Modeling and Design of a Novel 5-DOF AC–DC Hybrid Magnetic Bearing

: This paper investigates a novel integrated AC–DC hybrid magnetic bearing (HMB) to reduce the volume, weight, manufacturing, and operation cost of magnetic suspension motors. Two radial MBs and an axial MB are integrated with the proposed HMB. The ﬁve-degree-of-freedom (5-DOF) suspension is realized in one unit. Two axially polarized permanent magnets provide the radial and axial bias ﬂuxes. First, the HMB structure and the suspension mechanism are introduced. Second, based on the method of equivalent magnetic circuits, magnetic circuits are calculated. The mathematical models of suspension force are discussed. Third, the main parameters of the 5-DOF AC–DC HMB are given. The 3D ﬁnite element method (FEM) is adopted to analyze the proposed system’s electromagnetic characteristics, and the suspension mechanism of the 5-DOF is veriﬁed. The radial suspension forces versus the radial control current, the axial suspension forces versus the axial control current, the relationship between the axial suspension force and the axial control current with the X direction offset, and the relationship between the radial suspension force in the X direction and the axial control current with the Z direction offset are calculated. Based on the research results, it is shown that the HMB structure is compact and reasonable, and the mathematical models and suspension mechanism are correct.


Introduction
Currently, with the increase of rotor speed, the requirements for bearing performance are getting higher and higher.New types of bearings such as air-floated bearings, liquidfloated bearings, and magnetic bearings (MBs) have been employed.The friction between the rotor and bearing is reduced in high-speed electric machines supported by air-floated or liquid-floated bearings.However, special air-floated and liquid-floated equipment is required, which makes the motor system bulky, complicated in structure, and increased in cost.Moreover, there are inevitable problems of air and liquid leakage that prevent their use in pollution-free, ultra-clean special electric drives.MBs supported by electromagnetic force are new types of non-contact bearings.They have the advantages of no wear and lubrication, low vibration and noise [1][2][3][4][5][6][7][8][9], and can achieve higher rotor speed and power operation.Therefore, special machines use MBs, such as vacuum pumps, blood pumps, and heart pumps [10][11][12][13].
Commonly, thrust MBs are adopted to control the axial directional translation.The radial two-degree-of-freedom (2-DOF) MBs are employed to control the two radial directional translations [14].A 3-DOF MB is used to control the axial directional translation and two radial tilting motions [15,16].Recently, several studies have been proposed to develop the structures of MBs.In [8], a slice of 5-DOF four-pole MB was studied.The structure is compact.However, it is driven by DC power amplifiers, resulting in a bulky and costly system.In [17], a 3-DOF MB with a biased permanent magnet was designed and applied to a high-speed motor.Compared with the MBs using only solid magnetic materials, a laminated radial stator was applied in the 3-DOF MB to reduce eddy currents.Considering the eddy-current effects and leakage effects, an accurate mathematical model was established.Meanwhile, in [18], a 3-DOF MB was designed and optimized with a hierarchical multiobjective optimization structure to improve both the radial and axial suspension.However, a large thrust disk causes many problems such as speed limitation, increased rotor unbalance, and rotor dynamic performance degradation.
In [19], a novel four-pole 4-DOF MB with two active parts and one passive part was proposed.The structure of the 4-DOF MB was compact, so the size and power consumption were reduced.However, due to the application of four permanent magnets, the 4-DOF MB has high costs and is difficult to manufacture and assemble [20].Moreover, because the axial single DOF is passively stabilized by reluctance forces, the axial load-carrying capacity is limited, and the output power of the motor is low.It is only suitable for applications where the axial impact and motor output power are small.Commonly, a full magnetic suspension drive system needs axial MBs, radial MBs, or radial-axial MBs, and a highspeed electric machine [21][22][23].A combined structure of two or three MBs results in a rather long axial length of the rotor shaft and makes the structure complicated.It has low axial utilization, critical speed, and power density.Compared with bearingless motors such as bearingless permanent magnetic motors [24] and bearingless flux-switching permanent motors [25], combined structures have obvious advantages in terms of simple control and small coupling [26,27].
In addition, traditional MBs generally have an eight-pole or four-pole structure [28][29][30].The radial 2-DOF MB is driven by four unipolar or two bipolar DC power amplifiers.The whole MB system is bulky and costly [31,32].In order to reduce the size and save cost, a three-pole MB was proposed.At present, three-pole MBs mainly use two driving modes, namely a three-phase power inverter drive [33][34][35] and a DC power amplifier drive [36].In comparison with DC MBs, AC MBs can effectively reduce the number of inverters, system volume, and cost [37,38].
This research proposes a 5-DOF AC-DC HMB with permanent magnets to increase the critical speed of high-speed electric machines, especially in high-speed mechanical systems.The proposed HMB can realize 5-DOF in one unit.Three-phase power inverters are employed for the radial drive, while DC power amplifiers are used for the axial drive.The radial and axial control fluxes do not interfere with each other.The structure of the inclined axial magnetic poles reduces the axial length of the rotor and is beneficial to improving the critical speed of the rotor.It has a series of advantages such as low cost, low power consumption, and compact structure.The structure and working principle of the proposed 5-DOF AC-DC HMB are first introduced.Then, the equivalent magnetic circuit method is employed to deduce the mathematical models of suspension forces.Based on the finite element analysis (FEA), the electromagnetic properties of the 5-DOF AC-DC HMB are analyzed, including the radial suspension forces versus the radial control current, the axial suspension forces versus the axial control current, flux field distributions, and the relationship between the suspension force and the control current with the offset, all of which verify the correctness of the theoretical analysis.

