Motor Bearing Damage Induced by Bearing Current: A Review

Abstract

The occurrence of the motor shaft voltage and bearing current caused by the inverter will aggravate bearing damage and lead to the premature failure of bearings. Many types of equipment are being shut down due to bearing currents, such as filters, insulated bearings and grounding brushes. Traditional suppression measures cannot eliminate the bearing current and the bearing damage mechanism under the bearing current is not clear. In this paper, the damage caused by the bearing current to bearings is analyzed in detail. The influences of different working conditions on the bearing current and the damage caused are discussed. The source of bearing currents is introduced and the bearing current model under different working conditions is reviewed. An outlook for future studies is proposed, based on the current research status and challenges.

1. Introduction

The roller bearing is the crucial component of the motor and determines its performance. With the extensive use of the pulse width modulation (PWM) inverter in electric motors, the occurrence of the bearing current in the motor has become non-negligible. It will lead to corrosion damage to the bearing, shortening the motor service life and consequently causing motor failure [1,2].

Practically, the voltage frequency and amplitude on the output of the PWM inverter are controlled by the fast switching supply of the power electronics controls. The time taken to turn the thyristor on/off results in an unbalanced output from the inverter and generates a common-mode voltage with a high frequency at the neutral terminal of the motor windings. The motor winding, rotor, frame, bearing rollers and raceways act as capacitive electrodes, which form a coupling capacitance in the motor and provide a grounding path for the common-mode voltage [3,4,5,6]. In addition, the asymmetric magnetic flux inside the motor can induce the non-zero voltage at the shaft, which causes the circulating current to move through the “one end bearing-frame—other end bearing-rotor”.

The bearing currents will cause the electrical discharge machining (EDM) between the roller and raceway [7,8,9,10]. The surface materials of these components will melt and pitting defects will form [11,12,13,14]. Meanwhile, bearing currents will reduce the surface hardness of the bearing rollers and raceway. Combined with the stress concentration caused by corrosion pits, frosting damage will appear in the bearing raceways [15,16,17]. Under the action of the long-term bearing current, the surface of the bearing rollers and the raceway will present as bright and dark with corrugated damage [18,19,20,21,22]. Furthermore, the high temperature and the debris generated by the current discharge lead to the performance degradation of the bearing’s lubricating grease [23].

Currently, research on bearing current damage mainly includes the mechanism and the crucial influence parameters of the bearing current generation, the circuit model of bearing currents, and the damage to bearing performance [24,25]. In the study of current damage to bearings, many studies have shown that bearing current damage is mainly affected by the motor speed [26,27,28], load [29], lubricating grease composition [30,31] and insulation layer [32,33]. However, bearing failure is a diverse field under current investigation, and the mechanism analysis of the damage processing production is as yet unknown.

Based on a comprehensive literature survey, the bearing damage caused by bearing currents is reviewed in this paper. Section 2 introduces the different classifications of bearing current damage, the conditions of bearing current discharge and the process of bearing current damage. Section 3 describes the generation forms, types and circulation path of bearing currents, and the influences of motor structure, grease and operating parameters on the bearing current. Section 4 discusses the motor bearing current models, including the characteristics of the models, different ways to obtain model parameters and the establishment of equivalent models. Discussions on the current research status and outlooks for future investigations are highlighted in Section 5. Finally, conclusions of the present study are drawn. The relationship between the contents is shown in Figure 1.

2. Damage of Bearings Caused by Bearing Currents

Bearing current may flow through the bearing in several states: (1) When the bearing is in full film lubrication, the roller, raceway, and grease from an equivalent capacitance; there is a weak current flowing through the equivalent capacitance of the bearing from the motor shaft to the frame under the effect of rapid alternating common-mode voltage; (2) The bearing forms a resistance on the circuit when the roller and raceway are in contact. The current will flow through the contact area, which is extremely small, and the heat generated by the current may melt the surface material of the bearing rollers and raceways, making them stick together. (3) When the oil film is formed, the common-mode voltage will create an electric field between the inner and outer ring of the bearing until the bearing voltage exceeds the oil film breakdown voltage. In addition, the separation of the roller and raceway or the approach of the roller to the raceway may also cause the discharge phenomenon due to the electric field between the inner and outer rings of the bearing. Bearing current will cause early pitting damage to the bearing, medium-term frosting and eventually forms fluting as well as lubricant degradation.

In the process of bearing current damage, the bearing current density $J_b=i_b/A_H$ (where $i_b$ is bearing current, $A_H$ is Hertz contact area) can be used to determine the degree of bearing current harm [34]. References [35,36,37,38] concluded that small shaft voltages and bearing currents do not cause damage. However, Xie et al. [13] and Shoji et al. [39] found that weak currents can still cause damage to the bearings. Because the electrical contact area between the bearing roller and raceway is significantly smaller than the Hertz contact area when the bearing current passes through the bearing, the local current density flowing through the roller and raceway may be high [40]. Even if the current is very low, it will cause degradation of lubricating grease, shortening the bearing life.

2.1. Pitting

When the bearing is lubricated with a full film, the common mode voltage will form an electric field between the roller and the raceway, until the voltage value exceeds the oil film voltage threshold and a discharge occurs between the roller and the raceway [41,42], as shown in Figure 2. The discharge current generates a transient high temperature above the melting point of the bearing material, causing the bearing roller and raceway surface material to melt and form crater defects [11,12,14], and welding beads on the metal subsurface [17]. Xie et al. [13] applied gentle currents to a glycerin-lubricated bearing and discovered slope pits on the roller surface, presumably caused by the collapse of lubricant bubbles.

Pitting is an early type of bearing damage caused by bearing current, as shown in Figure 3. The current forms pits surrounded by melted material on the babbitt’s surface and material transfer between the two electrodes can be detected by energy dispersive spectroscopy [43]. This suggests that during the electrical corrosion process, the materials of the two electrodes (rollers and raceways) may be connected to create a resistance circuit. The pits generated by the current increase the roughness of the rollers and runways and increase bearing wear [44]. It is necessary to measure the degree of pit damage caused by bearing current to provide a basis for monitoring the running state and service life of bearings. The size of the pits can be estimated by the energy generated by the bearing current, and the dissipation of grease and energy in the air must be considered in the calculation. Prasad [45,46] investigated the energy accumulation and release of high-resistivity lubricants to determine the time it takes for electrical sparks to form a pitting on the raceway surface, and the time it takes to charge accumulation in grease [47]. Tischmacher et al. [16] also calculated the energy needed to evaporate the material. The electro-corrosion area showed a positive correlation with the discharge power, and the calculation of pitting was focused on a fault, which cannot predict the overall damage of the bearing raceway and roller.

2.2. Frosting

Under mechanical wear, as shown in Figure 4, the bearing raceway along the rolling direction shows light grey wear marks (frosting). The scanning electron microscope shows that there are many traces of bearing current damage in this area, and the hardness of this area is low [15,17]. The pit damage caused by the bearing current on the surface of the bearing roller and raceway will reduce the contact area in this area, and when the bearing current melts the surface material, it will soften the material near the pitting damage, and the frosting will be caused by mechanical wear.

The formation of frosting damage may be related to the load on the bearing. Tischmacher et al. [16,48] discovered that under heavier loads, the bearing current formed a light grey frosted raceway on the bearing raceway, with intensive material melting traces visible with the scanning electron microscopy; under lighter loads, the bearing raceway formed corrugated damage. This may be due to the bearing oil film thickening under lighter radial loads, allowing it to withstand higher shaft voltages and generate higher bearing currents, thus causing more damage. In addition, lighter loads may result in larger vibrations of the rollers in the raceway, which can cause the rollers to contact the raceway at a certain frequency and eventually form corrugated damage.

2.3. Corrugated Damage

The inner raceway surface of a 5.5 kw motor bearing is shown in Figure 5. As shown in Figure 5a, scanning electron microscopy was used to examine the microscopic morphology of the surface of the bright and dark areas. When compared to the bearing raceway without corrugated damage area (Figure 5d), it was discovered that the dark zone area organization structure was relatively dense, with low roughness. In contrast, the microstructure of the bright region was sparse and the roughness was high. When looking at the status of the surface material in the damaged region of the bearing, the raceway surface was covered with a layer of melted material, with the melted layer in the bright band area being thicker than the melted layer in the dark area, as shown in Figure 6.

Corrugated damage occurs in the middle and late life of bearings, accompanied by the phenomenon of increased bearing vibration. Corrugated damage is thought to be caused by the bearing current [21] or by the slip band [49]. However, corrugated damage is not caused by a single factor. EDM craters on the bearing raceway surface reduce the actual contact area between the rolling element and the raceway surface, resulting in higher contact pressure and the formation of corrugated damage [22]. Liu [50] discovered scratches from abrasive wear in the corrugated damage in the bearing raceway between bright and dark. As a result, the combined influence of mechanical wear and bearing currents cannot be isolated from the production of corrugated damage. In addition, corrugated damage has two forms; one is induced by sinusoidal AC bearing current and the high-frequency non-sinusoidal current [50]. The pitch of corrugations on the inner race is lower than on the outer race and is much greater on the rolling components, according to Prashad [19], and the breadth of corrugations on the bearing surface depends on the load. In 1991, Prashad [18] analyzed the corrugated damage width and spacing of bearing raceways under different operating conditions and found that the corrugated damage width depends on load, velocity, roller diameter, etc. In 1994, he studied the failure mechanism of bearings under the influence of an electric field and discovered that the load has a significant effect on the corrugated damage width [19]. In 1998, he computed the energy needed to generate corrugated damage in the bearing racetrack based on the bearing size and working conditions and established a model for calculating the corrugated damage width of the bearing raceway [20]. The degree of corrugated damage is related to the axial stress, the contact area of the roller and raceway, and the direction of current flow. In addition, the width and pitch of the corrugated damage may be related to the machine operating frequency and machine resonance.

2.4. Grease Degradation

When the bearing is in full-film lubrication, the common-mode voltage in the motor establishes a shaft voltage between the bearing inner and outer rings through a coupling circuit, and once the shaft voltage reaches the oil film breakdown threshold (dozens of volts [34]), the bearing current passes through the bearing. As the current passes through the gap between the bearing raceway and the rollers, heat is discharged into the surrounding area, causing the components in the grease to degrade and the grease to spoil [51]. As shown in Figure 7, increasing the shaft voltage can lead to more severe bearing grease degradation [52,53]. In addition, lubricant with discharge breakdown reduces dielectric strength; molecules in the grease may be polarized by the current, leading to lower dielectric constants and a lower threshold for shaft voltage penetration of the oil film [54]. Through X-ray analysis, Ost et al. [15] found a large amount of iron in the grease, indicating that bearing wear and bearing current cause the material on the raceway and roller surface to fall off and enter the grease, affecting grease performance and increasing friction between the roller and the raceway [55]. It can be observed that the bearing current degrades the lubrication grease in the bearing, thereby reducing its lubricating efficacy and reducing the bearing’s service life. So the motor’s bearing grease is required to withstand degradation from bearing wear and bearing currents [56].

