# Modelling and Experimental Testing of Passive Magnetic Bearings for Power Loss Reduction
## 4 Results and Discussion
### Dynamic Tests  
The dynamic tests consist of angular speed sweep down runs, performed at three PMB configurations (named A, B, and C, as shown in Table 2, where \(\Delta z\) refers to both the upper and the lower magnetic supports), with the aim of analysing the relation between the PMB ring offset and the system power loss reduction. Configuration A corresponds to a pure radial bearing with null axial offset and null rotor weight compensation, while the B and C configurations belong to the thrust-radial configuration offset range. Each test is performed according to the following procedure:  
* The rotor is accelerated, then it is decoupled from the actuation system and left freely rotating, so that its angular speed progressively decreases;
* The angular speed is acquired from 200 rad/s until the rotor stops. The angular speeds measured during three sets of tests performed for each PMB configuration listed in Table 2 are shown in Figure 8a. The evaluation of the rotor angular acceleration during the sweep down tests is a common practise to estimate the power losses [24]. Therefore, the angular accelerations in Figure 8b are obtained by computing the time derivative of the experimental angular speeds. The time history regions where the angular speed rapidly changes due to contacts between the rotor and the stator (clearly \begin{table}
\begin{tabular}{c c} \hline \hline
**Axial Offset \(\Delta z\)** & **Error \%** \\ \hline
**[mm]** & **[-]** \\ \hline
0 & - \\
0.6 & 1.9 \\
1.2 & 0.3 \\
1.8 & 0.1 \\
2.4 & 2.2 \\
2.9 & 5.0 \\
3.4 & 3.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relative error between the experimental measurements and the model. Figure 7: Axial force as a function of the PMB ring axial offset. visible in Conf. C), e.g., caused by critical speeds, superharmonic and subharmonic resonances [25; 26], are not considered. In fact, under these conditions the acceleration has high peaks, as shown in Figure 8b, that are not related only to Coulomb friction or aerodynamic drag losses. The torque equilibrium related to the rotation the rotor about its axis is considered in Equation (13). \[I_{p}\dot{\omega}+T_{r}=T_{d}, \tag{13}\]  
where \(T_{r}\) and \(T_{d}\) are the resisting and driving torques, respectively. The driving torque is null during the angular speed sweep down tests since the rotor is uncoupled from the motor. Thus, Equation (13) is rewritten in terms of powers as Equation (14) by multiplying both terms by the angular speed \(\omega\), obtaining the expression of the total power loss \(P_{total}\). \[P_{total}=T_{r}\omega=-I_{p}\dot{\omega}\omega. \tag{14}\]  
A polynomial fitting of the acceleration data of Figure 8b is performed in the selected ranges to evaluate the total power loss, with Equation (14), as a function of the angular speed excluding the stator-rotor contact effects. Figure 9a shows the total power loss for each configuration, while Figure 9b shows the percentage reduction in the total power loss of PMB configurations B and C, with respect to configuration A with null weight compensation, as a function of the angular speed. The aerodynamic and friction effects are the two main sources of system power loss, as expressed in Equation (15):  
\[P_{total}=P_{a}+P_{f}, \tag{15}\]  
where the aerodynamic losses are  
\[P_{a}=\sum_{k}a_{k}\omega^{k+1}, \tag{16}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**PMB** & **Axial Offset Az** & \
\begin{tabular}{c} **Experimental Contact** \\ **Reaction Force \(F_{N}\)** \\ **[m]** \\ \end{tabular} &
\begin{tabular}{c} **Weight Percentage** \\ **Compensation** \\ **[**N]** \\ \end{tabular} \\ \hline A & 0 & 49.0 & 0\% \\ B & 0.8 & 24.1 & 49.2\% \\ C & 1.6 & 7.3 & 85.1\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dynamic test PMB configurations. Figure 8: (**a**) Angular speed time histories; (**b**) angular acceleration fitting.