Modeling Finite-Sized Periodic Arrays Using NCCBF Technique

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Modeling Finite-Sized Periodic Arrays Using NCCBF Technique

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.

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 NF = Nx . Ny elements, and each radiating element involves a total of NB vectorial basis functions, the numerical solution of the problem will produce a system of N = NB. NFlinear equations. As an example, consider a rectangular patch antenna element that involves 240 X-directed and 240 Y-directed rooftop basis functions, i.e. NB= 480. Now consider a not-so-large, 8 × 8 array of these patch radiators, i.e. NF= 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.

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 NB= 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 = NB. NF = (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).

Running a NCCBF Simulation

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:

  1. Gap Sources
  2. Probe Sources
  3. Plane Wave Sources

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.

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.

To run an NCCBF simulation, open the Simulation Run Dialog, and then open thePlanar 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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Figure 1: Planar MoM's NCCBF Settings dialog.

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Figure 2: Planar MoM's "Add Unit Cell" dialog.

Symmetries, Array Objects & Composite Arrays

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 N2 elements. Instead of computing the entire N2 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.

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 ND domains, then a total of ND . (ND +1)/2 domain interactions (or matrix blocks) are computed. An EM.Cube array object consists of NF 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 NF identical domains of vectorial basis functions. A direct consequence of this is identification of only NF unique domain-pair interactions or matrix blocks. In the absence of these symmetries, a total of at least NF . (NF +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 × 8 array of patch radiators, i.e. NF= 64. It was previously assumed that each rectangular patch antenna element involves 240 X-directed and 240 Y-directed rooftop basis functions, i.e. NB= 480. The numerical solution of this structure produces a linear system of total size N = NB. NF= 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 NB2 = 230,400 elements, except for the self-domain interaction which requires NB . (NB +1)/2 = 115,440 integral computations. This amounts to a total of (NF -1) . NB2 + NB . (NB +1)/2 = 1.46E+07 integral computations, which is roughly NF(64) times fewer and faster than a brute-force matrix fill process.

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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Figure 1: Assigning gap sources to the elements of a composite array object.

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