## 3 HPMSM System with Output LC Filter  
In order to restrain the high-frequency harmonics, the three-phase output LC filter is connected between the inverter and the HPMSM. The single-phase circuit topology of the HPMSM drive system is illustrated in Figure 2.  
Based on the circuit topology, the following voltage and current equations can be obtained:  
\[u_{c}=L\frac{di_{a}}{dt}+Ri_{a}+e_{a}, \tag{11}\]  
\[i_{a1}=i_{a}+C\frac{du_{c}}{dt}=LC\frac{d^{2}i_{a}}{dt}+RC\frac{di_{a}}{dt}+i_ {a}+C\frac{de_{a}}{dt}, \tag{12}\]  
\[u_{a}=u_{c}+L_{1}\frac{di_{a1}}{dt}=LL_{1}C\frac{d^{3}i_{a}}{dt^{3}}+RL_{1}C \frac{d^{2}i_{a}}{dt^{2}}+(L+L_{1})\frac{di_{a}}{dt}+Ri_{a}+L_{1}C\frac{d^{2}e _{a}}{dt^{2}}+e_{a}. \tag{13}\]  
Depending on (11)-(13), the transfer function of the HPMSM with the LC filter can be written as  
\[\phi(s)=\frac{l_{a}(s)}{U_{a}(s)}=\frac{1}{LL_{1}Cs^{3}+RL_{1}Cs^{2}+(L+L_{1}) s+R}. \tag{14}\]  
From (14), it can be deduced that in the low-frequency bands, the above system can be simplified into a first-order inertial system, and the transfer function is equivalent to  
\[\phi(s)\approx\frac{1}{(L+L_{1})s+R}. \tag{15}\]  
According to (15), the cut-off frequency in the low-frequency bands can be calculated as  
\[\omega_{cut}\approx\frac{L+L_{1}-\sqrt{(L+L_{1})^{2}-4R^{2}L_{1}C}}{2 RL_{1}C} \approx\frac{R}{L+L_{1}}. \tag{16}\]  
The setting of the cut-off frequency needs to consider two aspects. Firstly, to filter out all the major harmonic components, the cut-off frequency should be set lower than the main harmonic of the lowest frequency, which is \(f_{s}-f_{b}\) according to (7). Secondly, to avoid adverse impact on the fundamental component, the cut-off frequency should be higher than the motor fundamental frequency and retain a certain margin, for example, not less than twice the fundamental frequency. Therefore, the cut-off frequency needs to satisfy:  
\[2f_{b}<\frac{\omega_{cut}}{2\pi}<f_{s}-f_{b}. \tag{17}\]  
It is worth noting that there is an optimal scope of application for this scheme. It can be seen from (17) that if the fundamental frequency of the motor is too high, the frequency band of the fundamental wave and the harmonics will cross, and the LC filter will not be  
Figure 2: Structure diagram of the single-phase drive system of HPMSM with LC filter. able to filter out some harmonics. Therefore, the condition under which this scheme can be effectively applied is:  
\[f_{b}<\frac{1}{3}f_{s}. \tag{18}\]  
Typically, \(f_{s}\) is 10 kHz, so \(f_{b}\) should be below 3.33 kHz, which encompasses most HPMSM operating ranges. In addition, the problem of voltage drop on LC filters should also be considered. The voltage drop will affect the transmission of electrical energy, resulting in additional reactive power loss. In general, the voltage drop should not be greater than 10% of the input voltage. As shown in Figure 2, the voltage drop can be calculated by obtaining the relationship between \(U_{c}\) and \(U_{a}\), and their relationship can be expressed as:  
\[u_{c}=L\frac{di_{a}}{dt}+Ri_{a}+e_{a}, \tag{11}\]0  
Since the inductance of the high-speed motor is very small, the required LC filter capacitance and inductance values are also small, and the following approximations can be made:  
\[\left\{ \begin{aligned} \omega^{2}CL_{1}\approx 0\\ \omega^{2}CLL_{1}\approx 0\end{aligned} \right. \tag{20}\]  
Therefore, (19) can be further simplified to  
\[\left|\frac{U_{c}}{U_{a}}\right|=\frac{\sqrt{R^{2}+\omega^{2}L^{2}}}{\sqrt{R^ {2}+\omega^{2}(L+L_{1})^{2}}} \tag{21}\]  
It can be seen from (21) that the voltage drop mainly depends on the inductance of the LC filter, and, the larger the inductance \(L_{1}\), the more severe the voltage drop. In summary, the selection of LC filter inductance needs to consider the setting of the cut-off frequency and the amplitude of voltage drop, and the specific calculation method is given from (16) to (21). Figure 3 shows the Bode diagram of HPMSM with the LC filter.