# Online High Frequency Impedance Identification Method of Inverter-Fed Electrical Machines for Stator Health Monitoring

Jérémy Creux, Najla Haje Obeid, Thierry Boileau, Farid Meibody-Tabar

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Jérémy Creux, Najla Haje Obeid, Thierry Boileau, Farid Meibody-Tabar. Online High Frequency Impedance Identification Method of Inverter-Fed Electrical Machines for Stator Health Monitoring. Applied Sciences, 2024, 14 (23), pp.10911. 10.3390/app142310911. hal-04821567

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Article

# Online High Frequency Impedance Identification Method of Inverter-Fed Electrical Machines for Stator Health Monitoring

Jérémy Creux<sup>1,2 ID</sup>, Najla Haje Obeid<sup>1 ID</sup>, Thierry Boileau<sup>2,* ID</sup> and Farid Meibody-Tabar<sup>2</sup>

**Abstract:** In electric powertrain traction applications, the adopted trend to improve the performance and efficiency of electromechanical power conversion systems is to increase supply voltages and inverter switching frequencies. As a result, electrical machine conductors are subjected to ever-increasing electrical stresses, leading to premature insulation degradation and eventual short-circuits. Winding condition monitoring is crucial to prevent such critical failures. Based on the scientific literature, several methods can be used for early identification of aging. A first solution is to monitor partial discharges. This method requires the use of a specific measurement device and an undisturbed test environment. A second solution is to monitor the inter-turn winding capacitance, which is directly related to the condition of the insulation and can cause a change in the stator impedance behavior. Several approaches can be used to estimate or characterize this impedance behavior. They must be performed on a machine at standstill, which limits their application. In this paper, a new characterization method is proposed to monitor the high-frequency stator impedance evolution of voltage source inverter-fed machines. This method can be applied at any time without removing the machine from its operating environment. The range and accuracy of the proposed frequency characterization depend in particular on the supply voltage level and the bandwidth of the measurement probes. The effects of parameters such as temperature, switching frequency, and DC voltage amplitude on the impedance characteristic were also studied and will be presented. Tests carried out on an automotive traction machine have shown that the first two series and parallel resonances of the high-frequency impedance can be accurately identified using the proposed technique. Therefore, by monitoring these resonances, it is possible to predict the aging rate of the conductor.

**Keywords:** winding health monitoring; frequency response analysis; online characterization; impedance; high-frequency behavior; inverter-fed electrical machines

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**Citation:** Creux, J.; Haje Obeid, N.; Boileau, T.; Meibody-Tabar, F. Online High Frequency Impedance Identification Method of Inverter-Fed Electrical Machines for Stator Health Monitoring. *Appl. Sci.* **2024**, *14*, 10911. https://doi.org/10.3390/app142310911

Academic Editors: Jose M. Guerrero, Araitz Iturregi and Sergio Toscani

Received: 31 October 2024

Revised: 19 November 2024

Accepted: 21 November 2024

Published: 25 November 2024

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**Copyright:** © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 1. Introduction

The health monitoring of stator windings in electrical machines is one of the most important challenges in the automotive traction industry. Increasing supply voltages and inverter switching frequencies lead to accelerated aging of these windings [1]. Such aging is characterized by the degradation of insulation layers between conductors. Factors such as the duration and level of exposure to electrical stress influence the quality of these insulators [2]. In order to reduce the risks of premature insulation degradation and to increase winding robustness, several techniques can be considered: improving the conductor impregnation or using enamels that are particularly resistant to partial discharge phenomena [3]; promoting the design of hairpin winding machines that help reduce vibration phenomena in the winding heads [4]. However, it remains essential to monitor the health of these windings to identify their aging as early as possible. Therefore, if a given signature were to exceed a certain threshold, preventive maintenance could be considered to protect the system and reduce maintenance and repair costs and times [5,6].

<footnote n="1">¹ IFP Energies Nouvelles, Institut Carnot IFPEN Transports Energie, 1 et 4 Avenue de Bois-Préau, 92852 Rueil-Malmaison, France; jeremy.creux@ifpen.fr (J.C.); najla.haje-obeid@ifpen.fr (N.H.O.)</footnote>

<footnote n="2">² Université de Lorraine, CNRS, LEMTA, F-54000 Nancy, France; farid.meibody-tabar@univ-lorraine.fr</footnote>

<footnote>* Correspondence: thierry.boileau@univ-lorraine.fr</footnote>
There are several approaches to monitoring the health of an electrical machine winding. First, partial discharge measurement can be highlighted [7–9]. Some strategies use insulation resistance measurement and polarization index calculation to perform the diagnosis [10,11]. Estimation of equivalent phase-to-ground or interturn capacitance [12,13] may also be conducted. The main drawback of these methods is that they are usually performed on a machine at standstill, disconnected from the power supply. In some applications, particularly industrial applications, these test conditions cannot be met, making aging monitoring impossible. Furthermore, resistance tests can give an indication of windings state of health [14], but since these tests are destructive, they cannot be applied industrially. Statistical studies are used in [15] to estimate the remaining useful life (RUL) of a system. Finally, in recent literature, many aging monitoring methods are based on the analysis of systems frequency response to determine the winding condition. The various approaches used are usually based on the variation of the frequency signatures of the machine impedance characteristic [16–19]. This characterization over a wide frequency band takes into account leakage and magnetization inductances, iron and eddy current losses, as well as interturn and phase-to-ground capacitive couplings. As a result, the different resonance frequencies reflect these different phenomena.

