A Review of Modeling and Mitigation Techniques for Bearing Currents in Electrical Machines With Variable-Frequency Drives

The converter switching in variable-frequency drives can generate high frequency common mode voltage between the machine winding and the converter ground, leading to high frequency parasitic currents which can flow through the machine bearings unless precautions are taken in design and installation. These parasitic and unintended currents in the bearings cause deterioration of the lubrication film and surface damage to the rolling elements of the bearings. These problems will be exacerbated as wide-bandgap semiconductors with faster switching rise times start becoming more widespread in variable-frequency drives. This paper reviews the modelling and mitigation techniques of high frequency bearing currents in inverter fed AC drives. It aims to provide a solid base for the research community to further understanding the bearing currents phenomenon and helping to find novel improved technique for their mitigation and measurement.


I. INTRODUCTION
In 1920s, bearing current were observed with coking of the oil, pitting of the bearing and scoring of the shaft [1]. At that time, which predated the emergence of modern power electronics, most AC machine were fed directly from sinusoidal line voltage. It is recognized that the magnetic field asymmetries, aligned with the machine shaft, induce voltage in it during operation, generating the ''shaft voltage''. The asymmetries are inherently caused by the manufacturing displacement of the machine even within the tolerances. Additional causes of these asymmetries are the anisotropy of magnet's remanence and permeability of iron core materials. Shaft voltage and bearing current fed by sine wave are somehow with the orders of the fundamental frequency. Low frequency The associate editor coordinating the review of this manuscript and approving it for publication was Giambattista Gruosso . bearing currents are not the focus of this paper, hence they will not be discussed in detail.
From the 1970s onwards, variable frequency drives (VFDs) have been deployed increasingly across many industry sectors. due to their advantages of flexibility, efficiency and high performance in torque and speed control. A growing proportion of AC machines, mainly induction machines (IMs) and permeant magnet synchronous machines (PMSMs), are now inverter-fed across a wide range of power levels. The typical switching frequency values of power electronic devices (IGBT, MOSFET, etc.) controlled using pulse width modulation (PWM) techniques, are several kHz up to tens of kHz. The high switching frequency generates a high frequency common mode voltage (CMV), which subsequently results in shaft voltage which can in turn give rise to a bearing current through the capacitive and inductive coupled paths in the AC machine.
Once the bearing current density exceeds a particular threshold, the metallic structure and lubricants of the bearings could be progressively damaged, hence significantly curtailing bearing life. According to the surveys in [1], around 30 percent of all motor failures are resulted from electrical bearing damage. A questionnaires-based survey of the bearing failures with 3934 samples of sinusoidal drive and 286 samples of variable speed drive showed that the drives fed by converters are 12 times more likely to exhibit bearing failures than the drives fed by line voltage [2]. Hence, additional attention should be paid to the bearing currents in machines fed by PWM inverters.
Nowadays, the emerging device technologies of SiC and GaN has allowed significant increases in switching frequency and given rise to dramatic increases in dv dt. Whereas the high switching frequencies and lower switching loss capability of SiC and GaN provide a route to more compact and power dense inverters, the increases in dv dt exacerbates the stress on the bearings.
Many studies have been carried out to understand the underlying mechanisms involved in bearing currents and corresponding mitigation methods. Four types of current damage on bearings are reviewed in [3], viz. frosting, spark tracks, pitting and welding. Actual field cases and solutions are reviewed in [4] but without detailed theoretical and quantitative analysis. A top-level introduction to bearing currents is presented in [5] and mainly focuses on presenting commercial, and straightforward techniques for mitigation of bearing current in classic PWM inverter drives, e.g. inverter output filters and insulated bearings. Methods requiring special hardware or motor modifications are not discussed. An evaluation of mitigation techniques for bearing currents, electromagnetic interference (EMI) and over-voltages in adjustable speed drive applications is carried out in [6]. It serves as a useful reference tool for to perform the necessary measurements and then determine if there is a potential VFD-related problem. If a problem is detected, one of the suggested mitigation techniques can be applied. These mitigation techniques include, inter-alia, shaft grounding system, insulated bearings and journals, ceramic bearings, conducting grease, a Faraday shield, and a new dual-bridge inverter designed to eliminate the common mode voltage. An overview of various modelling techniques for an AC machine and inverter system, specifically for the mid-and high-frequency ranges, is described in [7]. However, these models are mainly developed for EMI analysis and hance have limited read-across the bearing currents because the frequency of EMI phenomenon is in the range of several to tens of MHz, which are higher than the frequency of bearing currents. The mechanism and mitigation techniques of shaft voltages and bearing currents in electric vehicle and hybrid electric vehicle motors are reviewed in [8]. An overview of the techniques for monitoring the state of bearings as well as low frequency conventional and high frequency inverter generated shaft voltages is presented in [9]. However, neither [8] no [9] report on any thorough modelling or computation. In existing literature, there is arguably a lack of a systematic in-depth review on bearing currents in inverter driven machines which encompasses the underlying physical mechanisms, modelling, prediction and mitigation strategies and associated technologies. This paper aims to review these inter-linked aspects of research on bearing currents with a view to providing a comprehensive review of beating current phenomena.

