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Characteristic mode (CM) analysis is carried out to understand the underlying physics of two popular circularly polarized (CP) microstrip patch antennas, namely U-slot and E-shaped CP patch antennas, for achieving better performance. Modifications of these two CP antennas with optimal feed position and compact patch size are explicitly obtained through examining the CMs, modal significance, and characteristic angle. Compared to conventional designs, the two modified designs have better axial ratio (AR) and cross-polarization performance without introducing any additional design cost or complexity. It is also interesting to find that: 1) offset probe feed in CP U-slot patch antennas will provide excellent AR performance; 2) cutting off the redundant section from CP E-shaped patch antenna will yield low cross polarization and more compact configuration.