Suppression of Rotor-Grounding Bearing Currents Based on Matching Stator and Rotor Grounding Impedances

Abstract

Pulse-width modulation (PWM) inverter drive systems are widely applied in industry for their great speed-control characteristics and performance. However, the common-mode voltage of the inverter induces motor bearing voltage and bearing current, which result in premature bearing failure. In this paper, the influence of stator and rotor grounding impedances on the bearing voltage of the motor is investigated. A comprehensive bearing-current equivalent model circuit is proposed. Based on circuit analysis, the relationship between bearing voltage ratio (BVR) and grounding impedances is derived, showing that bearing voltage varies with the stator grounding impedance on a V-shaped surface. The minimum BVR appears at some specific combinations of the stator and rotor grounding impedance. Based on this, a method of suppression of rotor-grounding bearing currents based on matching the stator and rotor grounding impedances is proposed. The analytical results are verified by experimental results and show that the bearing voltage can be mitigated by matching the grounding impedances in a proper range.

1. Introduction

With the development of modern drive technology, PWM variable-frequency technology is widely used in AC motor drive systems. The applications of high-performance semiconductor switching devices and control strategies have shown the outstanding advantage of this frequency-conversion technology. However, PWM variable-frequency technology causes some negative effects, such as bearing currents [1,2].

Common-mode voltage from the inverter is one of the main causes of high-frequency bearing currents [3]. Through the coupling of the motor parasitic capacitances, the bearing voltage between bearing outer race and inner race is induced by the common-mode voltage. When the bearing voltage surpasses the threshold voltage of lubricating oil film of the bearing, oil-film breakdown appears, and bearing currents occur. A large amount of heat due to release of electric discharge in a very short time leads to melted metal near the breakdown point, and electric pitting and fluting occurs [4]. If the bearings are continuously stressed by pitting, electric erosion can shorten the bearing lifetime. Bearing currents can be divided into four types, according to [5], which are the dv/dt bearing currents, electric-discharge-machining (EDM) currents, rotor-grounding bearing currents and circulating bearing currents. dv/dt currents and the EDM currents are related to the common-mode voltage of the inverter. Circulating bearing currents and rotor-grounding bearing currents are related to ground currents. Presently, extensive research has been carried out on EDM currents and circulating bearing currents [6]. Various methods for suppressing the different types bearing currents have been put forward. For example, common-mode voltage-minimization techniques using semiconductor-based inverters is an important method to reduce bearing voltage and bearing current [7]. However, rotor-grounding bearing currents require further investigation. The mitigation methods adopted for the other types bearing currents are not suitable for suppression of rotor-grounding bearing currents.

Typically, the stator frame of motors should be well grounded for the sake of safety and the grounding impedance is usually neglected in bearing-current analysis. However, the frequency of grounding currents can reach tens of kilohertz or even megahertz. At high frequencies, the influence of grounding impedance cannot be ignored. At the same time, in some applications, there is also a situation where the stator frame and the ground are connected via impedance. For example, in electric locomotives, the traction-motor car body and the rail are connected through protective resistors, bus bars, carbon brushes, wheels and other devices. The high dv/dt grounding current caused by the inverter flows through the stator grounding impedance. At the same time, there is also a case where the shaft and the ground are connected via impedance. For example, in electric locomotives, the traction motor connects with gear boxes, and the gear boxes connect with the wheels and the rail. When the rotor grounding impedance is smaller than the stator grounding impedance, part of the grounding current flows through the bearing to form the rotor-grounding bearing current. Existing studies have shown that the grounding state of the motor has an effect on the bearing voltage and bearing currents of the motor fed by the PWM inverter, and corresponding bearing-current suppression measures have been proposed [8]. In [9], the influence of rotor grounding impedance on bearing current is discussed; the conclusion is that the ground impedance of the rotor affects the bearing voltage, and the bearing currents are obtained. The authors point out that a reasonable arrangement of grounding impedance can suppress the bearing current, but so far, no in-depth research has been carried out. The bearing current can be suppressed by grounding the rotor. Connection of the rotor side with the ground is a method to suppress the bearing current. The influence on bearing currents, grounding currents and common-mode currents caused by different lengths and types of cables is discussed [10,11,12]. For the case where the grounding impedance exists at the frame and the shaft, the equivalent circuit with the stator grounding impedance and the rotor grounding impedance is given [13,14,15], but there is no discussion on the changing bearing voltage with the variation of the grounding impedance [16]. Typically, electrically isolating plastic stator bushing is used, and the lowest bearing voltages are achieved if either the stator potential is adjusted to the rotor potential by adding a separate capacitor between the stator and ground or the stator is shorted to ground, but no in-depth quantitative research has been conducted. To research the rotor-grounding bearing current, the relationship between bearing voltage and grounding impedance needs to be quantitatively studied, but the current research is limited to qualitative analysis. Therefore, this paper will quantitatively analyze the relationship among the stator grounding impedance, rotor grounding impedance and bearing voltage, and search for a method to suppress the rotor-grounding bearing current.