A. COMMON MODE VOLTAGE
In this section, the classic configuration of a 3-phase full bridge PWM inverter is used as an example to demonstrate the mechanism by which common mode voltages are produced. The output voltage of each phase is switched between +V dc 2 and -V dc 2. In a star-connected machine, the neutral to ground voltage is the zero-sequence voltage, expressed in (1).
The frequency of the common mode voltage is 3 times the switching frequency and the switching action produces very high dv dt, typically up to 10kV /µs, as shown in Fig. 1. The works [10], [11] have demonstrated that the common mode voltages generated by a PWM technique, in an invertermotor system, couple through parasitic capacitances from stator windings to the rotor body and then return through the motor bearings to the grounded stator case as a closed loop circuit. This common mode voltage has been considered as the source of the bearing currents in inverter-motor systems.
The experimental study reported in [12] demonstrates that the RMS value of common mode voltage increases with an increase of switching frequency and a decrease of motor speed. However, the peak magnitude is affected by switching frequency only. Medium-voltage inverters (i.e. operating from DC links of 1kV and upwards) usually employ different power converter topologies and switching strategies compared low-voltage converter, but they still generate common-mode voltages and currents which can cause a very high dv dt [13]. The common mode voltage of the following 5 different PWM methods are compared in [14] in terms of the frequency, magnitude, modulations index, control index, voltage distortion factor, etc.: the fully controlled operation (FCO), the classical space vector modulation (SVM), the three active vector modulation (3AVM), the two additional space vectors (AZVC-2) and the one additional inverse voltage vector (AZVC-1). Since the common mode voltage is the fundamental source of bearing currents, considerable research efforts have been directed towards the reduction of the common mode voltage from the source. Proposed solutions are mainly focused on new topologies and new modulation technologies such as multilevel inverters and dual-bridge inverters.

B. FOUR TYPES OF BEARING CURRENT MECHANISMS
Traditional AC machine models tend to neglect high frequency phenomena. However, in the high frequency domain, which in the context of electrical machines corresponds to frequencies of tens of kHz or higher, the stray capacitances or parasitic capacitances in the machine provide a path for the bearing current between two conducting region which are separated by an insulating layer of material conductors naturally form a capacitor. In the conventional 3-node lumped parameter model, the main parasitic capacitances in the machine have been identified as winding-stator frame capacitor C wf , winding-rotor capacitor C wr , and stator-rotor capacitor C rf , as shown schematically in one stator slot in Fig. 2. An example of the equivalent circuit model was built based on the machine structure as shown in Fig. 3.
Muetze, Binder et al. have published numerous papers [15], [32] on systematic research into bearing currents in IMs. They noted that bearing currents can be classified into four types based on their different mechanisms [15].

1) CAPACITIVE BEARING CURRENTS
The common mode voltage has a divider V b across the bearings when the lubricant film formed in operation and the bearings act as capacitors, C b1 and C b2 . The bearing voltage ratio (BVR), which is a key indicator of the bearing voltage, is defined in (2).
The resulting capacitive bearing current can be calculated by (3).
If the speed is low (< 100rpm), the bearing ball may contact the races and hence the bearing is equivalent to a resistor. The amplitude of this bearing current is typically ≤ 200mA.

2) ELECTRIC DISCHARGE MACHINING (EDM) CURRENTS-i EDM
If the v b is high enough to cause electrical breakdown of the lubricant film, the energy stored in the rotor-frame capacitance discharges and generates an impulse current. The dielectric lubricant can flow into the channel and form a new insulation film, and thus the bearing capacitance charges again. This breakdown-recovery-breakdown process repeatedly continues. The so called EDM current damages the bearing with each impulse as the discharged energy damages the metal surface. This process progressively increases the presence of metallic particles in the lubricant, which exacerbates on-going electrical breakdown. The vicious circle accelerates bearing failure.

3) CIRCULATING BEARING CURRENTS-i CB
The circulating bearing currents were first described by S. Chen et al. in [33] and [34]. The common mode voltage generates a high frequency common mode current which flows in the winding→stator→ground path. This high frequency current induces a common mode flux which further generates a high frequency voltage along the shaft. The shaft voltage causes the circulating bearing currents in the loop: shaft → driving end bearing→stator frame→non driving end bearing→shaft.

