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<td>[[image:Cube-icon.png | link=Getting_Started_with_EM.CUBECube]] [[image:cad-ico.png | link=Building Geometrical Constructions in CubeCAD]] [[image:fdtd-ico.png | link=EM.Tempo]] [[image:prop-ico.png | link=EM.Terrano]] [[image:postatic-ico.png | link=EM.IlluminaFerma]] [[image:staticplanar-ico.png | link=EM.FermaPicasso]] [[image:planarmetal-ico.png | link=EM.PicassoLibera]] [[image:metalpo-ico.png | link=EM.LiberaIllumina]] </td>

[[Image:Back_icon.png|40px30px]] '''[[EM.Cube | Back to EM.Cube Main Page]]'''== An Overview of Computational Electromagnetics ==

[[Image:ship_image1.png|thumb|500px|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|Maxwell's equations]] in conjunction with subject to 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.

The numerical techniques used in computational electromagnetics (CEM) are generally divided into three categories: * '''Full-Wave Techniques''': These techniques solve the dynamic form of Maxwell's equations either in the time domain or in the frequency domain. They typically require a very fine discretization of the physical structure with a frequency-dependent mesh resolution. Some of the most well-know methods in this category include the finite difference time domain (FDTD) method, the method of moments (MoM) and the finite element method (FEM). Full-wave techniques provide the highest level of modeling accuracy among CEM methods. * '''Quasi-Static Techniques''': These techniques assume static DC or low-frequency conditions, which ignore wave retardation effects. Under these conditions, Maxwell's equations reduce to the electrostatic or magnetostatic forms of Laplace/Poisson equations. These methods are effective in solving lumped devices or structures with small electrical dimensions. * '''Asymptotic Techniques''': These techniques assume quasi-optical or high-frequency conditions, and solve the asymptotic forms of Maxwell'e equations. These methods are effective in solving structures or scenes with very large electrical dimensions. All the ray tracing techniques like the shoot-and Bounce-Rays (SBR) method fall into this category. Another example is the physical optics (PO) method.  [[Image:Info_icon.png|40px30px]] Click here for a brief review of '''[[A Review of Maxwell's Equations& Computational Electromagnetics (CEM)]]'''.  <table><tr><td> [[Image:ART GOLF Fig title.png|thumb|left|450px|The modified radiation pattern of a patch antenna installed on the front hood of a Volkswagen Golf automobile computed using the full-wave FDTD technique.]]</td> </tr></table> == Typical Steps of Computer Simulation of an Electromagnetic Problem ==

*#Geometrical construction of the physical structure and material assignments*#Material assignment to geometric objects#Definition of the computational domain and boundary conditions*#Definition of excitation sources*#Definition of simulation observables*#Geometrical reduction and mesh generation#Running the numerical solver#Post-processing and visualization of the output data

The above steps reduce first transform your original physical modeling problem to into a numerical computational 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 asked in conjunction with electromagnetic modelingis: "Does one really need more than one simulation engine? " Different numerical techniques have different strengths and weaknesses with respect to modeling versatility and breadth of scope, modeling accuracy and computational efficiency. There is no single numerical technique that can solve all the electromagnetic problems at all frequencies and involving all length scales from microns to miles. A true challenge of electromagnetic modeling is the right choice of numerical technique for any given problem. Depending on the electrical length scales and the physical nature of your problem, some modeling techniques may provide a more accurate or computationally more efficient solutions solution than the others. Full-wave techniques provide the most accurate solution of [[Maxwell's Equations|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 a 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. In addition, verifying and benchmarking different solvers in the same simulation environment helps you better strategize, formulate and validate a definitive solution.

== An Overview A Functional Comparison 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|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.

* ShootShooting-and-BounceBouncing-Rays (SBR) method* Physical Optics Finite Difference (POFD) method: Geometrical Optics for electrostatic, magnetostatic and steady- Physical Optics (GO-PO) method and Iterative Physical Optics (IPO) emthod state thermal Laplace/Poisson equations* Mixed Potential Integral Equation (MPIE) method for multilayer planar structuresalso known as Planar method of Moments (PMOM)

* Finite Difference Physical Optics (FDPO) method: Geometrical Optics - Physical Optics (GO-PO) method solution of electrostatic and magnetostatic Laplace/Poisson equationsIterative Physical Optics (IPO) method

 Of [[EM.Cube]]'s computational modules, [[EM.Tempo]] serves as a general-purpose electromagnetic simulator that can handle most types of modeling problems involving arbitrary geometries and complex material variations in both time and frequency domains. 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:

