# Electrical Monitoring under Transient Conditions:
## 2 Foundations of the Transient Analysis of Electrical Quantities
### Transient Analysis of Motor Currents  
The transient-based methodology described before was firstly applied to the analysis of transient motor currents. The procedure relies on capturing the current demanded by the motor under transient operation of the machine. Although the methodology has been successfully applied to different types of transient currents (load variations, plugging stop, deceleration, supply frequency changes...), the starting transient is the one that has summoned most of the attention. Many works in the last decade have deepened the investigation and characterization of the evolutions of the fault harmonics present in the starting current for the detection of different faults [8, 9, 10]. The main advantages of the analysis of starting currents are derived from the benefits that are inherent to the analysis of motor currents: unlike other quantities, the current can be measured without the need to access the motor (e.g., in the motor control center (M.C.C) or motor panelboard), hence enabling the remote monitoring of the motor condition. Moreover, the measurement can be carried out in the secondary windings of current transformers (CTs) that are employed for measurement devices with maximum safety and comfortability conditions. The equipment needed for the signal acquisition is simple: a single current clamp and an oscilloscope with minimum requirements are enough, and only one phase current is required. Finally, the current analysis enables a broad fault coverage (bearing always in mind that no quantity enables the integral monitoring of the whole motor condition) [18]. The analysis of starting currents begins with the registration of the current waveform under a starting of the electric motor; this signal is later transferred to a computer for its analysis. The application of suitable time-frequency transforms to this signal enables obtaining the t-f maps, where the possible evolutions of fault components will be identified. Moreover, appropriate fault severity indicators are computed based on certain parameters of these maps (for instance, the energy density at strategic t-f regions), which enable determining the level of failure. Depending on the fault that is diagnosed, different harmonic evolutions must be identified in the maps. For instance, in the case of rotor damages, the most relevant evolutions are those of the LSH (V-shaped pattern described above) and that of the _Upper Sideband Harmonic_ (USH), the frequency of which starts at \(3f\) and ends at near \(f\) at steady-state. However, apart from these, there are different families of harmonics that are also amplified by this failure, and since each harmonic has its own evolution, there will be additional patterns caused by these components that may serve to ratify the presence of the fault [8]. On the other hand, in case of eccentricities or misalignments, rather different components evolutions will appear. The most relevant are those of the components \(f-f_{r}\) and \(f+f_{r}\) that evolve from the supply frequency (\(f\)) toward their final frequencies at steady state (approximately \(f/2\) and \(3f/2\)) for a machine with two pole pairs) [19]. Figure 2 shows the expected evolutions for the most relevant harmonics associated to rotor damages (denoted by \(f_{\rm rot}\)) and eccentricities/misalignments (denoted by \(f_{\rm ecc}\)) that are present in the motor starting current, considering a direct on-line starting. Note that, similarly with what happened with rotor damages or eccentricities, other types of failures will yield their corresponding evolutions under starting. This is the case of bearing failures or even coupling system problems (faults linked to belt/pulleys systems or gear reducers) [20]. This confers an additional potential to the approach, since it may be applied to a wide range of potential motor faults. It is also important to remark that the aforementioned evolutions are valid for the case of direct on-line starting of electric motors.