A Procedure to Isolate the Electrical Capacitance of Tooth Contacts From Charging Curve Measurements of a Gear Stage

Inverters with high voltage gradients are used in electric drives as this enables dynamic operation with minimized switching losses [1–7]. These voltage gradients cause parasitic voltage buildup and current flow through the bearings and gearing of the drive train [8–11]. The resulting discharges can damage both the lubricant and the metallic surfaces of the drive components, which in turn leads to premature failure [11–14]. In addition to the tribological boundary conditions, the applied voltage and the current flow during a discharge are relevant for these damage mechanisms [15,16]. The voltage at which a discharge occurs in the contact is in the single- to double-digit range [17]. To be able to estimate the probability of damage, numerous studies on the prediction of the applied voltage, the resulting damage, and modeling of the tribo-electric contact have been developed [1,3,11,18–28]. Determining the electrical behavior of a tribo-electric contact is relevant if the system analysis of the entire drive train is to be carried out. For example, the contact capacitance of the bearings of an electric motor influences the so-called the bearing voltage ratio (BVR), which is defined as the ratio of the voltage applied to the bearing and the operating voltage applied to the motor [1,3,10,29]. Something similar can be assumed for the gear stages of a drive train. However while the BVR in a drivetrain lies between 0.5% and 10%, the gear voltage ratio can be as high as 30% [1,11]. The focus of research to date has mainly been on rolling bearings, while the established method for determining the electrical behavior of a bearing with the associated lubricant is to measure the impedance [30–33,27]. The capacitance of the rolling bearing is modeled based on the capacitance of the tribological contact. The contact is modeled as a capacitor, with the Hertzian contact area corresponding to the area of the capacitor plates and the central lubricating film thickness corresponding to the distance between them, see Fig. 1(a).

In addition to the Hertzian contact itself, the inlet and outlet are modeled as separate capacitors, Fig. 1(b). The capacitance of the contact results from a parallel connection of these three capacitors. Gonda et al. used elastohydrodynamic lubrication (EHL) simulation to calculate the capacitance of inlet, outlet, and the Hertzian contact of a thrust bearing and calculated the remaining capacitances of the bearing using finite element simulation [34]. Alternatively, the capacitance of the contact can be estimated by multiplying the capacitance of the Hertzian contact with a correction factor, which is introduced to take the contribution of the inlet and outlet of the Hertzian contact area into account. The bearing's total capacitance is in turn built up from the contact capacitances of all rolling elements with the inner ring and outer ring, Figs. 1(c) and 1(d).

Compared to rolling bearings, capacitance measurements of gear stages are more complex because the tribological and, therefore, also the electrical conditions along the line of action of a gear stage vary considerably, see Fig. 2.

Depending on the position on the meshing line of the gears shown in Fig. 8, one or two pairs of teeth are in mesh at the same time. In the case of double meshing, the torque is distributed over two pairs of teeth, which reduces the contact surface. In addition, the relative speeds of the two tooth flanks change along the meshing line, which leads to a changing lubricant film thickness. The changing contact surface and lubricating film thickness in turn lead to a changing capacitance of the contact. Furthermore, the double tooth mesh is represented by two parallel capacitors instead of a single one in the case of a single mesh. As a result, the capacitance of a gear stage changes significantly more over time than the capacitance of rolling bearings [36].

There exist several quite different assumptions of suitable values for the correction factor $kC$ for rolling bearings, but only a few studies on the electrical behavior of gears [11,35–40]. Some of these approaches factor in the lubricating film thickness [11,12,41–46], while the others do not. The wide spread of values for $kC$ suggests that $kC$ is not only depending on the bearing or gear geometry but also on the surrounding components in a test rig, or application. Results of Gonda et al. [34] support this finding. However up to now, no procedure has been proposed that considers test rig specific effects in the determination of $kC$⁠. To address this issue, this study provides a novel approach to determine the correction factor of a gear stage while eliminating all relevant test rig effects.

This study uses both experimental measurements and bearinx simulations. The article describes how electrical capacitance measurements of a single-stage gearbox can be divided into its individual capacitances for contact zone, inlet/outlet, trailing tooth flank, as well as test rig specific capacitances.