! scope| style="rowwidth:40px;"| [[image:fdtd-ico.png | link=[[EM.Tempo]]]]| style="width:80px;" | [[EM.Tempo]]| style="width:40px;" | [[Basic_FDTD_Theory Basic_Principles_of_The_Finite_Difference_Time_Domain_Method | FDTD]]| style="width:30px;" | Full-wave| style="width:90px;" | 3D volumetric solver| style="width:80px;" | Ultra-wideband time-domain | style="width:270px150px;" | Electric and magnetic fields in the entire domain | style="width:200px;" | General-purpose field simulator capable of handling complex geometrical and material variations

! scope| style="rowwidth:40px;"| [[image:prop-ico.png | link=[[EM.Terrano]]]]| style="width:80px;" | [[EM.Terrano]]| style="width:40px;" | [[SBR_Method Basic_Principles_of_SBR_Ray_Tracing | SBR]]| style="width:30px;" | Asymptotic| style="width:90px;" | 3D ray tracer| style="width:80px;" | High-frequency harmonic | style="width:270px150px;" | Electric field fields and ray tubes and power received at receiver locations| style="width:200px;" | Radio wave propagation in very large scenes

! scope| style="rowwidth:40px;"| [[image:static-ico.png | link=[[EM.IlluminaFerma]]]]| style="width:80px;" | [[Theory_of_Physical_Optics EM.Ferma]]| GO-PO & IPOstyle="width:40px;" | [[Electrostatic_%26_Magnetostatic_Field_Analysis | FD]]| Asymptoticstyle="width:30px;" | Static or quasi-static| Highstyle="width:90px;" | 3D volumetric solver| style="width:80px;" | DC or low-frequency harmonic | style="width:270px150px;" | Electric and or magnetic currents on surfacesfields or temperature in the entire domain | style="width:200px;" | Small-scale devices and structures

! scope| style="rowwidth:40px;"| [[image:planar-ico.png | link=[[EM.FermaPicasso]]]]| style="width:80px;" | [[Electrostatic_and_Magnetostatic_Methods EM.Picasso]]| style="width:40px;" | [[Basic_Principles_of_The_Method_of_Moments | FDMPIE (PMOM)]]| Static or quasistyle="width:30px;" | Full-staticwave| DC or low-frequency style="width:90px;" | 2.5D planar solver | style="width:270px80px;" | Arbitrary harmonic | style="width:150px;" | Electric or and magnetic fields in the entire domain currents on traces| style="width:200px;" | Multilayer planar circuits, antennas & arrays, FSS, homogeneous substrates

! scope| style="rowwidth:40px;"| [[image:metal-ico.png | link=[[EM.PicassoLibera]]]]| style="width:80px;" | [[EM.Libera]]| style="width:40px;" | [[Planar_Method_of_Moments Basic_Principles_of_The_Method_of_Moments | MPIEWMOM & SMOM]] | style="width:30px;" | Full-wave| style="width:90px;" | 3D wire & surface solvers | style="width:80px;" | Arbitrary harmonic | style="width:270px150px;" | Electric and magnetic currents on tracessurfaces or wires| style="width:200px;" | Radiation and scattering problems involving metals and homogeneous dielectric materials

! scope| style="rowwidth:40px;"| [[image:po-ico.png | link=[[EM.LiberaIllumina]]]]| style="width:80px;" | [[3D_Method_of_Moments EM.Illumina]]| WMOM style="width:40px;" | [[Basic_Principles_of_Physical_Optics | GO-PO & SMOMIPO]] | Full-wavestyle="width:30px;" | Asymptotic| Arbitrary style="width:90px;" | 3D surface solver| style="width:80px;" | High-frequency harmonic | style="width:270px150px;" | Electric and magnetic currents on surfaces or wires| style="width:200px;" | Scattering from very large surface structures & antenna-platform combinations

The table below compares A physical structure in [[EM.Cube]]'s computational modules with regards to their computational domain typeis typically made up of one or more geometric objects, boundary conditions, mesh type and solver type: which you either draw in its project workspace or import from an external CAD model file.

