## 2 Materials and Methods
### A Description of the Instantaneous Active Electrical Power  
These sinusoidal components that were mentioned earlier (also partially revealed in [34]) are not involved in the description of motor condition and (for the purpose of motor monitoring) should be eliminated from IEP (e.g., ideally by using a notch filter with multiple narrow stop bands centred on the frequency of these components). Removing these sinusoidal components from the IEP gives the instantaneous active electrical power (IAEP). The IAEP is mainly related to the active mechanical power used to rotate the motor rotor (here in no-load operation). The following is a proposal for a method to obtain a description of the IAEP (as IAEPf) through the filtering of the IEP. If we consider (as is undertaken in PC calculus) that the total IEP \(p(t)\) is approximated by numerical samples, with the \(s\)th sample written as \(p[s\cdot\Delta t]=3\cdot k_{VT}\cdot k_{CT}\cdot u_{VT}[s\cdot\Delta t]\cdot u _{CT}[s\cdot\Delta t]\), with sampling time \(\Delta t\) (with \(s=\lfloor t/\Delta t\rfloor\)), a simple way to eliminate all these unwanted components is to use a numerical backwards-moving average filter (BMAF) with a suitable number \(h\) of samples in the average. An output sample \(P_{f}[s\cdot\Delta t]\) from BMAF (as an IAEPf sample as well) is defined as  
\[P_{f}[s\cdot\Delta t]=\frac{1}{h}\sum_{j=0}^{h-1}p_{s}[(s-j)\cdot\Delta t]\text { with }s\geq h \tag{1}\]  
The number \(h\) should be determined in relation to the period (\(T_{IV}=50^{-1}\)\(s=0.02\) s) of fundamental component A (\(T_{IV}\) being the period of IV), as \(h=\lfloor T_{IV}/\Delta t\rfloor\). The IEP with the FFT spectrum partially described in Figure 5 has the sampling time \(\Delta t=10\) us, so \(h=2000\). In Figure 5, the transmissibility curve of a BMAF under these conditions is shown in red. Since it is superimposed on the FFT spectrum of IEP, it is clear that the transmissibility (_Tr(f)_) is zero exactly at the frequencies of the unwanted sinusoidal components, which are certainly removed by this averaging (with BMAF acting as multiple narrow stop band filter). Unfortunately, for many other frequencies (mainly related to the state of the motor), BMAF also acts as a strong attenuator. However, by multiplying the local (peak) FFT amplitude by the inverse of the transmissibility (\(1/\)_Tr(f)_, at the frequency of attenuated component), their real amplitude can be found. An important advantage of this IAEPf definition is that it maintains the same sampling time \(\Delta t\) as the IEP. In classical acceptance [44], the instantaneous active electrical power (IAEPc) is calculated as the average value of the IEP over a period of IV. The \(g\)th sample \(P_{c}[g\cdot T_{IV}]\) of IAEPc is calculated as the average of the \(h\) IEP samples during the time span of the \(g\)th period of IV as  
\[P_{c}[g\cdot T_{IV}]=P_{c}[g\cdot h\cdot\Delta t]=\frac{1}{h}\sum_{s=(g-1) \cdot h+1}^{g\cdot h}p_{s}[s\cdot\Delta t] \tag{2}\]  
The sampling time of IAEPc is \(T_{IV}\), which is \(h\) times higher than IEAPf (as \(T_{IV}=h\cdot\Delta t\)). In other words, the sampling rate (\(1/T_{IV}\)) of the IAEPc is \(h\) times smaller than the sampling time (\(1/\Delta t\)) of the IAEPf, which is a big disadvantage. In Figure 6, curve 1 represents a short sequence of IAEPf (as a result of IEP filtering using Equation (1)), curve 2 represents a short sequence of IAEPc (as a result of IEP averaging using Equation (2)), while a short sequence of IEP is represented by curve 3. Because the amplitude of the dominant component A\({}_{1}\) in IEP is large (1164 W, Figure 5), IEP appears here as a series of parallel lines. The graphical representations of the powers are made by a series of successive line segments bounded at the ends by two points, with each point corresponding to a sample. A high description rate (as for IAEPf) means segments of small length and vice versa (as for IAEPc). It is obvious that IAEPf is much more suitable for motor condition monitoring than IAEPc due to its higher sampling rate.