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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.Cube]]'s [[FDTD Module]] <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> :<mathtr> \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 </mathtd><!--[[Image:FDTD18(2)Tempo_L11_Fig2.png|thumb|left|720px|Moving an imported object from CubeCAD to EM.Tempo.]]--> where <math>\varepsilon_{\infty}</mathtd> is the value of the permittivity at infinite frequency, <math>\tau_p</mathtr> 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. Unmagnetized plasmas are typically modeled as Drude materials. The complex permittivity of a Drude material with N poles is given by: :<math> \varepsilon(\omega) = \varepsilon_{\infty} - \sum_{p=1}^N \dfrac{{\omega_p}^2}{\omega^2 - j\omega \nu_p} </math><!--[[Image:FDTD19(1).png]]--table>

where <math>\omega_p</math> and <math>\nu_p</math> are the angular plasma frequency and angular collision frequency corresponding to the p''th'' pole, respectively, and both are expressed in rad/s. For an unmagnetized plasma, <math>\varepsilon_{\infty} = 1</math>. The complex permittivity of a Lorentz material with N poles is given by: :<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}, \quad \Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty} </math><!--[[Image:FDTD20.png]]--> where <math>\omega _p</math> and <math>\delta_p</math> are the angular resonant frequency and angular damping frequency corresponding to the p''th'' pole, respectively, and both are expressed in rad/s. Similar to a Debye material, <math>\Delta \varepsilon_p = \varepsilon_{sp} - \varepsilon_{\infty}</math> represents the change in permittivity due to the p''th'' pole. {| border="0"|-| valign="top"|[[Image:FDTD2.png|thumb|250px|EM.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]]|-|} ===Geometrical Rules & Material Hierarchy=== [[Image:fdtd14_tn.png|thumb|400px|Geometric construction of a dielectric-coated metallic cylinder.]]The following rules apply to the definition of materials and objects in [[EM.Tempo]]: * Under 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.  [[EM.Tempo]] 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: # [[#Perfect Conductors|PEC]]# [[#Perfect Conductors|PMC]]# [[#Dispersive Materials|Dispersive]]# General [[#Anisotropic Materials|Anisotropic]]# Uniaxial [[#Anisotropic Materials|Anisotropic]]# [[#Dielectric Materials|Dielectric]] If planned carefully, taking advantage of EM.Tempo's material hierarchy rule would make the construction of complex objects easier. For example, a dielectric coated metallic cylinder can be modeled by two concentric cylinders: an inner PEC of smaller radius and an outer dielectric of larger radius as shown in the illustration below. The portion 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 hierarchy. Alternatively, you can model the same structure by an inner solid PEC cylinder enclosed by an outer hollow pipe-shaped dielectric cylinder.   ==Setting the Computational Domain & Boundary Conditions==

In [[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 (global bounding box). The offset is specified in free-space wavelengths. A "'''Custom'''" -type domain, on the other hand, is defined as a fixed-size and fixed-location 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.

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;₀). As soon as you draw your first object, 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 structure, the domain box grows accordingly to encompass your entire physical structure. When you delete objects from the project workspace, the domain box also shrinks accordingly.

* Click the '''Domain''' [[Image:domain_icon.png]] button of the '''Simulate ''' Toolbar or select the menu item '''Menu > 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.

===Settings the [[Image:Info_icon.png|30px]] Click here to learn more about '''[[Glossary_of_EM.Cube%27s_Simulation-Related_Operations#Domain_Settings | Domain Boundary Conditions===Settings]]'''.

<table><tr><td> [[Image:FDTD13FDTD14.png|thumb|300pxleft|[[FDTD Module]]480px|EM.Tempo's Boundary Conditions domain settings dialog.]]EM.Tempo supports four types of domain boundary conditions:</td></tr></table>

* PEC* PMC* Convolutional Perfectly Matched Layers (CPML)* Periodic ===Settings the Domain Boundary Conditions (PBC)===

[[EM.Tempo]] supports four types of domain boundary conditions: PEC, PMC, Convolutional Perfectly Matched Layers (CPML) and Periodic Boundary Conditions (PBC). 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.

* Select the menu item '''Menu > Simulate > &rarr; Computational Domain > &rarr; Boundary Conditions''' or right click on the '''Boundary Conditions''' item in the '''Computational Domain''' section of the Navigation Tree and select '''Boundary Conditions...''' from the contextual menu. The Boundary Conditions Dialog opens.

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<table><tr><td> [[Image:FDTD13. png|thumb|left|480px|EM.Tempo uses Convolutional Perfectly Matched Layers (CPML) for absorbing 's boundary conditionsdialog. The boundary CPML cells in the project workspace are transparent to the user. But, in effect, multiple rows of CPML cells are placed on the exterior side of each face of the visible domain box.]]</td></tr></table>

Click here to learn more about In open-boundary 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. [[Advanced CPML SetupEM.Tempo]]uses Convolutional Perfectly Matched Layers (CPML) for absorbing boundary conditions. Usually two or more ABC layers must be placed at the boundaries of the physical structure to maximize wave absorption. The boundary CPML cells in the project workspace are not visible to the user. But, in effect, multiple rows of CPML cells are placed on the exterior side of each face of the visible domain box.

===Modeling Planar Structures You can set the number of Infinite Extents===CPML layers as well as their order. This is done through the CPML Settings Dialog, which can be accessed by right clicking on the '''CPML''' item in the '''Computational Domain''' section of the navigation tree and selecting '''CPML Settings...''' from the contextual menu. By default, eight CPML layers of the third order are placed outside the FDTD problem domain. It is recommended that you always try a four-layer CPML first to assess the computational efficiency. The number of CPML layers may be increased if a very low reflection is required (<-40dB).

You can use {{Note|[[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 ]]'s default quarter wavelength offset for the domain box and set the lateral domain its 8-layer CPML walls are very conservative choices and can be relaxed in many cases. An offset values along the ±X and ±Y directions all equal to zero. If the planar structure ends in an infinite dielectric halfeight free-space from grid cells beyond the bottomlargest bounding box usually gives a more compact, but still valid, you must assign 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 with a nonzero value.}}

{{Note|The current release of [[EMImage:Info_icon.Tempopng|30px]] does not support anisotropic or dispersive layers Click here to learn more about the theory of laterally infinite extents'''[[Basic_Principles_of_The_Finite_Difference_Time_Domain_Method#CPML_vs. In other words, your anisotropic or dispersive material objects must not touch the CPML domain boundaries_PML | Perfectly Matched Layer Termination]]'''.}}

<td> [[Image:FDTD22(1)FDTD MAN10.png|thumb|300pxleft|360px|The computational boundary CPML cells placed outside the visible domain box of a metallic sphere with nonzero offset in all directions.]] </td><td> [[Image:FDTD24FDTD15.png|thumb|left|400px|The computational domain box of a laterally infinite planar structure with a PEC ground and zero ±X and ±Y and -Z domain offsetsCPML Settings dialog.]] </td>

== Generating You 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 FDTD Mesh ==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 bottom, you must assign 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 with a nonzero value.

[[Image:FDTD28.png|thumb|350px{{Note|The adaptive FDTD meshes current release of a metallic sphere.]][[EM.Tempo]]'s FDTD mesh is a rectangular Yee mesh that extends to the entire computational domaindoes not support full-anisotropic or dispersive or gyrotropic layers of laterally infinite extents. It is primarily constructed from three mesh grid profiles along the XYIn other words, YZ and ZX principal planes. These projections together create a 3D rectangular (voxel) mesh space. Straight lines, boxes and rectangular plates whose edges are aligned with the three principal axes are the simplest objects to mesh in EM.Tempo. Such objects preserve their exact shapes after discretization. All the objects with curved edges and curved surfaces your anisotropic or dispersive or gyrotropic material objects with straight edges and flat faces that are must not parallel to touch the principal axes or principal planes (such as oblique lines and slanted lateral faces of a pyramid) are discretized using a staircase (Yee) profileCPML domain boundaries.}}

EM<table><tr><td> [[Image:FDTD MAN8.Tempo's adaptive mesh generator uses a variable staircase profile, where the cell sizes png|thumb|left|360px|The domain box of grid line spacing vary a patch antenna with the curvature (derivative) of the edge or facea finite-sized substrate and ground. As ]] </td><td> [[Image:FDTD MAN9.png|thumb|left|360px|The domain box of a resultlaterally infinite patch antenna with zero ±X, ±Y and -Z domain offsets. Note that the bottom PEC plate can be replaced with a higher mesh resolution is achieved PEC boundary condition at "more curvy" areas to better capture the geometrical details-Z domain wall.]] </td></tr></table>

The default choice is the adaptive meshBefore you can run an FDTD simulation, which is you have to define a quite sophisticated meshsource to excite your project’s physical structure. The resolution of the adaptive FDTD mesh is driven by the '''Mesh Density''', expressed in cells per effective wavelengthEM. Since FDTD is Tempo offers a time-domain method and the variety of excitation waveform may have a wideband spectral content, the effective wavelength is calculated based mechanisms for your physical structure depending on the highest frequency your particular type of the projectmodeling problem or application: f_max = f₀ + &Delta;f/2, where f₀ is your project's center frequency and &Delta;f (or BW) is its specified bandwidth. In other words, the effective wavelength in the free space is &lambda;_0,eff = c / f_max, c being the speed of light in the free space.

The adaptive FDTD mesh{| 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, however_Sources, produces different grid cell sizes in the free space regions and inside dielectric regions. The effective wavelength in _Devices_%26_Other_Physical_Object_Types#Lumped Source |Lumped Source]]| style="width:250px;" | General-purpose point voltage source| style="width:200px;" | PEC or thin wire line parallel to a dielectric material with relative permittivity e_r and permeability µ_r is given by &lambdaprincipal axis| style="width:200px;_{d" | A single point| style="width:200px;" | None|-| style="width:30px;" | [[File:distrb_src_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,eff} _Sources,_Devices_%26_Other_Physical_Object_Types#Distributed Source |Distributed Source]]| style= &lambda"width:250px;_{0" | General-purpose distributed planar source with a uniform,eff} / &radicedge-singular or sinusoidal impressed field profile| style="width:200px;&epsilon" | Virtual rectangle strip parallel to a principal plane| style="width:200px;_r&mu" | A rectangular area| style="width:200px;_r" | None|-| style="width:30px;" | [[File:mstrip_icon.png]]| style="width:150px;" | [[Glossary_of_EM. ThereforeCube%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 rectangle strip parallel to a principal plane| style="width:200px;" | A vertical rectangular area underneath the average ratio of host strip| style="width:200px;" | Requires a PEC ground plane strip underneath the cell size host strip|-| style="width:30px;" | [[File:cpw_icon.png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_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 dielectric region principal plane| style="width:200px;" | Two parallel horizontal rectangular areas attached to the cell size in opposite lateral edges the free space is 1/&radichost center strip| style="width:200px;(&epsilon" | Requires two parallel PEC ground strips on the two sides of the host center strip|-| style="width:30px;_r&mu" | [[File:coax_icon.png]]| style="width:150px;_r)" | [[Glossary_of_EM. The adaptive FDTD mesh generator also takes note of 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| style="width:200px;" | PEC Cylinder oriented along a principal axis| style="width:200px;" | A circular ring area enveloping the geometrical features of the objects it discretizeshost inner conductor cylinder| style="width:200px;" | Requires a concentric hollow outer conductor cylinder|-| style="width:30px;" | [[File:wg_src_icon. This is more visible png]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials,_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 box oriented along a principal axis| style="width:200px;" | A rectangular area at the case cross section of curved solidsthe 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, curves surfaces _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 curved wires long wire current source with a uniform, triangular or obliquely oriented planes and lines which need to be approximated using a staircase sinusoidal current distribution profile. The mesh resolution varies with the slope | 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 geometrical shapes and tries to capture the curved segments in the best wayprincipal axes|-| style="width:30px;" | [[File:plane_wave_icon. Another important feature 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 the adaptive FDTD mesher is generation reflection/transmission characteristics of gradual grid transitions between lowperiodic surfaces | style="width:200px;" | None (stand-density and highalone source)| style="width:200px;" | Surface of a cube enclosing the physical structure| style="width:200px;" | None|-density mesh regions| style="width:30px;" | [[File:gauss_icon. For examplepng]]| style="width:150px;" | [[Glossary_of_EM.Cube%27s_Materials, this often happens around the interface between _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 free space and high permittivity dielectric objectsphysical structure| style="width:200px;" | None|-| style="width:30px;" | [[File:huyg_src_icon. Gradual mesh transitions provide better accuracy especially in the case png]]| style="width:150px;" | [[Glossary of highly resonant structuresEM.Cube's Materials, Sources, 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|}

According Click on each category 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 steps. This would then translate into a very long computation time. [[EM.Cube]] offers 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 learn more details about each source type and/or permeability based on the effective wavelength in exactly the same way as the adaptive mesh. Finally, [[EM.Cube]]'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 how to your physical structuredefine one.

{{Note[[Image:Info_icon.png|When choosing a mesh type for your FDTD simulation, keep in mind that adaptive and regular mesh 30px]] More information about all the source types are frequency-dependent and their density varies with can be found in the highest frequency '''[[Glossary of your specified bandwidth, while the uniform mesh type is always fixed and independent of your projectEM.Cube's frequency settingsMaterials, Sources, Devices & Other Physical Object Types]]'''.}}

===Viewing In the most general sense, one can consider two fundamental types of excitation sources for an FDTD Mesh===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.