{| class="wikitable"|-! scope="col"| Module Name ! scope="col"| Domain Type ! scope="col"| Domain Boundary Conditions ! scope="col"| Mesh Type ! scope="col"| Solver Type|-! scope="row"| When you start the [[EM.TempoCube]]| style="width:240px;" | Finite box | style="width:140px;" | PECapplication, PMC, PML | style="width:240px;" | Adaptive and fixed-cell volumetric brick mesh | style="width:140px;" | Volumetric solver|-! scope="row"| you land in its [[EM.TerranoBuilding Geometrical Constructions in CubeCAD | CubeCAD]]| style="width:240px;" | Open-boundary free space with optional half-space ground | style="width:140px;" | Radiation BC| style="width:240px;" | Triangular facet mesh| style="width:140px;" | Ray Tracer|-! scope="row"| [[EMmodule by default.Illumina]]| style="width:240px;" | Open-boundary free space | style="width:140px;" | Radiation BC| style="width:240px;" | Triangular surface mesh| style="width:140px;" | Surface solver|-! scope="row"| [[EM.Ferma]]Building Geometrical Constructions in CubeCAD | style="width:240px;" | Finite box | style="width:140px;" | Dirichlet & Neuman | style="width:240px;" | Fixed-cell volumetric brick mesh| style="width:140px;" | Volumetric solver|-! scope="row"| [[EM.PicassoCubeCAD]]| style="width:240px;" | Open-boundary is a comprehensive, parametric, 3D CAD modeler along with multilayer background medium | style="width:140px;" | Radiation BC| style="width:240px;" | Hybrid rectangular-triangular surface integrated mesh| style="width:140px;" | Planar solver |-! scope="row"| [[EMgeneration, data processing and visualization capabilities and a powerful Python scripting environment.Libera]]| style="width:240px;" | OpenWith the click of a few buttons, you can build complex 3D models and structures in seconds using a large variety of intuitive mouse-boundary free space | style="width:140px;" | Radiation BC| style="width:240px;" | Wireframe based object creation and triangular surface meshtransformation tools. Import of standard CAD formats allows you to fly in external CAD models with utmost ease. Imported structures can easily be augmented with | style="width:140px;" | Wire & surface solvers |}your own geometrical constructions created using the native standard geometric objects.

The table below compares [[EMimage:cad-ico.Cubepng | link=Building Geometrical Constructions in CubeCAD]]'s computational modules with regards Click here to their material variety, excitation source type and lumped device type: learn more about '''[[Building Geometrical Constructions in CubeCAD | CubeCAD]]'''.

{| class="wikitable"<table>|-<tr>! scope="col"| Module Name ! scope="col"| Material Capability! scope="col"| Excitation/Sources ! scope="col"| Lumped Devices |-! scope="row"| <td> [[EM.Tempo]]| style="widthImage:270px;" | PEC, PMC, dielectric, anisotropic, dispersive, complex materials | style="width:270px;" | Lumped and distributed sources, plane wave, Gaussian beam, arbitrary waveform | style="width:270px;" | Passive and active, linear and nonlinear devices and circuits |-! scope="row"| [[EMchopper1.Terrano]]png| style="width:270px;" thumb| Material surfaces, thin walls and material volumesleft| style="width:270px;" 360px| Transmitters, Hertzian dipoles| style="width:270px;" | N/A|-! scope="row"| [[EMAn Apache helicopter.Illumina]]| style="width:270px;" | PEC, PMC, impedance surfaces| style="width:270px;" | Hertzian dipole, plane wave, Huygens source| style="width:270px;" | N</Atd>|-! scope="row"| <td> [[EM.Ferma]]| style="widthImage:270px;" | PEC, dielectric or magnetic materials | style="width:270px;" | Charge, current and permanent magnet| style="width:270px;" | N/A|-! scope="row"| [[EMCAD_Apache_new.Picasso]]png| style="width:270px;" thumb| PEC and slot traces, short vias, infinite substrate layers left| style="width:270px;" 360px| Gap source, wave port, Hertzian dipole, plane wave, Huygens source | style="width:270px;" | Simple passive RLC lumped elements |-! scope="row"| [[EMThe imported model of the Apache helicopter.Libera]]</td> | style="width:270px;" | PEC, homogeneous dielectric </tr>| style="width:270px;" | Gap source, Hertzian dipole, plane wave, Huygens source | style="width:270px;" | Simple passive RLC lumped elements |}</table>

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

! scope="col"| Observables ! scope="col"| Applications Geometry Types

! scope| style="rowwidth:40px;"| [[image:fdtd-ico.png | link=[[EM.Tempo]]]]| style="width:360px;" | Near-field, far-field, RCS, periodic R/T, S/Z/Y parameters, port current/voltage/power [[EM.Tempo]]| style="width:360px450px;" | General-purpose field simulator capable of handling complex geometrical and material variations solid, surface & curve objects

! scope| style="rowwidth:40px;"| [[image:prop-ico.png | link=[[EM.Terrano]]]]| style="width:360px;" | Far-field & received power [[EM.Terrano]]| style="width:360px450px;" | Radio wave propagation in very large scenesGeneral solid & surface objects - no curve objects

! scope| style="rowwidth:40px;"| [[image:static-ico.png | link=[[EM.IlluminaFerma]]]]| style="width:360px;" | Far-field & RCS[[EM.Ferma]]| style="width:360px450px;" | Scattering from very large General solid, surface structures & antenna-platform combinationscurve objects

