Proof of Perturbational Duality Between Classical Cavities and Planar Resonators with Magnetic Side-walls

In this note a formal proof is given to an observation that the perturbational response of a planar resonator, which can be modelled as a cavity with magnetic side-walls, is the dual of the behaviour of a classical cavity with electric walls. This perturbational duality is observed even though the respective boundary types are not exactly dual. This observation is important as it provides the designer of planar resonators, such as filters and antennas, with a new ‘intuition’ to the pertubational reaction of such structures, and how it opposes that of classical cavities during perturbation. The formalism of the proof follows clear steps that parallel those found in the classic literature, and the observation is confirmed using numerical methods and measurements.