The Relationship between Cartesian Multipoles and Spherical Wavefunction Expansions with Application to Wireless Power Transfer

Spherical wavefunction expansions form the basis for spherical near field measurements and are useful for the analytical assessment of antenna performance, e.g. achievable quality factor and gain. On the other hand multipole expansions also have much utility. The cartesian multipole expansion provides physical insight to the behavior of compact sources. The relationship between cartesian multipoles and spherical wave function expansions is not one-to-one and is complicated as is evidenced by the multipole source configurations for low-order spherical wavefunctions previously given by Rusch. Here we provide a formal approach to the determination of the spherical wave function expansion for a particular cartesian multipole. Closed-form expressions for the spherical mode expansion coefficients are given for cartesian multipole sources. Cartesian multipole source representations have utility in representing inductive wireless power transfer couplers and the impetus for this effort is the modeling of such systems, in particular determination of the asymptotic decay of the near electromagnetic field components with distance.