## 5 Proposed TSFs
### Comparison of the Conventional and Proposed TSFs  
Figure 5 shows the typical profiles of four common conventional TSFs [25]. The turn-on angle, turn-off angle, and overlap angle, \(\theta_{on}\), \(\theta_{\textit{off}}\), and \(\theta_{ov}\) are set to \(5^{\circ}\), \(40^{\circ}\) and \(5^{\circ}\), respectively. And the torque reference \(T_{\textit{ref}}\) is set to 1 Nm. In Figure 5a-d, the profiles (top to bottom) are the torque reference, the current reference, the flux linkage reference, and the absolute value of RCFL respectively. From Figure 5d, the values of \(M_{R}\) for the linear, sinusoidal, exponential, and cubic TSFs are 11.78, 6.75, 5.05, and 19.10 Wb/rad, respectively. According to Equation (11), the values of maximum rotor speed \(\omega_{max}\) for a torque-ripple-free operation of four common conventional TSFs are 22.9, 40, 53.5, and 14.1 rad/s, which are equal to only 219, 382, 511, and 135 rpm, respectively. Figure 6a shows the change in the MAV-RCFL with the changing torque reference, indicating that the exponential TSF resulted in the lowest RCFL value. The peak values of the reference currents at each torque reference are compared in Figure 6b showing an increase in the currents as the torque reference increases. The peak current is the same for the linear TSF and the cubic TSF and it is lower than for the other two TSFs. This indicates that the power converter has much smaller pressure and better insulation for the linear and cubic TSFs. The rms currents at each torque reference are compared in Figure 6c, which indicates that the linear and the sinusoidal TSFs have almost the same value of rms current at the torque reference between 0.5 Nm and 3 Nm. The rms current is lower for cubic TSF than for the other TSFs. In summary, if the minimum \(M_{R}\) (when \(w_{f}=0\) in Equation (14)) is the only optimization target, the best choice of TSF is the exponential TSF, the second best choice is the sinusoidal TSF, and the worst choice is the cubic TSF. On the other hand, if the minimum copper loss \(J_{1}\) (when \(w_{f}=1\) in Equation (14)) is the only optimization objective, the cubic TSF represents a good choice, and it also has the lowest value for the peak current. Figure 7a-d show the typical profiles of several proposed TSFs for different \(r\) values. From Figure 7, when \(r\) tends to \(\infty\), the minimization of the copper loss is achieved theoretically because the torque reference tends to be zero during the commutation region. By decreasing \(r\), the torque reference increases during the commutation region, which results in an increase in the rms phase current. When \(r\) tends to 1, Figure 7b shows that the phase current reference of two adjacent excited phases in the commutation region is equal. And it is clear that the peak value of phase current is also the smallest when the TSF is defined by \(r\to 1\). Figure 5: Typical profiles of four conventional TSFs. (**a**) Torque Reference; (**b**) Current Reference; (**c**) Flux Linkage; (**d**) Absolute Value of RCFL. Figure 6: Comparison of four conventional TSFs: (**a**) absolute value of the rate-of-change of the flux linkage (RCFL); (**b**) peak phase current; (**c**) rms phase current. Figure 7: Typical profiles of the proposed TSF family. (**a**) Torque Reference; (**b**) Current Reference; (**c**) Flux Linkage; (**d**) Absolute Value of RCFL. Figure 8a shows the relationship between the MAV-RCFL and the exponent \(r\). As \(M_{R}\) reaches its minimum at \(r\approx 4\), it is reasonable to choose \(r=4\) as the exponent of the optimal TSF in Equation (16). When \(T_{r\prime}\) is set to 1 Nm and \(V_{dc}\) is set to 270 V in the case of \(r=4\), the maximum speed range \(\omega_{max}\) for operating torque ripple performance freely can be achieved at 30 rad/s, which is equal to 286.6 rpm. The equivalent level of the linear and the cubic TSFs are also marked with the dash-dotted line in the Figure 8a-c for comparison. Figure 8b shows the peak phase current reference as a function of the exponent \(r\).