[Objective] In order to improve the control performance of model predictive current control (MPCC) for permanent magnet synchronous motor (PMSM), an LC filter was added between the PMSM and the inverter. [Method] A third-order multi-step MPCC system with LC filter was established. Predictive control was applied to the inverter current, capacitor voltage, and motor current, with simulation comparisons made against a first-order multi-step MPCC and field orientation control (FOC). [Results] Comparison results showed that under the same switching frequency and number of prediction steps, the third-order MPCC significantly reduced torque ripple and current ripple in the PMSM compared to the first-order MPCC, while also achieving lower total harmonic distortion (THD) in the current. Additionally, as the number of prediction steps increased in the third-order MPCC, both current THD, current ripple, and torque ripple progressively decreased. [Conclusion] As the number of prediction steps increases, the control performance of the third-order MPCC system gradually improves. The third-order MPCC outperforms the first-order MPCC. At higher switching frequencies, the third-order MPCC with 4-5 prediction steps delivers better control performance than FOC.