4) ROTOR GROUND CURRENTS-i RG
The rotor can be grounded via load in some cases. If the impedance of rotor to ground is much lower than the stator to ground impedance, a proportion of the ground current will flow from the rotor through the bearing, since the current flows predominantly in the path with the lowest impedance. This type of current is noted as rotor ground current.
Since the capacitive bearing currents are usually too small to be harmful to the bearings, in some literature only the last 3 types of bearing currents are recognized [35], [36]. Fig. 4 shows schematic current flow loops of the common mode current i com and the other 3 types of bearing currents.
The rotor ground current should be investigated on a caseby-case basis as it is only present if the rotor is grounded, meaning it depends on the configuration of the load. The different load configurations are generally regarded out of scope of the ''inverter-cable-machine'' system modelling. Hence, the main interest in published studies has been on the modelling and prediction of the EDM and circulating currents.
In order to fully analyze bearing currents, a closed system modelling is needed, which includes inverter, cables, and an appropriate fidelity model of the machine. In some applications, the cables which connect the inverter to the machine could be tens of meters long. The impedance of these cables, including the grounding impedance, directly affects the common mode current i cm , and eventually influence the nature and magnitude of the bearing currents. Second order RLC-circuits, ladder-circuits and/or transmission-line representations are used to model the cables with due account the wave propagation phenomena. Some research on the EMI of electrical machines have put considerable effort into modelling the cables in the HF domain [37], [39]. These various studies also contribute to the understanding of bearing currents, in which only with the cable model is an important element in determining the overall system level behavior.

III. CAPACITIVE EDM CURRENT
The generic theoretical bearing current mechanisms fed by PWM inverters have been described in the previous sections. However, the lumped parameter circuit models require significant in-depth analysis to establish representative values of the circuit parameters which reflect the complex geometry and material composition of a typical electrical machine.

A. MODELING BASED ON THE MEASUREMENTS
For the purpose of extracting the circuit parameters, magnitude and phase of the impedance between the windings, the stator frame and the rotor were extracted from the measurements with signal generator excitations, often using an impedance analyzer. The measured output waveforms are used as a reference for calibrating the response of LCR circuit models [40], [41]. Indeed, in contrast to analytical methods based on representations of the physics of the actual geometry, measurement based methods treats the parasitic parameters as something of a black-box, and the equivalent circuit model and component values are formulated to match the measured responses.
A universal high-frequency lumped-parameter model which can represent the delta-connected IMs was proposed in [42]. The parameters were calculated by an iterative method to match the measured common and differentialmode impedance curve. Based on this model, the EDM current and common mode current can be predicted. Similar approaches were employed by [43], [44], [45], and [46], despite different complexity and data fitting algorithms, e.g. the trial-and-error method [43], the least-squares data fitting [44], the vector fitting technique [45] and the genetic algorithm [46].

B. ANALYTICAL MODELING
The use of measurement to characterize a particular machine is not the only method that can be used identify the capacitance parameters. They can be analytically and straightforwardly calculated by simplifying the machine geometry and drawing on well-established formulae for the capacitive of different electrode configuration. The plain plate and the cylinder model of capacitors are commonly used to calculate the rotor-to-frame capacitance C rf , the winding-to-frame capacitance C wf , and the winding-to-rotor capacitance C wr .
The rotor-to-frame capacitance C rf (Fig. 5) is considered as a cylinder capacitor with a Carter coefficient k c used to correct the opening slot effects. It has been shown that neglecting the cage rings can result in a 15%-30% underestimate of C rf in squirrel cage IMs [22]. The winding-to-frame capacitance C wf (Fig. 6) can be calculated by adopting an approximation to a plain parallel plate capacitor. The area of the electrodes is the circumference of stator slot multiplied by the axial length of stator lamination stack.
The winding-to-rotor capacitance C wr (Fig. 7) is modelled as Q s plate capacitors in parallel, where Q s is the number of slots. Again, each parallel plate capacitor is modelled as several capacitors in series, such as the capacitance of the air gap, the slot, the slot wedge, and the upper slot insulation. In Fig. 7, d ag is the air gap length, w so is the slot opening width, d st is the height of the teeth tip, d w is the height of the wedge, and d si is the height of the insulation layer.
Due to its simplicity and direct physical link to known dimensions, such simplified analytical methods are widely used to calculate the parasitic capacitances for both IMs and PMSMs [15], [23], [47], [59].
However, the aforementioned analytical methods treat the complex stranded windings as a single node. Hence, it cannot capture the unequal sharing of voltage between various turns in one winding coil, resulting in errors in the predictions of common mode current, shaft voltage and bearing current. Furthermore, the end-winding effect is usually neglected in such simplified analytical methods, which can underestimate the winding-rotor capacitance C wr by ∼90% as reported in [60].

C. 2D FINITE ELEMENT MODELING
To account for the stranded coil effect, researchers have employed electrostatic finite element methods (FEM) to perform detailed electrical field calculations and so extract values of machine capacitances. The mutual capacitances of multi-conductor system can be considered by the capacitance matrix method. Considering the stator-frame, the windings, the rotor as 3 nodes in the machine, the relationship between the voltage of windings V w , the voltage of rotor V r , the voltage of stator V s , the electric charge of windings Q w , the electric charge of rotor Q r , the electric charge of stator Q s can be expressed as: In [61], 2D FEM was used to calculate the capacitance, comparing the results with the analytical method. The comparison showed that there is a large difference between the methods in the calculation of the winding to magnet capacitance in a PMSM. This can be explained as the simplified model neglects the edge effects in which flat or cylindrical based capacitances cannot take such phenomena into consideration. A 2D FEM, which considers a double-layer winding wound from rectangular wires insulated with Kapton CR tape insulation, was investigated in [62] for a traction induction machine. The capacitance results were shown to be in accordance with measured values. Additional 2D FEM approaches to capacitance calculations can be found in [62] and [66].
The overall stray capacitance C wf is considered to be composed from a number of distributed ( turn-to-turn C tt and turn-to-core C tc ) capacitances connected in series-parallel combination [62]. The turn-to-turn detailed coil model is illustrated in Fig. 8 (a) whereas the random wound coil is illustrated in Fig. 8 (b).

