# Suppression of Harmonic Current in Magnetic Bearing-Rotor System with Redundant Structure
## 4 Suppression of Harmonic Current in AMB with Redundant Structures
### Design of the RC for AMB with Redundant Structures  
In this paper, as shown in Figure 7, \(K_{cx}\) = 5000 V/m, \(T_{s}\) = 0.0002 s, and \(G_{w}\) = 1 A/V were defined. The transfer functions of the controllers \(G_{cx}\) and \(G_{px}\) are described as  
\[G_{cx}(\mathrm{s})=k_{p}+\frac{k_{j}}{s}+k_{d}\mathrm{s}, \tag{45}\]\[G_{px}(\mathbf{s})=\frac{K_{ixx}}{ms^{2}-K_{xx}}, \tag{46}\]  
where \(k_{p}=10\), \(k_{i}=4\), and \(k_{\mathrm{d}}=0.01\). Considering the mode of the magnetic bearing with redundant structures and control system, the phase-frequency characteristic curve of the \(F\)(s) is presented in Figure 8, and it is shown that the phase is below \(90^{\circ}\) in the middle- and high-frequency range. Hence, the phase \(\theta(\omega)\) of the \(F\)(s) does not satisfy the stability requirement. In order to solve the problem, we can design the repetitive controller to adjust the phase of the entire system. (1) Design of low-pass filter \(Q(s)\)  
The main function of the low-pass filter (LPF) is to ensure the cutoff frequency of the system and eliminate the influence of high-frequency interference components that are applied to the entire system. To avoid the harmonic disturbance in the low-frequency region, the gain of the low-pass filter is usually required to be 0 dB, and the phase of the low-pass filter is required to be zero so as to ensure that the system is stable. Assuming a rotor speed of 60 Hz, and considering that the first, second, third, and fourth harmonic currents are required to be suppressed, the cutoff frequency of the low-pass filter was chosen as 1000 Hz. The conventional LPF is usually designed as \(Q_{1}(\mathbf{s})=\frac{1}{0.0000159s+1}\). However, this will lead to phase delay in the repetitive controller loop and degrade the performance of the system. To avoid this problem, was zero-phase low-pass filter was designed to replace the conventional LPF. Considering the cutoff frequency of the system, the zero-phase LPF was designed as follows:  
\[Q(z)=0.212z+0.575+0.212z^{-1}. \tag{47}\]  
In Figure 9, the blue and red lines indicate the amplitude and phase curve of the zero-phase low-pass filter and conventional LPF, respectively. By comparing the characteristics of \(Q(z)\) and \(Q_{1}(z)\), the following conclusions can be drawn: for a conventional LPF, although the gain is 0 dB, the phase changes in the middle- and high-frequency domain, but there is no phase change for the zero-phase low-pass filter. Hence, we adopted the zero-phase low-pass filter shown in Equation (47). (2) Design of phase compensator of \(F\)(s)  
According to the analysis of the previous sections, the phase compensator \(e^{T_{2}s^{2}}\) was used to improve the performance of \(F\)(s) in the high-frequency domain, and the \(C\)(s) was designed to compensate for the phase of the transfer function \(F\)(s) in the middle-frequency band. (a) Design of \(e^{T_{2}s}\) for \(F\)(s) in high-frequency domain  
For the magnetically levitated rotor with redundant structures in this paper, in order to suppress the noise of the system below the cutoff frequency, the phase-frequency characteristics in (0, 6283) rad/s should be considered. As shown in Figure 8, \(\theta(\omega)=86.5^{\circ}\)  
Figure 8: Phase–frequency diagram of \(F\)(s). when the frequency is 6283 rad/s. Thus, in order to satisfy Equations (44), \(e^{T_{2}s}\) should meet the following criterion:  
\[4.4^{\circ}<T_{2}\omega|_{\omega=\omega_{c}}<184.4^{\circ},(T_{2}=N_{2}T_{s}). \tag{48}\]  
The sampling time \(T_{s}=0.0002\) s, and \(N_{2}\) can be chosen as 1 or 2. The phase responses of \(\beta(\omega)\) with different \(N_{2}\) are shown in Figure 10. Obviously, when \(N_{2}\) was set to 1 or 2, the phase \(\theta(\omega)\) at \(\omega_{c}\) was changed to about \(150^{\circ}\) or \(230^{\circ}\). This result indicates that the system satisfies the stability condition. Considering sufficient stability margin of the system, the value of \(N_{2}\) was selected as 1.