It has been shown in [12] that the decrease of the winding breakdown voltage correlates with an increase of the machine equivalent capacitance. Furthermore, ref. [16] explains that this equivalent capacitance is particularly impacted by the evolution of the capacitance between windings. It can therefore be concluded that its monitoring will help to detect the aging of a machine inter-turn insulator at an early stage. Finally, the model developed in [20] shows that this inter-turn capacitance has a strong influence on the impedance frequency behavior of the machine. As a result, this characteristic will evolve as the machine age. Ref. [16] emphasizes that the first resonance frequency of the impedance characteristic decreases with time, indicating degradation of the winding insulation. It has also been observed in [18] when the windings are subjected to adverse environmental stresses, such as an exposure to a corrosive substance on their insulators. The resonance frequency decreases with prolonged exposure to the corrosive agent. The amplitude of this resonance also decreases as the conductors age. All of these results demonstrate the benefits of monitoring the evolution of frequency impedance over time to assess the aging of electrical machine windings.

Based on the above considerations, we decided to develop an online characterization and diagnostic method for monitoring the health of stator windings of electric traction machines. The advantage of an online approach is that it provides a continuous flow of information within the functional environment of the drive and without interrupting its operation. In particular, this approach can be used to estimate the “remaining useful life” (RUL) of the winding [21–23]. Therefore, an accurate and efficient condition-based maintenance strategy can be established to ensure the availability of the system.

This paper is organized as follows: Section 2 presents a review of the different types of impedance measurements in electrical machines and the associated characterization methods. Section 3 focuses on the broadband frequency behavior of these machines and reviews the different approaches found in the literature to model their high-frequency (HF) behavior. Section 4 explains the online strategy we have developed to characterize this frequency response without removing the machine from its operating environment. Section 5 examines the impact of various factors, such as operating temperature, supply voltage level, and the presence of the rotor in the machine, on the impedance frequency response of the machine. Finally, Section 6 draws conclusions from the paper and discusses future works.

## 2. Impedance Measurement in Electrical Drive Systems

### 2.1. *Electrical Machine Impedance Definition*

Several types of impedance measurements can be performed on a permanent magnet synchronous machine (PMSM). The most common types in the literature are common-
mode (CM) and differential-mode (DM) measurements. On an electrical machine with a star-connected winding, these CM and DM measurements are carried out by connecting the different phases of the machine, as shown in Figure 1a,b. For condition monitoring purposes and to locate the fault in the windings, it is possible to successively analyze the behavior of two of the machine phases, as shown in Figure 1c. Finally, with access to the neutral point, it is possible to characterize the behavior of each phase individually, as described in Figure 1d.

<figure><img src="image_4.png" /><figcaption>**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.</figcaption></figure>

Similarly, for machines with delta-connected windings, the CM and DM impedance behaviors are measured by connecting the machine phases as shown in Figure 2a,b. Figure 2c describes the connection scheme for characterizing the impedance behavior of each of the machine phases.

<figure><img src="image_5.png" /><figcaption>**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.</figcaption></figure>

Finally, in the case of an open-end winding, the behavior of each machine phase can be straightforwardly characterized according to the measurement scheme of Figure 3.

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Connection and measurement circuit for phase-to-phase impedance characterization in an open-end winding three-phase machine.</figcaption></figure>

## 2.2. Impedance Characterization Methods

Impedance measurement is traditionally performed on a stationary machine disconnected from any power source using an impedance analyzer [24–27], i.e., a measurement device dedicated to characterizing the winding impedance. Multiple instruments are
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.

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Connection and measurement scheme for impedance characterization, using a LFVG and an oscilloscope—redraw from [17].</figcaption></figure>

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><img src="image_8.png" /><figcaption>**Figure 5.** Equivalent circuit for online impedance measurement and characterization, using HF signal injection—taken from [29].</figcaption></figure>

Based on Kirchhoff laws, the impedance for each excitation frequency can be calculated. The derived expression is given by (1).

$$Z = \frac{V_2}{I} = R_{sh} \cdot \frac{V_2}{V_1 - V_2} \qquad (1)$$

where Z is the impedance amplitude of the system to be characterized, $V_1$ and $V_2$ are the voltage amplitudes measured at the HF generator and machine terminals respectively and $R_{sh}$ is the shunt resistor used to determine the amplitude of the current I.

One of the disadvantages of this approach is its invasiveness, as external HF injection has a significant impact on the nominal operation of the machine and can even accelerate the aging of its windings.

### 3. Broadband Stator Model for Data Storage

For health monitoring, comparison of a large number of data sets is needed. This requires the ability to store them in the form of models with a limited number of parameters. This would allow to reconstruct the impedance characteristic and keep only the essential information for aging assessment. In this section, the impedance frequency response will be first displayed to highlight the information that needs to be stored. Then, the various models suitable for condition monitoring that meet the aforementioned conditions are detailed.