Because a full 3D FDTD mesh A lumped source is difficult to visualize everywhere in the computational domain, only the discretized objects are displayed most commonly used way of exciting a structure in [[EM.Cube]]'s "'''Mesh View'''" modeTempo. In particular, only 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 outer boundary cells three principal axes. A lumped source is displayed as a small red arrow on the surface of [[Solid Objects|solid objects]] host line. Lumped sources are showntypically used to define ports and compute the port characteristics like S/Y/Z parameters. HoweverUsing simple lumped sources, you can view simulate a variety of transmission line structures including filters, couplers or antenna feeds. This approach may become less accurate at higher frequencies when the mesh grid planes across details of the domain. You can even step these planes back feed structure become important and forth inside 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 domain ports for calculation of the incident and view different mesh profiles reflected waves. Waveguide source is used to excite the dominant TE₁₀ mode of your physical structurea 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.

To generate When you create an FDTD mesh and view it array of an object type that can host one of the project workspaceabove source types, follow these steps:you can also associate a source array with that array object.

* First, click the '''Mesh Settings''' [[Image:mesh_settingsInfo_icon.png|30px]] button of the '''Simulate Toolbar''' or select '''Menu > Simulate > Discretization > Mesh Settings...''', or right click on the '''Yee Mesh''' item of the Navigation Tree and select '''Mesh Settings...''' from the contextual menu, or use the keyboard shortcut '''Ctrl+G'''. The Mesh Settings Dialog opens up, where you can set the values of the various mesh [[parameters]] including the '''Mesh Density'''.* After specifying the desired mesh density, you can examine the mesh grid plane. The XY, YZ, and ZX mesh grid planes can be displayed through '''Menu > Simulate > Discretization > Grid Planes > XY Plane''', '''YZ Plane''' or '''ZX Plane''' or by right clicking on one of the three '''XY Plane''', '''YZ Plane''' or '''ZX Plane''' items in the '''Discretization''' section of the Navigation Tree and selecting '''Show''' from the contextual menu. The mesh grid planes give you a good idea of what the mesh will look like once it is generated and its resolution along different planes. To remove a mesh grid plane from the project workspace, select '''Menu > Simulate > Discretization > Grid Planes >''' one Click here to learn more time and remove the check mark in front of the name of the currently displayed mesh grid plane, or right click on the name of the currently displayed mesh grid plane in the Navigation Tree and select '''Hide''' from the contextual menu.* To display the FDTD mesh, click the '''Show Meshabout ''' [[Image:mesh_tool.pngPreparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Finite-Sized_Source_Arrays | Modeling Finite-Sized Source Arrays]] button of the '''Simulate''' '''Toolbar '''or select '''Menu > Simulate > Discretization > Show Mesh''' or use the keyboard shortcut '''Ctrl+M'''. This takes [[EM.Cube]] into its "Mesh View" mode, and the Yee mesh of the whole structure is displayed in the project workspace. While the mesh view is enabled, the '''Show Mesh''' [[Image:mesh_tool.png]] button remains depressed. To get back to [[EM.Cube]]'s "Normal View" mode, click this button one more time, or deselect '''Menu > Simulate > Discretization > Show Mesh''' to remove its check mark or simply hit the '''Esc Key''' of the keyboard.

In [[EM.Cube]]'s "Mesh View" mode, you can rotate or pan A plane wave source is a popular excitation method that is used for calculation of the view radar cross section of targets or reflection and transmission characteristics of periodic surfaces. A Gaussian beam source is another source type that is highly localized as opposed to the project workspaceuniform plane wave. For both plane wave and Gaussian beam sources, but you cannot edit the objects[EM. '''"Show Mesh"''' generates Tempo requires a new mesh and displays it if there is none in finite incidence surface to calculate the memoryexcitation. When you create either of these sources, a plane wave box or it simply displays an existing mesh in a Gaussian beam box is created as part of their definition. A trident symbol on the memorybox shows the propagation vector as well as the E-field and H-field polarization vectors. This is a useful feature because generating an FDTD mesh may take a long The time depending domain plane wave or Gaussian beam excitation is calculated on the complexity surface of structure this box and the total size of injected into the computational domain. If you change The plane wave box is displayed in the project workspace as a purple wireframe box enclosing the structure or alter , while the mesh settings, Gaussian beam box appears as a new mesh is always generatedgreen wireframe box. You can ignore any mesh in the memory and force [[EMBoth boxes have an initial default size with an offset of 0.Cube]] to generate a fresh FDTD mesh 2&lambda;₀ from the ground up by selecting largest bounding box enclosing your entire physical structure. In both source dialogs, the radio button '''Menu > Simulate >Discretization > Regenerate MeshSize: Default''' or is selected by right clicking on the default. The radio button '''Yee MeshSize: Custom''' item of allows you to set the Navigation Tree excitation box manually. The values for the coordinates of '''Corner 1''' and selecting '''RegenerateCorner 2''' from can now be changed. Corner 1 is the contextual menufront lower left corner and Corner 2 is the rear upper right corner of the box.The corner coordinates are defined in the world coordinate system (WCS). <table><tr><td> [[Image:FDTD MAN11.png|thumb|360px|A plane wave box enclosing a PEC cylinder at oblique incidence: &theta; = 105&deg; and &phi; = 315&deg;.]] </td><td> [[Image:FDTD MAN12.png|thumb|360px|A Gaussian beam box enclosing a PEC cylinder at oblique incidence: &theta; = 105&deg; and &phi; = 315&deg;. The concentric circles represent the beam's focus point and radius.]] </td></tr></table>

{{twoimg|FDTD34.png|A human head model and === Simulating a cellular phone handset on its side.|FDTD33.png|The regular FDTD mesh of the human head model and the cellular phone handsetMultiport Structure in EM.}}Tempo ===

=== Changing 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 Settings ===location of the following types of sources:

*[[Image:FDTD80Glossary of EM.pngCube's Materials, Sources, Devices & Other Physical Object Types#Lumped Source |thumbLumped sources]]*[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Distributed Source |600pxDistributed sources]]*[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Microstrip Port |Microstrip port sources]]*[[Glossary of EM.TempoCube's Mesh Settings dialogMaterials, 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]]

[[EM.Cube]]'s [[FDTD Module|FDTD module]] discretizes objects using what is often referred to as Every time you create a new source with one of the “staircase approximation”. In this mesh generation schemeabove types, the structure is recreated using program asks if you want to initiate a large number of cubic cells carefully assembled in a way that approximates new port and associate it with the shape of newly created source. If the original physical structure. By default, a carefully calculated, "<u>'''Adaptive'''</u>" mesh of your physical structure is generated in project workspace has N sources, then N default ports are defined, with one port assigned to each source according to their order in the navigation tree. You can define any number of ports equal to satisfy or less than the following criteria:total number of sources in your project.

* 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 time, especially with the CPU version of the FDTD engine. Also, a very large mesh size requires If your physical structure has two or more RAM, which may exceed your GPU memory capacity. Set the '''Minimum Mesh Density''' to a moderately low value to keep the mesh size manageablesources, but be careful you have not to set it too low (see defined any ports, all the next item below).* Ensure simulation accuracy by requiring an acceptable minimum number of cells per wavelength through each object and in sources will excite the empty (free) space between them and structure simultaneously during the computational domain boundariessimulation. 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/ wavelength. But for some resonant structuresHowever, 25 or even 30 cells per wavelength may be required when you assign N ports to achieve acceptable accuracy. As you reduce the mesh densitysources, the simulation accuracy decreases.* Accurately represent and approximate the boundaries of edges or surfaces then you have a multiport structure that are not grid-aligned by closely adhering to their geometric contours. This is controlled characterized by the '''Minimum Grid Spacing Over Geometric Contours'''an N×N scattering matrix, which can be specified either as a fraction of the free space grid spacing or as an absolute length value in project unitsN×N impedance matrix, and an N×N admittance matrix.* Maximize the minimum grid spacing To calculate these matrices, EM.Tempo uses a binary excitation scheme in any dimension inside conjunction with the computational domain and thus maximize the simulation time stepprinciple of linear superposition. 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 stepIn this binary scheme, the larger the number of time steps required for convergence. This structure is controlled using the '''Absolute Minimum Grid Spacing''', which can be specified either as analyzed a fraction total of the free space grid spacing or as an absolute valueN times. It is critical to accurately represent and precisely maintain the object edge/surface boundaries in certain structures like resonant antennas and filters, as the phase Each time one of the reflected fields/waves N port-assigned sources is affected by excited, and all the object boundary positionsother port-assigned sources are turned off. When object boundaries are very close to each In otherwords, the mesh needs to represent them by two separate, but very closely spaced, grid linesFDTD solver runs a "port sweep" internally. To control the minimum allowed grid spacing, use When the ''j'Absolute Minimum Grid Spacing '''settingsth port is excited,* Maintain a smooth grid with no abrupt jumps from low-density to high-density regions. This feature is enabled with all the '''Create Gradual Grid Transitions '''check box (always checked by default).S_ij parameters are calculated together based on the following definition:

Occasionally, you may prefer a more regular FDTD mesh with almost equal grid line spacing everywhere, but still with a frequency-dependent cell size. In that case, you can select the "<u>'''Regular'''</u>" option of the '''Mesh Type '''dropdown list in the FDTD Mesh Settings dialog. The regular FDTD mesh enforces only two of the above [[parameters]]: '''Minimum Mesh Density''' and '''Absolute Minimum Grid Spacing'''. Or you may opt for an absolutely "<umath>'''Uniform'''S_{ij} = \sqrt{\frac{Re(Z_i)}{Re(Z_j)}} \cdot \frac{V_j - Z_j^*I_j}{V_i+Z_i I_i} </umath>" mesh type, for which you need to specify the '''Cell Size '''along the X, Y, Z directions in project units.

Click here to learn more about [[Advanced Meshing in EMwhere V_i is the voltage across Port i, I_i is the current flowing into Port i and Z_i is the characteristic impedance of Port i.Tempo]]The sweep loop then moves to the next port until all ports have been excited.

Before you can run an FDTD simulation, you have [[Image:Info_icon.png|30px]] Click here to define a source to excite your project’s physical structurelearn more about the '''[[Glossary_of_EM. EMCube%27s_Simulation_Observables_%26_Graph_Types#Port_Definition_Observable | Port Definition Observable]]'''.Tempo offers a variety of excitation mechanisms for your physical structure depending on your particular type of modeling problem or application:

# '''Ideal Source'''[[Image: A stand-alone localized voltage source with an internal resistanceInfo_icon.# '''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 of space parallel png|30px]] Click here to a principal plane.# learn more about '''Waveguide Source''': A distributed source that must be placed across a hollow PEC box object.[[Preparing_Physical_Structures_for_Electromagnetic_Simulation# '''Plane Wave Source''': A distributed source with a plane wave profile defined using a virtual box object enclosing the entire physical structureModeling_Coupled_Sources_. # 26_Ports | Modeling Coupled Sources & Ports]]'''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.

<table><tr><td> [[Image:FDTD 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> === Excitation Waveform & Frequency Domain Computations === When an FDTD simulation starts, your project'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, or the maximum number of time steps is 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:

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 The accuracy of the FDTD simulation results depends on the right choice of temporal waveform. EM.Tempo's default waveform choice is a modulated Gaussian pulse waveform to drive your FDTD source, after a certain number of time steps, the total energy of the computational domain drops to very negligible levels. At the point, you can consider your solution to have converged. If you drive your end of an FDTD source by a sinusoidal waveformsimulation, the total energy of time domain field data are transformed into the computational frequency domain will oscillate indefinitely, and you have at your specified frequency or bandwidth to force produce the time loop to terminate after a certain number of time steps assuming a steady state have been reacheddesired observables.

===Defining {{Note|All of EM.Tempo's excitation sources have a New Source===default modulated Gaussian pulse waveform unless you change them.}}

To create a new source, follow these steps[[Image:Info_icon.png|30px]] Click here to learn more about EM.Tempo's '''[[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]]'''.

* Right click on the name of the source type === Defining Custom Waveforms in the '''Sources''' section of the navigation tree and select '''Insert New Source...''' 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 necessaryEM.Tempo ===

===Ideal In some time-domain applications, you may want to simulate the propagation of a certain kind of waveform in a circuit or structure. In addition to the default waveforms, EM.Tempo allows you to define custom waveforms by either time or frequency specifications for each individual source in your project. If you open up the property dialog of any source type in EM.Tempo, you will see an {{key|Excitation Waveform...}} button located in the "Source===Properties" section of the dialog. Clicking this button opens up EM.Tempo's Excitation Waveform dialog. From this dialog, you can override EM.Tempo's default waveform and customize your own temporal waveform. The Excitation Waveform dialog offers three different options for defining the waveform:

A lumped source is Select the most commonly used way third option of exciting a structure in [[EMwaveform definition and then choose the '''Custom''' option from the '''Waveform Type''' dropdown list.Tempo]]. A lumped source is indeed an ideal source that must be placed on Enter a line object that is parallel to one mathematical expression for your custom waveform a function of the three principal axes time variable "T" or "t" in the box labeled '''Expression'''. You can use arithmetic operations, standard and shows up library functions as a small red arrow on the host linewell as user-defined Python functions. Lumped sources are typically used to define ports and compute the port characteristics like S/Y/Z [[parametersImage:Info_icon.png|30px]]. Lumped sources can also be place on line arrays. The property dialog of a lumped source has a drop-down list that contains the name of all the legitimate line objects (i.e. lines that are parallel Click here to one of the principal axes) and line arrays. The learn more about '''Offset[[Using Python to Create Functions, Models & Scripts#Creating Custom Python Functions | Creating Custom Python Functions]]''' 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 of the host line object.

