How should far-field computations based on PBC be interpreted?
Altair EmployeeAnswers
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When using periodic boundary conditions (either 1-D or 2-D infinite arrays) and requesting the far-field, this is the far-field as radiated from all the impressed sources and currents in a single unit cell only. The currents are of course obtained taking all the mutual coupling with all neighbouring elements into account, i.e. these currents (and hence the far-field) differ when just solving a single element without periodic boundary conditions. This approach of computing the far-field of a single element only (taking the mutual element coupling into account) has the advantage that in a second step the far-field pattern of also finite sized n x m arrays can be computed. This procedure will be illustrated and validated (by means of a full-wave MLFMM solution) with the following example.
The unit cell of an infinite array of strip dipoles is shown in Fig. 1. The dimensions of the unit cell (element spacing) are a = 0.5 λ and b = 0.3 λ. The dipole length L = 0.45 λ and width W = 0.02 λ. We feed the array so that we have a main beam at θ = 20°, requiring a phase increment of k*a*sinθ = 61.564°. The phase increment is set in CADFEKO under model where we define the periodic boundary condition.
Figure 1: Unit cell of a strip dipole Unit cell geometry of an infinite array of strip dipoles Far field of unit cell 
The example file 'strip_dipole_PBC' corresponds to Fig. 1 where we analyze the unit cell using the periodic boundary condition. We solve for the current distribution on the dipole. This is the current distribution on the dipole in an active array environment where all elements are excited with the same magnitude, but with the correct phase increment (Floquet excitation). From this current distribution the far-field is computed (FF-card). However, this far-field is that of the unit cell only (with the correct current distribution). So the far-field is computed from the correct current distribution on the dipole, but with the single dipole located in free-space. The computed far-field of the unit cell dipole is doughnut shaped and shown in Fig. 2.
With FEKO we can use the PBC to compute the far-field of a finite array of strip dipoles by setting on the advanced tab of the request far field dialog in CADFEKO as seen in Fig. 3. We choose to solve a 51x51 array of dipoles. In EDITFEKO the FF card contains this setting.
Figure 2: Setting a 51x51 array calculation with the PBC. 
In fig. 4 a truncated image of the finite array of dipoles is shown. By using the PBC to compute the pattern for the 51x51 array, we make the following approximation: the PBC assumes that the current distributions on the array elements are identical (except for the phase increment). In a finite array this is not the case, especially for the edge elements. However, for a large finite array the contribution of the edge elements to the total far-field pattern become smaller, enabling us to use the PBC to obtain accurate far-fields for large finite arrays.
Figure 3: Truncated view of finite array of strip dipoles. 
The Multilevel-Fast-Multipole-Method (MLFMM) will be taken as the reference solution where the complete array is solved (including edge effects). The reference MLFMM corresponds to example file 'strip_dipole_MLFMM_51x51.pre'. The 51x51 array can easily be created using the finite array tool in CADFEKO, or in EDITFEKO using the TG card to copy the elements and FOR loops to set the excitations. Fig. 4 compares the far-field cut at phi = 180 between the reference MLFMM solution and the PBC approximation. Agreement is very good for this 51x51 element array. The runtime for the PBC solution was less than 1 second making this the preferred method for large array synthesis.
Figure 4: Far field cut at phi = 180 degrees <?xml version="1.0" encoding="UTF-8"?> 
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