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[[Image:Splash-fdtd.jpg|right|720px]]<strong><font color="#961717" size="4">Fast Multicore & GPU-Accelerated FDTD Solvers for Simulating the Most Complex Electromagnetic Modeling Problems</font></strong><table><tr><td>[[image:Cube-icon.png | link=Getting_Started_with_EM.Cube]] [[image:cad-ico.png | link=Building_Geometrical_Constructions_in_CubeCAD]] [[image:prop-ico.png | link=An EM.Terrano]] [[image:static-ico.png | link=EM.Ferma]] [[image:planar-ico.png | link=EM.Picasso]] [[image:metal-ico.png | link=EM.Libera]] [[image:po-ico.png | link=EM.Illumina]]</td><tr></table>[[Image:Tutorial_icon.png|30px]] '''[[EM.Cube#EM.Tempo_Documentation | EM.Tempo Primer Tutorial Gateway]]''' [[Image:Back_icon.png|30px]] '''[[EM.Cube | Back to EM.Cube Main Page]]'''==Product Overview==

EM.Tempo is a powerful time-domain electromagnetic simulator for full-wave modeling of 3D radiation, scattering and propagation problems. It features a highly efficient Finite Difference Time Domain (FDTD) simulation engine that has been optimized for speed and memory usage. EM.Tempo brings to your desktop the ultimate in computational power. Its FDTD solver has been parallelized to take full advantage of multi-core processor architectures. With a large variety of geometrical, material and excitation features including open-boundary and periodic structures, you can use EM.Tempo as a general purpose 3D field simulator for most of your electromagnetic modeling needs. EM.Tempo's new advanced simulation capabilities are your the key to a thorough understanding of wave the interaction in of electromagnetic waves with complex media such as anisotropic composites, metamaterials or biological environmentsor with passive and active devices and nonlinear circuits.

=== Pros and Cons EM.Tempo has undergone several evolutionary development cycles since its inception in 2004. The original simulation engine utilized an FDTD formulation based on the uniaxial perfectly matched layer (UPML) boundary termination. Subsequently, a more advanced boundary termination based on the convolutional perfectly matched layer (CPML) was implemented with a far superior performance for all oblique wave incidences in different types of media. EM.Tempo now has the ability to model laterally infinite layered structures using CPML walls that touch material media. A novel formulation of periodic boundary conditions was implemented based on the constant transverse wavenumber method (or direct spectral FDTD Simulation ===). In 2013 we introduced an Open-MP optimized multi-core version of the FDTD engine as well as a hardware-accelerated solver that runs on CUDA-enabled graphical processing unit (GPU) platforms. Both of these fast solvers are now a standard part of the EM.Tempo Pro package.

A time domain simulation like FDTD offers several advantages over a frequency domain simulation[[Image:Info_icon. In certain applications, the time domain signature or behavior png|30px]] Click here for an overview of a system, e.g. the transient response '''[[Basic Principles of a circuit or an antenna, is sought. In other applications, you may need to determine the wideband frequency response of a system. In such cases, using a frequency domain technique, you have to run the simulation engine many times to adequately sample the specified frequency range. In contrast, using the FDTD method requires a single-run simulation. The temporal field data are transformed into the Fourier domain to obtain the wideband frequency response of the simulated system. Among other advantages of the Finite Difference Time Domain Method | Basic FDTD method are its versatility in handling complex material compositions as well as its superb numerical stability. It is worth noting that unlike frequency domain methods like the finite element method (FEM) or method of moments (MoM), the FDTD technique does not involve numerical solution of large ill-conditioned matrix equations that are often very sensitive to the mesh qualityTheory]]'''.

Like every numerical technique, the FDTD method has disadvantages, too<table><tr><td>[[Image:ART GOLF Fig title. Adding the fourth dimension, time, to the computations increases the size of the numerical problem significantly. Unfortunately, this translates to both larger memory usage and longer computation times. Note that the png|thumb|left|400px| The 3D far-field data are generated in both the 3D space and time. EM.Tempo uses a staircase "Yee" mesh to discretize the physical structure. This works perfectly fine for rectangular objects that are oriented along the three principal axes. In the case radiation pattern of highly curved structures or slanted surfaces and lines, however, this may compromise the geometrical fidelity of your structure. EM.Tempo provides a default adaptive FDTD mesher that can capture the fine details of geometric contours, slanted thin layers, surfaces, etc. to arbitrary precision. However, with smaller mesh cells, the stability criterion leads to smaller time steps; hence, longer computation times. Another disadvantage of the FDTD technique compared to naturally openvehicle-boundary methods like MoM is its finite-extent computational domain. This means that to model open boundary problems like radiation or scattering, absorbing boundary conditions are needed to dissipate the incident waves at the walls of the computational domain and prevent them from reflecting back into the domain. The accuracy of the FDTD simulation results depends on the quality of these absorbers and their distance from the actual physical mounted antenna structure. simulated by EM.Tempo provides high quality perfectly match layer (PML) terminations at the boundaries which can be placed fairly close your physical structure.]]</td></tr></table>

=== An Overview of EM.Tempo as the FDTD Modeling Module of EM.Cube ===

EM.Tempo is a general-purpose EM simulator than can solve most types of electromagnetic modeling problems involving arbitrary geometries and complex material variations in both time and frequency domains. It has also been integrated within the [[Image:FDTD93.png|thumb|300px|A metal ellipsoid object..EM.Cube]][[Image:FDTD94simulation environment as its full-wave "FDTD Module".png|thumb|300px|EM...and its Yee mesh.]]In Tempo shares the Finite Difference Time Domain (FDTD) methodvisual interface, a discretized form of Maxwell’s equations is solved numerically and simultaneously in both the 3D space and time. During this processparametric CAD modeler, data visualization tools, the electric and magnetic fields are computed everywhere in the computational domain many more utilities and features collectively known as a function [[Building Geometrical Constructions in CubeCAD | CubeCAD]] with all of time starting at t = 0. From knowledge of the primary fields in space and time, one can compute other secondary quantities including frequency domain characteristics like scattering [[parametersEM.Cube]], input impedance, far field radiation patterns, radar cross section, etc's other computational modules.

[[Image:Info_icon.png|30px]] Click here to learn more about the '''[[Differential Form of Maxwell's EquationsGetting_Started_with_EM.Cube | EM.Cube Modeling Environment]]'''.

Since FDTD is a finite domain numerical technique, the computational domain === The Advantages & Limitations of the problem must be truncatedEM. At the boundaries of the computational domain, proper boundary conditions must be enforced. In a shielded structure, all objects are enclosed within a perfect electric (or magnetic) conductor box. In an open boundary problem like an antenna, some kind of absorbing boundary conditions such as a perfectly matched layer (PML) must be used to emulate the free space. The absorbing boundaries should act such that the field propagates through them without any back reflection. The Tempo's FDTD simulation time depends directly on the size of the computational domain and on how close you can place the PML walls to the enclosed objects. Simulator ===

Click here A time domain simulation like FDTD offers several advantages over frequency domain simulations. In certain applications, the time domain signature or behavior of a system, e.g. the transient response of a circuit or an antenna, is sought. In other applications, you may need to learn more about EMdetermine the wideband frequency response of a system.Tempo's [[Perfectly Matched Layer Termination]]In such cases, using a frequency domain technique, you have to run the simulation engine many times to adequately sample the specified frequency range. In contrast, using the FDTD method requires a single-run simulation. The temporal field data are transformed into the Fourier domain to obtain the wideband frequency response of the simulated system. Among other advantages of the FDTD method are its versatility in handling complex material compositions as well as its superb numerical stability. It is worth noting that unlike most frequency domain methods, the FDTD technique does not involve numerical solution of large ill-conditioned matrix equations that are often very sensitive to the mesh quality.

The Like every numerical technique, the FDTD computational domain must be discretized using an appropriate meshing schememethod has disadvantages, too. Adding the fourth dimension, time, to the computations increases the size of the numerical problem significantly. Unfortunately, this translates to both larger memory usage and longer computation times. Note that the field data are generated in both the 3D space and time. EM.Tempo uses a non-uniform, variable, staircase (pixelated) "Yee " mesh with a mesh density that you can customizeto discretize the physical structure. A fixed-cell mesh generator is also available, where you can set constant cell dimensions This works perfectly fine for rectangular objects that are oriented along the three principal axes for the entire computational domain. The variable mesh density is specified in terms In the case of the effective wavelength inside material media. As a resulthighly curved structures or slanted surfaces and lines, however, this may compromise the mesh resolution and average mesh cell size differ in regions that are filled with different types geometrical fidelity of materialyour structure. [[EM.Cube]]'s non-uniform Tempo provides a default adaptive FDTD mesher generates more cells in the areas that are occupied by dielectric materialscan capture the fine details of geometric contours, fewer slanted thin layers, surfaces, etc. to arbitrary precision. However, with smaller mesh cells in , the free space regions and no cells inside (impenetrable) PEC regionsstability criterion leads to smaller time steps; hence, longer computation times. [[Another disadvantage of the FDTD Module]]'s default "adaptive" mesh generator also refines technique compared to naturally open-boundary methods like the mesh around curved segments method of lines, surface moments (MoM) is its finite-extent computational domain. This means that to model open boundary problems like radiation or solids scattering, absorbing boundary conditions are needed to produce a far more accurate representation dissipate the incident waves at the walls of your geometrythe computational domain and prevent them from reflecting back into the domain. The example accuracy of the FDTD simulation results depends on the right illustrates a metal ellipsoid quality of these absorbers and a 3D view their distance from the actual physical structure. EM.Tempo provides high quality perfectly matched layer (PML) terminations at the boundaries, which can be placed fairly close to your physical structure to reduce the total size of its Yee meshthe computational domain.

The FDTD method provides a wideband simulation of your physical structure. In order to produce sufficient spectral information, an appropriate wideband temporal waveform is needed to excite the physical structure. The choice of the waveform, its bandwidth and time delay all affect the convergence behavior of the FDTD time marching loop. By default, EM.Tempo uses a modulated Gaussian waveform with optimal <table><tr><td>[[parameters]]Image:Airplane Mesh. Another issue of concern is the numerical stability of the time marching scheme. You might expect to get better and more accurate results if you keep increasing the FDTD png|thumb|left|480px|The Yee mesh resolution. However, in order to satisfy the Courant-Friedrichs-Levy (CFL) stability condition, the time step must be inversely proportional to the maximum grid cell size . A high resolution mesh requires a smaller time step. To let the fields in the computational domain fully evolve over time, a smaller time step will require a larger number of time steps to convergean imported aircraft CAD model. [[EM.Cube]] automatically chooses a time step that satisfies the CFL condition.</td></tr></table>

For more detailed information, see [[Waveform, Bandwidth, Stability]]== EM.Tempo Features at a Glance ==

==Building the = Physical StructureDefinition ===

[[Image:FDTD1.png|thumb|200px|[[FDTD Module]]'s Navigation Tree.]]<ul>In [[EM.Tempo]] <li> PEC, a physical structure consists PMC and dielectric materials and thin wires</li> <li> Uniaxial and fully anisotropic materials with four complete constitutive tensors</li> <li> Dispersive materials of sets of objects that are grouped together Debye, Drude and identified by their material Lorentz types. All the objects belonging to the same material group share the same color with arbitrary number of poles</li> <li> Generalized uniaxial and same material properties. Materials are divided into seven categories that are listed under the '''Physical Structure''' node at the top doubly negative refractive index metamaterials with arbitrary numbers of the navigation treeboth electric and magnetic poles</li> <li> Two types of gyrotropic materials:ferrites and magnetoplasmas</li> <li> PEC, PMC and convolutional perfectly match layer (CPML) boundary conditions</li> <li> Doubly periodic structures</li></ul>

Under each material node<ul> <li> Lumped voltage sources with internal resistance placed on a PEC line or thin wire object with an arbitrary orientation</li> <li> Distributed sources with uniform, you can create new material groups of sinusoidal and edge-singular profiles</li> <li> Microstrip, coplanar Waveguide (CPW) and coaxial ports</li> <li> Waveguide sources with the same typedominant TE<sub>10</category but sub> modal profile</li> <li> Multi-port and coupled port definitions</li> <li> Two types of filamentary current sources: Hertzian short dipole radiators with different properties (color, texturearbitrary orientation and long wire current sources aligned along one of the principal axes with a uniform, triangular or electric sinusoidal current distribution profile</li> <li> Plane wave excitation with linear and magnetic constitutive circular polarizations</li> <li> Multi-ray excitation capability (ray data imported from [[parametersEM.Terrano]]). These material groups are used to organize the CAD objects you draw in the project workspace or import from external model files. When you create a new geometrical object such as a Box or a Sphere, it is inserted under the currently active material type. There is only one material group that is active at any time. It is recommended that you first create material groups)</li> <li> Gaussian beam excitation</li> <li> Huygens sources</li> <li> Source arrays with weight distribution &amp; phase progression</li> <li> Periodic sources with user defined beam scan angles</li> <li> Standard excitation waveforms (Gaussian pulse, modulated Gaussian and then draw new objects as part of the active material group. Howeversinusoidal) for optimal frequency&nbsp;domain computations&nbsp;</li> <li> Arbitrary user-defined temporal excitation waveforms using mathematical expressions and Python functions</li> <li> Passive lumped devices: R, if you start a new EM.Tempo project from scratchL, and start drawing a new object without having previously defined any material groupsC, a new default PEC group is created series RL and added to parallel RC and nonlinear diode device</li> <li> Active lumped one-port and two-port devices placed on PEC lines aligned along one of the navigation tree to hold your new CAD object.principal axes with arbitrary Netlist definitions</li> <li> Active distributed one-port and two-port devices placed under microstrip lines with arbitrary Netlist definitions</li></ul>

===Defining Material TypesMesh Generation ===

To define a new material group<ul> <li> Fast generation of Yee grid mesh of solids, follow these steps:surfaces and curves</li> <li> Geometry-aware and material-aware adaptive mesh generator with gradual grid transitions</li> <li> Fixed-cell uniform mesh generator with three unequal cell dimensions</li> <li> Mesh view with three principal grid profilers</li> <li> Manual control of mesh parameters and fixed grid points</li></ul>

Once <ul> <li> Wideband full-wave simulation of 3D structures</li> <li> Transient analysis with arbitrary user defined excitation waveforms</li> <li> Multi-frequency computation of frequency domain quantities in a new material node has been created single FDTD simulation run</li> <li> OpenMP-parallelized multi-core and multi-thread FDTD simulation engine</li> <li> GPU-accelerated FDTD simulation engine based on NVIDIA CUDA platforms</li> <li> Total-field-scattered-field analysis of plane wave and Gaussian beam excitation</li> <li> Full-wave analysis of periodic structures with arbitrary plane wave incidence angles using the navigation tree, it becomes the "Active" Direct Spectral FDTD method</li> <li> Infinite material group half-space Green&#39;s functions for calculation of the project workspace, which is always listed far fields in bold letters. Then you can start drawing new objects under that node. Any material can be made active by right clicking presence of a lossy ground</li> <li> Accelerated computation of S-parameters of resonant structures based on its name in the Navigation Tree and selecting the Prony'''Activate''' item s method of the contextual menu.exponential interpolation</li> <li> Parametric sweeps of variable object properties or source parameters including frequency and angular sweeps</li> <li> Multi-variable and multi-goal optimization of structures</li> <li> Automated generation of compact reduced order surrogate models from full-wave simulation data</li></ul>

===Moving Objects among Material GroupsData Generation &amp; Visualization ===

[[Image:FDTD21<ul> <li> Near-field intensity (1colorgrid).png|thumb|325px|Moving objects from one FDTD material group to another.]], contour and surface plots (vectorial - amplitude &amp; phase)</li>You can move one or more selected objects at a time among different material groups. The objects can be selected either <li> Near-field probes for monitoring field components in the project workspace, or their names can be selected from the navigation tree. Right click on the highlighted selection both time &amp; frequency domains</li> <li> Far-field radiation patterns: 3D pattern visualization and select '''Move To 2D polar and Cartesian graphs</li> FDTD <li>''' from the contextual menu. This opens up another sub Far-menu with a list field characteristics such as directivity, beam width, axial ratio, side lobe levels and null parameters, etc.</li> <li> Radiation pattern of arbitrary array configurations of all the available material groups already defined FDTD structure or periodic unit cell</li> <li> Bistatic and monostatic radar cross section</li> <li> Huygens surface data generation for use in your other [[EM.Tempo project. Select the desired material nodeCube]] modules</li> <li> Periodic reflection/transmission coefficients and k-&beta; diagrams</li> <li> Port characteristics: S/Y/Z parameters, VSWR and all the selected objects will move Smith chart</li> <li> Time and frequency domain port voltages, currents and powers</li> <li> Touchstone-style S-parameter text files for direct export to that material group[[RF. In the case Spice A/D]]</li> <li> Interanl node voltages and currents of a multiple selection from the navigation tree using the keyboard's '''Shift Key''' or '''Ctrl Key'''Netlist-based one-port and two-port networks</li> <li> Computation of electric, make sure that you continue to hold the keyboard's '''Shift Key''' magnetic and total energy densities, dissipated power density (Ohmic loss), specific absorption rate (SAR) density and complex Poynting vector on field sensor planes</li> <li> Animation of temporal evolution of fields</li> <li> Custom output parameters defined as mathematical expressions or '''Ctrl Key''' down while selecting the destination material group's name from the contextual menu.Python functions of standard outputs</li></ul>

In a similar way, you can move one or more objects from an FDTD material group to one of [[EM.Cube]]'s other modules. In this case, == Building the sub-[[menus]] of the '''Move To >''' item of the contextual menu will indicate all the [[EM.Cube]] modules that have valid groups for transfer of the selected objects. You can also move one or more objects from [[EM.Cube]]'s other modules to a material group Physical Structure in EM.Tempo. ==

{{Note|You can import external objects only to '''[[CubeCAD]]'''. You need to move the imported objects form [[CubeCAD]] to === Material Variety in EM.Tempo as described above.}}===

===Perfect Conductors===Your physical structure in EM.Tempo offers two types can be made up of perfect conductorsseveral geometric objects with different material compositions. In other words, the geometric objects you draw or import from external files are grouped together based on a common material composition. EM.Tempo's material types are divided into seven categories:

{| class="wikitable"|-! scope="col"| Icon! scope="col"| Material Type! scope="col"| Applications! scope="col"| Geometric Object Types Allowed|-| style="width:30px;" | [[File:pec_group_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types# '''Perfect Electric Conductor (PEC) Objects:''' The tangential electric field on the surface of this type of perfect conductor is zero. The electric and magnetic fields are assumed to vanish inside the volume of a PEC object. A PEC material is characterized by an infinite electric conductivity |Perfect Electric Conductor (&sigmaPEC)]]| style="width:300px; " | Modeling perfect metals| style= &infin"width:250px;). You can draw solid" | Solid, surface and curve objects|-| style="width:30px;" | [[Curve ObjectsFile:thin_group_icon.png]]|curve objectsstyle="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Thin Wire |Thin Wire]] as part | style="width:300px;" | Modeling wire radiators| style="width:250px;" | Lines parallel to one of a PEC groupthe three principal axes|-| style="width:30px;" | [[File:pmc_group_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types# '''Perfect Magnetic Conductor (PMC) Planes|Perfect Magnetic Conductor (PMC)]]| style="width:''' The tangential magnetic field on the surface of this type of 300px;" | Modeling perfect conductor is zero. The electric and magnetic fields are assumed sheets | style="width:250px;" | Rectangle strips parallel to vanish inside the volume one of a PMC objectthe three principal planes|-| style="width:30px;" | [[File:diel_group_icon. A PMC png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Dielectric Material |Dielectric Material]]| style="width:300px;" | Modeling any homogeneous material is characterized by an infinite magnetic conductivity (&sigma| style="width:250px;_m " | Solid objects|-| style= &infin"width:30px;)" | [[File:aniso_group_icon. EMpng]]| style="width:150px;" | [[Glossary_of_EM.Tempo currently allows only PMC plates (rectangle strips Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Anisotropic Material |Anisotropic Material]]| style="width:300px;" | Modeling unaxial or generalized anisotriopic materials| style="width:250px;" | Solid objects) parallel to one |-| style="width:30px;" | [[File:disp_group_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Dispersive Material |Dispersive Material]]| style="width:300px;" | Modeling Debye, Drude and Lorentz materials and generalized metamaterials | style="width:250px;" | Solid objects|-| style="width:30px;" | [[File:voxel_group_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Gyrotropic_Material |Gyrotropic Material]]| style="width:300px;" | Modeling ferrites and magnetoplasmas| style="width:250px;" | Solid objects|-| style="width:30px;" | [[File:Virt_group_icon.png]]| style="width:150px;" | [[Glossary of the three principal axesEM.Cube's Materials, Sources, Devices & Other Physical Object Types#Virtual_Object_Group | Virtual Object]]| style="width:300px;" | Used for representing non-physical items | style="width:250px;" | All types of objects|}

PEC and PMC materials do not have any constitutive material properties that you can modify except for their color or textureClick on each category to learn more details about it in the [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types]].