Structure and Working Principle
The configuration of the 5-DOF AC-DC HMB is shown in Figure 1.The HMB includes an axial control core, two axial control coils, two radial control cores, radial control coils, a rotor core, and two axially polarized permanent magnet rings.To reduce the axial length, the two ends of the rotor are designed as inclined structures.Three radial stator magnetic poles are evenly arranged around the circumference of the left and right radial control cores.Three radial control coils are, respectively, wound on the radial stator magnetic poles.For high-speed mechanical systems, high-speed electric machines can be installed between the two radial control cores and integrated with the HMB to improve the axial utilization ratio, critical speed, and power density.
installed between the two radial control cores and integrated wi the axial utilization ratio, critical speed, and power density.Figure 2 shows the magnetic flux paths of the 5-DOF AC-D path (solid lines with arrows) starts from the N-pole of the perm through the axial control core, axial air gap, rotor core, radial air core, and finally returns to the S-pole of the permanent magnet.flow through the axial control core, the axial air gap, and the rot radial control fluxes is formed by the radial control core, the radi core.The air-gap fluxes are composed of bias fluxes and contro netic resistance of the permanent magnet is large, the control flux the permanent magnet pole, and the demagnetization of the per control fluxes can be avoided.
In the common state, the rotor is in the equilibrium position is no control current, and the rotor is suspended under the m generated by the bias fluxes.If the rotor is offset, the displacemen dial and axial displacements.The axial DC power amplifier ad current to generate axial control fluxes, while the radial AC inv control fluxes generated by the three-phase symmetrical AC curr coils.In the direction of the rotor offset, due to the superposition o bias fluxes, air-gap fluxes are weakened.In the opposite direc air-gap fluxes are enhanced.The rotor is pulled back to the ba suspension forces in the opposite direction to the rotor offset dire Figure 2 shows the magnetic flux paths of the 5-DOF AC-DC HMB.The bias flux path (solid lines with arrows) starts from the N-pole of the permanent magnet, passing through the axial control core, axial air gap, rotor core, radial air gap, and radial control core, and finally returns to the S-pole of the permanent magnet.The axial control fluxes flow through the axial control core, the axial air gap, and the rotor core.The loop of the radial control fluxes is formed by the radial control core, the radial air gap, and the rotor core.The air-gap fluxes are composed of bias fluxes and control fluxes.Since the magnetic resistance of the permanent magnet is large, the control fluxes do not pass through the permanent magnet pole, and the demagnetization of the permanent magnet by the control fluxes can be avoided.Figure 2 shows the magnetic flux paths of the 5-DOF AC-DC HMB.T path (solid lines with arrows) starts from the N-pole of the permanent mag through the axial control core, axial air gap, rotor core, radial air gap, and ra core, and finally returns to the S-pole of the permanent magnet.The axial co flow through the axial control core, the axial air gap, and the rotor core.The radial control fluxes is formed by the radial control core, the radial air gap, a core.The air-gap fluxes are composed of bias fluxes and control fluxes.Sin netic resistance of the permanent magnet is large, the control fluxes do not p the permanent magnet pole, and the demagnetization of the permanent ma control fluxes can be avoided.
In the common state, the rotor is in the equilibrium position.At this mo is no control current, and the rotor is suspended under the magnetic attr generated by the bias fluxes.If the rotor is offset, the displacement sensors d dial and axial displacements.The axial DC power amplifier adjusts the a current to generate axial control fluxes, while the radial AC inverters adjus control fluxes generated by the three-phase symmetrical AC current in the ra coils.In the direction of the rotor offset, due to the superposition of the contro bias fluxes, air-gap fluxes are weakened.In the opposite direction of the air-gap fluxes are enhanced.The rotor is pulled back to the balanced pos suspension forces in the opposite direction to the rotor offset direction.In the common state, the rotor is in the equilibrium position.At this moment, there is no control current, and the rotor is suspended under the magnetic attractive force generated by the bias fluxes.If the rotor is offset, the displacement sensors detect the radial and axial displacements.The axial DC power amplifier adjusts the axial control current to generate axial control fluxes, while the radial AC inverters adjust the radial control fluxes generated by the three-phase symmetrical AC current in the radial control coils.In the direction of the rotor offset, due to the superposition of the control fluxes and bias fluxes, air-gap fluxes are weakened.In the opposite direction of the rotor offset, air-gap fluxes are enhanced.The rotor is pulled back to the balanced position by the suspension forces in the opposite direction to the rotor offset direction.

