## 1 Introduction
For continuous periodic signals (vibrations, variable part of instantaneous angular speed, currents and, rarely, instantaneous electrical power [34]), the conversion from the time domain to the frequency domain using the Fast Fourier Transform spectrum is widely used [2; 6; 8; 9; 17; 21; 34] to describe the sinusoidal components within (average amplitudes and frequencies). More commonly, the wavelet transform is used to detect and describe short (transient) sequences (based on local spectral information of components within these signals [2; 8; 9; 13; 14; 17; 18; 20; 26; 27; 28; 29; 37; 38; 39]. Actually, the use of artificial intelligence techniques (e.g., based on neural networks [2; 5; 11; 13; 18; 26; 38; 40], support vector machines [2; 13], machine learning algorithms [3; 5; 8; 10; 14; 29; 36], deep learning algorithms [11]) or the use of IoT [16; 38] for early detection [1; 4; 10; 14; 15; 41], possibly online [8; 9; 17], of motors states that may develop abnormally has become increasingly attractive. Table 1 briefly summarises the previously discussed topic of bibliographic references from the current state of scientific research in motor condition monitoring. There is a general opinion in the scientific literature that analysing the variation over time of the instantaneous current absorbed by a phase of an asynchronous three-phase induction motor (particularly with a squirrel-cage rotor) is one of the best ways of monitoring its condition. For incomprehensible reasons (with a few exceptions, e.g., [25; 34]), many scientists ignore the fact that there is an electrical input parameter that helps to describe the behaviour of this electric motor running with no load better, more accurately and more completely than the instantaneous current: the instantaneous electrical power (IEP). This IEP is defined as the product of the instantaneous voltage (IV) and the instantaneous current (IC); both (in the case of this motor running with no load) are periodic evolutions whose fundamentals are sinusoidal, time-shifted or phase-shifted. The IEP defines the instantaneous active electrical power (IAEP) absorbed by the motor (partiallyconverted into mechanical power required to turn the rotor), and the IAEP defines the active electrical energy absorbed by the motor (partially converted into mechanical energy). All the variable electrical and (especially) mechanical phenomena involved in the no-load operation of a three-phase AC induction motor should be mandatory and reflected in the variation over time of the IAEP. If the mechanical load on the rotor increases (e.g., due to a fault of its condition), the amplitude of the fundamental of the IC increases and--very importantly--the phase shift decreases; consequently (by cumulative effect), the IAEP also increases and vice versa. This cumulative effect explains why IEP (and also IAEP) is more suitable than IC for motor condition monitoring. Furthermore, IC alone cannot describe, for example, an operating situation with a negative IAEP flow. This situation occurs when the rotor of this motor is externally mechanically forced to rotate at a speed higher than the synchronous speed or during electrical braking periods. The motor acts as a generator, providing IAEP and active electrical energy. When we talk about describing variable mechanical phenomena through the evolution of IEP or IAEP, we must take into account the dynamic aspects (mainly damping and phase-shifting effects) due to the fact that the rotor (with mechanical inertia [19] and developing