{{Note<table><tr><td> [[Image:FDTD MAN13.png|In order to create a lumped source, you must have at least one line object or line array in thumb|left|720px|EM.Tempo's excitation waveform dialog showing the project workspacedefault standard modulated Gaussian pulse temporal waveform.}}]]</td></tr></table>

A real waveguide structure is usually excited using some type of strategically located probe mechanism[[Image:Info_icon. EM.Tempo also provides the png|30px]] Click here to learn more about '''Waveguide Source''', a special type of source that excites a prescribed TE₁₀ modal field distribution in a hollow rectangular waveguide structure. The scattering [[parameters]] are calculated from knowledge Glossary of incident and reflected fields at designated waveguide portsEM. Waveguide sources typically provide more accurate results for scattering [[parametersCube's Python Functions#Standard Python Functions | Python's Standard & Advanced Mathematical Functions]] compared to lumped ports as they represent the actual dominant propagating modes at the transmission line ports'''.

{{Note|In order to If you define a waveguide custom excitation waveform for your source, you must have none of the standard frequency domain output data and parameters will be computed at least one hollow box object with no caps or only one the end cap or a hollow box array in of your projectFDTD simulation.}}

A waveguide source must be placed across a rectangular waveguide which is oriented along one of the three principal axes<table><tr><td> [[Image:FDTD MAN14. In other words, the plane of the waveguide source must be parallel to one of the principal (XY, YZ or ZX) coordinate planespng|thumb|left|720px|EM. The property Tempo's excitation waveform dialog of the waveguide source provides showing a drop-down list containing custom modulated Bessel temporal waveform defined using the name of all the legitimate box objects or box arraysPython function sp. The waveguide source is displayed as an orange rectangle with a cross and a perpendicular small orange arrow across the host box object. The '''Offset''' parameter of a waveguide source is its distance from the base of the host box. A waveguide source by default is placed at the center of its host box. In other words, the default offset value is equal to half the longitudinal dimension of the host box objectj0(x).]]</td></tr></table>

=== Distributed Source=EM.Tempo's Active & Passive Devices ==

The sinusoidal type has the functional form cos(py/w)# '''[[Glossary of EM.Cube's Materials, and the edge-singular type has the functional form 1/v(1-(2y/w)^2)Sources, where y is the coordinate along the direction Devices & Other Physical Object Types#Resistor | Resistor]]''' # '''[[Glossary of field variation measured from the center EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Inductor | Inductor]]'''# '''[[Glossary of the rectangular area and w is its total widthEM.Cube's Materials, Sources, Devices & Other Physical Object Types#Capacitor | Capacitor]]''' # '''[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Series_RL_Device | Series RL Device]]''' # '''[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Parallel_RC_Device | Parallel RC Device]]''' # '''[[Glossary of EM.Cube's Materials, Sources, 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]]'''

[[Image:fdtd_src7_tnLumped 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.png|thumb|250px|A distributed source placed 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 two horizontal rectangular stripstheir location on the host line and its start point.]]

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: '''X, Y''' and '''Z''', which create planes parallel to the YZ, ZX and XY principal planes, respectively. Depending on the choice of the plane orientation, another dropdown list labeled '''Field Dir''' gives four options for the direction of the source field component. For example, the default plane orientation A lumped device is X (parallel to the YZcharacterized by a v-Plane) and the available field directions are +Y, -Y, +Z and -Z. Next, you have to enter the coordinates of two opposite corners i equation of the source planeform: 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 workspace.

:<math>i(t) ===Lumped Load===L \{ v(t) \} </math>

In [[EM.Tempo]] you can define four lumped load types: # '''Resistor''' with a Resistance value where V(Rt) in Ohms.# '''Capacitor''' with a Capacitance value is the voltage across the device, i(Ct) in pF.# is the current flowing through it and ''L'Inductor''' with is an Inductance value (L) in nH. # '''Nonlinear Diode''' with a Saturation Current (I_s)in fAoperator function, ambient temperature (T) in degree Kelvin, and a dimensionless ideality factor (n). The default values of these [[parameters]] are 100fA, 300&deg;K and 1, respectively.  Although lumped loads are not sources and do not excite a structure, their properties are similar to lumped sourceswhich may involve differential or integral operators. Lumped Loads devices are incorporated into the FDTD grid across two adjacent nodes in a similar manner to lumped sources. Likewise, lumped loads are defined on Line objects. In order to create At the location of a lumped loaddevice, you must have the FDTD solver enforces the device's governing equation by relating the device voltage and current to the electric and magnetic field components and updating the fields accordingly at least one line object in your projectevery time step. Lumped loads show up as small yellow arrows on their host line object [[Image:Info_icon. Similar to lumped png|30px]] Click here for a source, a lumped load has an offset parameter that determines its location on the host linegeneral discussion of '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#A_Review_of_Linear_.26_Nonlinear_Passive_.26_Active_Devices | Linear & Nonlinear Passive & Active Devices]]'''.

{| border="0"<table>|-| valign="top"|<tr><td> [[FileImage:FDTD53FDTD MAN17.png|thumb|left|250px480px|EM.Tempo's lumped device dialog for nonlinear diode.]]</td>| valign="top"|</tr><tr><td> [[FileImage:FDTD54FDTD MAN17A.png|thumb|left|250px|]]480px|EM.Tempo's lumped device dialog for active lumped two-| valign="top"|[[File:FDTD55port device.png|thumb|left|250px|]]</td>| valign="top"|</tr>|-|}</table>

{| border="0"|-| valign="top"|[[Image:FDTD42.png|thumb|250px|EM.Tempo's Ideal Source dialog]]| valign="top"|[[Image:FDTD43.png|thumb|250px| EM.Tempo's Lumped Source dialog]]| valignDefining Active Distributed Multiport Networks ="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.]]|-|}

# '''[[Image:FDTD48Glossary_of_EM.pngCube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Active_Distributed_One-Port_Device |thumb|200pxActive Distributed One-Port Device/Circuit]]''' # '''[[Glossary_of_EM.Cube%27s_Materials,_Sources,_Devices_%26_Other_Physical_Object_Types#Active_Distributed_Two-Port_Device |The Active Distributed Two-Port Definition dialogDevice/Circuit]]'''

Ports Unlike the active lumped devices, these devices are used to order rather distributed and index sources for circuit parameter calculations like S/Y/Z [[parameters]]. That their behavior is why they are defined in the '''Observables''' section of Navigation Treesimilar to a microstrip port source. In [[EM.Cube]]'s [[FDTD Module]]other words, you the active distributed one-port 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 define ports at choose one of the location edges of '''Lumped Sources''', '''Waveguide Sources''' and '''Distributed Sources'''the strip object for establishing the circuit port. In other words, ideal sources or other types the case of sources cannot be used to define ports or calculate a two-port characteristicsdevice, you need two parallel and end-to-end aligned strip objects.

Ports are The circuit behavior of these devices is defined in the '''Observables''' section of the Navigation Treeby a Netlist file. Right click on Their property dialog provides a text editor for simply writing the '''Port Definition''' item Netlist description of the Navigation Tree and select '''Insert New Port Definitiondevice.You can also import an existing external Netlist file with a ".CIR" or ".''' from TXT" file extension using the contextual menubutton labeled {{key|Load Netlist}}. The Port Definition Dialog opens up, showing the default port assignments. If you have N sources in your physical structure, then N default ports are defined, with one port assigned to each source according to their order on the Navigation Tree.

{{Note|[[Image:FDTD49.png|thumb|250px|Reassigning sources to ports and defining coupled portsRF.Spice A/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. }}

You can define any number of ports equal to or less than the total number of sources in your project<table><tr><td> [[Image:ActiveOnePort. The Port List of the dialog shows a list of all the ports in ascending order, with their associated sources and the portpng|thumb|left|480px|EM.Tempo's characteristic impedance, which is 50O by default. You can delete any port by selecting it from the Port List and clicking the '''Delete '''button of the dialog. Keep in mind that after deleting a port, you will have a source in your project without any port assignment. Make sure that is what you intend. When you delete active one or more ports in your project, their associated sources become free and "available" for either defining new ports or reassignment to the other ports. To define a new -port, click the '''Add '''button of the Port Definition dialog to open the "Add Port" device/circuit dialog. On the left side of this dialog, you will see a ]] </td></tr></table containing all the available sources. Select one or more ports and use the right arrow ('''--->''') button to move them to the table on the right side, labeled "Associated". These ports are now associated with the new port being defined. You can move sources from the "Associated" table back to the "Available" table on the left using the left arrow ('''<---''') button of the dialog. You can associate more than one source with the same port. In that case, you will have coupled sources, collectively representing a coupled port.

{{Note|In order to obtain correct results, the port impedance must equal the characteristic impedance of the transmission line on which the port is established. This is not done automatically in <table><tr><td> [[Image:ActiveTwoPort.png|thumb|left|720px|EM.CubeTempo's active two-port device/circuit dialog.]].}}</td></tr></table>

===Modeling Feeds When your physical structure contains an active device, EM.Tempo performs an EM-circuit co-simulation that involves both the full-wave FDTD EM solver and the SPICE circuit solver. In a global self-consistent co-simulation, at each time step of the FDTD time marching loop, the electric and magnetic fields at the location of the device ports are used to compute the port voltages and currents. These quantities are then used in Practical Applications===the SPCIE circuit solver to update all the voltages and currents at the internal nodes of the active device. The updated port voltages and currents are finally used to update the electric and magnetic fields in the physical mesh cells and the time marching loop proceeds to the next time step.

Using simple lumped sources, you can simulate a variety of transmission line structures in [[EM.Tempo]] including filters, couplers or antenna feeds and you can calculate their scattering [[parameters]]handle several active one-ports and two-ports simultaneously. This approach may become less accurate at very high frequencies when In that case, all the details devices are automatically compiled into a single Netlist that serves as the input of the feed structures become important and can no longer SPICE solver. The individual internal nodes of each device need to be modeled with highly localized lumped portsrenamed for the global Netlist. In such casesBesides the main circuit, it is recommended to use “Distributed Sources”, which utilize accurate modal field distributions at the ports for calculation Netlist of each device may contain several "subcircuits". Note that the incident and reflected wavessubcircuit nodes are not re-indexed for the global Netlist as is expected.

Click here {{Note|If you want to learn more about [[Using Lumped Sources to Model Transmission Line Feeds]]use a B-type nonlinear dependent source in the Netlist definition of an active one-port or two-port, it must be contained in a subcircuit definition rather than in the main circuit.}}

Click here to learn more about [[Using Sources & Loads in Antenna Arrays]]The figure below shows the geometry of a two-port amplifier device with microstrip input and output transmission lines.The Netlist of the two-port device is given below:

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. You select the polarization from the five radio buttons in the "Polarization" section of the dialog. 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 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''' = E1 2 0 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°).20

A plane wave box placed around a PEC sphere object. The trident at the corner of the box shows the propagation vector as well as the E-field and H-field polarization vectors.--

<table><tr><td> [[EM.Cube]] gives you an option to illuminate objects with a focused beam instead of a uniform plane waveImage:Amp circ. png|thumb|left|420px|The focused beam is a Gaussian beam, which is a solution schematic of the paraxial approximation to the Helmholtz equationamplifier circuit in RF. The fundamental Gaussian beam is rotationally-symmetric about its propagation axis, and its transverse field distribution follows a Gaussian function profile. The critical parameter is the beam radius w<sub>0<Spice A/sub>; it is the point where the field drops by 1/e from its value at the centerD. The beam opens up into a cone along the propagation direction, with a cone angle of tan &theta; = &lambda;<sub>0]] </subtd>/(&pi;.&omega;<sub>0</subtr>) (&lambda;<sub>0</subtable> is the free-space wavelength).