Mathematical Model
The mathematical model is the theoretical basis of MBs.To design the 5-DOF AC-DC HMB for high-speed mechanical systems, it is necessary to derive the mathematical model of the proposed MB.This paper focuses on a novel structure and suspension principle.Therefore, the influences of flux leakage, eddy current loss, and magnetic saturation are neglected.Only the air-gap permeances are considered.

Equivalent Magnetic Circuit
Figure 3 shows an equivalent bias magnetic circuit.If the rotor is displaced in the positive X, Y, and Z directions, the magnetic permeances of each air gap can be given as  The bias fluxes of permanent magnets at each air gap are written as Figure 4 shows the equivalent control magnetic circuit.The resultant fluxes at each air gap can be expressed as Appl.Sci.2022, 12, x FOR PEER REVIEW

Magnetic Analysis
To verify the suspension principle and mathematical models, the finite ele method (FEM) was employed to calculate the electromagnetic properties of the prop HMB, including its flux distribution, current, suspension forces, and coupling chara istics.Table 1 shows the main design parameters of the HMB.The magnetic flux density distribution is shown in Figure 5.The radial and axia flux distributions of the permanent magnets are shown in Figure 5a,b, respectively For three-phase AC systems, i lA , i lB , i lC , i rA , i rB , and i rC meet the condition of i lA + i lB + i lC = 0 and i rA + i rB + i rC = 0.
Therefore, the suspension force in the axial and radial directions can be written as The levitation force is nonlinear with the rotor offset and is also nonlinear with the current.The suspension force is linearized near the central position.The approximation can meet the engineering requirements.Therefore, the axial and radial suspension force can be further expressed as Appl.Sci.2022, 12, 8931 where The axial and radial suspension force can be further expressed as

Magnetic Analysis
To verify the suspension principle and mathematical models, the finite element method (FEM) was employed to calculate the electromagnetic properties of the proposed HMB, including its flux distribution, current, suspension forces, and coupling characteristics.Table 1 shows the main design parameters of the HMB.