2.5. Insulation Breakdown

For insulated bearings, bearing currents still occur. Malec et al. [57] and Zhang et al. [58] did a breakdown experiment on alumina and analyzed the breakdown channel morphology. When the current passes through the insulation, it creates a breakdown channel on the inside (as shown in Figure 8). Neusel et al. [59] showed that ceramic samples exhibit ohmic conduction at low pressure and space charge limited conduction at higher pressure. Insulation breakdown of insulated bearings may be related to inherent defects in the insulation and the electric field generated by local electron concentration. The original defects in the insulation layer are available for current flow and cause local Joule heating, according to Ref. [60], and when the applied electric field exceeds the insulation layer’s withstanding voltage value, the current increases until thermal runaway occurs, resulting in a material breakdown. Touzin et al. [61] analyzed the influence of the microstructure and temperature of materials on electron injection into alumina. The present breakdown of the insulation layer will result in permanent damage, which will degrade the insulation layer’s insulating effectiveness and may result in additional crack flaws around the breakdown channel. Malec et al. [62] examined the breakdown sites of alumina samples using spherical and flat electrodes and found that the breakdown was more frequent in areas with high electric field strength and initial electrode defects. Liebault et al. [63] investigated the variation curves of current absorption during the breakdown of alumina ceramics with different parameters and described the breakdown strength of alumina ceramics with different charge aggregation or diffusion parameters. Aside from the effect of the applied electric field, the thickness [57,59,60,61,62,64,65], porosity [64,65] and impurity [60] of the insulating layer directly impact the insulating properties of the insulating layer.

2.6. White Etching Cracks (WECs)

In recent years, it has been found that the formation of white etched cracks may be related to bearing discharges [38], and the high temperatures generated during discharge may lead to the production of hydrogen ions or hydrogen atoms in lubricating oil, making the material on the bearing surface fragile [54]. In addition, the defects caused by bearing currents can lead to the accumulation of local stresses, eventually causing cracks in the fragile material. A study showed that WECs were more likely to occur in statically loaded bearings than in those which are dynamically loaded [38]. In Ref. [66], the input DC amplitude was changed from 0 to 750 mA, and the time required to generate the WECs was decreased with the increase in the current amplitude. Gould et al. [67] found that the crack originated in a central inclusion and extends in other directions. In addition to the inclusions left inside the bearing material during the manufacturing process of the bearing, the welding beads created by the bearing current will also cause inclusions on the bearing subsurface [17].

The amplitude of the bearing current plays a role in the formation of WECs, primarily by producing initial defects on the bearing subsurface and embrittlement of the bearing surface material in some way, and the stress concentration under heavy bearing load causes the WECs to extend from the location of the initial defects to the bearing surface embrittlement.

3. Influencing Factors of Bearing Currents

The operating environment affects the inverter output voltage and current and the bearing voltage (bearing current) [68]. The magnitude of the current has a significant effect on the area of the bearing current damage [41,43,44]. However, bearing current is not the only measure of bearing damage; bearing type and size, application fields (high-power motors, generator), working conditions (grease, speed, load, size, inverter parameters), etc., can all affect bearing current damage [37]. Different discharge energy levels can cause different degrees of damage to the lubricated surface [55]. Shoji et al. [39,53] found that increasing the bearing voltage will exacerbate the current’s damage to the grease and bearing, and Plazenet et al. [69] revealed that the higher the shaft voltage, the less the discharge phenomenon occurs, but the energy of a single discharge increases. Modeling of bearing currents in motors can benefit in examining their amplitude, frequency, and other features, investigating the bearing current damage, and developing bearing current monitoring and suppression technologies. Using DC, AC and high-frequency pulsed currents to erode bearings and evaluate the damage to bearing raceways electrically, Zika et al. [70] found that high-frequency pulsed currents were more harmful to bearings than continuous currents, despite having less total dissipated energy.

3.1. Sources of Bearing Currents

Circulating currents are often seen in large motors, but the high-frequency current of inverters magnifies the motor’s structural asymmetry. Asymmetrical flux inside the motor will cause additional magnetic fields to induce voltages on the motor shaft [71]. Pratt et al. [72] and Torlay et al. [73] showed that the stator or rotor eccentricity and asymmetry in motor structure cause this flux. In addition, inverter unbalanced power output can also cause the asymmetric flux of current in the motor windings [74]. Finally, the shaft voltage driven by the circulating current generated by the magnetic circuit asymmetry passes through “one end of the bearing-frame-another end of bearing-rotor” (blue line in Figure 9) [75,76,77].

In a PWM inverter-driven motor, the motor inverter driver transmits sinusoidal voltages of different amplitudes and frequencies to the motor winding through a thyristor in the inverter circuit. Figure 10 shows the motor windings in a star configuration with three terminals linked to the driver output. Vdc is the DC bus voltage, V1–V6 are thyristors, and A, B, C denote the three-phase winding of the motor. The common-mode voltage at the motor winding’s neutral point is computed as [4]:

(1)${\mathrm{V}}_{\mathrm{com}}=\left({\mathrm{V}}_{\mathrm{A}}{+\mathrm{V}}_{\mathrm{B}}{+\mathrm{V}}_{\mathrm{C}}\right)/3$

where ${\mathrm{V}}_{\mathrm{A}}$,${\mathrm{V}}_{\mathrm{B}}$,${\mathrm{V}}_{\mathrm{C}}$ are the phase-to-ground-voltages of the inverter output.

If the inverter’s thyristor is switched on or off at the same time, the common-mode voltage is zero at the neutral point of the motor winding. Otherwise, due to the lost time of the thyristor switch, there will be a non-zero common mode voltage. The capacitance coupling in the motor with common-mode voltage is shown in Figure 11, where ${\mathrm{C}}_{\mathrm{wr}}$ is the capacitive coupling between the stator winding and the rotor, ${\mathrm{C}}_{\mathrm{wf}}$ is the capacitive coupling between the stator winding and the frame, and ${\mathrm{C}}_{\mathrm{rf}}$ is the capacitive coupling between the rotor and the frame [78,79]. The coupling capacitor inside the motor provides a low-impedance ground path for common-mode currents, as shown in Figure 9 [24]. The leakage current that flows from the motor windings to the ground through capacitance ${\mathrm{C}}_{\mathrm{wf}}$ (purple line in Figure 9) creates an encircling magnetic flux inside the motor which induces shaft voltage and the circulating current [80]. When the impedance of the winding to the ground through the rotor is low, the leakage current will reach the rotor through capacitive coupling between the rotor and the winding, and the bearing flows to the ground (red line in Figure 9) [81]. If the motor frame is poorly grounded or insulated bearings are used, current flows to the ground through the load end (green line and yellow in Figure 9) [82]. Boucenna et al. [83] analyzed the “zigzag” way on the motor stator laminations due to the skin effect of the high-frequency current.

3.2. Motor Structure Parameters

The structural characteristics of the motor will affect the equivalent capacitance inside the motor, which will affect the shaft voltage and bearing current. Schuster et al. [84] investigated the bearing current of two types of permanent magnet synchronous machines (PMSM, 90 kW) with different stator slots and geometric shapes, and discovered that the voltage division ratio (BVR) of motor bearings with the longer axial core length is higher and keeping the stator windings away from the rotor surface can reduce coupling capacitance and the peak value of bearing current. Peng et al. [85] investigated the impact combinations of magnetic poles and stator slots on the shaft voltage of magnet machines and discovered that stator slots can reduce the induced shaft voltage. It is worth noting that the 8-pole 12-slot combination has a high induction shaft voltage.

Han et al. [86] used a coupled field-circuit electromagnetic finite element analysis (FEA) to study the effect of winding distribution on bearing voltage and bearing current. Muetze et al. [87] found that the stator slot and cooling duct frame have a large effect on the common-mode flux, which will cause circulating bearing currents. By studying the influence of motor structure on shaft voltage and bearing current, the design and manufacture of the motor can be improved and the bearing current can be reduced.

3.3. Lubricant Grease

When the motor operates at high speeds, bearing rollers are separated from the raceway by non-conductive lubrication, and the non-zero common-mode voltage applies a voltage on the bearing, causing a small capacitive bearing current ${\mathrm{i}}_{\mathrm{bcap}}{=\mathrm{C}}_{\mathrm{b}}\mathrm{dv}/\mathrm{dt}$ [88]. Once the bearing voltage exceeds the oil film’s threshold value, EDM current will break through the oil film, and heat generated by electric sparks will melt the material on the roller and the raceway surface to form a defect [29,89,90].

Because of the non-conductive nature of lubricating grease, it makes the bearing grease directly affect the bearing voltage and bearing current. The base oil of grease has a significant impact on bearing current discharge and bearing damage [91,92]. By using low-resistivity lubricating grease, Prashad [93] detected the magnetic flux density (indicating current passing through) on bearing surfaces. The bearing acts electrically as an ohmic resistance when bearing rollers and raceways are in contact or there are conductive greases; the current flows directly through the bearing and heats the rollers and raceways [94]. According to Tischmacher et al. [95], the shaft voltage of conductive greases is lower than that of non-conductive greases, but their qualities are similar to non-conductive greases after the operation. Zheng et al. [96] found that the high-frequency dv/dt bearing current amplitude increased with the degradation of the bearing grease.

The dielectric strength of the grease and the minimum oil film thickness are now used to determine the breakdown voltage of the bearing oil film. Bhattacharya et al. [97] and Niskanen [98] discovered that the breakdown voltage of the bearing oil film decreases when the voltage frequency increases. Khan et al. [28] showed that high temperature reduces the viscosity of the grease, which thins the oil film and lowers the oil film breakdown voltage. The greater the bearing temperature, the more time the bearing spends in the ohmic state, which might be due to the temperature causing a reduction in the thickness of the oil coating, according to Refs. [26,95,98,99]. The rotational speed and temperature have effects on the oil film thickness and dielectric strength. The dielectric strength of the lubricating grease is higher at high temperatures than at low temperatures, and rotational speed decreases the dielectric strength while increasing the oil film thickness. Lin et al. [11] reported that when the concentration of MoS₂ (a lubricating oil ingredient that improves wear performance) and current amplitude increased, the insulating effect of the oil film decreased. Noguchi et al. [100] found that bearings with low raceway roughness vibrated more severely during the latter part of the experiment. This may be because a smooth surface is more likely to form a full oil film and therefore can withstand higher voltage, but a rough raceway can directly contact the oil film or reduce the breakdown voltage of the oil film, causing less bearing current damage than a smooth surface. Therefore, it may cause an error to calculate the breakdown voltage of the bearing lubricating film based on the thickness of the bearing lubricating film and its dielectric strength.