The effects of grounding status on the bearing currents will be discussed comprehensively. First, the equivalent bearing-current circuit is established, considering the stator grounding impedance and the rotor impedance. Through circuit analysis, the basic relationship between the bearing voltage and the grounding impedances is derived. Based on this, a method to suppress the rotor-grounding bearing current is put forward by adjusting the grounding impedance. Finally, the designed experiment is performed to verify the proposed method.

2. Modelling of the Motor Bearing Voltage in the PWM Inverter System

2.1. The Classic Equivalent Bearing-Current Model

The source of the bearing current is the common-mode voltage caused by the inverter. The coupling circuit of the common-mode current includes the stray capacitances of the motor.

For the squirrel-cage induction motor, the stator core and the frame can be regarded as the same electrode. The rotor core of the squirrel-cage motor and the rotor bars can be considered as another electrode, and the stator windings can be regarded as the third electrode. Stray capacitances exist among them, as shown in Figure 1.

C_wr is the capacitance between the stator winding and the rotor; C_wf is the capacitance between the stator winding and the frame; C_rf is the capacitance between the rotor and the frame; and C_b is the equivalent capacitance of the bearing oil film between the raceway and the rolling elements. It is assumed that the bearings on the drive end and the non-drive end are of the same type.

Figure 2 shows the classic equivalent bearing-current model in the two-level AC–DC–AC voltage source converter, which is often used in motor driving systems.

U_a, U_b and U_c are the output three-phase voltage; U_d is the DC link voltage; R and L are the winding common-mode resistance and inductance of each phase of the motor, respectively; V_com is the common-mode voltage; V_b is the bearing voltage; w, r and f represent the stator winding, the rotor and the frame, respectively. As shown in Figure 2, the stray capacitances in the motor constitute the common-mode path. As the motor is fed by the PWM inverter, the common-mode voltage is not zero, and the voltage, V_b, is induced between the motor shaft and the grounding.

The common-mode voltage at the output of the inverter is the average value of the voltage between the three-phase output and the ground, as shown in (1), where V_Uo, V_Vo and V_Wo are the output three-phase voltages.

$$V_{\mathrm{com}}=\frac{(V_{\mathrm{Uo}}+V_{\mathrm{Vo}}+V_{\mathrm{Wo}})}{3}$$

(1)

For the convenience of analysis, the circuit converter is omitted, and the voltage source, V_com, is used to express the common-mode voltage, as shown in Figure 3 [17,18].

As shown in Figure 3, the bearing voltage, V_b, is decided by stray capacitances of the motors and the common-mode voltage, V_com. We can obtain the ratio of the bearing voltage, V_b, and the common-mode voltage, V_com. The ratio is called bearing voltage ratio (BVR) [19].

$$BVR=\frac{V_{\mathrm{b}}}{V_{\mathrm{c}\mathrm{o}\mathrm{m}}}=\frac{C_{\mathrm{w}\mathrm{r}}}{C_{\mathrm{w}\mathrm{r}}+C_{\mathrm{r}\mathrm{f}}+2C_{\mathrm{b}}}$$

(2)

The above equation represents the relationship between the bearing voltage and the common-mode voltage without considering the stator and rotor grounding resistance. The bearing voltage, V_b, is an index of the bearing erosion failure. The higher the bearing voltage, the more serious the bearing erosion. In order to decrease the bearing voltage, V_b, under this condition, some methods could be adopted, such as inserting an electrostatic shielding layer between the stator winding and rotor in order to decrease the C_wr [20].