Magnetic Flux Analysis
The magnetic flux density distribution is shown in Figure 5.The radial and axial bias flux distributions of the permanent magnets are shown in Figure 5a,b, respectively.We can see from Figure 5a,b that the radial and axial bias fluxes were symmetrical, and the magnetic fields in the axial and radial air gaps were equal, so there was no resultant force on the rotor.
Assuming that the rotor moved in the −X direction, the length of the air gap in the −X direction became smaller.Therefore, the resultant forces caused the rotor to move in the −X direction further.To restore the rotor back to the central position, restoring forces in the +X direction were produced.Figure 5c shows the radial resultant fluxes.The radial control fluxes only formed a loop between the radial core and the rotor and had little effect on the bias flux distribution in the axial control core.The instantaneous fluxes at the three air gaps were different, so composite fluxes could be formed.By controlling the three-phase control currents, the fluxes were added or subtracted, and the corresponding radial restoring forces were generated to overcome the disturbance forces so that the rotor would be restored back to the central position.Therefore, the AC three-phase inverters can be used to control the radial 2-DOF and reduce the cost of the HMB system.Assuming that the rotor moved in the −X direction, the length of the air gap in th direction became smaller.Therefore, the resultant forces caused the rotor to move i −X direction further.To restore the rotor back to the central position, restoring forc the +X direction were produced.Figure 5c shows the radial resultant fluxes.The r control fluxes only formed a loop between the radial core and the rotor and had litt fect on the bias flux distribution in the axial control core.The instantaneous fluxes a three air gaps were different, so composite fluxes could be formed.By controllin three-phase control currents, the fluxes were added or subtracted, and the correspon radial restoring forces were generated to overcome the disturbance forces so that th tor would be restored back to the central position.Therefore, the AC three-phase in ers can be used to control the radial 2-DOF and reduce the cost of the HMB system.
Similarly, if the rotor was disturbed in the Z direction, Figure 5d shows the resultant fluxes.It can be found that the axial fluxes were reduced at one end, while a other end, they were increased and had little effect on the bias fluxes in the radial co core.Afterward, restoring axial forces were produced to move the rotor toward equilibrium position.
Figure 5e shows the flux distribution when the axial control fluxes and radial co fluxes were simultaneously generated.It was further proved that the axial-radial co fluxes did not interfere with each other, and the magnetic circuit had almost no coup

Suspension Force Analysis
For a better analysis of the relationship between current and force, three diff situations were analyzed through FEM and the equivalent magnetic circuit method.only the radial control current was applied, and the resulting FEM simulation equivalent magnetic circuit analysis were performed.Figure 6 shows the relation between forces in the radial direction and the radial current at the central position o rotor.Similarly, if the rotor was disturbed in the Z direction, Figure 5d shows the axial resultant fluxes.It can be found that the axial fluxes were reduced at one end, while at the other end, they were increased and had little effect on the bias fluxes in the radial control core.Afterward, restoring axial forces were produced to move the rotor toward the equilibrium position.
Figure 5e shows the flux distribution when the axial control fluxes and radial control fluxes were simultaneously generated.It was further proved that the axial-radial control fluxes did not interfere with each other, and the magnetic circuit had almost no coupling.

