All dielectrics, by nature are dispersive. This frequency dependence of materials must be modelled whenever microwave structures are expected to operate over broad bands of frequency. The Analytic Continuation Model of materials meets this need. As a compact, complete, ultrabroadband representation of all physically realizable dielectrics, this model can be used for the analysis and design of realistic electromagnetic engineering structures. The model represents dispersive materials as a sum of special complex functions reducible to analytic circuits. After a brief derivation of the model, an approach for obtaining the parameters of the circuits, in the case of dispersive radiofrequency materials, is disclosed. Its use is demonstrated by obtaining a model of a carbon loaded foam that is valid over a 2,000,000:1 bandwidth. The model is then applied to the problem of synthesizing materials with prescribed frequency dependent permittivity profiles for the modification of the electromagnetic properties of structures. Illustrative examples are solved using a Dispersive Finite Difference Time Domain computer code to demonstrate the simplicity and versatility of the model.