The following section describes the experimental test setup, the simulation software bearinx, a simulation tool of Schaeffler AG & Co. KG, Germany, as well as the general methodology for determining the different capacitances occurring in a single-stage gearbox. Further on, comparisons between experimental measurements and bearinx simulations are presented to show the validity of a correction factor approach in the simulation of gear capacitances.

$εr$ and $kC$ must be entered by the user. However, both parameters are often unknown. This study shows how the term $(εrkC)$ can be determined experimentally. For the simulations, $kC$ was preset to 1. This means, the calculated capacitance $CSimulation$ is identical to the contact capacitance $CContact$ and calculated accurately. $εr$ was also preset to 1 as this simplifies the described procedure. Once the value for $(εrkC)$ is found, it can be easily multiplied to $CSimulation$ to get the capacitance of the gear stage. bearinx was used to simulate 20 different positions or time-steps along the line of action for each of the ten parameter combinations listed in Table 3. The calculation of these 200 calculation points, including export of the results as a .csv file, takes less than a minute.

For the experimental investigations, a transmission test rig was used, which can tense the two gears with two electric motors. The bearing seats are insulated by hybrid bearings with ceramic balls. The shafts are insulated by claw couplings. The voltage is applied by mercury slip rings at the end of each shaft, see Fig. 3. The torque is measured by torque flanges on both shafts. The two spur gears have straight involute teeth, a module of 3 mm, and a tooth width of 12 mm. The detailed parameters of the gears are given in Table 1, while Table 2 shows the relevant properties of the test oil. At a standstill, both gears are equally submerged in the oil sump up to about 30% of their diameter. Preliminary to the test program of this study, an extended run-in of the gear stage has been performed over a period of several days. Capacitance measurements during this time did not exhibit a significant change due to roughness alteration or lubricant degeneration.

For the experimental tests, the varied parameters are speed, torque, temperature, and contact width of the gears. The variation of contact width was realized via an axial shift of one of the gears. Four test runs, each varying one parameter, were carried out. The selected parameters are shown in Table 3. Each individual measuring point was tested three times with three measurements of the capacitance each, so that nine measurements were carried out for each of the ten individual operating points. Each capacitance measurement consisted of around 2500 evaluated charging curves.

In this article, the gear capacitance was determined using a charging curve measurement, as like Wittek et al. [5]. A square-wave signal is applied via a function generator (HAMEG, Munich, Germany/Model: HM8030-5) and the voltage between the two gear shafts is measured with an oscilloscope (National Instruments, Austin, TX/Model: NI USB-5133) with a sample rate of 50 m/s. In this article, the charging curves are applied with a frequency of 100 kHz and a peak-to-peak voltage of 1.1 V. The circuit diagram of the measurement setup is shown in Fig. 4, where the capacitance of the gear stage C_Gearstage and the resistance of the gear stage R_Gearstage are the variables to be determined.

A non-linear least squares approach was used for the fit. The applied voltage should be as low as possible so that there are no breakdowns during the measurement, which would result in a faulty changing curve, unsuitable for further evaluation. An example of a suitable charging curve in comparison to a faulty curve is shown in Fig. 5.

When calculating the contact capacitance, any imperfect charging curves can be eliminated by considering the coefficient of determination value R² of the fit. Fits from faulty charging curves generally exhibit a lower R² value and can therefore be sorted out. For the following discussion, only charging curves with a fit with R² values above 0.99 were viewed as fault-free and further considered.

At least 100 charging curves should be measured over one tooth engagement for a sufficiently accurate resolution of the capacitance of the tooth contact over time. As different tooth engagements can show slightly different capacities, multiple engagements are measured in to order the capacitive behavior of the gear stage as shown in Fig. 6.

It is apparent that the capacitance varies significantly during each of the 20 tooth meshes. There are also some smaller differences between the meshes. Therefore, one full revolution of the larger gear was measured to account for the statistical variation of the capacitive behavior of the whole gear stage.

This article focuses on the gear stage as a whole and the general tendencies of the gearbox capacitance as a function of various operating parameters will be examined. To account for the variations along the line of action, the measured capacitances of one full revolution are evaluated in a box plot.

This methodology was followed for each individual measurement of the capacitance at each parameter combination given in Table 3 and is subsequently discussed.

