# 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
(b) Design of \(C(\mathrm{s})\) for \(F(\mathrm{s})\) in the middle-frequency domain  
Although, by inserting \(e^{T_{2}s}\), the phase characteristic of \(e^{T_{2}s}F(\mathrm{s})\) in the high-frequency domain is compensated for and improved, as shown in Figure 10, the value of the phase is still unchanged and below \(90^{\circ}\) in the middle-frequency domain; thus, the phase lag \(e^{T_{2}s}F(\mathrm{s})\) in the middle-frequency domain will lead to system instability. Therefore, it is necessary to design the phase compensator to improve the phase characteristic of the system in the middle-frequency range. To avoid the insertion of the phase lead compensator affecting the phase in the high-frequency range, the phase lead and phase lag were combined to form the phase compensator \(C(\mathrm{s})\) as  
\[C(\mathrm{s})=\frac{0.0015\mathrm{s}+1}{0.00526\mathrm{s}+1}\bullet\frac{0.0 068\mathrm{s}+1}{0.00051\mathrm{s}+1}. \tag{49}\]  
Figure 10: Bode plot of \(F(\mathrm{s})e^{T_{2}s}\). Figure 9: Bode plot of \(Q(z)\) with different filters. After introducing the phase compensator \(C(s)\), the phase-frequency curve is as shown in Figure 11.  
As shown in Figure 11, the black line denotes the phase-frequency characteristic of \(F(s)\), and the red dotted line denotes the phase-frequency characteristic of \(C(s)F(s)e^{T_{\rm s}s}\). It can be observed that the phase range was about (\(100^{\circ}\), \(175^{\circ}\)) within (0, \(\omega_{c}\)). Thus, it can be seen that, when the phase compensator \(e^{T_{\rm 25}}\) and \(C(s)\) are applied in the system, Equation (44) is satisfied. (3) Design of fractional delay filters  
In practice, the repetitive controller generally is adopted in discrete form. The discrete form of delay \(e^{-T_{\rm 1}s}\) in repetitive control is \(z^{-N}\), and \(N\) is an integer. It depends on the ratio of the sampling frequency \(f_{\rm s}\) to the fundamental frequency \(f\). The value of \(f_{\rm s}\) is usually fixed, and it is possible to make the value of \(N\) a non-integer for sampling frequency \(f_{\rm s}\) with different values. In this paper, fundamental frequency \(f=60\) Hz, sampling frequency \(f_{\rm s}=5000\) Hz, and \(N=83.333\). To solve this problem, Lagrange interpolation-based fractional delay (FD) filters were adopted to approximate the fractional part. For the standard fractional delay function \(H_{\rm F}(z)=z^{-F}\), it can be described as  
\[ \begin{array}{l}H_{\rm F}(z)\approx\sum\limits_{n=0}^{N_{\rm m}}h(z)z^{-n} \\ h(z)=\prod\limits_{\begin{array}{c}N_{\rm m}\\ \prod\\ k\
eq n\end{array}}\frac{F-k}{n-k},\,(n=0,1\cdots N_{n})\\ \end{array}  \tag{50}\]  
In this paper, because \(N_{2}=1\) and \(N_{1}\)= \(N-N_{2}\) = 82.333, the discrete form of \(e^{-T_{\rm 1}s}\) can be written as \(z^{-82.333}\), which can be divided into integer and decimal parts, i.e., \(z^{-82.333}=z^{-81}\times z^{-1.333}\). According to Equation (50) and \(N_{m}=3\), we can obtain \(h(0)=-0.0617\), \(h(1)=0.7411\), \(h(2)=0.3699\), and \(h(3)=-0.0493\). Therefore, \(z^{-1.333}\) can be rewritten as  
\[z^{-1.333}\approx h(0)+h(1)z^{-1}+h(2)z^{-2}+h(3)z^{-3}. \tag{51}\]  
Furthermore, according to Equation (44), \(K_{\rm rc}\) = 0.001 in this paper. Accordingly, combining the designed modules, the structure of the proposed repetitive controller for the magnetically suspended rotor with redundant structures in the \(x\)-direction can be depicted as in Figure 12.