#### 3.1. Impedance Frequency Response

As mentioned in the previous section, common-mode and differential-mode impedance behaviors are generally characterized and modeled. These works address electromagnetic interference and compatibility issues [34–36], particularly during the interconnection of the various subsystems that form an electromechanical powertrain. The impedance characteristics outlined are intended to evaluate the CM and DM currents that will circulate and then to size the components that will limit them. However, there are also approaches to characterize the impedant or admittant behavior of the machine, carried out from a health monitoring perspective [17,37].

The impedant behavior is generally displayed as a Bode diagram over a wide frequency range (from hundreds of hertz to several tens of megahertz), as shown in the example in Figure 6 taken from [25]. In this example, the low-frequency behavior of CM is purely capacitive, while that of DM is mostly inductive. Moreover, at higher frequencies, there are numerous resonant frequencies that represent the wealth of the information provided by this characteristic.

Ref. [24] explains that the impedance characteristics of mechatronic systems are largely impacted by parasitic effects such as inductive loops, capacitive couplings, or dissipative phenomena. Consequently, it is difficult to accurately predict and model this behavior during the design phase of these systems because there are too many manufacturing
uncertainties. Therefore, most studies are based on actual measurements of the impedance
behavior and attempt to capture and represent it in a simplified manner.

<figure><img src="image_9.png" /><figcaption>**Figure 6.** Example of Bode diagram depicting CM (a) and DM (b) impedance behaviors—taken from [25].</figcaption></figure>

3.2. *High-Frequency Modeling of PMSM*

Classical electrical equivalent circuit (EEC) models of electrical machines, either sound [38,39] or tailored to the study of interturn short-circuit faults [40], are external machine models. They can also be described as “low-frequency”, since high-frequency phenomena, which are of less importance for the purpose of the modeling, are generally neglected. As mentioned in the previous section and recalled in [41], the HF behavior is particularly sensitive to “parasitic” effects linked to the quality of the conductor insulation, which means that it must be properly modeled to monitor the evolution of the winding state of health. There are several approaches in the literature to model and/or characterize the HF behavior of a PMSM. These have interesting properties for identifying behavioral deviations associated with insulator aging. They can be divided into two categories: those based on a physical model of the machine phases and those based on an experimental frequency response of the machine behavior. The most popular are HF EEC models [19,25,34,42–45]. There are many such methods in the literature, mainly because they can be used to evaluate either overvoltages at the terminals of inverter-fed machines [46], CM currents circulating through bearings [47], or electromagnetic interference phenomena within power conversion chains [35]. The accuracy of these methods depends on their complexity, i.e., the number of parameters to be evaluated. Depending on the purpose of the model and the accuracy required, these parameters are calculated using hybrid methods that combine analytical and numerical calculations with experimental characterization of the machine behavior [48].

There are also transfer function approaches to the impedance frequency response of PMSM, as presented in [17,35]. Finally, there is an HF approach based on the physical design of the machine phases, which takes into account capacitive coupling phenomena between coils and with the machine ground [13].

The next parts will describe these different HF modeling approaches in detail.

3.2.1. PMSM HF Electrical Equivalent Circuit Model

This HF approach of EEC models is similar to conventional low-frequency circuits,
but the proposed EEC models are much more complex and take into account a number
of phenomena that are considered parasitic and which are neglected in the standard
approach. Thus, the complex association of electrical components (resistance, inductance,
and capacitance) connected in series or in parallel means that the following phenomena
can be taken into account:

* Interturn capacitive couplings;

* Phase-to-ground capacitive couplings;

* Leakage inductances in the first turns of the windings;

* Magnetizing inductances;

* Iron and eddy current losses.

The model complexity is determined by the purpose of the modeling and the accuracy required. The first approaches identified in the literature are based on π modeling of each machine phase, similar to what is used to describe the behavior of transmission lines [49]. For example, we can identify the HF EEC models developed in [34,43,50] and presented in Figure 7.

<figure><img src="image_10.png" /><figcaption>**Figure 7.** Examples of π-based HF EEC models of machine phases—taken from [50] (a), [34] (b), and [43] (c).</figcaption></figure>

Figure 8a compares the impedance frequency response estimated using the model pre-
sented in Figure 7a with its experimental profile for the three parallelized phases of the 750 W
induction machine studied in [50]. Figure 8b compares the experimental and estimated fre-
quency response characteristics, using the model described in Figure 7b of the CM (curve
with the lowest mean impedance over the considered frequency range) and DM (curve with
the highest mean impedance over the considered frequency range) impedances, of the 15 kW
induction machine tested in [34]. Finally, Figure 8c compares the experimental and estimated
CM (left curve) and DM (right curve) impedance profiles calculated using the model shown in
Figure 7c for the 2.2 kW electric machine studied in [43]. It should be noted that the accuracy
of the various models presented remains rather limited and does not perfectly characterize the
impedant frequency behavior of these machines over the entire frequency range considered.
Building on this postulate, more recent work has sought to construct more complex HF EEC
models capable of taking into account more HF phenomena. This is particularly the case
in [44,51], whose models have been improved in [45].

Figure 9 shows the complete HF EEC model developed in [45] to characterize the
impedant frequency behavior of an 8 kW, star-connected PMSM. Note the following in
this model:

* $R_g$ is the resistance of each phase.

