiation BC

Module Name	Mesh Type
EM.Tempo	Adaptive and fixed-cell volumetric brick mesh
EM.Terrano	Triangular facet mesh
EM.Illumina	Triangular surface mesh
EM.Ferma	Fixed-cell volumetric brick mesh
EM.Picasso	Hybrid rectangular-triangular surface mesh
EM.Libera	Wireframe and triangular surface mesh

The table below compares EM.Cube's computational modules with regards to their material variety, excitation source type and lumped device type:

Module Name	Material Capability
EM.Tempo	PEC, PMC, dielectric, anisotropic, dispersive, complex materials
EM.Terrano	Material surfaces, thin walls and material volumes
EM.Illumina	PEC, PMC, impedance surfaces
EM.Ferma	PEC, dielectric or magnetic materials
EM.Picasso	PEC and slot traces, short vias, infinite substrate layers
EM.Libera	PEC, homogeneous dielectric

Module Name	Excitation/Sources	Lumped Devices
EM.Tempo	Lumped and distributed sources, plane wave, Gaussian beam, arbitrary waveform	Passive and active, linear and nonlinear devices and circuits
EM.Terrano	Transmitters, Hertzian dipoles	N/A
EM.Illumina	Hertzian dipole, plane wave, Huygens source	N/A
EM.Ferma	Charge, current and permanent magnet	N/A
EM.Picasso	Gap source, wave port, Hertzian dipole, plane wave, Huygens source	Simple passive RLC lumped elements
EM.Libera	Gap source, Hertzian dipole, plane wave, Huygens source	Simple passive RLC lumped elements

The table below compares EM.Cube's computational modules with regards to their observable types and the general types of applications they are suitable for:

Module Name	Observables
EM.Tempo	Near-field, far-field, RCS, periodic R/T, S/Z/Y parameters, port current/voltage/power
EM.Terrano	Far-field & received power
EM.Illumina	Far-field & RCS
EM.Ferma	Electric or magnetic field & potential, voltage, current, energy, power
EM.Picasso	Current distribution, far-field, periodic R/T, S/Z/Y parameters
EM.Libera	Current distribution, far-field, RCS, S/Z/Y parameters

The Material Composition of the Physical Structure

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

A physical structure in EM.Cube in made up of a number of geometric objects you either draw in the project workspace or import from an external CAD model file. In CubeCAD, geometric objects are simply grouped together by their color. They do not have any physical properties. However, in all of EM.Cube's computational modules, you need to assign physical properties to each geometric object. In these modules, geometric objects are grouped together by their common physical properties as well as their color. The types of physical properties may differ in different computational modules, but they are typically related to the material properties or boundary conditions.

From an electromagnetic modeling point of view, materials are categorized by the constitutive relations or boundary conditions that relate electric and magnetic fields. 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} = \epsilon^{\prime}_r (1 - j \tan \delta ) [/math]

[math] \mu_r = \mu^{\prime}_r -j\mu^{\prime\prime}_r = \mu^{\prime}_r - j\frac{\sigma_m}{\omega \mu_0} = \mu^{\prime}_r (1 - j \tan \delta_m)[/math]

where ω = 2πf, and f is the operational frequency, and the electric and magnetic loss tangents are defined 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 are frequently encountered in electromagnetic problems:

Medium	εr	μr	σ	σm
Free Space	1.0	1.0	0.0	0.0
Perfect Electric Conductor (PEC)	1.0	1.0	∞	0.0
Perfect Magnetic Conductor (PMC)	1.0	1.0	0.0	∞

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
Thin Wire	EM.Tempo, 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

Exciting a Physical Structure Using Sources or Devices in EM.Cube

In order to perform an electromagnetic simulation in any of EM.Cube's computational modules, you need to excite your physical structure using some kind of source. In most cases, you can define more than one source if necessary. In EM.Tempo, a source pumps energy into your FDTD computational domain in the form of a temporal waveform varying as a function time. In the MoM-based modules, EM.picasso and EM.Libera, a source provides the "right-hand-side (RHS)" vector of the MoM linear system resulting from the integral equation formulation of your boundary value problem. In EM.Illuumina, a source is used to illuminate your surfaces. In EM.Terrano, a source acts as a transmitter that launches the broadcast signal into the free space. In EM.Ferma, you need either an electric or magnetic source to set the boundary conditions for the Laplace equation or provide the source term for the Poisson equation.

In each module, you should choose the right source type depending on the purpose of your simulation and based on the observables you define for your project. For example, for computing the radar cross section (RCS) of a target, you need a plane wave source. If you are interested in computing the S/Z/Y parameters of your structure, then you have to choose a source type like a gap or lumped source that supports a "Port Definition" observable.

Current distribution on a metallic plate excited by a plane wave source.

Current distribution on a metallic plate excited by a short horizontal dipole source above it.