3.4. Rotational Speed

In the motor operation process, the speed of the motor changes the running state of the bearings and the average output voltage of the inverter, which in turn affects the bearing current damage to the bearings. Some research showed that when motor speed increased, the peak value of bearing current increased as well and that at a particular speed, the peak value reached its maximum and subsequently fell [26,27,28]. When testing the peak discharge current of a motor, Magdun et al. [29] found that as the load increased, the rotational speed at which the highest bearing current appeared decreased. Khan et al. [28] found that the discharge activity of the motor is more significant at low speeds than at high speeds. They reasoned that when the rotational speed rose after passing a certain threshold, the oil film thickness grew, preventing the discharge phenomena and supplying more resistance to the bearing current loop, lowering the peak bearing current. This was also demonstrated by the frequency of discharge phenomena seen in [29,95] at varied rotating speeds.

During the motor operation, it may be in a state of frequent speed change and start-stop due to the evolution of load or application environment, which will also contribute to the formation of damage to the bearing current. In comparison to constant speed operations, Tischmacher et al. [16,69] indicated that variable speed operations that frequently start and stop cause more bearing current damage pits for the motor than constant speed operations.

3.5. Inverter Parameters

The power parameters determine the frequency and the motor’s neutral voltage. The carrier frequency of the inverter, the thyristor’s switching frequency and the bus voltage all affect the bearing voltage and current of the motor. Statistical data on bearing currents collected by Smolenski et al. [101] revealed that increasing the inverter carrier frequency also increases the number of bearing current discharges as well as the current amplitude. Inverter switching frequency and inverter DC bus voltage are positively linked with the number of discharges [26,28]. Xu et al. [102] also pointed out that the increase in switching frequency of the inverter increases the dv/dt current amplitude, but it has a nonlinear effect on the current discharge frequency of EDM. It is surmised that the increase in switching frequency may increase the common-mode voltage amplitude and the electric field of the oil film. Moreover, the increase in frequency reduces the time required for the electric field to break through the oil film and cause discharge.

To determine the impact of inverter types on shaft voltage and bearing current, Reddy et al. [103] and Chandrashekar et al. [104] compared the common-mode voltage, shaft voltage, and bearing current generated by two-level and three-level inverters and found that the common-mode voltage, shaft voltage, and bearing current generated by three-level inverters were smaller than those caused by two-level inverters. Collin et al. [105] compared the shaft voltage and bearing current amplitudes of silicon- and silicon carbide-based PWM inverters, and discovered that SiC inverters have higher shaft voltage and bearing current than Si inverters.

3.6. Connection Cable

The use of long cable connections between frequency converters and the motor can easily lead to transient overvoltage at the motor terminals [106]. Weicker et al. [107] measured the common-mode voltage of motors at different lengths of three types of cables, and data revealed that the 300-m-long cable contains more high-frequency components. Compared with unshielded cables, shielded cables showed lower common-mode voltage amplitudes, smaller bearing current peaks, and a part of the common-mode voltage waveform was suppressed. Muetze et al. [108] measured the influence of cable shielding on EDM currents, high-frequency circulating currents, and rotor ground currents, and found that using shielded cables only affects the high-frequency circulating currents of large motors. Because of the long cable voltage reflection phenomena and the asymmetry of the long cable, the motor bearings may be exposed to higher voltages, increasing the risk of bearing current damage.

4. Modeling

An electrical model is established for different states of bearings to describe the states of shaft voltage and bearing current. The existing shaft current models are summarized in Table 1. The bearing current lumped parameter model can predict the shaft voltage and shaft current. Based on this model, the three-phase windings model of the motor can be studied independently to reveal the effect of each phase winding on the shaft current. The motor distribution parameter model divides the motor winding into several pieces and establishes the electrical connection between each motor winding and the motor rotor and frame. The high-frequency model can analyze the high-frequency component of the motor shaft current and the bearing current when the motor is connected to a long cable, but the models are more complex and have more parameters.

4.1. Lumped Parameter Model

Based on the distribution of coupling capacitance inside the motor and the electrical characteristics of the bearing under common-mode voltage, Chen et al. [25] developed equivalent lumped parameter circuit models to describe the motor ground current and bearing current generated by the coupling from the windings to the frame and the rotor. As shown in Figure 12, Erdman et al. [24] built a model of the current equivalent circuit of the motor bearing by considering capacitance coupling between the motor rotor, winding and frame. Park et al. [7] considered the electrical effects of the three-phase windings of the motor to establish an equivalent circuit model. In addition to the motor structure, Mäki-Ontto et al. [111] constructed a model of the motor system including the power supply, inverter, wires and filters. Mirafzal et al. [112] further considered stator resistance, core loss resistance, magnetizing inductance, etc., to build a three-phase motor model. Tagami et al. [113] developed a motor shaft voltage model describing the imbalance between the stator winding and the frame and rotor. Duan et al. [110] modelled the shaft voltage of the surface permanent magnet (SPM) motor and the interior permanent magnet (IPM) motor. Wang et al. [114] developed a model of the motor that includes the insulation of the outer ring and the wire that connects the outer ring of the bearing to the frame.

According to the parameters in the bearing current model, researchers [7,109,139] have calculated the value of coupling capacitance in the machine based on the machine structure parameters. Jun et al. [140,141] analyzed the impedance of the motor windings, rotor and frame, and the coupling capacitance in the motor by the finite element model. Magdun et al. [142] studied the equivalent capacitance between winding and frame for random wound round wire winding motors and wound profile copper winding motors.

4.2. Circulating Current Model

The leakage current in the motor and the axial flux generated by the unbalanced flux cause a voltage at the motor shaft, forming a high-frequency circulating current that causes damage to the bearings at both ends of the motor [117], as shown in Figure 13. Muetze et al. [116] and Postariu et al. [119] proposed an eddy-current model for the circulating bearing current based on the circulation loop of circulating bearing currents. Magdun et al. [120,143] also consider the motor rotor impedance in the circulating bearing current model. Maki-Ontto et al. [115,144] investigated the phenomenon of the skin effect causing the high-frequency current to propagate in a zigzag pattern on the motor stator laminations by refining the electrical model of the motor stator core and windings. Jaritz et al. [121] considered the electromagnetic coupling between the common mode current and the circulating current in their circulating bearing current model. Muetze et al. [118] developed a circulating bearing current model for insulated bearings and insulated housings. The status of the bearings at both ends of the motor during operation is not always the same, and the above model may be used to investigate the difference in bearing damage at both ends caused by a circulating current.

4.3. High-Frequency Model

The influence of common mode and differential mode voltages on bearing current at a high-frequency state may be studied by using a high-frequency motor model, which can then be used to develop shaft voltage suppression strategies. Naik et al. [79] developed RLC circuits for the electrical connection between the motor winding and frame and rotor, the air-gap impedance, and the bearing to meet the response of the motor model at different frequencies. Cacciato et al. [122] refined the connection of each winding coil to the frame and rotor to form a high-frequency motor model. Bubert and Quabeck et al. [124,125] characterized the electrical characteristics of motor windings and built a more comprehensive model for motor windings. Magdun et al. [123] compared several motor winding high-frequency models and found that the equivalent capacitance between the winding and the frame, the stator leakage inductance, and the eddy currents loss resistance have a large impact on the accuracy of the model and proposed the corresponding motor model.

Because the voltage and current distribution in the motor winding are not constant in high-frequency alternating voltage [116,145,146], Gersem and Vivo et al. [126,127,128] developed the transmission line model to describe the effects of electromagnetic wave-propagation and reflection in cables and motor windings, as well as the frequency-dependent distributed losses in the system. However, the transmission line model must account for skin and proximity effects and dielectric losses, which adds to the complexity of the model.

The transmission line effect of the cable may raise the values of common-mode voltage, shaft voltage, and bearing current in systems when the inverter is connected to the motor with a long cable. Moreira et al. [129] modelled the high frequency per unit length of the cable. In Refs. [111,130,147] established the motor equivalent electrical model with long cables. Asymmetrical cables connecting the motor to the inverter will result in a high frequency zero system voltage at the motor terminals [106]. Magdun et al. [132] developed transmission line models for the motor-inverter cable (long cable, cable asymmetry). Dahl et al. [131] modelled the system circuit for back-feeding bearing currents due to cable asymmetry.

4.4. Bearing Model

The equivalent bearing model can be described as an equivalent circuit model with a capacitor composed of an oil film and a bearing resistor connected in series and parallel [133]. The circuit models of insulated bearing and non-insulated bearing are shown in Figure 14, where R_b is the bearing resistance, R_break is the nonlinear resistance when the bearing breaks down, C-oil is the equivalent capacitance of bearing oil film, C_ins is the equivalent capacitance of the bearing insulation layer, S1 and S2 represent the state switching when the bearing is broken. Ren et al. [134] pointed out that the bearing resistance is not constant when the bearing current is generated. Erdman et al. [24] and Bubert et al. [124] added the bearing nonlinear resistance to the model. In addition, for insulated bearings, Wang et al. [135] and Quabeck et al. [125] added to the bearing model the equivalent capacitance used to describe the insulation layer. Bearing equivalent capacitance is calculated by the Hertzian contact area [148] of the roller and raceway and the thickness of the oil film [149]. Moreover, the grease near the Hertzian contact area between the roller and the raceway is also involved in the composition of the equivalent capacitance of the bearing [150,151]. In addition, the bearing equivalent model will be affected by motor speed [151,152] and load [153,154], the temperature-dependent dielectric constant of the lubricant [133,150], raceway surface roughness [151,155], and bearing type [156]. Prediction of bearing currents for different parameters of bearing rollers and raceway surfaces can be performed by the finite element software [136,137,138].

5. Discussion and Outlook

Failures associated with significant bearing damage induced by bearing currents have been commonly reported. Some studies have focused on bearing current damage to bearings. For example, Chiou et al. [41] and Xie et al. [13] studied the craters generated on the lubricated surfaces of bearings under bearing currents. Tischmacher et al. [16] and Raadnui et al. [17] analyzed the frosting damage caused by bearing currents to the bearings. Prashad [18,49] studied the corrugated damage caused by bearing currents to bearings and developed a model for calculating the width of corrugated damage. However, few studies have focused on modeling the early microdamage of bearing currents, and most of them tend to observe and analyze the damage of bearing currents rather than develop mathematical models of the damage of bearing currents. This leads to the incapability of existing models to effectively predict the length of a bearing life under the bearing currents. The reason for the existence of the bearing currents research status is that there are still more difficulties in the study of early micro-damage of bearing currents, which are summarized as follows:

• (1). The mechanism of bearing current generation is not clear. Many existing studies have used the bearing oil film breakdown voltage threshold proposed by Doyle et al. [90] as the condition for bearing current generation. However, insulated bearings with an insulating layer are not compatible with this method. The insulating properties of each bearing grease may vary considerably and will change as the grease deteriorates. In addition, there are factors such as load changes and grid disturbances during motor operation, and the bearing currents appear for very short periods and are more difficult to monitor. Therefore, it is very challenging to identify the conditions in which bearing currents occur.

