## 2 Materials and Methods
### A Description of the Instantaneous Active Electrical Power
The proposed FASS acts as a frequency-selective pass filter since it eliminates all the variable components of the VIAEPf that are not harmonically correlated with the fundamental frequency (\(1/T_{p}\)) of the periodic phenomenon being analysed. Obviously, the pattern of a variable periodic phenomenon within the VIAEPf can only be correctly extracted if (at least) its fundamental frequency is constant. For practical reasons, it is appropriate to calculate the samples of PCRRF using FASS according to Equation (3), where \(k=1\) corresponds to the moment when an angular marker on the rotor (e.g., the marker 2 from Figure 3) is passed through the origin (line 3 of the incident laser beam, the same Figure 3). Thus, by graphically superimposing the PCRRF patterns at different times or under different conditions of motor running with no load, similarities and possible differences can be seen. Also, in this way, a PCRRF can be described and studied not in terms of time but in terms of the angular position of the rotor (the pattern is described graphically with IAEPf on the \(y\)-axis and the angular position \(\varphi_{pTp}[k\cdot\Delta t]\) of the rotor related by the angular origin on the \(x\)-axis, with \(\varphi_{pTp}[k\cdot\Delta t]=2\pi\cdot k\cdot\Delta t/T_{p}\)). The effectiveness of this FASS procedure can be demonstrated experimentally using a simulated VIAEPf signal. Let us suppose, for example, that this simulated signal can be described as the sum of 2000 sinusoidal components, having frequencies uniformly distributed between 0 and 1000 Hz, with the same amplitude (1W) and uniformly distributed phases at the origin of time. A sample of this simulated signal is generically described as follows:  
\[P_{fp}[s\cdot\Delta t]=\sum_{j=0}^{2000}\sin(2\cdot\pi\cdot\frac{j}{2}\cdot s \cdot\Delta t+\frac{j}{2}\ ) \tag{4}\]  
The variation of this simulated VIAEPf signal  
The samples of a pattern of a periodic component (from simulated VIAEPf) with the fundamental frequency of 33.5 Hz (and harmonics at 2-33.5 Hz, 3-33.5 Hz,... 29-33.5 Hz) can be found using the FASS method shown in Equation (3), with \(T_{p}\) = 1/33.5 s, \(n\) = 2985 and \(m\) = 600 (an optional value). Figure 8 shows this pattern (in blue). Overlaid is the simulated pattern (in red), with a generic sample described with  
\[P_{fvsp}[s\cdot\Delta t]=\sum_{l=0}^{29}\sin[33.5\cdot l\cdot(2\cdot\pi\cdot s \cdot\Delta t+1)]\text{ with }s=1\div n \tag{5}\]  
As a result of substitution, \(j/2\) = 33.5\(\cdot l\) in Equation (4). There appears to be a very good fit between the two patterns (simulated in Equation (5) and found by FASS of the signal generated with Equation (4)). As expected, due to the numerical description of VIAEPf, the overlap is not perfect (as shown in detail B, zoomed in at the top left of Figure 8). The bottom left area of this figure shows the variation over time of the difference between the simulated pattern (Equation (5)) and the pattern found by FASS (Equation (3)) of simulated VIAEPf (Equation (4)). This difference is not zero (at any moment in time), mainly because of the phase shift between the two patterns. This convincing result leads to the conclusion that the FASS method can also be successfully used to determine the shape of any available periodic pattern within a variable signal (in particular, the PCRRf) within a real VIAEPf absorbed by a motor running with no load. In order to apply the FASS (from Equation (3)), it is necessary to solve two essential requirements:  
* The exact determination of the value of \(T_{p}\), and \(n\), with \(n=\left\lfloor T_{p}/\Delta t\right\rfloor\). Of course, \(T_{p}\) is the period of a complete rotation of the rotor;
* The exact determination of a moment of time when the angular mark placed on the rotating rotor (2 in Figure 3) passes through the angular origin (when \(k\) = 1). This moment of time should preferably be chosen at the very beginning of the VIAEPf signal.