===Dielectric MaterialsOrganizing the Physical Structure by Material Groups ===[[Image:FDTD5.png|thumb|450px|[[EM.Cube]]'s material list]]In [[EM.Tempo]], a dielectric material represents a general isotropic, homogeneous material with both electric and magnetic properties. The constitutive [[parameters]] of a dielectric material include permittivity (&epsilon;), permeability (&mu;), electric conductivity (&sigma;) and magnetic conductivity (&sigma;_m):

:<math> \mathbf{D} = \epsilon \mathbf{E}EM.Tempo groups your geometric objects in the project workspace based on their material type. All the objects belonging to the same material group share the same color and same material properties. Under each material node in the navigation tree, \quad \quad \mathbf{J} = \sigma \mathbf{E} </math>you can create new material groups of the same type but with different properties such as color, texture, or electric and magnetic constitutive parameters.

:<math> \mathbf{B} = \epsilon \mathbf{H}Once a new material node has been created on the navigation tree, \quad \quad \mathbf{M} = \sigma_m \mathbf{H} </math>it becomes the "Active" material group of the project workspace, which is always listed in bold letters. When you draw a new geometric object such as a box or a sphere, its name is added under the currently active material type. There is only one material group that is active at any time. Any material can be made active by right clicking on its name in the navigation tree and selecting the '''Activate''' item of the contextual menu. It is recommended that you first create material groups, and then draw new objects under the active material group. However, if you start a new EM.Tempo project from scratch, and start drawing a new object without having previously defined any material groups, a new default PEC group is created and added to the navigation tree to hold your new object.

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{{Note|You can import external objects only to CubeCAD. For example, an imperfect metal You can be represented by a dielectric material that has a large, finite, electric conductivitythen move the imported objects form CubeCAD to EM. PEC and PMC, therefore, are the limiting cases of an isotropic dielectric material when &sigma; &rarr; &infin; or &sigma;_m &rarr; &infin;, respectivelyTempo.}}

You may also choose from [[EMImage:Info_icon.Cubepng|30px]]'s list of preloaded material types. Click here to access the button labeled '''Material''' to open [[Glossary of EM.Cube]]'s Material List dialog. Select the desired material from the list or type the first letter of a material to find it. For exampleMaterials, typing Sources, Devices & Other Physical Object Types]]'''V''' selects '''Vacuum '''in the list. Once you close the dialog by clicking '''OK''', the selected material properties fill the parameter fields automatically.

[[=== Material Hierarchy in EM.Tempo]] allows you to define a general anisotropic material, whose constitutive [[parameters]], i.e. permittivity ('''&epsilon;'''), permeability ('''&mu;'''), electrical conductivity ('''&sigma;''') and magnetic conductivity ('''&sigma;_m'''), are all tensorial in nature. Each constitutive parameter in this case is represented by a 3×3 matrix:===

[[File:FDTD16EM.png|600pxTempo]]allows overlapping objects although it is generally recommended that object overlaps be avoided in favor of clearly defined geometries and object boundaries. If two or more objects of the same material type and group overlap, they are merged using the Boolean union operation during the mesh generation process. If two overlapping objects belong to two different material categories, then the material properties of the FDTD cells in the overlap region will follow the [[EM.Tempo]]'s material hierarchy rule. In that case, the overlap area cells will always be regarded as having the material type of the higher priority. According to this rule, the material types are ordered from the highest priority to the lowest in the following manner:

A "'''# PEC# PMC# Dispersive# Gyrotropic# General Anisotropic# Uniaxial'''" material is a special case of an anisotropic material whose constitutive [[parameters]] are all diagonal matrices. Specifying an anisotropic material as <u>'''Uniaxial'''</u> in the [[FDTD Module]] has a very important computational implication. There are six field update equations for uniaxial materials at each time steps: three for the electric field and three for the magnetic field. In this respect, a uniaxial material is similar to an isotropic dielectric material. On the other hand, a fully anisotropic material with non-zero off-diagonal constitutive matrix elements requires twelve update equations at each time step: three equations for the three components of each of the four vector fields '''E''', '''D''', '''H''' and '''B'''. As a result, the time loop for fully anisotropic materials takes much longer time than uniaxial materials.Anisotropic# Dielectric

<table><tr><td> [[FileImage:FDTD7FDTD_MAN2.png|thumb|250pxleft|Debye Add Pole Dialog]][[#Perfect Conductors|PEC]], [[#Perfect Conductors|PMC]], [[#Dielectric Materials360px|The geometric construction of a dielectric]] and [[#Anisotropic Materials|anisotropic]] materials are non-dispersive. In other words, their constitutive [[parameters]] do not vary coated metallic cylinder with frequencya conformal foil. Most of the materials used in the design of RF and microwave circuits, antennas and systems fall into this frequency-independent category. However, there are other types of materials whose constitutive [[parameters]] exhibit frequency-dependent behaviors. [[EM.Cube]]'s [[FDTD Module]] currently offers four types of dispersive material:</td></tr></table>

# Debye === Moving Objects Among Different Material# Drude Material (Unmagnetized Plasma)# Lorentz Material# Left-handed Metamaterial Groups or EM.Cube Modules ===

The FDTD simulation engine uses the Auxiliary Differential Equation (ADE) method You can move any geometric object or a selection of objects from one material group to model dispersive materialsanother. You can also transfer objects among [[EM.Cube]] allows 's different modules. For example, you often need to define an arbitrary number of poles for each of the above dispersive material typesmove imported CAD models from CubeCAD to [[EM. Keep Tempo]]. To transfer objects, first select them in mind that all the objects belonging to project workspace or select their names in the same dispersive navigation tree. Then right-click on them and select <b>Move To &rarr; Module Name &rarr; Object Group</b> from the contextual menu. For example, if you want to move a selected object to a material group called "Dielectric_1" in [[EM.Tempo]], then you have to select the same dispersion propertiesmenu item '''Move To &rarr; [[EM.Tempo]] &rarr; Dielectric_1''' as shown in the figure below. Note that you can transfer several objects altogether using the keyboards's {{key|Ctrl}} or {{key|Shift}} keys to make multiple selections.

The complex permittivity of a Debye material with N poles is given by<table><tr><td>[[Image:Tempo_L11_Fig2.png|thumb|left|720px|Moving an imported object from CubeCAD to EM.Tempo.]]</td></tr></table>

:<math> \varepsilon (\omega) = \varepsilon_\infty + \sum_{p=1}^N \dfrac{\Delta \varepsilon_p}{1 + j\omega \tau_p}, \quad \Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_\infty </math><!--[[Image:FDTD18(2)EM.png]]-->Tempo's Computational Domain & Boundary Conditions ==

where <math>\varepsilon_{\infty}</math> is the value of the permittivity at infinite frequency, <math>\tau_p</math> is the relaxation time corresponding to the p''th'' pole having the unit of seconds, and <math>\varepsilon_{sp}</math> is the value of the static permittivity (at DC) corresponding to the p''th'' pole. <math>\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}</math> represents the change in permittivity due to the p''th'' pole.==The FDTD Solution Domain===

Unmagnetized plasmas are typically modeled as Drude materialsThe FDTD method requires a finite-extent solution domain. This is rather straightforward for shielded structures, where a typical PEC enclosure box defines the computational domain. For open-boundary structures like antennas and scatterers, the computational domain must be truncated using appropriate termination boundary conditions. The complex permittivity objective of a Drude material with N poles termination boundary conditions is given by:to eliminate the reflections from the walls of the domain box back to the computational domain.

:<math> \varepsilonIn [[EM.Tempo]], you can define two types of domain box. A '''Default'''-type domain is a box that is placed at a specified offset distance from the largest extents of your physical structure (\omegaglobal bounding box) = \varepsilon_{\infty} . The offset is specified in free- \sum_{p=1}^N \dfrac{{\omega_p}^2}{\omega^2 space wavelengths. A '''Custom'''- j\omega \nu_p} </math><!type domain, on the other hand, is defined as a fixed-size and fixed-[[Image:FDTD19location box in the World Coordinate System (WCS). In this case, you have to specify the coordinates of the lower left front corner (Corner 1)and upper right back corner (Corner 2) of the domain box.png]]-->

where When you start a new project in [[EM.Tempo]], a default-type domain is automatically created with a default offset value set equal to a quarter free-space wavelength (0.25&lambda;<mathsub>\omega_p0</mathsub> and <math>\nu_p</math> are ). As soon as you draw your first object, a blue domain box shows up in the angular plasma frequency project workspace and angular collision frequency corresponding to encloses your object. As you add more objects and increase the p''th'' poleoverall size of your structure, respectively, and both are expressed in rad/sthe domain box grows accordingly to encompass your entire physical structure. For an unmagnetized plasmaWhen you delete objects from the project workspace, <math>\varepsilon_{\infty} = 1</math>the domain box also shrinks accordingly.

:<math> \varepsilon(\omega) = \varepsilon_{\infty} - \sum_{p=1}^N \dfrac{\Delta \varepsilon_p {\omega_p}^2}{\omega^2 - 2j\omega \delta_p - {\omega_p}^2}To set the solution domain of your FDTD project, \quad \Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty} </math><!--[[Imagefollow these steps:FDTD20.png]]-->

where <math>\omega _p</math> * Click the '''Domain''' [[Image:domain_icon.png]] button of the '''Simulate ''' Toolbar or select the menu item '''Simulate &rarr; Computational Domain &rarr; Domain Settings...''' or right click on the '''FDTD Domain''' item of the Navigation Tree and select '''Domain Settings...''' from the contextual menu, or use the keyboard shortcut '''Ctrl+A'''. The Domain Settings Dialog opens up, showing the current domain type selection.* Select one of the two options for '''Domain Type'''<mathnowiki>\delta_p: </mathnowiki> are '''Default''' or '''Custom'''.* If you select the angular resonant frequency "Default" domain type, the domain box is defined in terms of the offsets along the X, Y and angular damping Z directions from the largest extents of your physical structure. Select one of the two options for '''Offset Units: Grid''' and '''Wavelength'''. In the section titled '''"Domain Size"''', enter the amount of domain extension beyond the largest extents of the structure along the ±X, ±Y and ±Z directions. Note that in the case of a default-type domain box, the offset values based on your current project settings (frequency corresponding to and units).* When the p''th'Wavelength''' option is selected for '''Offset Units''' pole, respectivelyadditional free space is added around the structure by the specified ±X, ±Y and both ±Z offsets in free space wavelengths. Note that the free space wavelength for this purpose is calculated at the center frequency of the project. The default value of the offset in this case is a quarter free space wavelength. Note that with this option, the number of the additional cells and their cell size is not fixed; they vary from structure to structure.* When the '''Grid''' option is selected for '''Offset Units''', the six offset values represent the number of additional free-space mesh cells that are expressed placed in rad/seach direction beyond the largest bounding box around the physical structure. Similar The default value of the offset in this case is eight grid cells along the ±X, ±Y and ±Z directions.* If you select the "Custom" domain type instead, you need to a Debye materialenter values for the coordinates of the lower-left-front corner, <math>\Delta \varepsilon_p = \varepsilon_{sp} ''' Corner 1''', and the upper- \varepsilon_{\infty}</math> represents right-back corner, '''Corner 2''', of the domain box.* After you change in permittivity due values or settings, click the '''Apply''' button to make the changes effective. To recover the default values, click the p''th'Defaults' pole'' button of the dialog. Click '''OK''' to save the settings and close the dialog.

{| border="0"|-| valign="top"|[[Image:FDTD2By default, the domain box is shown as a wireframe box with blue lines.png|thumb|250px|EMYou can change the color of the domain box or hide it.Tempo's PEC Dialog]]| valign="top"|[[Image:FDTD3.png|thumb|250px|EM.Tempo's PMC Dialog]]| valign="top"|[[Image:FDTD4.png|thumb|250px|Dielectric Material dialog]]| valign="top"|[[Image:FDTD6.png|thumb|250px|[[FDTD Module]]'s Anisotropic Material dialog]]| valign="top"|[[Image:FDTD8.png|thumb|250px|Debye Material Dialog]]|-|}

<table><tr><td> [[Image:fdtd14_tnFDTD14.png|thumb|400pxleft|480px|Geometric construction of a dielectric-coated metallic cylinder.]]The following rules apply to the definition of materials and objects in [[EM.Tempo's domain settings dialog.]]:</td></tr></table>

* Under ===Settings the [[#Perfect Conductors|PEC]] category, you can define all types of solid, and surface and [[Curve Objects|curve objects]].* Under the [[#Perfect Conductors|PMC]] category, you can define only define rectangle strip objects parallel to the principal planes. * Under the [[#Dielectric Materials|Dielectric]], [[#Anisotropic Materials|Anisotropic]] and [[#Dispersive Materials|Dispersive]] material categories, you can define only [[Solid Objects|solid objects]].* Under the Inhomogeneous Material category, you can only import a Cartesian ".CAR" data file.* Under the Thin Wire category, you can only define line objects parallel to the principal axes. Domain Boundary Conditions===

[[EM.Tempo]] allows overlapping objects, although it is generally recommended that object overlaps be avoided in favor supports four types of clearly defined geometries domain boundary conditions: PEC, PMC, Convolutional Perfectly Matched Layers (CPML) and object boundariesPeriodic Boundary Conditions (PBC). If two or more objects By default, all the six sides of the same material type and group overlapcomputational domain box are set to CPML, they are merged using the Boolean union operation during the mesh generation processrepresenting a completely open-boundary structure. If two overlapping objects belong Different boundary conditions can be assigned to two different material categories, then each of the material properties six walls of the FDTD cells in the overlap region will follow the domain box. The periodic boundary conditions are special ones that are assigned through [[EM.Tempo]]'s material hierarchy rule. In that case, the overlap area cells Periodicity Dialog and will always be regarded as having discussed later under modeling of periodic structures. The current release of [[EM.Cube]] allows periodic boundary conditions only on the material type side walls of the higher priority. According to this rulecomputational domain, and not on the material types are ordered from the highest priority to the lowest in the following manner:top or bottom walls.

If planned carefully, taking advantage of EM.Tempo* Select the menu item 's material hierarchy rule would make ''Simulate &rarr; Computational Domain &rarr; Boundary Conditions''' or right click on the construction '''Boundary Conditions''' item in the '''Computational Domain''' section of complex objects easier. For example, a dielectric coated metallic cylinder can be modeled by two concentric cylinders: an inner PEC of smaller radius the Navigation Tree and an outer dielectric of larger radius as shown in select '''Boundary Conditions...''' from the illustration belowcontextual menu. The portion Boundary Conditions Dialog opens.* You need to assign the type of boundary condition on each of the dielectric cylinder that overlaps the inner PEC cylinder is ignored by the FDTD engine because the PEC cylinder takes precedence over the dielectric in the material hierarchysix domain boundaries: ±X, ±Y and ±Z. AlternativelyFor each face, you can model choose one of the same structure by an inner solid three options available: '''PEC cylinder enclosed by an outer hollow pipe-shaped dielectric cylinder''', '''PMC '''or '''PML'''.

===The FDTD Solution DomainAdvanced CPML Setup ===

The FDTD method requires a finiteIn open-extent solution domain. This is rather straightforward for shielded structuresboundary electromagnetic modeling problems, where you need a typical PEC enclosure box defines boundary condition that simply absorbs all the computational domainincoming radiation. For open-boundary structures like antennas and scatterersproblems of this nature, an absorbing boundary condition (ABC) is often chosen that effectively minimizes wave reflections at the computational domain must be truncated using appropriate termination boundary. [[EM.Tempo]] uses Convolutional Perfectly Matched Layers (CPML) for absorbing boundary conditions. The objective Usually two or more ABC layers must be placed at the boundaries of termination the physical structure to maximize wave absorption. The boundary conditions is CPML cells in the project workspace are not visible to eliminate the reflections from user. But, in effect, multiple rows of CPML cells are placed on the walls exterior side of each face of the visible domain box back to the computational domain.

In EMYou can set the number of CPML layers as well as their order.TempoThis is done through the CPML Settings Dialog, you which can define two types of domain box. A "be accessed by right clicking on the '''DefaultCPML'''" type domain is a box that is placed at a specified offset distance from item in the largest extents '''Computational Domain''' section of your physical structure (global bounding box). The offset is specified in free-space wavelengths. A "the navigation tree and selecting '''CustomCPML Settings...'''" type domainfrom the contextual menu. By default, on eight CPML layers of the other hand, third order are placed outside the FDTD problem domain. It is defined as recommended that you always try a fixedfour-size and fixed-location box in layer CPML first to assess the World Coordinate System (WCS)computational efficiency. In this case, you have to specify the coordinates The number of the lower left front corner CPML layers may be increased if a very low reflection is required (Corner 1<-40dB) and upper right back corner (Corner 2) of the domain box.

When you start a new project in {{Note|[[EM.Tempo, a ]]'s defaultquarter wavelength offset for the domain box and its 8-type domain is automatically created with a default layer CPML walls are very conservative choices and can be relaxed in many cases. An offset value set equal to a quarter eight free-space wavelength (0.25&lambda;₀). As soon as you draw your first object, grid cells beyond the largest bounding box usually gives a blue domain box shows up in the project workspace and encloses your object. As you add more objects and increase the overall size of your structurecompact, the domain box grows accordingly to encompass your entire physical structure. When you delete objects from the project workspacebut still valid, the domain box also shrinks accordingly.}}

[[Image:FDTD14Info_icon.png|thumb|300px|[[FDTD Module30px]]Click here to learn more about the theory of 's Domain Settings dialog''[[Basic_Principles_of_The_Finite_Difference_Time_Domain_Method#CPML_vs._PML | Perfectly Matched Layer Termination]]===Changing the Domain Settings==='''.

To set <table><tr><td> [[Image:FDTD MAN10.png|thumb|left|360px|The boundary CPML cells placed outside the solution visible domain of your FDTD project, follow these stepsbox.]] </td><td> [[Image:FDTD15.png|thumb|left|400px|CPML Settings dialog.]] </td></tr></table>

* Click the '''Domain''' [[Image:domain_icon.png]] button of the '''Simulate ''' Toolbar or select '''Menu > Simulate > Computational Domain > Domain Settings...''' or right click on the '''FDTD Domain''' item of the Navigation Tree and select '''Domain Settings...''' from the contextual menu, or use the keyboard shortcut '''Ctrl+A'''. The Domain Settings Dialog opens up, showing the current domain type selection.* Select one of the two options for '''Domain Type'''<nowiki>: </nowiki>'''Default''' or '''Custom'''.* If you select the "Default" domain type, the domain box is defined in terms of the offsets along the X, Y and Z directions from the largest extents of your physical structure. Select one of the two options for '''Offset Units: Grid''' and '''Wavelength'''. In the section titled '''"Domain Size"''', enter the amount of domain extension beyond the largest extents of the structure along the ±X, ±Y and ±Z directions. Note that in the case of a default-type domain box, the offset values based on your current project settings (frequency and units).* When the '''Wavelength''' option is selected for '''Offset Units''', additional free space is added around the structure by the specified ±X, ±Y and ±Z offsets in free space wavelengths. Note that the free space wavelength for this purpose is calculated at the center frequency of the project. The default value of the offset in this case is a quarter free space wavelength. Note that with this option, the number of the additional cells and their cell size is not fixed; they vary from structure === Using CPML to structure.* When the '''Grid''' option is selected for '''Offset Units''', the six offset values represent the number Model Structures of additional free-space mesh cells that are placed in each direction beyond the largest bounding box around the physical structure. The default value of the offset in this case is eight grid cells along the ±X, ±Y and ±Z directions.* If you select the "Custom" domain type instead, you need to enter values for the coordinates of the lower-left-front corner,''' Corner 1''', and the upper-right-back corner, '''Corner 2''', of the domain box.* After you change values or settings, click the '''Apply''' button to make the changes effective. To recover the default values, click the '''Defaults''' button of the dialog. Click '''OK''' to save the settings and close the dialog.Infinite Extents ===

By defaultYou can use EM.Tempo to model planar structures of infinite extents. A planar substrate usually consists of one or more dielectric layers, possibly with a PEC ground plane at its bottom. To model a laterally infinite dielectric substrate, you must assign a PML boundary condition to the four lateral sides of the domain box is shown as a wireframe box with blue linesand set the lateral domain offset values along the ±X and ±Y directions all equal to zero. You can change If the color planar structure ends in an infinite dielectric half-space from the bottom, you must assign a PML boundary condition to the bottom side of the domain box or hide itand set the -Z offset equal to zero. This leaves only the +Z offset with a nonzero value.

===When a domain boundary wall is designated as CPML and its has a zero domain offset, meaning it touches a material block, the CPML cells outside the domain wall are reflected back inside the computational domain. In other words, the effective number of CPML layers will be twice the one specified in the CPML Settings dialog. This will effectively extend the material block infinitely beyond the boundary wall and will create an open boundary effect in the specified direction. It goes without saying that only "substrate" objects are supposed to touch the boundary walls in such a scenario. Because of the rolled-back CPML cells inside the domain, it is very important to make sure that other finite-sized parts and objects stay clear from the domain walls as well as from the Domain Boundary Conditions===invisible "interior" CPML cells.

