# Suppression of Harmonic Current in Magnetic Bearing-Rotor System with Redundant Structure
## 4 Suppression of Harmonic Current in AMB with Redundant Structures
### Analysis of Magnetic Bearing System with RC  
Taking the control loop of the magnetic bearing-rotor system in the \(x\)-direction as an example, the repetitive controller \(G_{xRC}\) is added into the original system. The control block diagram with RC is depicted in Figure 7, where \(Q\)(s) is the low-pass filter, \(C\)(s) is the phase compensator for the middle and low frequencies, the phase compensator \(e^{-T_{2}s}\) is used for high frequency, \(e^{-T_{1}s}\) is the time delay element, \(K_{\rm rc}\) is the gain of RC and ensures the robustness of the system, \(T_{1}+T_{2}=T=1\)/_f_, and \(f\) is the fundamental frequency of the disturbance signal. The transfer function of the proposed RC can be expressed as  
\[G_{xRC}(\mathsf{s})=\frac{e^{-T_{1}s}}{1-e^{-(T_{1}+T_{2})s}}Q(\mathsf{s})K_{ rc}(\mathsf{s})C(\mathsf{s}). \tag{33}\]  
\(S_{0}\) is the sensitivity function of the system without RC, and it can be expressed as  
\[S_{0}(\mathsf{s})=\frac{G_{c}(\mathsf{s})G_{w}(\mathsf{s})}{1+G_{c}(\mathsf{s })G_{w}(\mathsf{s})G_{p}(\mathsf{s})K_{s}(\mathsf{s})}. \tag{34}\]  
The logical current \(i_{\rm x}\) is as the virtual output of the magnetic bearing, according to Figure 7. The logical current \(i_{\rm x}\) can be written as  
\[i_{\rm x}(\mathsf{s})=\frac{S_{0}(\mathsf{s})\big{[}1-e^{-T_{\rm S}}\big{]} \big{[}R(\mathsf{s})-D(\mathsf{s})K(\mathsf{s})\big{]}}{1-e^{-T_{\rm S}}-K_{ rc}C(\mathsf{s})Q(\mathsf{s})\frac{S_{0}(\mathsf{s})}{G_{w}(\mathsf{s})}e^{-T_{1}s}}. \tag{35}\]  
While the frequency \(w\) of disturbance signal is lower than the cut-off frequency \(w_{c}\omega=\frac{2\pi}{\pi}n\) (\(n=1,2,3,\dots\) ). Obviously, according to [PERSON]'s formula, we can conclude that \(\big{|}e^{-T_{\rm S}}\big{|}_{s=j\omega}=1\) and \(\arg\big{[}e^{-T_{\rm S}}\big{]}_{s=j\omega}=-2n\pi\); therefore,  
\[1-e^{-T_{\rm S}}=0. \tag{36}\]  
Equation (36) denotes that the proposed RC can eliminate the disturbances which consist of multiples of the fundamental frequency \(f\). However, the system with the RC  
Figure 7: Control schematic diagram in \(x\)-direction with RC. should satisfy the stability conditions of the system. The transfer function of the whole system is  
\[G(\mathrm{s})=\frac{G_{cx}(\mathrm{s})G_{w}(\mathrm{s})G_{p}(\mathrm{s})}{1-G_{cx }(\mathrm{s})G_{xRC}(\mathrm{s})+G_{cx}(\mathrm{s})G_{w}(\mathrm{s})G_{p}( \mathrm{s})K_{s}(\mathrm{s})}. \tag{37}\]  
The closed-loop characteristic equation can be obtained as  
\[1-G_{cx}(\mathrm{s})G_{xRC}(\mathrm{s})+G_{cx}(\mathrm{s})G_{w}(\mathrm{s})G_{ p}(\mathrm{s})K_{s1}(\mathrm{s})=0. \tag{38}\]  
Equation (38) can be rearranged as  
\[M(\mathrm{s})-e^{-\mathrm{Ts}}N(\mathrm{s})=0, \tag{39}\]  
where  
\[ \begin{array}{c}M(\mathrm{s})=1+G_{c}(\mathrm{s})G_{cp}(\mathrm{s})G_{p}( \mathrm{s})K_{s}(\mathrm{s}),\\ N(\mathrm{s})=1+G_{c}(\mathrm{s})K_{rc}C(\mathrm{s})Q(\mathrm{s})e^{T_{2} \mathrm{s}}+G_{c}(\mathrm{s})G_{w}(\mathrm{s})G_{p}(\mathrm{s})K_{s}(\mathrm{ s}).\end{array} \]  
The regeneration spectrum of the whole system can be obtained as  
\[R(\omega)=\left|\frac{N(\mathrm{s})}{M(\mathrm{s})}\right|_{j\omega}=\left|1+ K_{rc}C(\mathrm{s})Q(\mathrm{s})e^{T_{2}\mathrm{s}}\times\frac{G_{c}(\mathrm{s})}{1+G_{c}( \mathrm{s})G_{w}(\mathrm{s})G_{p}(\mathrm{s})K_{s}(\mathrm{s})}\right|_{s=j \omega}. \tag{40}\]  
Generally, a smaller value of the regeneration spectrum function denotes better relative stability of the system. According to the small gain theory, the sufficient condition of the closed-loop system depicted in Figure 7 is given by  
\((1)\)\(F(\mathrm{s})=\frac{G_{c}(\mathrm{s})}{1+G_{c}(\mathrm{s})G_{w}(\mathrm{s}) G_{p}(\mathrm{s})K_{s}(\mathrm{s})}\) is stability, (2) \(R(\omega)<1\). It is assumed that the original system is stable, and that the system has met condition 1. Therefore, the structure of the repetitive controller needs to be designed to make the system meet condition 2.