Electromagnetic boundary-value problems supported by closed-form solutions provide a framework for efficient and optimised design methods. Closed-form Green's theorem models are used when a well-defined source is coupled to a well-defined space or structure. Here the Green's theorem is applied to describe the relationship between the current in a coaxial probe and the electromagnetic fields inside an overmoded coaxial airspace. Several single-port boundary-value problems are illustrated before extending this to two-port problems with an application to a cavity filter. Each single and multi-port example is supported by experimental data that is in close agreement with the closed-form models in all cases. © The Institution of Engineering and Technology 2008.