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A novel beam-supported membrane (BSM) structure for the fiber optic extrinsic Fabry-Perot interferometer (EFPI) sensors showing an enhanced performance and an improved resistance to the temperature change was proposed for detecting partial discharges (PDs). The fundamental frequency, sensitivity, linear range, and flatness of the BSM structure were investigated by employing the finite element simulations. Compared with the intact membrane (IM) structure commonly used by EFPI sensors, BSM structure provides extra geometrical parameters to define the fundamental frequency when the diameter of the whole membrane and its thickness is determined, resulting in an enhanced design flexibility of the sensor structure. According to the simulation results, it is noted that BSM structure not only shows a much higher sensitivity (increased by almost four times for some cases), and a wider working range of fundamental frequency to choose, but also an improved linear range, making the system development much easier. In addition, BSM structure presents a better flatness than its IM counterpart, providing an increased signal-to-noise ratio (SNR). A further improvement of performance is thought to be possible with a step-forward structural optimization. The BSM structure shows a great potential to design the EFPI sensors, as well as others for detecting the acoustic signals.