[[Image:FDTD13.png{{Note|thumb|300px|[[FDTD Module]]'s Boundary Conditions dialog]]The current release of EM.Tempo supports four types does not support full-anisotropic or dispersive or gyrotropic layers of laterally infinite extents. In other words, your anisotropic or dispersive or gyrotropic material objects must not touch the CPML domain boundary conditions:boundaries.}}

By default, all the six sides of the computational domain box are set to CPML, representing a completely open-boundary structure. Different boundary conditions can be assigned to each of the six walls of the domain box. The periodic boundary conditions are special ones that are assigned through == EM.Tempo's Periodicity Dialog and will be discussed later under modeling of periodic structures. The current release of [[EM.Cube]] allows periodic boundary conditions only on the side walls of the computational domain, and not on the top or bottom walls.Excitation Sources ==

* Select '''Menu > Simulate > Computational Domain > Boundary Conditions''' or right click on the '''Boundary Conditions''' item in the '''Computational Domain''' section of the Navigation Tree and select '''Boundary ConditionsBefore you can run an FDTD simulation, you have to define a source to excite your project’s physical structure.EM..''' from the contextual menu. The Boundary Conditions Dialog opens.* You need to assign the type Tempo offers a variety of boundary condition excitation mechanisms for your physical structure depending on each your particular type of the six domain boundariesmodeling problem or application: ±X, ±Y and ±Z. For each face, choose one of the three options available: '''PEC''', '''PMC '''or '''PML'''.

The {| class="wikitable"|-! scope="col"| Icon! scope="col"| Source Type! scope="col"| Applications! scope="col"| Host Object! scope="col"| Spatial Domain! scope="col"| Restrictions / Additional Requirements|-| style="width:30px;" | [[File:lumped_src_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Lumped Source |Lumped Source]]| style="width:250px;" | General-purpose point voltage source| style="width:200px;" | PEC and PMC boundary conditions are the most straightforward or thin wire line parallel to set up and usea principal axis| style="width:200px;" | A single point| style="width:200px;" | None|-| style="width:30px;" | [[File:distrb_src_icon. Assigning the png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Distributed Source |Distributed Source]]| style="width:250px;" | General-purpose distributed planar source with a uniform, edge-singular or sinusoidal impressed field profile| style="width:200px;" | Virtual rectangle strip parallel to a principal plane| style="width:200px;" | A rectangular area| style="width:200px;" | None|-| style="width:30px;" | [[File:mstrip_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Microstrip Port |Microstrip Port Source]]| style="width:250px;" | Used for S-parameter computations in microstrip-type structures| style="width:200px;" | PEC boundary rectangle strip parallel to one of a principal plane| style="width:200px;" | A vertical rectangular area underneath the bounding walls of the solution domain simply forces the tangential component of host strip| style="width:200px;" | Requires a PEC ground plane strip underneath the electric field to vanish at all points along that wallhost strip|-| style="width:30px;" | [[File:cpw_icon. Similarlypng]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials, assigning the PMC boundary _Sources,_Devices_%26_Other_Physical_Object_Types#Coplanar Waveguide (CPW) Port |Coplanar Waveguide (CPW) Port Source]]| style="width:250px;" | Used for S-parameter computations in CPW-type structures| style="width:200px;" | PEC rectangle strip parallel to a principal plane| style="width:200px;" | Two parallel horizontal rectangular areas attached to one of the bounding walls of opposite lateral edges the solution domain forces host center strip| style="width:200px;" | Requires two parallel PEC ground strips on the tangential component two sides of the magnetic field to vanish at all points along that wallhost center strip|-| style="width:30px;" | [[File:coax_icon. For planar png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Coaxial Port |Coaxial Port Source]]| style="width:250px;" | Used for S-parameter computations in coaxial-type structures with | style="width:200px;" | PEC Cylinder oriented along a principal axis| style="width:200px;" | A circular ring area enveloping the host inner conductorcylinder| style="width:200px;" | Requires a concentric hollow outer conductor cylinder|-backed substrate| style="width:30px;" | [[File:wg_src_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials, you can use the _Sources,_Devices_%26_Other_Physical_Object_Types#Waveguide Port |Waveguide Port Source]]| style="width:250px;" | Used for S-parameter computations in waveguide structures| style="width:200px;" | Hollow PEC boundary condition to designate box oriented along a principal axis| style="width:200px;" | A rectangular area at the bottom cross section of the substrate host hollow box| style="width:200px;" | The host box object can have one capped end at most. |-| style="width:30px;" | [[File:hertz_src_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Filamentary_Current_Source |Filamentary Current Source]]| style="width:250px;" | General-purpose wire current source of two types: Hertzian short dipole radiator and long wire current source with a uniform, triangular or sinusoidal current distribution profile| style="width:200px;" | None (stand-alone source)| style="width:200px;" | A line| style="width:200px;" | Hertzian short dipole radiators can have an arbitrary orientation, but long wire current sources must be aligned along one of the principal axes|-Z Domain Wall| style="width:30px;" | [[File:plane_wave_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Plane Wave |Plane Wave Source]]| style="width:250px;" | Used for modeling electromagnetic scattering & computation of reflection/transmission characteristics of periodic surfaces | style="width:200px;" | None (stand-alone source) as | style="width:200px;" | Surface of a PEC groundcube enclosing the physical structure| style="width:200px;" | None|-| style="width:30px;" | [[File:gauss_icon. For shielded waveguide structurespng]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials, you can designate all _Sources,_Devices_%26_Other_Physical_Object_Types#Gaussian Beam |Gaussian Beam Source]]| style="width:250px;" | Used for modeling focused beams | style="width:200px;" | None (stand-alone source)| style="width:200px;" | Surface of a cube enclosing the lateral walls as PECphysical structure| style="width:200px;" | None|-| style="width:30px;" | [[File:huyg_src_icon. Similarly to model shielded cavity resonatorspng]]| style="width:150px;" | [[Glossary of EM.Cube's Materials, you designate all the six walls as PECSources, Devices & Other Physical Object Types#Huygens Source |Huygens Source]]| style="width:250px;" | Used for modeling equivalent sources imported from other [[EM.Cube]] modules | style="width:200px;" | None (stand-alone source)| style="width:200px;" | Surface of a cube | style="width:200px;" | Imported from a Huygens surface data file|}

In many electromagnetic modeling problems you need a boundary condition that simply absorbs all the incoming radiation. For problems of this nature, an absorbing boundary condition (ABC) is often chosen that effectively minimizes wave reflections at the boundary. EM.Tempo uses Convolutional Perfectly Matched Layers (CPML) for absorbing boundary conditions. The boundary CPML cells in the project workspace are transparent to the user. But, in effect, multiple rows of CPML cells are placed Click on the exterior side of each face of the visible domain boxcategory to learn more details about each source type and how to define one.

Click here to learn more [[Image:Info_icon.png|30px]] More information about all the theory of source types can be found in the '''[[Perfectly Matched Layer TerminationGlossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types]]'''.

Click here to learn more about [[Advanced CPML Setup]]In the most general sense, one can consider two fundamental types of excitation sources for an FDTD simulation: a lumped source and a distributed source. A lumped sources is localized at a single mesh point in the computational domain, while a distributed source is spread over several mesh cells. Among the source types of the above list, the microstrip port, CPW port, coaxial port, waveguide port, plane wave and Gaussian beam sources are indeed special cases of a distributed source for specific applications.

===Modeling Planar Structures A lumped source is the most commonly used way of Infinite Extents===exciting a structure in EM.Tempo. A lumped source is a voltage source with a series internal resistor that must be placed on a PEC or thin wire line object that is parallel to one of the three principal axes. A lumped source is displayed as a small red arrow on the host line. Lumped sources are typically used to define ports and compute the port characteristics like S/Y/Z parameters. Using simple lumped sources, you can simulate a variety of transmission line structures including filters, couplers or antenna feeds. This approach may become less accurate at higher frequencies when the details of the feed structure become important and can no longer be modeled with highly localized lumped ports. In such cases, it is recommended to use “Distributed Sources”, which utilize accurate modal field distributions at the ports for calculation of the incident and reflected waves. Waveguide source is used to excite the dominant TE₁₀ mode of a hollow rectangular waveguide. Other special types of distributed sources are microstrip port, CPW port and coaxial ports that can be used effectively to excite their respective transmission line structures.

You When you create an array of an object type that can use EM.Tempo to model planar structures of infinite extents. A planar substrate usually consists of host one or more dielectric layers, possibly with a PEC ground plane at its bottom. To model a laterally infinite dielectric substrate, you must assign a PML boundary condition to the four lateral sides of the domain box and set the lateral domain offset values along the ±X and ±Y directions all equal to zero. If the planar structure ends in an infinite dielectric half-space from the bottomabove source types, you must assign can also associate a PML boundary condition to the bottom side of the domain box and set the -Z offset equal to zero. This leaves only the +Z offset source array with a nonzero valuethat array object.

{{Note|The current release of [[EMImage:Info_icon.Tempopng|30px]] does not support anisotropic or dispersive layers of laterally infinite extents. In other words, your anisotropic or dispersive material objects must not touch the CPML domain boundariesClick here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Finite-Sized_Source_Arrays | Modeling Finite-Sized Source Arrays]]'''.}}

<td> [[Image:FDTD22(1)FDTD MAN11.png|thumb|300px360px|The computational domain A plane wave box of enclosing a metallic sphere with nonzero offset in all directionsPEC cylinder at oblique incidence: &theta; = 105&deg; and &phi; = 315&deg;.]] </td><td> [[Image:FDTD24FDTD MAN12.png|thumb|400px360px|The computational domain A Gaussian beam box of a laterally infinite planar structure with enclosing a PEC ground cylinder at oblique incidence: &theta; = 105&deg; and zero ±X and ±Y &phi; = 315&deg;. The concentric circles represent the beam's focus point and -Z domain offsetsradius.]] </td>

== Generating Ports are used to order and index sources for circuit parameter calculations like S/Y/Z parameters. In EM.Tempo, you can define ports at the FDTD Mesh ==location of the following types of sources:

=== The FDTD Mesh *[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types ===#Lumped Source |Lumped sources]]*[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Distributed Source |Distributed sources]]*[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Microstrip Port |Microstrip port sources]]*[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Coplanar Waveguide (CPW) Port |CPW port sources]]*[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Coaxial Port |Coaxial port sources]]*[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Waveguide Port |Waveguide port sources]]

[[Image:FDTD28.png|thumb|350px|The adaptive FDTD meshes Every time you create a new source with one of the above types, the program asks if you want to initiate a metallic sphere.]][[EM.Tempo]]'s FDTD mesh is a rectangular Yee mesh that extends to new port and associate it with the entire computational domainnewly created source. It is primarily constructed from three mesh grid profiles along If the XYphysical structure of your project workspace has N sources, YZ and ZX principal planes. These projections together create a 3D rectangular (voxel) mesh space. Straight linesthen N default ports are defined, boxes and rectangular plates whose edges are aligned with the three principal axes are the simplest objects one port assigned to mesh each source according to their order in EMthe navigation tree.Tempo. Such objects preserve their exact shapes after discretization. All the objects with curved edges and curved surfaces You can define any number of ports equal to or objects with straight edges and flat faces that are not parallel to less than the principal axes or principal planes (such as oblique lines and slanted lateral faces total number of a pyramid) are discretized using a staircase (Yee) profilesources in your project.

If your physical structure has two or more sources, but you have not defined any ports, all the sources will excite the structure simultaneously during the simulation. However, when you assign N ports to the sources, then you have a multiport structure that is characterized by an N×N scattering matrix, an N×N impedance matrix, and an N×N admittance matrix. To calculate these matrices, EM.Tempo's adaptive mesh generator uses a variable staircase profile, where binary excitation scheme in conjunction with the cell sizes principle of grid line spacing vary with linear superposition. In this binary scheme, the curvature (derivative) structure is analyzed a total of N times. Each time one of the edge or faceN port-assigned sources is excited, and all the other port-assigned sources are turned off. As a resultIn other words, the FDTD solver runs a higher mesh resolution is achieved at "more curvyport sweep" areas to better capture the geometrical detailsinternally.When the ''j''th port is excited, all the S_ij parameters are calculated together based on the following definition:

You have the option to choose one of the three FDTD mesh types:<math> S_{ij} = \sqrt{\frac{Re(Z_i)}{Re(Z_j)}} \cdot \frac{V_j - Z_j^*I_j}{V_i+Z_i I_i} </math>

The default choice is the adaptive meshIn summary, which is a quite sophisticated mesh. The resolution of the adaptive FDTD mesh is driven by the '''Mesh Density'''to analyze an N-port structure, expressed in cells per effective wavelengthEM. Since Tempo runs N separate FDTD is a timemarching loops. The S/Z/Y parameters are frequency-domain method quantities. The port voltages and currents are Fourier-transformed to the excitation waveform may have a wideband spectral content, the effective wavelength is calculated based on the highest frequency of domain over the project: f<sub>max<frequency range [fc-bw/sub> = f₀ 2, fc+ &Delta;fbw/2], where f₀ fc is your project's the center frequency and &Delta;f (or BW) bw is its specified the bandwidthof your project. In other words, You can reduce the effective wavelength frequency range of the Fourier transform by settings new values for '''Start''' and '''End''' frequencies in the free space is &lambda;<sub>0"Port Definition" dialog as long as these are within the range [fc-bw/2,eff<fc+bw/sub> = c / f_max2]. By default, c being 200 frequency samples are taken over the speed of light in specified frequency range. This number can be modified from the free spaceFDTD simulation engine settings dialog.

The adaptive FDTD mesh{{Note|In order to obtain correct results, however, produces different grid cell sizes in the free space regions and inside dielectric regions. The effective wavelength in a dielectric material with relative permittivity e_r and permeability µ_r is given by &lambda;_d,eff = &lambda;_0,eff / &radic;&epsilon;_r&mu;_r. Therefore, port impedance must equal the average ratio characteristic impedance of the cell size in a dielectric region to transmission line on which the cell size in the free space port is 1/&radic;(&epsilon;_r&mu;_r). The adaptive FDTD mesh generator also takes note of the geometrical features of the objects it discretizesestablished. This is more visible in the case not automatically taken care of curved solids, curves surfaces and curved wires or obliquely oriented planes and lines which need to be approximated using a staircase profileby EM. The mesh resolution varies with the slope of the geometrical shapes and tries to capture the curved segments in the best way. Another important feature of the adaptive FDTD mesher is generation of gradual grid transitions between low-density and high-density mesh regions. For example, this often happens around the interface between the free space and high permittivity dielectric objects. Gradual mesh transitions provide better accuracy especially in the case of highly resonant structuresTempo.}}

According to the Courant-Friedrichs-Levy (CFL) stability criterion, the FDTD time step is determined by the smallest cell size in your FDTD mesh. Occasionally, [[FDTD Module]]'s adaptive mesh generator may create extremely tiny grid cells that would result in extremely small time stepsImage:Info_icon. This would then translate into a very long computation time. [[EM.Cubepng|30px]] offers Click here to learn more about the "Regular" FDTD mesh generator, which is a simplified version of the adaptive mesh generator. In a regular FDTD mesh, the grid cell sizes stay rather the same in objects of the same material composition. The mesh resolution increases in materials of higher permittivity and/or permeability based on the effective wavelength in exactly the same way as the adaptive mesh. Finally, '''[[EMGlossary_of_EM.Cube%27s_Simulation_Observables_%26_Graph_Types#Port_Definition_Observable | Port Definition Observable]]'s FDTD Modules offers a "Uniform" FDTD mesh generator. The uniform mesh consists of three uniform grids along the XY, YZ and ZX principal planes. In other words, the grid cell sizes &Delta;x, &Delta;y and &Delta;z are fixed throughout the entire computational domain. In this case, the uniform mesh generator has to fit your physical structure to the fixed mesh, rather than adapting the mesh to your physical structure''.

{{Note[[Image:Info_icon.png|When choosing a mesh type for your FDTD simulation, keep in mind that adaptive and regular mesh types are frequency-dependent and their density varies with the highest frequency of your specified bandwidth, while the uniform mesh type is always fixed and independent of your project30px]] Click here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Coupled_Sources_.26_Ports | Modeling Coupled Sources & Ports]]'''s frequency settings.}}

===Viewing the <table><tr><td> [[Image:FDTD Mesh===MAN15.png|thumb|left|640px|A two-port CWP transmission line segment.]] </td></tr><tr><td> [[Image:FDTD MAN16.png|thumb|left|480px|EM.Tempo's port definition dialog.]] </td></tr></table>

To generate When an FDTD mesh and view it the simulation starts, your project workspace's source starts pumping energy into the computational domain at t > 0. Maxwell's equations are solved in all cells at every time step until the solution converges, follow these or the maximum number of time stepsis reached. A physical source has a zero value at t = 0, but it rises from zero at t > 0 according to a specified waveform. EM.Tempo currently offers four types of temporal waveform:

In [[EMA sinusoidal waveform is single-tone and periodic.Cube]]Its spectrum is concentrated around a single frequency, which is equal to your project's "Mesh View" mode, you can rotate or pan the view of the project workspacecenter frequency. A Gaussian pulse decays exponentially as t &rarr; &infin;, but you cannot edit the objects. '''"Show Mesh"''' generates it has a new mesh and displays it if there lowpass frequency spectrum which is none in the memoryconcentrated around f = 0. A modulated Gaussian pulse decays exponentially as t &rarr; &infin;, or and it simply displays an existing mesh in the memory. This is has a useful feature because generating an FDTD mesh may take a long time depending on the complexity of structure and the total size of the computational domainbandpass frequency spectrum concentrated around your project's center frequency. If you change the structure or alter the mesh settingsFor most practical problems, a new mesh is always generated. You can ignore any mesh in the memory and force [[modulated Gaussian pulse waveform with EM.Cube]] to generate a fresh FDTD mesh from the ground up by selecting Tempo'''Menu > Simulate >Discretization > Regenerate Mesh''' or by right clicking on the '''Yee Mesh''' item of the Navigation Tree and selecting '''Regenerate''' from the contextual menus default parameters provides an adequate performance.

{{twoimg|FDTD34.png|A human head model and a cellular phone handset The accuracy of the FDTD simulation results depends on its sidethe right choice of temporal waveform.|FDTD33EM.png|The regular FDTD mesh Tempo's default waveform choice is a modulated Gaussian pulse. At the end of an FDTD simulation, the time domain field data are transformed into the human head model and frequency domain at your specified frequency or bandwidth to produce the cellular phone handsetdesired observables.}}

[[Image:FDTD80Info_icon.png|thumb|600px|30px]] Click here to learn more about EM.Tempo's Mesh Settings dialog'''[[Basic_Principles_of_The_Finite_Difference_Time_Domain_Method#The_Relationship_Between_Excitation_Waveform_and_Frequency-Domain_Characteristics | Standard & Custom Waveforms and Discrete Fourier Transforms]]'''.

[[=== Defining Custom Waveforms in EM.Cube]]'s [[FDTD Module|FDTD module]] discretizes objects using what is often referred to as the “staircase approximation”. In this mesh generation scheme, the structure is recreated using a large number of cubic cells carefully assembled in a way that approximates the shape of the original structure. By default, a carefully calculated, "<u>'''Adaptive'''</u>" mesh of your physical structure is generated in order to satisfy the following criteria:Tempo ===

* Optimize the number of mesh cells in each dimension. The product of the number of cells in each dimension determines the total mesh size. The larger the mesh size, the longer the simulation In some time-domain applications, especially with the CPU version of the FDTD engine. Also, a very large mesh size requires more RAM, which you may exceed your GPU memory capacity. Set the '''Minimum Mesh Density''' to a moderately low value want to keep simulate the mesh size manageable, but be careful not to set it too low (see the next item below).* Ensure simulation accuracy by requiring an acceptable minimum number propagation of cells per wavelength through each object and a certain kind of waveform in the empty (free) space between them and the computational domain boundaries. An effective wavelength is defined for each material at the highest frequency of the project's specified spectrum. We recommend a '''Minimum Mesh Density '''of at least 15-20 cells/ wavelengthcircuit or structure. But for some resonant structures, 25 or even 30 cells per wavelength may be required In addition to achieve acceptable accuracy. As you reduce the mesh densitydefault waveforms, the simulation accuracy decreasesEM.* Accurately represent and approximate the boundaries of edges or surfaces that are not grid-aligned by closely adhering Tempo allows you to their geometric contours. This is controlled define custom waveforms by the '''Minimum Grid Spacing Over Geometric Contours''', which can be specified either as a fraction of the free space grid spacing time or as an absolute length value frequency specifications for each individual source in your project units.* Maximize If you open up the minimum grid spacing in property dialog of any dimension inside the computational domain and thus maximize the simulation time step. The time step size is dictated by the CFL stability criterion and is driven by the smallest grid spacing source type in each dimensionEM. The smaller the time stepTempo, the larger the number of time steps required for convergence. This is controlled using the '''Absolute Minimum Grid Spacing''', which can be specified either as a fraction of the free space grid spacing or as you will see an absolute value{{key|Excitation Waveform. It is critical to accurately represent and precisely maintain the object edge/surface boundaries ..}} button located in certain structures like resonant antennas and filters, as the phase "Source Properties" section of the reflected fields/waves is affected by the object boundary positionsdialog. When object boundaries are very close to each other, the mesh needs to represent them by two separate, but very closely spaced, grid linesClicking this button opens up EM. To control the minimum allowed grid spacing, use the Tempo'''Absolute Minimum Grid Spacing '''settingss Excitation Waveform dialog. From this dialog,* Maintain a smooth grid with no abrupt jumps from low-density to high-density regionsyou can override EM. This feature is enabled with the Tempo'''Create Gradual Grid Transitions '''check box (always checked by s default)waveform and customize your own temporal waveform.The Excitation Waveform dialog offers three different options for defining the waveform:

Click here to learn more about [[Advanced Meshing The first option, which is also the default option, constructs an optimal modulated Gaussian pulse waveform based on your project's specified center frequency and bandwidth. This optimal waveform guarantees the most accurate frequency domain computations for your simulation. The second option gives you a choice of the three standard waveforms and lets you define their waveform parameters in EMterms of frequency domain characteristics like center frequency and bandwidth and spectral contents.Tempo]]The third option lets you define a completely arbitrary temporal waveform for your source.

