## How to interpret far fields calculated from a Periodic Boundary Condition (PBC) solution

This article helps to interpret far-fields (total and scattered) calculated with the Periodic Boundary Condition (PBC) solution.

For the PBC solution, the default far field is calculated from the solution of a single unit cell radiating in free space. The PBC unit cell solution is obtained for an infinite array, meaning all the array element currents/fields are the same.

For a PBC unit cell, consider a truncated patch (derived from the component library) that radiates left-hand circular (LHC) polarisation. Truncated patch PBC unit cell geometry

Consider two cases:

• The PBC patch is exited with a voltage source with a specified beam pointing angle.
• The PBC patch is excited with a plane wave, and the port is loaded with 50 Ohm.

#### Voltage source excitation

The PBC solution is obtained by exciting the patch with a voltage to have a beam pointing angle {theta, phi} = {30, 0} degrees.

Note: The unit cell beam will not necessarily point to the beam pointing angle. LHC far field directivity pattern of the unit cell

It is only when requesting the far field for a large finite array that the beam pattern will align with the specified pointing angle.

Note: Only the unit cell geometry is displayed in the 3D view. LHC far field directivity of a 21x21 element array excited uniformly

#### Plane wave excitation

The PBC solution is obtained for two sets of incident plane waves: one with left-hand circular polarisation (LHC) and another with right-hand circular (RHC) polarisation. Request the bi-static far field for the unit cell (default) and a 21x21 element array.

The figure below shows the bi-static RCS pattern for a 21x21 array. Again, only the unit cell geometry is displayed in the 3D view. The bi-static RCS includes only the scattered field from the array. It cannot include the plane-wave field contribution as the plane-wave E-field does not decay with 1/r. This is the reason for the large forward scatter beam below the array. A large forward scatter beam is needed to cancel the incident field.

The total near field, which includes the plane wave field contribution, below the unit cell will be zero. Numerically it will have some finite (but insignificant small) level due to the discrete nature of the current solution, typically >40 dB below the incident field level.

The patch array reflects less power when the incident polarisation is matched (LHC), compared to when the incident polarisation is mismatched (RHC). Co-polarised bi-static RCS for the 21x21 array with incident angle (theta_i, phi_i} = {30, 0}.

Left: LHC polarised plane wave; Right: RHC polarised plane wave

This is consistent with the received power in the 50 Ohm load versus incident angle for LHC and RHC incident polarisation. Received load power versus incident angle for LHC polarisation (blue curve) and RHC polarisation (green curve).

More power is received when the incident polarisation (LHC) is matched to pattern polarisation (LHC) when the patch is excited.