Sum rules and constraints on passive systems with applications in electromagnetics

A passive system is one that cannot produce energy, a property that naturally poses constraints on the system. In this paper there is a review of some results on linear, time translational invariant, continuous, causal and passive systems, where it turns out that Herglotz functions are related to the Fourier transform of the impulse response of such systems. Some well known facts of this function class is considered, and a set of integral identities and an outline of the proof of these are presented. The identities may be used to derive sum rules and constraints on various physical systems. The theory is illuminated with two examples from electromagnetics: the first revisits Fano's maching equations, while the latter makes a link to the Kramers-Kronig relations and discusses physical limitations on metamaterials.