We introduce the large-signal and small-signal equivalent circuit models for a capacitive micromachined ultrasonic transducer (CMUT) cell, which has radiating plates on both sides. We present the diffraction coefficient of baffled and unbaffled CMUT cells. We show that the substrate can be modeled as a very thick radiating plate on one side, which can be readily incorporated in the introduced model. In the limiting case, the reactance of this backing impedance is entirely compliant for substrate materials with a Poisson's ratio less than 1/3. We assess the dependence of the radiation performance of the front plate on the thickness of the back plate by simulating an array of bilateral CMUT cells. We find that the small-signal linear model is sufficiently accurate for large-signal excitation, for the purpose of the determining the fundamental component. To determine harmonic distortion, the large-signal model must be used with harmonic balance analysis. Rayleigh-Bloch waves are excited at the front and back surfaces similar to conventional CMUT arrays.