Analysis of the cross-coupling effect and magnetic force nonlinearity in the 6-pole radial hybrid magnetic bearing

Abstract

A hybrid magnetic bearing (HMB) with six poles is an effective construction of the magnetic bearing. It is energy efficient due to the presence of permanent magnets and it can be supplied from a three phase inverter. Unfortunately, the cross-coupling effect and nonlinearity of magnetic force occur in the 6-pole construction of the HMB. The purpose of this paper is to investigate the cross-coupling effect and nonlinearity of magnetic force as well as to propose two methods of reducing them. The proposed methods include division of the stator joke into three parts and exciting an additional current in one winding. Presented HMB was analysed with the 3D finite element method. The numerical model was verified by measurement.

Keywords

Hybrid magnetic bearing electromagnetic calculations cross-coupling effect magnetic nonlinearity

1. Introduction

Magnetic bearings (MBs) offer outstanding features like no-contamination, no-lubrication, long-life operation, online fault diagnosis and low power loss [1,2]. Due to their features, MBs have been applied in many industrial applications like compressors, flywheel energy storage, blood pumps, turbo-molecular vacuum pumps, high speed milling machines, etc. [3].

Presented in this paper 6-pole radial MB requires a less complex suppling system in relation to 8-pole MB because it has three windings. Even it is possible to use a traditional three-phase voltage source inverter [4]. Additionally, the 6-pole MB excels 8-pole MBs by lower rotation losses and lower copper losses. Unfortunately, presented MB with permanent magnets has disadvantages like the cross-coupling effect and the magnetic force nonlineariety. Several studies were carried out to investigate the nonlinearities and cross-couplings of different magnetic bearings types. Polajzer et al. [5] studied the impact of magnetic nonlineariety and the cross-coupling effect on properties of 8-pole radial magnetic bearings. Based on the finite element model authors calculated that due to the magnetic nonlineariety and cross-coupling effects the electromotive force varies up to 12%, while magnetic force varies by 13%. Meeker and Maslen [6] investigated a 3-pole active radial magnetic bearing, where the cross-coupling effect occurs due to the construction of the stator. Authors derived equations for the magnetic force and proposed three control approaches. However, authors did not investigate the cross-coupling effect at all. Zhu et al. [7] established a mathematical model of a three-pole radial homopolar magnetic bearing with permanent magnets based on a magnetic equivalent circuit. Authors proposed an improved mathematical model that factors in an edge effect, but they did not analyse the cross-coupling effect between axes. Wang et al. [8] studied magnetic force coupling between axes of a 3 degree of freedom conical magnetic bearing based on the magnetic equivalent circuit. Authors proposed coupling factors to evaluate the degree of magnetic force coupling. However, authors do not propose a method of reducing the cross-coupling effect. Zhong et al. [9] investigated the impact of the cross-coupling effect between radial and axial magnetic bearing in a 3 degree of freedom magnetic bearing with a permanent magnet. Authors used the improved magnetic equivalent circuit to highlight the impact of the control current and rotor position in axial bearing on magnetic force and stiffness in the radial bearing. The cross-coupling effects were quantified by calculation of normalized stiffness in x and z axis. Moreover, authors proposed an additional coil as one method of reducing the cross-coupling effect that has a positive impact on the reduction of the cross-coupling effect. All discussed examples do not consider the cross-coupling effect and magnetic force nonlineariety in a 6-pole radial magnetic bearing with permanent magnets and do not propose methods of reducing these effects. Only short information about the cross-coupling effect and magnetic force nonlinearity in 6-pole HMB were mentioned in two papers [4] and [10].

The aim of this paper is to investigate the cross-coupling effect and the magnetic force nonlinearity in the 6-pole radial hybrid magnetic bearing as well as to propose methods of reducing them. Two methods were introduced: division of the stator yoke into three parts and exciting an additional current in the first winding that depends on the control current i _x and position of the rotor in the x-axis. Root mean square errors were calculated in order to evaluate the cross-coupling effect and the nonlinearity of the magnetic force.