• (2). The damage process of the bearing current to the bearing cannot be directly observed. The bearings are installed inside the motor and the discharge phenomenon of the bearing current occurs between the bearing rollers and the raceway, so the motor components and the roller cage of the bearings will limit visibility. In addition, the emergence of the bearing current process is very short. Thus, it is difficult to observe the damage process of bearing current on the bearing. Because of this, it is hard to determine how bearing current damage occurs.

• (3). The single damage area caused by the bearing current to the bearing is at the micron level and cannot be directly observed. Therefore, scanning electron microscopy is needed to observe the micro damage. Moreover, in the initial bearing current damage bearing roller and raceway surface will not be covered with bearing current damage traces, which adds difficulty to the observation of damage.

In order to establish early micro-damage models of bearing currents and to achieve effective prediction of the length of motor bearing life under the action of bearing currents, further research should be conducted in the following areas:

• (1). Consider the effect of power supply frequency on the breakdown voltage of the bearing oil film and insulation layer. The high-frequency component of the common-mode voltage and the extremely high rate of voltage change accelerate the injection of electrons into the oil film and insulation layer, increasing the density of high-energy charged particles in the oil film and insulation layer and increasing the chance of breakdown.

• (2). Accurate modeling of the bearing currents helps to analyze the extent of bearing current damage to the bearings. Electric vehicles have used permanent magnet synchronous motors instead of induction motors, but at high speeds, the permanent magnets in the rotor may be close to the stator winding and cut the magnetic field, thus creating the back electromagnetic fields (EMF). Inside the motor, electromagnetic interference may affect the bearing currents. Therefore, when modeling the bearing current, the electromagnetic interference between the current in the winding, and the leakage current should be considered.

• (3). Establish the heat source model of bearing current discharge using the heat source model of electrical discharge machining (EDM). The bearing damage caused by bearing current may be calculated by energy analysis and with the help of finite element software.

6. Concluding Remarks

According to the literature, the effects of different operating conditions on bearing currents have been analyzed and the corresponding models of bearing currents have been developed to describe the bearing current characteristics. However, the bearing current generation conditions in these models do not apply to the commonly used insulated bearings. Furthermore, the damage morphology and extent of bearing current damage at different periods have been investigated. Although models for late corrugated damage are available, they have limitations for predicting bearing service life. In general, it is still difficult to predict the length of bearing life due to the absence of a clear mechanism of bearing current generation and early micro-damage models.

The mechanism of bearing current generation and bearing current damage process to bearings are not clear. As an important part of electric motors, the prediction of bearing life under the action of currents is still a necessary research direction in this field, and the above problems will be gradually solved. So, the purpose of this paper was to give a review of the bearing currents, from the origins of bearing current, and bearing fault caused by bearing current, to the model of bearing current. The content of this paper will assist researchers in quickly grasping the hotspots and challenges of current research.

Author Contributions

Conceptualization, J.M. and X.L.; methodology, J.M. and Y.X.; software, J.M. and Y.X.; validation, X.L. and Q.H.; formal analysis, J.M. and C.Y.; investigation, J.M. and Y.X.; resources, Q.H. and C.Y.; writing—original draft preparation, Y.X. and J.M.; writing—review and editing J.M.; supervision, Q.H.; project administration, X.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Table

Figure 1. Relationship diagram of bearing current research content.

Figure 2. The process of discharge between the bearing roller and the raceway (a) ignition; (b) formation; (c) discharge; (d) ejection [22].

Figure 3. Pitting damage micromorphology (a) SEM image of the rolling elements in the bearing with the passage of the electrical current; (b) the magnified image of a typical pit [13].

Figure 4. Frosting on the inner raceway ring and a magnified view of the running track [17].

Figure 5. Bearing corrugated damage and its microstructure (a) Corrugated damage of raceway surface (b) Micro-morphology of dark area (c) Micro-morphology of bright area (d) No damage area.

Figure 6. Bearing corrugated damage area subsurface.

Figure 7. Changes in bearing grease after 500 h of operation at different voltages [53].

Figure 8. Breakdown channel in a single-crystal Al2O3 [59].

Figure 9. Bearing current paths in an inverter-driven induction motor system.

Figure 10. Motor and driver connection circuit diagram.

Figure 11. Capacitance coupling in an induction motor.

Figure 12. Inverter and motor shaft voltage model [24].

Figure 13. Circulating current due to magnetic flux asymmetry [117].

Figure 14. Models of insulated and non-insulated bearings.

Table 1

Classification of motor shaft current model.

Type		Modeling Methods	Bearing Type	Parameter Acquisition	References
Motor model	Lumped parameter model	RLC circuit	Non-insulated bearings	Analytical calculation	[24,109]
				Finite element method, simulation, analytical calculation	[110]
	Three-Phase Model	RLC circuit	Non-insulated bearings	Analytical calculation	[7,111,112,113]
			insulated bearings	Analytical calculation	[114]
	Distributional parameter model	RLC circuit	Non-insulated bearings	Measurement	[25]
Circulating current model		Mathematical model		Analytical calculation	[115,116,117]
		RLC circuit	insulated bearings	Analytical calculation	[118,119,120]
		RLC circuit	Non-insulated bearings	Finite element method	[121]
High-frequency model	High-frequency model for winding	RLC circuit	Non-insulated bearings	Measurement	[79,122,123,124]
		RLC circuit	insulated bearings	Measurement, finite element method	[125]
	Transmission-line models	Finite element method			[126,127,128]
	Models with power cables	RLC circuit		Analytical calculation, measurement	[111,129,130]
				Analytical calculation, measurements, finite element methods	[131,132]
Bearing model	Non-insulated bearings	RLC circuit		Analytical calculation	[24,25,133]
	Nonlinear resistance model at the breakdown	RLC circuit		Analytical calculation	[24,124,134]
	Insulated bearings	RLC circuit		Analytical calculation	[125,135]
	Discharge model	Finite element method			[136,137,138]

References

1. Son, D.W.; Zhang, T.; Lee, G. Study on Electrical Pitting Prevention Device of a Rotating Shaft Using Automatic Control Potential Balancing. Materials; 2022; 15, 4510. [DOI: https://dx.doi.org/10.3390/ma15134510] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/35806635]

2. Schneider, V.; Behrendt, C.; Höltje, P.; Cornel, D.; Becker-Dombrowsky, F.M.; Puchtler, S.; Gutiérrez Guzmán, F.; Ponick, B.; Jacobs, G.; Kirchner, E. Electrical Bearing Damage, A Problem in the Nano- and Macro-Range. Lubricants; 2022; 10, 194. [DOI: https://dx.doi.org/10.3390/lubricants10080194]

3. Chen, S.; Lipo, T.A.; Fitzgerald, D. Source of induction motor bearing currents caused by PWM inverters. IEEE Trans. Energy Convers.; 1996; 11, pp. 25-32. [DOI: https://dx.doi.org/10.1109/60.486572]

4. Muetze, A. Bearing Currents in Inverter-Fed AC-Motors; Shaker-Verlag GmbH: Düren, Germany, 2004.

5. Im, J.-H.; Lee, Y.-K.; Park, J.-K.; Hur, J. Shaft Voltage Reduction Method Using Carrier Wave Phase Shift in IPMSM. Energies; 2021; 14, 6924. [DOI: https://dx.doi.org/10.3390/en14216924]

6. Collin, R.; Yokochi, A.; von Jouanne, A. Novel Characterization of Si- and SiC-Based PWM Inverter Bearing Currents Using Probability Density Functions. Energies; 2022; 15, 3043. [DOI: https://dx.doi.org/10.3390/en15093043]

7. Park, J.-K.; Wellawatta, T.R.; Ullah, Z.; Hur, J. New equivalent circuit of the IPM-type BLDC motor for calculation of shaft voltage by considering electric and magnetic fields. IEEE Trans. Ind. Appl.; 2016; 52, pp. 3763-3771. [DOI: https://dx.doi.org/10.1109/TIA.2016.2562009]

8. Han, P.; Heins, G.; Patterson, D.; Thiele, M.; Ionel, D.M. Modeling of Bearing Voltage in Electric Machines Based on Electromagnetic FEA and Measured Bearing Capacitance. IEEE Trans. Ind. Appl.; 2021; 57, pp. 4765-4775. [DOI: https://dx.doi.org/10.1109/TIA.2021.3092700]

9. Berhausen, S.; Jarek, T. Method of Limiting Shaft Voltages in AC Electric Machines. Energies; 2021; 14, 3326. [DOI: https://dx.doi.org/10.3390/en14113326]

10. Stockbrügger, J.O.; Ponick, B. Analytical Determination of the Slot and the End-Winding Portion of the Winding-to-Rotor Capacitance for the Prediction of Shaft Voltage in Electrical Machines. Energies; 2021; 14, 174. [DOI: https://dx.doi.org/10.3390/en14010174]

11. Lin, C.-M.; Chiou, Y.-C.; Lee, R.-T. Effect of MoS2 additive on electrical pitting mechanism of lubricated surface for Babbitt alloy/bearing steel pair under ac electric field. Wear; 2004; 257, pp. 833-842. [DOI: https://dx.doi.org/10.1016/j.wear.2004.05.002]

12. Chiou, Y.-C.; Lee, R.-T.; Lin, S.-M. Formation mechanism of electrical damage on sliding lubricated contacts for steel pair under DC electric field. Wear; 2009; 266, pp. 110-118. [DOI: https://dx.doi.org/10.1016/j.wear.2008.06.001]

13. Xie, G.; Luo, J.; Guo, D.; Liu, S.; Li, G. Damages on the lubricated surfaces in bearings under the influence of weak electrical currents. Sci. China Technol. Sci.; 2013; 56, pp. 2979-2987. [DOI: https://dx.doi.org/10.1007/s11431-013-5399-7]

14. von Jouanne, A.; Collin, R.; Stephens, M.; Miao, Y.; Thayil, B.; Li, C.; Agamloh, E.; Yokochi, A. Motor Bearing Current Characterization in SiC-based Variable Frequency Drive Applications. Proceedings of the 2020 IEEE Energy Conversion Congress and Exposition (ECCE); Detroit, MI, USA, 11–15 October 2020; pp. 2718-2725.