* $C_{g1}$ represents the parasitic capacitances between the windings and the stator frame.

* “EMF” blocks describe the electromotive forces generated in each of the machine phases during operation.

* “Part 0” blocks characterize the CM and the DM behavior around resonance and anti-resonance frequencies at very high frequencies (between one and a few tens of megahertz).

* “Part 1” blocks characterize the CM and the DM behavior around resonance and anti-resonance frequencies at high frequency (between 10 kHz and 1 MHz).
* "Part 2" blocks model magnetizing inductances and eddy-current losses in the ferro-magnetic core.

* "Part 3" blocks characterize the DM impedant behavior at high frequencies (between 10 kHz and 1 MHz).

* "Part 4" blocks model the effects of leakage inductances in the first turns of coils.

<figure><img src="image_11.png" /><figcaption>**Figure 8.** Experimental and modeled impedance frequency response of three parallel phases in [50] (a), of CM and DM in [34] (b), and of CM and DM in [43] (c).</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 9.** Example of a complete HF EEC model of a star-connected PMSM—taken from [45].</figcaption></figure>

Figure 10 compares the experimental and estimated frequency response of the CM (a) and DM (b) impedances of the machine studied in [45], using the model shown in Figure 9.
Comparing the accuracy of these plots with those of Figure 8, a much better correlation of the impedance behavior is observed over the entire frequency range considered, especially at very high frequencies (beyond the megahertz). The various resonances are perfectly described, both in terms of frequency and amplitude.

<figure><img src="image_13.png" /><figcaption>Figure 10. Comparison of experimental and modeled CM (a) and DM (b) impedance frequency behavior—taken from [45].</figcaption></figure>

To evaluate the different parameters of the models presented in Figures 7 and 9, the usual approach is to experimentally characterize the impedance frequency response of the machine under test, then to simplify certain frequency ranges according to the assumptions made, and finally to estimate the various equivalent parameters, as explained by M. Schinkel in [34]. Such a model cannot be considered generic and must be developed for each machine topology. Finally, when the model becomes more complex, it may be necessary to couple this approach with analytical and/or numerical calculations in order to evaluate all the different parameters of the HF EEC model designed [45,48].

### 3.2.2. Transfer Function Approach to PMSM Behavior

In the interest of early diagnosis of interturn short-circuit faults in PMSM, N. Haje Obeid proposes in [52] to develop a transfer function multi-model to accurately model the behavior of a machine stator winding over a wide frequency range from a few kilohertz to over ten megahertz. The idea is to determine the transfer function that matches the experimental behavior of the winding. The method is based on a graphical model that uses the experimental admittance plot (inverse of the impedance) to determine the linear, time-invariant transfer function characteristic of the machine behavior. The transfer function of such a system is described by the Equation (2) below:

$$H(s) = \frac{a_n s^n + a_{n-1} s^{n-1} + \dots + a_0}{b_m s^m + b_{m-1} s^{m-1} + \dots + b_0} \quad (2)$$

where n and m are the order of the numerator and denominator polynomials, respectively, while the coefficients $a_i$ and $b_j$ that characterize the frequency response are calculated by solving an optimization problem.

In the example developed in [52], the optimization problem is solved using a genetic algorithm. This type of algorithm uses the principle of natural selection to ensure that the initial population of the problem evolves towards the best possible solution after a given number of generations $N_g$. To find an optimal solution, the order of the transfer function is increased if the transfer function of the best individual from the previous generation does not satisfy the system frequency response. The solution is adopted when this condition is satisfied. It should be noted that if the optimization problem is solved by a relatively high-order transfer function, it is possible to split the study into two different
frequency ranges in order to obtain simpler solutions that describe the desired behavior as accurately. The resulting multi-model transfer function can then be stored and reused in a diagnostic process.

Another approach is to use vector fitting techniques to extract a generic transfer function for characterizing the CM impedance of an electrical machine [53]. This transfer function is defined by the relation (3).

$$H(s) = A + B.s + \sum_{k=1}^{N} \frac{r_k}{s - p_k} \quad (3)$$

where A denotes a resistive behavior, B refers to an inductance type of behavior, N is a natural integer that characterizes the order of the polynomial, and r and p are the various residues and poles of the transfer function, respectively.

To go further, ref. [35] proposes to use this transfer function as a basis for an equivalent circuit model, where the number of parameters to be estimated depends on the order of the resulting transfer function. Two approaches are then considered to evaluate these parameters:

* A constrained optimization, to minimize the error between the model and the experimental measurement of the machine CM impedance [35];

* An analytical calculation of the various parameters, taking into account the type and design of the winding as well as the material and method of impregnation of the conductors [24].

The author argues that for the machine studied, a 4th-order transfer function satisfies the conditions to accurately model the CM impedance behavior. The resulting model is compared in Figure 11 with the impedance frequency response measured on the machine.

