A degeneracy study in the circulant and bordered-circulant approach to birdcage and planar coils

A method for finding closed-form solutions for the normal mode frequencies of systems with circulant $$ \left( {{{2\pi } \over n}} \right) $$ symmetry was investigated. This method is particularly useful for questions of degeneracy that arise when one considers parallel imaging techniques like SENSE and SMASH in MRI. It is applicable to systems that include birdcage coils as well as planar coils with the appropriate rotational symmetry. A proof is given that complete degeneracy of all normal mode frequencies is impossible when all mutual inductive couplings are included. We tested the method against measurements made on a planar coil array and on an 8-element birdcage coil. The inclusion of the co-rotating end-ring mode changes the fundamental symmetry of the system from circulant to 'bordered circulant.' Closed-form solutions for the normal mode frequencies of a bordered circulant system are also given.