{{note|The beam radius has to same Netlist can be at least &lambda;₀/&pi;; otherwise, strong fields appear outside the excitation box}}written using a B-type nonlinear dependent source as follows:

* Right click on the '''Gaussian Beam''' item in the '''Sources''' section of the Navigation Tree and select '''Insert New Source...''' This opens up the Gaussian Beam Dialog.* Similar to the plane wave, a default Excitation Box three cells away from the bounding box of the geometry is suggested, i.e. the radio button '''Size: Default''' is selected by default. The radio button '''Size: Custom''' allows you to set the excitation box manually by modifying the coordinates of '''Corner C1 1''' (front lower left) and '''Corner 2''' (back upper right) of the box in the grid coordinate system (GCS).* In the Field Definition section of the dialog, you can enter the Amplitude of incident electric field in V/m. The default field '''Amplitude''' is 1 V/m. Note that you do not specify the phase of a Gaussian beam because the beam focus already contains the phase information.* The direction of the Gaussian Beam is determined by the incident '''Theta''' and '''Phi''' angles in degrees. You can also set the '''Polarization''' of the Gaussian Beam and choose from the three options: '''TM_z''', '''TE_z''' and '''User Defined'''.* Unlike plane waves, a Gaussian beam is a localized field. Therefore, you need to specify the '''Beam Properties'''. This includes the coordinates of the beam's '''Focus''', which is the beam's waist center in the world coordinate system as well as the beam's '''Radius''' in project units.0 1p

{| border="0"|-| valign="top"|[[Image:FDTD46.png|thumb|250px|[[FDTD Module]]'s Plane Wave dialog]]| valign="top"|[[Image:FDTD47.png|thumb|250px|[[FDTD Module]]'s Gaussian Beam dialog]]|-|}subckt amp_dev 1 2 3 4

B1 3 4 v ==Running FDTD Simulations==20*v(1,2)

<table><tr><td> [[Image:FDTD58Amp ex.png|thumb|300pxleft|[[FDTD Module]]'s Engine Settings dialog550px|The geometry of a microstrip-based amplifier with an active two-port device.]]</td></tr></table>

An FDTD simulation involves a number of numerical [[parameters]] that can be accessed and modified from the FDTD Engine Settings Dialog== EM. To open this dialog, select Tempo'''Menu > Simulate > s Observables & Simulation Engine Settings... '''or open the '''Run Dialog''', and click the '''Settings''' button next to the engine dropdown list.Data Types==

In the " '''Convergence''' " section of the dialog, you can set the '''Termination Criterion''' for === Understanding 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_n Observable Types = &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 = 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 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 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 = 0 until the end of the time marching loop. However, in mind that for highly resonant structuresorder to save memory usage, you may have the engine discards the temporal field data from each time step to increase the maximum number 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 steps to very large values above 20-consuming,000post-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.}}

The "'''Acceleration'''" section EM.Tempo offers the following types of the FDTD Simulation Engine Settings dialog give three options for the FDTD kerneloutput 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# Serial CPU SolverTemporal_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# MultiTemporal_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|-Core CPU Solver| 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# GPU SolverNear-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|-| style="width:30px;" | [[File:huyg_surf_icon.png]]| style="width:150px;" | Equivalent Electric and Magnetic Surface Currents| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Huygens_Surface_Observable |Huygens Surface]]| style="width:300px;" | Collecting tangential field data on a box to be used later as a Huygens source in other [[EM.Cube]] modules| style="width:250px;" | None|-| style="width:30px;" | [[File:CartData_icon.png]]| style="width:150px;" | Generic 3D Cartesian Spatial Data| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#3D_Cartesian_Data_Observable | 3D Cartesian Data]]| style="width:300px;" | Visualizing the contents of generic 3D Cartesian spatial data files overlaid on the project workspace | style="width:250px;" | Requires import of an existing ".CAR" data file|}

The serial CPU solver is Click on each category to learn more details about it in the [[Glossary of EM.Cube's Simulation Observables & Graph Types]]'s basic FDTD kernel that run the time marching loop on a single central processing unit (CPU) of your computer. The default option is the multi-core CPU solver. This is a highly parallelized version of the FDTD kernel based on the Open-MP framework. It takes full advantage of a multi-core, multi-CPU architecture, if your computer does have one. The GPU solver is a hardware-accelerated FDTD kernel optimized for CUDA-enabled graphical processing unit (GPU) cards. If your computer has a fast NVIDIA GPU card with enough onboard RAM, the GPU kernel can speed up your FDTD simulations up to 50 times or more over the single CPU solver.

For structures excited with a plane wave source, there are two standard FDTD formulations: '''Scattered Field '''(SF) formulation and '''Total Field - Scattered Field''' (TF-SF) formulation. [[Of EM.Cube]]Tempo's [[FDTD Module|FDTD module]] offers both formulations. The TF-SF solver is frequency domain observables, the default choice near fields, far fields and all of their associated parameters like directivity, RCS, etc., are calculated at a certain single frequency that is typically much faster than specified as part of the SF solver definition of the observable. To compute those frequency domain data at several frequencies, you need to define multiple observables, one for most problemseach frequency. In two casesOn the other hand, when port characteristics like S/Y/Z parameters and VSWR are calculated over the structure has periodic boundary conditions entire specified bandwidth of your project. Of EM.Tempo's source types, lumped sources, waveguide sources and distributed sources let you define one or infinite CPML boundary conditions (zero more ports for your physical structure and compute its port characteristics. One of EM.Tempo's real advantages over frequency-domain offsets), only the SF solver solvers is available. The other sections its ability of the FDTD Simulation Engine Settings dialog will be described next generate wideband S/Z/Y parameter data in the context of [[Waveforms and Discrete Fourier Transforms]]a single simulation run.

===Running A Wideband FDTD SimulationExamining the Near Fields in Time and Frequency Domains ===

Once you build your physical structure in EM.Tempo's FDTD time marching loop computes all the project workspace six electric and define an excitation sourcemagnetic 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 computationally very inefficient to store. Instead, you are ready can instruct EM.Tempo to run an save a small potion of these data for visualization and plotting purposes. Using a '''Field Probe''' at a specified point, you can record the a time-domain field component over the entire FDTD simulationloop. The simulation engine will run even if you have not defined any observables. Obviously, no simulation data will be generated in that casetime-domain results are also transformed to the frequency domain within the specified bandwidth using a discrete Fourier transform (DFT). <table><tr><td> [[EMImage:FDTD77.png|thumb|left|480px|Time-domain evolution of the electric field at a given point.Cube]]'s [[FDTD Module]] currently offers several different simulation modes as follows:</td></tr></table>

<table><tr><td> [[Image:FDTD57FDTD_FS2.png|thumb|250pxleft|Figure 1420px|EM.Tempo's Field Sensor dialog.]] </td></tr><tr><td> [[Image: FDTD_FS1_new.png|thumb|left|480px|Three field sensor planes defined around a PEC ellipsoid illuminated by a plane wave source.]] </td></tr></table><table><tr><td> [[EMImage:FDTD_FS3_new.png|thumb|left|360px|Electric field distribution above the PEC plate.Cube]]'s FDTD Simulation dialog</td><td> [[Image:FDTD_FS4_new.png|thumb|left|360px|Magnetic field distribution above the PEC plate.]]</td></tr></table>

Analysis is the simplest and most straightforward simulation mode of the [[FDTD Module]]. It runs the FDTD time marching loop once. At the end of the simulation, the time=== Computing Far-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 Field Characteristics in this manual.FDTD ===

To open Far fields are the Simulation Run Dialog, click the '''Run''' [[Image:run_icon.png]] button of asymptotic form the '''Simulate Toolbar''' fields when r &rarr; &infin; or select '''Menu k_{Simulate 0} Runr >> 1...''' from Under these assumptions, the menu bar or use the keyboard shortcut '''Ctrl+R'''.fields propagate outward as transverse electromagnetic (TEM) waves:

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-cells per seconds, and the value of the convergence ratio U<submath>n\mathbf{H^{ff}(r)} = \frac{1}{\eta_0} \mathbf{ \hat{k} \times E^{ff}(r)} </submath>/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.

{{isoimg|FDTD66Far fields are typically computed in the spherical coordinate system as functions of the elevation and azimuth observation angles &theta; and &phi;.png|[[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 far fields represent the radiation pattern of your source(s) in the far zone. In that case, you need to define a '''Radiation Pattern - Far Field Observable''' for your project. When your physical structure is illuminated by a plane wave source or 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 '''RCS - Far Field Observable''' for your project. In the FDTD Module]]method, 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's output windowdefault 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 entire 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.}}

The default radiation box is placed at an offset of 0.1&lambda;₀ from the largest bounding box of your physical structure. You can change the offset value from the "Far Field Acceleration" dialog, which can be accessed by clicking the {{mainpagekey|[[Waveforms and Discrete Fourier Transforms]]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 accuracy of the FDTD simulation results depends on the right choice of temporal waveform. <table><tr><td> [[Image:FDTD_FF2.png|thumb|left|480px|EM.Cube]]Tempo's default waveform choice is a modulated Gaussian pulse. At the end of an FDTD simulation, the time domain far field data are transformed into the frequency domain at your specified frequency or bandwidth to produce the desired observablesacceleration dialog. ]] </td></tr></table>

In addition to the default waveforms, [[EM.Cube|EM.CUBE]] allows the ability to define custom waveforms by either time or frequency specifications on === Radiation Pattern Above a per source basis.Half-Space Medium ===

Of [[FDTD Module]]'s observablesIn EM.Tempo, you can use CPML boundary conditions with zero offsets to model a structure with infinite lateral extents. The calculation of the near fields, far fields and all using the near-field-to-far-field transformation requires the dyadic Green's function of their associated [[parameters]] like directivity, RCS, etcthe background structure.By default, are calculated at a certain frequency that is specified as part of the definition of FDTD engine uses the observable. On free space dyadic Green's function for the other hand, port characteristics like S/Y/Z [[parameters]], VSWR and periodic characteristics like reflection and transmission coefficientsfar field calculation. In general, are calculated over the entire specified bandwidth of your projectEM.Tempo provides the dyadic Green's functions for four scenarios:

In [[EMThe 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.Cube]], project observables are If you set the simulation data that are generated by lateral domain offset values along the simulation engine at the end of each simulation run. [[EM.Cube]]'s FDTD simulation engine calculates all the six electric ±X and magnetic field components (E_x±Y directions equal to zero, E_ytoo, E_z, H_xthen your structure is, Hin effect, terminated at an infinite half-space dielectric medium. In that case, you have to specify the permittivity &epsilon;_yr and H_z) at every mesh grid node at all time steps from t = 0 until the end electric conductivity &sigma; of the time loop. However, terminating medium in order to save memory space, the engine has Background Medium dialog. You may additionally want to destroy set the temporal field data from each time step to Z-coordinate of the next top of that dielectric layer as the position of the interface between the free space and reuse the memorylower dielectric half-space. Storage, manipulation and visualization Note that the current version of 3D data can become overwhelming for complex structures and larger computational domainsEM. Furthermore, calculation Tempo does not calculate the far-field Green's function of some field characteristics such as radiation patterns or radar cross section (RCS) can be sizable, timea conductor-consumingbacked, post-processing tasksdielectric substrate with a finite layer thickness. That is why [[To use the background medium feature of EM.Cube]] asks you to define project observables to instruct why types of simulation data you seek in each simulation effortTempo, your structure can have either an infinite PEC/PMC ground or a dielectric half-space termination.

<table><tr><td> [[EMImage:fdtd_out36_tn.png|thumb|left|360px|Radiation pattern of a vertical dipole above PEC ground.Cube]]'s FDTD Modules currently offers the following types </td><td> [[Image:fdtd_out37_tn.png|thumb|left|360px|Radiation pattern of observablea vertical dipole above PMC ground.]] </td></tr><tr><td> [[Image:fdtd_out38_tn.png|thumb|left|360px|Radiation pattern of a horizontal dipole above PEC ground.]] </td><td> [[Image:fdtd_out39_tn.png|thumb|left|360px|Radiation pattern of a horizontal dipole above PMC ground.]] </td></tr></table>

* Field Probes* Field Sensors* Domain Energy* Far Field === Generating and Working with Multi- Radiation Patterns* Far Field - RCS* Huygens Surface Frequency Simulation Data* Port Characteristics (S/Y/Z [[Parameters]] and VSWR)* Reflection and Transmission Coefficients===

Field probes monitor One of the field components at a certain point in primary advantages of the computational FDTD method is its ability to run wideband EM simulations. The frequency domain. They record data are computed by transforming the time-domain field data during to the entire time loop and compute their frequency spectrum using a discrete Fourier transformdomain. Field sensors are primarily intended for observation of near field maps on a certain cross section of This is done automatically when EM.Tempo computes the computational domainport characteristics such as S/Z/Y parameters. The field sensor planes following frequency-domain observables are parallel to one of the three principal XY, YZ or ZX planes. When you run defined at a single frequency sweep or parametric sweep, multiple maps are generated for each sample of your sweep variable, and you can animate these maps. You can also animate the evolution of the near fields in the time domain over the course of the simulated time loop. [[EM.Cube]] can also keep track of the electric, magnetic and total energy of the computational domain as functions of the time step.:

Using asymptotic near* Near-toField Sensor* Far-far-field transformations, [[EM.Cube]] calculates the far fields of your physical structure in the standard spherical coordinate system. The radiation patterns are indeed the spherical electric field components E_&theta; and E_&phi; expressed as functions of the observation angles &theta; and &phi; over a unit sphere. The far field data are calculated in the frequency domain at a specified frequency, which is equal to your project's center frequency by default. When your excitation source is a plane wave or a Gaussian beam, the far field data actually represent the scattering behavior of your "target". In the case of a plane wave source, the FDTD simulation engine can also compute the radar cross section of you target. If your structure is periodic, then the reflection and transmission coefficients of the periodic surface are also calculated over the entire bandwidth of your project.Radiation Pattern* RCS* Huygens Surface

