How do I set up a periodic boundary condition for simulating an infinite phased array?

Periodic boundary conditions (PBCs) enable the simulation of an infinite phased array by modeling only a single unit cell, dramatically reducing computational cost. The setup in HFSS: (1) Define the unit cell: the smallest repeating element of the array, including the radiating element (patch, dipole, slot), the substrate and ground plane, and the feed structure. The unit cell boundaries are placed at the array lattice planes (typically rectangular or triangular lattice). (2) Apply master-slave boundary conditions: on opposing faces of the unit cell, assign master and slave boundaries. The slave boundary field is related to the master boundary field by a phase shift: E_slave = E_master × exp(-j × k × d × sin(theta) × cos(phi)), where k = 2*pi/lambda, d is the element spacing, and (theta, phi) is the scan angle. This phase relationship simulates the beam-steering effect of the phased array. In HFSS: set up two pairs of master-slave boundaries (one for each periodic direction: x and y for a rectangular lattice). (3) Define Floquet port: a special port type for periodic structures that decomposes the radiated field into a set of Floquet modes (plane waves propagating at angles determined by the array lattice and frequency). The dominant Floquet mode corresponds to the main beam; higher-order modes correspond to grating lobes and higher-order interactions. Set the Floquet port to include enough modes to capture all propagating and the first few evanescent modes. (4) Parametric scan: sweep the scan angle (theta) from 0° (broadside) to the maximum scan angle (typically 60-70°) to analyze the array performance as a function of scan direction. Plot active S11 (the reflection coefficient of an element in the infinite array environment) vs frequency for each scan angle. The active S11 includes mutual coupling effects from all surrounding elements.