D. 3D FINITE ELEMENT MODELING
One straightforward method to account the end-winding effects is to use 3D model. A 3D FEM model of an induction machine with end-windings (shown in Fig. 9) was constructed in [67], for the purposes of comparing the 2D and 3D models without end windings. In [60], 3D FEM models were built with end-windings to investigate the electrostatic shielding effects and compare the winding-rotor capacitances of models with and without end-windings. The results showed that the end-windings have significant effects on  the winding-to-rotor capacitance, and neglecting their contribution can underestimate the winding-rotor capacitance by 40%-90%.
3D FEM analysis has also applied to salient pole synchronous machine in [68], taking into consideration the endwinding effect The parameters of capacitances, extracted from the 3D FEM model, were fed into the equivalent circuit model. The calculated common mode impedance curve matches the measured values over a wide frequency range [69].
Additionally, it is also possible to include the influence of the rotor end-plate into the 3D model. Hence, the rotor-toplate capacitance can be analyzed. A comparison of merits and drawbacks of different parameter identification methods for parasitic capacitances in electrical machines is summarized in TABLE 1.
The calculated or measured capacitances can be incorporated into the common mode circuit (Fig. 3). When used in combination with a bearing circuit model, the EDM current can be predicted.

IV. INDUCTIVE CIRCULATING BEARING CURRENT
The circulating bearing current i CB whose path was shown schematically in Fig. 4 is generated by the induced shaft voltage which results from the common mode current i com . The circulating bearing current and the common mode current are coupled via the common mode flux. The mechanism can be considered as a transformer with the primary side having common mode current and the secondary side with  circulating bearing current. Fig. 10 shows the path of common mode current and the distribution of circumferential flux [36].

A. CIRCUIT MODEL OF CIRCULATING BEARING CURRENT
Due to skin effect, the high frequency circulating current follows a zigzag path in the stator laminated core. One equivalent circuit model which can be used to represent this behavior is a frequency dependent mutual inductance M ω or a current controlled voltage source [56], which are key elements for modelling the circulating bearing current and the shaft voltage, as shown in Fig. 11.
The common mode voltage generates the common mode current which flows via the winding-to-stator capacitor into the stator laminations. The impedance of the stator core is strongly frequency-dependent due to the skin effect. The skin and proximity effects are taken into account by modeling the resistances and inductances by ladder circuits such as those shown in Fig. 12 [70]. Typically, 3-4 orders of ladder circuits are used to represent the measured impedance curve with a wide frequency band.

B. ANALYTICAL CALCULATION OF i CB
In a similar manner to the parasitic capacitance, analytical methods can be applied to calculate the impedance of the circulating current loop based on a few simplified assumptions.  An approach to common mode circumferential flux calculation is presented in [71] and [72]. Based on the common mode circumferential flux, an analytical calculation of the mutual inductance and the resulting circulating current path impedance was reported in [22]. This study noted that the calculated maximum bearing current ratio (BCR), which the ratio of the circulating bearing to common mode current, is approximately 0.35.
In this study reported in [22], the analytical calculation of circumferential flux and induced voltage can be performed using: where N Fe is the number of sheets of stator core, d se is the outer diameters of stator lamination, d si is the inner diameters of stator lamination, h s is the height of stator slot, δ s is the skin depth and f is the frequency of common mode current. These analytical formulas are derived based on the assumption that the common mode current flows into each stator lamination sheet equally, denote as i sheet , and flows into the neighboring sheet through a zig-zag path in the surface of laminations, due to skin effects, until the very last sheet as shown in Fig. 13. The analytical method and the measured data fitting were combined to develop a 3 phase IM model to analyze the capacitive and inductive shaft voltage simultaneously in [73].

C. FINITE ELEMENT CALCULATION OF i CB
The investigation reported in [63] analyzed the resistances and inductances based on a FEM eddy current model which proposed a high frequency lumped model for PMSM. The parameters were extracted by using a 2D FEM model. The resistance and inductance of each individual winding turn were calculated by time harmonic finite element analysis. The self-capacitance of each turn and the mutual capacitances between turns were evaluated by electrostatic FEM calculations. The distributed parameter winding circuit was reduced to a π model branch by Kron reduction technique.
In a study reported in [74] the common mode flux was calculated by 2D axisymmetric FEM and combined with an analytical method. It was demonstrated that the rotor impedance cannot be neglected at high frequency as it is often comparable to the stator impedance. Moreover, it was shown VOLUME 10, 2022 that the circulating bearing currents can be mitigated if the rotor impedance is significantly increased.
An elaborate 3D model of the induction motor in [75] takes into account the nonlinear B-H characteristics of ferromagnetic cores, skin effect in cage bars, magnetic field space harmonics, rotational motion of the rotor as well as the anisotropy of the laminations.
A full-3D FEM field analysis including inductive and capacitive couplings would be too demanding in terms of computational resources. It also presents stability problems as the frequency range of interest varies from a few Hz to MHz. To avoid these difficulties, a 10 • section of the PMSM was modeled in 3D FEM in [76]. The mutual inductance M ω , and the circulating bearing-current loop inductance were obtained by solving the 3D magnetic stationary current distribution boundary value problem.
Once the induced flux and impedance of the laminations are calculated, the circulating bearing currents can be predicted.

