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The Sievenpiper high-impedance surface is a periodic structure characterized by a substrate filled with an array of vertical vias, capped by a capacitive frequency selective surface (FSS). It functions as the ideal antenna groundplane for wireless applications because it simultaneously enhances the gain of the antenna as it suppresses the surface waves associated with it (thus reducing the undesired back-lobe and the reactive coupling to nearby circuits). These two properties are known to occur approximately over the frequency bandwidth where the phase of the reflection coefficient of the surface changes from +90° to -90°. Since this behavior takes place at frequencies where the unit cell of the structure is small compared to the wavelength, it can be modeled in terms of a layered homogeneous material where each layer has an anisotropic magneto-dielectric tensor. These tensors, readily derived using an effective medium model, can be designed to obtain independent control of the bandwidths of gain increase and surface wave suppression. Based on a transverse resonance model of the effective medium material model, it is shown that Sievenpiper high-impedance surfaces exist that can suppress TE surface waves alone or TM surface waves alone, or both TE and TM surface waves at the same time. Maximum TM surface wave suppression bandwidth is obtained when the distance between the vias in the via array is as close as possible to λ/2. Maximum TE bandwidth is obtained when the conductors of the capacitive FSS offer maximum blockage to the normal magnetic field of the wave. A reduction of the transverse resonance solution to nearly closed form is used to obtain a simple picture of the design space available when the desired operating frequency is fixed.