Before When you can run an FDTD simulation, you have to define a source custom waveform in the Excitation Waveform dialog, make sure to click the {{key|Accept}} button of the dialog to excite make your project’s physical structurechanges effective. EM.Tempo offers a variety A graph of excitation mechanisms your custom waveform is plotted in the right panel of the dialog for your physical structure depending review. It is important to keep in mind that typical time scales in the FDTD simulation of RF structures are on your particular type the order of modeling problem nanosecond or application:smaller. Using the variable "fc" in the expression of your waveform definition usually takes care of this required scaling. Otherwise, you need to use scaling factors like 1e-9 explicitly in your expression. For example, in the figure below, we have defined a modulated Bessel waveform in the form of "sp.j0(t/2e-9)*sin(2*pi*fc*t)", where sp.j0(x) denotes the zeroth-order Bessel function of the first kind burrowed from Python's special functions module.

# '''Ideal Source'''[[Image: A stand-alone localized voltage source with an internal resistanceInfo_icon.# png|30px]] Click here to learn more about '''Lumped Source''': An ideal source that must be place on a wire (a PEC line object).# '''Distributed Source''': A source with a prescribed impressed field component that is defined on a rectangular region [[Glossary of space parallel to a principal planeEM.# Cube'''Waveguide Source''': A distributed source that must be placed across a hollow PEC box object.s Python Functions# Standard Python Functions | Python's Standard & Advanced Mathematical Functions]]''Plane Wave Source''': A distributed source with a plane wave profile defined using a virtual box object enclosing the entire physical structure. # '''Gaussian Beam Source''': A distributed source with a complex-valued focused Gaussian beam profile defined using a virtual box object enclosing the entire physical structure. # '''Huygens Source''': A distributed source defined based on know tangential electric and magnetic field components on the surface of a virtual box object.

===Defining {{Note| If you define a New Source===custom excitation waveform for your source, none of the standard frequency domain output data and parameters will be computed at the end of your FDTD simulation.}}

To create a new source, follow these steps<table><tr><td> [[Image:FDTD MAN14.png|thumb|left|720px|EM.Tempo's excitation waveform dialog showing a custom modulated Bessel temporal waveform defined using the Python function sp.j0(x).]]</td></tr></table>

* Right click on the name of the source type in the '''Sources''' section of the navigation tree and select '''Insert New Source..== EM.Tempo''' from the contextual menu. This opens up the respective Source Dialog.* You can change the default name of the source as well as its color.* Change the location of the source, if necessary, by the changing the values of the supplied coordinate fields. * Change the polarization of the source, if necessary. * In the '''Source Properties''' section, you can specify the '''Source Amplitude''' in Volts and the '''Phase''' in Degrees.* To change the amplitude and/or phase of a source, click the button labeled '''Excitation Waveform''' to open the Waveform Dialog.* From the waveform dialog, you can also change the waveform type, if necessary.s Active & Passive Devices ==

An ideal source acts as a voltage source in series with an internal resistance that can be placed between any two adjacent mesh grid nodes anywhere in the computational domain# '''[[Glossary of EM. The ideal source is displayed as a small orange arrow in the project workspaceCube's Materials, Sources, Devices & Other Physical Object Types#Resistor | Resistor]]''' # '''[[Glossary of EM. By defaultCube's Materials, Sources, Devices & Other Physical Object Types#Inductor | Inductor]]'''# '''[[Glossary of EM.TempoCube's Materials, Sources, Devices & Other Physical Object Types#Capacitor | Capacitor]] creates a +Z-directed ideal source located at the origin ''' # '''[[Glossary of coordinates (0EM.Cube's Materials, 0Sources, 0)Devices & Other Physical Object Types#Series_RL_Device | Series RL Device]]''' # '''[[Glossary of EM. You can change the direction Cube's Materials, Sources, Devices & Other Physical Object Types#Parallel_RC_Device | Parallel RC Device]]''' # '''[[Glossary of the ideal source to ±XEM.Cube's Materials, ±Y or ±ZSources, Devices & Other Physical Object Types#Diode | Nonlinear Diode]]''' # '''[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Active_Lumped_One-Port_Device | Active Lumped One-Port Device]]''' # '''[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Active_Lumped_Two-Port_Device | Active Lumped Two-Port Device]]'''

===Lumped Source===devices are connected between two adjacent FDTD mesh nodes. Although lumped devices are not sources and the passive types do not excite a structure, their properties are similar to lumped sources. That is why they are listed under the '''Sources''' section of the navigation tree. A lumped device has to be associated with a PEC line object that is parallel to one of the three principal axes. Similar to lumped sources, lumped devices have an '''Offset''' parameter that is equal to the distance between their location on the host line and its start point.

A lumped source device is the most commonly used way of exciting characterized by a structure in [[EM.Tempo]]. A lumped source is indeed an ideal source that must be placed on a line object that is parallel to one of the three principal axes and shows up as a small red arrow on the host line. Lumped sources are typically used to define ports and compute the port characteristics like S/Y/Z [[parameters]]. Lumped sources can also be place on line arrays. The property dialog of a lumped source has a dropv-down list that contains the name of all the legitimate line objects (i.e. lines that are parallel to one of the principal axes) and line arrays. The '''Offset''' parameter of a lumped source is its distance from the start point of the host line. A lumped source by default is placed at the center of its host line. In other words, the default offset value is equal to half the length equation of the host line object.form:

:<math>i(t) = L \{{Note|In order to create a lumped source, you must have at least one line object or line array in the project workspace.}v(t) \}</math>

A real waveguide structure is usually excited using some type {{Note|Small values of strategically located probe mechanism. EM.Tempo also provides inductance may result in the '''Waveguide Source''', a special type divergence of source that excites a prescribed TE₁₀ modal field distribution in a hollow rectangular waveguide structurethe FDTD numerical scheme. The scattering [[parameters]] are calculated from knowledge of incident To avoid this problem, you need to increase the mesh resolution and reflected fields at designated waveguide portsadopt a higher mesh density. Waveguide sources typically provide more accurate results for scattering [[parameters]] compared This, of course, may lead to lumped ports as they represent the actual dominant propagating modes at the transmission line portsa much longer computation time.}}

{{Note<table><tr><td> [[Image:FDTD MAN17.png|In order to define a waveguide source, you must have at least one hollow box object with no caps or only one end cap or a hollow box array in your projectthumb|left|480px|EM.Tempo's lumped device dialog for nonlinear diode.]] </td></tr><tr><td> [[Image:FDTD MAN17A.png|thumb|left|480px|EM.Tempo's lumped device dialog for active lumped two-port device.}}]] </td></tr></table>

Waveguide sources are a special case of distributed sources in # '''[[EMGlossary_of_EM.Tempo]]. A Distributed Source is defined in a rectangular plane of finite extentsCube%27s_Materials, parallel to one of the three principal coordinate planes. An impressed electric field component is assumed across the specified rectangular area_Sources, which pumps energy into the computational domain. The current version of _Devices_%26_Other_Physical_Object_Types#Active_Distributed_One-Port_Device | Active Distributed One-Port Device/Circuit]]''' # '''[[EMGlossary_of_EM.TempoCube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Active_Distributed_Two-Port_Device | Active Distributed Two-Port Device/Circuit]] provides three spatial field profiles for a distributed source:'''

# Uniform# Sinusoidal# EdgeUnlike the active lumped devices, these devices are rather distributed and their behavior is similar to a microstrip port source. In other words, the active distributed one-Singularport device requires a rectangle strip object as a host, while the active distributed two-port device requires two rectangle strip objects for its definition. You can choose one of the edges of the strip object for establishing the circuit port. In the case of a two-port device, you need two parallel and end-to-end aligned strip objects.

The sinusoidal type has the functional form cos(py/w), and the edge-singular type has the functional form 1/v(1-(2y/w)^2), where y circuit behavior of these devices is defined by a Netlist file. Their property dialog provides a text editor for simply writing the coordinate along the direction Netlist description of field variation measured from the center of device. You can also import an existing external Netlist file with a ".CIR" or ".TXT" file extension using the rectangular area and w is its total widthbutton labeled {{key|Load Netlist}}..

{{Note|[[Image:fdtd_src7_tnRF.png|thumb|250px|Spice A distributed source placed between two horizontal rectangular strips./D]]can generate a Netlist file corresponding to an existing circuit project, which can then be saved to a text file with a ".TXT" file extension. }}

In the '''Excitation Plane''' section of the dialog, first you have to select the orientation of the source plane. The dropdown list labeled '''Direction''' gives three options<table><tr><td> [[Image: '''X, Y''' and '''Z''', which create planes parallel to the YZ, ZX and XY principal planes, respectivelyActiveOnePort. Depending on the choice of the plane orientation, another dropdown list labeled png|thumb|left|480px|EM.Tempo'''Field Dir''' gives four options for the direction of the source field component. For example, the default plane orientation is X (parallel to the YZs active one-Plane) and the available field directions are +Y, -Y, +Z and -Z. Next, you have to enter the coordinates of two opposite corners of the source plane: the lower left and upper right corners. You can type in values for the X, Y, Z coordinates or you can use the spin buttons to slide the default source planes in the project workspaceport device/circuit dialog.]] </td></tr></table>

# '''Resistor''' with a Resistance value (R) in OhmsWhen your physical structure contains an active device, EM.# '''Capacitor''' with a Capacitance value (C) in pF.# '''Inductor''' with Tempo performs an Inductance value (L) in nHEM-circuit co-simulation that involves both the full-wave FDTD EM solver and the SPICE circuit solver. # '''Nonlinear Diode''' with In a Saturation Current (I_s)in fAglobal self-consistent co-simulation, ambient temperature (T) in degree Kelvinat each time step of the FDTD time marching loop, the electric and a dimensionless ideality factor (n)magnetic fields at the location of the device ports are used to compute the port voltages and currents. The default values These quantities are then used in the SPCIE circuit solver to update all the voltages and currents at the internal nodes of these [[parameters]] the active device. The updated port voltages and currents are 100fA, 300&deg;K finally used to update the electric and 1, respectivelymagnetic fields in the physical mesh cells and the time marching loop proceeds to the next time step.

Although lumped loads are not sources EM.Tempo can handle several active one-ports and do not excite a structuretwo-ports simultaneously. In that case, their properties all the devices are similar to lumped sources. Lumped Loads are incorporated automatically compiled into a single Netlist that serves as the input of the FDTD grid across two adjacent SPICE solver. The individual internal nodes in a similar manner of each device need to lumped sourcesbe renamed for the global Netlist. LikewiseBesides the main circuit, lumped loads are defined on Line objectsthe Netlist of each device may contain several "subcircuits". In order to create a lumped load, you must have at least one line object in your project. Lumped loads show up as small yellow arrows on their host line object. Similar to lumped a source, a lumped load has an offset parameter Note that determines its location on the host linesubcircuit nodes are not re-indexed for the global Netlist as is expected.

{{Note|Small values of inductance may result in the divergence of the FDTD numerical scheme. To avoid this problem, If you need want to increase use a B-type nonlinear dependent source in the mesh resolution and adopt a higher mesh density. This, Netlist definition of coursean active one-port or two-port, may lead to it must be contained in a much longer computation timesubcircuit definition rather than in the main circuit.}}

{| border="0"|The figure below shows the geometry of a two-| valign="top"|[[Image:FDTD42port amplifier device with microstrip input and output transmission lines.png|thumb|250px|EM.Tempo's Ideal Source dialog]]| valign="top"|[[ImageThe Netlist of the two-port device is given below:FDTD43.png|thumb|250px| EM.Tempo's Lumped Source dialog]]| valign="top"|[[Image:FDTD44.png|thumb|250px| EM.Tempo's Waveguide Source dialog.]]| valign="top"|[[Image:FDTD45.png|thumb|250px| EM.Tempo's Distributed Source dialog]]| valign="top"|[[File:FDTD56.png|thumb|250px|EM.Tempo's Lumped Load dialog.]]|-|}

Using simple lumped sources, you can simulate a variety of transmission line structures in <table><tr><td> [[EMImage:Amp circ.Tempo]] including filters, couplers or antenna feeds and you can calculate their scattering [[parameters]]. This approach may become less accurate at very high frequencies when the details png|thumb|left|420px|The schematic of the feed structures become important and can no longer be modeled with highly localized lumped portsamplifier circuit in RF. In such cases, it is recommended to use “Distributed Sources”, which utilize accurate modal field distributions at the ports for calculation of the incident and reflected wavesSpice A/D. ]] </td></tr></table>

The direction of incidence is defined through the &theta; and &phi; angles of the unit propagation vector in the spherical coordinate system. The values of these angles are set in degrees in the boxes labeled '''Theta''' and '''Phi'''. The default incidence angles are &theta; = 180° and &phi; = 0° representing a normally incident plane wave propagating along the -Z direction with a +X-polarized E-vector. In the TM_z and TE_z polarization cases, the magnetic and electric fields are parallel to the XY plane, respectively. The components of the unit propagation vector and normalized E- and H-field vectors are displayed in the source dialog. This way of defining a plane wave source is more convenient when the structure is laid out along the XY plane and Z-axis such as layered and periodic structures. In the more general case of custom linear polarization, besides the incidence angles, you have to enter the components of the unit electric '''Field Vector'''. However, two requirements must be satisfied: '''ê . ê''' = R1 1 and '''ê × k''' = 0 . This can be enforced using the '''Validate''' button at the bottom of the dialog. If these conditions are not met, an error message is generated. The left-hand (LCP) and right-hand (RCP) circular polarization cases are restricted to normal incidences only (&theta; = 180°).2 50

EM.Tempo requires a finite plane wave incidence surface to calculate the excitation. When you create a plane wave sourceB1 3 4 v = 20*v(1, a plane wave box is created as part of its definition. A trident symbol on the box shows the propagation vector as well as the E-field and H-field polarization vectors. The time domain plane wave excitation is calculated on the surface of this box and injected into the computational domain. The plane wave box is displayed in the project workspace as a purple wireframe box enclosing the structure. Initially, the radio button '''Size: Default''' is selected. With this option, the boundaries of the excitation box always have a distance of three cells from the bounding box of the geometry and cannot be changed. The radio button '''Size: Custom''' allows you to set the excitation box manually. The values for the coordinates of '''Corner 1''' and '''Corner 2''' can now be changed. Corner 1 is the front lower left corner and Corner 2 is the rear upper right corner of the box. The box has to be defined in the world coordinate system (WCS).

{{Note| borderYou can use active one-ports to define custom voltage or current sources for your entire physical structure rather than using one of the physical excitation source types of the navigation tree.}} <table><tr><td> [[Image:Amp ex.png|thumb|left|550px|The geometry of a microstrip-based amplifier with an active two-port device.]] </td></tr></table> == EM.Tempo's Observables & Simulation Data Types== === Understanding the FDTD Observable Types === EM.Tempo's FDTD simulation engine calculates all the six electric and magnetic field components (E_x, E_y, E_z, H_x, H_y and H_z) at every mesh grid node at all time steps from t ="0until the end of the time marching loop. However, in order to save memory usage, the engine discards the temporal field data from each time step to the next. Storage, manipulation and visualization of 3D data can become overwhelming for complex structures and larger computational domains. Furthermore, calculation of some field characteristics such as radiation patterns or radar cross section (RCS) can be sizable, time-consuming, post-processing tasks. That is why EM.Tempo asks you to define project observables to instruct what types of output data you want in each simulation process. EM.Tempo offers the following types of output simulation data:  {| class="wikitable"|-! scope="col"| Icon! scope="col"| Simulation Data Type! scope="col"| Associated Observable Type! scope="col"| Applications! scope="col"| Restrictions|-| style="width:30px;" | [[File:fieldprobe_icon.png]]| style="width:150px;" | Temporal Waveforms| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Temporal_Field_Probe_Observable |Temporal Field Probe]]| style="width:300px;" | Computing electric and magnetic field components at a fixed location in the time domain| style="width:250px;" | None|-| style="width:30px;" | [[File:fieldprobe_icon.png]]| style="width:150px;" | Point Fields| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Temporal_Field_Probe_Observable |Temporal Field Probe]]| style="width:300px;" | Computing the amplitude and phase of electric and magnetic field components at a fixed location in the frequency domain| style="width:250px;" | None|-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Near-Field Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]] | style="width:300px;" | Computing the amplitude and phase of electric and magnetic field components on a planar cross section of the computational domain in the frequency domain| style="width:250px;" | None|-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Time-Domain Near-Field Animation| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]] | style="width:300px;" | Computing either total electric or total magnetic field distribution on a planar cross section of the computational domain in the time domain| style="width:250px;" | The field maps are generated at certain specified time intervals|-| style="width:30px;" | [[File:farfield_icon.png]]| style="width:150px;" | Far-Field Radiation Patterns| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Far-Field_Radiation_Pattern_Observable |Far-Field Radiation Pattern]]| style="width:300px;" | Computing the 3D radiation pattern in spherical coordinates | style="width:250px;" | Requires one of these source types: lumped, distributed, microstrip, CPW, coaxial or waveguide port|-| style="width:30px;" | [[File:farfield_icon.png]]| style="width:150px;" | Far-Field Radiation Characteristics| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Far-Field_Radiation_Pattern_Observable |Far-Field Radiation Pattern]]| style="width:300px;" | Computing additional radiation characteristics such as directivity, axial ratio, side lobe levels, etc. | style="width:250px;" | Requires one of these source types: lumped, distributed, microstrip, CPW, coaxial or waveguide port|-| style="width:30px;" | [[File:farfield_icon.png]]| style="width:150px;" | Far-Field Scattering Patterns| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Far-Field_Radiation_Pattern_Observable |Far-Field Radiation Pattern]]| style="width:300px;" | Computing the 3D scattering pattern in spherical coordinates | style="width:250px;" | Requires a plane wave or Gaussian beam source|-| style="width:30px;" | [[File:rcs_icon.png]]| style="width:150px;" | Radar Cross Section| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Radar_Cross_Section_(RCS)_Observable | RCS]] | style="width:300px;" | Computing the bistatic and monostatic RCS of a target| style="width:250px;" | Requires a plane wave source|-| style="width:30px;" | [[File:rcs_icon.png]]| style="width:150px;" | Polarimetric Scattering Matrix Data| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Radar_Cross_Section_(RCS)_Observable | RCS]] | style="width:300px;" | Computing the scattering matrix of a target for various plane wave source incident angles| style="width:250px;" | Requires a plane wave source|-| style="width:30px;" | [[File:port_icon.png]]| style="width:150px;" | Port Characteristics| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Port_Definition_Observable |Port Definition]] | style="width:300px;" | Computing the S/Y/Z parameters and voltage standing wave ratio (VSWR)| style="width:250px;" | Requires one of these source types: lumped, distributed, microstrip, CPW, coaxial or waveguide port|-| style="width:30px;" | [[File:port_icon.png]]| style="width:150px;" | Port Voltages, Currents & Powers| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Port_Definition_Observable |Port Definition]] | style="width:300px;" | Computing the port voltages, port currents and total port powers in both time and frequency domains| style="width:250px;" | Requires one of these source types: lumped, distributed, microstrip, CPW, coaxial or waveguide port|-| style="width:30px;" | [[File:period_icon.png]]| style="width:150px;" | Periodic Reflection & Transmission Coefficients| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Periodic Characteristics |Periodic Characteristics]] (No observable definition required) | style="width:300px;" | Computing the reflection and transmission coefficients of a periodic surface| style="width:250px;" | Requires a plane wave source and periodic boundary conditions |-| style="width:30px;" | [[File:energy_icon.png]]| style="width:150px;" | Electric and Magnetic Energy| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Energy-Power_Observable | Energy-Power]]| style="width:300px;" | Computing the electric, magnetic and total energy inside the entire computational domain in the time domain| style="width:250px;" | None|-| style="width:30px;" | [[File:energy_icon.png]]| style="width:150px;" | Dissipated Power| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Energy-Power_Observable | Energy-Power]]| style="width:300px;" | Computing the total dissipated power inside the entire computational domain in the time domain| style="width:250px;" | None|-| style="width:30px;" | [[File:energy_icon.png]]| style="width:150px;" | Electric and Magnetic Energy Density| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Energy-Power_Observable | Energy-Power]]| style="width:300px;" | Computing the electric, magnetic and total energy density on a field sensor plane in the frequency domain| style="width:250px;" | Requires at least one field sensor observable|-| style="width:30px;" | [[File:energy_icon.png]]| style="width:150px;" | Dissipated Power (Ohmic Loss) Density and Specific Absorption Rate (SAR) Density| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Energy-Power_Observable | Energy-Power]]| style="width:300px;" | Computing the dissipated power density and SAR density on a field sensor plane in the frequency domain| style="width:250px;" | Requires at least one field sensor observable|-| style="width:30px;" | [[File:energy_icon.png]]| style="width:150px;" | Poynting Vector| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Energy-Power_Observable | Energy-Power]]| style="width:300px;" | Computing the complex Poynting vector on a field sensor plane in the frequency domain| style="width:250px;" | Requires at least one field sensor observable

| valignstyle="topwidth:30px;"|[[ImageFile:FDTD46huyg_surf_icon.png]]|thumbstyle="width:150px;" |250pxEquivalent Electric and Magnetic Surface Currents| style="width:150px;" |[[FDTD Module]]Glossary of EM.Cube's Plane Wave dialogSimulation Observables & Graph Types#Huygens_Surface_Observable |Huygens Surface]]| valignstyle="topwidth:300px;"|Collecting tangential field data on a box to be used later as a Huygens source in other [[Image:FDTD47EM.png|thumbCube]] modules|style="width:250px;" |[[FDTD Module]]'s Gaussian Beam dialog]]None

===Strategy For An Accurate & Efficient FDTD SimulationExamining the Near Fields in Time and Frequency Domains ===

The EM.Tempo's FDTD method time marching loop computes all the six electric and magnetic field components at every Yee cell of your structure's mesh at every time step. This amounts to a formidable amount of data that is one computationally very inefficient to store. Instead, you can instruct EM.Tempo to save a small potion of the most versatile numerical techniques these data for solving electromagnetic modeling problemsvisualization and plotting purposes. Choosing Using a '''Field Probe''' at a specified point, you can record the right settings and optimal values for certain numerical [[parameters]] will have a significant impact on both accuracy and computational efficiency of an time-domain field component over the entire FDTD simulationloop. Below The time-domain results are also transformed to the frequency domain within the specified bandwidth using a number of steps that you should typically follow by order when planning your FDTD simulationdiscrete Fourier transform (DFT). <table><tr><td> [[Image:FDTD77.png|thumb|left|480px|Time-domain evolution of the electric field at a given point.]]</td></tr></table>

* Identify material types and proper domain boundary conditionsIn EM.* Identify Tempo, you can visualize the source type and excitation mechanism.* Define near fields at a specific frequency in a specific plane of the project observablescomputational domain.* Mesh To do so, you need to define a '''Field Sensor''' observable. EM.Tempo's field sensor defines a plane across the physical structure entire computational domain parallel to one of the three principal planes. The magnitude and examine phase of all the quality six components of the generated mesh electric and it geometric fidelity.* Determine magnetic fields on the proper temporal waveform.* Select mesh grid points on the simulation mode sensor plane are computed and run the FDTD enginedisplayed.

