The Poynting theorems and the potential for electrically small antennas

Analyses of power in radiation fields are nearly universally based upon the complex Poynting theorem (CPT), yet the scientific basis for its use is not well established. The time dependent Poynting theorem (TDPT), is seldom used yet its scientific basis is well established. We use Hansen's method to obtain a generalized modal expansion for all possible electromagnetic fields that can originate within a confined volume and travel infinitely outward, combine the modal expansions and both Poynting theorems to determine the power expressions, and contrast results. The TDPT shows that the characterization of power in a sinusoidal field requires three numbers. Therefore it cannot be fully described by only real and reactive values. We conclude that the CPT is intrinsically an insufficient basis for a full description of power in a radiation field. We compare TDPT and CPT solutions for several idealized antenna designs. Comparative analyses show that the TDPT provides solutions that correspond to reality in all cases and the CPT does not. Proofs that electrically small antennas cannot be efficient are based upon the CPT; we re-examine these proofs using the TDPT and find that certain mixed modal configurations surpass the generally accepted small antenna rules.