You can define ports for lumped sources, waveguide sources and distributed sources. In that case, The default computation frequency of the FDTD simulation engine calculates above observables is the scattering project's center frequency (Sfc) [[parameters]] of your multiport network over the entire bandwidth specified in your project. From You can change the scattering matrix, [[EM.Cube]] determines observable frequency from the impedance observable's property dialog and admittance matrices of your network over the operational bandwidth. You can plot the S/Y/Z [[parameters]] enter any frequency in EMHz.GridThe reason these types of simulation data are computed at a single frequency is their typically very large size. If your project has more than one portHowever, the FDTD time loop will be run you can define as many times as the number instances of ports, Nthese observables and set different frequency values for each one. In each time loop run j (j = 1, 2the case of radiation pattern and RCS, there are two dialogs that can be accessed from the navigation tree..., N), Right-click on the source(s) associated with "Fer-Field Radiation Patterns" or "Radar Cross Sections" items of the jth port is (are) excited with a unit amplitude navigation tree and all the other sources are turned offselect '''Insert Multi-Frequency Radiation Pattern. In this run, all the S_ij parameters (i = 1, 2, ..''' or '''Insert Multi-Frequency RCS., N) are calculated. At .''' from the end of the Nth run, the entire S matrix is completedcontextual menu.===Probing Fields in Time and Frequency Domains===

<table><tr><td> [[Image:FDTD75RadPattern multi.png|thumb|300pxleft|FDTD Field Probe Dialog360px|EM.Tempo's Multi-frequency Radiation Pattern dialog.]]</td>By computing the time domain fields at a certain location, you can examine the transient response of a system at that location. This is also very useful for monitoring the convergence of FDTD time marching loop. <td> [[Image:RCS multi.png|thumb|left|360px|EM.Cube]]Tempo's field probes allow you to save the temporal values of a field component at a specified point in the computational domain during the entire time marching loop. You can plot the time domain field components as a function of the time step index. You can also plot the spectral contents of those field components, i.e. their Fourier transform, over the project's specified Multi-frequency bandwidthRadar Cross Section dialog. To define a new field probe, follow these steps:]] </td></tr></table>

* Right click on Using the '''Field Probe''' item in multi-frequency dialogs, you can set the '''Observables''' section value of the Navigation Tree Start Frequency, Stop Frequency and select '''Insert New ObservableStep Frequency in Hz...'''* You can change also set the default name values of the probe as well as its colorTheta Angle Increment and Phi Angle Increment in degrees. 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 values of coordinates (0,0,0)both quantities are 5&deg;. You can move the probe to any location by changing its X, Y and Z coordinates.* In the Probe Location section case of the dialogRCS, you can also set have choose one of the two options: '''DirectionBistatic RCS''' or '''Monostatic RCS''' of the probe from a dropdown list that contains ±X, ±Y and ±Z options. The default direction is +Z.

[[Image:FDTD76.png|thumb|300px|An XTo facilitate the process of all the defining multi-directed probe placed above a PEC plate illuminated by a normally incident plane wavefrequency observables in EM.]]Tempo, you can also use the following Python functions at the command line:

At the end of an 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 to the last time step and their frequency domain spectrum are recorded. 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. 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 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 the selected field component as a function of time step. The frequency-domain probe contains two Cartesian graphs: amplitude and phase of the selected field component over the project's frequency range.

[[Image:FDTD71emag_rcs_bistatic_multi_freq(1f1,f2,df,theta_incr,phi_incr).png|thumb|300px|[[FDTD Module]]'s Field Sensor dialog]]

In [[EM.Cube]] you can visualize the near fields at a specific frequency in a specific plane of the computational domain. At the end of an FDTD simulationemag_rcs_monostatic_multi_freq(f1, all the time domain electric and magnetic field values are available at all mesh nodes. These temporal quantities are transformed into the frequency domain using discrete Fourier transforms to calculate the electric and magnetic fields on a specified sensor plane. To define a new Field Sensorf2, follow these steps:df,theta_incr,phi_incr)

* Right click on the '''Field Sensors''' item in the '''Observables''' section of the Navigation Tree and select '''Insert New Observable...'''* The '''Label''' box allows you to change the sensor’s name.* Set the '''Direction''' of the field sensor. This is specified by the normal vector of the sensor plane. The available options are '''X'''emag_huygens_surface_multi_freq(f1, '''Y''' and '''Z'''f2, with the last being the default option.* By default [[EM.Cube]] creates a field sensor plane passing through the origin of coordinates (0df,0x1,0) 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 Xy1, Y and Z coordinates. Keep in mind that you can move a sensor plane only along the specified direction of the sensor. Thereforez1, only one coordinate can effectively be changed. As you increment or decrement this coordinatex2, 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 defaulty2, this is equal to the project's center frequency.z2)

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. 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 to 0 of the color map, maximum field value corresponding to 1 of the color map, and the unit of the field quantity, which is V/m for E-field and A/m for H-field. The values of phase plots are always shown in Radians between -p and p. 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, etc.

{{twoimg|FDTD72In the above Python functions, f1 and f2 are the start and stop frequencies, respectively, and df is the frequency increment, all expressed in Hz.png|Field Sensor (E-field) |FDTD74Note that the above commands simply create and insert the specified observables in the navigation tree.png|Field Sensor (HThey do not run perform a simulation. The 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, ...

[[Image:FDTD73EM.png|thumb|300px|Cartesian graph of total magnetic field vs. YTempo also provides some additional Python functions for the far-index along the crosshair in the field senor planeradiation patterns and RCS observables.]]

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 the specified bandwidth at any point within the computational domain. 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.

In the course of the FDTD time marching processemag_rcs_consolidate(x1, a tremendous amount of data are generated that include all the six E/H field components at every Yee cell and at every time step. The temporal field values at a sensor plane are of particular interest. Such plots show the evolution of the fields as a function of time starting from time t = 0x2, when all the fields are zero everywhere in the computational domain. [[EM.Cube]] can record snapshots of the field sensor data as the time loop marches forward. When you define a field sensor for the first timedx, by default it displays the frequency domain near field data. In order to record and save the time domain data, you have to open the field sensor's property dialog by right clicking on the field sensor's name in the Navigation Tree and selecting '''Properties...'''from the contextual menu. In the section titled '''Sensor Domain''', select the radio button labeled '''Time Domain'''. Also, in the section titled "Field Display - Multiple Plots", select one of the two radio buttons labeled '''E-Field''' or '''H-Field'''. By default, the time domain field data are saved every 100 time steps. To change this setting, right click on the '''Field Sensors''' item in the Navigation Tree and select '''Time Domain Settings...''' from the contextual menu. In the Time Domain Settings Dialog, change the value of the box labeled '''Sampling Interval (in time stepsbase_name)'''.

:<math> S_{ij} = \sqrt{\frac{Re(Z_i)}{Re(Z_j)}} \cdot \frac{V_j The two "consolidate" Python functions take the results of multi- Z_j^*I_j}{V_i+Z_i I_i} </math><!frequency simulation observables and merge them into a single data file. The base name in the case of far-field radiation patterns is "Multi_FF" as pointed out earlier. The name of the resulting consolidated data file is the same as the base name with a "_All" suffix and a ".DAT" file extension. In the case of far-[[Image:FDTD82(1)field radiation patterns, it is "Multi_FF_All.png]]DAT". The 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, the two "average" Python functions take several radiation pattern or RCS files with a common base name in the current project folder, compute their average and save the results to a new data file named "base_name_ave" with a ".RAD" or ".RCS" file extensions, respectively.

where V_i is the voltage across Port i, I_i is the current flowing into Port i and Z_i is the characteristic impedance of Port i. The sweep loop then moves to the next port until all ports have been excited. After == Generating 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 Mesh in complex data files with '''.CPX''' file extensionsEM. The following relationships are used:Tempo ==

:<math>\mathbf{ [Z] = [\sqrt{Z_0}] \cdot ([U]+[S]) \cdot ([U]-[S])^{-1} \cdot [\sqrt{Z_0}] }</math>== EM.Tempo's Mesh Types ===

:<math> \mathbf{ [Y] = [Z]^{-1} } </math><!--[[Image:FDTD83EM.Tempo generates a brick volume mesh for FDTD simulation. The FDTD mesh is a rectangular Yee mesh that extends to the entire computational domain. It is primarily constructed from three mesh grid profiles in the XY, YZ and ZX principal planes. These projections together create a 3D mesh space consisting of a large number of cubic volume cells (voxels) carefully assembled in a way that approximates the shape of the original structure.png]]-->

where <math>\mathbf{[U]}</math> is the identity matrix of order NIn EM.Tempo, and <math>\mathbf{\sqrt{Z_0}}</math> is a diagonal matrix whose diagonal elements are the square roots of port characteristic impedances. The voltage standing wave ratio (VSWR) you can choose one of the structure at the first port is also computed and saved to a real data '''.DAT''' file. The following definition is usedthree FDTD mesh types:

:<math> \text{VSWR} = \frac{|V_{max}|}{|V_{min}|} = \frac{1+|S_{11}|}{1-|S_{11}|} </math>* Adaptive Mesh<!--[[Image:FDTD84.png]]-* Regular Mesh* Fixed->Cell Mesh

You can plot EM.Tempo's default mesh generator produces an adaptive brick mesh of your physical structure, whose mesh resolution varies with the port characteristics from the Navigation Treefrequency. Right click on As the '''Port Definition''' item in operating frequency of your project increases, the default '''ObservablesAdaptive''' section FDTD 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 Navigation Tree geometric details, curvatures and select one thin slanted plates or sheets, which often pose a challenge to staircase meshing. It usually provides a reasonably accurate discretization of the items: '''Plot S [[Parameters]]'''most complex structures. Occasionally, '''Plot Y [[Parameters]]'''you may opt for a more regularized FDTD mesh with almost equal grid line spacings everywhere, '''Plot Z [[Parameters]]''', or '''Plot VSWR'''but still with a frequency-dependent cell size. In the first three cases, another sub-menu gives a list of individual port [[parameters]]. Keep in mind that in multi-port structurescase, each individual port parameter has its own graph. You you can also see a list of all the port characteristics data files in [[use EM.Cube]]Tempo's data manager. To open data manager, click the '''Data ManagerRegular''' [[Image:data_manager_icon.png]] button FDTD mesh generator, which is indeed a simplified version 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 clicking and highlighting its row in the table and then click the '''Plot''' button to plot the graph in '''EMadaptive mesh generator.Grid'''. By default, the S [[parameters]] are plotted as dual magnitude-phase graphs, while the Y The regular FDTD mesh enforces only two criteria: minimum mesh density and Z [[parameters]] are plotted as dual real-imaginary part graphsabsolute minimum grid spacing. The VSWR data grid cell sizes in this mesh are plotted on a Cartesian graph. You change almost uniform in objects of the format of complex data plotssame material composition or in free-space regions. In general complex data can be plotted in three forms:

# Magnitude EM.Tempo also offers a uniform, frequency-independent, '''Fixed-Cell''' FDTD mesh generator. The fixed-cell mesh consists of three uniform grids in the XY, YZ and Phase# Real ZX principal planes. However, the uniform mesh cell dimensions along the three direction, i.e. &Delta;x, &Delta;y and Imaginary Parts# Smith Chart&Delta;z do not have to be equal. The fixed-cell mesh generator tries to fit your physical structure to the mesh grid rather than adapting the mesh to your physical structure.

In particular{{Note|When choosing a mesh type for your FDTD simulation, it may be useful to plot keep in mind that adaptive and regular mesh types are frequency-dependent and their density varies with the S_ii highest frequency of your specified bandwidth, while the uniform mesh type is always fixed and independent of your project's frequency settings.}} [[parametersImage:Info_icon.png|30px]] on a Smith chart. To change the format of a data plot, select it and click the Click here to learn more about '''Edit [[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Working_with_EM.Cube.27s_Mesh_Generators | Working with Mesh Generator]]'''button of Data Manager and select one of the available graph type options.<br />

{{twoimg|FDTD114.png|[[EMImage:Info_icon.Cubepng|30px]]'s data manager showing a list Click here to learn more about the properties of complex data files available for plotting in EM.Grid'''[[Glossary_of_EM.Cube%27s_Simulation-Related_Operations#Adaptive_Yee_Mesh |FDTD115.png|Plot of S₂₁ of a filter in EM.GridTempo's Adaptive Brick Mesh Generator]]'''.}}

{{mainpage<table><tr><td> [[Image:Tempo L11 Fig5.png|thumb|left|550px|A human head model and a cellular phone handset on its side.]] </td></tr><tr><td> [[Farfield Calculations in EMImage:Tempo L11 Fig7.png|thumb|left|550px|The FDTD mesh of the human head model and the cellular phone handset.]] </td></tr><tr><td> [[Image:TempoL11 Fig8.png|thumb|left|550px|Another view of the FDTD mesh of the human head model and the cellular phone handset.]]}}</td></tr></table>

For radiating structures or scatterers, === Discretizing the far field quantities are of primary interest. [[EM.Cube]]'s [[FDTD Module]] can calculate Physical Structure Using the far field radiation patterns of an antenna or the radar cross section (RCS) of a target. In general, by far fields we mean the electric fields evaluated in the far zone of a physical structure, which satisfies the following condition:Adaptive Yee Mesh ===

In the FDTD methodEM.Tempo's default mesh generator creates an adaptive brick volume mesh that uses a variable staircase profile, where the far fields are calculated using a near-field-to-far-field transformation grid line spacings vary with the curvature (derivative) of the field quantities on a given closed surfaceobject edges or faces. [[EM.Cube]] uses rectangular boxes As a result, a higher mesh resolution is produced at "curved" areas to define these closed surfacesbetter capture the geometrical details. You can use [[EMThe resolution of the adaptive FDTD mesh is driven by the '''Mesh Density''', expressed in cells per effective wavelength.Cube]]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 = f₀ + &Delta;f/2, where f₀ (or fc) is your project's default radiation box center frequency and &Delta;f (or define your ownbw) is its specified bandwidth. NormallyIn other words, the radiation box should enclose effective wavelength in the entire FDTD structure. In this casefree space is &lambda;_{0, eff} = c / f_max, c being the calculated radiation pattern corresponds to speed of light in the entire radiating structurefree space. The radiation box may also contain only parts of effective wavelength in a structuredielectric material with relative permittivity &epsilon;_r and permeability &mu;_r is given by &lambda;_{d, which results in partial radiation patternseff} = &lambda;_0,eff / &radic;&epsilon;_r&mu;_r.