<figure><img src="image_14.png" /><figcaption>Figure 11. Comparison of the experimental frequency response and the proposed transfer function model of CM impedance—taken from [35].</figcaption></figure>

### 3.2.3. HF Approach Based on PMSM Phase Design

The approach based on the machine phase design developed in [13] takes into account capacitive coupling phenomena between the coils and with the machine ground, which are neglected when building a traditional EEC model. The consideration of these couplings increases the accuracy of the model. Furthermore, the formulation in state equations makes the numerical resolution of the problem fast and straightforward. Finally, because the model is based on a portion of the winding, its symmetry properties provide a simple and effective tool for representing the behavior of a single coil or of the entire stator winding by placing state models in series or in parallel.
W. Liu explains in [13] the problem formulation for three healthy turns of a winding. The drawing of the studied structure helps to consider the different phenomena encountered (capacitive phase-to-ground couplings, couplings between two adjacent turns, and inductive couplings between adjacent half-turns). This structure is described in Figure 12.

<figure><img src="image_15.png" /><figcaption>**Figure 12.** Structure of a three-turn coil model with inductive and capacitive couplings—taken from [13].</figcaption></figure>

To simplify the formulation, an equivalent circuit of the three machine turns is defined. Then, the state variables of the model are the voltages across the equivalent capacitors and the currents flowing in each of the equivalent inductances. For a three-turn coil, as in [13], there are 13 such variables. The state vector aggregates these different variables, allowing us to formulate the problem as (4). Finally, after linear transformations, we obtain a state model (5) capable of describing the behavior of 3 inductively and capacitively coupled PMSM turns.

$$ \dot{X} = G.X + H.X + K.U \quad (4) $$

$$ \left\{ \begin{array}{l} \dot{X} = A.X + B.U \\ Y = C.X + D.U \end{array} \right. \quad (5) $$

where X is the state vector, Y is the output vector, and U is the input vector.

Therefore, thanks to this state-space formulation of the behavior of the coupled turns, the precise behavior of both the low-frequency and high-frequency behavior of the complete MSAP can be evaluated with fast numerical resolution. The intermediate step is the estimation of the various model parameters. This estimation is done in [13] using an output error optimization algorithm. The advantage of this formulation, which is based on the machine design, is that it takes into account the winding topology and capacitive couplings, which are generally neglected. As explained earlier, these couplings are representative of the quality of the insulation between the conductors and consequently characteristic of the winding aging and the development of interturn short circuits.

## 4. Proposed Online Method for Impedance Characterization of Electrical Machines

The impedance characterization method developed in this section has been designed to monitor the aging of windings in electrical machines powered by voltage inverters. The aim of this online approach is to observe, at each moment of the machine operation, the evolution of the different resonance frequencies, which are typical of insulation degradation.
tion [16,18]. To do this, we assumed that a supply through a voltage source inverter would intrinsically excite the stator windings of electric machines over a very wide frequency range. Actually, as W. Chen points out in [54], every multiple of the switching frequency is found in the voltage signal applied to the machine terminals. Therefore, unlike the online methods presented in Section 2.2, the proposed method does not require any dedicated HF injection device. However, for accurate broadband impedance characterization of the windings, it relies on wideband gap probes and a dedicated data acquisition system.

In the following, we will present the system under study and the different measurements required for the characterization. Then, we will describe the proposed characterization process, step by step, to obtain the information needed to monitor the winding state of health.

## 4.1. *Studied System and Measurements*

The system under investigation is described in Figure 13. The electrical machine is powered by a voltage source inverter (VSI), made of SiC power components, and driven by pulse-width modulation (PWM). The only variables required for frequency impedance characterization are the voltages at machine terminals and the currents circulating in the windings. As mentioned above, these signals are evaluated using wideband gap probes and processed to calculate the frequency response of the windings. Finally, the resistance value of each machine phase is also required to implement the proposed method.

The experimental test bench used in this study is shown in Figure 14. The prototype in our experimental setup is star-connected. Also, we have made the neutral point accessible in order to measure the voltage at this point of the winding. Therefore, two types of characterization presented in Section 2.1 can be undertaken with our method: phase-to-phase impedance and phase-to-neutral impedance.

<figure><img src="image_16.png" /><figcaption>**Figure 13.** System description and measured signals.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Figure 14.** Experimental test bench.</figcaption></figure>

## 4.2. Flow Chart of the Proposed Characterization Method

The online impedance characterization method developed in this work is based on the flowchart presented in Figure 15.

<figure><img src="image_18.png" /><figcaption>**Figure 15.** Flowchart of the proposed characterization method.</figcaption></figure>

The various steps in this process are detailed in the following paragraphs.

### Step 1—Measurements

Voltages are measured at points A, B, and C (and N, if the neutral point is accessible) of the system described in Figure 13. The currents flowing in each phase of the machine are also measured. It should be noted that the frequency range of the characterization depends directly on the bandwidth (BW) of these sensors. In our study, their BW is [12 Hz, 50 MHZ]. To measure these data, the acquisition is performed using an oscilloscope with a sampling frequency of 62.5 MHz. Therefore, the Nyquist theorem states that only
frequency components up to 31.25 MHz ($f_{max}$) are properly displayed in the discrete signals obtained [55].

The machine under test is fed by a VSI with a switching frequency between 10 kHz and 35 kHz. The electrical fundamental frequency ($f_e$) of this traction machine is in the frequency range [100 Hz, 500 Hz]. Also, in these applications, the DC bus voltage ranges from 300 V to 750 V. When the neutral point is not accessible, we can consider a simple voltage system by applying the relation 6.

$$V_{abc} = \frac{U_{abc}}{\sqrt{3}} \qquad (6)$$

where $U_{abc}$ are phase-to-phase voltages and $V_{abc}$ are simple voltages.

