In this article, the design of a single element elliptical microstrip patch antenna (EMSPA) operating at the X-band frequencies is proposed using the Computer Simulation Technology (CST) simulator. The inset-fed technique is employed in the design to feed the patch. Also, various patch eccentricity with different substrate materials of various thickness have been investigated in order to obtain the optimum realized gain and bandwidth (BW) of the antenna. The simulated results have shown that an elliptical patch having a size of 48.71 mm2 provides a gain of 3.94 dBi and a BW of 389 MHz at the center frequency 9.8 GHz. Moreover, the efficiency of more than 71.6% is predicted when the Taconic_RF-60A substrates of thickness h=1.0 mm and patch eccentricity values of e=0.5 are utilized. Additionally, the gain and the BW of the EMSPA can be further enhanced when a Rogers_RT5880 substrate of thickness h=1.4 mm and patch eccentricity values of e=0.7 are utilized. Comparing the simulated antenna response to the available theoretical and practical results obtained by other researchers previously, a very good agreement is observed. The proposed EMSPA is low profile and very efficient which could be of interest in some radar applications.