The adaptive FDTD mesh, by default, produces different grid cell sizes in the free space regions than 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 ===Defining &lambda;_0,eff / &radic;&epsilon;_r&mu;_r. Therefore, the average ratio of the cell size in a dielectric region to the cell size in the free space is 1/&radic;(&epsilon;_r&mu;_r). The Far Field Box===adaptive FDTD mesh generator also takes note of the geometrical features of the objects it discretizes. This is more visible in the case of curved solids, curves surfaces and curved wires or obliquely oriented planes and lines which need to be approximated using a staircase profile. 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 structures.

[[Image:FDTD116.png|thumb|250px|[[FDTD Module]]A carefully calculated, "<u>'s Radiation Pattern dialog]]''Adaptive'''</u>" mesh of your physical structure is generated in order to satisfy the following criteria:

For any far field calculations * Optimize the number of mesh cells in [[EMeach dimension.Cube]], first you have to define a far field observable The product of the number of cells in all the Navigation Treethree dimension determines the total mesh size. In [[The larger the mesh size, the longer the simulation time, especially with the CPU version of the FDTD Module]]engine. Also, defining a far field observable also initiates very large mesh size requires more RAM, which may exceed your GPU memory capacity. Set the '''Minimum Mesh Density''' to a far field box 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 empty (free) space between them and the computational domainboundaries. This box An effective wavelength is used to perform defined for each material at the nearhighest frequency of the project's 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 toachieve acceptable accuracy. As you reduce the mesh density, the simulation accuracy decreases.* Accurately represent and approximate the boundaries of edges or surfaces that are not grid-far-field transformation at aligned by closely adhering to their geometric contours. This is controlled by the end '''Minimum Grid Spacing Over Geometric Contours''', which can be specified either as a fraction of the free space grid spacing or as an FDTD absolute length value in project units.* Maximize the minimum grid spacing in any dimension inside the computational domain and thus maximize the simulationtime 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 '''Absolute Minimum Grid Spacing''', which can be specified either as a fraction of the free space grid spacing or as an absolute value. It is critical to accurately represent and precisely maintain the 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 mesh needs to represent them by two separate, but very closely spaced, grid lines. To insert control the minimum allowed grid spacing, use the '''Absolute Minimum Grid Spacing '''settings,* Maintain a new far field 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, follow these steps:(always checked by default).

* Right click on the '''Far Fields''' item in the '''Observables''' section When [[EM.Cube]] generates an FDTD mesh, a large number of the Navigation Tree and select '''Insert New Radiation Patterngeometrical considerations are taken into account...''' to open These include the Radiation Pattern Dialog.* Use the '''Label''' bounding box to change the name of each object and its corners, the far field or change the color ends of a line, the far field box using the '''Color''' button.* The frequency apex of radiation pattern calculation can be specified in the box labeled '''Far Field Frequency'''. By defaulta cone or pyramid, this is equal to or the center frequency locations of the project. Howeverlumped sources, you can calculate the far field data at any other frequency within the project's frequency range.* The resolution probes and sensors, vertices of plane wave or far field calculations is specified by '''Angle Increment''' expressed in degreesboxes, to name a few examples. By default, the &theta; and &phi; angles These points are incremented by 5 degrees.* Define the desired box for far field calculations “locked” as fixed grid nodes in the '''Radiation Box''' section of FDTD mesh. [[EM.Cube]] determines these points internally to generate a mesh that best approximates the dialogoriginal structure. As in the case of plane waves and Gaussian beamsyou saw earlier, there are two options available, 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 can use the bounding box of the entire structure. Select '''Size: Custom''' FDTD mesh settings to set control the far field box manually. The values for the coordinates of '''Corner 1''' shape and '''Corner 2''' can now be changed. '''Corner 1''' is the lower-front-left corner and '''Corner 2''' is the upper-rear-right corner resolution of the radiation box. The dimensions are specified in the world coordinate system (WCS).* At mesh, for example, around the end curved portions of an FDTD simulationyour structure, besides calculating the radiation data over the entire (spherical) 3D spaceor on slanted lines or faces, a number of 2D pattern graphs are also generatedetc. These settings are indeed pattern cuts at certain planes, which include the three principal XY, YZ global 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 in the box labeled '''Non-Principal Phi Plane'''. Also, the 2D radiation pattern graphs are normalized by default. You can instruct [[EM.Cube]] apply to plot all the 2D pattern graphs un-normalized (as calculated) by removing the check mark from the box labeled '''Normalize 2D Patterns'''objects making up your physical structure.

After closing You can control the Far Field global mesh more selectively using the Advanced FDTD Mesh Settings Dialog. To open this dialog, a far field entry immediately appears with its given name under click the '''Far FieldsAdvanced ''' item 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 '''ObservablesMax Adjacent Cell Size Ratio''' section in the Navigation Tree. A far field box shows up as The default value of this parameter is 1.3, which maintains a light blue wireframe box in the project workspacesmooth grid line spacing scheme with no more than 1:1. You can right click on the far field item's name in the navigation tree 3 ratio for adjacent cells. By default, grid lines are enforced at all source and select '''Propertiesobservable locations...''' You have the option to open disable this feature and round up source locations to their closest grid lines. You may also uncheck the radiation pattern dialog for further editingbox labeled "Adapt mesh resolution to material properties". Bear in mind In that a full 3D radiation pattern calculation with a high angular resolution might case, the same effective wavelength will be very timeused to determine the mesh resolution inside all materials as well as the free-consumingspace regions.

Once an FDTD simulation is finished, three far field items are added to The figures below compare the Far Field section three types of the Navigation Tree. These are the far-zone E-field component along FDTD mesh for a dielectric ellipsoid with &phiepsilon; direction, _r = 4. Note that the far-zone E-field component along &phi; direction and cell size inside the total far-zone E-field defined as:dielectric region is half the cell size in the air region.

:<mathtable><tr><td> [[Image:FDTD MAN21.png|\mathbf{E_{ff,tot}}thumb| = \sqrt{left| E_{\theta}360px|^2 + |E_{\phi}|^2 }The geometry of a dielectric ellipsoid with &epsilon;<sub>r</mathsub>= 4.]]<!--/td><td> [[Image:FDTD129FDTD MAN22.png|thumb|left|360px|The adaptive mesh of the dielectric ellipsoid.]]--</td></tr></table>

<table><tr><td> [[Image:FDTD MAN18.png|thumb|left|360px|The 3D plots can be viewed in top view of the project workspace by clicking on each itemadaptive FDTD mesh of the dielectric ellipsoid. ]]</td><td> [[Image:FDTD MAN19.png|thumb|left|360px|The top view of the 3D far field plot can be changed regular FDTD mesh of the dielectric ellipsoid with the available view operations such as rotate, pan and zoomsame mesh density. A legend box appears in ]]</td></tr><tr><td> [[Image:FDTD MAN20A.png|thumb|left|360px|The top view of the upper right corner fixed-cell FDTD mesh of the 3D radiation pattern plot, which can be dragged around with dielectric ellipsoid using the left mouse buttonlarger cell size inside the air region. ]]</td><td> [[Image:FDTD MAN20.png|thumb|left|360px|The (maximum) '''Directivity''' top view of the radiating structure is displayed at the bottom fixed-cell FDTD mesh of the legend box and is calculated dielectric ellipsoid using the definition:smaller cell size inside the dielectric region.]]</td></tr></table>

:<math> D_0 = \frac{4\pi [S(\theta,\phi)]_{max}}{P_{rad}} = \frac{ 4\pi \big| \mathbf{E^{ff}}(\theta,\phi) \big|^2 |_{max} } { \int\limits_0^{2\pi} \int\limits_0^{\pi} \big| \mathbf{E^{ff}}(\theta,\phi) \big|^2 \sin\theta \,d\theta \,d\phi } </math><!--[[Image:FDTD113The 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.png]]-->

You can change the type of the 3D radiation pattern plot through the '''Radiation Pattern Dialog'''. In the '''3D Display Type''' section of this dialog you can choose from three options<table><tr><td> [[Image: '''3D Polar''', which is the default choice, '''Spherical Map''' and '''Cone'''FDTD MAN23. In the case png|thumb|left|450px|The geometry of cone type, you can also set the size of the cones that are used for a vectorial visualization of the far field data. If the structure blocks the view of the radiation pattern, you can simply hide or freeze the entire physical structure or parts of it. Note that 3D radiation patterns are always positioned at the origin (0,0,0) of the spherical world coordinate system even though the radiation center of the structure may not be located at that point. Sometimes, it might be a good idea to hide the physical structure when you are viewing the 3D radiation patterns to avoid any confusion. In a 3D radiation pattern visualization, the fields are always normalized to the maximum of the total far field. For this reason, sometimes the cross-polarization component might get lost compared to the co-polarization component and you have to zoom in to make it visiblePEC parabolic reflector.]]</td></tr></table>

You can also change the properties <table><tr><td> [[Image:FDTD MAN24.png|thumb|left|360px|The low-resolution adaptive mesh of the 3D radiation pattern plot by selecting the '''PropertiesPEC parabolic reflector.]]</td><td> [[Image:FDTD MAN27..''' item in the right click menu png|thumb|left|360px|The high-resolution adaptive mesh of the plot's name in the Navigation Tree or by double-clicking the legend boxPEC parabolic reflector. This opens up the '''Output Plot Settings Dialog''']]</td></tr><tr><td> [[Image:FDTD MAN26. In general, there are two scale options: Linear png|thumb|left|360px|The top (which is the default optionXY) and dB. In view of the case low-resolution adaptive mesh of a linear plot, the plot range varies between 0 and 1PEC parabolic reflector. In the case ]]</td><td> [[Image:FDTD MAN25.png|thumb|left|360px|The right (YZ) view of a dB plot, the range is fixed from low-50 to 0dB. You can change the '''Color Map''' option as well the foreground and background colors resolution adaptive mesh of the legend boxPEC parabolic reflector.]]</td></tr></table>

At * Open the end of an FDTD simulation, the radiation pattern data E<subFixed Grid Points Dialog by selecting '''Menu >&theta;</subSimulate >, E<subDiscretization >&phi;</sub> and E_tot in the three principal XY, YZ and ZX planes plus one additional user defined phi plane cut are available for plotting on 2D graphs in '''EM.Fixed Grid'''Points. There are a total of eight 2D pattern graphs in the data manager: 4 polar graphs and 4 Cartesian graphs of the same pattern data. To open data manager, click the '''Data Manager''' [[Image:data_manager_icon.png]] button of the '''Simulate Toolbar''' or select by right-clicking on the '''Simulate > Data ManagerFDTD''' from the menu bar or right click on the '''Data ManagerMesh''' item of the Navigation Tree navigation tree and select selecting '''Open Data ManagerFixed Grid Points Settings...''' from * Click the contextual menu or use {{key|Add/Edit}} button to open the keyboard shortcut '''Ctrl+D'''"Add Fixed Grid Point" dialog. In * Enter the Data manager Dialog(X, you will see a list Y, Z) coordinates of all the data files available for plotting. These include new fixed point in the four polar pattern data files with a '''.ANG''' file extension coordinate boxes and click the four Cartesian pattern data file with a '''{{key|OK}} button.DAT''' file extension. Select any data file by highlighting its row in * To modify the coordinates of an existing fixed grid point, select it from the table and then click the '''Plot''' {{key|Add/Edit}} button to plot .* You can also remove a fix grid point from the graphFDTD mesh using the {{key|Delete}} button.