2.2. Bearing-Current Circuit Considering the Stator and Rotor Grounding Impedance

In practical applications, there are multiple grounding forms for motors. The influence of grounding impedances should be considered in circuit analysis. The following cases are the main grounding conditions in practical applications.

Case 1: the motor stator frame is well grounded, and the rotor shaft is not grounded. The stator grounding impedance can be neglected, as shown in Figure 2. The BVR can be determined by (2).

Case 2: The motor stator frame is poorly grounded or not grounded, and the shaft is grounded through the load. Poor stator grounding status means that the stator grounding impedance is very big and can sometimes reach MΩ levels. However, the equivalent impedance of the rotor to the grounding is relatively small, and the stator frame is clamped at a high voltage, so the bearing voltage is significantly increased and the bearing current will be very big, resulting in serious electric erosion of the bearing.

Case 3: both the stator and the rotor are not grounded, and the bearing voltage does not increase, which induces potential safety problems.

Case 4: The stator frame is grounded through an earthing device, but the grounding impedance cannot be neglected, whereas the rotor shaft is grounded through the impedance. For example, in rail-transit traction motors, the motor shaft connects the gear box and the wheels to the rails. The rotor grounding impedance provides a path for the flow of the rotor grounding impedance current. In this case, the grounding impedance of the motor frame and the grounding impedance of the rotor exist simultaneously. The topology of the bearing voltage circuit is changed, as shown in Figure 4.

Z_fg is the stator grounding impedance, and Z_fg is the equivalent impedance of the stator grounding.

The equivalent circuit model in Figure 4 considers both the rotor and the stator grounding impedances. This means that the grounding impedances on both sides are neither zero nor infinite, as is the situation in Case 4.

The stator and rotor grounding impedances and the stray capacitances of the motor have effect on the bearing voltage. The relationship between the bearing voltage and the parameters in the circuit will be analyzed by solving the circuit.

3. Influence of Motor Grounding Impedances on Bearing Voltage

In Figure 4, C_rf and the two bearing capacitances, C_b, are parallelly connected for convenience. These capacitances are taken as C’_rf, as shown in (3)

$${C^{\prime }}_{\mathrm{rf}}=C_{\mathrm{rf}}+2C_{\mathrm{b}}$$

(3)

The composition of the grounding impedance of the stator and the rotor depends on the actual grounding situation and often includes resistance and reactance. In order to simplify the analysis, the common-mode voltage is treated as the sinusoidal high-frequency AC voltage, and the circuit can be analyzed through the phasor method. The phasor circuit is shown in Figure 5. There are three loops in this circuit. In loop 1, the current, ${\dot{I}}_1$, passes through V_com, C_wf and Z_fg; in loop 2, the current, ${\dot{I}}_2$, passes through V_com, C_wr and Z_rg; and in loop 3, the current, ${\dot{I}}_3$, passes through C’_rf, Z_rg and Z_fg.

The voltage equations of the above circuit are shown as (4).

$$\{\begin{array}{l}{\dot{V}}_{\mathrm{com}}={\dot{I}}_1({\mathrm{Z}}_{\mathrm{fg}}+1/\mathrm{j}\omega C_{\mathrm{wf}})+0-{\mathrm{Z}}_{\mathrm{fg}}{\dot{I}}_3\hfill \\ {\dot{V}}_{\mathrm{com}}=0+{\dot{I}}_2({\mathrm{Z}}_{\mathrm{r}\mathrm{g}}+1/\mathrm{j}\omega C_{\mathrm{wr}})+{\mathrm{Z}}_{\mathrm{r}\mathrm{g}}{\dot{I}}_3\hfill \\ 0=-{\mathrm{Z}}_{\mathrm{fg}}{\dot{I}}_1+{\mathrm{Z}}_{\mathrm{r}\mathrm{g}}{\dot{I}}_2+({\mathrm{Z}}_{\mathrm{fg}}+{\mathrm{Z}}_{\mathrm{r}\mathrm{g}}+1/\mathrm{j}\omega {C^{\prime }}_{\mathrm{rf}}){\dot{I}}_3\hfill \end{array}$$