The measured capacitance of a gear stage can be subdivided into different parts, which depend on the geometry of its individual components, as well as on the operating conditions. The following describes a procedure to quantify these parts using both measurements and bearinx simulations with various operating parameters. In this study, the measured capacitance consists of the following components:

• The capacitance C_Contact represents the Hertzian contact area between the contacting teeth. C_Contact results from Eq. (1) if the correction factor $kC$ is set to 1 and it varies with speed, load, and temperature, see Fig. 7.

• The capacitance C_In/Outlet covers the inlet and outlet zone of the tooth contact. C_In/Outlet is expected to be linearly proportional to C_Contact and therefore also varies with speed, load, and temperature. It is considered in Eq. (1) by a correction factor $kC$ greater than 1. For rolling bearings, $kC$ is often assumed to be 3.5, Ref. [48].

• The capacitance C_Gear contains the remaining capacitance between the two gear wheels, which has not been attributed to C_Contact or C_In/Outlet. Mostly, C_Gear consists of the capacitance of the trailing tooth flanks and is therefore independent of C_Contact or C_In/Outlet.

• The capacitance C_Oilsump reflects the oil sump in the gearbox. This capacitance varies with the amount of lubricant and its permittivity. As both parameters were kept constant in the experiments, C_Oilsump is considered to be constant as well.

• The capacitance C_Rig encapsulates all other capacitances of the rig that affect the measurement and are not covered by aforementioned capacitances. C_Rig is also assumed to be constant in all experiments.

Again, a linear fit is employed to receive $(CRig+COilsump)$⁠. Finally, measurements of the stationary test rig with and without oil allow for a separation of $CRig$ and $COilsump$⁠. An illustration of the dependencies of the individual capacitances on operating parameters or tooth width is given in Fig. 8.

With a variation of speed, force, and temperature, the capacitances $CContact$ and $CIn/Outlet$ vary due to a change in the lubricating film thickness and contact area, whereas the capacitances $CGear$⁠, $COilsump$⁠, and $CRig$ remain constant. If the tooth width changes at a constant pressure, temperature, and speed, the lubricant film thickness remains the same, while the contact area and the area of the trailing tooth flank change linearly with the tooth width. Accordingly, the capacitances $CContact$⁠, $CIn/Outlet$⁠, and $CGear$ change linearly, while the capacitances of the $COilsump$ and $CRig$ remain constant.

The results of the gearbox capacitance measurements for a variation of the operating conditions speed, temperature, and load are discussed in the following. The goal now is to capture the effect of these operating conditions on the capacitance of the gear stage in terms of a statistical evaluation. As previously described, each operating point was run three times and one revolution was measured three times in each run. These nine measurements consisted of around 2500 charging curves each. The results are shown as boxplots in Fig. 9. The value of the median, the upper and lower quartiles, the whiskers, and the highest and lowest outlier values are shown.

The measurements show that the capacitance tends to decrease with increasing speed, while an increase in load or temperature causes an increase in capacitance. However, the change in capacitance between operating points is small in relation to the capacitance change along the line of action, which can be seen in Fig. 10, where the magnitude of the capacitance variation over a single tooth mesh is compared to the boxplot of the corresponding measuring point.

It can be seen that the change in capacitance over the tooth mesh matches the interval within the lower and upper whiskers. This explains the magnitude of the scattering and shows that the variation of the capacitance along the line of action has a significant impact on the capacitance and should not be neglected.

The capacitance depends not just on the permittivity of the lubricant but also on the mixture of air and oil between the tooth flanks. Both effects are contained in the term $(εrkC)$⁠, which is experimentally determined.

For this purpose, the test cases given in Table 3 were simulated in bearinx with $kC$ and $εr$ preset to 1. A comparison of these simulations with the corresponding measurements depicted in Fig. 9 is shown in Fig. 11.

Figure 11 shows just as Eq. (5) suggests that a linear fit can be applied to connect the measured capacitance with the contact capacitance. This results in a slope of 11 which equals ε_rk_C and an intercept of 162.3 pF that equals $(CGear+CRig+COilsump)$⁠. This is an important indicator that the usage of correction factors is an appropriate method.

Figure 12 depicts the comparison of measured capacitances for the variation of tooth width while speed, contact pressure, and temperature were kept constant.

The intercept in Fig. 12 amounts to 124.5 pF, which represents $(CRig+COilsump)$⁠. The difference between the two found intercepts results in $CGear=37.8pF$⁠.