For certain problems, more than one combination or choice of settings and <table><tr><td> [[parameters]] may still give acceptable resultsImage:FDTD_FS2. In most cases, [[png|thumb|left|420px|EM.CubeTempo's Field Sensor dialog.]] tries to make these choices convenient for you by suggesting default settings or default parameter values. For example, </td></tr><tr><td> [[EMImage:FDTD_FS1_new.Cube]] png|thumb|left|480px|Three field sensor planes defined around a PEC ellipsoid illuminated by default generated am "adaptive" type mesh with a default density of 20 cells per effective wavelengthplane wave source. The default computational domain features CPML walls placed a quarter free-space wavelength away from ]] </td></tr></table><table><tr><td> [[Image:FDTD_FS3_new.png|thumb|left|360px|Electric field distribution above the large bounding box of the entire physical structurePEC plate. A modulated Gaussian waveform with certain optimal ]] </td><td> [[parameters]] is used to drive the project's excitation source by defaultImage:FDTD_FS4_new. You can change most of these settings arbitrarily. For example, you can set up your own computational domain with different types of boundary conditions, customize png|thumb|left|360px|Magnetic field distribution above the FDTD mesh by modifying a large number of mesh settings and use other types of excitation waveformsPEC plate.]] </td></tr></table>

{{Note|Keep === Computing Far-Field Characteristics in mind that you are always responsible for the choice of excitation source and the project observables. In other words, [[EM.Cube]] does not automatically provide a default excitation source or does not suggest default observables.}}FDTD ===

An FDTD simulation involves a number Far fields are typically computed in the spherical coordinate system as functions of numerical [[parameters]] that can be accessed the elevation and modified from azimuth observation angles &theta; and &phi;. Only far-zone electric fields are normally considered. When your physical structure is excited using a lumped source, a waveguide source, a distributed source, a short dipole source, or an array of such sources, the FDTD Engine Settings Dialogfar fields represent the radiation pattern of your source(s) in the far zone. To open this dialogIn that case, select you need to define a '''Menu > Simulate > Simulation Engine Settings... Radiation Pattern - Far Field Observable'''for your project. When your physical structure is illuminated by a plane wave source or open a Gaussian beam source, the far fields represent the scattered fields. In the case of a plane source, you can compute the radar cross section (RCS) of your target structure. In that case, you need to define an '''Run DialogRCS - Far Field Observable'''for your project. In the FDTD method, and click the far fields are calculated using a near-field-to-far-field transformation of the field quantities on a given closed surface. EM.Tempo uses rectangular boxes to define these closed surfaces. You can use EM.Tempo'''Settings''' button next s default radiation box or define your own custom box. Normally, the radiation box must enclose the entire FDTD structure. In this case, the calculated radiation pattern corresponds to the engine dropdown listentire radiating structure. Alternatively, you can define a custom radiation box that may contain only parts of a structure, which results in a partial radiation pattern.

In the " '''Convergence''' " section of the dialog, you can set the '''Termination Criterion''' for the FDTD time loop. The time loop must stop after a certain point in time. If you use a decaying waveform like a Gaussian pulse or a Modulated Gaussian pulse, after certain number of time steps, the total energy of the computational domain drops to very negligible values, and continuing the time loop thereafter would not generate any new information about your physical structure. By contrast, a sinusoidal waveform will keep pumping energy into the computational domain forever, and you have to force the simulation engine to exit the time loop. [[EM.Cube]]'s [[FDTD Module]] provides two mechanism to terminated the time loop. In the first approach, an energy-like quantity defined as U<subtable>n</subtr> = &Sigma; [ &epsilon;<sub>0</subtd>[[Image:FDTD_FF1.png|'''E_i,n'''thumb|² + &mu;₀left|'''H_i,n'''720px|² ]EM.&Delta;V_i is calculated and recorded at a large random set of points in the computational domainTempo's Radiation Pattern dialog. Here i is the space index and n is the time index. The quantity U_n]] has a zero value at t = 0 (i.e. n = 0), and its value starts to build up over time. With a Gaussian or Modulated Gaussian pulse waveform, U<subtd>n</subtr> reach a maximum value U<subtr>max</subtd> at some time step and starts to decline thereafter[[Image:FDTD_FF3. The ratio 10png|thumb|left|600px|EM.log( U<sub>nTempo's Radar Cross Section dialog.]] </subtd>/ U<sub>max</subtr>) expressed in dB is used as the convergence criterion. When its value drops below certain '''Power Threshold''', the time loop is exited. The default value of Power Threshold is -30dB, meaning that the FDTD engine will exit the time loop if the quantity U<sub>n</subtable> drops to 1/1000 of its maximum value ever. The second termination criterion is simply reaching a '''Maximum Number of Time Steps''' , whose default value set to 10,000. A third option, which is [[EM.Cube]]'s default setting (labeled "'''Both'''"), terminates the simulation as soon as either of the first two criteria is met first.

{{Note|Keep in mind that for highly resonant structures, you may have to increase The default radiation box is placed at an offset of 0.1&lambda;₀ from the maximum number largest bounding box of time steps to very large values above 20your physical structure. You can change the offset value from the "Far Field Acceleration" dialog,000which can be accessed by clicking the {{key|Acceleration...}}button of EM.Tempo's Radiation Pattern dialog. Calculation of far-field characteristics at high angular resolutions can be a very time consuming computational task. You can accelerate this process by setting a lower '''Max. Far Field Sampling Rate''' from the same dialog. The default sampling rate is 30 samples per wavelength. A low sampling rate will under-sample the mesh grid points on the radiation box.

The "<table><tr><td> [[Image:FDTD_FF2.png|thumb|left|480px|EM.Tempo'''Acceleration'''" section of the FDTD Simulation Engine Settings s far field acceleration dialog give three options for the FDTD kernel:.]] </td></tr></table>

# Serial CPU Solver# Multi=== Radiation Pattern Above a Half-Core CPU Solver# GPU SolverSpace Medium ===

The serial CPU solver is [[In EM.Cube]]'s basic FDTD kernel that run the time marching loop on Tempo, you can use CPML boundary conditions with zero offsets to model a single central processing unit (CPU) of your computerstructure with infinite lateral extents. The default option is the multi-core CPU solver. This is a highly parallelized version calculation of the FDTD kernel based on far fields using the Opennear-MP framework. It takes full advantage of a multifield-core, multito-CPU architecturefar-field transformation requires the dyadic Green's function of the background structure. By default, if your computer does have one. The GPU solver is a hardware-accelerated the FDTD kernel optimized engine uses the free space dyadic Green's function for CUDA-enabled graphical processing unit (GPU) cardsthe far field calculation. If your computer has a fast NVIDIA GPU card with enough onboard RAMIn general, the GPU kernel can speed up your FDTD simulations up to 50 times or more over the single CPU solverEM.Tempo provides the dyadic Green's functions for four scenarios:

For structures excited with a # Free space background# Free space background terminated in an infinite PEC ground plane wave source, there are two standard FDTD formulations: '''Scattered Field '''(SF) formulation and '''Total Field - Scattered Field''' (TF-SF) formulation. [[EM.Cube]]'s [[FDTD Module|FDTD module]] offers both formulations. The TF-SF solver is at the default choice and is typically much faster than the SF solver for most problems. In two cases, when the structure has periodic boundary conditions or bottom# Free space background terminated in an infinite CPML boundary conditions (zero domain offsets), only PMC ground plane at the SF solver is available. The other sections of the FDTD Simulation Engine Settings dialog will be described next bottom# Free space background terminated in the context of [[Waveforms and Discrete Fourier Transforms]].an infinite dielectric half-space medium

Once In other words, EM.Tempo lets you build your physical calculate the far field radiation pattern of a structure in the project workspace presence of any of the above four background structure types. You can set these choices in EM.Tempo's "Far Field Background Medium" dialog. To access this dialog, open the Radiation Pattern dialog and define an excitation sourceclick the button labeled {{key|Background...}}. From this dialog, you are ready to run an FDTD simulationcan also set the Z-coordinate of the top of the terminating half-space medium. The simulation engine will run even if If you have not defined any observables. Obviouslyset the -Z boundary condition of your computational domain to PEC or PMC types, no simulation data the cases of infinite PEC or PMC ground planes from the above list are automatically selected, respectively, and the Z-coordinates of the ground plane and the bottom face of the computational domain will be generated in that caseidentical. [[EM.Cube]]'s [[FDTD Module]] currently offers several different simulation modes as follows:

# Analysis# Frequency Sweep# Parametric Sweep# Angular Sweep# RThe fourth case applies when your computational domain ends from the bottom in a dielectric layer with a CPML -Z boundary along with a -Z domain offset equal to zero. If you set the lateral domain offset values along the ±X and ±Y directions equal to zero, too, , then your structure is, in effect, terminated at an infinite half-space dielectric medium. In that case, you have to specify the permittivity &epsilon;<sub>r</T Macromodel# Dispersion Sweep# Huygens Sweep# [[Optimization]]# HDMRsub> and electric conductivity &sigma; of the terminating medium in the Background Medium dialog. You may additionally want to set the Z-coordinate of the top of that dielectric layer as the position of the interface between the free space and the lower dielectric half-space. Note that the current version of EM.Tempo does not calculate the far-field Green's function of a conductor-backed, dielectric substrate with a finite layer thickness. To use the background medium feature of EM.Tempo, your structure can have either an infinite PEC/PMC ground or a dielectric half-space termination.

<table><tr><td> [[Image:FDTD57fdtd_out36_tn.png|thumb|250pxleft|Figure 1360px|Radiation pattern of a vertical dipole above PEC ground.]] </td><td> [[Image: fdtd_out37_tn.png|thumb|left|360px|Radiation pattern of a vertical dipole above PMC ground.]] </td></tr><tr><td> [[EMImage:fdtd_out38_tn.png|thumb|left|360px|Radiation pattern of a horizontal dipole above PEC ground.Cube]]'s FDTD Simulation dialog</td><td> [[Image:fdtd_out39_tn.png|thumb|left|360px|Radiation pattern of a horizontal dipole above PMC ground.]]</td></tr></table>

Analysis is the simplest === Generating and most straightforward simulation mode of the [[FDTD Module]]. It runs the FDTD time marching loop once. At the end of the simulation, the timeWorking with Multi-domain field data are transformed into the frequency domain using a discrete Fourier transform (DFT). As a result, you can generate wideband frequency data from a single time-domain simulation run. The other simulation modes will be explained later in this manual.Frequency Simulation Data ===

To open One of the Simulation Run Dialog, click primary advantages of the '''Run''' [[Image:run_iconFDTD method is its ability to run wideband EM simulations.png]] button of The frequency domain data are computed by transforming the '''Simulate Toolbar''' or select '''Menu > Simulate > Runtime-domain data to the Fourier domain.This is done automatically when EM..''' from Tempo computes the menu bar or use the keyboard shortcut '''Ctrl+R'''port characteristics such as S/Z/Y parameters.The following frequency-domain observables are defined at a single frequency:

To start the FDTD simulation, click the '''Run''' button at the bottom of this dialog. Once the simulation starts, the "'''Output Window'''" pops up and reports messages during the different stages of the FDTD simulation. During the FDTD time marching loop, after every 10th time step, the output window updates the values of the time step, elapsed time, the engine performance in Mega* Near-cells per seconds, and the value of the convergence ratio U_n/U_max in dB. An [[EM.Cube]] FDTD simulation is terminated when the ratio U_n/U_max falls below the specified power threshold or when the maximum number of time steps is reached. You can, however, terminate the FDTD engine earlier by clicking the '''Abort Simulation''' button.Field Sensor* Far-field Radiation Pattern* RCS* Huygens Surface

{{isoimg|FDTD66The default computation frequency of the above observables is the project's center frequency (fc).png|[[FDTD Module]]You can change the observable frequency from the observable's output windowproperty dialog and enter any frequency in Hz. The reason these types of simulation data are computed at a single frequency is their typically very large size. However, you can define as many instances of these observables and set different frequency values for each one. In the case of radiation pattern and RCS, there are two dialogs that can be accessed from the navigation tree. Right-click on the "Fer-Field Radiation Patterns" or "Radar Cross Sections" items of the navigation tree and select '''Insert Multi-Frequency Radiation Pattern...''' or '''Insert Multi-Frequency RCS...''' from the contextual menu.}}

When an FDTD simulation starts, your project's source starts pumping energy into Using the computational domain at t > 0. Maxwell's equations are solved in all cells at every time step until the solution convergesmulti-frequency dialogs, or you can set the maximum number of time steps is reached. A physical source has a zero value at t = 0of Start Frequency, but it rises from zero at t > 0 according to a specified waveformStop Frequency and Step Frequency in Hz. [[EMYou can also set the values of Theta Angle Increment and Phi Angle Increment in degrees.Tempo]] currently offers four types The default values of temporal waveformboth quantities are 5&deg;. In the case of RCS, you have choose one of the two options:'''Bistatic RCS''' or '''Monostatic RCS'''.

# Sinusoidal# Gaussian Pulse# Modulated Gaussian Pulse# Arbitrary UserTo facilitate the process of all the defining multi-Defined Functionfrequency observables in EM.Tempo, you can also use the following Python functions at the command line:

A sinusoidal waveform is single-tone and periodic. Its spectrum is concentrated around a single frequency, which is equal to your project's center frequency. A Gaussian pulse decays exponentially as t &rarr; &infin;, but it has a lowpass frequency spectrum which is concentrated around f = 0. A modulated Gaussian pulse decays exponentially as t &rarr; &infin;, and it has a bandpass frequency spectrum concentrated around your project's center frequency. For most practical problems, a modulated Gaussian pulse waveform with EM.Tempo's default [[parameters]] provides an adequate performance. ---

If you use a Gaussian pulse or a modulated Gaussian pulse waveform to drive your FDTD sourceemag_field_sensor_multi_freq(f1, after a certain number of time stepsf2, the total energy of the computational domain drops to very negligible levels. At the pointdf, you can consider your solution to have converged. If you drive your FDTD source by a sinusoidal waveformdir_coordinate, the total energy of the computational domain will oscillate indefinitelyx0, and you have to force the time loop to terminate after a certain number of time steps assuming a steady state have been reached.y0,z0)

The accuracy of the FDTD simulation results depends on the right choice of temporal waveform. [[EM.Cube]]'s default waveform choice is a modulated Gaussian pulse. At the end of an FDTD simulationemag_farfield_multi_freq(f1, the time domain field data are transformed into the frequency domain at your specified frequency or bandwidth to produce the desired observables. f2,df,theta_incr,phi_incr)

In addition to the default waveformsemag_rcs_bistatic_multi_freq(f1, [[EM.Cube]] allows the ability to define custom waveforms by either time or frequency specifications on a per source basis.f2,df,theta_incr,phi_incr)

===In the above Python functions, f1 and f2 are the start and stop frequencies, respectively, and df is the frequency increment, all expressed in Hz. Note that the above commands simply create and insert the specified observables in the navigation tree. They do not run perform a simulation. The FDTD Observable Types===created observables have the same "base name" with ordered numeric indices. For example, far-field radiation patterns are names as Multi_FF_1, Multi_FF_2, ...

In [[EM.Cube]], project observables are Tempo also provides some additional Python functions for the simulation data that are generated by the simulation engine at the end of each simulation run. [[EM.Cube]]'s FDTD simulation engine calculates all the six electric and magnetic far-field components (E_x, E_y, E_z, H_x, H_y and H_z) at every mesh grid node at all time steps from t = 0 until the end of the time loop. However, in order to save memory space, the engine has to destroy the temporal field data from each time step to the next and reuse the memory. Storage, manipulation and visualization of 3D data can become overwhelming for complex structures and larger computational domains. Furthermore, calculation of some field characteristics such as radiation patterns or radar cross section (and RCS) can be sizable, time-consuming, post-processing tasks. That is why [[EM.Cube]] asks you to define project observables to instruct why types of simulation data you seek in each simulation effort.

* '''Field Probe''' for monitoring E- and H-field components at a fixed location in both time and frequency domains. * '''Field Sensor''' for monitoring E- and H-field components on a cross section of the computational domain in both time and frequency domains.* '''Far Field Radiation Pattern''' for monitoring the radiation behavior of your structure.* '''Far Field RCS''' for monitoring the scattering behavior of your structure.* '''Huygens Surface''' for collecting tangential field data on a box. * '''Port Definition''' for calculating the S/Y/Z [[parameters]] and voltage standing wave ratio emag_farfield_consolidate(VSWRx1,x2,dx,base_name). * '''Domain Energy''' for calculating the total electric and magnetic energy in the computational domain.* '''Periodic Characteristics''' for calculating the reflection and transmission coefficients when your periodic structure is excited by a plane wave source.

Of [[EM.Tempo]]'s frequency domain observablesemag_rcs_consolidate(x1, the near fieldsx2, far fields and all of their associated [[parameters]] like directivitydx, RCS, etc., are calculated at a certain single frequency that is specified as part of the definition of the observable. On the other hand, port characteristics like S/Y/Z [[parameters]], VSWR and periodic characteristics like reflection and transmission coefficients, are calculated over the entire specified bandwidth of your project.base_name)

* Right click on the name of the observable type in the '''Observables''' section of the navigation tree and select '''Insert New Observable...''' from the contextual menu. This opens up the respective Observable Dialog.* You can change the default name of the observable as well as its color.* Change the location of the observableemag_farfield_average(n, if necessary, by the changing the values of the supplied coordinate fields. * Change the orientation of the observable, if necessary. * In the case of frequency domain observables, change the observed frequency or frequency range, if necessary. base_name)

Once you define an observableemag_rcs_average(n, you can always changes its [[parameters]] later from its property dialog, which can be accessed from its right-click contextual menu. You can also delete observables. base_name)

[[Image:FDTD75.png|thumb|300px|FDTD Field Probe Dialog]]Field probes monitor The two "consolidate" Python functions take the field components at a certain point in the computational domain. They record the timeresults of multi-domain field data during the entire time loop frequency simulation observables and compute their frequency spectrum using merge them into a discrete Fourier transformsingle data file. By computing The base name in the time domain fields at a certain location, you can examine the transient response case of a system at that location. This far-field radiation patterns is also very useful for monitoring the convergence of FDTD time marching loop"Multi_FF" as pointed out earlier. [[EM.Cube]]'s field probes allow you to save the temporal values The name of a field component at a specified point in the computational domain during resulting consolidated data file is the entire time marching loop. You can plot same as the time domain field components as base name with a function of the time step index"_All" suffix and a ". You can also plot DAT" file extension. In the spectral contents case of those far-field componentsradiation patterns, iit is "Multi_FF_All.eDAT". their Fourier transformThe two "explode" Python functions take a consolidated data file names as "base_name_All.DAT" and break it up into several single-frequency ".RAD" or ".RCS" data files. Finally, over the two "average" Python functions take several radiation pattern or RCS files with a common base name in the current project's specified frequency bandwidth. To define folder, compute their average and save the results to a new field probedata file named "base_name_ave" with a ".RAD" or ".RCS" file extensions, follow these steps:respectively.