EM.Cube provides a large variety of source types listed in the table below:

Source Type	Supporting Module(s)
Lumped Source	EM.Tempo
Waveguide Source	EM.Tempo
Distributed Source	EM.Tempo
Gap Source	EM.Picasso, EM.Libera
Probe Source	EM.Picasso
De-embedded Source	EM.Picasso
Hertzian Dipole Source	EM.Tempo, EM.Illumina, EM.Picasso, EM.Libera, EM.Terrano
Plane Wave Source	EM.Tempo, EM.Illumina, EM.Picasso, EM.Libera
Gaussian Beam Source	EM.Tempo
Huygens Source	EM.Illumina, EM.Picasso, EM.Libera
Transmitter Set	EM.Terrano
Fixed-Potential PEC with Nonzero Voltage	EM.Ferma
Volume Charge Source	EM.Ferma
Wire Current Source	EM.Ferma
Volume Current Source	EM.Ferma
Permanent Magnet with Nonzero Magnetization	EM.Ferma

Defining Simulation Observables in EM.Cube

In most of EM.Cube's computational modules, you have to define one or more observables to generate any output data at the end of a simulation. In other words, no simulation data is generated by itself. EM.Cube provides a large variety of simulation data and observable types as listed in the table below:

Simulation Data Type	Required Observable	Supporting Module(s)
Electric and Magnetic Field Distributions	Field Sensor	EM.Tempo, Em.Terrano, EM.Illumina, EM.Ferma, EM.Picasso, EM.Libera
Electric and Magnetic Current Distributions	Current Distribution	EM.Illumina, EM.Picasso, EM.Libera
Temporal Fields	Field Probe	EM.Tempo
Far-Field Radiation Patterns	Far Fields - Radiation Pattern	EM.Tempo, Em.Terrano, EM.Illumina, EM.Picasso, EM.Libera
Radar Cross Section (RCS)	Far Fields - RCS	EM.Tempo, EM.Illumina, EM.Picasso, EM.Libera
Huygens Surface Data	Huygens Surface	EM.Tempo, Em.Terrano, EM.Illumina, EM.Picasso, EM.Libera
Port Characteristics (S/Z/Y Parameters)	Port Definition	EM.Tempo, EM.Picasso, EM.Libera
Periodic Reflection and Transmission Coefficients	No Observables Required	EM.Tempo, EM.Picasso
Temporal Electric and Magnetic Energy	Domain Energy	EM.Ferma
Static Electric and Magnetic Energy & Ohmic Losses	Field Integral	EM.Ferma
Voltage and Current	Field Integrals	EM.Ferma
Electric and Magnetic Flux	Field Integrals	EM.Ferma
Resistance, Capacitance, Inductance	Field Integrals	EM.Ferma
Received Power	Receiver Set	EM.Terrano
Singal-to-Noise Ratio (SNR)	Receiver Set	EM.Terrano
Channel Path Loss	Receiver Set	EM.Terrano

Discretizing a Physical Structure Using a Mesh Generator in EM.Cube

EM.Cube's computational modules use a number of different mesh generation schemes to discretize your physical structure. Even CubeCAD provides several tools for object discretization. In general, all of EM.Cube's mesh generation schemes can be grouped into three categories representing their dimensionality:

1. Linear Mesh
2. Surface Mesh
3. Volume Mesh

The linear mesh is also known as wireframe mesh and is used by EM.Libera to discretize the physical structure for Wire MoM simulation. EM.Cube offer two surface mesh types: triangular surface mesh and hybrid surface mesh. As its name implies, a triangular surface mesh is made up of interconnected triangular cells. EM.Terrano, EM.Illumina, EM.Libera and EM.Picasso all use triangular surface mesh generators to discretized surface CAD objects and the surface of solid CAD objects. The hybrid surface mesh is EM.Picasso's default mesh. It combines rectangular and triangular cells to discretize planar structures. The hybrid surface mesh generator tries to produce as many identical rectangular cells as possible in rectangular regions of your planar structure.

EM.Cube provides two types of brick meshes to discretize the volume of your computational domain. Brick meshes are entire-domain volume meshes and are made up of cubic cells. Brick meshes are indeed generated by a three-dimensional arrangement of grid lines along the X, Y and Z dimensions. EM.Tempo offers an Adaptive brick mesh as well as a fixed-cell brick mesh for the FDTD simulation of your physical structure. EM.Ferma offers only a fixed-mesh brick mesh for the solution of electrostatic and magnetostatic Poisson equations.

The geometry of a metallic torus.

The brick volume mesh of the metallic torus.

The triangular surface mesh of the metallic torus.

The Significance of Mesh Resolution

The objects of your physical structure are discretized based on a specified mesh density. The default mesh densities of EM.Tempo, EM.Picasso, EM.Libera and EM.Illumina are expressed as the number of cells per effective wavelength. Therefore, the resolution of the default mesh in these modules are frequency-dependent. You can also define the mesh resolution using a fixed cell size or fixed edge length specified in project units. The mesh density of EM.Terrano is always expressed in terms of cell edge length. The mesh resolution of EM.Ferma is always specified as the fixed cell size. All of EM.Cube's computational modules have default mesh settings that usually work well for most simulations.

The accuracy of the numerical solution of an electromagnet problem depends greatly on the quality and resolution of the generated mesh. As a rule of thumb, a mesh density of about 10-30 cells per effective wavelength usually yields satisfactory results. Yet, for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required. Also, the particular simulation data that you seek in a project also influence your choice of mesh resolution. For example, far field characteristics like radiation patterns are less sensitive to the mesh density than the near-field distributions on a structure with a highly irregular shape and a rugged boundary.