15. Ost, W.; De Baets, P. Failure analysis of the deep groove ball bearings of an electric motor. Eng. Fail. Anal.; 2005; 12, pp. 772-783. [DOI: https://dx.doi.org/10.1016/j.engfailanal.2004.12.003]

16. Tischmacher, H.; Gattermann, S. Bearing currents in converter operation. Proceedings of the XIX International Conference on Electrical Machines-ICEM 2010; Rome, Italy, 6–8 September 2010; pp. 1-8.

17. Raadnui, S.; Kleesuwan, S. Electrical pitting wear debris analysis of grease-lubricated rolling element bearings. Wear; 2011; 271, pp. 1707-1718. [DOI: https://dx.doi.org/10.1016/j.wear.2011.01.040]

18. Prashad, H. Theoretical and experimental investigations on the pitch and width of corrugations on the surfaces of ball bearings. Wear; 1991; 143, pp. 1-14. [DOI: https://dx.doi.org/10.1016/0043-1648(91)90081-5]

19. Prashad, H. Investigation of damaged rolling-element bearings and deterioration of lubricants under the influence of electric fields. Wear; 1994; 176, pp. 151-161. [DOI: https://dx.doi.org/10.1016/0043-1648(94)90143-0]

20. Prashad, H. Determination of Time Span for the Appearance of Flutes on the Track Surface of Rolling-Element Bearings Under the Influence of Electric Current. Tribol. Trans.; 1998; 41, pp. 103-109. [DOI: https://dx.doi.org/10.1080/10402009808983727]

21. Charoy, A.; Dunand, P. Bearing currents induced by a power drive. Automot. Power Electron.; 2007; 9, pp. 26-27.

22. Didenko, T.; Pridemore, W.D. Electrical Fluting Failure of a Tri-Lobe Roller Bearing. J. Fail. Anal. Prev.; 2012; 12, pp. 575-580. [DOI: https://dx.doi.org/10.1007/s11668-012-9598-5]

23. Kudelina, K.; Vaimann, T.; Rassolkin, A.; Kallaste, A.; Asad, B.; Demidova, G. Induction Motor Bearing Currents—Causes and Damages. Proceedings of the 2021 28th International Workshop on Electric Drives: Improving Reliability of Electric Drives (IWED); Moscow, Russia, 27–29 January 2021; pp. 1-5.

24. Erdman, J.M.; Kerkman, R.J.; Schlegel, D.W.; Skibinski, G.L. Effect of PWM inverters on AC motor bearing currents and shaft voltages. IEEE Trans. Ind. Appl.; 1996; 32, pp. 250-259. [DOI: https://dx.doi.org/10.1109/28.491472]

25. Chen, S.; Lipo, T.A.; Fitzgerald, D. Modeling of motor bearing currents in PWM inverter drives. IEEE Trans. Ind. Appl.; 1996; 32, pp. 1365-1370. [DOI: https://dx.doi.org/10.1109/28.556640]

26. Muetze, A.; Tamminen, J.; Ahola, J. Influence of Motor Operating Parameters on Discharge Bearing Current Activity. IEEE Trans. Ind. Appl.; 2011; 47, pp. 1767-1777. [DOI: https://dx.doi.org/10.1109/TIA.2011.2154353]

27. Schuster, M.; Springer, J.; Binder, A. Comparison of a 1.1 kW-induction machine and a 1.5 kW-PMSM regarding common-mode bearing currents. Proceedings of the 2018 International Symposium on Power Electronics, Electrical Drives, Automation and Motion (SPEEDAM); Amalfi, Italy, 20–22 June 2018; pp. 1-6.

28. Khan, K.; Gyllensten, F. Experimental investigation of bearing currents in low voltage motors. Proceedings of the 2018 XIII International Conference on Electrical Machines (ICEM); Alexandroupoli, Greece, 3–6 September 2018; pp. 218-224.

29. Magdun, O.; Gemeinder, Y.; Binder, A. Investigation of influence of bearing load and bearing temperature on EDM bearing currents. Proceedings of the 2010 IEEE Energy Conversion Congress and Exposition; Atlanta, GA, USA, 12–16 September 2010; pp. 2733-2738.

30. Suzumura, J. Prevention of electrical pitting on rolling bearings by electrically conductive grease. Q. Rep. RTRI; 2016; 57, pp. 42-47. [DOI: https://dx.doi.org/10.2219/rtriqr.57.1_42] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/36450716]

31. Gonda, A.; Capan, R.; Bechev, D.; Sauer, B. The Influence of Lubricant Conductivity on Bearing Currents in the Case of Rolling Bearing Greases. Lubricants; 2019; 7, 108. [DOI: https://dx.doi.org/10.3390/lubricants7120108]

32. Touzin, M.; Goeuriot, D.; Guerret-Piécourt, C.; Juvé, D.; Fitting, H.J. Alumina based ceramics for high-voltage insulation. J. Eur. Ceram. Soc.; 2010; 30, pp. 805-817. [DOI: https://dx.doi.org/10.1016/j.jeurceramsoc.2009.09.025]

33. Toma, F.-L.; Scheitz, S.; Berger, L.-M.; Sauchuk, V.; Kusnezoff, M.; Thiele, S. Comparative Study of the Electrical Properties and Characteristics of Thermally Sprayed Alumina and Spinel Coatings. J. Therm. Spray Technol.; 2010; 20, pp. 195-204. [DOI: https://dx.doi.org/10.1007/s11666-010-9580-2]

34. Busse, D.F.; Erdman, J.M.; Kerkman, R.J.; Schlegel, D.W.; Skibinski, G.L. The effects of PWM voltage source inverters on the mechanical performance of rolling bearings. IEEE Trans. Ind. Appl.; 1997; 33, pp. 567-576. [DOI: https://dx.doi.org/10.1109/28.568024]

35. Pietroff, H. Wälzlager im elektrischen Stromkreis (Bearings in the electrical circuit). Elektr. Bahnen; 1968; 3, pp. 54-61.

36. Busse, D.; Erdman, J.; Kerkman, R.J.; Schlegel, D.; Skibinski, G. System electrical parameters and their effects on bearing currents. IEEE Trans. Ind. Appl.; 1997; 33, pp. 577-584. [DOI: https://dx.doi.org/10.1109/28.568025]

37. Mütze, A.; Binder, A.; Vogel, H.; Hering, J. Experimental evaluation of the endangerment of ball bearings due to inverter-induced bearing currents. Proceedings of the Conference Record of the 2004 IEEE Industry Applications Conference, 2004. 39th IAS Annual Meeting; Seattle, WA, USA, 3–7 October 2004; pp. 1989-1995.

38. Loos, J.; Bergmann, I.; Goss, M. Influence of High Electrical Currents on WEC Formation in Rolling Bearings. Tribol. Trans.; 2021; 64, pp. 708-720. [DOI: https://dx.doi.org/10.1080/10402004.2021.1909789]

39. Noguchi, S.; Korenaga, A.; Kanada, T. Occurrence Condition of Electric Current Density in Electrical Pitting. J. Adv. Mech. Des. Syst. Manuf.; 2010; 4, pp. 469-479. [DOI: https://dx.doi.org/10.1299/jamdsm.4.469]

40. Safdarzadeh, O.; Weicker, M.; Binder, A. Steady-state Thermal Analysis of the Contact in Bearings Exposed to Electrical Currents. Proceedings of the IECON 2021—47th Annual Conference of the IEEE Industrial Electronics Society; Toronto, ON, Canada, 13–16 October 2021; pp. 1-6.

41. Chiou, Y.-C.; Lee, R.-T.; Lin, C.-M. Formation criterion and mechanism of electrical pitting on the lubricated surface under AC electric field. Wear; 1999; 236, pp. 62-72. [DOI: https://dx.doi.org/10.1016/S0043-1648(99)00264-1]

42. Magdun, O.; Gemeinder, Y.; Binder, A.; Reis, K. Calculation of bearing and common-mode voltages for the prediction of bearing failures caused by EDM currents. Proceedings of the 8th IEEE Symposium on Diagnostics for Electrical Machines, Power Electronics & Drives; Bologna, Italy, 6–8 September 2011; pp. 462-467.

43. Lin, C.-M.; Chiou, Y.-C.; Lee, R.-T. Pitting mechanism on lubricated surface of babbitt alloy/bearing steel pair under ac electric field. Wear; 2001; 249, pp. 132-141. [DOI: https://dx.doi.org/10.1016/S0043-1648(01)00528-2]

44. Kaufman, H.N.; Boyd, J. The Conduction of Current in Bearings. ASLE Trans.; 1959; 2, pp. 67-77. [DOI: https://dx.doi.org/10.1080/05698195908972359]

45. Prashad, H. Theoretical analysis of the effects of instantaneous charge leakage on roller tracks of roller bearings lubricated with high resistivity lubricants under the influence of electric current. J. Tribol.; 1990; 112, pp. 37-43. [DOI: https://dx.doi.org/10.1115/1.2920228]

46. Prashad, H. Appearance of craters on track surface of rolling element bearings by spark erosion. Tribol. Int.; 2001; 34, pp. 39-47. [DOI: https://dx.doi.org/10.1016/S0301-679X(00)00133-X]

47. Prashad, H. Theoretical analysis of capacitive effect of roller bearings on repeated starts and stops of a machine operating under the influence of shaft voltages. J. Tribol.; 1992; 114, pp. 818-822. [DOI: https://dx.doi.org/10.1115/1.2920954]

48. Tischmacher, H.; Gattermann, S.; Kriese, M.; Wittek, E. Bearing wear caused by converter-induced bearing currents. Proceedings of the IECON 2010–36th Annual Conference on IEEE Industrial Electronics Society; Glendale, USA, 7–10 November 2010; pp. 784-791.

49. Prashad, H. Analysis of the effects of an electric current on contact temperature, contact stresses and slip band initiation on the roller tracks of roller bearings. Wear; 1989; 131, pp. 1-14. [DOI: https://dx.doi.org/10.1016/0043-1648(89)90242-1]

50. Liu, W. The prevalent motor bearing premature failures due to the high frequency electric current passage. Eng. Fail. Anal.; 2014; 45, pp. 118-127. [DOI: https://dx.doi.org/10.1016/j.engfailanal.2014.06.021]

51. Romanenko, A.; Ahola, J.; Muetze, A. Influence of electric discharge activity on bearing lubricating grease degradation. Proceedings of the 2015 IEEE Energy Conversion Congress and Exposition (ECCE); Montreal, QC, Canada, 20–24 September 2015; pp. 4851-4852.