* Right click on == Generating the '''Field Probe''' item FDTD Mesh in the '''Observables''' section of the Navigation Tree and select '''Insert New Observable...'''* You can change the default name of the probe as well as its color. The field probe is displayed as a small green arrow in the Project Workspace.* By default [[EM.Cube]] creates a field probe located at the origin of coordinates (0,0,0). You can move the probe to any location by changing its X, Y and Z coordinates.* In the Probe Location section of the dialog, you can also set the '''Direction''' of the probe from a dropdown list that contains ±X, ±Y and ±Z options. The default direction is +Z.Tempo ==

[[Image:FDTD76=== EM.png|thumb|300px|An X-directed probe placed above a PEC plate illuminated by a normally incident plane wave.]]Tempo's Mesh Types ===

At the end of an EM.Tempo generates a brick volume mesh for FDTD simulation, the electric and magnetic field components along the specified probe direction are saved at the probe's location. Both the time domain fields from t = 0 The FDTD mesh is a rectangular Yee mesh that extends to the last time step and their frequency entire computational domain spectrum are recorded. You can plot these data It is primarily constructed from three mesh grid profiles in EM.Grid, which can be accessed from [[EM.Cube]]'s Data Manager. To open data manager, click the '''Data Manager''' [[Image:data_manager_icon.png]] button of the '''Simulate Toolbar'''XY, or select '''Simulate > Data Manager''' from the menu bar, or right click on the '''Data Manager''' item of the Navigation Tree YZ and select Open Data ManagerZX principal planes... from the contextual menu, or use the keyboard shortcut '''Ctrl+D'''. In the Data manager Dialog, you see These projections together create a list 3D mesh space consisting of all the data files available for plotting. These include the time-domain and frequency-domain probe data files with '''.DAT''' and '''.CPX''' file extensions, respectively. Select any data file by clicking and highlighting its row a large number of cubic volume cells (voxels) carefully assembled in the table and then click the '''Plot''' button to plot the graph. The time-domain field probe is plotted on a Cartesian graph showing way that approximates the selected field component as a function shape of time step. The frequency-domain probe contains two Cartesian graphs: amplitude and phase of the selected field component over the project's frequency rangeoriginal structure.

{{twoimg|FDTD77In EM.png|Time domain component plotted vs. time|FDTD78.png|Probed field plotted vs. frequency.}}Tempo, you can choose one of the three FDTD mesh types:

===Frequency* Adaptive Mesh* Regular Mesh* Fixed-Domain Near Field Visualization===Cell Mesh

[[Image:FDTD71(1)EM.png|thumb|300px|[[Tempo's default mesh generator produces an adaptive brick mesh of your physical structure, whose mesh resolution varies with the frequency. As the operating frequency of your project increases, the default '''Adaptive''' FDTD Module]]mesh generator creates a larger number of smaller voxels for a given physical structure. The adaptive mesh is optimized in such a way as to capture all the geometric details, curvatures and thin slanted plates or sheets, which often pose a challenge to staircase meshing. It usually provides a reasonably accurate discretization of most complex structures. Occasionally, you may opt for a more regularized FDTD mesh with almost equal grid line spacings everywhere, but still with a frequency-dependent cell size. In that case, you can use EM.Tempo's Field Sensor dialog]] '''Regular''' FDTD mesh generator, which is indeed a simplified version of its adaptive mesh generator. The regular FDTD mesh enforces only two criteria: minimum mesh density and absolute minimum grid spacing. The grid cell sizes in this mesh are almost uniform in objects of the same material composition or in free-space regions.

In [[EM.Cube]] you can visualize the near fields at Tempo also offers a specific uniform, frequency in a specific plane -independent, '''Fixed-Cell''' FDTD mesh generator. The fixed-cell mesh consists of three uniform grids in the computational domainXY, YZ and ZX principal planes. At the end of an FDTD simulationHowever, all the time domain electric and magnetic field values are available at all uniform mesh nodes. These temporal quantities are transformed into cell dimensions along the frequency domain using discrete Fourier transforms three direction, i.e. &Delta;x, &Delta;y and &Delta;z do not have to be equal. The fixed-cell mesh generator tries to fit your physical structure to calculate the electric and magnetic fields on a specified sensor planemesh grid rather than adapting the mesh to your physical structure. To define a new Field Sensor, follow these steps:

* Right click on the '''Field Sensors''' item {{Note|When choosing a mesh type for your FDTD simulation, keep in mind that adaptive and regular mesh types are frequency-dependent and their density varies with the '''Observables''' section highest frequency of your specified bandwidth, while the Navigation Tree uniform mesh type is always fixed and select independent of your project'''Insert New Observables frequency settings...'''}}* The '''Label''' box allows you to change the sensor’s name. * Set the '''Direction''' of the field sensor[[Image:Info_icon. This is specified by the normal vector of the sensor plane. The available options are png|30px]] Click here to learn more about '''X''', '''Y''' and '''Z''', with the last being the default option.* By default [[EMPreparing_Physical_Structures_for_Electromagnetic_Simulation#Working_with_EM.Cube.27s_Mesh_Generators | Working with Mesh Generator]] creates a field sensor plane passing through the origin of coordinates (0,0,0) and coinciding with the XY plane. Note that the sensor plane extends across the entire computational domain. You can change the location of the sensor plane to any point by typing in new values for the X, Y and Z coordinates. Keep in mind that you can move a sensor plane only along the specified direction of the sensor. Therefore, only one coordinate can effectively be changed. As you increment or decrement this coordinate, you can observe the sensor plane moving along that direction in the project workspace.* The frequency at which the field is evaluated has to be specified in the box labeled '''Near Field Frequency''' in the project's frequency unit. By default, this is equal to the project's center frequency.

After closing the Field Sensor Dialog, the a new field sensor item immediately appears under the '''Observables''' section in the Navigation Tree and can be right clicked for additional editing. Once an FDTD simulation is finished, a total of 14 plots are added to every Field Sensor node in the Navigation Tree. These include the magnitude and phase of all three components of E and H fields and the total electric and magnetic field values at the specified frequency[[Image:Info_icon. png|30px]] Click on any of these items and a color-coded intensity plot of it is visualized in the project workspace. A legend box appears in the upper right corner of the field plot, which can be dragged around using the left mouse button. The values of the magnitude plots are normalized between 0 and 1. The legend box contains the minimum field value corresponding here to 0 of learn more about the color map, maximum field value corresponding to 1 properties of the color map, and the unit of the field quantity, which is V/m for E-field and A/m for H-field'''[[Glossary_of_EM. The values of phase plots are always shown in Radians between Cube%27s_Simulation-p and pRelated_Operations#Adaptive_Yee_Mesh | EM. To display the fields properly, the structure is cut through the field sensor plane, and only part of it is shown. If the structure still blocks your view, you can simply hide or freeze it. You can change the view of the field plot with the available view operations such as rotating, panning, zooming, etcTempo's Adaptive Brick Mesh Generator]]'''.

{{twoimg|FDTD72[[Image:Info_icon.png|Field Sensor (E30px]] Click here to learn more about the properties of '''[[Glossary_of_EM.Cube%27s_Simulation-field) Related_Operations#Fixed-Cell_Brick_Mesh |FDTD74EM.png|Field Sensor (HTempo's Fixed-field)}}Cell Brick Mesh Generator]]'''.

<table><tr><td> [[Image:FDTD73Tempo L11 Fig5.png|thumb|300pxleft|Cartesian graph 550px|A human head model and a cellular phone handset on its side.]] </td></tr><tr><td> [[Image:Tempo L11 Fig7.png|thumb|left|550px|The FDTD mesh of total magnetic field vsthe human head model and the cellular phone handset. Y-index along ]] </td></tr><tr><td> [[Image:Tempo L11 Fig8.png|thumb|left|550px|Another view of the FDTD mesh of the crosshair in human head model and the field senor planecellular phone handset.]]</td></tr></table>

You can plot frequency domain fields in EM.Grid on 2D Cartesian graphs. Using field probes, you can plot any frequency domain field component as a function of frequency over === Discretizing the specified bandwidth at any point within the computational domain. Physical Structure Using field sensors, you can plot the total frequency domain fields as a function of position (spatial coordinates) across the computational domain. Every field sensor has a crosshair made up of two perpendicular lines parallel to the boundaries of the sensor plane. When you define a field sensor for the first time, the crosshair passes through the origin of coordinates. You can change the location of the crosshair on the sensor plane using the other two coordinate boxes besides the one that moves the location of the sensor plane. At the end of an FDTD simulation, in addition to the 3D near field maps, [[EM.Cube]] also generates 2D Cartesian graphs of the total electric and magnetic fields along the two perpendicular crosshair lines. A total of four Cartesian data files are generated, two for total E-field and two for total H-field along the two lines. You can plot these data in EM.Grid, which can be accessed from [[EM.Cube]]'s Data Manager. To open data manager, click the '''Data Manager''' [[Image:data_manager_icon.png]] button of the '''Simulate Toolbar''', or select '''Simulate > Data Manager''' from the menu bar, or right click on the '''Data Manager''' item of the Navigation Tree and select Open Data Manager... from the contextual menu, or use the keyboard shortcut '''Ctrl+D'''. In the Data Manager dialog, you see a list of all the data files available for plotting including the frequency-domain sensor data files with a '''.DAT''' file extension. Select any data file by clicking and highlighting its row in the table and then click the '''Plot''' button to plot the graph. Frequency domain field sensor graphs show the total field as a function of cell index along one of the principal axes. If the FDTD mesh is uniform in that direction, the position is found by multiplying the cell index by the cell dimension and offsetting with respect to lower-front-left corner of the computational domain.Adaptive Yee Mesh ===

EM.Tempo's default mesh generator creates an adaptive brick volume mesh that uses a variable staircase profile, where the grid line spacings vary with the curvature (derivative) of the object edges or faces. As a result, a higher mesh resolution is produced at "curved" areas to better capture the geometrical details. The resolution of the adaptive FDTD mesh is driven by the '''Mesh Density''', expressed in cells per effective wavelength. Since FDTD is a time-domain method and the excitation waveform may have a wideband spectral content, the effective wavelength is calculated based on the highest frequency of the project: f_max ===Visualizing Field Evolution f₀ + &Delta;f/2, where f₀ (or fc) is your project's center frequency and &Delta;f (or bw) is its specified bandwidth. In other words, the effective wavelength in Time Domain=the free space is &lambda;_0,eff =c / f_max, c being the speed of light in the free space. The effective wavelength in a dielectric material with relative permittivity &epsilon;_r and permeability &mu;_r is given by &lambda;_d,eff =&lambda;_0,eff / &radic;&epsilon;_r&mu;_r.

In the course of the The adaptive FDTD time marching processmesh, a tremendous amount of data are generated that include all by default, produces different grid cell sizes in the six E/H field components at every Yee cell and at every time stepfree space regions than inside dielectric regions. The temporal field values at effective wavelength in a sensor plane are of particular interestdielectric material with relative permittivity e_r and permeability µ_r is given by &lambda;_d,eff = &lambda;_0,eff / &radic;&epsilon;_r&mu;_r. Such plots show Therefore, the evolution average ratio of the fields as cell size in a function of time starting from time t = 0, when all dielectric region to the fields are zero everywhere cell size in the computational domainfree space is 1/&radic;(&epsilon;_r&mu;_r). [[EM.Cube]] can record snapshots The adaptive FDTD mesh generator also takes note of the field sensor data as geometrical features of the time loop marches forwardobjects it discretizes. When you define a field sensor for This is more visible in the first timecase of curved solids, by default it displays the frequency domain near field data. In order to record curves surfaces and save the time domain data, you have curved wires or obliquely oriented planes and lines which need to open be approximated using a staircase profile. The mesh resolution varies with the field sensor's property dialog by right clicking on the field sensor's name in slope of the Navigation Tree geometrical shapes and selecting '''Properties...'''from tries to capture the contextual menu. In the section titled '''Sensor Domain''', select curved segments in the radio button labeled '''Time Domain'''best way. Also, in Another important feature of the section titled "Field Display - Multiple Plots", select one adaptive FDTD mesher is generation of the two radio buttons labeled '''Egradual grid transitions between low-Field''' or '''Hdensity and high-Field'''density mesh regions. By defaultFor example, the time domain field data are saved every 100 time steps. To change this setting, right click on often happens around the '''Field Sensors''' item in interface between the Navigation Tree free space and select '''Time Domain Settingshigh permittivity dielectric objects...''' from Gradual mesh transitions provide better accuracy especially in the contextual menu. In the Time Domain Settings Dialog, change the value case of the box labeled '''Sampling Interval (in time steps)'''highly resonant structures.

Time domain [[animation]] is available only for FDTD simulations * Optimize the number of "Analysis" typemesh cells in each dimension. It cannot be used The product of the number of cells in conjunction with sweep simulationsall the three dimension determines the total mesh size. Once The larger the FDTD Analysis is finishedmesh size, you can click any the longer the simulation time, especially with the CPU version of the field plots and visualize FDTD engine. Also, a very large mesh size requires more RAM, which may exceed your GPU memory capacity. Set the '''Minimum Mesh Density''' to a moderately low value to keep the mesh size manageable, but be careful not to set it too low (see the next item below).* Ensure simulation accuracy by requiring an acceptable minimum number of cells per wavelength through each object and in the main window or you can animate empty (free) space between them by right clicking on and the field sensorcomputational domain boundaries. An effective wavelength is defined for each material at the highest frequency of the project's name in specified spectrum. We recommend a '''Minimum Mesh Density '''of at least 15-20 cells/ wavelength. But for some resonant structures, 25 or even 30 cells per wavelength may be required to achieve acceptable accuracy. As you reduce the Navigation Tree mesh density, the simulation accuracy decreases.* Accurately represent and selecting approximate the boundaries of edges or surfaces that are not grid-aligned by closely adhering to their geometric contours. This is controlled by the '''[[Animation]]Minimum Grid Spacing Over Geometric Contours''' from , which can be specified either as a fraction of the contextual menufree space grid spacing or as an absolute length value in project units. You can change * Maximize the [[animation]] settings from minimum grid spacing in any dimension inside the computational domain and thus maximize the simulation time step. The time step size is dictated by the CFL stability criterion and is driven by the smallest grid spacing in each dimension. The smaller the time step, the larger the number of time steps required for convergence. This is controlled using the '''[[Animation]] Controls DialogAbsolute Minimum Grid Spacing''', which can be specified either as a fraction of the free space grid spacing or as an absolute value. Note that It is critical to accurately represent and precisely maintain the [[animation]] loop repeats itself indefinitely until you object edge/surface boundaries in certain structures like resonant antennas and filters, as the phase of the reflected fields/waves is affected by the object boundary positions. When object boundaries are very close to each other, the [[Animation]] Controls dialog or hit mesh needs to represent them by two separate, but very closely spaced, grid lines. To control the minimum allowed grid spacing, use the keyboard’s '''Esc KeyAbsolute Minimum Grid Spacing '''settings,* Maintain a smooth grid with no abrupt jumps from low-density to high-density regions. This feature is enabled with the '''Create Gradual Grid Transitions '''check box (always checked by default).

{{twoimg|FDTD121When [[EM.png|Field sensor setup Cube]] generates an FDTD mesh, a large number of geometrical considerations are taken into account. These include the bounding box of each object and its corners, the ends of a line, the apex of a cone or pyramid, or the locations of lumped sources, field probes and sensors, vertices of plane wave or far field boxes, to name a few examples. These points are “locked” as fixed grid nodes in the FDTD mesh. [[EM.Cube]] determines these points internally to generate a mesh that best approximates the original structure. As you saw earlier, you can use the FDTD mesh settings to control the shape and resolution of the mesh, for time-domain output|FDTD126example, around the curved portions of your structure, or on slanted lines or faces, etc.png|Time interval These settings}}are global and apply to all the objects making up your physical structure.

===Scattering Parameters You can control the global mesh more selectively using the Advanced FDTD Mesh Settings Dialog. To open this dialog, click the '''Advanced '''button at the bottom of the FDTD Mesh Settings dialog. For example, you can control the quality of the gradual grid transitions by setting the value of '''Max Adjacent Cell Size Ratio'''. The default value of this parameter is 1.3, which maintains a smooth grid line spacing scheme with no more than 1:1.3 ratio for adjacent cells. By default, grid lines are enforced at all source and Port Characteristics===observable locations. You have the option to disable this feature and round up source locations to their closest grid lines. You may also uncheck the box labeled "Adapt mesh resolution to material properties". In that case, the same effective wavelength will be used to determine the mesh resolution inside all materials as well as the free-space regions.

If your physical structure is excited by a Lumped Source or a Waveguide Source or a Distributed Source, and one or more ports have been defined, the FDTD engine calculates the scattering (S) <table><tr><td> [[parameters]], impedance (Z) [[parameters]] and admittance (Y) [[parameters]] of the selected portsImage:FDTD80. The S [[parameters]] are calculated based on the port impedances specified in the projectpng|thumb|left|720px|EM.Tempo's "Port Definition"mesh settings dialog. If more than one port has been defined in the project, the FDTD engine runs an internal port sweep. Each port is excited separately with all the other ports turned off. When the ''j''th port is excited, all the S]]<sub/td>ij</subtr></table> [[parameters]] are calculated together based on the following definition:

:The figures below compare the three types of the FDTD mesh for a dielectric ellipsoid with &epsilon;<mathsub> S_{ij} = \sqrt{\frac{Re(Z_i)}{Re(Z_j)}} \cdot \frac{V_j - Z_j^*I_j}{V_i+Z_i I_i} r</mathsub><!--[[Image:FDTD82(1)= 4. Note that the cell size inside the dielectric region is half the cell size in the air region.png]]-->

where V<subtable>i</subtr> is the voltage across Port i, I<subtd>i[[Image:FDTD MAN21.png|thumb|left|360px|The geometry of a dielectric ellipsoid with &epsilon;</sub> is the current flowing into Port i and Zr</sub>i= 4.]]</subtd> is the characteristic impedance of Port i<td> [[Image:FDTD MAN22. png|thumb|left|360px|The sweep loop then moves to adaptive mesh of the next port until all ports have been exciteddielectric ellipsoid. After the FDTD simulation is finished, the S [[parameters]] are written into output ASCII data files. Since these data are complex, they are stored as '''.CPX''' files. Every file begins with a header starting with "#". Besides the scattering [[parameters]], the admittance (Y) and impedance (Z) [[parameters]] are also calculated and saved in complex data files with '''.CPX''' file extensions. </td></tr></table>

Click here for more details on <table><tr><td> [[Image:FDTD MAN18.png|thumb|left|360px|The top view of the computation adaptive FDTD mesh of the dielectric ellipsoid.]]</td><td> [[Data_Visualization_and_Processing#Port_Characteristics Image:FDTD MAN19.png| Port Characteristicsthumb|left|360px|The top view of the regular FDTD mesh of the dielectric ellipsoid with the same mesh density.]]</td></tr><tr><td> [[Image:FDTD MAN20A.png|thumb|left|360px|The top view of the fixed-cell FDTD mesh of the dielectric ellipsoid using the larger cell size inside the air region.]]</td><td> [[Image:FDTD MAN20.png|thumb|left|360px|The top view of the fixed-cell FDTD mesh of the dielectric ellipsoid using the smaller cell size inside the dielectric region.]]</td></tr></table>

===Far Field Calculations in The figures below compare the low resolution and high resolution adaptive FDTD===meshes of a PEC parabolic reflector. This structure involves both a curved surface and a very thin surface.

{{mainpage|<table><tr><td> [[Farfield Calculations in EMImage:FDTD MAN23.png|thumb|left|450px|The geometry of a PEC parabolic reflector.Tempo]]}}</td></tr></table>

For radiating structures or scatterers, the far field quantities are of primary interest. <table><tr><td> [[EMImage:FDTD MAN24.png|thumb|left|360px|The low-resolution adaptive mesh of the PEC parabolic reflector.Cube]]'s </td><td> [[Image:FDTD ModuleMAN27.png|thumb|left|360px|The high-resolution adaptive mesh of the PEC parabolic reflector.]] can calculate </td></tr><tr><td> [[Image:FDTD MAN26.png|thumb|left|360px|The top (XY) view of the far field radiation patterns low-resolution adaptive mesh of an antenna or the radar cross section PEC parabolic reflector.]]</td><td> [[Image:FDTD MAN25.png|thumb|left|360px|The right (RCSYZ) view of a target. In general, by far fields we mean the electric fields evaluated in the far zone low-resolution adaptive mesh of a physical structure, which satisfies the following condition:PEC parabolic reflector.]]</td></tr></table>

In the FDTD method, the far fields are calculated using a near-field-to-far-field transformation of the field quantities on a given closed surface. [[EM.Cube]] uses rectangular boxes to define these closed surfaces. You can use [[EM.Cube]]'s default radiation box or define your own. Normally, the radiation box should enclose the entire FDTD structure. In this case, the calculated radiation pattern corresponds === Adding Fixed Grid Points to the entire radiating structure. The radiation box may also contain only parts of a structure, which results in partial radiation patterns.Adaptive Yee Mesh ===

[[Image:FDTD116* Open the Fixed Grid Points Dialog by selecting '''Menu > Simulate > Discretization > Fixed Grid Points.png|thumb|250px|[[..''' or by right-clicking on the '''FDTD Module]]'s Radiation Pattern dialog]]'' '''Mesh''' item of the navigation tree and selecting '''Fixed Grid Points Settings...'''[[Image:fdtd_out26_tn* Click the {{key|Add/Edit}} button to open the "Add Fixed Grid Point" dialog.png|thumb* Enter the (X, Y, Z) coordinates of the new fixed point in the coordinate boxes and click the {{key|400px|The 3D total radiation pattern OK}} button.* To modify the coordinates of an existing fixed grid point, select it from the table and click the {{key|Add/Edit}} button.* You can also remove a dipole antenna: polar typefix grid point from the FDTD mesh using the {{key|Delete}} button.]]