To find out, whether a simulation based on correction factors is capable of capturing the effects of speed, load, or temperature correctly, simulation and measurement results are plotted in relation to each operating parameter. Therefore, the constant capacitances or 162.3 pF representing $CGear$⁠, $CRig$⁠, and $COilsump$ are subtracted from the measurements, while a value of 11.0 for (ε_rk_C) has been applied into the simulation. Figure 13 shows the results of this comparison for a variation of load, speed, and temperature. It also shows the asperity ratio calculated in bearinx, which is an indicator, whether surface asperities are in direct contact.

Simulation and experiment match well here and follow the same characteristic trends for each operating parameter. It can also be seen that the scatter of the measurement increases for higher temperatures. As the temperature rises, the surfaces are brought closer together due to the decreasing lubricant film thickness. As a result, some roughness peaks of the two surfaces come into contact. This corresponds with a slight increase of the asperity ratio which indicates the onset of mixed friction. As already illustrated in Fig. 10, the measured values represent the median over a full rotation of the gears. The changes in capacitance and thus high scattering are due to the significant change in contact conditions along the line of action and therefore attributable to the geometry of the gears.

According to Table 2, the relative permittivity ε_r of the oil is 2.3. This results in a correction factor k_C of 4.8. The two measurements with and without oil in the stationary test rig resulted in $COilsump=4.2pF$⁠.

Finally, we can dismantle the measured capacitance C_Measurement into its individual components C_Contact, C_In/Outlet, C_Rig, C_Sump, and C_Gear according to Eq. (4). For the investigated operating points, the individual capacitances are given in Table 4 

• The contribution of the capacitances C_Rig, C_Sump, and C_Gear, which are unaffected by the operating conditions, is in the range of 59.6–70.3%, 2.1–2.5%, and 18.7–22.1% for the setup examined.

• The contribution of the capacitances C_In/Outlet and C_Contact, which change depending on the operating conditions, are in the range of 4.1–17.2% and 1.1% and 3.5%.

It becomes obvious that the capacitance of the contact and the areas close to the contact are relatively small in relation to the other capacitances. This, however, is specific to the test setup, which consisted of a relatively small gear stage in a fairly massive housing. Nevertheless, it underlines the importance of a thorough evaluation of measured capacitances, as it was done in this study. The huge scatter of the measured capacitance due to the changing contact situation along the meshing path is characteristic for tooth contacts and expected to be present in other gear applications as well.

In this article, the electrical capacitance of a gear stage is investigated using both experimental measurements and analytical simulations with the simulation tool bearinx.

First, a measurement setup to determine the capacitive behavior of a gear with high temporal resolution by means of charging curve measurements was explained. In this way, areas that show no clear capacitive behavior due to electrical breakdown or the transition to conductive behavior can be excluded from the measurement results. Second, a method was presented that derives the relative permittivity ε_r of the lubricant and the correction factor k_C, which are required coefficients for a capacitance simulation in bearinx from a series of capacitance measurements with varied operating parameters speed, temperature, or load. Further measurements with a varied tooth width allowed to dismantling the measured capacitance into its individual parts. The measurements showed that the capacitance varied significantly more along the line of action than due to the selected parameter variations.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

b =

width of the contact surface

n =

rotational speed

t =

time

A =

surface area of the capacitor

F =

contact force in the tooth flank

T =

temperature

h₀ =

central lubricant film thickness

k_C =

correction factor

A_Contact =

contact area

C_Simulation =

capacitance of the contact, calculated in bearinx

C_Contact =

capacitance of the contact

C_In/Outlet =

capacitance of the near-contact areas

C_Gear =

capacitance of the far-contact areas

C_Gearstage =

measured capacitance of the gear stage

C_Measurement =

measured capacitance

C_Rig =

capacitance of the rig

C_Oilsump =

capacitance of the oil in the rig

R_Gearstage =

resistance of the gear stage

R_V =

resistance in series of the tribo-electrical contact

U₀ =

source voltage

U_{0 max} =

maximum source voltage

U_Gear =

voltage measured at the gear

U_{Gearstage max} =

maximum voltage measured at the gear

ε₀ =

permittivity of the vacuum

ε_r =

relative permittivity of the medium between the capacitor plates

λ =

asperity ratio

τ =

time constant of the capacitor