V. COMBINED EDM AND CIRCULATING CURRENT MODELING
Distinguishing between different mechanisms and contributions to the overall bearing currents helps to understand their individual cause-and-effect chains, which in turn provides insight into effective mitigation methods. However, such a superposition approach with separate modelling of EDM current or circulating currents may lead to an underestimation of total bearing currents due to the fact that the bearings could be in a deeper breakdown condition with a smaller resistance under the two types of bearing currents than only one type of bearing currents modelled. A combined model which can predict the EDM and circulating currents simultaneously is necessary to evaluate the risk of bearing damage.
Relatively few published studies have quantified the two bearing current components simultaneously.
A comprehensive high frequency model of an induction machine was proposed in [69], employing parasitic capacitances derived from 3D FEM. The stator core impedance is derived from a 3D FEM model of the eddy current  distribution in a representative model based on three laminations. A measured impedance curve fitting approach was used to identify the remaining parameters. The current path of 3D FEM model, which is shown in Fig. 14, matches with the assumption made in the analytical method (Fig. 13). Then the ladder circuit is used to represent the frequency-dependent impedance, shown as the ''surface impedance'' in Fig. 15, to complete the circuit model of EDM and circulating current. Fig. 16 shows a lumped parameter circuit model proposed in [76] in which the paths of i com , i EDM and i CB are clearly identified, and the parameters are derived by FEM.
The basic idea of the combined EDM and circulating bearing currents modelling is to connect the two different current paths into one complete circuit model. The capacitance of the circuit would be mainly geometry dependent, and the resistance and inductance depend on frequency (due to skin effects) and permeability (due to saturation of soft magnetic materials).

VI. MODELING OF A ROLLING ELEMENT BEARING
The bearing itself exhibits a complex nonlinear electrical impedance. The load, speed of revolution, temperature and lubricant conditions all affect the impendence of a rolling element bearing [77]. To predict the magnitude of the bearing currents, a detailed electrical bearing model is essential.

A. BREAKDOWN OF BEARINGS AND EQUIVALENT CIRCUIT MODEL
When stationary or rotating at low speed, the balls or rollers make metallic contact with the races and the impedance is predominantly resistive. With the shaft speed increasing, the bearing lubricant creates a dielectric oil film between balls and raceways. The thickness of the lubricant film, which is highly dependent on the viscosity, can be estimated by hydrodynamic theory. In this condition, the bearing acts essentially as a capacitor.
If the bearing voltage exceeds a certain threshold value, the electrical field strength is sufficient to generate a discharge path in the lubricating oil film between the metallic balls and the inner/outer races. This process results in electrical breakdown of bearings. During the process of breakdown, the bearing impedance changes its behavior from a predominantly capacitive state to being predominantly resistive. The typical threshold electric field strength is approximately 15 V/µ m [49]. Considering that in most beating the lubricant film thickness ranges from 0.1 − 2µ m, the reference breakdown voltage is therefore typically in the range 1.5 − 30V.
It is well known that the breakdown voltage varies with the machine speed. Increasing the shaft speed tends to increase the oil film thickness and hence on the one hand it become more difficult to initiate breakdown for a particular bearing voltage. However, at the same time, residual metallic particles in the lubricants pass the breakdown channel more frequently, allowing the discharge to happen more readily at a particular voltage. These two counteracting effects result in a more nuanced, and application specific, variation in breakdown voltage with speed variation.
Several different bearing equivalent circuit models have been proposed in literature, albeit they share many of the same principles and approaches. Bearings are commonly represented as a capacitor and resistors in a parallel/series arrangement with a switch to represent the breakdown or nonbreakdown state. Fig. 17 illustrates four examples of bearing equivalent circuit presented in literature.

B. ANALYTICAL MODELING OF BEARINGS
Busse et al. [49] proposed an analytical method for a-priori parameters estimation for a given bearing design. The model . Impendence models for a rolling element bearing (a [11];b [77], [78];c [79];d [80]). employed for ball bearings assumes several pairs of concentric spheres. Each capacitor pair includes an inner sphere (representing the balls) within an outer sphere (modeling the raceways). Furthermore, a theory of Hertzian contact area from [48] was drawn on in order to model the capacitance of bearings. The Hertzian contact area A H is taken as the corresponding electrode area [23]. This method is now widely accepted as an effective and convenient, is rather simplified, model for bearing capacitance.
Gemeinder et al. [79] developed an electrical bearing model based on detailed mechanical calculations for radial ball bearings. The bearing load, contact area and lubrication film thickness were taken into the consideration. The model demonstrates a good fit between the measured and calculated impedance in the frequency range from 10 kHz to 10 MHz. However, for frequencies outside this range, there were notable discrepancies between the model and measurements.