At the end of an FDTD sweep simulation, other radiation characteristics are also computed as a function of the sweep variable (frequency, angle, or any other <table><tr><td> [[Image:FDTD36.png|thumb|left|480px|A user -defined variable). These include the '''Directivity (D0)''', '''Total Radiated Power (PRAD)''' and '''Directive Gain (DG)''' as a function of the &theta; and &phi; angles. Another radiation characteristic of interest especially fixed grid point in circularly polarized scenarios is the Axial Ratioan FDTD mesh. In ]] </td></tr><tr><td> [[Image:FDTD38.png|thumb|left|480px|Adding a new fixed grid point in EM.CubeTempo's fixed grid points settings dialog.]], the axial ratio is always defined in the LCP<sub/td>z</subtr> or RCP<subtr>z</subtd> sense based on the X- and Y-components of the electric field[[Image:FDTD39. 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 png|thumb|left|480px|The "Additional Radiation CharacteristicsAdd Fixed Grid Point" 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 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 zenithdialog.]] </td></tr></table>

{| border="0"|According to the Courant-| valign="top"|[[File:FDTD119Friedrichs-Levy (CFL) stability criterion, the FDTD time step is determined by the smallest cell size in your FDTD mesh.png|thumb|left|250px|[[Occasionally, EM.Cube]]Tempo's Data Manager dialog showing adaptive mesh generator may create extremely tiny grid cells that would result in extremely small time steps. This would then translate into a list of 2D polar and Cartesian radiation pattern graphsvery long computation time.]]| valign="top"|[[File:FDTD118.png|thumb|left|250px|A 2D Cartesian radiation pattern in the ZX plane cutEM.Cube]]| valign=offers the "topRegular"|[[File:FDTD117FDTD mesh generator, which is a simplified version of the adaptive mesh generator.png|thumb|left|250px|A 2D Cartesian radiation pattern In a regular FDTD mesh, the grid cell sizes stay rather the same in objects of the ZY plane cutsame 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.]]|-|}

===Radiation Pattern Above A Half-Space MediumProfiling the Brick Mesh ===

{{mainpage|A volumetric brick mesh is overwhelming for visualization in the 3D space. For this reason, [[Radiation Pattern Above A Half Space MediumEM.Cube]]}}'s mesh view shows only the outline of the cells on exterior surface of the (staircased) meshed objects. The mesh grid planes provide a 2D profile of the mesh cells along the principal coordinate planes. To display a mesh grid plane, select '''Menu > Simulate > Discretization > Grid Planes >''' and pick one of the three options: '''XY Plane''', '''YZ Plane''' or '''ZX Plane'''. You may also right click on one of the '''XY Plane''', '''YZ Plane''' or '''ZX Plane''' items in the '''Discretization''' section of the navigation tree and select '''Show''' from the contextual menu.

As mentioned earlier when discussing boundary conditions and computational domainWhile a mesh grid plane is visible, you can use CPML move it back and forth between the two boundary conditions with zero offsets to model a structure with infinite lateral extents. At planes at the end two opposite sides of the FDTD simulation, the far fields are calculated using the near-field-to-far-field transformationcomputational domain. This calculation requires the dyadic Green's function You can do this in one of the background structure. By default, the FDTD engine uses the free space dyadic Green's function for the far field calculation. In general, the [[FDTD Module]] features dyadic Green's functions for following four scenariosways:

# Free space background# Free space background terminated in an infinite PEC ground plane at * Using the bottomkeyboard's Page Up {{key|PgUp}} key and Page Down {{key|PgDn}} key.# Free space background terminated * By selecting '''Menu > Simulate > Discretization > Grid Planes > Increment Grid''' or ''' Decrement Grid'''.* By right clicking on one of the '''XY Plane''', '''YZ Plane''' or '''ZX Plane''' items in an infinite PMC ground plane at the bottom'''Discretization''' section of the navigation tree and selecting '''Increment Grid''' or ''' Decrement Grid''' from the contextual menu.# Free space background terminated in an infinite dielectric half-space medium* Using the keyboard shortcut {{key|>}} or {{key|<}}.

<table><tr><td> [[Image:FDTD131Tempo L1 Fig11.png|thumb|300pxleft|[[FDTD Module360px|The XY mesh grid plane.]]'s RCS dialog</td><td> [[Image:Tempo L1 Fig12.png|thumb|left|360px|The YZ mesh grid plane.]]</td></tr></table>

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 === The FDTD Grid Coordinate System (RCSGCS) of a target defined as:===

:When 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 to specify any location in the computational domain in terms of node indices on the mesh grid. [[EM.Cube]] displays the total number of mesh grid lines of the simulation domain (N<mathsub>\sigma_{\theta} x</sub> × N_y × N_z) along the three principal axes on the '''Status Bar'''. Therefore, the number of cells in each direction is one less than the number of grid lines, i.e. (N_x-1)× (N_y-1) × (N_z-1). The lower left front corner of the domain box (Xmin, Ymin, Zmin) becomes the origin of the mesh grid coordinate system (I = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{\theta}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}0, \quad \sigma_{\phi} J = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{\phi}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}0, \quad \sigma K = \sigma_{\theta} + \sigma_{\phi} 0). The upper right back corner of the domain box (Xmax, Ymax, Zmax) therefore becomes (I = 4\pi r^2 \dfrac{ \big| \mathbf{E}_{tot}^{scat} \big| ^2} {\big| \mathbf{E}^{inc} \big|^2}N_{x</mathsub>-1, J = N<!sub>y}-1, K = N_z-[[Image:FDTD1301).png]]-->

To compute [[EM.Cube]] allows you to navigate through the RCS mesh grid and evaluate the grid points individually. Every time you display one of your physical structurethe three mesh grid planes, the "'''Grid Coordinate System (GCS)'''" is automatically activated. On the Status Bar, you must define an RCS observable 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 grid coordinate system. The current grid point is displayed by a radiation patternsmall white circle on the current mesh grid plane, and it always starts from (I = 0, J = 0, K = 0). Using the 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 lower left front corner of the computational domain and measure distances in project unit just like the WCS. Follow these steps:

* 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.<table>* 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.<tr>* 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.<td> * 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, a default radiation box (radio button '''Size[[Image: Default''') or a user defined radiation box FDTD35(radio buttons '''Size: Custom'''1). If you check '''Size: Default''', no radiation box corner coordinates need to be specified. png|thumb|left|480px|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 grid cursor on the XY grid plane and its grid 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 FDTD simulationI, besides calculating the RCS data over the entire (sphericalJ, K) 3D space, a number of 2D RCS graphs are also generated. These are indeed RCS cuts at certain planes, which include displayed on the three principal XY, YZ and ZX planes plus one additional constant &phi;-cut. This latter cut is at &phi; = 45° by default. You can assign another &phi; angle in degrees in the box labeled '''Non-Principal Phi Plane'''status bar.]]</td></tr></table>

[[Image:FDTD132=== EM.png|thumb|300px|An example of the 3D radar cross section of a PEC plate.]]Tempo's Simulation Modes ===

At Once you build your physical structure in the end of project workspace and define an FDTD simulation, in the far field section of the Navigation Treeexcitation source, 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 ready to the maximum RCS value, which is displayed in the legend boxrun an FDTD simulation. The 2D RCS graphs can be plotted in '''EM.Grid '''exactly in the same way that simulation engine will run even if you plot 2D radiation pattern graphs. A total of eight 2D RCS graphs are available: 4 polar and 4 Cartesian graphs for the XY, YZ, ZX and user have not defined plane cuts. at the end of a sweep simulation, [[EM.Cube]] calculates some other quantities including the backscatter RCS (BRCS), forward-scatter RCS (FRCS) and the maximum RCS (MRCS) as functions of the sweep variable (frequency, angle, or any user defined variable)observables. In this caseObviously, the RCS needs to no simulation data will be computed at a fixed pair of &phi; and &theta; angles. These angles are specified generated 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 zeniththat case.  {{Note|Unlike [[EM.Cube]]'s Planar, MoM3D and Physical Optics Modules, the [[FDTD ModuleTempo]] currently does not support 3D mono-static RCS calculation due to the enormous amount of computational work needed. Only the bi-static RCS is calculated for a given plane wave source.}} ===Angular Sweeps===If your FDTD project has a plane wave excitation, then you can also run an angular sweep. In this sweep, the values of the incidence angles &theta; and &phi; are varied at each sweep run. To run an angular sweep, open the FDTD '''Run Dialog''' and from the '''Simulation Mode '''dropdown list select the '''Angular Sweep''' option. Click the '''Settings''' button next to this dropdown list to open up the Angle Settings Dialog. In an angular sweep, only one of the two angles, &theta; and &phi;, can be varied at a time. Choose the radio button corresponding to the angle that you want to sweep. Then, set the values of the '''Start Angle''' and '''End Angle''' as well as the '''Number of Samples'''. Under normal circumstances, you would sweep &theta; from 180°to 90° backward and sweep &phi; from zero to 360° forward.<br /> {{isoimg|FDTD128.png|[[FDTD Module]]'s Angle Settings dialog.}} ===Defining Custom Output Parameters=== {{mainpage|Custom Output}} At the end of an FDTD offers several different simulation, a number of computed quantities are designated modes as "Standard Output" [[parameters]] and can be used for various post-processing data operations. For example, you can define design objectives based on them, which you need for [[optimization]]. The table below gives a list of all the currently available standard output [[parameters]] in [[EM.Cube]]'s [[FDTD Module]]follows:

! scope="col"| FNLZXSimulation Mode! scope="col"| First Null Level in ZX PlaneUsage! scope="col"| Number of Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions

| FNLUstyle="width:120px;" | [[#Running a Wideband FDTD Analysis | Wideband Analysis]]| First Null Level in User Defined Planestyle="width:270px;" | Simulates the physical structure "As Is"| style="width:100px;" | Single run| style="width:200px;" | Generates data for many frequency samples| style="width:150px;" | None

| BRCSstyle="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]| Back-Scatter RCSstyle="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

| FRCSstyle="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]]| Forward-Scatter RCS along User Defined Incident Directionstyle="width:270px;" | Optimizes the value(s) of one or more project variables to achieve a design goal | style="width:100px;" | Multiple runs | style="width:200px;" | Runs at the center frequency fc| style="width:150px;" | None

| MRCSstyle="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Generating_Surrogate_Models | HDMR Sweep]]| Maximum Bi-static RCSstyle="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

| RCMstyle="width:120px;" | Magnitude of Reflection Coefficient|-| RCI| Phase of Reflection Coefficient ([[#Running a Dispersion Sweep in radians)EM.Tempo |-Dispersion Sweep]]| RCRstyle="width:270px;" | Real Part Varies the value of Reflection Coefficient|-| RCI| Imaginary Part of Reflection Coefficient|-| TCM| Magnitude of Transmission Coefficient|-| TCP| Phase of Transmission Coefficient (wavenumber in radians)a periodic structure |-style="width:100px;" | TCRMultiple runs | Real Part of Transmission Coefficientstyle="width:200px;" |-Runs at multiple frequency points corresponding to constant wavenumber values| TCIstyle="width:150px;" | Imaginary Part of Transmission CoefficientOnly for periodic structures excited by a plane wave source

All the radiation- and scattering-related standard outputs are available only if you have defined === Running a radiation pattern far field observable or an RCS far field observable, respectively. The standard output [[parameters]] DGU and ARU are the directive gain and axial ratio calculated at the certain user defined direction with spherical observation angles (&theta;, &phi;). These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radiation Pattern Dialog'''. The standard output [[parameters]] HPBWU, SLLU, FNBU and FNLU are determined at a user defined f-plane cut. This azimuth angle is specified in degrees as '''Non-Principal Phi Plane''' in the "Output Settings" section of the '''Radiation Pattern Dialog''', and its default value is 45°. The standard output [[parameters]] BRCS and MRCS are the total back-scatter RCS and the maximum total RCS of your planar structure when it is excited by an incident plane wave source at the specified &theta;_s and &phi;_s source angles. FRCS, on the other hand, is the total forward-scatter RCS measured at the predetermined &theta;_o and &phi;_o observation angles. These angles are specified 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 zenith.Wideband FDTD Analysis ===

==Modeling 3D Periodic Structures Using * Identify material types and proper domain boundary conditions.* Identify the source type and excitation mechanism.* Define the project observables.* Mesh the physical structure and examine the quality of the generated mesh and it geometric fidelity.* Determine the proper temporal waveform.* Select the simulation mode and run the FDTD==engine.

Wideband analysis is [[EM.Tempo allows you to simulate doubly periodic structures with periodicities along the X ]]'s simplest and Y directionsmost straightforward simulation mode. Many interesting structures such as frequency selective surfaces (FSS)It runs the FDTD time marching loop once. At the end of the simulation, electromagnetic bandthe time-gap domain field data are transformed into the frequency domain using a discrete Fourier transform (EBGDFT) structures and metamaterial structures can be modeled using periodic geometries. In the case of an infinitely extended periodic structureAs a result, it is sufficient to analyze only you can generate wideband frequency data from a unit cellsingle time-domain simulation run. In the FDTD method, The other simulation modes will be explained later in this is accomplished by applying periodic boundary conditions (PBC) at the side walls of the computational domainmanual.

Click here to learn more about To open the Simulation Run Dialog, click the '''Run''' [[Time Domain Simulation Image:run_icon.png]] button of Periodic Structuresthe '''Simulate Toolbar''' or select the menu item '''Simulate &rarr; Run...''' from the menu bar or use the keyboard shortcut {{key|Ctrl+R}}. To start the FDTD simulation, click the {{key|Run}} button at the bottom of this dialog. Once the simulation starts, the "Output Message 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-cells per seconds, and the value of the convergence ratio U_n/U_max in dB. An [[EM.Tempo]]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.

Using [[EM.Cube]]'s [[=== The FDTD Module]], you can simulate complex 3D periodic 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 have to instruct [[EM.Cube]] to turn it into a periodic structure through [[FDTD Module]]'s Periodicity Dialog. By designating a structure as periodic, 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 lines.Simulation Engine Settings ===

[[Image:FDTD134.png|250px|thumb|[[An FDTD Module]]'s Periodicity simulation involves a number of 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.