Note that the proposed method can be performed with simple voltages or phase-to-phase voltages. For the remainder of this study, we will only consider simple voltages. Figure 16 depicts the waveforms of the simple voltages (a) and the currents circulating in the stator windings (b), respectively. Per unit has been used for confidentiality reasons. For time variables, the signal amplitude is divided by the amplitude of the fundamental component. However, for frequential spectrums, the signal amplitude is divided by the maximum amplitude over the considered frequency range.

<figure><img src="image_19.png" /><figcaption>Figure 16. Waveforms of the simple voltages (a) and measured currents (b).</figcaption></figure>

### Step 2—Frequency analysis

To perform the frequency analysis of voltages and currents, we considered these signals to be stationary, i.e., measured on a machine in steady-state operation. Therefore, we decided to use the Fast Fourier Transform (FFT) to extract their frequency spectrum. However, under transient conditions, time-frequency transformations should be used to extract the frequency spectrum of these signals. These include Short-Time Fourier Transform (STFT); Wavelet Transform (WT); Wigner-Ville Distribution (WVD); Choi-Williams Distribution (CWD); Hilbert-Huang Transform (HHT); etc.

This transformation is expressed mathematically by the relation 7 [56].

$$ \left[ \begin{array}{l} f[n] = \frac{2n}{N\Delta t} \quad , \quad y[n] = \sum_{k=0}^{N/2-1} x[k].e^{-j\frac{4\pi}{N}kn} \end{array} \right] \qquad (7) $$

where $f[n]$ is the frequency associated with the nth point of the transform, $y[n]$ is the result at the nth point of the transform, $N$ is the maximum number of points of the transform to satisfy the Nyquist criterion, and $\Delta t$ is the sampling period of the measured time signal.

The frequency content of the various signals analyzed is shown in Figure 17. The logarithmic scale on the x-axis and the transformation of the processed signal into decibel amplitude have been used for visual clarity.
<figure><img src="image_20.png" /><figcaption>Figure 17. Frequential analysis of the simple voltages (a) and measured currents (b).</figcaption></figure>

### Step 3—Search for dominant frequency components

Then, the idea is to extract the frequency components whose amplitude is predominant depending on the frequency range considered. For this purpose, we have defined adaptive thresholds for the frequency spectrum of voltages and currents. These thresholds are characterized by relations (8) and (9), respectively. The graphical representation of this search for the dominant frequency components is explained for the simple voltages in Figure 18. Note that these thresholds have been defined using the DC bus voltage ($U_{DC}$, which for this excitation corresponds to the amplitude at the switching frequency) and the maximum sampling frequency (to satisfy the Nyquist criterion).

$$S_V(f) = \frac{U_{DC}}{\sqrt{3}} \cdot \frac{f_{max}}{f_{PWM}} \cdot \frac{1}{f} \qquad (8)$$

$$S_I(f) = S_V(f) \cdot R_{ph} \qquad (9)$$

where $S_V$ and $S_I$ are the thresholds for simple voltages and currents, respectively, $f$ is the given frequency, $U_{DC}$ is the DC bus voltage, $f_{max}$ is the maximum sampling frequency, $f_{PWM}$ is the switching frequency, and $R_{ph}$ is the electrical resistance of the machine phases.

<figure><img src="image_21.png" /><figcaption>Figure 18. Graphical representation of the adaptive threshold defined to extract the predominant frequency components within single voltages.</figcaption></figure>
Finally, for each of the processed signals (simple voltages and currents), once the amplitude of the given component exceeds the threshold set at a given frequency ($f_{prep}$), the following quintet is stored:

$$Q = [ f_{prep} , |V_{abc}(f_{prep})| , |I_{abc}(f_{prep})| , \angle V_{abc}(f_{prep}) , \angle I_{abc}(f_{prep}) ]$$

### Step 4—Impedance calculation

To characterize the impedance behavior as a function of the excitation frequency, we must determine the corresponding impedance for each frequency $f_{prep}$ contained in the vectors $Q_V$ and $Q_I$. The Ohm law is used to calculate the amplitude and the phase of this impedance, as described in Equations (10) and (11).

$$|Z(f_{prep})| = \frac{|V_{abc}(f_{prep})|}{|I_{abc}(f_{prep})|} \qquad (10)$$

$$\angle Z(f_{prep}) = \angle V_{abc}(f_{prep}) - \angle I_{abc}(f_{prep}) \qquad (11)$$

The raw impedance characteristics obtained independently from the two thresholding methods are shown in Figure 19. In particular, it can be seen that the point densities are not evenly distributed depending on the threshold used. On the one hand, around the so-called “parallel” resonances, i.e., when the inductive reactance is equal to the capacitive reactance, the current amplitude is minimal and its signal is in phase with the applied voltage. This results in a local impedance maximum and more points with the threshold at simple voltages. On the other hand, around the so-called “series” resonances, the applied AC voltage generates the maximum current amplitude. These two signals are then in phase. This results in a local impedance minimum and more points with the threshold at the currents.

