# Loss Research and Thermal Analysis of BLDC Hollow-Cup Motor Under Reactor Suppression
## 3 Improvement Design Study
### RMS of Load Line Voltage Analysis Considering Inductance Value  
By the time the reactor inductance reaches 15 uH, the loss variation is stabilized. As the inductance of the controller's voltage output continues to increase, the electrical time constant of the entire system will also increase, affecting the system's dynamic performance. The following equations are analyzed. The reactor in series between the BLDC hollow-cup motor and the controller acts as a suppressor of the efficiency of the controller. The differential equation for the voltage at the input of the hollow-cup motor is given in Ref. [20], expressed as follows:  
\[U_{\text{a}}=Ri_{\text{a}}+(L-M)\frac{\text{d}i_{\text{a}}}{\text{d}t}+E_{ \text{a}}, \tag{10}\]  
\[U_{\text{b}}=Ri_{\text{b}}+(L-M)\frac{\text{d}i_{\text{b}}}{\text{d}t}+E_{ \text{b}}, \tag{11}\]  
Figure 3: Variation of core loss with inductance of reactor. \[U_{\rm c}=Ri_{\rm c}+(L-M)\frac{\rm d\dot{\it{c}}_{\rm c}}{\rm d\it{t}}+E_{\rm c}, \tag{12}\]  
where \(U_{\rm a}\), \(U_{\rm b}\), and \(U_{\rm c}\) represent the three-phase voltage, \(E_{\rm a}\), \(E_{\rm b}\), and \(E_{\rm c}\) represent three opposite electromotive forces, \(i_{\rm a}\), \(i_{\rm b}\), and \(i_{\rm c}\) represent the three-phase current, \(R\) is the resistance of the three-phase circuit winding, \(L\) is the three-phase circuit inductor, and \(M\) is the mutual inductance in a three-phase circuit. In the following steps, the calculations for the motor load line voltage are performed, the changes in value are analyzed, and the appropriate reactor value is selected. Therefore, loss suppression is achieved, while avoiding the degradation of the dynamic performance of the entire motor system due to excessive inductance values. The trend of the load line voltage RMS is analyzed for each 5 \(\upmu\)H increase in reactor inductance in the range of 0 \(\upmu\)H to 25 \(\upmu\)H. The variation of line voltage RMS with the value of inductance of the reactor is presented in Figure 5.  
According to the FE calculation, the motor load line voltage tends to decrease with the increase in reactor inductance, and the decreasing trend is more obvious after the  
Figure 4: Variation of core loss at different values of inductance: (**a**) core loss at inductance of 0 \(\upmu\)H; (**b**) core loss at inductance of 15 \(\upmu\)H.  
Figure 5: Variation of line voltage RMS with inductance of reactor. inductance value of 15 \(\upmu\)H. The variation of line voltage RMS for the reactor inductance values of 0 \(\upmu\)H and 15 \(\upmu\)H is presented in Figure 6.  
After comparative analysis, the core loss of the BLDC hollow-cup motor shows a decreasing trend with the increase in the inductance value of the reactor, and this trend tends to stabilize when the inductance value reaches 15 \(\upmu\)H. However, the increase in inductance value will have a dampening effect on the motor load line voltage RMS. When the inductance reaches 15 \(\upmu\)H, the motor load line voltage RMS decreases more significantly, while the output efficiency of the controller is reduced. In summary, a 15 \(\upmu\)H reactor in series between the BLDC hollow-cup motor and the controller improves the efficiency of the motor, without affecting the voltage at the output of the controller. A comparative analysis is shown in Figure 7.  
Similarly, the calculation results for current and torque also confirm this conclusion. As the effective value of the motor terminal voltage decreases (the inductance value of the reactor in the series motor system increases), to ensure that the output torque of the motor remains unchanged under rated conditions, the input current of the motor will increase accordingly. However, when the inductance value of the reactor exceeds 15 \(\upmu\)H, the current increases from 2.5 A to around 3 A, which significantly increases the energy consumption of the motor system.