A novel modular stator surface-mounted permanent magnet (SSPM) machine is proposed, which features E-type stator modular units with surface-mounted permanent magnets (PMs) on alternate teeth, as well as the tangential flux gaps (FGs) sandwiched by the stator units. The bipolar sinusoidal phase flux-linkage can be obtained in the machines with specific slot/pole number combinations, albeit with the unipolar coil flux-linkages. Based on the theoretical and finite element analyses, the influences of FGs on electromagnetic performances of modular SSPM machines, including winding factors, airgap flux densities, back-EMFs, cogging torques, and on-load torques, are comprehensively investigated. It is revealed that the appropriate stator FGs not only enhance the manufacturability and phase independence, but also improve the back-EMF and average torque by 21.6% and 7.1% respectively. In addition, it is identified that the multiple working harmonics co-exist for torque generations in the modular SSPM machines.