Lumped element isolator with lower-symmetrical configuration of three windings

In a lumped element isolator having three windings with general cross-angle θ and φ, the ideal isolator conditions are derived using circuit analysis. The angle θ can be arbitrary, however φ is restricted to on the bisector of cross-angle θ or compensation angle of θ. We found that the isolator with θ=φ=60 [deg] behaves the same as the one of conventional structure with θ=φ=120 [deg]. However, the former has 0-degree phase shift and the later 180-degree phase shift between input and output. The theoretical prediction was experimentally confirmed with 800MHz-band isolator.