! scope| style="rowwidth:40px;"| [[image:planar-ico.png | link=[[EM.FermaPicasso]]]]| style="width:360px;" | Electric or magnetic field & potential, voltage, current, energy, power [[EM.Picasso]]| style="width:360px450px;" | SmallPlanar surface objects only - no solid or curve objects (vias are automatically constructed from cross-scale devices and structures sectional planar objects as vertical prisms)

! scope| style="rowwidth:40px;"| [[image:metal-ico.png | link=[[EM.PicassoLibera]]]]| style="width:360px;" | Current distribution, far-field, periodic R/T, S/Z/Y parameters[[EM.Libera]]| style="width:360px450px;" | Multilayer planar circuits, antennas General solid & arrays, FSSsurface objects for Surface MoM solver, homogeneous substratesgeneral curve objects and wireframe structures for Wire MoM solver

! scope| style="rowwidth:40px;"| [[image:po-ico.png | link=[[EM.LiberaIllumina]]]]| style="width:360px;" | Current distribution, far-field, RCS, S/Z/Y parameters[[EM.Illumina]]| style="width:360px450px;" | Radiation and scattering problems involving metals and homogeneous dielectric materials General solid & surface objects - no curve objects

== The Material Composition of the Physical Structure ==

In [[Image:MAT1.png|thumb|400px|A structure made up of a PEC plate and different dielectric materials.]] A physical structure Building Geometrical Constructions 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 | CubeCAD]], geometric objects are simply grouped together simply by their color. They do not have any physical properties. However, in In all of [[EM.Cube]]'s computational modules, however, you need to assign physical properties to each geometric object. From an electromagnetic modeling point of view, the difference between a material block and a free-space region is the constitutive relations that govern the electric and magnetic fields in these media and/or their boundary conditions. In these [[EM.Cube]]'s computational modules, geometric objects are grouped together by their common physical propertiesas well as their color. The types of physical properties may differ in different computational modules, but they are typically the same as related to the material properties or related 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.

| style="width:320px;" | Perfect Electric Conductor (PEC)| [[EM.Tempo]], [[EM.IlluminaFerma]], [[EM.FermaPicasso]], [[EM.PicassoLibera]], [[EM.LiberaIllumina]]

| style="width:320px;" | Thin Wire

| style="width:320px;" | Perfect Magnetic Conductor (PMC)| [[EM.Tempo]], [[EM.IlluminaPicasso]], [[EM.PicassoIllumina]]

| style="width:320px;" | Perfect Thermal Conductor (PTC)| [[EM.Ferma]]|-| style="width:320px;" | Dielectric(or Magnetic) Material

| style="width:320px;" | Insulator Material| [[EM.Ferma]]|-| style="width:320px;" | Impedance Surface

| style="width:320px;" | Conductive Sheet

| style="width:320px;" | Anisotropic Material

| style="width:320px;" | Dispersive Material(Debye, Drude, Lorentz, Generalized Metamaterial)

| Inhomogeneous style="width:320px;" | Gyrotropic Material(Ferrite, Magnetoplasma)

In general{| class="wikitable"|-! scope="col"| ! scope="col"| Module Name ! scope="col"| Material Types|-| style="width:40px;" | [[image:fdtd-ico.png | link=[[EM.Tempo]]]] | [[EM.Tempo]]| style="width:450px;" | PEC, an isotropic thin wire, PMC, dielectric, anisotropic, dispersive, gyrotropic |-| style="width:40px;" | [[image:prop-ico.png | link=[[EM.Terrano]]]] | [[EM.Terrano]]| style="width:450px;" | Material surfaces, thin walls and material medium is macroscopically characterized by four constitutive parametersvolumes|-| style="width:40px;" | [[image:static-ico.png | link=[[EM.Ferma]]]] | [[EM.Ferma]]| style="width:450px;" | PEC, PTC, dielectric or magnetic or insulator materials |-| style="width:40px;" | [[image:planar-ico.png | link=[[EM.Picasso]]]] | [[EM.Picasso]]| style="width:450px;" | PEC and slot traces, short vias, infinite substrate layers |-| style="width:40px;" | [[image:metal-ico.png | link=[[EM.Libera]]]] | [[EM.Libera]]| style="width:450px;" | PEC, thin wire, homogeneous dielectric |-| style="width:40px;" | [[image:po-ico.png | link=[[EM.Illumina]]]] | [[EM.Illumina]]| style="width:450px;" | PEC, PMC, impedance surfaces|}

* Permittivity (&epsilon;) having units [[Image:Info_icon.png|30px]] Click here for a more detailed discussion of F/m* Permeability (&mu;) having units of H/m * Electric conductivity (&sigma;) having units of S/m* Magnetic conductivity (&sigma;_m) having units of &Omega;/m'''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Assigning_Material_Properties_to_the_Physical_Structure | Assigning Material Properties to the Physical Structure]]'''.

