Fractional-slot permanent magnet (PM) machines with an odd number of stator slots are attractive candidates for industrial applications due to their high torque density and low torque ripple. However, the unbalanced magnetic force is detrimental to the bearing and yields noise and vibration. This paper determines the existence criterion of UMF on both open-circuit and on-load conditions by analysis of air gap field and magnetic force waves. Only when the machine periodicity is one, will the open-circuit UMF be generated due to the mutual interaction between the modulated PM harmonics and the original PM harmonics. In addition, the existence of on-load UMF in machines with a periodicity of one is confirmed even when the slotting effect is ignored. Furthermore, among the five investigated machines with a given odd slot number of 9 but different pole numbers, the UMF in a machine with two poles is the smallest since the magnitude of the 2nd harmonic in the armature reaction is the smallest compared with other contributing harmonics. Meanwhile, the UMF in a machine with pole number 2p = N-s-1 is the largest due to the relatively larger magnitude of the (p + 1)(th) armature reaction harmonic and the additive effect between the first order radial and tangential stresses.