## 2 Materials and Methods
### A Description of the Instantaneous Active Electrical Power
A Method of Determining a Pattern in IAEPf Variation over Time Useful in Motor Condition Characterisation  
There are two very simple ways of using IAEPf to monitor the condition of a three-phase AC induction asynchronous motor: 1, based on the variation of IAEPf over time at the start (more precisely, calculation of the active electrical energy absorbed during the start phase, ESP), and 2, based on the determination of the average IAEPf value running with no load (as constant part of IAEPf, or CAEPf). There is also a third, more interesting approach. The mechanical power required to rotate the rotor of the motor with no load has a constant part (firmly reflected in CAEPf) and a variable part (also reflected in IAEPf), VIAEPf. The constant and variable parts of the IAEPf (CAEPf and VIAEPf) associated with the rotation of the rotor are absorbed by the motor from the electrical supply and converted partially into mechanical power, which is used to rotate the rotor with no load. Obviously, this VIEPf is a sum of variable components, some of which are periodic. One of these components is expected to have a period equal to the period of rotation of the motor rotor. The time evolution of this periodic component reflects all the electrical and mechanical phenomena associated with the rotation of the rotor. One period of this component can be considered a pattern (PCRRf, periodic component on rotor rotation), useful (by its shape and size) for monitoring and diagnosing the state of the motor. Assuming that there are no electrical phenomena associated with the rotation of the rotor, the peak-to-peak amplitude of this PCRRf should be zero if the rotor is in perfect mechanical condition (in terms of bearings, dynamic mechanical balance, vibration, etc.). Otherwise, this amplitude of the PCRRf is not zero, and the pattern shape and size can be an important indicator of the mechanical condition of the motor (from the point of view of its rotor rotation). A method for determining and synthetically describing the PCRRf (as pattern recognition) is proposed below. The synthetic description of the pattern of a periodic phenomenon (in particular the PCRRf) within a periodic signal (in particular the IAEPf), with the period of this phenomenon \(T_{p}\) and \(n\) samples per period (where \(n\) is defined as \(n=\left\lfloor T_{p}/\Delta t\right\rfloor\)), is obtained by a special type of filtering by averaging of several selected samples (as FASS) of the variable part of the IAEPf (as VIAEPf, obtained by removing CAEPf from IAEPf, with samples represented as \(P_{p}\!\!\left[s\cdot\Delta t\right]\)). A sample \(P_{pTp}\!\!\left[s\cdot\Delta t\right]\) (with \(k=1\div n\)) of PCRRf is obtained by arithmetic averaging of \(m\) uniformly selected samples of VIAEPf or as the average of these VIAEPf samples: \(P_{p}\!\!\left[k\cdot\Delta t\right]\), \(P_{pfl}\!\!\left[(n+k)\cdot\Delta t\right]\), \(P_{p}\!\!\left[(2\cdot n+k)\cdot\Delta t\right]\), \(P_{p}\!\!\left[(3\cdot n+k)\cdot\Delta t\right]\), \(P_{pfl}\!\!\left[(3\cdot n+k)\cdot\Delta t\right]\), \(\ldots\ldots\), \(P_{f\!\!\left[l(m-1)\cdot n+k\right]\cdot\Delta t\right]\). There are \(n\) unselected VIAEPf samples between each of the two  
Figure 6: A short sequence showing the variation over time: 1—IAEPf; 2—IAEPc; 3—IEP. successive samples selected for FASS. In other words, if VIAEPf has at least \(m\cdot n\) samples, a sample \(P_{pTp}[k\cdot\Delta t]\) of PCRRF is mathematically calculated by FASS as  
\[P_{pTp}[k\cdot\Delta t]=\frac{1}{m}\sum_{i=1}^{m}P_{fp}\{[(i-1)\cdot n+k]\cdot \Delta t\}\text{ with }k=1\div n \tag{3}\]  
The samples \(P_{pTp}[k\cdot\Delta t]\) of any pattern of a periodic phenomenon within the IAEPf (in particular, the PCRRF) depend rigorously on the values of the period \(T_{p}\) and the sampling time \(\Delta t\) and are calculated on the basis of the number \(n\) (with \(n=\left\lfloor T_{p}/\Delta t\right\rfloor\)) and the optional value of \(m\) (in a first approach, the bigger, the better), according to Equation (3).