The permittivity and permeability of a material are typically related [[Image:Info_icon.png|30px]] Click here to the permittivity and permeability access '''[[Glossary of the free space as follows:EM.Cube's Materials, Sources, Devices & Other Physical Object Types]]'''.

:<math> \epsilon = \epsilon_r \epsilon_0 </math>= Defining the Computational Domain & Boundary Conditions in EM.Cube ==

The table below compares [[EM.Cube]]'s computational modules with regards to their computational domain and boundary condition types:<math> \mu = \mu_r \mu_0, \quad \quad </math>

where &epsilon{| class="wikitable"|-! scope="col"| ! scope="col"| Module Name ! scope="col"| Domain Type ! scope="col"| Domain Boundary Conditions |-| style="width:40px;₀ " | [[image:fdtd-ico.png | link= 8[[EM.854e-12 F/mTempo]]]] | [[EM.Tempo]]| style="width:350px;" | Finite box | style="width:250px;" | PEC, &muPMC, PML, Periodic |-| style="width:40px;_r " | [[image:prop-ico.png | link= 1[[EM.257eTerrano]]]] | [[EM.Terrano]]| style="width:350px;" | Open-6 H/mboundary free space with optional half-space ground | style="width:250px;" | Radiation BC|-| style="width:40px;" | [[image:static-ico.png | link=[[EM.Ferma]]]] | [[EM.Ferma]]| style="width:350px;" | Finite box | style="width:250px;" | Dirichlet, Neumann, Adiabatic and &epsilonConvective (for thermal simulation) |-| style="width:40px;_r and &mu" | [[image:planar-ico.png | link=[[EM.Picasso]]]] | [[EM.Picasso]]| style="width:350px;_r are called relative permittivity and permeability of the material, respectively" | Open-boundary with multilayer background medium | style="width:250px;" | Radiation BC|-| style="width:40px;" | [[image:metal-ico.png | link=[[EM.Libera]]]] | [[EM.Libera]]| style="width:350px;" | Open-boundary free space | style="width:250px;" | Radiation BC|-| style="width:40px;" | [[image:po-ico.png | link=[[EM.Illumina]]]] | [[EM.Illumina]]| style="width:350px;" | Open-boundary free space | style="width:250px;" | Radiation BC|}

The constitutive parameters relate For more information, refer to the field quantities in the material medium:manuals of individual computational modules.

:<math> \mathbf{D} = \epsilon \mathbf{E}, \quad \quad \mathbf{J} = \sigma \mathbf{E} </math>Exciting a Physical Structure Using Sources or Devices in EM.Cube ==

:<math> \mathbf{B} = \epsilon \mathbf{H}In order to perform an electromagnetic simulation in any of [[EM.Cube]]'s computational modules, \quad \quad \mathbf{M} = \sigma_m \mathbf{H} </math>you need to excite your physical structure using some kind of source. In most cases, you can define more than one source if necessary.

where '''E''' and '''H''' are the electric and magnetic fields, respectively*In [[EM.Tempo]], '''D''' is a source pumps energy into your FDTD computational domain in the electric flux density, also known form of a temporal waveform varying as a function time. *In the electric displacement MoM-based modules, [[EM.Picasso|EM.picasso]] and [[EM.Libera]], a source provides the "right-hand-side (RHS)" vectorof the MoM linear system resulting from the integral equation formulation of your boundary value problem. *In [[EM.Illumina]], '''B''' a source is used to illuminate your surfaces, and the magnetic flux densityscattrering of the incident fields from those surfaces are then computed. *In [[EM.Terrano]], also known a source acts as a transmitter that launches the magnetic induction vectorbroadcast signal into the free space. The transmitter rays travel in the free space until they reach a receiver or are intercepted by an obstructing surface. *In [[EM.Ferma]], sources like volume charges and '''J '''and '''M '''are volume currents provide the electric and magnetic current densitiessource term for the Poisson equation, respectivelywhile fixed-potential PEC objects set the boundary conditions for the Laplace equation.

The electric conductivity and magnetic conductivity parameters represent the material losses. In frequency-domain simulations under a time-harmonic (e^j&omega;t) 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 &omega; = 2&pi;f, 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''': &epsilon;_r = &mu;_r = 1 and &sigma; = &sigma;_m = 0* '''Perfect Electric Conductor (PEC)''': &epsilon;_r = &mu;_r = 1, &sigma; = &infin;, &sigma;_m = 0* '''Perfect Magnetic Conductor (PMC)''': &epsilon;_r = &mu;_r = 1, &sigma; = 0, &sigma;_m = &infin; == Variety of Source Types 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|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.