2. Description of the basic version of the hybrid magnetic bearing

Figure 1 presents the 6-pole radial magnetic bearing with permanent magnets, also called hybrid magnetic bearing (HMB) due to the presence of permanent magnets. The HMB consists of the stator and rotor that are fabricated from silicon steel sheets M400-50A. The laminated magnetic circuit significantly reduces eddy currents effects and therefore losses in the magnetic material. The stator includes three NdFeB permanent magnets and three windings. Permanent magnets are magnetized along the shortest edge and generate the magnetic flux towards the rotor. Each winding has 100 turns. Figure 1 indicates the main dimensions of the HMB and the main parameters of the bearing are shown in Table 1.

Fig. 1.

Geometry and dimensions of the hybrid magnetic bearing.

Table 1

Main parameters of the 6-pole hybrid magnetic bearing

Parameters	Value
The outer diameter of the stator yoke (mm)	43.0
The inner diameter of the stator yoke (mm)	35.0
The outer diameter of the rotor (mm)	19.7
The inner diameter of the rotor (mm)	10.0
The width of the pole (mm)	12.0
Length of the air gap (μm)	300
The axial length of the stator (mm)	10
The magnetic material of stator and rotor	M400-50A
Number of winding turns	100
Coercive force of permanent magnet (kA/m)	937.4
Remanence of the permanent magnet (T)	1.24
Cross-section area of permanent magnet (mm²)	200
The thickness of the permanent magnet (mm)	2

The length of the air gap between a stator and rotor in the prototype of the HMB is equal to 0.3 mm, whereas in the simulation model was increased to 0.39 mm. The reason for that is the manufacturing process that reduces the magnetic permeability of the narrow layer of the rotor and stator poles. The length of the nonmagnetic layer varies and takes value 40 μm [11], 50 μm [12], as well as 70 μm [11]. The axial length of the HMB stator is equal to 10 mm. The HMB is controlled through control currents in both axes i _x, i _y. Winding currents i ₁, i ₂, i ₃ are calculated from the control currents: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle i_{1}=i_{y}\end{eqnarray}$ (1a) $\begin{eqnarray}\displaystyle \text{}&\text{}& i_{2}=-\displaystyle \frac{1}{2}i_{y}+ \displaystyle \frac{\sqrt{3}}{2}i_{x}\end{eqnarray}$ (1b) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle i_{3}=-\displaystyle \frac{1}{2}i_{y}-\frac{\sqrt{3}}{2}i_{x}.\end{eqnarray}$ (1c) Figure 2 presents paths of the bias flux and control flux in the analysed HMB. The bias flux provides the linearization of the magnetic force characteristics, what simplifies the control procedure of the rotor position. The control flux is used to regulate the magnetic force generated by the HMB. The bias flux, generated by permanent magnets goes through the rotor, two nearest poles with windings and returns through the stator joke. For the central position of the rotor, the magnetic field in every air gap is equal and the HMB does not generate the magnetic force. Change of the rotor position without control current results in the magnetic force generation. The magnetic field simulation and working principle of the presented HMB are described in the paper [13].

Fig. 2.

Paths of magnetic flux in the HMB.

3. Simulation model and description of the cross-coupling effect and magnetic force nonlinearity

Simulation of the magnetic field distribution was performed by a 3 dimensional finite element analysis. Figure 3 shows a finite element model prepared in the software Maxwell 3D. Due to lamination of the magnetic circuit, the simulation model accounts the stationary magnetic field with a nonlinear characteristic of the magnetic material M400-50A [14]. Presented in Fig. 4 a characteristic of the dynamo steel sheet M400-50A was tested with closed magnetic circuit method [15]. In order to limit the number of elements and reduce calculation time, the simulation model represents only half of the real object geometry. The simulation model contains approximately 110 000 tetrahedral elements and execution time of the magnetic field takes about 20 minutes. Due to the significant nonlinearity of the magnetic field in the air gap, that region contains a high density of finite elements. The boundary of the simulation model was set 40 mm from the stator and rotor in every direction, apart from the symmetry plane that was set in the middle of the stator length. The Dirichlet boundary condition was set on the outer faces of the model, while the Neumann boundary condition was set on the symmetry plane. Simulations were carried out in the operating range of the HMB, i.e. x, y ∈ (−0.2 mm, 0.2 mm) and i _x, i _y ∈ (−2 A, 2 A).

Fig. 3.

Finite element model showing a 3D mesh for half of the object.

Fig. 4.

B-H characteristic of steel sheet M400-50A.

