## 2 Efficiency Estimation Methods
### Efficiency Estimation from IM Sound Analysis  
The method proposed in this study consists of performing the digital processing of the sound signal emitted by an in-service motor. The firmware is fed with the IM nameplate data. First, the shaft rotational speed of the motor is estimated, then the torque, and lastly, the efficiency can be obtained. The noise of the rotating motor tends to repeat when the engine is kept at a constant speed during a certain time period, originating multiple harmonics in the frequency domain of the shaft rotational \begin{table}
\begin{tabular}{c c c} \hline \hline
**Nominal Power (kW)** & **Additional Losses (\% of Nominal)** & **Mechanical Losses (\% of Nominal)** \\ \hline
1–90 & 1.8\% & 1.7\% \\
91–375 & 1.5\% & 2.0\% \\
376–1850 & 1.2\% & 2.3\% \\
1851 and higher & 0.9\% & 2.6\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Additional and mechanical losses in the induction motors (IM) [4; 6]. speed fr. Therefore, the peaks found by a Fourier Fast Transform (FFT), due to the shaft rotation, are located in multiple frequency components of the rotational frequency, according to  
\[\left(\mathrm{f_{p}}\right)_{\mathrm{k}}=\mathrm{k}\mathrm{f_{r}} \tag{4}\]  
where k = 1, 2, 3,.... Figure 1 presents the FFT of a sound signal (highlighted f\({}_{\mathrm{p}}\) frequency peaks) for a motor with a shaft rotational speed close to 30 Hz. As observed, at 30 Hz, the rotation harmonics are not very clear because the frequency is outside the passband of the microphone used. The shaft rotational speed can be accurately estimated using the weighted average of the frequencies of the 2\({}^{\mathrm{nd}}\) and 3\({}^{\mathrm{rd}}\) harmonics, according to  
\[\hat{\omega}=\pi\Bigg{[}\frac{3\left(f_{p}\right)_{2}+2\left(f_{p}\right)_{3}} {3}\Bigg{]} \tag{5}\]  
Determining the motor speed from the sound does not depend on the motor type/class. The analysis is carried out in the frequency domain and it is based on the repeatability of the sound emitted by the motor under stationary or quasi-stationary conditions, reflecting on multiple harmonics of speed in the signal spectrum, as it is illustrated in Figure 1. Thus, it can be affirmed that the harmonic peaks in the spectrum of the speed signal Equation (4) must be in evidence in relation to other noises that were presented, therefore, independent of the motor class or type, the speed is detectable by the proposed method. The torque can also be estimated by a sound analysis of the three-phase IM, which follows the estimation of the motor shaft rotational speed. The curve of the torque vs. speed relationship illustrated in Figure 2 is linearized to consider two aspects: the first aspect regards the synchronous speed and zero torque, and the second aspect regards the nominal speed and the nominal torque. The information on the synchronous and nominal speeds is obtained from the motor nameplate. Thus, the torque can be estimated using  
\[\hat{\Upsilon}=\frac{P_{nom}}{\omega_{nom}}\Bigg{(}\frac{\hat{\omega}-\omega_{ syn}}{\omega_{nom}-\omega_{syn}}\Bigg{)} \tag{6}\]  
where \(P_{nom}\) is the nominal motor power, \({}_{nom}\) is the nominal speed, and \({}_{syn}\) the is the synchronous speed of the three-phase IM. Figure 1: Fourier Fast Transform (FFT) of a sound signal, highlighting the peaks of the rotation harmonics in the shaft. The power obtained in the shaft is the estimated output power \(\hat{\text{P}}_{\text{out}}\), which can be expressed by  
\[\hat{\text{P}}_{\text{out}}=\hat{T}\omega. \tag{7}\]  
Based on the estimated output power in the shaft, curves provided by the manufacturer can be used for the in-service motor efficiency estimation. In the absence of data from the motor manufacturer, an alternative method is to use typical efficiency values for motors of the same efficiency category. Studies conducted by [9] with three EFF1 induction motors of 200 kW made by different manufacturers show that the discrepancy of the maximum efficiency of the medium curve using the pattern IEEE 112 B is only 0.7% (answer 1).