## 3 Measurement Improvement by Time Aggregation
### The Analysis of Time Aggregation to Mitigate the Effects on Amplitude  
In the example proposed in Section 2.1, the appropriate number of short windows is three (\(p=0.6\) s/\(0.2\) s = 3 windows). The only interharmonic at 253.3 Hz produces the spectral leakage. Its representative vector will rotate, relative to the position of the vector of the other component, at an average rate of (253.3-250) Hz \(\times\) 0.2 s/_tw_\(\times\) 360/cycle = 240 per short window _tw_. As it can be observed in Figure 2, the initial position of the leakage vector is repeated every three windows; that is, in 3 _tw_\(\times\) 240/_tw_ = 720 (two complete turns). The speed of rotation between any two tones is just the difference of frequencies between them, which, multiplied by the ideal period _Tw_, gives the number of complete turns of one tone with respect to the other until the same relative position between them is reached (3.3 Hz \(\times\) 0.6 s/_tw_ = 2, in the example). For this example, Table 1 shows the values obtained with each acquisition window, together with the corresponding aggregated values calculated with the modules accumulated up to each window considered, for the spectral bars corresponding to the 5 th and 6 th harmonics, interharmonic group 5, and the total energy (or sum of all the spectral bars, according to [PERSON]'s theorem). It can be observed how every three windows, all the values of the modules of each window repeat, and the value aggregated after this third window is repeated after three other windows and so on. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Window** & \multicolumn{2}{c}{**5 th Harmonic (250 Hz)**} & \multicolumn{2}{c}{**6 th Harmonic (300 Hz)**} & \multicolumn{2}{c}{**gIH 5 (255-295 Hz)**} & \multicolumn{2}{c}{**Total Vrms**} \\ \hline  & \multicolumn{2}{c}{**Aggregation**} & \multicolumn{2}{c}{**Aggregation**} & \multicolumn{2}{c}{**Aggregation**} \\ \hline  & **No** & **Yes** & **No** & **Yes** & **No** & **Yes** & **No** & **Yes** \\ \hline
**tw1** & 9.372 & 9.372 & 10.303 & 10.303 & 8.574 & 8.574 & 16.550 & 16.550 \\ \hline
**tw2** & 16.303 & 13.297 & 9.780 & 10.045 & 8.537 & 8.555 & 21.000 & 18.906 \\ \hline
**tw3** & 14.777 & 13.808 & 9.922 & 10.004 & 8.497 & 8.536 & 19.898 & 19.243 \\ \hline
**tw4** & 9.372 & 12.843 & 10.303 & 10.080 & 8.574 & 8.545 & 16.550 & 18.606 \\ \hline
**tw5** & 16.303 & 13.606 & 9.780 & 10.021 & 8.537 & 8.544 & 21.000 & 19.109 \\ \hline
**tw6** & 14.777 & 13.808 & 9.922 & 10.004 & 8.497 & 8.536 & 19.898 & 19.243 \\ \hline
**tw7** & 9.372 & 13.265 & 10.303 & 10.048 & 8.574 & 8.541 & 16.550 & 18.882 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modules obtained for each acquisition window for the total signal of Figure 1, for different frequency groupings, with and without time aggregation. The aggregated values are separated again from the ideal, common RMS value if more windows are added, but while decreasing the error, without it being necessary to stop exactly in one of the windows' multiples of _Tw_ to obtain a good result. The final aggregated value is still an average that approximates the desired common RMS value in all groupings, including that of the total RMS. The approximation is better, the longer the time of the aggregation. In practice, in the signals obtained from real tests, there is a limitation of this time due to the thermal constants of this type of test. Reaching the true common RMS value does not mean that an interharmonic group such as gIH5 encompasses all the energy released by the interharmonic leakage of 253.3 Hz (9.8 V), as this is only possible if a frequency grouping of the entire spectrum is considered. It means that this common RMS value is correctly measured, as the square root of the sum of squares of the modules of the components that are added in each bar are (for 5 th harmonic: \(\sqrt{13.2^{2}+4.05^{2}}=13.808\)\(V\), as seen in Section 2.1).