(4)

The bearing voltage, V_b, is expressed as (5).

$${\dot{V}}_{\mathrm{b}}=-{\dot{I}}_3/\mathrm{j}\omega {C^{\prime }}_{\mathrm{rf}}$$

(5)

By simultaneous solving the three equations in (4), ${\dot{I}}_3$ can be derived as (6).

$${\dot{I}}_3=\frac{\frac{{\mathrm{Z}}_{\mathrm{rg}}}{{\mathrm{Z}}_{\mathrm{rg}}+1/\mathrm{j}\omega C_{\mathrm{wr}}}-\frac{{\mathrm{Z}}_{\mathrm{fg}}}{{\mathrm{Z}}_{\mathrm{fg}}+1/\mathrm{j}\omega C_{\mathrm{wf}}}}{1/\mathrm{j}\omega {C^{\prime }}_{\mathrm{r}f}+\frac{{\mathrm{Z}}_{\mathrm{fg}}/\mathrm{j}\omega C_{\mathrm{wf}}}{{\mathrm{Z}}_{\mathrm{fg}}+1/\mathrm{j}\omega C_{\mathrm{wf}}}+\frac{{\mathrm{Z}}_{\mathrm{rg}}/\mathrm{j}\omega C_{\mathrm{wr}}}{{\mathrm{Z}}_{\mathrm{rg}}+1/\mathrm{j}\omega C_{\mathrm{wr}}}}{\dot{V}}_{\mathrm{com}}$$

(6)

By combining (5) and (6), the bearing voltage V_b can be derived as (7).

$${\dot{V}}_{\mathrm{b}}=\frac{1/\mathrm{j}\omega {C^{\prime }}_{\mathrm{rf}}(\frac{{\mathrm{Z}}_{\mathrm{rg}}}{{\mathrm{Z}}_{\mathrm{rg}}+1/\mathrm{j}\omega C_{\mathrm{wr}}}-\frac{{\mathrm{Z}}_{\mathrm{fg}}}{{\mathrm{Z}}_{\mathrm{fg}}+1/\mathrm{j}\omega C_{\mathrm{wf}}})}{1/\mathrm{j}\omega {C^{\prime }}_{\mathrm{r}f}+\frac{{\mathrm{Z}}_{\mathrm{fg}}/\mathrm{j}\omega C_{\mathrm{wf}}}{{\mathrm{Z}}_{\mathrm{fg}}+1/\mathrm{j}\omega C_{\mathrm{wf}}}+\frac{{\mathrm{Z}}_{\mathrm{rg}}/\mathrm{j}\omega C_{\mathrm{wr}}}{{\mathrm{Z}}_{\mathrm{rg}}+1/\mathrm{j}\omega C_{\mathrm{wr}}}}{\dot{V}}_{\mathrm{com}}$$

(7)

From (6), the bearing voltage ratio under this condition, BVR’, can be derived as (8).

$$BVR\prime =\frac{V_{\mathrm{b}}}{V_{\mathrm{com}}}=\left|\frac{1/\mathrm{j}\omega {C^{\prime }}_{\mathrm{rf}}(\frac{{\mathrm{Z}}_{\mathrm{fg}}}{{\mathrm{Z}}_{\mathrm{fg}}+1/\mathrm{j}\omega C_{\mathrm{wf}}}-\frac{{\mathrm{Z}}_{\mathrm{rg}}}{{\mathrm{Z}}_{\mathrm{rg}}+1/\mathrm{j}\omega C_{\mathrm{wr}}})}{1/\mathrm{j}\omega {C^{\prime }}_{\mathrm{r}f}+\frac{{\mathrm{Z}}_{\mathrm{fg}}/\mathrm{j}\omega C_{\mathrm{wf}}}{{\mathrm{Z}}_{\mathrm{fg}}+1/\mathrm{j}\omega C_{\mathrm{wf}}}+\frac{{\mathrm{Z}}_{\mathrm{rg}}/\mathrm{j}\omega C_{\mathrm{wr}}}{{\mathrm{Z}}_{\mathrm{rg}}+1/\mathrm{j}\omega C_{\mathrm{wr}}}}\right|$$