52. Muetze, A.; Binder, A.; Vogel, H.; Hering, J. What can bearings bear?. IEEE Ind. Appl. Mag.; 2006; 12, pp. 57-64. [DOI: https://dx.doi.org/10.1109/IA-M.2006.248014]

53. Noguchi, S.; Kakinuma, S.-n.; Kanada, T. Measurement of Direct Current Voltage Causing Electrical Pitting. J. Adv. Mech. Des. Syst. Manuf.; 2010; 4, pp. 1084-1094. [DOI: https://dx.doi.org/10.1299/jamdsm.4.1084]

54. Romanenko, A.; Muetze, A.; Ahola, J. Effects of Electrostatic Discharges on Bearing Grease Dielectric Strength and Composition. IEEE Trans. Ind. Appl.; 2016; 52, pp. 4835-4842. [DOI: https://dx.doi.org/10.1109/TIA.2016.2596239]

55. Matsumoto, K.; Sasaki, A.; Yoshida, N.; Watanabe, H. Surface Damage under Tribological Conditions Induced by Electrical Discharge. Tribol. Online; 2019; 14, pp. 214-219. [DOI: https://dx.doi.org/10.2474/trol.14.214]

56. Komatsuzaki, S.; Uematsu, T.; Nakano, F. Bearing damage by electrical wear and its effect on deterioration of lubricating greases. Lubr. Eng.; 1987; 43, pp. 25-30.

57. Malec, D.; Bley, V.; Talbi, F.; Lalam, F. Contribution to the understanding of the relationship between mechanical and dielectric strengths of Alumina. J. Eur. Ceram. Soc.; 2010; 30, pp. 3117-3123. [DOI: https://dx.doi.org/10.1016/j.jeurceramsoc.2010.07.024]

58. Zhang, T.; Du, J.; Lei, Y.; Yi, X.; Yin, J.; Yu, P. Contributing to the understanding of the dielectric breakdown channel of alumina under high voltage. Ceram. Int.; 2019; 45, pp. 16608-16611. [DOI: https://dx.doi.org/10.1016/j.ceramint.2019.05.200]

59. Neusel, C.; Jelitto, H.; Schneider, G.A. Electrical conduction mechanism in bulk ceramic insulators at high voltages until dielectric breakdown. J. Appl. Phys.; 2015; 117, 154902. [DOI: https://dx.doi.org/10.1063/1.4917208]

60. Yoshimura, M.; Bowen, H. Electrical breakdown strength of alumina at high temperatures. J. Am. Ceram. Soc.; 1981; 64, pp. 404-410. [DOI: https://dx.doi.org/10.1111/j.1151-2916.1981.tb09879.x]

61. Touzin, M.; Gœuriot, D.; Fitting, H.-J.; Guerret-Piecourt, C.; Juvé, D.; Tréheux, D. Relationships between dielectric breakdown resistance and charge transport in alumina materials—Effects of the microstructure. J. Eur. Ceram. Soc.; 2007; 27, pp. 1193-1197. [DOI: https://dx.doi.org/10.1016/j.jeurceramsoc.2006.05.047]

62. Malec, D.; Bley, V.; Lebey, T. Investigations on dielectric breakdown of ceramic materials. Proceedings of the CEIDP’05. 2005 Annual Report Conference on Electrical Insulation and Dielectric Phenomena; Nashville, TN, USA, 16–19 October 2005; pp. 63-66.

63. Liebault, J.; Vallayer, J.; Goeuriot, D.; Treheux, D.; Thevenot, F. How the trapping of charges can explain the dielectric breakdown performance of alumina ceramics. J. Eur. Ceram. Soc.; 2001; 21, pp. 389-397. [DOI: https://dx.doi.org/10.1016/S0955-2219(00)00186-2]

64. Yin, Z.; Tao, S.; Zhou, X. Effect of the thickness on properties of Al2O3 coatings deposited by plasma spraying. Mater. Charact.; 2011; 62, pp. 90-93. [DOI: https://dx.doi.org/10.1016/j.matchar.2010.11.002]

65. Neusel, C.; Jelitto, H.; Schmidt, D.; Janßen, R.; Felten, F.; Schneider, G.A. Thickness-dependence of the breakdown strength: Analysis of the dielectric and mechanical failure. J. Eur. Ceram. Soc.; 2015; 35, pp. 113-123. [DOI: https://dx.doi.org/10.1016/j.jeurceramsoc.2014.08.028]

66. Gould, B.; Demas, N.; Erck, R.; Lorenzo-Martin, M.C.; Ajayi, O.; Greco, A. The effect of electrical current on premature failures and microstructural degradation in bearing steel. Int. J. Fatigue; 2021; 145, 106078. [DOI: https://dx.doi.org/10.1016/j.ijfatigue.2020.106078]

67. Gould, B.; Greco, A.; Stadler, K.; Vegter, E.; Xiao, X. Using advanced tomography techniques to investigate the development of White Etching Cracks in a prematurely failed field bearing. Tribol. Int.; 2017; 116, pp. 362-370. [DOI: https://dx.doi.org/10.1016/j.triboint.2017.07.028]

68. Katayama, T.; Ogitsu, T.; Kokumai, H.; Takemura, H.; Nakayama, T.; Mizoguchi, H. Study on effect of neutral voltages on shaft voltages causing bearing currents. Proceedings of the IECON 2013-39th Annual Conference of the IEEE Industrial Electronics Society; Vienna, Austria, 10–13 November 2013; pp. 4690-4693.

69. Plazenet, T.; Boileau, T.; Caironi, C.C.; Nahid-Mobarakeh, B. Influencing Parameters on Discharge Bearing Currents in Inverter-Fed Induction Motors. IEEE Trans. Energy Convers.; 2021; 36, pp. 940-949. [DOI: https://dx.doi.org/10.1109/TEC.2020.3018630]

70. Zika, T.; Gebeshuber, I.C.; Buschbeck, F.; Preisinger, G.; Gröschl, M. Surface analysis on rolling bearings after exposure to defined electric stress. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.; 2009; 223, pp. 787-797. [DOI: https://dx.doi.org/10.1243/13506501JET538]

71. Alger, P.; Samson, H. Shaft currents in electric machines. Trans. Am. Inst. Electr. Eng.; 1924; 43, pp. 235-245. [DOI: https://dx.doi.org/10.1109/T-AIEE.1924.5060981]

72. Pratt, J. Shaft voltages caused by alternating flux encircling the shaft. Proceedings of the 1995 Seventh International Conference on Electrical Machines and Drives; Durham, UK, 11–13 September 1995; pp. 198-202.

73. Torlay, J.-É.; Corenwinder, C.; Audoli, A.; Herigault, J.; Foggia, A. Analysis of shaft voltages in large synchronous generators. Proceedings of the IEEE International Electric Machines and Drives Conference; Seattle, WA, USA, 9–12 May 1999; pp. 607-609.

74. Chen, S.; Lipo, T.A. Circulating type motor bearing current in inverter drives. IEEE Ind. Appl. Mag.; 1998; 4, pp. 32-38. [DOI: https://dx.doi.org/10.1109/2943.644884]

75. Ong Raymond, K.J. An Investigation of Shaft Current in a Large Sleeve Bearing Induction Machine. Ph.D. Thesis; McMaster University: Hamilton, ON, Canada, 1999.

76. Schiferl, R.; Melfi, M. Bearing current remediation options. IEEE Ind. Appl. Mag.; 2004; 10, pp. 40-50. [DOI: https://dx.doi.org/10.1109/MIA.2004.1311162]

77. Vaimann, T.; Belahcen, A.; Kallaste, A. Changing of magnetic flux density distribution in a squirrel-cage induction motor with broken rotor bars. Elektron. Ir Elektrotechnika; 2014; 20, pp. 11-14. [DOI: https://dx.doi.org/10.5755/j01.eee.20.7.8018]

78. Boyanton, H.E.; Hodges, G. Bearing fluting [motors]. IEEE Ind. Appl. Mag.; 2002; 8, pp. 53-57. [DOI: https://dx.doi.org/10.1109/MIA.2002.1028391]

79. Naik, R.; Nondahl, T.A.; Melfi, M.J.; Schiferl, R.; Jian-She, W. Circuit model for shaft voltage prediction in induction motors fed by PWM-based AC drives. IEEE Trans. Ind. Appl.; 2003; 39, pp. 1294-1299. [DOI: https://dx.doi.org/10.1109/TIA.2003.816504]

80. Adabi, J.; Zare, F.; Ledwich, G.; Ghosh, A. Leakage current and common mode voltage issues in modern AC drive systems. Proceedings of the 2007 Australasian Universities Power Engineering Conference; Perth, Australia, 9–12 December 2007; pp. 1-6.

81. Ollila, J.; Hammar, T.; Iisakkala, J.; Tuusa, H. On the bearing currents in medium power variable speed AC drives. Proceedings of the 1997 IEEE International Electric Machines and Drives Conference Record; Milwaukee, Wl, USA, 18–21 May 1997; pp. MD1/1.1-MD1/1.3.

82. Muetze, A.; Binder, A. Practical Rules for Assessment of Inverter-Induced Bearing Currents in Inverter-Fed AC Motors up to 500 kW. IEEE Trans. Ind. Electron.; 2007; 54, pp. 1614-1622. [DOI: https://dx.doi.org/10.1109/TIE.2007.894698]

83. Boucenna, N.; Hlioui, S.; Revol, B.; Costa, F. A detailed analysis of the propagation paths of high-frequency common mode currents in AC motors. Proceedings of the 2013 15th European Conference on Power Electronics and Applications (EPE); Lille, France, 2–6 September 2013; pp. 1-6.

84. Schuster, M.; Masendorf, D.; Binder, A. Two PMSMs and the influence of their geometry on common-mode bearing currents. Proceedings of the 2016 XXII International Conference on Electrical Machines (ICEM); Lausanne, Switzerland, 4–7 September 2016; pp. 2126-2132.

85. Peng, B.; Wang, X.; Zhao, W.; Ren, J. Study on Shaft Voltage in Fractional Slot Permanent Magnet Machine With Different Pole and Slot Number Combinations. IEEE Trans. Magn.; 2019; 55, pp. 1-5. [DOI: https://dx.doi.org/10.1109/TMAG.2019.2898566]

86. Han, P.; Zhang, Y.; Kesgin, M.G.; Heins, G.; Patterson, D.; Thiele, M.; Ionel, D.M. On the Modeling of Bearing Voltage and Current in PWM Converter-fed Electric Machines Using Electromagnetic Finite Element Analysis. Proceedings of the 2021 IEEE Energy Conversion Congress and Exposition (ECCE); Vancouver, BC, Canada, 10–14 October 2021; pp. 4606-4610.

87. Muetze, A.; De Gersem, H.; Weiland, T. Influence of teeth and cooling ducts on the high-frequency common mode flux of inverter-fed ac machines. Proceedings of the Fourtieth IAS Annual Meeting. Conference Record of the 2005 Industry Applications Conference; Hong Kong, China, 2–6 October 2005; pp. 1350-1356.

