## 2 Impedance Measurement in Electrical Drive Systems
### Impedance Characterization Methods  
Impedance measurement is traditionally performed on a stationary machine disconnected from any power source using an impedance analyzer [24; 25; 26; 27], i.e., a measurement device dedicated to characterizing the winding impedance. Multiple instruments are  
Figure 1: Connection and measurement circuit for CM (**a**), DM (**b**), phase-to-phase (**c**), and phase-to-neutral (**d**) impedance characterization in a star-connected three-phase machine. Figure 3: Connection and measurement circuit for phase-to-phase impedance characterization in an open-end winding three-phase machine. Figure 2: Connection and measurement circuit for CM (**a**), DM (**b**), and phase-to-phase (**c**) impedance characterization in a delta-connected three-phase machine. usually required to cover a wide frequency range. The reason for this is that different calculation approaches must be implemented for certain frequency ranges in order to guarantee the best possible accuracy. It should be noted that the calibration process is a very important step in obtaining consistent and accurate results. The vector network analyzer we used in our study performs a so-called \"Gain/Phase\" calculation over a frequency range [5 Hz-30 MHz] to determine the impedance characteristic of a machine [28]. The approach consists of the application of a sine wave excitation and the measurement of voltages at two different points of a system to evaluate the impedance of a circuit at a given frequency. The instrument then scans multiple excitation frequencies while storing the various input and output signals obtained in order to construct a Bode diagram of the impedance over the selected frequency range. Another approach to offline characterization of stator windings requires an oscilloscope to measure the amplitudes of the voltage across a winding and the current flowing through it, as well as the temporal phase shift between these two signals. Then, by applying Ohm law, we can calculate the amplitude and the phase of the winding impedance at a given frequency [17]. The winding must be connected to a low-frequency voltage generator (LFVG), which provides a sweep of frequencies from 1 kHz to 30 MHz to extract the Bode diagram of this impedance. The connection and measurement scheme of this method are shown in Figure 4. This approach can therefore be implemented using fairly common electrical engineering equipment. However, it is a time-consuming process that is difficult to automate and requires numerous measurement points, especially for frequencies above megahertz, to ensure the same accuracy as the results obtained with the impedance analyzer. Moreover, the accuracy of the characterization around certain resonances may be limited by the current amplitude that the LFVG can deliver. The main disadvantage of these characterization methods is that they must be performed on a machine that is disconnected from the power supply. This can be particularly challenging in an industrial application. However, online approaches have the advantage of providing continuous information measurement without interrupting the drive's nominal operation. When monitoring the health of windings in an electrical machine, behavioral deviations synonymous with aging can therefore be identified as early as possible. The online approach to impedance characterization most widely used in the literature involves superimposing an HF voltage signal on the machine's main power supply. For this purpose, active [29; 30], passive [31; 32], or inductive [33] coupling systems are used. An equivalent circuit of the injection and measurement principle taken from [29] is shown in Figure 5.  
Figure 4: Connection and measurement scheme for impedance characterization, using a LFVG and an oscilloscope—redraw from [17]. Based on Kirchhoff laws, the impedance for each excitation frequency can be calculated. The derived expression is given by (1).