C. FEM MODELING OF BEARINGS
In addition to the analytical method, FEM has also been applied to bearing modelling. Jun et al. [81] modelled a bearing equipped with 14 rollers using 2D FEM. Each roller was considered as being electrically in parallel with the remaining rollers. The capacitance between outer race and roller (C bo ) and the capacitance between inner race and roller (C bi ) are series connected. The capacitance between outer race and inner race (C o ) is in parallel to the branch of C bo and C bi , as shown in Fig. 18.
The influence of gravity and/or an applied load can give rise to both an asymmetric ball position and an asymmetric shaft position, especially at low speeds. A 2D FEM analysis for extracting the bearing capacitance was performed in [82]. The bearing capacitance is not constant but has a nonlinear relationship with load and speed. 3D electrostatic field solver analysis was implemented by Cay et al. [83], where they calculated the bearing capacitance in pre-breakdown state.

D. EXPERIMENTAL TESTS
Prashad [84] carried out an experimental study on bearing discharge with different operating parameters. It focused on the threshold voltages and impedance response of non-insulated rolling element bearings under the influence of various levels of electrical currents. However, the test was undertaken with a low frequency source rather than a high frequency switching source.
During operation, the capacitances of the inner/outer raceway and balls are not constant but vary with the motor shaft speed and load. Zhu et al. [85] described the effect of load distribution (viz. gravity, unilateral magnetic pull and centrifugal force) and speed on the contact area and oil film thickness between each rolling element and the raceways and concluded that the speed has a marked influence on the film thickness.
Krzemien [86] performed an investigation of the electrical properties of ball bearings by testing the discharge current pulse number per revolution in DC, low frequency AC and high frequency AC bearing voltages. The test results showed that the increase in bearing voltage values causes the value of the bearing critical temperature (at which the bearing current pulse number per shaft revolution reaches its maximum) to decrease.
In contrast to the conventional constant resistance model which is commonly used to represent the breakdown state, a variable resistance model of bearings was proposed by Ren et al. [87]. It suggested that the breakdown process can be divided into 3 intervals: the starting interval, the stable interval, and the recovery interval. The resistance was described as 3 different time dependent values according to the arc discharge model principle.
A statistical approach was applied in [88] to study the EDM current since it appears intermittently and seems to depend on many complex factors such as a level of shaft voltage, motor speed and temperature. It was observed that the EDM currents occur within a reasonably steady window of shaft voltage, even though most of events occur at the higher levels of shaft voltage [89]. It was reported in [90] that the incidences of EDM current pulses are not necessarily synchronized with the edges of the common mode voltage. This phenomenon cannot be explained straightforwardly by the classical equivalent bearing model which is based on a comparison between the instantaneous bearing voltage and the fixed value of threshold breakdown voltage. However, it is recognized that this approach has a shortcoming in that the threshold breakdown voltage varies during operation. Hence, if the switch in the model can be activated by the energy stored in the lubricant instead of by the threshold voltage, the prediction of EDM current can be improved [80].
A statistical or enhanced model which considers the influence of the operating conditions, viz. shaft speed, temperature, load, lubricant decomposition, etc. on the bearing breakdown could provide the basis for improved switch control in the model for an improved EDM current prediction.

VII. MITIGATION METHODS
By using physically based models, it is possible to calculate the bearing current magnitude and hence devise and analyze the effectiveness of different mitigation methods within a drive system. At the system level, mitigation methods can be applied on the motor side, the inverter side, and on the connections. Schiferl et al. [91] summarized 7 different mitigation options which can be applied on the motor side. It was noted that effective remediation methods depend upon a thorough understanding of the potential current paths in each installation. Bjekic et al. [5] reviewed the mitigation methods applicable to existing installations with the straightforward addition of equipment. The influence of the cable type was discussed in [17] which also summarized different mitigation techniques such as filters, insulated bearings, and hybrid bearings on the bearing current phenomena for machines at 11 kW to 500 kW power levels.

