Inhomogeneous ferrite microstrip circulator: theory and numerical calculations using a recursive Green's function

A two dimensional recursive Green's function G is developed which is suitable for determining the electromagnetic fields anywhere within a microstrip (or stripline) ferrite circulator. The recursive nature of G is a reflection of the inhomogeneous region being broken up into one inner disk containing a singularity and N annuli. G has the correct properties to allow matching to the external ports, and to allow computing the 3-port s-parameters of a circulator. Because of the general nature of the problem construction, the ports may be located at arbitrary azimuthal angle ϕ and possess arbitrary line widths. Non-uniformities can occur in the applied magnetic field H_app, magnetization 4πMs, and demagnetizing factor N_d. All inhomogeneity effects can be put into the frequency dependent tensor elements of the anisotropic permeability tensor. Numerical results are presented for the simpler but immensely practical case of symmetrically disposed ports of equal widths taking into account these radial inhomogeneities. Studies of breaking up the area into various numbers of annuli are undertaken to display the approximation levels required to treat inhomogeneous problems. Convergence properties of the recursive Green's function are presented, and computation speed is shown.