# Periodic Analysis

Analysis of Periodic Structures

### Feature Introduction

FEKO includes a solution method for the analysis of infinite periodic structures. The MoM is used in the periodic analysis, employing a periodic version of the free space Green function, such that a PBC is realized.  A typical application of this method is to analyse FSS structures, as will be shown in the validation results to follow.

### Capabilities

The PBC feature has the following capabilities:

• Periodic boundaries in both one and two dimensions is supported.
• In the case of two dimensional periodicity, arbitrary (non-orthogonal) lattice vectors are supported.
• The unit cell can include conducting surfaces, thin dielectric sheets, wire segments and simple SEP dielectric structures (e.g. substrate).
• The geometry of the unit cell may intersect its boundaries (half basis functions are created on the relevant elements in the discretized model), as long as it lines up with the geometry edge on the opposite side of the unit cell.
• In the case of solving a radiation problem, the periodic phase shift between cell excitations can be specified, or alternatively the beam pointing (squint) angle can be specified.
• In the case of solving a scattering problem, the periodic phase shift follows from the incident plane wave definition and cannot be specified independently.
• PBC analysis is part of the MoM kernel and can be executed sequentially or in parallel. However, it is not available in conjuction with MLFMM, VEP, PO, UTD, GO or FEM.
• Periodicity in three dimensions is not supported.

These features allow the user to analyse a very large class of problems, though there are some restrictions, as noted.  An example is the solution of far-field gain patterns for large, but finite, periodic arrays.

### Usage

In CADFEKO, a PBC analysis is requested via the Define periodic boundary condition dialog under Model in the main menu. The dialog enables the user to:

• Choose between one and two dimensional periodicity
• Define lattice vector(s)
• Define directional phase shifts for local excitations, if applicable
• Visualize the PBC
 1D periodic cell. The phase change can be specified by the user in the direction u1. The points S1 and S2 define the lattice vector. 2D periodic unit cell. The phase change can be specified by the user in the directions u1 and u2. The points S1, S2 and S3 define the lattice vectors. PBC analysis dialog in CADFEKO

Transmission and reflection coefficients can be directly computed by FEKO for surfaces that were defined using the PBC formulation.

### Validation Results

Figure 1 shows the CADFEKO model of a Jerusalem cross FSS unit cell.  The geometry is from [1].  To verify the PBC results the MLFMM was used to analyse a large finite array of 51x51 cells.  Given a normally incident plane wave excitation at 7 GHz, Figure 2 compares the PBC current distribution to that of the central element in the MLFMM analysis.  Excellent agreement can be observed, verifying the validity of the PBC analysis method.

 Figure 1: Visualization in POSTFEKO of the unit cell geometry of the Jerusalem Cross FSS. Planes of periodicity, lattice vectors and the incident plane wave source are also shown. Figure 2(a): Current distribution obtained with PBC method for an infinite array. Figure 2(b): Current distribution obtained with MLFMM for a very large, finite array

Further consider the reflection coefficient of this FSS over the frequency range 2-12 GHz. In Figure 3, results calculated with the PBC method is compared to the results published in [1].  For the FEKO results, 328 triangles were used in the discretization of the unit cell. Excellent agreement is again observed.

 Figure 3 (a): Magnitude Figure 3 (b): Phase Reflection coefficient of Jerusalem cross FSS

### References

 [1] Ivica Stevanovic, Pedro Crespo-Valero, Katarina Blagovic, Frederic Bongard and Juan R. Mosig, “Integral-Equation Analysis of 3-D Metallic Objects Arranged in 2-D Lattices Using the Ewald Transformation,” IEEE Trans. Microwave Theory and Techniques, vol. 54, no. 10, pp. 3688-3697, October 2006.