  A common variant of a metasurface antenna consists of an array of
metamaterial elements coupled to a waveguide feed. The guided wave excites the
metamaterial elements, coupling energy from the waveguide mode to radiation.
Under appropriate conditions, each sub-wavelength metamaterial element can be
modeled as a polarizable dipole, with the polarizability determined by an
extraction procedure from the computed or measured waveguide scattering
parameters. Here we establish the equivalence of this polarizability
description of a metamaterial element with an equivalent circuit model,
providing an additional tool for metasurface design that offers significant
insight and a path towards efficiently modeling very large apertures. With this
equivalence established, more complicated external circuits that include lumped
elements and devices such as diodes and transistors can be integrated into the
metamaterial element, which can then be transformed into an equivalent
polarizability for modeling in the coupled dipole framework. We derive
appropriate circuit models for several basic metamaterial elements, which
provide direct relationships between the equivalent circuit parameters of an
element and its effective polarizability. These expressions are confirmed using
scattering parameters for several example structures obtained via full-wave
simulations.
