## 5 Estimation of Demagnetization Level  
It was clarified by the measurement and FE analysis in the previous section that the amplitude of vibration is approximately proportional to the load torque and demagnetization level. This section proposes an estimation method for the demagnetization level using the pre-measured characteristics shown in Figure 9. Let us explain how to estimate the demagnetization level using Figure 14. When a motor with demagnetized PMs, the demagnetization level of which is unknown, is driven under a vector control strategy, the vibration and the load torque are measured. Let the vibration be \(V\) and the load torque be \(T\). Four pre-measured points (\(T_{i}\), \(V_{j}\)), (\(T_{i+1}\), \(V_{j}\)), (\(T_{i}\), \(V_{j+1}\)), and (\(T_{i+1}\), \(V_{j+1}\)) are chosen which are close to (\(T\), \(V\)). The demagnetization level \(\alpha\) can be estimated as follows:  
\[\alpha=\frac{V-V_{1}}{V_{2}-V_{1}}\big{(}\alpha_{j+1}-\alpha_{j}\big{)}+\alpha_ {j} \tag{1}\]  
where  
\[V_{1}=\frac{T-T_{i}}{T_{i+1}-T_{i}}\big{(}V_{i+1,j}-V_{i,j}\big{)}+V_{i,j} \tag{2}\]  
and  
\[V_{2}=\frac{T-T_{i}}{T_{i+1}-T_{i}}\big{(}V_{i+1,j+1}-V_{i,j+1}\big{)}+V_{i,j+1} \tag{3}\]  
In order to verify the method for estimation of the demagnetization level, a new PM with a demagnetization level of \(-3.75\%\) was prepared. We estimated this demagnetization level by using the proposed method. Table 4 shows the estimated results of the demagnetization level for the radially demagnetized PMs using the 12 th vibration component shown in Figure 9a. It was found that the estimated demagnetization level was \(-0.8\%\) at no load and \(-3.2\%\) through \(-4.1\%\) for load torque larger than \(0.48\) Nm. Since the actual demagnetization level was \(-3.75\%\), it was verified that the demagnetization level can be estimated by measuring the load torque and the 12 th component of vibration. Using the other vibration components, namely, the 11 th, 13 th, 24 th, and 25 th components, \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & **Vibration (mm/s\({}^{2}\))** & \\ \hline
**Frequency (Hz)** & **Condition** & **Healthy** & **R-7.5\%** & **Z-7.5\%** \\ \hline
440 (11 th) & No load & 0.070 & 0.235 & 0.162 \\ \hline
480 (12 th) & No load & 0.205 & 0.294 & 0.194 \\ \hline
520 (13 th) & No load & 0.192 & 0.256 & 0.160 \\ \hline
920 (23 rd) & No load & 0.173 & 0.257 & 0.176 \\ \hline
960 (24 th) & No load & 0.197 & 0.297 & 0.153 \\ \hline
1000 (25 th) & No load & 0.177 & 0.244 & 0.183 \\ \hline
440 (11 th) & 70\% load & 0.090 & 1.14 & 0.915 \\ \hline
480 (12 th) & 70\% load & 0.263 & 1.43 & 1.10 \\ \hline
520 (13 th) & 70\% load & 0.245 & 1.25 & 0.903 \\ \hline
920 (23 rd) & 70\% load & 0.221 & 1.25 & 0.997 \\ \hline
960 (24 th) & 70\% load & 0.252 & 1.44 & 0.866 \\ \hline
1000 (25 th) & 70\% load & 0.226 & 1.19 & 1.03 \\ \hline
207 (1 st eigenvalue) & 70\% load & 0.006 & 0.008 & 0.001 \\ \hline
472 (2 nd eigenvalue) & 70\% load & 0.022 & 0.007 & 0.068 \\ \hline
593 (3 rd eigenvalue) & 70\% load & 0.022 & 0.009 & 0.001 \\ \hline
952 (4 th eigenvalue) & 70\% load & 0.051 & 0.18 & 0.018 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Calculated amplitudes of vibration. the demagnetization level can also be estimated. Table 5 shows the estimated demagnetization level for the axially demagnetized PMs using the 12 th vibration component shown in Figure 9b. Since it was found, as shown in Table 5, that the estimated magnetization level was +1.12% at no load and -3.0% through -3.6% for load torque larger than 0.48 Nm, the demagnetization level for PMs with axial demagnetization can also be estimated, except when under no-load conditions. Therefore, the measurement of vibration and torque can be used to estimate the magnetization level of PMs under vector control, except for when under very light load.