Figure 5.a presents a magnetic field distribution for the central position of the rotor and lack of control currents. This figure indicates that saturation of the magnetic circuit occurs in the neighbourhood of permanent magnets. Therefore, the control flux goes only through the poles with windings. This phenomenon is responsible for the cross-coupling effect in the HMB. Figure 4.b shows magnetic field distribution for central position of the rotor and control current i _x = 2 A. It can be seen that the magnetic field density inside a pole of the 3rd winding is almost equal to 0 T. For this condition, the HMB generates magnetic forces F _x = 39.45 N and F _y = −10.76 N.

Fig. 5.

Magnetic field distribution for (a) central position of the rotor and lack of control currents and for (b) central position of the rotor and control current i _x = 2 A.

Figures 6.a and 6.b present magnetic force F _x and F _y in function of the rotor position x and control current i _x. Figure 6.a shows that change of the rotor position x, as well as control current i_x, causes an almost linear change of the magnetic force F _x. Figure 6.b indicates that movement of the rotor position in x-axis or change of the control current i _x causes generation of the magnetic force in the y-axis. For the position x equals to −200 μm and control current i _x equals to −2 A, the magnetic force F _y is equal to −28.41 N. This behaviour is not desirable in magnetic bearings because restricts the control procedure.

Fig. 6.

Magnetic force F _x (a) and magnetic force F _y (b) in function of the rotor position x and control current i _x.

Figures 7.a and 7.b present magnetic force F _x and F _y in function of the rotor position y and control current i _y. It can be seen from Fig. 7.a that the HMB does not generate magnetic force F _x due to movement of the rotor position in y-axis or change of the control current i _y. Figure 7.b points out that the HMB generates nonlinear magnetic force in y-axis, because for position y equals to 200 μm and control current i _y equals to 2 A, the magnetic force F _y is equal to 101 N. On the other hand, for position y equals to −200 μm and control current i _y equals to −2 A the magnetic force F _y is equal to −53.57 N. Occurrence of the cross-coupling effect and magnetic force nonlinearity is due to 6-pole construction of the HMB.

Fig. 7.

Magnetic force F _x (a) and magnetic force F _y (b) in function of the rotor position y and control current i _y.

The accuracy of the simulation model was verified by measurement of the magnetic forces in x- and y-axis. Magnetic forces were measured similarly to the procedure described in the paper [16]. That means, the HMB was normally operated and the rotor was stopped. During the measurement procedure, the external force was exerted on the rotor that caused a change of control currents. Value of the force was measured by a strain gauge, while control currents were measured by the control system. In Figs 8.a and 8.b are presented magnetic force F _x and F _y in function of control currents i _x and i _y, respectively.

Fig. 8.

Simulation and measurement comparison of the magnetic force F _x(i _x) (a) and magnetic force F _y(i _y) (b).

Magnetic bearings are described by the following parameters: position stiffness k _s, current stiffness k _i, dynamic inductance L _d and velocity induce voltage e _v [8]. The position stiffness k _sx and k _sy are calculated as derivative of magnetic force with respect to the rotor position: $\begin{eqnarray}\displaystyle k_{sx}=\displaystyle \left.\frac{\partial F_{x}}{\partial x}\right|_{x=0,i_{x}=0} & & \displaystyle\end{eqnarray}$ (2a) $\begin{eqnarray}\displaystyle k_{sy}=\left.\displaystyle \frac{\partial F_{y}}{\partial y}\right|_{y=0,i_{y}=0}. & & \displaystyle\end{eqnarray}$ (2b) The current stiffness k _ix and k _iy are calculated as derivative of magnetic force with respect to the control current: $\begin{eqnarray}\displaystyle k_{ix}=\displaystyle \left.\frac{\partial F_{x}}{\partial i_{x}}\right|_{x=0,i_{x}=0} & & \displaystyle\end{eqnarray}$ (3a) $\begin{eqnarray}\displaystyle k_{iy}=\displaystyle \left.\frac{\partial F_{y}}{\partial i_{y}}\right|_{y=0,i_{y}=0}. & & \displaystyle\end{eqnarray}$ (3b) The dynamic inductance L _d is calculated as derivative of linkage flux with respect to the winding current: $\begin{eqnarray}\displaystyle L_{d}=\displaystyle \frac{\partial {\Psi}}{\partial i}. & & \displaystyle\end{eqnarray}$ (4) The velocity induced voltage e _v is calculated as derivative of linkage flux with respect to the rotor position: $\begin{eqnarray}\displaystyle e_{v}=\displaystyle \frac{\partial {\Psi}}{\partial y}. & & \displaystyle\end{eqnarray}$ (5) The presented simulation model was used to calculate the parameters of the HMB, which are listed in Table 2.