A. MITIGATION WITHIN THE INVERTER
In order to reduce the bearing current from the source, mitigation methods can be used on the inverter side. These can include new modulation technologies, new topologies, or a combination thereof, with the aim of reducing the amplitude and/or dv/dt of the common mode voltage.
A simple PWM scheme which alleviates the common mode voltage problems in three level inverter applications was proposed in [92]. Zhao et al. [93] proposed a hybrid selective harmonic elimination pulse width modulation scheme to reduce the common-mode voltage in a three-level neutralpoint-clamped inverter-based induction motor drive for pump and fan applications. Chen et al. [94] performed a comparative study based on both experimental measurements and theoretical analysis on a 2.1kW induction machine. It is showed that the soft switching inverter did not assist in bearing currents mitigation compared to an equivalent hard switched converter. De Broe et al. [95] developed a space vector PWM control strategy for a full controlled rectifier and inverter combination to reduce the common mode neutral-toground voltage magnitude in induction machines from V dc to 2V dc /3. However, it requires a fully controlled rectifier, and hence is not suitable for drives fed with a diode rectifier. Moreover, the switching frequency must be synchronous between the rectifier and inverter, which is challenging to achieve in practice without compromising other aspects of performance.
A single-phase, multi-level, half-bridge inverter with opposite polarity to the main 3 phase inverter was used as active common mode voltage compensator in [96]. An additional 4th leg was introduced into a traditional 3 phase inverter along with a four phase LC filter in [97]. By appropriate modulation strategies, this configuration can eliminate the common mode voltage while outputting a sinusoidal lineto-line voltage. Von Jouanne et al. [98] proposed a dualbridge inverter that was found to virtually eliminate the shaft voltage and bearing currents. Additionally, it significantly reduced the leakage current with an appropriate modulation strategy. However, it must be used in combination with a dual-voltage motor operating at the lower voltage rating. The system complexity increases as a result of the additional 3 bridges, which adds the cost of 6 more IGBTs. Tekwani et al. [99] proposed a dual five-level inverter topology and switching strategy to eliminate the common mode voltage for open ended winding induction machines. Chandrashekar et al. [100] concluded that the common mode voltage, shaft voltage and bearing current of 3-level inverters are lower than equivalent 2-level converter. Furthermore, Sunitha et al. [101] note that a multi-level inverter generates lower v b and i b than a 3-level converter due to the reduction of dv/dt and increase of rise time. Further examples of modulation techniques can be found in [102] and [125] while a variety of novel converter topologies can be found in [126] and [154].
In summary, for conventional 3 phase 2-level inverters, the space vector can be selected to minimize the common mode voltage, but it cannot be eliminated completely by this approach. In order to eliminate the common mode voltage, new topologies with additional circuit elements are necessary. These new topologies include the 4th leg or the dual inverter to cancel the common mode voltage in the opposing polarity, cascade multi-level inverters to reduce the dv/dt, auxiliary circuits in the DC bus to actively control the DC voltage (i.e., reducing the common mode voltage amplitude). These inverter side mitigation methods act directly on the source so that they are effective for all four bearing current mechanisms.

B. MITIGATION ON THE CONNECTION
This subsection presents a review of mitigation methods applied to the connections between the machine and its associated inverter. These methods include passive/active filters, common mode chokes and shielded cables.
Passive filters, reactors and sinusoidal filters all act to reduce the dv/dt of the motor terminals. Employing filters can result in the reduction of ground current and circulating current for 30%-50%, as demonstrated to experimentally in [17]. As the filters do not eliminate the common mode voltage, the EDM current is more of less unaffected [17].
Additional examples of studies investigation the role of filters can be found in [155] and [183]. An active common-noise canceler (ACC) fed from the DC bus and fed into the connecting wires was designed by Ogasawara et al. [184], [185] to efficiently reduce conducted EMI as well as motor shaft voltage and bearing current.
The common mode choke at the inverter output is able to suppress the common mode current as suggested in [186]. Hence the ground current as well as the circulating current can be reduced. The EDM current remains unaffected as the common mode choke does not modify the common mode voltage.
The use of shielded cables can reduce the ground impedance between stator to ground, resulting in a reduction in the rotor ground current. On the other hand, it increases the stator ground current, which can lead to an increase in the circulating bearing current. This approach does not have any influence on the EDM current [17], [187].

C. MITIGATION IN THE MOTOR
Modifications on the motor can be carried out at design stage or following installation. Bearing current can be suppressed by increasing the magnitude of the impedance or reduced the electromagnetic coupling between winding, stator, and rotor.
Insulated bearings or coated bearings have been a longstanding and effective means of suppressing the low frequency ''classic'' bearing currents. With a discrete polymer sleeve or aluminum oxide layer on the outer races, additional resistance and capacitance is introduced into the bearing current path. In this way, all various bearing current mechanisms are reduced. The use of hybrid bearings with ceramic balls can also suppress entirely all types of bearing currents. However, hybrid bearings tend to be more expensive than the insulated bearings [17].
An electrostatic shielded induction motor (ESIM) was proposed in [188]. This motor was capable of reducing the shaft voltages below the bearing lubricant breakdown threshold. The insertion of a Faraday shield into the airgap can attenuate the electric field coupling between stator and rotor whereas its insertion in the slot can eliminate the electric coupling between windings and stator. This can suppress the bearing currents. Further electrostatic shield designs can be found in [189] and [193].
As the design stage, the motor can be modified to reduce the bearing currents with only marginal influence on its performance. Oblique slots can be incorporated with a view to reducing the winding-to-rotor capacitance, in turn reducing the BVR and hence reducing the bearing currents without affecting the machine performance. This approach has been investigated using 2D FEM but not experimentally [194]. If the slot fill factor is relatively low, there is space for re-arranging the winding configuration to reduce the winding-to-rotor capacitance [58]. Other machine design modifications can be found in literature, including inter-alia, insulated rotor [195,196], adding capacitors between the brackets and the stator core [197], adding capacitors between VOLUME 10, 2022 brackets and the neutral line of the inverter dc link [198], optimizing the magnetic flux-path in BLDC [52] and insulating circular comb-like coil in the stator slots [199].
Conducting brushes can be used to connect the shaft and the frame, creating a short circuit path in parallel with the bearings to limit the bearing voltage and hence prevent bearing breakdown. However, the brushes require periodic maintenance and occasional replacement. The use of multiple conductive microfibers to bridge the bearings using the electrical field emission effect was proposed in [27]. This acts as a mean of bypassing the bearings. This allows the bearing voltage to discharge while also delivering ultra-low friction. This solution, which was detailed in [28], proved effective in suppressing the EDM current.