(8)

Comparing (7) with (1), we find that the relationship between the common-mode voltage and the bearing voltage in Case 4 is more complex than that in Case 1. If the stray capacitances of the motor are constant, the change of grounding impedance, Z_rg and Z_fg, respectively, affect the voltage between the frame and the shaft, which, in turn, affects the bearing voltage and the bearing current.

If the bearing voltage, V_b, is zero, the bearing current will not appear. If V_b is zero, the numerator of the right side of Equation (7) should be zero, and we can derive the following.

$$\frac{{\mathrm{Z}}_{\mathrm{fg}}}{{\mathrm{Z}}_{\mathrm{fg}}+1/\mathrm{j}\omega C_{\mathrm{wf}}}-\frac{{\mathrm{Z}}_{\mathrm{rg}}}{{\mathrm{Z}}_{\mathrm{rg}}+1/\mathrm{j}\omega C_{\mathrm{wr}}}=0$$

(9)

Then,

$$\frac{{\mathrm{Z}}_{\mathrm{fg}}}{{\mathrm{Z}}_{\mathrm{fg}}+\frac{1}{\mathrm{j}\omega C_{\mathrm{wf}}}}=\frac{{\mathrm{Z}}_{\mathrm{rg}}}{{\mathrm{Z}}_{\mathrm{rg}}+\frac{1}{\mathrm{j}\omega C_{\mathrm{wr}}}}$$

(10)

If the numerator and denominator of the left side are divided by Z_fg and the numerator and denominator of the right side are divided by Z_rg, we get:

$$\frac{1}{1+\frac{1}{{\mathrm{Z}}_{\mathrm{fg}}\mathrm{j}\omega C_{\mathrm{wf}}}}=\frac{1}{1+\frac{1}{{\mathrm{Z}}_{\mathrm{rg}}\mathrm{j}\omega C_{\mathrm{wr}}}}$$

(11)

The denominators of both sides should be same, i.e.,

$${\mathrm{Z}}_{\mathrm{fg}}{C}_{\mathrm{wf}}={\mathrm{Z}}_{\mathrm{rg}}{C}_{\mathrm{wr}}$$

(12)

Then, we get:

$$\frac{{\mathrm{Z}}_{\mathrm{fg}}}{{\mathrm{Z}}_{rg}}=\frac{C_{\mathrm{wr}}}{C_{\mathrm{wf}}}$$

(13)

The above impedance balance condition can be explained from the perspective of the circuit topology. Due to the simultaneous existence of the stator grounding impedance and the rotor grounding impedance, the equivalent circuit in Figure 5 forms a Wheatstone bridge circuit. The Wheatstone bridge circuit is shown in Figure 6.

In Figure 6, Z₁, Z₂, Z₃ and Z₄ are the impedances of the four arms; the meter, G, between nodes c and d is the galvanometer; and U_s is the voltage source. According to the principle of the Wheatstone bridge circuit, if the magnitude and the phase angle of the impedances of the four arms meet the relationships in (11), the voltage between c and d is zero, and there is no current passing through the galvanometer G [21].

$$\{\begin{array}{c}\left|{\mathrm{Z}}_1\right|\left|{\mathrm{Z}}_3\right|=\left|{\mathrm{Z}}_2\right|\left|{\mathrm{Z}}_4\right|\\ {\phi }_1+{\phi }_3={\phi }_2+{\phi }_4\end{array}$$

(14)

where |Z₁|, |Z₂|, |Z₃| and |Z₄| are the amplitudes of the impedances of arm 1, 2, 3 and 4 respectively, and φ₁, φ₂, φ₃,and φ₄ is the phase angle of the impedance of arm 1, 2, 3 and 4 respectively.

