You couple two or more sources using the '''Port Definition Dialog'''. To do so, you need to change the default port assignments. First, delete all the ports that are to be coupled from the Port List of the dialog. Then, define a new port by clicking the '''Add''' button of the dialog. This opens up the Add Port dialog, which consists of two tables: '''Available''' sources on the left and '''Associated''' sources on the right. A right arrow ('''-->''') button and a left arrow ('''<--''') button let you move the sources freely between these two tables. You will see in the "Available" table a list of all the sources that you deleted earlier. You may even see more available sources. Select all the sources that you want to couple and move them to the "Associated" table on the right. You can make multiple selections using the keyboard's '''Shift''' and '''Ctrl''' keys. Closing the Add Port dialog returns you to the Port Definition dialog, where you will now see the names of all the coupled sources next to the name of the newly added port.
{{Note|It is your responsibility to set up coupled ports and coupled [[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|transmission lines]]]]]]]]]] properly. For example, to excite the desirable odd mode of a coplanar waveguide (CPW), you need to create two rectangular slots parallel to and aligned with each other and place two gap sources on them with the same offsets and opposite polarities. To excite the even mode of the CPW, you use the same polarity for the two collocated gap sources. Whether you define a coupled port for the CPW or not, the right definition of sources will excite the proper mode. The couple ports are needed only for correct calculation of the port characteristics.}}
[[File:PMOM51(2).png|800px]]
Figure 2: Viewing the contents of a mesh-type 3D data file in Data Manager.
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=== Standard vs. Custom Output ===
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At the end of a planar MoM simulation, a number of computed quantities are designated 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 [[Planar Module]]:
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{| class="wikitable"
!scope="col"| Standard Output Name / Syntax
!scope="col"| Description
|-
| SijM
| Magnitude of (i,j)-th Scattering Parameter
|-
| SijP
| Phase of (i,j)-th Scattering Parameter (in radians)
|-
| SijR
| Real Part of (i,j)-th Scattering Parameter
|-
| SijI
| Imaginary Part of (i,j)-th Scattering Parameter
|-
| ZijM
| Magnitude of (i,j)-th Impedance Parameter
|-
| ZijP
| Phase of (i,j)-th Impedance Parameter (in radians)
|-
| ZijR
| Real Part of (i,j)-th Impedance Parameter
|-
| ZijI
| Imaginary Part of (i,j)-th Impedance Parameter
|-
| YijM
| Magnitude of (i,j)-th Admittance Parameter
|-
| YijP
| Phase of (i,j)-th Admittance Parameter (in radians)
|-
| YijR
| Real Part of (i,j)-th Admittance Parameter
|-
| YijI
| Imaginary Part of (i,j)-th Admittance Parameter
|-
| VSWR
| Voltage Standing Wave Ratio
|-
| D0
| Directivity
|-
| PRAD
| Total Radiated Power
|-
| THM
| Main Beam Theta
|-
| PHM
| Main Beam Phi
|-
| DGU
| Directive Gain along User Defined Direction
|-
| ARU
| Axial Ratio along User Defined Direction
|-
| FBR
| Front-to-Back Ratio
|-
| HPBWXY
| Half Power Beam Width in XY Plane
|-
| HPBWYZ
| Half Power Beam Width in YZ Plane
|-
| HPBWZX
| Half Power Beam Width in ZX Plane
|-
| HPBWU
| Half Power Beam Width in User Defined Plane
|-
| SLLXY
| Maximum Side Lobe Level in XY Plane
|-
| SLLYZ
| Maximum Side Lobe Level in YZ Plane
|-
| SLLZX
| Maximum Side Lobe Level in ZX Plane
|-
| SLLU
| Maximum Side Lobe Level in User Defined Plane
|-
| FNBXY
| First Null Beam Width in XY Plane
|-
| FNBYZ
| First Null Beam Width in YZ Plane
|-
| FNBZX
| First Null Beam Width in ZX Plane
|-
| FNBU
| First Null Beam Width in User Defined Plane
|-
| FNLXY
| First Null Level in XY Plane
|-
| FNLYZ
| First Null Level in YZ Plane
|-
| FNLZX
| First Null Level in ZX Plane
|-
| FNLU
| First Null Level in User Defined Plane
|-
| BRCS
| Back-Scatter RCS
|-
| FRCS
| Forward-Scatter RCS along User Defined Incident Direction
|-
| MRCS
| Maximum Bi-static RCS
|-
| RCM
| Magnitude of Reflection Coefficient
|-
| RCI
| Phase of Reflection Coefficient (in radians)
|-
| RCR
