Liquid crystal compounds are increasingly used as tunable materials for a plethora of microwave and millimeter-wave devices such as phase shifters and printed antennas. Modeling of liquid crystals mandate the solution of the directors' field under an externally biased electric field which is governed by the Oseen-Frank free-energy functional. Minimization of this functional results in a nonlinear partial differential equation which is often simplified by applying the one-constant approximation, where the splay and bend elastic constants are set equal to each other. The effects of this approximation on the radiation characteristics of a microstrip patch antenna built on top of a liquid-crystal substrate are not well-studied. In this work, we adopt this approximation, along with neglecting the off-diagonal entries of the corresponding dielectric tensor, and compare the results with the original model. The reduced model results in a more computationally efficient algorithm for the characterization of liquid crystal materials; however, there are substantial discrepancies in the simulated antenna figures of merit for intermediate bias voltages.