Table 2

Parameters of the basic version of the HMB

Position stiffness k _sx	122.97 (N/mm)
Current stiffness k _ix	20.67 (N/A)
Position stiffness k _sy	116.11 (N/mm)
Current stiffness k _iy	20.61 (N/A)
Dynamic inductance L _d	4.65 (mH)
Velocity induce voltage e _v	14.36 (Vs/m)

Magnetic forces generated by the HMB are described by the following linear equations: $\begin{eqnarray}\displaystyle F_{x}(x,i_{x})=k_{sx}x+k_{ix}i_{x} & & \displaystyle\end{eqnarray}$ (6a) $\begin{eqnarray}\displaystyle F_{y}(y,i_{y})=k_{sy}y+k_{iy}i_{y} & & \displaystyle\end{eqnarray}$ (6b) Equations (6a) and (6b) are often used for developing a control system of the magnetic bearing. Unfortunately, these equations omit the cross-coupling effect and the magnetic force nonlinearity of the 6-pole HMB. Therefore, root mean square errors (RMSEs) between the magnetic forces calculated from FEM and their linear approximation Eq. (6a and 6b) were proposed to quantify the magnetic force nonlinearity. RMSEs are calculated from equations: $\begin{eqnarray}\displaystyle RMSE_{F_{x}(x,i_{x})}=\sqrt{\frac{1}{n}\mathop{\sum }_{x=-0.2∼\text{mm}}^{0.2∼\text{mm}}\mathop{\sum }_{i_{x}=-2∼\text{A}}^{2∼\text{A}}[F_{x}(x,i_{x})-(k_{sx}x+k_{ix}i_{x})]^{2}} & & \displaystyle\end{eqnarray}$ (7a) $\begin{eqnarray}\displaystyle RMSE_{F_{y}(y,i_{y})}=\sqrt{\frac{1}{n}\mathop{\sum }_{y=-0.2∼\text{mm}}^{0.2∼\text{mm}}\mathop{\sum }_{i_{y}=-2∼\text{A}}^{2∼\text{A}}[F_{y}(y,i_{y})-(k_{sy}y+k_{iy}i_{y})]^{2}} & & \displaystyle\end{eqnarray}$ (7b) where n denotes numbers of comparison points (n = 121).

The cross-coupling effect was quantified by RMSEs calculated from the following expressions: $\begin{eqnarray}\displaystyle RMSE_{F_{x}(y,i_{y})}=\sqrt[]{\frac{1}{n}\mathop{\sum }_{y=-0.2∼\text{mm}}^{0.2∼\text{mm}}\mathop{\sum }_{i_{y}=-2∼\text{A}}^{2∼\text{A}}F_{x}(y,i_{y})^{2}} & & \displaystyle\end{eqnarray}$ (8a) $\begin{eqnarray}\displaystyle RMSE_{F_{y}(x,i_{x})}=\sqrt[]{\frac{1}{n}\mathop{\sum }_{x=-0.2∼\text{mm}}^{0.2∼\text{mm}}\mathop{\sum }_{i_{x}=-2∼\text{A}}^{2∼\text{A}}F_{y}(x,i_{x})^{2}} & & \displaystyle\end{eqnarray}$ (8b) In Table 3 are listed RMSEs for described 6-pole HMB. The highest error occurs for the characteristic of the magnetic force F _y(y, i _y). The second highest error occurs for the magnetic force characteristic F _y(x, i _x).

Table 3

Root mean square errors for the basic version of the HMB

RMSE _{Fx (x, ix)}	5.77
RMSE _{Fy (x, ix)}	9.99
RMSE _{Fx (y, iy)}	0.20
RMSE _{Fy (y, iy)}	10.71

4. The first method of reduction of the cross-coupling effect and magnetic force nonlinearity

The first method of reducing the cross-coupling effect and magnetic force nonlinearity relays on the division of the stator yoke into three separate parts what creates three electromagnets with permanent magnets. Figure 9.a presents magnetic field distribution for the central position of the rotor and lack of control currents. In comparison to the basic version of the HMB, the value of the magnetic field density in the stator yoke has increased. The reason for this phenomenon is a smaller cross-section of the flux path in the HMB with the fragmented stator. Figure 9.b shows magnetic field distribution for the central position of the rotor and control current i _x = 2 A. For this condition, the HMB generates magnetic forces F _x = 21.95 N and F _y = −7.51 N. In comparison to the basic version of the HMB, the value of the magnetic force F _y decreased from −10.76 N to −7.51 N, but simultaneously the value of magnetic force F _x reduced from 39.45 N to 21.95 N.