| Real Part of Reflection Coefficient
|-
| RCI
| Imaginary Part of Reflection Coefficient
|-
| TCM
| Magnitude of Transmission Coefficient
|-
| TCP
| Phase of Transmission Coefficient (in radians)
|-
| TCR
| Real Part of Transmission Coefficient
|-
| TCI
| Imaginary Part of Transmission Coefficient
|}
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In the table above, SijM, etc. means the scattering parameter observed at port i due to a source excited at port j. Similar definitions apply to all the S, Z and Y [[parameters]]. If your planar structure has N ports, there will be a total of N<sup>2</sup> scattering [[parameters]], a total of N<sup>2</sup> impedance [[parameters]], and a total of N<sup>2</sup> admittance [[parameters]]. Additionally, there are four standard output [[parameters]] associated with each of the individual S/Z/Y [[parameters]]: magnitude, phase (in radians), real part and imaginary part. The same is true for the reflection and transmission coefficients of a periodic planar structure excited by a plane wave source. Each coefficient has four associated standard output [[parameters]]. These [[parameters]], of course, are available only if your planar structure has a periodic domain and is also excited by a plane wave source incident at the specified θ and φ angles.
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All the radiation- and scattering-related standard outputs are available only if you have defined 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 (θ, φ). 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 θ<sub>s</sub> and φ<sub>s</sub> source angles. FRCS, on the other hand, is the total forward-scatter RCS measured at the predetermined θ<sub>o</sub> and φ<sub>o</sub> 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.
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If you are interested in calculating certain quantities at the end of a simulation, which you do not find among [[EM.Cube]]'s standard output data, you can define your own custom output. [[EM.Cube]] allows you to define new custom output as any mathematical expression that involves the available standard output [[parameters]], numbers, [[variables]] and all of [[EM.Cube]]'s mathematical functions. For a list of legitimate mathematical functions, click the '''Functions [[File:functions_icon.png]]'''button of the '''Simulate Toolbar''' or select '''Simulate > Functions...'''from the menu bar, or use the keyboard shortcut '''Ctrl+I''' to open the Function Dialog. Here you can see a list of all the available [[EM.Cube]] functions with their syntax and a brief description. To define a custom output, click the '''Custom Output [[File:custom_icon.png]]'''button of the '''Simulate Toolbar''' or select '''Simulate > Custom Output...'''from the menu bar, or use the keyboard shortcut '''Ctrl+K''' to open the Custom Output Dialog. This dialog has a list of all of your custom output [[parameters]]. Initially, the list empty. You can define a new custom output by clicking the '''Add''' button of the dialog to open up the '''Add Custom Output Dialog'''. In this dialog, first you have to choose a new label for your new parameter and then define a mathematical expression for it. At the bottom of the dialog you can see a list of all the available standard output [[parameters]], whose number and variety depends on your project's source type as well as the defined project observables. When you close the Add Custom Output dialog, it returns you to the Custom Output dialog, where the parameter list now reflects your newly defined custom output. You can edit an existing parameter by selecting its row in the table and clicking the '''Edit''' button, or you can delete any parameter from the list using the '''Delete''' button.
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[[File:PMOM141.png]]
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Figure 1: [[EM.Cube]]'s Custom Output dialog.
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[[File:PMOM140.png]]
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Figure 2: Defining a new custom output using the available standard output [[parameters]].
=== Viewing & Visualizing Various Output Data Types ===
Click here to learn about [[Modeling Finite-Sized Periodic Arrays Using NCCBF Technique]].