Suspension Force Analysis
For a better analysis of the relationship between current and force, three different situations were analyzed through FEM and the equivalent magnetic circuit method.First, only the radial control current was applied, and the resulting FEM simulation and equivalent magnetic circuit analysis were performed.Figure 6 shows the relationship between forces in the radial direction and the radial current at the central position of the rotor.
From the FEM results, it was found that the radial suspension force changed from −523.4 to 523.4 N with the control current changing from −5 to 5 A, and the result of the equivalent magnetic circuit indicated that the radial suspension force changed from −561.2 to 561.2 N. Thus, the results of FEM were in accordance with the results of the equivalent magnetic circuit method.It is clear that the relationship was linear between the radial force F x and the radial current.
In the second experiment, the excitation of the axial current was only used, and the radial current was zero.The characteristics of the axial forces versus axial current at the equilibrium position of the rotor were analyzed using the FEM and equivalent magnetic circuit; the results are shown in Figure 7. Furthermore, the FEM results showed that the axial suspension force changed from −1183.6 to 1183.6 N with the axial current changing from −5 to 5 A. The calculation result of the equivalent magnetic circuit revealed that the axial suspension force changed from −1195.5 to 1195.5 N. We can see that when the axial current was small, the axial suspension force was linear with the axial current, and the FEM analysis result was consistent with the theoretical result.With the increase in the axial current, the axial force of the FEM was less than that of the equivalent magnetic circuit due to the magnetic saturation.Therefore, the linearity and consistency were ideal when working in a small current range, which can meet the actual operation requirements.From the FEM results, it was found that the radial suspensio −523.4 to 523.4 N with the control current changing from −5 to 5 A equivalent magnetic circuit indicated that the radial suspensio −561.2 to 561.2 N. Thus, the results of FEM were in accordance equivalent magnetic circuit method.It is clear that the relationsh the radial force Fx and the radial current.
In the second experiment, the excitation of the axial current w radial current was zero.The characteristics of the axial forces ver equilibrium position of the rotor were analyzed using the FEM an circuit; the results are shown in Figure 7. Furthermore, the FEM r axial suspension force changed from −1183.6 to 1183.6 N with the from −5 to 5 A. The calculation result of the equivalent magnetic c axial suspension force changed from −1195.5 to 1195.5 N. We can current was small, the axial suspension force was linear with the FEM analysis result was consistent with the theoretical result.W axial current, the axial force of the FEM was less than that of th circuit due to the magnetic saturation.Therefore, the linearity and al when working in a small current range, which can meet the ac ments.From the FEM results, it was found that the radial suspensi −523.4 to 523.4 N with the control current changing from −5 to 5 equivalent magnetic circuit indicated that the radial suspensio −561.2 to 561.2 N. Thus, the results of FEM were in accordance equivalent magnetic circuit method.It is clear that the relationsh the radial force Fx and the radial current.
In the second experiment, the excitation of the axial current radial current was zero.The characteristics of the axial forces ver equilibrium position of the rotor were analyzed using the FEM an circuit; the results are shown in Figure 7. Furthermore, the FEM r axial suspension force changed from −1183.6 to 1183.6 N with the from −5 to 5 A. The calculation result of the equivalent magnetic c axial suspension force changed from −1195.5 to 1195.5 N. We can current was small, the axial suspension force was linear with the FEM analysis result was consistent with the theoretical result.W axial current, the axial force of the FEM was less than that of th circuit due to the magnetic saturation.Therefore, the linearity and al when working in a small current range, which can meet the ac ments.In the third experiment, under the condition of a simultaneo and axial current, Figure 8 shows the variation between the force directions and the current at the central position.There are six cu of which represent Fx, Fy, and Fz calculated using FEM, and the o Fy, and Fz calculated using the equivalent magnetic circuit.The showed that the suspension forces of the combined action of a currents were basically the same as those of the separate action rents, which means that the radial control fluxes and axial contro uncoupled.In the third experiment, under the condition of a simultaneous application of radial and axial current, Figure 8 shows the variation between the force in the radial and axial directions and the current at the central position.There are six curves in Figure 8, three of which represent F x , F y, and F z calculated using FEM, and the other three represent F x , F y , and F z calculated using the equivalent magnetic circuit.The results of the analysis showed that the suspension forces of the combined action of axial and radial control currents were basically the same as those of the separate action of the two control currents, which means that the radial control fluxes and axial control fluxes were basically uncoupled.
The difference between the two methods mainly results from the consideration of flux leakage, eddy current loss, and magnetic saturation.The flux leakage, eddy current loss, and magnetic saturation were considered in the FEM analysis but neglected in the equivalent magnetic circuit model.Nevertheless, the correctness of theoretical analysis was proved by a comparison of its results with those found using the FEM analysis.The difference between the two methods mainly results fro flux leakage, eddy current loss, and magnetic saturation.The flux loss, and magnetic saturation were considered in the FEM analys equivalent magnetic circuit model.Nevertheless, the correctness was proved by a comparison of its results with those found using

Displacement-Force-Current Analysis
Figure 9 shows the relationship between the axial suspensi control current with the rotor offset in the X direction.The rotor position in the Y and Z directions, and the control current was zer tions.The axial current changed from −5 to 5 A. The rotor moved in the X direction.When the axial current was a fixed value, the c sion force Fz was very small.This indicated that the change in the the X direction had little effect on the axial suspension force Fz.D uration, the slope of the force-current curve decreased when However, the relationship was basically linear in the effective wo control current.

