How do I design a waveguide phase shifter for a phased array feed network?

Designing a waveguide phase shifter for a phased array feed network creates a controllable delay element in the waveguide that adjusts the phase of the signal passing through it, enabling electronic beam steering of the phased array antenna. The phase shift required for beam steering: phi_n = n × d × sin(theta) × 2 × pi / lambda, where n is the element number, d is the element spacing, and theta is the desired steering angle. Waveguide phase shifter types: ferrite phase shifter (a ferrite toroid or slab is placed inside the waveguide; the ferrite's permeability (and therefore the waveguide's propagation constant) is controlled by an external magnetic field from a coil wrapped around the waveguide; the phase shift is proportional to the applied magnetic field: delta_phi = (beta(H) - beta(0)) × L, where L is the ferrite length; ferrite phase shifters can be: latching (the ferrite retains its magnetization state after the control current is removed, consuming no holding power; ideal for phased arrays that switch beams infrequently), and non-latching (continuous control but requires continuous bias current)), diode-based digital phase shifter (PIN diodes switch waveguide sections of specific lengths in and out of the signal path; each switched section provides a fixed phase increment (e.g., 180 degrees, 90 degrees, 45 degrees, 22.5 degrees); cascading N bits provides 2^N phase states; a 5-bit phase shifter provides 32 states with 11.25 degree resolution), and mechanical phase shifter (a sliding short or dielectric insert is mechanically moved to change the waveguide's electrical length; very accurate but slow; used for calibration and non-real-time applications).