# Measurement of the Speed of Induction Motors Based on Vibration with a Smartphone
## 4 Discussion  
The rather basic, low-featured smartphone used in this work only allowed the sampling of the axial components of the motor vibration with a maximum rate of 200 samples per second, that is, with a sampling frequency of \(\omega_{S}\) = 200 Hz (as a reference, in [26], the vibration signal acquired from an external accelerometer, was sampled at 10 kHz). Consequently, the Nyquist-Shannon sampling theorem limits to \(\omega_{F\_MAX}\) = 100 Hz, the maximum value of the theoretical rotor shaft mechanical frequency that could be measured by the proposed method. In fact, the practical limit will be somewhat less than that theoretical value. Fortunately, this limitation is of little practical importance since industrial induction motors, for various considerations, are rarely operated at such high speeds. In a real case, this limitation would only affect two-pole motors, operating near twice the rated frequency (50-60 Hz), which is not very common in the industrial field. The synchronous speed of a four, six, or eight8-pole motor is 1/2, 1/3, or 1/4 times less than the synchronous speed of a two-pole motor, so the mechanical rotation frequency of the rotor will not exceed the maximum limit of the accelerometer built into the smartphone. Anyway, in case it was necessary to measure the speed of motors with higher rotor mechanical frequencies, it would be necessary to use a smartphone that allows acquiring the components of the motor vibration with a higher sampling frequency. Today's smartphones allow sampling frequencies of 400-500 Hz. To compare the performance of the proposed methodology, Table 7 shows the results reported in [26], corresponding to three different two-pole induction motors fed from the grid (380 V, 50 Hz), whose speeds are estimated at no load and at full load by means of an external accelerometer sampled at 10 kHz (10 s duration). As can be seen, the errors ranged from 0.07% to 0.55%. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Resistance \({}^{*}\)**} & \multicolumn{2}{c}{**Experimental**} & \multicolumn{2}{c}{**Algorithm**} & \multicolumn{2}{c}{**Relative Error**} \\ \cline{2-6}  & \(\Omega_{m}\) **(r/min)** & \(\omega_{m}\) **(Hz)** & \(\Omega\) **(r/min)** & \(\omega_{F}\) **(Hz)** & \(\epsilon\) **(\%)** \\ \hline R2 & 3790 & 63.17 & 3791.0 & 63.18 & 0.0268 \\ R1 & 3922 & 65.37 & 3925.8 & 65.43 & 0.0964 \\ R0 & 4142 & 69.03 & 4139.6 & 68.99 & 0.0568 \\ \hline \hline \end{tabular} \({}^{*}\)The capability of the converter is exceeded for the cases corresponding to R3-R7.  
\end{table} 
Table 6: Motor fed from a frequency converter, driving a DC generator. Measured and estimated values of rotor speed and frequencies, as well as relative errors at a frequency supply of 70 Hz \({}^{*}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Resistance \({}^{*}\)**} & \multicolumn{2}{c}{**Experimental**} & \multicolumn{2}{c}{**Algorithm**} & \multicolumn{2}{c}{**Relative Error**} \\ \cline{2-6}  & \(\Omega_{m}\) **(r/min)** & \(\omega_{m}\) **(Hz)** & \(\Omega\) **(r/min)** & \(\omega_{F}\) **(Hz)** & \(\epsilon\) **(\%)** \\ \hline R3 & 3220 & 53.67 & 3222.7 & 53.71 & 0.0825 \\ R2 & 3400 & 56.67 & 3401.4 & 56.69 & 0.0402 \\ R1 & 3451 & 57.52 & 3451.2 & 57.52 & 0.0050 \\ R0 & 3564 & 59.40 & 3562.5 & 59.38 & 0.0421 \\ \hline \hline \end{tabular} \({}^{*}\)The capability of the converter is exceeded for the cases corresponding to R4-R7.  
\end{table} 
Table 5: Motor fed from a frequency converter, driving a DC generator. Measured and estimated values of rotor speed and frequencies, as well as relative errors at a frequency supply of 60 Hz \({}^{*}\). The results in Table 1 show that, with the proposed methodology, when the smartphone is placed in _Pos1_ (vibration signal sampled at 200 Hz for 10 s) the relative errors are always under 0.09% (ranging from 0.0029% to 0.0831%), while Table 2 shows that for other smartphone positions and sample durations, the maximum error is lower than 0.14% (from 0.0131% to 0.1311%).