Fig. 9.

Magnetic field distribution for central position of the rotor and lack of control currents (a) and for central position of the rotor and control current i _x = 2 A.

Fig. 10.

Magnetic force F _x (a) and magnetic force F _y (b) in function of the rotor position x and control current i _x.

Figures 10.a and 10.b present magnetic force F _x and F _y in function of the rotor position x and control current i _x. One can notice that the maximal value of the magnetic force F _x has significantly decreased. Also characteristic of the magnetic force F _y in function of the rotor position x and control current i _x has changed in comparison to the basic version of the HMB. For the version of the HMB with fragmented stator the maximal value of the magnetic force F _y is equal to 7.099 N, while the minimal value is equal to −12.23 N.

Figures 11.a and 11.b present magnetic force F _x and F _y in function of the rotor position y and control current i _y. In comparison with the basic version of the HMB, magnetic force F _x in function of the rotor position and control current i _y varies from −6.72 N to 7.18 N. Similarly to the characteristic of the magnetic force F _x(x, i _x), values of the magnetic force F _y in function of the rotor position y and control current i _y has decreased.

Fig. 11.

Magnetic force F _x (a) and magnetic force F _y (b) in function of the rotor position y and control current i _y.

Parameters of the HMB with fragmented stator are listed in Table 4. Root mean square errors for the HMB with fragmented stator are listed in Table 5.

Table 4

Parameters of the HMB with the fragmented stator

Position stiffness k _sx	92.93 (N/mm)
Current stiffness k _ix	11.47 (N/A)
Position stiffness k _sy	92.29 (N/mm)
Current stiffness k _iy	11.43 (N/A)
Dynamic inductance L _d	2.21 (mH)
Velocity induce voltage e _v	8.13 (Vs/m)

Table 5

Root mean square errors for the HMB with the fragmented stator

RMSE _{Fx (x, ix)}	2.23
RMSE _{Fy (x, ix)}	4.04
RMSE _{Fx (y, iy)}	3.44
RMSE _{Fy (y, iy)}	2.96

Division of the HMB stator has a positive impact on reducing the cross-coupling effect and the magnetic force nonlinearity. Three of the four root mean square errors decreased its value, only the value of error RMSE _{Fx (y, iy)} has increased from 0.20 to 3.44. Unfortunately, this method of reducing nonlinearities has one disadvantage. Values of the position stiffness and current stiffness have decreased 24% and 44%, respectively.

5. The second method of reduction of the cross-coupling effect and magnetic force nonlinearity

Figure 6.b indicates that the most negative values of the magnetic force F _y occur for the rotor position x equals to −0.2 mm and control current i _x equals to −2 A as well as for the rotor position x equals to 0.2 mm and control current i _x equals to 2 A. Therefore, the second method of reducing the cross-coupling effect and magnetic force nonlinearity relays on exciting an additional current in the first winding in order to compensate the negative value of the magnetic force F _y. The additional current i _add is described by the following equation: $\begin{eqnarray}\displaystyle i_{add}=abs\left(\frac{1}{2}i_{x}+5000x\right) & & \displaystyle\end{eqnarray}$ (9) where x is the rotor position in the x-axis.

The additional current is excited only in the first winding, hence winding currents i ₁, i ₂, i ₃ are described by the following equations: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle i_{1}=i_{y}+abs\left(\frac{1}{2}i_{x}+5000x\right)\end{eqnarray}$ (10a) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle i_{2}=-\displaystyle \frac{1}{2}i_{y}+\frac{\sqrt[]{3}}{2}i_{x}\end{eqnarray}$ (10b) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle i_{3}=-\displaystyle \frac{1}{2}i_{y}-\frac{\sqrt[]{3}}{2}i_{x}\end{eqnarray}$ (10c) Figure 12 depicts values of the additional current i _add in function of the rotor position x and control current i _x. The additional current obtains value 2A for the rotor position x equals to 0.2 mm and the control current i _x equals to 2 A as well as for the rotor position x equals to −0.2 mm and control current i _x equals to −2 A.