<figure><img src="image_22.png" /><figcaption>Figure 19. Raw frequency impedance characterization obtained by thresholding simple voltages (a) and currents (b).</figcaption></figure>

Also, as expected, the maximum frequency of these characteristics is equal to $f_{max}/2$, i.e., around 15.6 MHz. Finally, these two profiles are particularly noisy at very high frequencies (over a dozen megahertz).

### Step 5—Combining impedances

To take advantage of the precision of both plots around the different resonance frequencies, we will concatenate these two characterizations. The superposition of the two curves is shown in Figure 20. Despite the relative clarity of the full impedance characterization, we can see that it remains fairly noisy, which is why it is necessary to perform a cleanup step on the raw data.
<figure><img src="image_23.png" /><figcaption>**Figure 20.** Complete raw frequency impedance characterization.</figcaption></figure>

### Step 6—Cleaning the raw impedance characterization

To extract a more precise impedance characteristic from the raw pattern generated in the previous step, we need to perform a "cleaning" step. The strategy adopted is to cluster the various points gravitating around multiples of the switching frequency. In this study, an interval of 5% on either side of these multiples was considered. Thus, the impedance amplitude and phase for each of these multiples is defined as the average of the values within this interval. If no point is close to one of the switching frequency multiples, then no value is assigned to it.

By applying this strategy to the raw impedance characterization obtained in Figure 20, we obtain a much cleaner frequency spectrum, especially at very high frequencies. To complete the resulting spectrum, the value of the impedance calculated at the fundamental excitation frequency ($f_e$) is added. This new characterization is shown in Figure 21.

<figure><img src="image_24.png" /><figcaption>**Figure 21.** Improved frequency impedance characterization.</figcaption></figure>

### Step 7—Model storage and aging data collection

Finally, for aging monitoring purposes, it is essential to accurately evaluate the various parallel and series resonances of this improved impedance characterization. Since each point represents a multiple of the switching frequency, it is not possible to precisely assess these
resonance values. There are two ways to recover these informations. The first is to interpolate these profiles and extract local extremes in order to analyze their evolution. The second is to model the impedance frequency behavior of the machine phases using the various approaches presented in Section 3.2 and look for behavioral deviations synonymous with aging.

Figure 22 shows the impedance behavior obtained by interpolating the improved characterization calculated in step 6. The type of interpolation used is the “Akima spline” [57]. The first two parallel and series resonances can be properly characterized, and data can be stored to monitor the health of the winding. Finally, the overall behavior at very high frequencies is more erratic and fails to accurately identify the other resonances.

<figure><img src="image_25.png" /><figcaption>**Figure 22.** Interpolation of the impedance frequency response.</figcaption></figure>

### 4.3. Validation and Interest of the Proposed Method

The objective was to verify the reliability and consistency of our strategy. For this purpose, we performed about 15 measurements on the same test machine at different times during its operation (at different operating points and on different days). Interpolation was used to determine the amplitude and the frequency of the various resonances found.

Figure 23 shows the different characteristics acquired during these tests. It can be seen that the different profiles are superimposed regardless of the machine operating point. The first two resonances can be identified. Based on the different results, we highlight that the uncertainty to characterize the first parallel resonance is less than or equal to 12.33 kHz and 4.4 dBΩ (<3%). The frequency uncertainty to characterize the first series resonance is identical to that observed for the first parallel resonance. However, in terms of amplitude, it is less than or equal to 233 kHz.

Furthermore, in order to validate the proposed online characterization approach, we compared the resulting plot with that measured offline using a vector network analyzer. The results are shown in Figure 24.

There is a strong correlation between the two frequency behaviors. The relative difference between the two characteristic frequencies of the first parallel resonance is less than 10%. In terms of amplitude, the two approaches give the same result. On the other hand, there is a deviation between the behavior observed with our online method and with the impedance analyzer around the first series resonance. This is intrinsically related to the use of fixture adapters and a measurement cable between the impedance analyzer and the test machine, as explained in [58] and confirmed by Figure 25, which compares the impedance response of a traction machine as a function of the cable used. Based on these results, it is essential to emphasize the importance of using the same measurement conditions if we want to properly exploit the results of these frequency analyses and draw the correct conclusions about the health of the windings.
<figure><img src="image_26.png" /><figcaption>**Figure 23.** Comparison of impedance interpolations obtained at different times and operating points.</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>**Figure 24.** Comparison of impedance characterizations obtained with the proposed online method and with an impedance analyzer.</figcaption></figure>

Finally, the benefits of this online impedance characterization approach can be summarized in three points:

* The HF excitation is naturally provided by the voltage source inverter. The method is therefore non-invasive and has no effect on actuator operation or aging.

* No measurement cables or fixture adapters are required, eliminating the need for a calibration phase and making the characterization process highly reliable.

* Finally, health monitoring is performed at all times without removing the machine from its operating environment, allowing early detection of insulation degradation in the windings.
<figure><img src="image_28.png" /><figcaption>Figure 25. Measurement cable effect on the frequency response measured by the impedance analyzer.</figcaption></figure>

**5. Parameters Influencing Impedance Frequency Response**

To extend our study, we wanted to understand the influence of some parameters on the impedance characteristics obtained using the proposed strategy. The three parameters investigated are the machine operating temperature, the inverter switching frequency, and the DC bus voltage amplitude. Findings are discussed below.

