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A Nematic Liquid Crystal (N-LC) compound may be injected into a cavity beneath a printed patch antenna to act as a tunable material whose dielectric properties are controlled by an externally applied electric field. The strength and direction of the applied low-frequency field affects the orientation of the LC molecules known as directors. The orientation of the directors determines the dielectric tensor entries of the LC compound which, in general, is anisotropic and lossy. The directors' tilt angle is governed by a Partial Differential Equation (PDE) which is obtained through minimization of the Oseen-Frank free-energy functional and solved using a Finite-Difference (FD) scheme. The profile of the directors' tilt angle underneath the patch follows a flattened sinusoidal shape along the normal-to-the-patch direction. In previous work by the authors, this non-uniform dielectric profile was averaged out, thus treating the LC substrate as a homogeneous but lossy material. In this paper, we are investigating the accuracy of this model as compared to a more realistic representation of the sinusoidal dielectric profile using multiple homogeneous layers. In addition, we are proposing ways to improve the existing design of the tunable LC patch antenna in order to obtain enhanced radiation characteristics such as gain and radiation efficiency.