## 2 Materials and Methods  
### Experimental Setup  
There is a very simple way to obtain the information needed to find the IEP absorbed by a three-phase AC induction asynchronous motor with squirrel-cage rotor, which is based on the considerations shown in Figure 1 (describing a setup partially used previously in [42] and similar to [25; 34]).  
It is reasonable to assume that the three-phase network (with theoretically pure sinusoidal voltages, 50 Hz frequency) is electrically balanced and that the IEP passing through the motor on each phase (A, B and C) is the same (one-third of the total IEP). The total IEP absorbed by the motor can be determined as three times the instantaneous power on a single phase (A, in Figure 1).  
The primary winding of an instantaneous current transformer (ICT, labelled with 1 in Figure 2, which shows a partial photo of the setup) is supplied by phase A. It is reasonable to suppose that this transformer (acting as a current sensor) provides an IC description of the current (\(i(t)\) that passes through phase A) on the secondary winding (as a proportional voltage \(u_{CT}(t)\)), which is delivered through the channel 1 (Ch1) of a PCB USB oscilloscope PicoScope 4824A (2 in Figure 2[43]) to a personal computer (PC, 3 in Figure 2). The primary winding of an instantaneous voltage transformer (IVT, 4 in Figure 2) is placed between phase A and the null wire (neutral line) N. This transformer (acting as a voltage sensor) provides the IV description of the voltage (\(u(t)\), between phase A and neutral wire) on the secondary winding (as a proportional voltage \(u_{VT}(t)\)), which is delivered through the channel 2 (Ch2) of the oscilloscope to PC. The PC also receives a signal \(u_{SL}(t)\) from a laser position sensor ILD 2000-20 (1 in Figure 3) through the third channel (Ch3) of the oscilloscope. This signal is used to accurately describe the moment in time when an angular marker (2 in Figure 3) passes through the angular rotation origin (line 3 of the incident laser  
Figure 1: A conceptual description of the experimental setup.  
beam, Figure 3) during its rotation. This angular marker is placed on a jaw coupling hub 4. In Figure 3, a permanent magnet (5) is also placed on this jaw coupling hub. In some experiments, it is used to create a temporary mechanical dynamic imbalance of the rotor.  
Two different theoretical synchronous speeds (750 or 1500 rpm obtained with 2 or 4 pairs of poles at 50 Hz frequency) are available for this motor, with nominal active electrical powers of 2.2 KW or 1.1 KW.  
Figure 4 shows a view with the three signals acquired simultaneously during no-load operation at the theoretical synchronous speed of 1500 rpm: 1--a description \(u_{VT}(t)\) involved in IV (\(u(t)\)) description as \(u(t)=k_{VT}\cdot u_{VT}(t)\); 2--a description \(u_{CT}(t)\) involved in IC (\(i(t)\)) description as \(i(t)=k_{CT}\cdot u_{CT}(t)\); and 3--the voltage \(u_{SL}(t)\) delivered by laser sensor.  
As expected, the IC description is delayed compared to the IV description; there is a negative phase shift between the fundamentals of IV and IC (the stator winding acts as an inductive consumer). It is obvious that IV and especially IC are not pure sine waves, partly because of the power supply network but mainly because of the behaviour (condition) of the motor running. Each passing of the angular mark placed on the rotor through the angular origin produces a strong negative peak on \(u_{SL}(t)\) signal (e.g., A or B); the time interval between two successive peaks corresponds to the period of a full rotation of the rotor.  
Figure 3: A view of the jaw coupling hub placed on the motor rotor.  
Figure 2: A view on transformers ICT (1) and IVT (4) on digital oscilloscope (2) and personal computer (3).  
In this work, the processing of the data and the signals was carried out using MATLAB.