Variational nonquasi-static formulations for the impedance of planar transmission lines

This paper investigates the possible variational behavior of transmission-l ine parameters established for planar structures by using a nonquasi-transv erse-electromagnetic full-wave formalism. A variational principle for the p ropagation constant is linked to the power flow and magnetic and electric d ensities of energy by means of a functional expressing the complex power ba lance for an infinitesimal line length. It is first shown that using this v ariational principle for the propagation constant renders the power balance functional stationary, but does not ensure a variational behavior for the power and energies. For this reason, the stationarity of the power flow is investigated by defining an equivalent reaction for the Poynting vector. By doing so, a full-wave variational expression for the characteristic impeda nce is derived, which is a main and new result. The theoretical observation s are confirmed by simulations carried out on three different topologies of planar lossy guiding structures. II is observed that the new formulation f or the characteristic impedance is extremum for the value of the trial fiel d components, which renders the propagation constant stationary.