# A Low Torque Ripple Direct Torque Control Method for Interior Permanent Magnet Motor
## 1 Introduction  
The direct torque control (DTC) was first proposed for control of inverter-fed induction motors (IMs) by [PERSON] et al. [1] and [PERSON] [2] decades ago. In the conventional DTC, an active voltage vector or a zero-voltage vector is applied to electric motors in one control action and the voltage vector is determined through a switching table based on the output of the flux linkage and torque hysteresis comparators. The conventional DTC possesses advantages of simple realization and robustness; however, its high torque ripples and flux ripples are critical drawbacks caused by the variable switching frequency. Many previous studies have been dedicated to resolve the high torque ripple problem of DTC. For example, the duty-cycle-based DTC methods were introduced [3; 4; 5], where one active voltage vector and one zero voltage vector are applied to motors in a fixed cycle to achieve better torque control. In calculation of time duration, only torque ripple reduction is considered but not flux linkage and therefore, for operating points with high flux linkage, the reduction in flux linkage ripple may not be effective. Consequently, this may result in high current harmonics. In [6; 7; 8], the space vector modulation (SVM) is employed in DTC. The voltage vector is decided by varying the load angle which is obtained through an estimator. The torque ripple is reduced through the space vector modulation direct torque control (SVM-DTC) with fixed switching frequency; however, the dynamic response is weaker than the conventional DTC. In the work conducted by [PERSON] regarding SVM-DTC [9],the \(\alpha\)- and \(\beta\)-components of the voltage command is calculated through the magnitude and angle of the flux linkage vector. However, the estimation of flux linkage angle often contains errors that are possibly caused by the noise of the current, flux linkage estimator, or even motor vibration. This would affect the calculation of voltage command and cause fluctuation, which further produces torque ripple. In [10], the relationship between electromagnetic torque and voltage vector is analyzed. Based on the analysis, minimum torque ripple and flux ripple in steady state, and fast dynamic response can be obtained by precise control of the magnitude, phase, and time duration of the voltage vector applied to the motor. However, the control algorithm could be too complex for a digital signal processor (DSP) to realize. In summary, there would still be room for improvement from the above-mentioned studies:  
1. Torque ripples may be effectively reduced but not flux linkage ripple that can result in significant iron loss. Moreover, dynamic response may be sacrificed. 2. Both torque ripple and flux linkage ripple can be reduced, and dynamic response is maintained; however, the control algorithm may be so complex that the implementation could be difficult. Therefore, this paper proposes a simple approach to determine the voltage command vector for decreasing the torque ripple and improving the response based on SVM-DTC. This proposed method is called the \"revised SVM-DTC (rSVM-DTC)\". In this work, the performance in both transient state and steady state is investigated. Based on [10], the relationship between the voltage vector and the torque ripple for a permanent magnet synchronous motor (PMSM) driven with SVM-DTC is further analyzed. Through the analysis, a proper voltage command vector determined by the proposed method is fed to the voltage source inverter (VSI) for torque ripple reduction. Unlike other methods that rely on both flux linkage magnitude and angle for calculation of voltage command [6; 7; 8; 9], the proposed method does not require the flux linkage angle for calculation of the magnitude of voltage command. This significantly reduces the voltage command fluctuation (will be detailed later). The proposed method is simple and easily implemented. This is treated as the major contribution in this paper. The result is compared to that of the conventional SVM-DTC.