Fig. 12.

The additional current i _add in function of the rotor position x and control current i _x.

Figure 13 presents magnetic field distribution for the central position of the rotor and control current i _x = 2 A. For this condition, the HMB generates magnetic forces F _x = 34.93 N and F _y = 4.74 N. In comparison to the basic version of the HMB, the value of the magnetic force F _y changed from −10.76 N to 4.74 N, but simultaneously the value of magnetic force F _x has the same value.

Fig. 13.

Magnetic field distribution for central position of the rotor and control current i _x = 2 A.

Figures 14.a and 14.b present magnetic force F _x and F _y in function of the rotor position x and control current i _x. It is visible from those figures that the additional current doesn’t change the characteristic of the magnetic force F _x(x, i _x) in comparison with the basic version of the HMB. But, it significantly changes the characteristic of the magnetic force F _y(x, i _x).

Fig. 14.

Magnetic force F _x (a) and magnetic force F _y (b) in function of the rotor position x and control current i _x.

Characteristics of the magnetic force F _x(y, i _y) and F _y(y, i _y) (Fig. 15.a and 15.b) are the same as counterpart characteristics for the basic version of the HMB (Fig. 7.a and Fig. 7.b) because the additional current i _add does not depend on the rotor position and control current in the y-axis.

Fig. 15.

Magnetic force F _x (a) and magnetic force F _y (b) in function of the rotor position y and control current i _y.

Parameters of the HMB with the additional current i _add are listed in Table 6. Root mean square errors for the HMB with additional current are listed in Table 7.

Table 6

Parameters of the HMB with an additional current i _dd in the first winding

Position stiffness k _sx	109.84 (N/mm)
Current stiffness k _ix	20.06 (N/A)
Position stiffness k _sy	116.11 (N/mm)
Current stiffness k _iy	20.61 (N/A)
Dynamic inductance L _d	4.65 (mH)
Velocity induce voltage e _v	14.36 (Vs/m)

Table 7

Root mean square errors for the HMB with an additional current i _dd in the first winding

RMSE _{Fx (x, ix)}	3.95
RMSE _{Fy (x, ix)}	6.34
RMSE _{Fx (y, iy)}	0.20
RMSE _{Fy (y, iy)}	10.71

An additional current i _add excited in the first winding has a positive impact on reducing the cross-coupling effect. The value of the error RMSE _{F
_y(y, i
_y)} decreased from 9.99 to 6.34 as well as the value of the error RMSE _{F
_x(x, i
_x)} decreased from 5.77 to 3.95. The value of errors RMSE _{F
_y(y, i
_y)} and RMSE _{F
_x(y, i
_y)} are the same as for the basic version of the HMB, because the additional current depends only from the rotor position and control current in x-axis. Significant advantage of this method is that values of the HMB parameters are almost the same as for the basic version.

6. Conclusions

This paper investigates the cross-coupling effect and magnetic nonlinearity of the 6-pole hybrid magnetic bearing. Results obtained from the 3D finite element model illustrate that the rotor position and control current in x-axis negatively influence the magnetic force in the y-axis. Moreover, it can be observed the nonlinearity of the magnetic force characteristic F _y in function of the rotor position y and control current i _y.

Two methods of reducing the cross-coupling effect and the magnetic force nonlinearity are proposed. The first method relays on the division of the stator yoke into three parts that causes the different flux distribution in comparison to the basic version of the HMB. This approach limits the cross-coupling effect and reduces the nonlinearity of the magnetic force characteristic F _y(y, i _y), but simultaneously it causes decreasing of the position and current stiffness values. The second method relays on the excitation of an additional current in the first winding, which produces additional flux in the first winding. This approach constitutes the best method of reducing the cross-coupling effect because it almost eliminates the force coupling between x- and y-axis, while values of the position and current stiffness are slightly smaller in comparison to the basic version of the HMB.

The position of the permanent magnets in the stator, as well as their dimensions, can influence the cross-coupling effect and the nonlinearity of the magnetic force, therefore further research will concentrate on this aspect of the stator design.

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