On the stored energy of planar apertures

The volume integrals in the complex Poynting theorem are usually assumed to represent stored magnetic and electric energy. But for a planar aperture radiating into a lossless homogeneous medium it is shown that the only physically observable part of the volume integrals is twice that produced in the invisible region of the pattern space factor by those field components that vanish in the plane outside of the aperture. The inductive and capacitive reactive powers corresponding to the observable parts Wₘ Wₑ , respectively, of the volume integrals are found to be given by the following integrals over the invisible regionky² + kz² > k²,2ω Wₘ = {(2π)²}{2kZ₀} ∫ ∫_{ky²+kz² >k²} {|kzFy-kyFz|²} {√{ky²+kz²-k²} dkydkz,2ω Wₑ = {(2π)²}{2kZ₀} ∫ ∫_{ky²+kz² >k²} {k²}(|Fy|²+|Fz|²)} {√{ky²+kz²-k²} dkydkz, for an arbitrary aperture consisting of holes in a conducting plane. They are expressed in terms of the rectangular componentsFyandFzof the pattern space factor obtained from the tangential componentsEyandEzof electric field over the holesS,Fy(ky,kz,k) = {1}{(2π)²} ∫ ∫ₛ Ey(0,y,z,k)e^{i (kyy+kzz)}dydzFz(ky,kz,k) = {1}{(2π)²} ∫ ∫ₛ Ez(0,y,z,k)e^{i (kyy+kzz)}dydzwhereZ₀ = √μ/ε, andkyandkzare two components of the vector propagation constant of magnitudek=ω√με. For the complementary aperture whose space factor is obtained fromZ₀times the tangential components of magnetic field in the aperture, the formulas for inductive and capacitive reactive power simply interchange. The new formulas are used to make the first known test of the widely held assumption that the reciprocal relationship betweenQand bandwidth of nonradiating systems applies also to radiating systems. By direct computation ofQand of bandwidth independently it is shown that the reciprocal relationship does hold for the test case treated (the planar dipole).