Comparing Figure 6 with Figure 5, the capacitances C_wf and C_wr constitute two upside arms, Z₁ and Z₄; and the stator grounding impedances, Z_fg, and the rotor grounding impedance, Z_rg, constitute two downside arms, Z₂ and Z₃. The bearing capacitance, C_b, and its parallel capacitance, C_rf, constitute the bridge arm. The bridge voltage is the bearing voltage, V_b. If the impedances of the two downside arms change, the corresponding bridge voltage varies with them. If the impedances of the four arms cannot satisfy the magnitude and phase-balance relationship in (13), the bearing voltage can be deduced through circuit analysis.

When the circuit impedances reach balance, as in (14), the value of BVR should be zero. The relationship between grounding impedances and motor stray capacitances is shown in (13). The impedance balance relationship inspires a method for suppressing the bearing voltage induced by the grounding impedances.

4. Bearing-Current Suppression Method Based on Matching Grounding Impedances

According to (13), if the stray capacitances are known, the value of the balanced impedance, Z_fg, can be obtained, which allows for the determination of the minimum value of bearing voltage, V_b. According to Figure 5, when the impedance, Z_fg, equals the balance impedance, the BVR is minimum. Then, the bearing voltage can be suppressed by adjusting the grounding impedance. Based on this, a rotor-grounding-current suppression method based on matching grounding impedance can be proposed.

In practical applications, the type and value of the grounding impedance on one side can usually be determined with the aid of an impedance analyzer, and the stray capacitances of the motor can be acquired by calculation or measurement [1,5]. Then, the appropriate grounding impedance on the other side can be determined according to (13) in order to minimize the bearing voltage

In following discussion, the rotor grounding impedance is assumed as fixed or unchangeable. The rotor grounding impedances can be divided into three types, as shown in Figure 7. Type a: the rotor is grounded through the pure resistance, R_rg, as in Figure 7a; type (b): the rotor is grounded through the resistance, R_rg, in series with the inductance, L_rg, as in Figure 7b; and type (c): the rotor is grounded through the resistance, R_rg, in series with the capacitance, C_rg, as in Figure 7c.

In Figure 7a, the rotor grounding impedance is pure resistance, R_rg. According to (13), the balanced stator grounding impedance should also be pure resistance, and its value, R_fg, is shown in (15).

$$R_{\mathrm{fg}}=\frac{C_{\mathrm{wr}}}{C_{\mathrm{wf}}}{R}_{\mathrm{rg}}$$

(15)

In Figure 7b, the rotor grounding impedance is the resistance, R_rg, connected in series with the inductance, L_rg. According to (13), the type of the balanced stator grounding impedance should also be series connection of the resistance, R_fg, and the inductance, L_fg, the values of which are as shown in (16).

$$\{\begin{array}{l}{R}_{\mathrm{fg}}=\frac{C_{\mathrm{wr}}}{C_{\mathrm{wf}}}{R}_{\mathrm{rg}}\\ L_{\mathrm{fg}}=\frac{C_{\mathrm{wr}}}{C_{\mathrm{wf}}}{L}_{\mathrm{rg}}\end{array}$$

(16)

In Figure 7c, the rotor grounding impedance is the resistance, R_rg, connected in series with the capacitance, C_rg. According to (13), the type of the balanced stator grounding impedance should be series connection of resistance, R_fg, and the capacitance, C_fg, the values of which are as shown in (17).

$$\{\begin{array}{l}{R}_{\mathrm{fg}}=\frac{C_{\mathrm{wr}}}{C_{\mathrm{wf}}}{R}_{\mathrm{rg}}\\ C_{\mathrm{fg}}=\frac{C_{\mathrm{wf}}}{C_{\mathrm{wr}}}{C}_{\mathrm{rg}}\end{array}$$

(17)

In general, when the grounding impedance on one side is known, adjusting the grounding impedance on the other side to a proper balanced value could suppress the bearing voltage. When the grounding impedance on both sides meets the balance relationship, the bearing voltage reaches the minimum at the balance impedance. In order to verify the rationality of the above analysis, a simulation and an experimental platform were set up. The experimental scheme was designed for testing.