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=== Modeling Finite-Sized Periodic Arrays Using NCCBF Technique ===
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Previously, you saw how the concept of "Array Factor" is used to approximate the far field radiation pattern of a finite-sized array of radiators. The total radiation pattern can be expressed as the product of the array factor and the "Element Pattern". The array factor captures the topology of the array lattice and depends on the number of elements along the X and Y directions as well as the element spacing along those directions. As for the choice of element pattern, you saw two extreme cases. In the "'''Isolated Element'''" option, you compute the radiation pattern of a single stand-alone radiator and completely ignore any coupling effects from the neighboring elements. This option is readily available in the Radiation Pattern Dialog of the Far Field observable. In the "'''Periodic Element'''" option, you analyze a periodic version of the radiating element with periods equal to the element spacing. The computed radiation pattern of the periodic unit cell in this case captures the coupling effects from an infinite number of elements.
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The array factor approach works well when the inter-element coupling is negligible or when the array contains a very large number of elements. However, finite edge and corner effects cannot be modeled accurately using a simple array factor. A full-wave approach is needed where all the elements are discretized properly, and their interactions are incorporated into the final solution. [[EM.Cube]]'s Planar MoM simulation engine lets you analyze finite-sized antenna arrays in a rigorous, full-wave manner. As you saw earlier, you can even introduce a source array with arbitrary (amplitude and phase) weights and realize a complex, non-uniformly excited, finite-sized antenna array. Note that if the array contains a total of N<sub>F</sub> = N<sub>x</sub> . N<sub>y</sub> elements, and each radiating element involves a total of N<sub>B</sub> vectorial basis functions, the numerical solution of the problem will produce a system of N = N<sub>B</sub>. N<sub>F</sub>linear equations. As an example, consider a rectangular patch antenna element that involves 240 X-directed and 240 Y-directed rooftop basis functions, i.e. N<sub>B</sub>= 480. Now consider a not-so-large, 8 Ã 8 array of these patch radiators, i.e. N<sub>F</sub>= 64. The resulting linear system will have an enormous size of N = 30,720. Keep in mind that, unlike the sparse matrices of the Finite Element Method (FEM), MoM linear systems are dense by nature and typically ill-conditioned. The inversion of dense matrices of such sizes or larger takes a significant amount of computation time even if you use fast iterative solvers.
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EMAG Technologies Inc. has recently developed a novel technique, called '''Numerically Constructed Characteristic Basis Functions (NCCBF)''', which generates physics-based entire-domain basis functions for the elements of a finite-sized array. These "sophisticated" basis functions are linear combinations of the "'''Isolated Element'''" solutions and "'''Periodic Element'''" solutions. Unlike the array factor method, which is a post-processing calculation of far-field data, the NCCBF method generate a full-wave MoM solution with entire-domain basis functions. Considering the same example of the patch antenna array discussed earlier, the NCCBF method generates a total of N<sub>B</sub>= 4 entire-domain basis functions on each patch element: an isolated X-directed solution, a periodic X-directed solution, an isolated Y-directed solution, and a periodic Y-directed solution. The same approach applies equally well to triangular RWG basis functions and is not limited to rectangular cells. As a result, the new MoM linear system has a dimension of N = N<sub>B</sub>. N<sub>F</sub> = (4)(64) = 256. In other words, the NCCBF method compresses the original MoM matrix of size N = 30,720 to one of significantly reduced size N = 256 (i.e. a compression factor of 120x).
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=== Running a NCCBF Simulation ===
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In the current release of [[EM.Cube]]'s [[Planar Module]], the NCCBF MoM solver works with any number of distinct, finite-sized arrays if they are excited with one of the following three source types:
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# Gap Sources
# Probe Sources
# Plane Wave Sources
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Note that you can have several coexisting finite arrays with different element spacings (or different periodicities). You can also have regular (aperiodic) objects coexisting with your collection of finite arrays. In that case, the NCCBF process will create entire-domain basis functions for the elements of the finite arrays, while the regular method of moments will apply to the aperiodic portions of your planar structure. This flexibility makes NCCBF a very versatile and powerful technique.