Displacement-Force-Current Analysis
Figure 9 shows the relationship between the axial suspension force and the axial control current with the rotor offset in the X direction.The rotor was in an equilibrium position in the Y and Z directions, and the control current was zero in the X and Y directions.The axial current changed from −5 to 5 A. The rotor moved from −0.25 to 0.25 mm in the X direction.When the axial current was a fixed value, the change in axial suspension force F z was very small.This indicated that the change in the rotor displacement in the X direction had little effect on the axial suspension force F z .Due to the magnetic saturation, the slope of the force-current curve decreased when the current was large.However, the relationship was basically linear in the effective working range of the axial control current.The difference between the two methods mainly results from the co flux leakage, eddy current loss, and magnetic saturation.The flux leakage, loss, and magnetic saturation were considered in the FEM analysis but ne equivalent magnetic circuit model.Nevertheless, the correctness of theor was proved by a comparison of its results with those found using the FEM

Displacement-Force-Current Analysis
Figure 9 shows the relationship between the axial suspension force control current with the rotor offset in the X direction.The rotor was in a position in the Y and Z directions, and the control current was zero in the X tions.The axial current changed from −5 to 5 A. The rotor moved from −0. in the X direction.When the axial current was a fixed value, the change in sion force Fz was very small.This indicated that the change in the rotor di the X direction had little effect on the axial suspension force Fz.Due to the uration, the slope of the force-current curve decreased when the curre However, the relationship was basically linear in the effective working ran control current.Figure 10 shows the relationship between the radial suspension force tion and the axial current with the rotor offset in the Z direction.The ro equilibrium position in the X and Y directions, and the control current was and Z directions.The radial control current in the X direction changed fr The rotor was displaced from −0.25 to 0.25 mm in the Z direction.When t trol current in the X direction was a fixed value, we can see that the displa Figure 10 shows the relationship between the radial suspension force in the X direction and the axial current with the rotor offset in the Z direction.The rotor was in an equilibrium position in the X and Y directions, and the control current was zero in the Y and Z directions.The radial control current in the X direction changed from −5 to 5 A. The rotor was displaced from −0.25 to 0.25 mm in the Z direction.When the radial control current in the X direction was a fixed value, we can see that the displacement in the Z direction had little effect on the force in the X direction.Although with the increase in the axial displacement, the relationship between the force in the X direction and the radial current still maintained good linearity.
ppl.Sci.2022, 12, x FOR PEER REVIEW Z direction had little effect on the force in the X direction.Although with t the axial displacement, the relationship between the force in the X directio dial current still maintained good linearity.Finally, from Figures 9 and 10, it can be concluded that the axial co the radial control flux were decoupled.

Conclusions
In this paper, a novel integrated AC-DC HMB was proposed, which 2-DOF MBs and one single DOF MB into one unit to realize 5-DOF.Comp traditional 5-DOF MB, its structure is more compact, which makes the hi tors with this kind of MB have a higher axial utilization ratio, critical spee density.The mathematical models of the 5-DOF AC-DC HMB were bui characteristics.The results of the comparison of 3D FEM and equivalent m methods in the proposed 5-DOF HMB revealed a good linear relationship al suspension forces versus the radial control current and between axi forces versus the axial control current.The analysis of displacement-force cates that there was almost no coupling between the radial control fluxes trol fluxes.The FEM analysis results verified the correctness of the theoreti the novel 5-DOF AC-DC HMB proposed in this study.Finally, from Figures 9 and 10, it can be concluded that the axial control flux and the radial control flux were decoupled.

Conclusions
In this paper, a novel integrated AC-DC HMB was proposed, which combines two 2-DOF MBs and one single DOF MB into one unit to realize 5-DOF.Compared with the traditional 5-DOF MB, its structure is more compact, which makes the high-speed motors with this kind of MB have a higher axial utilization ratio, critical speed, and power density.The mathematical models of the 5-DOF AC-DC HMB were built to study its characteristics.The results of the comparison of 3D FEM and equivalent magnetic circuit methods in the proposed 5-DOF HMB revealed a good linear relationship between radial suspension forces versus the radial control current and between axial suspension forces versus the axial control current.The analysis of displacement-force-current indicates that there was almost no coupling between the radial control fluxes and axial control fluxes.The FEM analysis results verified the correctness of the theoretical analysis of the novel 5-DOF AC-DC HMB proposed in this study.

