## 2 The Effects on Amplitude Due to Interactions between Tones
### Analysis of the Effects of the Interaction between Tones  
Spectral leakage is produced by interharmonic components of the signal or by desynchronization of the fundamental frequency (and all their integer multiples) to the sample window. To avoid spectral leakage when applying DFT, the sampling window must contain an integer multiple of all the periods of the frequency components of the signal. Optimally, this window is calculated as the minimum common multiple of all the contained periods and the resolution specified by IEC standards (5 Hz). The interharmonics can be due to the desynchronization of the fundamental with the sampling _tw_ window (and consequently of all its multiples or harmonics), or due to actual interharmonic components, such as those generated by variable and non-linear loads, such as frequency converters. If the existing harmonics and interharmonics in the signal are all multiples of the spectral resolution (of 5 Hz, for example), or if only one tone is taken into account, the possible time aggregation benefits are not appreciated, since the same absolute values are obtained with the DFT applied to every window, as in the cases analyzed in [16; 17; 19; 20]. However, it will then be proven that the absolute value of the total vector changes if successive sampling windows of a signal with several nearby tones are analyzed. The spectral leakage of these tones may interact, as it is the case in signals composed by a harmonic tone and an interharmonic tone generating leakage. This variability is explained by the addition of the leakage vector to the fixed harmonic tone vector. The leakage vector rotates because its offset at the beginning of each window change. An example was proposed, based on one described in the standard IEC 61000-4-7, Appendix C.4, to analyze the interaction between harmonic tones and the leakage produced by nearby interharmonics, and their relationship with time aggregation. A voltage waveform (Figure 1) with a 5 th harmonic of 13.2 V, a 6 th one of 10 V, and an interharmonic of 9.8 V at 253.3 Hz is considered. This interharmonic was added to the example of the standard since it is close to the 5 th harmonic and far from the 6 th one. This way, the effects of the relative situation between the leakage receiver harmonics and the interharmonics producing the leakage could be better observed. Figure 1 shows the waveform of the signal in the time domain and the spectrum obtained using the DFT with a 0.6 s rectangular window. As the window used contained complete periods of all the components of the signal, no spectral leakage was observed. This ideal window was selected considering that the minimum common multiple of 250, 253.3, 300, and 5 Hz is 1.6 Hz. This ideal period (_Tw_) was also chosen as a multiple of 1/5 Hz = 0.2 s so it could be synchronized with the window size proposed in the standard. However, the harmonic composition of a signal is usually unknown, and following IEC standards, it was analyzed in successive short windows of 0.2 s, with leakage generally appearing in each of them. Therefore, total vectors with varying amplitudes in each successive short window analyzed were obtained, due to the combination of representative vectors, which rotate at different speeds. The reason for choosing this short window size was the non-stationary nature of the signals analyzed and the time-resolution improvement at the cost of worsening the frequency resolution (uncertainty principle). These changes of amplitude are illustrated in Figure 2a with the spectra for the first three short windows of the example of Figure 1. It can be observed how the total amplitude measured at the 250 Hz spectral bar changed since in that position the 5 th harmonic and the leakage received from the close interharmonic at 253.3 Hz were added. The resulting vector oscillated between the sum of the modules or their difference, depending on the phase angle between the leakage vectors and the harmonic.