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There are a few rules that must be followed and observed when planning a NCCBF simulation. '''Each finite-sized array must be constructed using an [[EM.Cube]] "Array Object". Additionally, each array object must stand alone in a dedicated trace or embedded object set of its own.''' In other words, if an array object belongs to a trace or embedded object set that contains other objects, it will be excluded from the NCCBF process and will get a regular MoM treatment. Keep in mind that [[Planar Module]] allows you to define different traces located at the same Z-plane, although the objects belonging to these separate traces cannot be connected to one another according to the planar meshing rules. Similarly, you can define two or more PEC via sets hosted by the same substrate layer. Therefore, if your planar structure contains finite arrays and aperiodic objects, you have to group them into separate traces or embedded object sets.
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To run an NCCBF simulation, open the '''Simulation Run Dialog''', and then open the'''Planar MoM Engine Settings Dialog'''. In the "'''Finite Array Simulation'''" section of the latter dialog, check the box labeled "'''NCCBF Matrix Compression'''". This box is unchecked by default. Checking it enables the NCCBF Settings button. Click this button to open the NCCBF Settings Dialog. The dialog features a "List of Unit Cells Used for NCCBF Matrix Compression". This list initially empty. To add unit cells to it, click the '''Add''' button of the dialog to open the "'''Add Unit Cell Dialog'''". This dialog has two tables: Available Unit Cells on the left side and Associated Unit Cells on the right side. The left table shows a list of all the available, legitimate array objects in your project workspace. Remember that for an array object to be eligible for NCCBF compression, it has to stand alone on a dedicated trace or embedded object set, whichever applies. Select an array object from the left table and use the right arrow button (-->) to move it to the right table to associate it with the new NCCBF unit cell. You can associate more than one array object with the same NCCBF unit cell. In this case, the parent elements of all the associated array objects collectively constitute the NCCBF unit cell. The NCCBF unit cell is the planar structure that is analyzed separately, first, as a stand-alone isolated element, and next, as a periodic unit cell, to generate the NCCBF entire-domain basis function solutions. It is therefore very important that the array objects be positioned carefully with respect to the origin of coordinated and relative to one another to form the correct NCCBF unit cell. Once you move one or more array object names to the "Associated" table on the right, you can move them back to the "Available" table on the left using the left arrow (<--) button. You can also instruct [[EM.Cube]] to use only the isolated element solution by unchecking the box labeled "'''Include Periodic Solution of Unit Cell'''". Once you are satisfied with the definition of your NCCBF unit cell, close the dialog to return to the NCCBF Settings dialog. Here you see the name of the newly added NCCBF unit cell in the list along with the Number of Solutions and the names of all the associated array objects for each NCCBF unit cell. You can modify each row using the '''Edit''' button or remove it from the list using the '''Delete''' button. Close the NCCBF Settings dialog to return to the Planar MoM Engine Settings dialog, and close the latter to return to the Simulation Run dialog, where you can now start the NCCBF simulation by clicking the '''Run''' button.
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[[File:PMOM163.png]]
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Figure 1: Planar MoM's NCCBF Settings dialog.
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[[File:PMOM162.png]]
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Figure 2: Planar MoM's "Add Unit Cell" dialog.
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=== Symmetries, Array Objects & Composite Arrays ===
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[[EM.Cube]]'s [[Planar Module]] treats array objects in a special way. That is why you need to use array objects with certain rules for NCCBF simulations. In general, if the mesh of your planar structure involves a total of N vectorial basis functions, the MoM matrix will contain a total of N<sup>2</sup> elements. Instead of computing the entire N<sup>2</sup> basis interactions, the Planar MoM simulation engine takes advantage of the inherent symmetry properties of the dyadic Green's functions and camputes the diagonal elements of the matrix and all the elements below the diagonal. This amounts to N.(N+1)/2 basis interactions. In many cases, the MoM matrix is symmetric, and the elements above the diagonal are simply mirror-image of the below-diagonal elements. In planar structures that involve both metal and slot traces, there will be sign reversals for some interactions.