88. Wang, F. Motor shaft voltages and bearing currents and their reduction in multilevel medium-voltage PWM voltage-source-inverter drive applications. IEEE Trans. Ind. Appl.; 2000; 36, pp. 1336-1341. [DOI: https://dx.doi.org/10.1109/28.871282]

89. Fu, C.; Zhu, W.; Zheng, Z.; Sun, C.; Yang, Y.; Lu, K. Nonlinear responses of a dual-rotor system with rub-impact fault subject to interval uncertain parameters. Mech. Syst. Signal Process.; 2022; 170, 108827. [DOI: https://dx.doi.org/10.1016/j.ymssp.2022.108827]

90. Busse, D.; Erdman, J.; Kerkman, R.J.; Schlegel, D.; Skibinski, G. Bearing currents and their relationship to PWM drives. IEEE Trans. Power Electron.; 1997; 12, pp. 243-252. [DOI: https://dx.doi.org/10.1109/63.558735]

91. Jagenbrein, A.; Buschbeck, F.; Gröschl, M.; Preisinger, G. Investigation of the physical mechanisms in rolling bearings during the passage of electric current. Tribotest; 2005; 11, pp. 295-306. [DOI: https://dx.doi.org/10.1002/tt.3020110402]

92. Sunahara, K.; Ishida, Y.; Yamashita, S.; Yamamoto, M.; Nishikawa, H.; Matsuda, K.; Kaneta, M. Preliminary Measurements of Electrical Micropitting in Grease-Lubricated Point Contacts. Tribol. Trans.; 2011; 54, pp. 730-735. [DOI: https://dx.doi.org/10.1080/10402004.2011.597543]

93. Prashad, H. The effects of current leakage on electroadhesion forces in rolling friction and magnetic flux density distribution on the surface of rolling element bearings. J. Tribol.; 1988; 110, pp. 448-455. [DOI: https://dx.doi.org/10.1115/1.3261649]

94. Hausberg, V.; Seinsch, H. Kapazitive Lagerspannungen und-ströme bei umrichtergespeisten Induktionsmaschinen. Electr. Eng.; 2000; 82, pp. 153-162. [DOI: https://dx.doi.org/10.1007/s002020050006]

95. Tischmacher, H.; Gattermann, S. Investigations on bearing currents in converter-fed electrical motors. Proceedings of the 2012 XXth International Conference on Electrical Machines; Marseille, France, 2–5 September 2012; pp. 1764-1770.

96. Zheng, J.; Xiang, D.; Li, H.; Quach, D.-C. An Investigation into the Effect of Bearing Grease Degradation on the High-frequency $ dv/dt $ Bearing Current in an Inverter-fed Motor System. Proceedings of the 2021 6th International Conference on Power and Renewable Energy (ICPRE); Shanghai, China, 17–20 September 2021; pp. 543-547.

97. Bhattacharya, S.; Resta, L.; Divan, D.M.; Novotny, D.W.; Lipo, T.A. Experimental comparison of motor bearing currents with PWM hard and soft switched voltage source inverters. Proceedings of the PESC Record. 27th Annual IEEE Power Electronics Specialists Conference; Baveno, Italy, 23–27 June 1996; pp. 1528-1534.

98. Niskanen, V.; Muetze, A.; Ahola, J. Study on Bearing Impedance Properties at Several Hundred Kilohertz for Different Electric Machine Operating Parameters. IEEE Trans. Ind. Appl.; 2014; 50, pp. 3438-3447. [DOI: https://dx.doi.org/10.1109/TIA.2014.2308392]

99. Kriese, M.; Wittek, E.; Gattermann, S.; Tischmacher, H.; Poll, G.; Ponick, B. Prediction of motor bearing currents for converter operation. Proceedings of the XIX International Conference on Electrical Machines-ICEM 2010; Rome, Italy, 6–8 September 2010; pp. 1-6.

100. Noguchi, S.; Fukuda, E.; Kanada, T. Effect of Oil Film Parameter on Vibration Acceleration and Electrical Pitting of Small Ball Bearing. Tribol. Online; 2012; 7, pp. 33-40. [DOI: https://dx.doi.org/10.2474/trol.7.33]

101. Smolenski, R.; Kempski, A.; Bojarski, J. Statistical approach to discharge bearing currents. COMPEL—Int. J. Comput. Math. Electr. Electron. Eng.; 2010; 29, pp. 647-666. [DOI: https://dx.doi.org/10.1108/03321641011028233]

102. Xu, Y.; Liang, Y.; Yuan, X.; Wu, X.; Li, Y. Experimental Assessment of High Frequency Bearing Currents in an Induction Motor Driven by a SiC Inverter. IEEE Access; 2021; 9, pp. 40540-40549. [DOI: https://dx.doi.org/10.1109/ACCESS.2021.3064685]

103. Reddy, S.; Basavaraja, B. Simulation and analysis of common mode voltage, bearing voltage and bearing current in two-level and three-level PWM inverter fed induction motor drive with long cable. Proceedings of the 2015 International Conference on Power and Advanced Control Engineering (ICPACE); Bangalore, India, 12–14 August 2015; pp. 221-226.

104. Chandrashekar, S.; Ramachandran, A.; Reddy, M.C. Simulation and experimental measurement of shaft voltage, bearing current in induction motor drive. Proceedings of the 2017 IEEE International Conference on Power, Control, Signals and Instrumentation Engineering (ICPCSI); Chennai, India, 21–22 September 2017; pp. 732-737.

105. Collin, R.; Yokochi, A.; Jouanne, A.V. Novel Characterization of Si- and SiC-based PWM Inverter Bearing Currents Using Probability Density Functions. Proceedings of the 2021 IEEE Energy Conversion Congress and Exposition (ECCE); Vancouver, BC, Canada, 10–14 October 2021; pp. 5146-5153.

106. Kriese, M.; Wittek, E.; Gattermann, S.; Tischmacher, H.; Poll, G.; Ponick, B. Influence of bearing currents on the bearing lifetime for converter driven machines. Proceedings of the 2012 XXth International Conference on Electrical Machines; Marseille, France, 2–5 September 2012; pp. 1735-1739.

107. Weicker, M.; Binder, A. Review on system parameters in variable speed AC-induction motor drives with parasitic electric bearing currents. Proceedings of the 2021 23rd European Conference on Power Electronics and Applications (EPE’21 ECCE Europe); Ghent, Belgium, 6–10 September 2021; pp. 1-8.

108. Mütze, A.; Binder, A. Experimental evaluation of mitigation techniques for bearing currents in inverter-supplied drive-systems-investigations on induction motors up to 500 kW. Proceedings of the IEEE International Electric Machines and Drives Conference, IEMDC’03; Madison, WI, USA, 1–4 June 2003; pp. 1859-1865.

109. Adabi, J.; Zare, F.; Ghosh, A.; Lorenz, R.D. Calculations of capacitive couplings in induction generators to analyse shaft voltage. IET Power Electron.; 2010; 3, pp. 379-390. [DOI: https://dx.doi.org/10.1049/iet-pel.2008.0332]

110. Duan, M.; Ou, Z.; Deng, C. Analysis of Shaft Voltage in Rotor Permanent Magnet Synchronous Motor System for Traction. Proceedings of the 2020 15th IEEE Conference on Industrial Electronics and Applications (ICIEA); Kristiansand, Norway, 9–13 November 2020; pp. 1908-1911.

111. Maki-Ontto, P.; Kinnunen, H.; Luomi, J. Three-phase model for the circuit simulation of common-mode phenomena and shaft voltages in ac motor drive systems. Proceedings of the IEEE International Conference on Electric Machines and Drives; San Antonio, TX, USA, 15 May 2005; pp. 437-443.

112. Mirafzal, B.; Skibinski, G.L.; Tallam, R.M.; Schlegel, D.W.; Lukaszewski, R.A. Universal Induction Motor Model With Low-to-High Frequency-Response Characteristics. IEEE Trans. Ind. Appl.; 2007; 43, pp. 1233-1246. [DOI: https://dx.doi.org/10.1109/TIA.2007.904401]

113. Tagami, K.; Ogasawara, S.; Funato, H.; Kanazawa, H.; Uesugi, M.; Moizumi, K. Analysis of shaft voltage and bearing current in an inverter-fed nongrounded motor. IEEJ Trans. Ind. Appl.; 2007; 127, pp. 286-292. [DOI: https://dx.doi.org/10.1541/ieejias.127.286]

114. Wang, Y.; Bai, B.; Liu, W. Research on discharging bearing currents of PWM inverter-fed variable frequency induction motor. Proceedings of the 2014 IEEE Conference and Expo Transportation Electrification Asia-Pacific (ITEC Asia-Pacific); Hangzhou, China, 22–25 October 2014; pp. 1-6.

115. Maki-Ontto, P.; Luomi, J. Induction motor model for the analysis of capacitive and induced shaft voltages. Proceedings of the IEEE International Conference on Electric Machines and Drives; San Antonio, TX, USA, 15 May 2005; pp. 1653-1660.

116. Muetze, A.; Binder, A. Calculation of circulating bearing currents in machines of inverter-based drive systems. IEEE Trans. Ind. Electron.; 2007; 54, pp. 932-938. [DOI: https://dx.doi.org/10.1109/TIE.2007.892001]

117. Kovanen, T.; Kettunen, L.; Söderlund, L. Mechanism behind high-frequency circulating bearing currents in view of motor structures. IET Sci. Meas. Technol.; 2008; 2, pp. 233-243. [DOI: https://dx.doi.org/10.1049/iet-smt:20070099]

118. Muetze, A.; Binder, A. Calculation of influence of insulated bearings and insulated inner bearing seats on circulating bearing currents in machines of inverter-based drive systems. IEEE Trans. Ind. Appl.; 2006; 42, pp. 965-972. [DOI: https://dx.doi.org/10.1109/TIA.2006.876083]

119. Postariu, D.; Chillet, C.; Roudet, J.; Boualem, B.; Périot, R. Experimental validation of the circulating-bearing-currents mathematical theory in the 20 MHz frequency range. Proceedings of the 2009 8th International Symposium on Advanced Electromechanical Motion Systems & Electric Drives Joint Symposium; Lillie, France, 1–3 July 2009; pp. 1-4.

120. Magdun, O.; Gemeinder, Y.; Binder, A. Rotor impedance of the high frequency circulating bearing current path in inverter-fed AC machines. Proceedings of the 2013 IEEE Energy Conversion Congress and Exposition; Denver, CO, USA, 15–19 September 2013; pp. 3512-3519.

121. Jaritz, M.; Jaeger, C.; Bucher, M.; Smajic, J.; Vukovic, D.; Blume, S. An improved model for circulating bearing currents in inverter-fed AC machines. Proceedings of the 2019 IEEE International Conference on Industrial Technology (ICIT); Melbourne, VIC, Australia, 13–15 February 2019; pp. 225-230.