5.1. *Temperature*

To investigate the effect of temperature on the HF impedance frequency response of the machine, we carried out two characterizations at different temperatures. The first at room temperature (T = 25 °C), after a few moments of operation, and the second after a few tens of minutes of circulating a current in the windings, so that the thermal regime is established. The second test was performed at a temperature of 80 °C. The results are shown in Figure 26. From the characterization plot after cleaning, as well as its interpolation, we can see that the HF impedance behavior remains identical, regardless of the operating temperature.

<figure><img src="image_29.png" /><figcaption>**Figure 26.** Comparison of the impact of the operating temperature on the cleaned impedance characteristic (a) and its interpolation (b).</figcaption></figure>
## 5.2. Switching Frequency

We have also studied the influence of the inverter switching frequency on the accuracy of the proposed impedance identification method. Since it is the various multiples of the switching frequency that are intrinsically excited, this means that for a fixed measurement frequency range, the characteristic obtained is enriched or depleted, depending on the switching frequency used. This assertion is confirmed by comparing the cleaned impedance characteristics shown in Figure 27a, obtained with $f_{PWM} = 15$ kHz and $f_{PWM} = 30$ kHz. The number of blue points is twice as high when the switching frequency is set to 15 kHz.

<figure><img src="image_30.png" /><figcaption>**Figure 27.** Comparison of the impact of the switching frequency on the cleaned impedance characteristic (a) and its interpolation (b).</figcaption></figure>

This increased point density provides a better approximation of the different series and parallel resonance frequencies. As a result, their estimation is all the more accurate and reliable. However, the interpolations presented in Figure 27b show that the difference in accuracy obtained with these two switching frequencies is not significant enough to bias the characterization, as the deviation is less than 3% (the uncertainty range calculated in Section 4.3 during reliability tests of the method).

## 5.3. DC Voltage

Finally, we have investigated the impact of the DC bus voltage on the impedance frequency response of the machine. For this purpose, we performed two characterizations with $U_{DC} = 300$ V and $U_{DC} = 600$ V. The results presented in Figure 28 show that the amplitude of the DC bus voltage has little effect on the accuracy of the impedance frequency response method. The reason for this lies in the very definition of the proposed method, since the adaptive threshold used to identify the various dominant frequency components that are used to derive the frequency characterization is defined using this parameter. The approach is therefore inherently independent of the DC bus voltage amplitude.
<figure><img src="image_31.png" /><figcaption>**Figure 28.** Comparison of the impact of the DC bus voltage on the cleaned impedance characteristic (a) and its interpolation (b).</figcaption></figure>

**6. Conclusions**

In this paper, after reviewing the state of the art in frequency characterization and modeling of stator winding impedance behavior in electrical machines, we have presented the online characterization principle we have developed for monitoring conductor aging. This characterization method is applicable to any machine fed by a voltage inverter. When the inverter is controlled by PWM, it naturally excites all the multiples of its switching frequency up to very high frequencies (i.e., above tens of megahertz), and thus we are able to extract the impedance frequency response of the machine phases using time measurements of voltages and currents.

The proposed method consists of several steps. First, a frequency analysis of the signals is performed. Then we look for frequency components whose amplitude is above a defined threshold. Then, using these data, we apply Ohm law to calculate the impedance amplitude and phase for each highlighted frequency. To reduce noise and uncertainties around multiples of the switching frequency, we must refine the raw characterization. Finally, modeling or interpolation is performed to store this impedance behavior and thus gather useful information for aging monitoring (frequencies and amplitudes of the first parallel and series resonances).

We have demonstrated the reliability and accuracy of our characterization method compared with other offline approaches available in the literature. We have also studied the influence of various parameters, such as operating temperature, inverter switching frequency, or DC bus voltage, on the impedance frequency response of the machine windings.

Finally, the advantage of the proposed broadband impedance characterization approach designed for aging monitoring lies in the fact that unlike other online methods in the literature, it does not require any additional injection system to excite the windings. The invasiveness of the method is therefore limited since it does not affect the operation or aging of the machine. Calibration is the most critical step to ensure good data analysis when using an impedance analyzer. Our approach eliminates this step, making it all the more reliable. Finally, aging monitoring can be done at any time without removing the machine from its operating environment, allowing early detection of insulation degradation in the windings.

**Author Contributions:** Conceptualization, J.C. and N.H.O.; methodology, J.C., N.H.O., T.B. and F.M.-T.; software, J.C.; validation, J.C., N.H.O., T.B. and F.M.-T.; formal analysis, J.C. and N.H.O.; investigation, J.C. and N.H.O.; resources, N.H.O.; data curation, J.C.; writing—original draft preparation
ration, J.C.; writing—review and editing, N.H.O., T.B. and F.M.-T.; visualization, J.C.; supervision, N.H.O., T.B. and F.M.-T.; project administration, N.H.O., T.B. and F.M.-T.; funding acquisition, N.H.O., T.B. and F.M.-T. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy concerns for the participants involved in the study.

**Conflicts of Interest:** The authors declare no conflicts of interest.

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