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Ideal isolator conditions are derived using circuit analysis in a lumped-element isolator with three windings at angles of θ and φ. The angle θ can be arbitrary; however, φ is restricted to lie on the bisector of cross-angle θ or the supplementary angle of θ. We found that the isolator with θ=φ=60 [deg] behaves the same as the one with conventional structure with θ=φ=120 [deg]. However, the former has a 0° phase shift and the latter a 180° phase shift between the input and output. The theoretical prediction was experimentally confirmed with an 800-MHz-band isolator.