<td> [[Image:Source12Source12_new.png|thumb|left|360px|Current distribution on a metallic plate excited by a plane wave source.]] </td><td> [[Image:Source13Source13_new.png|thumb|left|360px|Current distribution on a metallic plate excited by a short horizontal dipole source above it.]] </td>

| [[EM.Tempo]]style="width:250px;" |General-purpose point voltage source| Waveguide Sourcestyle="width:250px;" | Associated with a PEC or thin wire line| style="width:200px;" | [[EM.Tempo]]

| style="width:250px;" | General-purpose distributed voltage source| style="width:250px;" | Associated with a virtual rectangle strip| style="width:200px;" | [[EM.Tempo]]

| Gap Microstrip Port Source| style="width:250px;" | Used for S-parameter computations| style="width:250px;" | Associated with a PEC rectangle strip| style="width:200px;" | [[EM.Picasso]], [[EM.LiberaTempo]]

| Probe CPW Port Source| style="width:250px;" | Used for S-parameter computations| style="width:250px;" | Associated with a PEC rectangle strip| style="width:200px;" | [[EM.PicassoTempo]]

| DeCoaxial Port Source| style="width:250px;" | Used for S-parameter computations| style="width:250px;" | Associated with a PEC Cylinder| style="width:200px;" | [[EM.Tempo]]|-| Waveguide Port Source| style="width:250px;" | Used for S-parameter computations| style="width:250px;" | Associated with a hollow PEC box| style="width:200px;" | [[EM.Tempo]]|-| Wire Current Source| style="width:250px;" | General-purpose current source| style="width:250px;" | Stand-alone source| style="width:200px;" | [[EM.Tempo]]|-| Wire Gap Circuit Source| style="width:250px;" | General-purpose point voltage source| style="width:250px;" | Associated with a PEC or thin wire line| style="width:200px;" | [[EM.Libera]]|-| Strip Gap Circuit Source| style="width:250px;" | General-purpose point voltage source| style="width:250px;" | Associated with a PEC rectangle strip| style="width:200px;" | [[EM.Picasso]], [[EM.Libera]]|-| Probe Gap Circuit Source| style="width:250px;" | Used for modeling short coaxial probes | style="width:250px;" | Associated with a PEC embedded via | style="width:200px;" | [[EM.Picasso]] |-| Scattering Wave Port Source| style="width:250px;" | Used for S-parameter computations| style="width:250px;" | Associated with a PEC rectangle strip| style="width:200px;" | [[EM.Picasso]]

| style="width:250px;" | General-purpose short filament current source| style="width:250px;" | Stand-alone source| style="width:200px;" | [[EM.Tempo]], [[EM.IlluminaTerrano]], [[EM.Picasso]], [[EM.Libera]], [[EM.TerranoIllumina]]

| style="width:250px;" | Used for scattering computations| style="width:250px;" | Stand-alone source| style="width:200px;" | [[EM.Tempo]], [[EM.IlluminaPicasso]], [[EM.PicassoLibera]], [[EM.LiberaIllumina]]

| style="width:250px;" | Used for modeling focused beams | style="width:250px;" | Stand-alone source| style="width:200px;" | [[EM.Tempo]]

| style="width:250px;" | Used as a distributed equivalent | style="width:250px;" | Imported from a Huygens data file| style="width:200px;" | [[EM.IlluminaTempo]], [[EM.Terrano]], [[EM.Picasso]], [[EM.Libera]], [[EM.Illumina]]

| Point Transmitter Set| style="width:250px;" | Used as a point source with an imported radiation pattern | style="width:250px;" | Associated with a point or point array| style="width:200px;" | [[EM.Terrano]]

| Fixed-Potential PEC with Nonzero Voltage| style="width:250px;" | Used as a distributed DC voltage source | style="width:250px;" | Requires drawing geometric objects| style="width:200px;" | [[EM.Ferma]]

| Volume Charge Source| style="width:250px;" | Used as a distributed electric charge source | style="width:250px;" | Requires drawing geometric objects| style="width:200px;" | [[EM.Ferma]]

| Wire Current Source| style="width:250px;" | Used as a linear filament current source | style="width:250px;" | Requires drawing geometric objects| style="width:200px;" | [[EM.Ferma]]