5. Simulation and Experiment

5.1. Simulation

In order to specifically study the influence of grounding impedances on the BVR, a 5.5 kW induction motor was taken as an example. The values of the stray capacitances of a 5.5 kW motor are shown in

Table 1. Stray capacitances of a 5.5 kW induction motor.

Cwf/pF	Cwr/pF	Crf/pF	Cb/pF
5098.4	35.8	782.7	110.0

. The stray capacitances were calculated through the finite element method, as in [22].

For the motor capacitances in Table 1, the ratio of C_wr and C_wf is 0.007. According to the capacitances of the motor and the balance equations (16), when the ratio of the stator grounding inductance, L_fg, to the rotor grounding inductance, L_rg, is 0.007, the bearing voltage is zero, and BVR is the minimum.

It is assumed that the motor frame and the shaft are respectively grounded through the inductances. The variation range of the motor-frame grounding inductance is from 0 μH to 8 μH, and the shaft grounding inductance is from 0 μH to 1000 μH. Through Matlab analysis of Equation (8), we can get the bearing voltage ratio under the combination of different stator grounding inductances and rotor grounding inductances. The bearing voltage ratio with the stator and rotor grounding inductances at a frequency of 100 kHz is shown in Figure 8.

Figure 8 shows that bearing voltage varies with the stator grounding impedance in a V-shaped surface. When the grounding impedance on one side is fixed, the bearing voltage ratio will first decrease and then increase as the grounding impedance on the other side increases. This means that there is a minimum bearing voltage ratio when grounding impedances exist on both sides. The impedance in this scenario is called the balanced impedance. We can observe that when L_rg and L_fg take different values, BVR fluctuates from 0 to 2.5, which has a great influence on the bearing voltage. When L_rg and L_fg are reasonably selected, BVR could be located at the bottom of V-shaped surface.. At this time, theoretically, BVR is the lowest, and the bearing voltage is the smallest. Under a given rotor grounding inductance, we can find the proper stator grounding inductance to achieve the minimum bearing voltage.

5.2. Experimental Platform

The test induction motor was a 5.5 kW type Y2-135S. The motor was driven by an E2000-0055T3 inverter from EURA Drives. The motor-frame grounding inductance and the shaft grounding inductance used grounding wires and sliding rheostats, respectively. A schematic diagram of the test platform is shown in Figure 9. The measured waveform was observed through a RIGOL DS1204B oscilloscope with real-time sampling of 2 GSa/s and a bandwidth of 200 MHz. The bearing voltage and common-mode voltage under different grounding impedances were recorded.

The grounding of the inverter was set to a reference connected to the laboratory power-supply grounding with a short line. Therefore, the common-mode voltage, V_com, is the voltage between the stator winding-star connection neutral point and the reference grounding. V_com was measured with a Tektronix P520A high-voltage differential probe. V_b is the potential difference between the shaft and stator frame. V_b was measured with an oscilloscope probe. In order to measure the bearing current, one bearing was isolated from the motor frame. A 5 mm insulation layer was inserted between the bearing outer raceway and the bearing housing of one bearing [23].

In the experiment, the change in grounding impedance of the stator was realized by changing the length of the grounding wire. The rotor grounding impedance was unchanged, so the scale of the sliding rheostat was unchanged. At high frequency, the sliding rheostat can be regarded as a solenoid, and the main component is the inductance. The relevant parameters of the grounding wire and the rheostat were measured separately with an RLC meter (measurement frequency: 100 kHz). The test rig is shown in Figure 10.

The stator grounding wire length, l, ranges from 0.5 m to 5 m and was measured every 0.5 m. The scale, k, of the sliding wire rheostat ranges from 5 to 50 and was is measured every 5 scales. At l = 4 and k = 10, the waveforms of the common-mode voltage, V_com, the bearing voltage, V_b, and the bearing current, i_b, are shown in Figure 11. It can be seen from Figure 11 that the common–mode voltage is a four–step wave, which is consistent with the theory. The measured bearing voltage is not a four–step wave, similar to the common-mode voltage, and is not mapped from the common-mode voltage.

