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A type of electromagnetic bandgap structure is described that is easily parameterized and can produce a range of square and spiral geometries. Individual electromagnetic bandgap (EBG) geometries are defined on a cell-by-cell basis in terms of their convolution factor k, which defines the extent to which the elements are interleaved and controls the coupling slot length between adjacent elements. Polar equations are used to define the slot locus which also incorporate a transformation which ensures the slot extends into the corners of the square unit cell and hence extends the maximum slot length achievable. The electromagnetic properties of the so-called polar EBG are evaluated by means of numerical simulation and measurements and dispersion diagrams are presented. Finally, the performance is compared with other similar miniaturized EBG cell geometries. It is shown that the polar EBG has better angular stability than the equivalent square patch design and is comparable in terms of performance to other low frequency EBG elements. At the same time it retains the ability to fine tune the response by adjusting k.