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In many cases, especially in the areas that contain sizable numbers of rectangular mesh cells, the basis functions are naturally grouped into distinct sets that are called domains. As you saw earlier in the discussion of planar mesh generation, uniform domains with identical rectangular cells bring significant savings during the matrix fill process. Using the concept of domains renders the MoM matrix as a block matrix, whose blocks represent the interactions among the domains. The diagonal blocks therefore correspond to self-domain interactions. By a similar argument, if your planar structure is made up of N<sub>D</sub> domains, then a total of N<sub>D</sub> . (N<sub>D</sub> +1)/2 domain interactions (or matrix blocks) are computed. An [[EM.Cube]] array object consists of N<sub>F</sub> identical geometrical elements. If the array object belongs to a trace that has other objects in it, then by the planar mesh generator's rules, the elements of the array object are merged with the other objects on the same trace using the "Union" Boolean operation. If some array elements possibly have connections with other objects, such connections are taken care of in the meshing process. '''However, if an array object stands alone in a dedicated trace, then only the parent (first) element is meshed, and it mesh is copied and cloned for all the other elements of the array.''' This produces a total of N<sub>F</sub> identical domains of vectorial basis functions. A direct consequence of this is identification of only N<sub>F</sub> unique domain-pair interactions or matrix blocks. In the absence of these symmetries, a total of at least N<sub>F</sub> . (N<sub>F</sub> +1)/2 domain interactions (or matrix blocks) must be computed. To better illustrate such matrix fill savings, let us consider the previous, not-so-large, 8 Ã 8 array of patch radiators, i.e. N<sub>F</sub>= 64. It was previously assumed that each rectangular patch antenna element involves 240 X-directed and 240 Y-directed rooftop basis functions, i.e. N<sub>B</sub>= 480. The numerical solution of this structure produces a linear system of total size N = N<sub>B</sub>. N<sub>F</sub>= 30,720. The total number of complex-valued elements of this matrix is 9.44E+08. This is the total number of highly sophisticated multi-dimensional integrals that you need to compute during a brute-force matrix fill process. For the sake of generality of the argument, here we ignore the huge additional savings that rectangular cells offer, and we assume that each unique domain-pair interaction involves N<sub>B</sub><sup>2</sup> = 230,400 elements, except for the self-domain interaction which requires N<sub>B</sub> . (N<sub>B</sub> +1)/2 = 115,440 integral computations. This amounts to a total of (N<sub>F</sub> -1) . N<sub>B</sub><sup>2</sup> + N<sub>B</sub> . (N<sub>B</sub> +1)/2 = 1.46E+07 integral computations, which is roughly N<sub>F</sub>(64) times fewer and faster than a brute-force matrix fill process.
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To remedy the limitation that stand-alone array objects on dedicated traces cannot be connected to anything and therefore would severely limit the geometrical complexity of individual elements, [[EM.Cube]] allows you to build arrays of composite objects. A "'''Composite Object'''" in [[EM.Cube]] is a group of objects that are tied together only for the purpose of organization. As a result, the mesh of a composite object is the same as that of its constituent member objects (including any possible connections), had they not been grouped together. In a similar manner to simpler array objects, if a composite array stands alone in a dedicated trace or embedded object set, then only the parent composite object is meshed, and its mesh is copied and cloned for all the other composite elements of the array object. Furthermore, all the unique domain-pair interactions are identified during the matrix fill process, and lead to a major saving in computation time. '''Note that you can assign gap or probe sources to composite arrays in a similar way as you excite simpler array objects.''' If your planar structure involves a composite array, whose composite parent element has constituent members: Object1, Object2, ..., then the Gap Source dialog or Probe Source dialog will include the names of all the eligible constituent members (rectangle strips or PEC via objects) that can host the respective source types. '''Complex composite arrays hosting gap or probe source arrays or illuminated by plane wave sources are great candidates for NCCBF simulation.'''
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[[File:PMOM165.png|800px]]
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Figure 1: Assigning gap sources to the elements of a composite array object.
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