6. Measurement Result and Discussion

The common-mode voltage and bearing voltage of the motor were measured according to the above experimental scheme. The bearing voltage, V_b, and the common-mode voltage, V_com, were recorded under 100 different combinations of the stator and the rotor grounding impedance. The bearing voltage ratios were acquired according to the peak–peak value of V_b and V_com.

Table 2. Bearing voltage ratio percentage results of the test.

l (Stator)	k (Rotor)
	10	20	30	40	50
	64.9 μH	211.1 μH	414.8 μH	611.2 μH	830.4 μH
	21.4 mΩ	47.6 mΩ	70.4 mΩ	83.1 mΩ	99.7 mΩ
0.5 m 17.5 mΩ 0.6 μH	0.371	0.536	0.603	0.683	0.763
1 m 35.0 mΩ 1.2 μH	0.256	0.475	0.560	0.592	0.683
1.5 m 52.5 mΩ 1.8 μH	0.205	0.328	0.469	0.571	0.635
2 m 70 mΩ 2.4 μH	0.309	0.352	0.496	0.603	0.629
2.5 m 87.5 mΩ 3.0 μH	0.491	0.355	0.533	0.592	0.613
3 m 105 mΩ 3.6 μH	0.485	0.283	0.416	0.560	0.608
3.5 m 22.5 mΩ 4.2 μH	0.576	0.237	0.453	0.517	0.580
4 m 140 mΩ 4.8 μH	0.704	0.189	0.416	0.512	0.560
4.5 m 157.5 mΩ 5.4 μH	1.056	0.651	0.389	0.453	0.544
5 m 175 mΩ 6.0 μH	1.269	0.704	0.272	0.555	0.571

shows partial data about the rotor grounding scale, k, with the grounding resistance and inductance; the stator grounding cable, l, with the grounding resistance and inductance; and bearing voltage ratio percentage.

The variation of BVR with the stator grounding cable length, l, and the rotor ground impedance scale, k, is shown in Figure 12.

The variation of BVR in the experiment shows there is a band where the BVR is lower than other grounding conditions. This means that under such grounding conditions, the bearing voltage can be suppressed.

There are differences between the simulation results in Figure 8 and the measurement results in Figure 12. In the experiment, the stator grounding inductance was 4.8 μH to obtain the minimum bearing voltage value; however, in Figure 8, the minimum bearing voltage is obtained when the stator grounding inductance is 4.28 μH. The minimum BVR in Figure 12 is not zero. This is because the structure of the motor is relatively complicated. Using the circuit to estimate the BVR cannot fully reflect the complex structure of the motor and its electromagnetic relationship. The actual state of the motor grounding is more complicated than the circuit calculation; the lowest point of the test result is an oblique band but not zero. On the other side, the common-mode voltage is not a single-frequency source but is made up of many harmonic components, which does not satisfy the assumption in Section 3, so the analysis of BVR in (8) can be taken as a reference.

Since the theoretical analysis and experimental results reflect that the law is basically consistent, this method is feasible within the allowable range of error. When the grounding impedance type and the value of one side of the motor are known, it is effective to suppress the bearing current by matching the grounding impedance on the other side of the motor.

7. Conclusions

The influence of motor grounding conditions on the bearing currents in PWM inverter driving system was discussed. The equivalent circuit model of bearing current of variable frequency motors was established, considering the stator and rotor grounding impedance. Through circuit analysis, we deduced that the bearing voltage ratio changes with the stator grounding impedance. Analysis results show that when the grounding impedance of the stator and the grounding impedance of the rotor exist simultaneously, the common-mode equivalent circuit of the PWM frequency-conversion system constitutes a Wheatstone bridge circuit. If the grounding impedance changes, the bearing voltage will change to a V shape.

In practice, the traction-motor car body and the rail in electric locomotives are connected through protective resistors, bus bars, carbon brushes, wheels and other devices. We can design those impedances to match the rotor ground impedance according to the impedance balancing restrain to reduce the bearing voltage and suppress bearing electric erosion.