# Quaternion Signal Analysis for Detection of Broken Rotor Fault Degrees in Induction Motors
## 2 Methodology
### Experimental Setup  
Our work is tested with a dataset of the workbench presented in [27] to validate its effectiveness. The experimental setup, illustrated in Figure 4, includes a three-phase induction motor and a direct current machine acting as a generator to simulate load torque. They are connected by a shaft with a rotary torque wrench. Additionally, various sensors are connected at different locations within the setup.  
The workbench consists of vibration, current, and voltage measurements from which we selected four vibration measurements. These are taken from various induction motor parts to analyze its mechanical behavior. This analysis enables the adaptation of four signals into a quaternion format. In past studies [22; 23], a combination of current and mechanical vibration signals was utilized due to a scarcity of signals. However, this work aims to use signals of the same type for a more consistent analysis.  
These four signals are acquired by four vibrocontrol uniaxial accelerometers model PU2001, located on the housing, on the driven side, and on the non-driven side to obtain tangential, radial, and axial vibration. The sensors present a sensitivity of 10 mV/mm/s, with a frequency range from 5 to 2000 Hz. The data acquisition system utilized in the experimental setup is a National Instruments (Austin, TX, USA) data acquisition card, model NI DAQmx PCIe-6259. This card features 32 analog input channels, each with a resolution of 16 bits and an acquisition rate of 1.25 MHz. Additionally, it includes four channels of 16-bit analog output, operating at a speed of 2.8 MHz, and 48 digital I/O channels capable of functioning at a clock frequency of up to 10 MHz.  
Figure 4: Experimental setup of the induction motor with one, two, and three BRBs [27].  
Figure 3: Polynomial and Fourier regressions.  
The signals \(V_{h}\), \(V_{d}\), \(V_{a}\), and \(V_{s}\) are acquired from a three phase induction motor of the W22 standard line from manufacturer WEG, with 1 hp, 220 V/380 V, 3.02 A/1.75 A, four poles, 34 rotor bars in the squirrel cage rotor, a rated speed of 1715 rpm, and a nominal torque of 4.1 Nm. The database includes two additional vibration signals. These signals were tested using various combinations, but the resulting statistics were insufficient for proper characterization. Therefore, the signals that yielded the best results were those recorded from the engine chassis. Different directional measurements were selected to enhance the results. The placement of the sensors and the orientation of the signals are shown in Figure 5.  
The induction motor is driven by a control panel, for which the type of drive can be selected as star or triangle, and the type of supply as direct mains voltage or via a three-phase inverter. A shaft containing a rotary torque wrench Transtec model MT-103 (Arlington, VA, USA) connects the motor with a direct-current machine, simulating the load in the motor of 100% of full load in all tests.  
The workbench shows between one and four adjacent rotor bar failures caused by 6 mm diameter drilling in the squirrel cage rotor to ensure that the rotor bar was completely drilled alongside the healthy state by means of a bench drill mounted with the tip centered at half the longitudinal length of the rotor, as is shown in Figure 6.  
We selected the sets of healthy and from one to three BRBs because they presented a gradual increase in the results, while the four BRB states provoked regressions with instability.  
The dataset presents ten repetitions of each state acquired during 18 s with a sample rate of 8.326 kHz. We selected thirty thousand samples in a steady state of the four vibration signals to test.