# A Low Torque Ripple Direct Torque Control Method for Interior Permanent Magnet Motor
## 3 Computer Simulation  
In this section, the effectiveness of the proposed method, rSVM-DTC was first verified by a simulation conducted with Simulink/MATLAB. The parameters of the PMSM for all the simulations are shown in Table 1. The performance of the proposed method was investigated and then compared with the conventional SVM-DTC proposed in [8]. In addition, two different switching frequencies were applied to the conventional SVM-DTC, namely 10 kHz and 11 kHz, to investigate the effect of switching frequency on torque ripple, even with such a small increment. For simplicity, the conventional SVM-DTC methods with 10 kHz and 11 kHz are respectively referred to as cSVM-DTC1 and cSVM-DTC2 in what follows. Note that 10 kHz can be treated as a common switching frequency for motor drives (e.g., with insulated-gate bipolar transistors, IGBTs) to avoid high switching loss, and this frequency was used in this study. Some transistors (e.g., silicon carbide metal-oxide semiconductor field-effect transistors, SiC MOSFETs) can be switched at a higher frequency but the cost may be increased if such devices are employed. Another objective of the proposed method is to make use of a low switching frequency for  
Figure 4: Overall block diagram of conventional SVM-DTC for PMSM [19]. Figure 5: Architecture of the proposed method, rSVM-DTC. \(\Psi_{s,cmd}\) here is \(\Psi_{s}^{\prime}\). motor drive so that less expensive transistors can be used, and less switching loss can be achieved, while keeping the torque ripple low. For 11 kHz, it was chosen only for comparison. The switching frequencies were swept from low to high and it was found that the motor torque ripple at 11 kHz switching with conventional SVM-DTC was similar to that of the proposed method at 10 kHz. This is the reason why 10 kHz and 11 kHz were used and compared. In the steady state simulation, as the PMSM operated at 2500 rpm with a 40 Nm rated load, the torque ripples for cSVM-DTC1, cSVM-DTC2 and the proposed rSVM-DTC were 27.5%, 14.8% and 17.3%, respectively; the torque waveforms for the three cases are shown in Figures 6-8. It can be observed that the torque ripples for the proposed method were smaller than that of cSVM-DTC1 and similar to that of cSVM-DTC2 (at a higher switching frequency, 11 kHz). As shown in Figures 9-11, the variation of magnitude of voltage command vector for cSVM-DTC1 was larger than that of the proposed rSVM-DTC and the higher-switching-frequency cSVM-DTC2. Referring to Figures 6-8, it can be seen that lower torque ripple could be obtained when the fluctuation of the magnitude of the voltage command vector was smaller. The torque ripple can be analyzed based on Equations (13)-(16). As previously mentioned and indicated in (15), the third torque variation term \(\Delta T_{3k}\) can be affected by voltage vector. Figures 12-14 show the sole contribution of \(\Delta T_{3k}\) in (15) calculated using the simulation results. As can be seen, the torque component varies when active voltage vector is applied but is kept at zero when zero vector is applied. Obviously, the number of switchings within a period depends also on the switching frequency. In order to analyze the influence of the third torque variation term, the integral of this torque component in one control cycle was conducted. As shown in Figures 15-17, the third torque variation term provoked significant torque ripple for cSVM-DTC1 but with less impact on cSVM-DTC2 and the proposed rSVM-DTC. As indicated in Equation (16), the fourth torque variation term is related to flux linkage, which is in turn affected by voltage vector. That is, the fourth torque variation is also influenced by the voltage vector, which causes flux ripple. As can be observed in Figures 18-20, the torque variations due  
Figure 16: Influence of the third torque variation on torque output on cSVM-DTC2. Figure 14: The third torque variation term for proposed rSVM-DTC (10 kHz). Figure 17: Influence of the third torque variation on torque output on proposed rSVM-DTC.