122. Cacciato, M.; Consoli, A.; Finocchiaro, L.; Testa, A. High frequency modeling of bearing currents and shaft voltage on electrical motors. Proceedings of the 2005 International Conference on Electrical Machines and Systems; Nanjing, China, 27–29 September 2005; pp. 2065-2070.

123. Magdun, O.; Binder, A. High-Frequency Induction Machine Modeling for Common Mode Current and Bearing Voltage Calculation. IEEE Trans. Ind. Appl.; 2014; 50, pp. 1780-1790. [DOI: https://dx.doi.org/10.1109/TIA.2013.2284301]

124. Bubert, A.; Zhang, J.; De Doncker, R.W. Modeling and measurement of capacitive and inductive bearing current in electrical machines. Proceedings of the 2017 Brazilian Power Electronics Conference (COBEP); Juiz de Fora, Brazil, 19–22 November 2017; pp. 1-6.

125. Quabeck, S.; Grau, V.; De Doncker, R.W. Modeling and Mitigation of Bearing Currents in Electrical Traction Drives. Proceedings of the 2020 23rd International Conference on Electrical Machines and Systems (ICEMS); Hamamatsu, Japan, 24–27 November 2020; pp. 1101-1106.

126. De Gersem, H.; Henze, O.; Weiland, T.; Binder, A. Simulation of wave propagation effects in machine windings. COMPEL—Int. J. Comput. Math. Electr. Electron. Eng.; 2010; 29, pp. 23-38. [DOI: https://dx.doi.org/10.1108/03321641011007948]

127. De Vivo, B.; Lamberti, P.; Raimo, R.; Tucci, V.; Petrarca, C. Evaluation of the bearing voltage and the overshoot phase voltage in PWM inverter-fed by means of a multiconductor transmission line model. Proceedings of the 2014 International Symposium on Power Electronics, Electrical Drives, Automation and Motion; Ischia, Italy, 18–20 June 2014; pp. 1060-1064.

128. De Gersem, H.; Muetze, A. Finite-Element Supported Transmission-Line Models for Calculating High-Frequency Effects in Machine Windings. IEEE Trans. Magn.; 2012; 48, pp. 787-790. [DOI: https://dx.doi.org/10.1109/TMAG.2011.2172197]

129. Moreira, A.F.; Lipo, T.A.; Venkataramanan, G.; Bernet, S. High-frequency modeling for cable and induction motor overvoltage studies in long cable drives. IEEE Trans. Ind. Appl.; 2002; 38, pp. 1297-1306. [DOI: https://dx.doi.org/10.1109/TIA.2002.802920]

130. Svimonishvili, T.; Fan, F.; See, K.Y.; Liu, X.; Zagrodnik, M.A.; Gupta, A.K. High-frequency model and simulation for the investigation of bearing current in inverter-driven induction machines. Proceedings of the 2016 IEEE Region 10 Conference (TENCON); Singapore, 22–25 November 2016; pp. 55-59.

131. Dahl, D.; Sosnowski, D.; Schlegel, D.; Kerkman, R.; Pennings, M. Field experience identifying electrically induced bearing failures. Proceedings of the Conference Record of 2007 Annual Pulp and Paper Industry Technical Conference; Williamsburg, VA, USA, 24–28 June 2007; pp. 155-163.

132. Magdun, O.; Binder, A.; Purcarea, C.; Rocks, A.; Funieru, B. Modeling of asymmetrical cables for an accurate calculation of common mode ground currents. Proceedings of the 2009 IEEE Energy Conversion Congress and Exposition; San Jose, CA, USA, 20–24 September 2009; pp. 1075-1082.

133. Han, P.; Heins, G.; Patterson, D.; Thiele, M.; Ionel, D.M. Combined Numerical and Experimental Determination of Ball Bearing Capacitances for Bearing Current Prediction. Proceedings of the 2020 IEEE Energy Conversion Congress and Exposition (ECCE); Detroit, MI, USA, 11–15 October 2020; pp. 5590-5594.

134. Ren, X.; Liu, R.; Yang, E. Modelling of the Bearing Breakdown Resistance in Bearing Currents Problem of AC Motors. Energies; 2019; 12, 1121. [DOI: https://dx.doi.org/10.3390/en12061121]

135. Wang, Y.; Liu, W.; Chen, Z.; Bai, B. Calculation of high frequency bearing currents of PWM inverter-fed VF induction motor. Proceedings of the 2014 International Power Electronics and Application Conference and Exposition; Shanghai, China, 5–8 November 2014; pp. 1428-1433.

136. Kolbe, H.; Muetze, A.; Hameyer, K. Modelling of impulse currents in mechanical rolling element bearings. COMPEL—Int. J. Comput. Math. Electr. Electron. Eng.; 2012; 31, pp. 1575-1589. [DOI: https://dx.doi.org/10.1108/03321641211267001]

137. Birjukovs, M.; Jakovics, A.; Holweger, W. Modelling of thermal stresses in bearing steel structure generated by electrical current impulses. Proceedings of the IOP Conference Series Materials Science and Engineering; Riga, Latvia, 21–22 September 2017; 355, 012017.

138. Wang, Q.; Liu, R.; Ren, X. A Method to Determine the Critical Current Value of the Bearing Electrical Corrosion. Proceedings of the 2019 IEEE 3rd International Electrical and Energy Conference (CIEEC); Beijing, China, 7–9 September 2019; pp. 991-995.

139. Muetze, A.; Binder, A. Calculation of Motor Capacitances for Prediction of the Voltage Across the Bearings in Machines of Inverter-Based Drive Systems. IEEE Trans. Ind. Appl.; 2007; 43, pp. 665-672. [DOI: https://dx.doi.org/10.1109/TIA.2007.895734]

140. Jun, J.-h.; Lee, C.-k.; Kwon, B.-i. The analysis of bearing current using common mode equivalent circuit parameters by FEM. Proceedings of the 2005 International Conference on Electrical Machines and Systems; Nanjing, China, 29 September 2005; pp. 49-51.

141. Jaeger, C.; Grinbaum, I.; Smajic, J. Numerical simulation and measurement of common-mode and circulating bearing currents. Proceedings of the 2016 XXII International Conference on Electrical Machines (ICEM); Lausanne, Switzerland, 4–7 September 2016; pp. 486-491.

142. Magdun, O.; Binder, A.; Gemeinder, Y. Representation of iron core and dielectric losses for calculation of common mode stator ground currents in inverter-fed AC machines. Proceedings of the 2010 12th International Conference on Optimization of Electrical and Electronic Equipment; Brasov, Romania, 20–22 May 2010; pp. 371-376.

143. Magdun, O.; Binder, A. An iron core impedance model for calculating high frequency common mode currents and shaft voltages in inverter-fed AC machines. Proceedings of the International Symposium on Power Electronics Power Electronics, Electrical Drives, Automation and Motion; Sorrento, Italy, 20–22 June 2012; pp. 135-140.

144. Boucenna, N.; Hlioui, S.; Revol, B.; Costa, F. Modeling of the propagation of high-frequency currents in AC motors. Proceedings of the International Symposium on Electromagnetic Compatibility-EMC EUROPE; Rome, Italy, 17–21 September 2012; pp. 1-6.

145. Gubbala, L.; Von Jouanne, A.; Enjeti, P.; Singh, C.; Toliyat, H. Voltage Distribution in the Windings of an AC Motor Subjected to High dv/dt PWM Voltages. Proceedings of the PESC’95-Power Electronics Specialist Conference; Atlanta, GA, USA, 18–22 June 1995; pp. 579-585.

146. Oyegoke, B. Voltage distribution in the stator winding of an induction motor following a voltage surge. Electr. Eng.; 2000; 82, pp. 199-205. [DOI: https://dx.doi.org/10.1007/s002020050011]

147. Sunitha, P.; Banakara, B.; Reddy, S. Modeling, simulation and analysis of common mode voltage, bearing voltage and bearing current in PWM multilevel inverter fed induction motor with long cable. Proceedings of the 2017 2nd IEEE International Conference on Recent Trends in Electronics, Information & Communication Technology (RTEICT); Bangalore, India, 19–20 May 2017; pp. 1161-1167.

148. Hamrock, B.J.; Dowson, D. Ball bearing lubrication: The elastohydrodynamics of elliptical contacts. J. Lubr. Tech.; 1982; 104, pp. 279-281. [DOI: https://dx.doi.org/10.1115/1.3253193]

149. Prashad, H. Theoretical evaluation of impedance, capacitance and charge accumulation on roller bearings operated under electrical fields. Wear; 1988; 125, pp. 223-239. [DOI: https://dx.doi.org/10.1016/0043-1648(88)90115-9]

150. Wittek, E.; Kriese, M.; Tischmacher, H.; Gattermann, S.; Ponick, B.; Poll, G. Capacitances and lubricant film thicknesses of motor bearings under different operating conditions. Proceedings of the XIX International Conference on Electrical Machines-ICEM 2010; Rome, Italy, 6–8 September 2010; pp. 1-6.

151. Gemeinder, Y.; Schuster, M.; Radnai, B.; Sauer, B.; Binder, A. Calculation and validation of a bearing impedance model for ball bearings and the influence on EDM-currents. Proceedings of the 2014 International Conference on Electrical Machines (ICEM); Berlin, Germany, 2–5 September 2014; pp. 1804-1810.

152. Safdarzadeh, O.; Weicker, M.; Binder, A. Measuring of electrical currents, voltage and resistance of an axial bearing. Proceedings of the 2021 International Aegean Conference on Electrical Machines and Power Electronics (ACEMP) & 2021 International Conference on Optimization of Electrical and Electronic Equipment (OPTIM); Brasov, Romania, 2–3 September 2021; pp. 376-382.

153. Adabi, J.; Zare, F.; Ledwich, G.; Ghosh, A.; Lorenz, R.D. Bearing damage analysis by calculation of capacitive coupling between inner and outer races of a ball bearing. Proceedings of the 2008 13th International Power Electronics and Motion Control Conference; Poznan, Poland, 1–3 September 2008; pp. 903-907.

154. Wittek, E.; Kriese, M.; Tischmacher, H.; Gattermann, S.; Ponick, B.; Poll, G. Capacitance of bearings for electric motors at variable mechanical loads. Proceedings of the 2012 XXth International Conference on Electrical Machines; Marseille, France, 2–5 September 2012; pp. 1602-1607.

155. Liu, J.Y.; Tallian, T.E.; McCool, J.I. Dependence of Bearing Fatigue Life on Film Thickness to Surface Roughness Ratio. A S L E Trans.; 1975; 18, pp. 144-152. [DOI: https://dx.doi.org/10.1080/05698197508982757]

156. Magdun, O.; Binder, A. Calculation of roller and ball bearing capacitances and prediction of EDM currents. Proceedings of the 2009 35th Annual Conference of IEEE Industrial Electronics; Porto, Portugal, 3–5 November 2009; pp. 1051-1056.