| Volume Current | style="width:250px;" | Used as a distributed elctric current source | style="width:250px;" | Requires drawing a line or polyline| style="width:200px;" | [[EM.Ferma]]|-| Permanent Magnet| style="width:250px;" | Used as a distributed magnetization source | style="width:250px;" | Requires drawing geometric objects| style="width:200px;" | [[EM.Ferma]]|-| Fixed-Temperature PTC| style="width:250px;" | Used as an isothermal surface heat source for thermal simulation| style="width:250px;" | Requires drawing geometric objects| style="width:200px;" | [[EM.Ferma]]|-| Volume Heat Source| style="width:250px;" | Used as a distributed volume heat source for thermal simulation | style="width:250px;" | Requires drawing geometric objects| style="width:200px;" | [[EM.Ferma]]|} The table below compares [[EM.Cube]]'s computational modules with regards to their excitation source and lumped device types:  {| class="wikitable"|-! scope="col"| ! scope="col"| Module Name ! scope="col"| Excitation/Sources ! scope="col"| Lumped Devices |-| style="width:40px;" | [[image:fdtd-ico.png | link=[[EM.Tempo]]]] | [[EM.Tempo]]| style="width:350px;" | Lumped source, distributed source, microstrip, CPW, coaxial and waveguide port sources, Hertzian dipole, wire current source, plane wave, Gaussian beam, Huygens source, arbitrary waveform | style="width:250px;" | Resistor, capacitor, inductor, series RL, parallel RC, nonlinear diode, active lumped one-port and two-port devices, active distributed one-port and two-port devices |-| style="width:40px;" | [[image:prop-ico.png | link=[[EM.Terrano]]]] | [[EM.Terrano]]| style="width:350px;" | Point transmitter set, Hertzian dipole, Huygens source| style="width:250px;" | N/A

| Permanent Magnet with Nonzero Magnetizationstyle="width:40px;" | [[image:static-ico.png | link=[[EM.Ferma]]]]

[[Image:Info_icon.png|30px]] Click here for a more detailed discussion of '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Defining_an_Excitation_Source | Defining an Excitation Source]]'''. [[Image:Info_icon.png|30px]] Click here to access '''[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types]]'''. == Variety of Defining Simulation Data Observables in EM.Cube ==

| style="width:250px;" | Electric and Magnetic Field Distributions

| style="width:250px;" | [[EM.Tempo]], [[EM.Terrano|Em.Terrano]], [[EM.IlluminaFerma]], [[EM.FermaPicasso]], [[EM.PicassoLibera]], [[EM.LiberaIllumina]]

| style="width:250px;" | Electric and Magnetic Current Distributions

| style="width:250px;" | [[EM.IlluminaPicasso]], [[EM.PicassoLibera]], [[EM.LiberaIllumina]]

| style="width:250px;" | Temporal Fields

| style="width:250px;" | [[EM.Tempo]]

| style="width:250px;" | Far-Field Radiation Patterns| Far Fields - Field Radiation Pattern| style="width:250px;" | [[EM.Tempo]], [[EM.Terrano|Em.TerranoPicasso]], [[EM.IlluminaLibera]], [[EM.PicassoIllumina]], [[EM.LiberaTerrano]]

| Radar Cross Section style="width:250px;" | Radiation Characteristics (RCSD0, HPBW, SLL, AR, etc.)| Far Fields - RCSField Radiation Pattern| style="width:250px;" | [[EM.Tempo]], [[EM.IlluminaPicasso]], [[EM.PicassoLibera]], [[EM.LiberaIllumina]], [[EM.Terrano]]

| style="width:250px;" | Radar Cross Section (RCS)| RCS| style="width:250px;" | [[EM.Tempo]], [[EM.Picasso]], [[EM.Libera]], [[EM.Illumina]]|-| style="width:250px;" | Huygens Surface Data

| style="width:250px;" | [[EM.Tempo]], [[EM.Terrano|Em.TerranoPicasso]], [[EM.IlluminaLibera]], [[EM.PicassoIllumina]], [[EM.LiberaTerrano]]

| style="width:250px;" | Port Characteristics (S/Z/Y Parameters)

| style="width:250px;" | [[EM.Tempo]], [[EM.Picasso]], [[EM.Libera]]

| style="width:250px;" | Port Voltags, Currents & Powers| Port Definition| style="width:250px;" | [[EM.Tempo]]|-| style="width:250px;" | Periodic Reflection and Transmission Coefficients

| style="width:250px;" | [[EM.Tempo]], [[EM.Picasso]]

| style="width:250px;" | Temporal Electric and Magnetic EnergyEnergies and Dissipated Power| Domain Energy-Power| style="width:250px;" | [[EM.FermaTempo]]

| style="width:250px;" | Electric and Magnetic Field Densities and Dissipated Power Density | Field Sensor & Energy-Power| style="width:250px;" | [[EM.Tempo]], [[EM.Ferma]]|-| style="width:250px;" | Specific Absorption Rate (SAR) Density| Field Sensor & Energy-Power| style="width:250px;" | [[EM.Tempo]]|-| style="width:250px;" | Complex Poynting Vector| Field Sensor & Energy-Power| style="width:250px;" | [[EM.Tempo]]|-| style="width:250px;" | Static Electric and Magnetic Energy & Ohmic Losses

| style="width:250px;" | [[EM.Ferma]]