D. SUMMARY
In this subsection, a summary of mitigation methods has been presented. Fig. 19 categories these methods based on their application: on the inverter, on the connection, on the motor. Improved modulation strategies and new topologies of inverters reduce the amplitude of the common mode voltage and/or reduce the dv/dt. They are considered to reduce the bearing currents from the perspective of the source.
Other mitigation methods are implemented along the various conduction paths either by increasing the impedance of the bearing current loop or by providing a low impedance path for the induced current to bypass the bearings. TABLE 2 presents a summary of comparison of merits and drawbacks of various mitigation methods. As indicated in TABLE 2, mitigation methods are not all effective for reduction of all the types of bearing current. Some methods could even lead to increment of certain type of bearing currents. This requires an understanding of bearing current mechanism before the selection of appropriate mitigation methods. Most mitigation methods come with additional cost, which again requires high accuracy modelling of bearing current to determine necessity of implementing the mitigation.

VIII. DISCUSSION AND FUTURE PERSPECTIVES
In literature, existing high frequency bearing current models solve the EDM and circulating bearing current components separately [22], [23]. These approaches are based on an arguably unproven assumption that the electric field and magnetic field behavior can be fully decoupled. Furthermore, these do not consider the nonlinearity of soft magnetic materials. Relatively few published studies have quantified the two bearing current components simultaneously [69], [76]. However, even in these cases, the windings were modelled as a single homogenous region, neglecting the turn-to-turn capacitance and mutual inductance, without accounting for magnetic saturation effect on high frequency eddy-currents in the stator and rotor laminations. The drawbacks of linearized representations can be illustrated by considering the classical skin depth equation and the magnetization curve of a typical stator core material (M270-35A). If the material non-linearity is not accounted for by adopting a single relative permeability of 700 at 1.5T [200], the skin depth is underestimated by 58% compared to the calculation which uses the relative permeability of 123 at 1.8T [200]. Thus, the corresponding resistance and inductance are overestimated by 138%. In machine cores, the flux densities vary spatially, in time, and with fundamental load current. Hence, using the permeability at an a-priori fixed flux density for skin depth and impedance can lead to significant error.
Based on the above discussion, future perspectives on bearing currents modelling may include: • Build a detailed winding model which includes endwinding, turn-to-turn capacitance, and turn-to-frame capacitance; • Consider the uneven distribution of voltage stress between turns, hence the different contribution of each turn to the total common mode current; • Develop a uniform model that can predict all bearing current types simultaneously; • Consider the non-linearity saturation of stator and rotor core caused by fundamental current and its influence on the bearing currents; • Include the effects of the connection cables into the bearing current model. After an accurate modelling of bearing currents with a inverter-cable-machine system, the bearing breakdown and lifetime may be evaluated, which includes: • A solid understanding of bearing breakdown: fluid dynamics to depict the lubricant oil film, molecular ionization process at microscopic level, and statistics description of bearing behaviors at macroscopic level; • Lifetime and maintenance cycle prediction; • Online condition monitoring of bearing operation. For the applications which requires high reliability or have difficulty to replace the bearings, proper mitigation methods should be selected carefully with a balance of effectiveness/reliability and additional cost/complexity. Possible works in the mitigation includes: • Novel modulation technologies of multi-phase or multi-level inverters to reduce CMV; • Novel machine designs to reduce the high frequency coupling but maintain the same level of electromagnetic performance; • Low-cost components to insulate or bypass the bearings.

IX. CONCLUSION
In this paper, the mechanisms of the different types of bearing currents in VFDs are introduced in detail. It is demonstrated that the common mode voltage generated by PWM switching is the fundamental source of the high frequency bearing currents. Even through the causes of bearing currents are clearly demonstrated in literature, the modelling of AC machines in high frequency and prediction of bearing currents is far from straightforward. A range of modelling and parameter identification methods have been introduced and compared, including curve fitting of measured data, analytical calculations, and 2D/3D FEM. Continually improving the accuracy of models while maintaining a competitive level of computational efficiency is challenging. This review has demonstrated that advances still need to be made in models which can accurately calculate the capacitive EDM current as well as the inductive circulating current. Some of the factors that need to be accounted for are end winding effects, turn-toturn capacitance, skin and proximity effects in stator and rotor lamination.