For any far field calculations in <table><tr><td> [[EMImage:FDTD36.Cube]], first you have to define a far field observable png|thumb|left|480px|A user-defined fixed grid point in the Navigation Tree. In [[an FDTD Modulemesh.]], defining </td></tr><tr><td> [[Image:FDTD38.png|thumb|left|480px|Adding a far field observable also initiates a far field box new fixed grid point in the computational domainEM. This box is used to perform the near-to-far-field transformation at the end of an FDTD simulationTempo's fixed grid points settings dialog. To insert a new far field box, follow these steps]] </td></tr><tr><td> [[Image:FDTD39.png|thumb|left|480px|The "Add Fixed Grid Point" dialog.]] </td></tr></table>

* Right click on the '''Far Fields''' item in the '''Observables''' section of the Navigation Tree and select '''Insert New Radiation Pattern...''' According to open the Radiation Pattern Dialog.* Use Courant-Friedrichs-Levy (CFL) stability criterion, the '''Label''' box to change FDTD time step is determined by the name of the far field or change the color of the far field box using the '''Color''' button.* The frequency of radiation pattern calculation can be specified smallest cell size in the box labeled '''Far Field Frequency'''your FDTD mesh. By defaultOccasionally, this is equal to the center frequency of the projectEM. However, you can calculate the far field data at any other frequency within the projectTempo's frequency range.* The resolution of far field calculations is specified by '''Angle Increment''' expressed adaptive mesh generator may create extremely tiny grid cells that would result in degreesextremely small time steps. By default, the &theta; and &phi; angles are incremented by 5 degreesThis would then translate into a very long computation time.* Define the desired box for far field calculations in the '''Radiation Box''' section of the dialog[[EM. As in Cube]] offers the case of plane waves and Gaussian beams, there are two options available"Regular" FDTD mesh generator, which is a default radiation box (radio button '''Size: Default''') or a user defined radiation box (radio buttons '''Size: Custom'''). If you check '''Size: Default''', no radiation box corner coordinates need to be specified. The radiation box will always be 0.1 free space wavelength away from the bounding box simplified version of the entire structureadaptive mesh generator. Select '''Size: Custom''' to set In a regular FDTD mesh, the far field box manually. The values for grid cell sizes stay rather the coordinates same in objects of '''Corner 1''' and '''Corner 2''' can now be changed. '''Corner 1''' is the lower-front-left corner and '''Corner 2''' is the upper-rear-right corner of the radiation boxsame material composition. The dimensions are specified mesh resolution increases in the world coordinate system (WCS).* At the end materials of an FDTD simulation, besides calculating higher permittivity and/or permeability based on the radiation data over the entire (spherical) 3D space, a number of 2D pattern graphs are also generated. These are indeed pattern cuts at certain planes, which include the three principal XY, YZ and ZX planes plus one additional constant f-cut. This latter cut is at &phi; = 45° by default. You can assign another phi angle in degrees effective wavelength in exactly the box labeled '''Non-Principal Phi Plane'''. Also, the 2D radiation pattern graphs are normalized by default. You can instruct [[EM.Cube]] to plot the 2D pattern graphs un-normalized (same way as calculated) by removing the check mark from the box labeled '''Normalize 2D Patterns'''adaptive mesh.

After closing === Profiling the Far Field Dialog, a far field entry immediately appears with its given name under the '''Far Fields''' item of the '''Observables''' section in the Navigation Tree. A far field box shows up as a light blue wireframe box in the project workspace. You can right click on the far field item's name in the navigation tree and select '''Properties...''' to open up the radiation pattern dialog for further editing. Bear in mind that a full 3D radiation pattern calculation with a high angular resolution might be very time-consuming.Brick Mesh ===

Once an FDTD simulation A volumetric brick mesh is finishedoverwhelming for visualization in the 3D space. For this reason, three far field items are added to [[EM.Cube]]'s mesh view shows only the Far Field section outline of the Navigation Treecells on exterior surface of the (staircased) meshed objects. These are The mesh grid planes provide a 2D profile of the far-zone E-field component mesh cells along &phi; direction, the far-zone E-field component along &phi; direction principal coordinate planes. To display a mesh grid plane, select '''Menu > Simulate > Discretization > Grid Planes >''' and pick one of the total far-zone E-fieldthree options: '''XY Plane''', '''YZ Plane''' or '''ZX Plane'''.The 3D plots can be viewed You may also right click on one of the '''XY Plane''', '''YZ Plane''' or '''ZX Plane''' items in the project workspace by clicking on each item'''Discretization''' section of the navigation tree and select '''Show''' from the contextual menu.

The view of the 3D far field plot can be changed with the available view operations such as rotate, pan and zoom. A legend box appears in the upper right corner of the 3D radiation pattern plot, which can be dragged around with the left mouse button. If the structure blocks the view of the radiation patternWhile a mesh grid plane is visible, you can simply hide or freeze move it back and forth between the entire physical structure or parts of it. Note that 3D radiation patterns are always positioned two boundary planes at the origin (0,0,0) two opposite sides of the spherical world coordinate system even though the radiation center of the structure may not be located at that pointcomputational domain. The (maximum) '''Directivity''' You can do this in one of the radiating structure is displayed at the bottom of the legend box and is calculated using the definitionfollowing four ways:

At * Using the end of an FDTD simulation, the radiation pattern data E<subkeyboard's Page Up {{key|PgUp}} key and Page Down {{key|PgDn}} key.* By selecting '''Menu >&theta;</subSimulate >, E<subDiscretization >&phi;</subGrid Planes > and E_tot in Increment Grid''' or ''' Decrement Grid'''.* By right clicking on one of the three principal '''XYPlane''', '''YZ and Plane''' or '''ZX planes plus one additional user defined phi plane cut are available for plotting on 2D graphs Plane''' items in the '''EM.GridDiscretization'''. There are a total section of eight 2D pattern graphs in the data manager: 4 polar graphs navigation tree and 4 Cartesian graphs of selecting '''Increment Grid''' or ''' Decrement Grid''' from the same pattern datacontextual menu.* Using the keyboard shortcut {{key|>}} or {{key|<}}.

At the end of an FDTD sweep simulation, other radiation characteristics are also computed as a function of the sweep variable (frequency, angle, As you “step through” or any other user defined variable). These include profile the '''Directivity (D0)'''mesh grid, '''Total Radiated Power (PRAD)''' and '''Directive Gain (DG)''' as a function of you can see how the &theta; and &phi; angles. Another radiation characteristic of interest especially in circularly polarized scenarios structure is the Axial Ratio. In [[EM.Cube]], the axial ratio is always defined in the LCP_z or RCP_z sense based on the X- and Y-components of the electric field. In order to calculate the directive gain or axial ratio, you have to check the boxes labeled '''Axial Ratio (AR)''' or '''Directive Gain (DG)''' in the "Additional Radiation Characteristics" section of the '''Radiation Pattern Dialog'''. Four 2D Cartesian graphs of the axial ratio as functions of the theta angle are generated in the three principal XY, YZ and ZX discretized along internal planes as well as the additional user defined phi plane cut. At the end of an FDTD sweep simulation, the directive gain and axial ratio can also be plotted as functions of the sweep variable. In that case, either quantity needs to be computed at a fixed pair of &theta; and &phi; angles. These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radiation Pattern Dialog'''. The default values of the user defined azimuth and elevation are both zero corresponding to the zenithcomputational domain.

As mentioned earlier when discussing boundary conditions and computational domainWhen your physical structure is discretized using the brick mesh generator, a second coordinate system becomes available to you. The mesh grid coordinate system allows you can use CPML boundary conditions with zero offsets to model a structure with infinite lateral extentsspecify any location in the computational domain in terms of node indices on the mesh grid. At [[EM.Cube]] displays the end total number of mesh grid lines of the FDTD simulationdomain (N_x × N_y × N_z) along the three principal axes on the '''Status Bar'''. Therefore, the far fields are calculated using number of cells in each direction is one less than the nearnumber of grid lines, i.e. (N_x-field-to1)× (N_y-far1) × (N_z-field transformation1). This calculation requires the dyadic Green's function The lower left front corner of the background structure. By defaultdomain box (Xmin, Ymin, Zmin) becomes the FDTD engine uses origin of the free space dyadic Green's function for the far field calculation. In generalmesh grid coordinate system (I = 0, J = 0, K = 0). The upper right back corner of the [[FDTD Module]] features dyadic Green's functions for four scenarios:domain box (Xmax, Ymax, Zmax) therefore becomes (I = N_x-1, J = N_y-1, K = N_z-1).

# Free space background# Free space background terminated [[EM.Cube]] allows you to navigate through the mesh grid and evaluate the grid points individually. Every time you display one of the three mesh grid planes, the "'''Grid Coordinate System (GCS)'''" is automatically activated. On the Status Bar, you will see [[Image:statusgrid.png]] instead of the default [[Image:statusworld.png]]. This means that the current coordinates reported on Status Bar are now expressed in an infinite PEC ground grid coordinate system. The current grid point is displayed by a small white circle on the current mesh grid plane at , and it always starts from (I = 0, J = 0, K = 0). Using the bottom# Free space background terminated in an infinite PMC ground keyboard's '''Arrow Keys''', you can move the white circle through the mesh grid plane and read the current node's (I, J, K) indices on the status bar. You can switch back to the "'''World Coordinate System (WCS)'''" or change to the "'''Domain Coordinate System'''" by double-clicking the status bar box that shows the current coordinate system and cycling through the three options. The domain coordinate system is one that establishes its origin at the bottom# Free space background terminated lower left front corner of the computational domain and measure distances in an infinite dielectric half-space mediumproject unit just like the WCS.

[[Image:FDTD131.png|thumb|300px|[[== Running FDTD Module]]'s RCS dialog]]Simulations in EM.Tempo ==

When the physical structure is illuminated by a plane wave source, the calculated far field data indeed represent the scattered fields. In that case, the incident and scattered fields can be separated. [[=== EM.Cube]] can calculate the radar cross section (RCS) of a target defined as:Tempo's Simulation Modes ===

:<math>\sigma_{\theta} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{\theta}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}Once you build your physical structure in the project workspace and define an excitation source, \quad \sigma_{\phi} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{\phi}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}you are ready to run an FDTD simulation. The simulation engine will run even if you have not defined any observables. Obviously, \quad \sigma = \sigma_{\theta} + \sigma_{\phi} = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{tot}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}</math><!--no simulation data will be generated in that case. [[Image:FDTD130EM.pngTempo]]-->currently offers several different simulation modes as follows:

To compute the RCS {| class="wikitable"|-! scope="col"| Simulation Mode! scope="col"| Usage! scope="col"| Number of your Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-| style="width:120px;" | [[#Running a Wideband FDTD Analysis | Wideband Analysis]]| style="width:270px;" | Simulates the physical structure, you must define an RCS observable instead "As Is"| style="width:100px;" | Single run| style="width:200px;" | Generates data for many frequency samples| style="width:150px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]| style="width:270px;" | Varies the value(s) of one or more project variables| style="width:100px;" | Multiple runs| style="width:200px;" | Runs at the center frequency fc| style="width:150px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]]| style="width:270px;" | Optimizes the value(s) of one or more project variables to achieve a radiation patterndesign goal | style="width:100px;" | Multiple runs | style="width:200px;" | Runs at the center frequency fc| style="width:150px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM. Follow these stepsCube#Generating_Surrogate_Models | HDMR Sweep]]| style="width:270px;" | Varies the value(s) of one or more project variables to generate a compact model| style="width:100px;" | Multiple runs | style="width:200px;" | Runs at the center frequency fc| style="width:150px;" | None|-| style="width:120px;" | [[#Running a Dispersion Sweep in EM.Tempo | Dispersion Sweep]]| style="width:270px;" | Varies the value of wavenumber in a periodic structure | style="width:100px;" | Multiple runs | style="width:200px;" | Runs at multiple frequency points corresponding to constant wavenumber values| style="width:150px;" | Only for periodic structures excited by a plane wave source|}

* Right click on the '''Far Fields''' item in the '''Observables''' section of the Navigation Tree and select '''Insert New RCS...''' to open the Radar Cross Section Dialog.* Use the '''Label''' box to change the name of the far field or change the color of the far field box using the '''Color''' button.* The frequency of RCS calculation can be specified in the box labeled '''Far Field Frequency'''. By default, this is equal to the center frequency of the project. However, you can calculate the far field data at any other frequency within the project's frequency range.* The resolution of RCS calculation is specified by '''Angle Increment''' expressed in degrees. By default, the &theta; and &phi; angles are incremented by 5 degrees.* Define the desired box for far field calculations in the '''Scattering Box''' section of the dialog. As in the case of radiation pattern, there are two options available, === Running a default radiation box (radio button '''Size: Default''') or a user defined radiation box (radio buttons '''Size: Custom'''). If you check '''Size: Default''', no radiation box corner coordinates need to be specified. The radiation box will always be 0.1 free space wavelength away from the bounding box of the entire physical structure. Select '''Size: Custom''' to set the far field box manually. The values for the coordinates of '''Corner 1''' and '''Corner 2''' can now be changed. '''Corner 1''' is the lower-front-left corner and '''Corner 2''' is the upper-rear-right corner of the radiation box. The dimensions are entered in world coordinate system (WCS).* At the end of an Wideband FDTD simulation, besides calculating the RCS data over the entire (spherical) 3D space, a number of 2D RCS graphs are also generated. These are indeed RCS cuts at certain planes, which include the three principal XY, YZ and ZX planes plus one additional constant &phi;-cut. This latter cut is at &phi; Analysis === 45° by default. You can assign another &phi; angle in degrees in the box labeled '''Non-Principal Phi Plane'''.

[[Image:FDTD132* Identify material types and proper domain boundary conditions.png|thumb|300px|An example of * Identify the 3D radar cross section source type and excitation mechanism.* Define the project observables.* Mesh the physical structure and examine the quality of a PEC platethe generated mesh and it geometric fidelity.* Determine the proper temporal waveform.* Select the simulation mode and run the FDTD engine.]]

At the end of an FDTD simulation, in the far field section of the Navigation Tree, you will have the &theta; and &phi; components of RCS as well as the total radar cross section: &sigma;_&theta;, &sigma;_&phi;, and &sigma;_tot. You can view a 3D visualization of these quantities by clicking on their entries in the Navigation Tree. The RCS values (&sigma;) are expressed in m². The 3D plots are normalized to the maximum RCS value, which Wideband analysis is displayed in the legend box. The 2D RCS graphs can be plotted in '''[[EM.Grid Tempo]]'''exactly in the same way that you plot 2D radiation pattern graphss simplest and most straightforward simulation mode. A total of eight 2D RCS graphs are available: 4 polar and 4 Cartesian graphs for It runs the XY, YZ, ZX and user defined plane cutsFDTD time marching loop once. at At the end of a sweep the simulation, [[EM.Cube]] calculates some other quantities including the backscatter RCS (BRCS), forwardtime-scatter RCS (FRCS) and domain field data are transformed into the maximum RCS frequency domain using a discrete Fourier transform (MRCS) as functions of the sweep variable (frequency, angle, or any user defined variableDFT). In this caseAs a result, the RCS needs to be computed at you can generate wideband frequency data from a fixed pair of &phi; and &theta; anglessingle time-domain simulation run. These angles are specified The other simulation modes will be explained later in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radar Cross Section Dialog'''. The default values of the user defined azimuth and elevation are both zero corresponding to the zeniththis manual.

EM.Tempo allows you to simulate doubly periodic structures with periodicities along the X and Y directions. Many interesting structures such as frequency selective surfaces (FSS), electromagnetic band-gap (EBG) structures and metamaterial structures can be modeled using periodic geometries. In the case of an infinitely extended periodic structure, it is sufficient to analyze only a unit cell. In the === The FDTD method, this is accomplished by applying periodic boundary conditions (PBC) at the side walls of the computational domain. Simulation Engine Settings ===

Click here to learn more about [[Time Domain Simulation An FDTD simulation involves a number of Periodic Structures]]numerical parameters that can be accessed and modified from the FDTD Engine Settings Dialog. To open this dialog, select '''Menu > Simulate > Simulation Engine Settings... '''or open the '''Run Dialog''', and click the {{key|Settings}} button next to the engine dropdown list.

In the " '''Convergence''' " section of the dialog, you can set the '''Termination Criterion''' for the FDTD time loop. The time loop must stop after a certain point in time. If you use a decaying waveform like a Gaussian pulse or a Modulated Gaussian pulse, after certain number of time steps, the total energy of the computational domain drops to very negligible values, and continuing the time loop thereafter would not generate any new information about your physical structure. By contrast, a sinusoidal waveform will keep pumping energy into the computational domain forever, and you have to force the simulation engine to exit the time loop. [[EM.Tempo]] provides two mechanism to terminated the time loop. In the first approach, an energy-like quantity defined as U_n =&Sigma; [ &epsilon;₀|'''E_i,n'''|² + &mu;₀|'''H_i,n'''|² ].&Delta;V_i is calculated and recorded at a large random set of points in the computational domain. Here i is the space index and n is the time index. The quantity U_n has a zero value at t =0 (i.e. n =Setting Up 0), and its value starts to build up over time. With a Gaussian or Modulated Gaussian pulse waveform, U_n reach a maximum value U_max at some time step and starts to decline thereafter. The ratio 10.log( U_n/ U_max) expressed in dB is used as the convergence criterion. When its value drops below certain '''Power Threshold''', the time loop is exited. The default value of Power Threshold is -30dB, meaning that the FDTD engine will exit the time loop if the quantity U_n drops to 1/1000 of its maximum value ever. The second termination criterion is simply reaching a '''Maximum Number of Time Steps''' , whose default value set to 10,000. A Periodic Unit Cell===third option, which is [[EM.Tempo]]'s default setting (labeled "'''Both'''"), terminates the simulation as soon as either of the first two criteria is met first.

Using [[EM.Cube]]'s [[FDTD Module]], you can simulate complex 3D periodic {{Note|Keep in mind that for highly resonant structures. A periodic structure is one that repeats itself infinitely along one, two or three directions. In this release of [[EM.Cube]]'s [[FDTD Module]], the periodicity is limited to the X-Y plane. In other words, the periodic structure repeats itself along the X- and Y-axes, but not along the Z-axis. By default, your physical structure is not periodic, and you may have to instruct [[EM.Cube]] increase the maximum number of time steps to turn it into a periodic structure through [[FDTD Module]]'s Periodicity Dialog. By designating a structure as periodicvery large values above 20, you enforce periodic boundary conditions (PBC) on the side walls of its computational domain. Your structure in the project workspace then turns into a periodic unit cell. The periodic side walls are displayed with dashed blues lines000.}}

[[Image:FDTD134.png|250px|thumb|[[FDTD Module]]The "'s Periodicity ''Acceleration'''" section of the FDTD Simulation Engine Settings dialog]]give three options for the FDTD kernel:

* Select '''Menu > Simulate > Computational Domain > Periodicity Settings..The serial CPU solver is [[EM.Tempo]]''' or right click s basic FDTD kernel that run the time marching loop on the '''Periodicity''' item in the '''Computational Domain''' section a single central processing unit (CPU) of the Navigation Tree and select '''Periodicity Settingsyour computer...''' from The default option is the contextual menumulti-core CPU solver. This open up is a highly parallelized version of the Periodicity Settings Dialog.* Check FDTD kernel based on the box labeled '''Periodic Structure''' and click the '''Apply''' button of this dialogOpen-MP framework. The default domain box initially shrinks to the edges It takes full advantage of the physical structure in the project workspacea multi-core, multi-CPU architecture, if your computer does have one. The default periods along the X and Y axes appear in the dialog, which are equal to the dimensions of the structure's bounding box.* Enter new values GPU solver is a hardware-accelerated FDTD kernel optimized for '''X Spacing''' and '''Y Spacing '''in project units and close the dialog.* Periodic boundary conditions CUDA-enabled graphical processing unit (PBCGPU) are established on the ±X and ±Y faces of the domain boxcards. You still have to designate If your computer has a fast NVIDIA GPU card with enough onboard RAM, the boundary conditions on the ±Z faces of the computational domain. These are CPML by default. But you GPU kernel can change them speed up your FDTD simulations up to PEC 50 times or PMCmore over the single CPU solver.

In <table><tr><td> [[Image:FDTD58.png|thumb|left|720px|EM.Cube]]Tempo's [[FDTD Modulesimulation engine settings dialog.]], a periodic structure can be excited using various source types. Exciting the unit cell structure using a lumped source, a waveguide source, an ideal source or a distributed source, you can model an infinite periodic antenna array. For most practical antenna types, you will excite your periodic structure with a lumped source or waveguide source. In this case, you can define a port for the lumped source or waveguide source and calculate the S<sub/td>11</subtr></table> parameter or input impedance of the periodic antenna array. You can also compute the near-field and far-field data.