| style="width:250px;" | Voltage and Current

| style="width:250px;" | [[EM.Ferma]]

| style="width:250px;" | Electric and Magnetic Flux

| style="width:250px;" | [[EM.Ferma]]

| style="width:250px;" | Resistance, Capacitance, Self-Inductance and Mutual Inductance

| style="width:250px;" | [[EM.Ferma]]

| style="width:250px;" | Temperature and heat flux distributions| Field Sensor| style="width:250px;" | [[EM.Ferma]]|-| style="width:250px;" | Thermal energy density | Field Sensor| style="width:250px;" | [[EM.Ferma]]|-| style="width:250px;" | Thermal Flux and Thermal Energy| Field Integrals| style="width:250px;" | [[EM.Ferma]]|-| style="width:250px;" | Received Power

| style="width:250px;" | [[EM.Terrano]]

| Singal-to-Noise Ratio (SNR) style="width:250px;" | Channel Path Loss

| style="width:250px;" | [[EM.Terrano]]

| Channel Path Loss style="width:250px;" | Singal-to-Noise Ratio (SNR)

== [[Image:Info_icon.png|30px]] Click here to access '''[[Glossary of EM.Cube Mesh 's Simulation Observables & Graph Types ==]]'''.

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

The linear mesh, also known as the wireframe mesh, is used by [[EM.Libera]] to discretize the physical structure for Wire MoM simulation. [[EM.Cube]] offers two types of surface mesh: 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 discretize surface geometric objects as well as the surface of solid geometric 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 mesh, also known as voxel mesh, to discretize the volume of your computational domain. Brick meshes are entire-domain volume meshes and are made up of cubic cells. Brick meshes They are indeed generated by a three-dimensional arrangement of grid lines along the X, Y and Z dimensionsaxes. [[EM.Tempo]] offers an "Adaptive " brick mesh as well as a fixed"Fixed-cell 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 Laplace/Poisson equations.

<td> [[Image:Mesh1Mesh1_new.png|thumb|left|240px|The geometry of a metallic torus.]] </td><td> [[Image:Mesh2Mesh2_new.png|thumb|left|240px|The brick volume mesh of the metallic torus.]] </td><td> [[Image:Mesh3Mesh3_new.png|thumb|200pxleft|240px|The triangular surface mesh of the metallic torus.]] </td>

== The Significance 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 is 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 very much on the quality and resolution of the generated mesh. As a rule of thumb, a mesh density of about 10-25 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. The particular simulation data you seek in a project also influences 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. The table below compares [[EM.Cube]]'s computational modules with regards to their mesh generator types:  {| class="wikitable"|-! scope="col"| ! scope="col"| Module Name ! scope="col"| Mesh Resolution Type |-| style="width:40px;" | [[image:fdtd-ico.png | link=[[EM.Tempo]]]] | [[EM.Tempo]]| style="width:450px;" | Adaptive and fixed-cell volumetric brick (voxel) mesh |-| style="width:40px;" | [[image:prop-ico.png | link=[[EM.Terrano]]]] | [[EM.Terrano]]| style="width:450px;" | Triangular facet mesh|-| style="width:40px;" | [[image:static-ico.png | link=[[EM.Ferma]]]] | [[EM.Ferma]]| style="width:450px;" | Fixed-cell volumetric brick mesh|-| style="width:40px;" | [[image:planar-ico.png | link=[[EM.Picasso]]]] | [[EM.Picasso]]| style="width:450px;" | Hybrid rectangular-triangular surface mesh|-| style="width:40px;" | [[image:metal-ico.png | link=[[EM.Libera]]]] | [[EM.Libera]]| style="width:450px;" | Wireframe and triangular surface mesh|-| style="width:40px;" | [[image:po-ico.png | link=[[EM.Illumina]]]] | [[EM.Illumina]]| style="width:450px;" | Triangular surface mesh|} [[Image:Info_icon.png|30px]] Click here to access '''[[Glossary of EM.Cube's Simulation-Related Operations]]'''. <br /> <hr>

The objects of your physical structure are discretized based on a specified mesh density. The default mesh densities of [[EMImage:Top_icon.Tempopng|30px]], '''[[EMNumerical_Modeling_of_Electromagnetic_Problems_Using_EM.Picasso]], [[EM.Libera]] and [[EM.Illumina]] are expressed as Cube#An_Overview_of_Computational_Electromagnetics | Back to the number Top 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.CubePage]]'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[[Image:Back_icon. As a rule of thumb, a mesh density of about 10-30 cells per effective wavelength usually yields satisfactory resultspng|30px]] '''[[EM. 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 Cube | Back to the mesh density than the near-field distributions on a structure with a highly irregular shape and a rugged boundaryEM.Cube Main Page]]'''