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The characteristic basis function method (CBFM) is extended to apply to efficient analysis of dual-polarized connected-patch arrays where neighboring patches are connected via two-wire transmission lines and a groundplane. Comparison with a reference solution shows that the use of tertiary basis functions can significantly improve the CBFM approximation to the array currents and radiation patterns. A simple procedure for estimating the error in the CBFM current is shown to produce good approximations to the true error and therefore may be useful more generally in determining suitable parameters of the CBFM basis. The technique is applied to a prototype phased array feed for a next-generation radio telescope. A thin dielectric supporting the array patches is shown to produce a loading effect that modifies the radiation patterns. Good agreement with the results of other techniques and measured data has been obtained.