[[EM.Cube]]'s periodic FDTD simulator uses periodic boundary conditions (PBC) to model an infinite periodic array. All the periodic replicas of the unit cell structure are excited. In this case, you can impose a phase progression across the infinite array to steer its beam. You can do this from the property dialog of the lumped source or waveguide source. At the bottom of the '''Lumped Source Dialog''' or '''Waveguide Source Dialog''', there is a section titled '''==Modeling 3D Periodic Beam Scan Angles'''. This section is grayed out when the project structure is not periodic. You can enter desired beam scan angle values for both '''Theta''' and '''Phi''' Structures in degrees. At the end of the periodic FDTD simulation, the radiation pattern of the unit cell is calculated and stored in a radiation data file with a '''.RAD''' file extension. The 3D radiation patterns that you normally visualize in [[EM.Cube]], in this case, correspond to the single unit cell, not the infinite array. Therefore, they do not show the beam scanning even if you have entered nonzero values for the &theta; and/or &phi; scan angles. For this purpose, you have to define a finite-sized array factor. You do this in the "Impose Array Factor" section of the '''Radiation Pattern Dialog'''. In the case of a periodic structure, when you define a new far field item in the Navigation Tree, the values of element spacing along the X and Y directions are automatically set equal to the values of the periodic lattice spacing along those directions. Set the number of elements along the X and Y directions to any desired values. [[EM.Cube]] will then compute the radiation pattern of the specified finite-sized periodic array, and the beam scanning will appear in the radiation pattern plots, if any.Tempo==

{{Note|For large &theta; scan angles, the [[EM.Tempo]] allows you to simulate doubly periodic structures with periodicities along the X and Y directions. In the FDTD time matching loop may take far more time steps to convergemethod, this is accomplished by applying periodic boundary conditions (PBC) at the side walls of the computational domain.}}

{{twoimgNote|FDTD137[[EM.png|Setting Tempo]] can only handle regular, non-skewed periodic scan angles in the lumped source dialog|FDTD138lattices with no secondary offsets.}} [[Image:Info_icon.png| Setting 30px]] Click here to learn more about the array factor in radiation pattern dialogtheory of '''[[Basic_Principles_of_The_Finite_Difference_Time_Domain_Method#Time_Domain_Simulation_of_Periodic_Structures | Time Domain Simulation of Periodic Structures]]'''.}}

{{twoimg|FDTD135.png|Radiation pattern of a 8×8 finite-sized periodic dipole array with scan angles with phi and theta equal to 0 degrees.|FDTD136.png| Radiation pattern of ===Defining a 8×8 finite-sized periodic dipole array with scan angles theta equal to 45 degrees, and phi equal to 0 degreesPeriodic Structure in EM.}}Tempo===

Real antenna arrays have finite extents, that is, finite numbers of elements along the X and Y directions. Earlier, you saw how to excite an array of line objects using an array of lumped sources or an array of rectangular waveguides (hollow boxes) using an array of waveguide sources. Setting up array structures of this kind using [[EM.Cube]]'s '''Array Tool '''and exciting the individual elements using individual lumped or waveguide sources results in an accurate full-wave analysis of your antenna array. This type of simulation takes into account all the inter-element coupling effects as well as the finite edge and corner effects of the finite-sized array. At the end of the FDTD simulation of your antenna array, you can plot the radiation patterns and other far field characteristics of the array just like any other FDTD structure. However, depending on the total size of your array, To define a full-wave simulation like this may easily lead to a very large computational problem. As the number of elements grow very large, the array starts to look like an infinite periodic structure. In that case, it is possible to consider and analyze a periodic unit cell of the array structure and use an "Array Factor" representing the finite-extent topology of the array grid to calculate the radiation pattern of your antenna array. This approach works well for most large arrays. However, it ignores the finite edge and corner effects, which may be important for certain array architectures. In that case we recommend that you use [[EM.Cube]]'s [[Planar Module]]. Also, note that using an array factor for far field calculations, you cannot assign non-uniform amplitude or phase distributions to the array elements. For this purpose, you have to define an array object.follow these steps:

[[Image:FDTD146(1)* Select '''Menu > Simulate > Computational Domain > Periodicity Settings.png|thumb|250px|Defining additional radiation characteristics ..''' or right click on the '''Periodicity''' item in [[FDTD Module]]the '''Computational Domain''' section of the Navigation Tree and select '''Periodicity Settings...''' from the contextual menu. This open up the Periodicity Settings Dialog.* Check the box labeled '''Periodic Structure''' and click the '''Apply''' button of this dialog. The default domain box initially shrinks to the edges of the physical structure in the project workspace. The default periods along the X and Y axes appear in the dialog, which are equal to the dimensions of the structure's Radiation Pattern bounding box.* Enter new values for '''X Spacing''' and '''Y Spacing '''in project units and close the dialog.]]* Periodic boundary conditions (PBC) are established on the ±X and ±Y faces of the domain box. You still have to designate the boundary conditions on the ±Z faces of the computational domain. These are CPML by default. But you can change them to PEC or PMC.

In the previous section, you saw how to excite a periodic unit cell using a lumped source or a waveguide source<table><tr><td> [[Image:FDTD134. You can specify the beam scan angles in the source dialogspng|thumb|360px|EM. The finite array factor is defined in the radiation pattern Tempo's periodicity settings dialog. At the end of the periodic FDTD simulation, you can visualize the 3D radiation patterns in the project workspace and plot the 2D Cartesian and polar pattern graphs in EM.Grid. [[EM.Cube]] also calculates the '''Directive Gain (DG)''' as a function of the &theta; and &phi; angles. This is defined as:</td></tr></table>

<math>D(\theta,\phi) = \dfrac{4\pi [S(\theta,\phi)]}{P_{rad}} = \dfrac{4\pi \big| \mathbf{E}^{ff}(\theta,\phi) \big|^2} {\int\limits_0^{2\pi} \int\limits_0^{\pi} \big| \mathbf{E}^{ff}(\theta,\phi) \big|^2 \sin\theta \, d\theta \, d\phi}</math>=Exciting Periodic Structures as Radiators in EM.Tempo===

The directivity D₀ is the maximum value of the directive gain. In [[EM.CubeTempo]] generates four Cartesian graphs of directive gain in the three principal XY, YZ, ZX planes as well as in the user defined f-plane cut. The radiation patterns of antenna arrays usually have a main beam and several side lobesperiodic structure can be excited using various source types. Some [[parameters]] of interest in such structures include Exciting the '''Half Power Beam Width (HPBW)'''unit cell structure using a lumped source, '''Maximum Side Lobe Level (SLL)''' and '''First Null [[Parameters]]''' (i.e. first null level and first null beam width). You a waveguide source, or a distributed source, you can have [[EMmodel an infinite periodic antenna array.Cube]] calculate all such [[parameters]] if For most practical antenna types, you check the relevant boxes in the "Additional Radiation Characteristics" section of the '''Radiation Pattern Dialog'''. These quantities are saved into ASCII data files of similar names excite your periodic structure with '''.DAT''' file extensions. You can plot graphs of such data files at the end of a sweep simulation in''' '''EM.Grid. You can also plot the directive gain as a function of the sweep variable at the end of an FDTD sweep simulationlumped source or waveguide source. In that this case, you can define a port for the directive gain is computed at a fixed pair of &theta; lumped source or waveguide source and &phi; angles. These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in calculate the "Output Settings" section S₁₁ parameter or input impedance of the Radiation Pattern dialogperiodic antenna array. The default values of You can also compute the user defined azimuth near-field and elevation are both zero corresponding to the zenith. The results are saved to an ASCII far-field data file called "DGU.DAT". Note that DGU is also one of [[EM.Cube]]'s standard output [[parameters]] and can be used to define custom output or design objectives.

===Exciting A [[EM.Tempo]]'s periodic FDTD simulator uses periodic boundary conditions (PBC) to model an infinite periodic array. All the periodic replicas of the unit cell structure are excited. In this case, you can impose a phase progression across the infinite array to steer its beam. You can do this from the property dialog of the lumped source or waveguide source. At the bottom of the '''Lumped Source Dialog''' or '''Waveguide Source Dialog''', there is a section titled '''Periodic Surface With A Plane Wave===Beam Scan Angles'''. This section is grayed out when the project structure is not periodic. You can enter desired beam scan angle values for both '''Theta''' and '''Phi''' in degrees. To visualize the radiation pattern of the beam-steered array, you have to define a finite-sized array factor. You do this in the "Impose Array Factor" section of the '''Radiation Pattern Dialog'''.

Using a plane wave source to excite a periodic structure in [[EM.Cube]]'s [[FDTD Module]], you can model frequency selective surfaces, electromagnetic band-gap (EBG) structures, metamaterials, etc. Exciting periodic structures with plane wave sources requires careful attention. [[EM.Cube]]'s FDTD simulation engine uses the direct spectral domain FDTD or constant transverse wavenumber method for analyzing periodic structures. In this technique, instead of a plane wave box, one defines a plane wave surface parallel to the X-Y plane. If the plane wave source illuminates the periodic unit cell from the top (90&deg; < {{Note|For large &theta; < 180&deg;)scan angles, the excitation surface is placed above the structure's bounding box. If the plane wave source illuminates the periodic unit cell from the bottom up (0&deg; < &theta; < 90&deg;), the excitation surface is placed below the structure's bounding box. In either case, the plane wave must intercept the excitation surface before hitting the unit cell's physical structure. It is highly recommended that you accept [[EM.Cube]]'s default settings for the plane wave box of periodic structures. Nevertheless, you can change the location of the excitation surface if you wish. To do so, you have FDTD time marching loop may take far more time steps to open the '''Plane Wave Dialog'''. In the Excitation Box section of the dialog, select the '''Size: Custom''' option. Only the '''Z Coordinate''' of '''Corner 1''' is available for editing. The rest of the coordinates are enforced by the periodic domain. You can enter the incidence angles '''Theta''' and '''Phi''' in degrees. For periodic structures, only the '''TM_z''' and '''TE_z''' polarization options are availableconverge.}}

One of the pitfalls of the direct spectral FDTD method is the possibility of horizontal resonances, which may lead to indefinite oscillation or even divergence of field values during the time marching loop. This happens in the case of oblique plane wave incidence when &theta; <table><tr><td> 0&deg;. [[Image:Period1.png|thumb|350px|Setting periodic scan angles in EM.Cube]]Tempo's FDTD engine automatically detects such cases and avoids those resonances by shifting the modulation frequency of the modulated Gaussian pulse waveform away from the resonant frequency. However, in some cases, the size of oscillations may still remain large after a large number of time steps. Occasionally, a late-time diverging behavior may appear. To avoid situations like these, it is highly recommended that you place a time-domain field probe above your structure and monitor the temporal field behavior during the time marching loop as shown in the figure belowLumped Source dialog.]] </td></tr></tr></table>

{{twoimg|FDTD140(1)<table><tr><tr><td> [[Image:Period2.png|Setting a custom plane wave source planethumb|FDTD139.png720px|Plane Wave visualization Setting the array factor in the sceneEM.Tempo's Radiation Pattern dialog.}}]] </td></tr></table>

===Reflection <table><tr><td> [[Image:Period3.png|thumb|360px|Radiation pattern of an 8×8 finite-sized periodic wire dipole array with 0& Transmission Characteristics===deg; phi and theta scan angles.]] </td><td> [[Image:Period4.png|thumb|360px|Radiation pattern of a beam-steered 8×8 finite-sized periodic wire dipole array with 45&deg; phi and theta scan angles.]] </td></tr></table>

At the end of the FDTD simulation of a periodic structure with plane wave excitation, the reflection and transmission coefficients of the structure are calculated and saved into two complex data files with '''.CPX''' file extensions. These coefficients behave like the S₁₁ and S₂₁ [[parameters]] of a two-port network. You can think of the upper half-space as Port 1 and the lower half-space as Port 2 of this network. The reflection and transmission (R/T) coefficients can be plotted on 2D graphs in '''EM.Grid '''similar to the scattering [[parameters]]. You can plot them from the Navigation Tree. To do so, right click on the '''===Exciting Periodic Characteristics''' item Structures Using Plane Waves in the '''Observables''' section of the Navigation Tree and select '''Plot Reflection Coefficients''' or '''Plot Transmission Coefficients'''. The complex data files are also listed in [[EM.Cube]]'s data manager. To open data manager, click the '''Data Manager''' [[Image:data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate > Data Manager''' from the menu bar or right click on the '''Data Manager''' item of the Navigation Tree and select Open Data Manager... from the contextual menu or use the keyboard shortcut '''Ctrl+D'''. Select any data file by selecting its row in the table and then click the '''Plot''' button to plot the graph in EM.Grid.Tempo===

{{Note|It is very important Using a plane wave source to keep excite a periodic structure in mind that only in the case of normal incidence does [[EM.CubeTempo]] compute the reflection and transmission coefficients over the entire specified bandwidth of the project. At oblique incidences when &theta; > 0, the computed R/T coefficients after the discrete Fourier transformation are valid only at the center you can model frequency of selective surfaces, electromagnetic band-gap (EBG) structures, metamaterials, etc. Exciting periodic structures with plane wave sources requires careful attention. [[EM.Tempo]]'s FDTD simulation engine uses the project direct spectral domain FDTD or constant transverse wavenumber method for the given value of the incident &theta;₀ angleanalyzing periodic structures. In other wordsthis technique, instead of a plane wave box, one defines a plane wave surface parallel to the computed R/T coefficients at all X-Y plane. At the other frequencies away from end of the center frequency correspond to different values FDTD simulation of the incident &theta; angle. As a resultperiodic structure with plane wave excitation, [[EM.Cube]] only saves the reflection and transmission coefficients at of the center frequency structure are calculated and saved into the output ASCII data files "reflection_coefficient.CPX" and "transmission_coefficient.CPX".}}

{| border="0"<table>|-| valign="top"|<tr><td> [[FileImage:FDTD141Period11.png|400px|thumb|Magnitude and Phase 380px|Geometry of reflection coefficient from a periodic surface plotted vsprinted strip FSS in EM. frequencyTempo.]]</td>| valign="bottom"|<td> [[FileImage:FDTD142Period12.png|400px|thumb|Magnitude and Phase of transmission coefficient from 340px|Define a custom periodic surface plotted vsplane wave box in EM. frequencyTempo.]]</td>|-</tr>|}</table>

===Periodic Using a plane wave source to excite a periodic structure in [[EM.Tempo]], you can model frequency selective surfaces, electromagnetic band-gap (EBG) structures, metamaterials, etc. Exciting periodic structures with plane wave sources requires careful attention. [[EM.Tempo]]'s FDTD Simulation Types===simulation engine uses the direct spectral domain FDTD or constant transverse wavenumber method for analyzing periodic structures. In this technique, instead of a plane wave box, one defines a plane wave surface parallel to the X-Y plane. If the plane wave source illuminates the periodic unit cell from the top (90&deg; < &theta; < 180&deg;), the excitation surface is placed above the structure's bounding box. If the plane wave source illuminates the periodic unit cell from the bottom up (0&deg; < &theta; < 90&deg;), the excitation surface is placed below the structure's bounding box. In either case, the plane wave must intercept the excitation surface before hitting the unit cell's physical structure. It is highly recommended that you accept [[EM.Tempo]]'s default settings for the plane wave box of periodic structures. Nevertheless, you can change the location of the excitation surface if you wish. To do so, you have to open the '''Plane Wave Dialog'''. In the Excitation Box section of the dialog, select the '''Size: Custom''' option. Only the '''Z Coordinate''' of '''Corner 1''' is available for editing. The rest of the coordinates are enforced by the periodic domain. You can enter the incidence angles '''Theta''' and '''Phi''' in degrees. For periodic structures, only the '''TM_z''' and '''TE_z''' polarization options are available.

{{Note|It is very important to keep in mind that only in the case of normal incidence does [[File:FDTD143EM.png|thumb|200px|[ Cube]] compute the reflection and transmission coefficients over the entire specified bandwidth of the project. At oblique incidences when &theta; > 0, the computed R/T Macromodel Settings Dialogcoefficients after the discrete Fourier transformation are valid only at the center frequency of the project for the given value of the incident &theta;₀ angle. In other words, the computed R/T coefficients at all the other frequencies away from the center frequency correspond to different values of the incident &theta; angle. As a result, [[EM.Cube]]only saves the reflection and transmission coefficients at the center frequency into the output data files "reflection_coefficient.CPX" and "transmission_coefficient.CPX".}}

Besides analyzing === Running a periodic structure in a single-run simulation, [[EM.Cube]]'s [[FDTD Module]] offers a number of sweep simulations for periodic structures. These include '''Frequency Sweep''', '''Angular Sweep''', '''R/T Macromodel Sweep '''and '''Dispersion Sweep'''. These options are available from the '''Simulation Mode''' dropdown list of the [[FDTD Module]]'s '''Run Dialog'''. Of these, frequency sweep and angular sweep are similar to the non-periodic case as discussed earlier. Keep in mind that in this release of [[EM.Cube]]'s [[FDTD Module]], for oblique plane wave incidences, you need to run a frequency sweep to get wideband reflection/transmission coefficient data. Similarly, you need to run an angular sweep to plot R/T coefficients vs. the incident angle.Tempo ===

The '''Dispersion Sweep '''option of the Simulation Mode drop-down list performs a sweep of constant k_l wavenumber values. This is a specialized sweep for the constant transverse wavenumber method that [[File:FDTD144EM.png|thumb|200px|Tempo]] uses to model periodic structures illuminated by a plane wave source. The real advantage of a dispersion sweep is that through a one-dimensional sweep of k_li, you can find the reflection and transmission coefficients for all combinations of frequency f_j and incident angle &theta;_j such that (2&pi;/c) . f_j. sin &theta;_j = k_li. This provides a complete picture of the dispersion behavior of your periodic structure. The sweep data can be graphed as a wavenumber-frequency intensity plot (also known as beta-k diagram) that projects the eigenvalues of the periodic structure. The horizontal axis represents the constant transverse wavenumber k_l (or beta). The vertical axis represents frequency. Sometimes, the free space wave number k₀ = (2&pi;/c).f is used as the vertical axis, hence, the term beta-k diagram. However, [[FDTD ModuleEM.Cube]]'s Dispersion Sweep Settings dialogplots frequency vs.wavenumber. Both the horizontal and vertical axes start from 0 and extend to f_max and k_l,max, respectively, where f_max = f₀ + &Delta;f/2, and &Delta;f is the specified bandwidth of the project. For this sweep option you have to specify the number of wavenumber samples. Note that the dispersion sweep is run for a fixed given value of the plane wave incident angle &phi; as specified in [[EM.Tempo]]'s Plane Wave Dialog.

The '''R/T Macromodel Sweep''' option of the Simulation Mode dropdown list is only available for periodic structures. It is used to generate a lookup <table model for the reflection and transmission coefficients of a periodic surface for both TM and TE polarizations. The results are written into a file named "PW_UserDefinedMacroData.mat". Through the Macromodel ><tr><td>[[Image:KBT Settings dialog you can set the start and end value and number of samples for both the Theta (&theta;) and Phi (&phi;) angles of the incident plane wave. The R/T macormodels can be used by png|thumb|360px| [[EM.CubeTempo]]'s Dispersion Sweep Settings dialog.]]</td></tr></table> <table><tr><td>[[Propagation ModuleImage:KBT R.png|thumb|360px|A typical reflection coefficient dispersion diagram of a periodic structure.]] to calculate the reflection and </td><td>[[Image:KBT T.png|thumb|360px|A typical transmission coefficients coefficient dispersion diagram of incident rays at the facets of obstructing blocks with "non-standard" a periodic surfacesstructure.]]</td></tr></table>

The '''Dispersion Sweep '''option of the Simulation Mode dropdown list performs a sweep of constant k<sub>l<br /sub> wavenumber values. This is a specialized sweep for the constant transverse wavenumber method that [[EM.Cube]]'s [[FDTD Module]] uses to model periodic structures illuminated by a plane wave source. The real advantage of a dispersion sweep is that through a one-dimensional sweep of k_li, you can find the reflection and transmission coefficients for all combinations of frequency f_j and incident angle &theta;_j such that (2&pi;/c) . f_j. sin &theta;_j = k_li. This provides a complete picture of the dispersion behavior of your periodic structure. The sweep data can be graphed as a wavenumber-frequency intensity plot (also known as beta-k diagram) that projects the eigenvalues of the periodic structure. The horizontal axis represents the constant transverse wavenumber k_l (or beta). The vertical axis represents frequency. Sometimes, the free space wave number k₀ = (2&pi;/c).f is used as the vertical axis, hence, the term beta-k diagram. However, [[EM.Cube]] plots frequency vs. wavenumber. Both the horizontal and vertical axes start from 0 and extend to f_max and k_l,max, respectively, where f_max = f₀ + &Delta;f/2, and &Delta;f is the specified bandwidth of the project. For this sweep option you have to specify the number of wavenumber samples. Note that the dispersion sweep is run for a fixed given value of the plane wave incident angle &phi; as specified in [[FDTD Module]]'s Plane Wave Dialog.

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