Inverse source problem with reactive power constraint

This paper reports a general Lagrangian formulation for constrained electromagnetic inverse source problems. The formulation is applicable to different forms of inverse source problems having different constraints. Two possibilities are emphasized in the paper. One yields the usual minimum energy solution. The other establishes a new minimum energy solution with the additional constraint that the source has a prescribed reactive power (which can be zero). The latter solution incorporates the important reactive energy issues of an antenna which had not been considered before in the context of the inverse source problem. The new solution is shown to obey a homogeneous wave equation in the interior of the source volume, and expressions for the associated interior field and interaction power are also derived. The derived theory is illustrated for a canonical dipolar source and an alternative approach where the reactive power is minimized for a prescribed maximum functional energy is also presented. Advantage is taken of some of the results to briefly discuss some questions about fundamental antenna limits.