The model predictive current control (MPCC) technique has been an effective alternative for motor drive applications requiring fast dynamic response. However, the conventional MPCC suffers from high switching losses due to high switching frequency of the inverter. This study proposes a low average switching frequency MPCC method based on boundary‐constrained quadratically extrapolated current trajectory for a permanent magnet synchronous motor drive. First, a quadratic function‐based current prediction model is constructed to obtain the current trajectory in the long‐term future. Second, the boundary constraint is designed to obtain the prediction horizon which is changing in real‐time in order to ensure the validity of the predicted current trajectory. Finally, the cost function established according to the value of the prediction horizon is used to reduce the average switching frequency. In comparison with the conventional MPCC method, the proposed method can significantly reduce the switching frequency of the inverter. In the meantime, fast dynamic response and high steady‐state performance of the conventional MPCC method are preserved. Both simulation and experimental results of the proposed method and the conventional MPCC method are presented to verify the feasibility and advantages of the proposed method.