To define 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 periodic structurecertain point in time. If you use a decaying waveform like a Gaussian pulse or a Modulated Gaussian pulse, follow these 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 = 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 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.

* Select '''Menu > Simulate > Computational Domain > Periodicity Settings...''' or right click on the '''Periodicity''' item {{Note|Keep in 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 dialogmind that for highly resonant structures, which are equal to the dimensions of the structure's 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 you may have to designate increase the boundary conditions on the ±Z faces maximum number of the computational domain. These are CPML by default. But you can change them time steps to PEC or PMCvery large values above 20,000.}}

In [[EM.Cube]]'s [[FDTD Module]], 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 S₁₁ parameter or input impedance of the periodic antenna array. You can also compute the near# Serial CPU Solver# Multi-field and far-field data.Core CPU Solver# GPU Solver

The serial CPU solver is [[EM.CubeTempo]]'s periodic basic FDTD simulator uses periodic boundary conditions kernel that run the time marching loop on a single central processing unit (PBCCPU) to model an infinite periodic array. All the periodic replicas of the unit cell structure are excitedyour computer. In this case, you can impose a phase progression across The default option is the infinite array to steer its beammulti-core CPU solver. 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 This is a section titled '''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''' in degrees. At the end highly parallelized version of the periodic FDTD simulation, kernel based on the radiation pattern Open-MP framework. It takes full advantage 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]]multi-core, in this casemulti-CPU architecture, correspond to the single unit cell, not the infinite array. Therefore, they do not show the beam scanning even if you your computer does have entered nonzero values for the &theta; and/or &phi; scan anglesone. For this purpose, you have to define The GPU solver is a finitehardware-sized array factoraccelerated FDTD kernel optimized for CUDA-enabled graphical processing unit (GPU) cards. You do this in the "Impose Array Factor" section of the '''Radiation Pattern Dialog'''. In the case of If your computer has a periodic structurefast NVIDIA GPU card with enough onboard RAM, 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 GPU kernel can speed up your FDTD simulations up to 50 times or more over 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 anysingle CPU solver.

{{Note|For large &theta; scan anglesstructures excited with a plane wave source, there are two standard FDTD formulations: '''Scattered Field '''(SF) formulation and '''Total Field - Scattered Field''' (TF-SF) formulation. [[EM.Tempo]] offers both formulations. The TF-SF solver is the default choice and is typically much faster than the SF solver for most problems. In two cases, when the structure has periodic FDTD time matching loop may take far more time steps to convergeboundary conditions or infinite CPML boundary conditions (zero domain offsets), only the SF solver is available.}}

{{twoimg|FDTD137<table><tr><td> [[Image:FDTD58.png|Setting periodic scan angles in the lumped source dialogthumb|FDTD138left|720px|EM.png| Setting the array factor in radiation pattern Tempo's simulation engine settings dialog.}}]]</td></tr></table>

{{twoimg|FDTD135==Modeling 3D Periodic Structures in EM.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 a 8×8 finite-sized periodic dipole array with scan angles theta equal to 45 degrees, and phi equal to 0 degrees.}}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 {{Note| [[EM.CubeTempo]]'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. Howeveronly handle regular, depending on the total size of your array, a fullnon-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 skewed periodic structurelattices with no secondary offsets. 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 }} [[EMImage:Info_icon.Cubepng|30px]]Click here to learn more about the theory of '''s [[Planar ModuleBasic_Principles_of_The_Finite_Difference_Time_Domain_Method#Time_Domain_Simulation_of_Periodic_Structures | Time Domain Simulation of Periodic Structures]]. 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'''.

[[Image:FDTD146(1).png|thumb|250px|===Defining additional radiation characteristics a Periodic Structure in [[FDTD Module]]'s Radiation Pattern dialogEM.]]Tempo===

In By default, your physical structure in the previous sectionproject workspace is not periodic, and you saw how have to excite instruct [[EM.Tempo]] to turn it into a periodic unit cell structure using its Periodicity Dialog. By designating a lumped source or a waveguide source. You can specify the beam scan angles in the source dialogs. The finite array factor is defined in the radiation pattern dialog. At the end of the structure as periodic FDTD simulation, you can visualize enforce periodic boundary conditions (PBC) on the 3D radiation patterns side walls of its computational domain. Your structure in the project workspace and plot the 2D Cartesian and polar pattern graphs in EMthen turns into a periodic unit cell.GridThe periodic side walls are displayed with dashed blues lines. [[EM.Cube]] also calculates the '''Directive Gain (DG)''' as a function of the &theta; and &phi; angles. This is defined as:

<math>D(\thetaTo define a periodic structure,\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>follow these steps:

The directivity D<sub* Select '''Menu >0</subSimulate > is the maximum value of the directive gainComputational Domain > Periodicity Settings. [[EM.Cube]] 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 lobes. Some [[parameters]] of interest in such structures include the '''Half Power Beam Width (HPBW)or right click on the ''', Periodicity'''Maximum Side Lobe Level (SLL)item in the ''' and Computational Domain'''First Null [[Parameters]]section of the Navigation Tree and select ''' (iPeriodicity Settings.e. first null level and first null beam width). You can have [[EM.Cube]] calculate all such [[parameters]] if you check the relevant boxes in the "Additional Radiation Characteristics" section of the '''Radiation Pattern from the contextual menu. This open up the Periodicity Settings Dialog'''. These quantities are saved into ASCII data files of similar names with * Check the box labeled '''.DATPeriodic Structure''' file extensions. You can plot graphs of such data files at and click the end of a sweep simulation in''' Apply'''EMbutton of this dialog.Grid. You can also plot The default domain box initially shrinks to the directive gain as a function edges of the sweep variable at physical structure in the end of an FDTD sweep simulationproject workspace. In that caseThe default periods along the X and Y axes appear in the dialog, which are equal to the directive gain is computed at a fixed pair dimensions of &theta; and &phi; anglesthe structure's bounding box. These angles are specified in degrees as * Enter new values for '''User Defined Azimuth & ElevationX Spacing''' and '''Y Spacing ''' in project units and close the "Output Settings" section of the Radiation Pattern dialog. The default values * Periodic boundary conditions (PBC) are established on the ±X and ±Y faces of the user defined azimuth and elevation are both zero corresponding domain box. You still have to designate the zenithboundary conditions on the ±Z faces of the computational domain. The results These are saved to an ASCII data file called "DGUCPML by default.DAT". Note that DGU is also one of [[EM.Cube]]'s standard output [[parameters]] and But you can be used change them to define custom output PEC or design objectivesPMC.

Using a plane wave source to excite a periodic structure ===Exciting Periodic Structures as Radiators 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; < &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.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 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.Tempo===

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; > 0&deg;. In [[EM.CubeTempo]]'s FDTD engine automatically detects such cases and avoids those resonances by shifting , a periodic structure can be excited using various source types. Exciting the modulation frequency of the modulated Gaussian pulse waveform away from the resonant frequency. Howeverunit cell structure using a lumped source, in some casesa waveguide source, the size of oscillations may still remain large after or a large number of time stepsdistributed source, you can model an infinite periodic antenna array. OccasionallyFor most practical antenna types, you excite your periodic structure with a late-time diverging behavior may appearlumped source or waveguide source. To avoid situations like theseIn this case, it is highly recommended that you place can define a timeport for the lumped source or waveguide source and calculate the S₁₁ parameter or input impedance of the periodic antenna array. You can also compute the near-domain field probe above your structure and monitor the temporal far-field behavior during the time marching loop as shown in the figure belowdata.

{{twoimg|FDTD140[[EM.Tempo]]'s periodic FDTD simulator uses periodic boundary conditions (1PBC)to model an infinite periodic array.png|Setting All the periodic replicas of the unit cell structure are excited. In this case, you can impose a custom plane wave 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 plane|FDTD139.png|Plane Wave visualization At the bottom of the '''Lumped Source Dialog''' or '''Waveguide Source Dialog''', there is a section titled '''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''' 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 scene'''Radiation Pattern Dialog'''.}}

===Reflection {{Note|For large & Transmission Characteristics===theta; scan angles, the periodic FDTD time marching loop may take far more time steps to converge.}}

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 S<subtable>11</subtr> and S<sub>21</subtd> [[parameters]] of a two-port networkImage:Period1. 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 png|thumb|350px|Setting periodic scan angles in '''EM.Grid '''similar to the scattering [[parameters]]. You can plot them from the Navigation Tree. To do so, right click on the '''Periodic Characteristics''' item 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]]Tempo's data managerLumped Source dialog. To open data manager, click the '''Data Manager''' [[Image:data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate </td> 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 </tr></tr></table and then click the '''Plot''' button to plot the graph in EM.Grid.>

{{Note<table><tr><tr><td> [[Image:Period2.png|It is very important to keep thumb|720px|Setting the array factor in mind that only in the case of normal incidence does [[EM.CubeTempo's Radiation Pattern dialog.]] compute the reflection and transmission coefficients over the entire specified bandwidth of the project. At oblique incidences when &theta; </td> 0, the computed R</T coefficients after the discrete Fourier transformation are valid only at the center frequency of the project for the given value of the incident &theta;<subtr>0</subtable> 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".}}

{| border="0"<table>|-| valign="top"|<tr><td> [[FileImage:FDTD141Period3.png|400px|thumb|Magnitude and Phase 360px|Radiation pattern of reflection coefficient from a an 8×8 finite-sized periodic surface plotted vs. frequencywire dipole array with 0&deg; phi and theta scan angles.]]</td>| valign="bottom"|<td> [[FileImage:FDTD142Period4.png|400px|thumb|Magnitude and Phase 360px|Radiation pattern of transmission coefficient from a beam-steered 8×8 finite-sized periodic surface plotted vs. frequencywire dipole array with 45&deg; phi and theta scan angles.]]</td>|-</tr>|}</table>

===Exciting Periodic FDTD Simulation TypesStructures Using Plane Waves in EM.Tempo===

<table><tr><td> [[FileImage:FDTD143Period11.png|thumb|200px380px|[ RGeometry of a periodic printed strip FSS in EM.Tempo.]] </T Macromodel Settings Dialogtd><td> [[Image:Period12.png|thumb|340px|Define a custom periodic plane wave box in EM.Tempo.]]</td></tr></table>

Besides analyzing Using a plane wave source to excite a periodic structure in a single[[EM.Tempo]], you can model frequency selective surfaces, electromagnetic band-run simulationgap (EBG) structures, metamaterials, etc. Exciting periodic structures with plane wave sources requires careful attention. [[EM.CubeTempo]]'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; < &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 [[FDTD ModuleEM.Tempo]] offers a number of sweep simulations 's default settings for the plane wave box of periodic structures. These include Nevertheless, you can change the location of the excitation surface if you wish. To do so, you have to open the '''Frequency SweepPlane Wave Dialog'''. In the Excitation Box section of the dialog, select the '''Angular SweepSize: Custom''', option. Only the '''R/T Macromodel Sweep Z Coordinate'''and of '''Dispersion SweepCorner 1'''is available for editing. These options The rest of the coordinates are available from enforced by the periodic domain. You can enter the incidence angles '''Simulation ModeTheta''' dropdown list of the [[FDTD Module]]and 's '''Run DialogPhi'''in degrees. Of theseFor periodic structures, frequency sweep and angular sweep are similar to only 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''TM<sub>z</transmission coefficient data. Similarly, you need to run an angular sweep to plot Rsub>''' and '''TE<sub>z</T coefficients vs. the incident anglesub>''' polarization options are available.

[[File:FDTD144One 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; > 0&deg;.png|thumb|200px|[[FDTD ModuleEM.Cube]]'s Dispersion Sweep Settings dialogFDTD 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 below.]]

{{Note|It is very important to keep in mind that only in the case of normal incidence does [[EM.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 coefficients 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".}} === Running a Dispersion Sweep in EM.Tempo === The '''R/T Macromodel Dispersion Sweep''' option of the Simulation Mode dropdown drop-down list performs a sweep of constant k_l wavenumber values. This is only available a specialized sweep for the constant transverse wavenumber method that [[EM.Tempo]] uses to model periodic structuresilluminated by a plane wave source. It The real advantage of a dispersion sweep is used to generate that through a lookup table model for 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 surface for both TM and TE polarizationsstructure. The results are written into sweep data can be graphed as a file named "PW_UserDefinedMacroDatawavenumber-frequency intensity plot (also known as beta-k diagram) that projects the eigenvalues of the periodic structure.mat"The horizontal axis represents the constant transverse wavenumber k_l (or beta). The vertical axis represents frequency. Through Sometimes, the Macromodel Settings dialog you can set 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 end value extend to f_max and number of samples for both the Theta (k_l,max, respectively, where f_max = f₀ + &thetaDelta;) f/2, and Phi (&phiDelta;) angles 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 incident plane wave. The R/T macormodels can be used by incident angle &phi; as specified in [[EM.CubeTempo]]'s Plane Wave Dialog. <table><tr><td>[[Propagation ModuleImage:KBT Settings.png|thumb|360px| [[EM.Tempo]] to calculate the 's Dispersion Sweep Settings dialog.]]</td></tr></table> <table><tr><td>[[Image:KBT R.png|thumb|360px|A typical reflection and coefficient dispersion diagram of a periodic structure.]]</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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