Figure 3 .
Figure 3. Equivalent bias magnetic circuit: (a) left bias fluxes and (b) right bias fluxes.Due to the symmetrical structure of the left and right 2-DOF MBs, the geometric dimensions are the same.The bias fluxes of permanent magnets at each air gap are written as

Figure 4 .
Figure 4. Equivalent control magnetic circuit: (a) left radial control fluxes, (b) right radial control fluxes, and (c) axial control fluxes.

Figure 6 .
Figure 6.The curves show the correlation between radial suspension forc

Figure 6 .
Figure 6.The curves show the correlation between radial suspension forces and radial current.

Figure 7 .
Figure 7.The curves show the correlation between axial suspension force

Figure 7 .
Figure 7.The curves show the correlation between axial suspension forces and axial current.

Figure 8 .
Figure 8. Variations in radial and axial force with current.

Figure 8 .
Figure 8. Variations in radial and axial force with current.

Figure 8 .
Figure 8. Variations in radial and axial force with current.

Figure 9 .
Figure 9.The influence of radial displacement on the axial suspension force-current

Figure 9 .
Figure 9.The influence of radial displacement on the axial suspension force-current characteristics.

Figure 10 .
Figure 10.The influence of axial displacement on the radial suspension force-curr tics in X direction.

Figure 10 .
Figure 10.The influence of axial displacement on the radial suspension force-current characteristics in X direction.
Magnetomotive force of the left and right permanent magnets (A) φ lm Total fluxes of the left permanent magnets (Wb) φ rm Total fluxes of the right permanent magnets (Wb) G lpz Left-inclined axial air-gap permeances (H) G rpz Right-inclined axial air-gap permeances (H) G lA Left radial air-gap permeances in the A direction (H) G lB Left radial air-gap permeances in the B direction (H) G lC Left radial air-gap permeances in the C direction (H) G ra Right radial air-gap permeances in the A direction (H) G rB Right radial air-gap permeances in the B direction (H) G rC Right radial air-gap permeances in the C direction (H) φ lpz Left axial bias fluxes (Wb) φ rpz Right axial bias fluxes (Wb) φ lpA Left radial bias fluxes in the A direction (Wb) φ lpB Left radial bias fluxes in the B direction (Wb) φ lpC Left radial bias fluxes in the C direction (Wb) φ rpA Right radial bias fluxes in the A direction (Wb) φ rpB Right radial bias fluxes in the B direction (Wb) φ rpC Right radial bias fluxes in the C direction (Wb) x l Left displacements in the X direction (m) y l Left displacements in the Y direction (m) x r Right displacements in the X direction (m) y r Right displacements in the Y direction (m) z Displacement in the Z direction (m) µ 0 Vacuum permeability (md) S r Radial pole face area( m 2 ) S a Axial pole face area (m 2 ) d r Average radial air gap (m) d a Axial inclined air-gap length (m) θ Angle of the inclined axial pole φ lkA Left radial control fluxes in the A direction (Wb) φ lkB Left radial control fluxes in the B direction (Wb) φ lkC Left radial control fluxes in the C direction (Wb) φ rkA Right radial control fluxes in the A direction (Wb) φ rkB Right radial control fluxes in the B direction (Wb) φ rkC Right radial control fluxes in the C direction (Wb) φ lkz Left axial control fluxes (Wb) φ rkz Right axial control fluxes (Wb) N l Numbers of the left radial coil turns N r Numbers of the right radial coil turns i lA Current in the A direction left radial coil (A) i lB Current in the B direction left radial coil (A) i lC Current in the C direction left radial coil (A) i rA Current in the A direction right radial coil (A) i rB Current in the B direction right radial coil (A) i rC Current in the C direction right radial coil (A) N z Number of the axial coil turns on one side i z Axial control currents (A) φ lA Left radial air-gap fluxes in the A direction (Wb) φ lB Left radial air-gap fluxes in the B direction (Wb) φ lC Left radial air-gap fluxes in the C direction (Wb) φ rA Right radial air-gap fluxes in the A direction (Wb) φ rB Right radial air-gap fluxes in the B direction (Wb) φ rC Right radial air-gap fluxes in the C direction (Wb) φ z1 Left axial air-gap fluxes (Wb) φ z2 Right axial air-gap fluxes (Wb) F z Force in the Z direction (N) F lA Left forces in the A direction (N) F lB Left forces in the B direction (N) F lC Left forces in the C direction (N) F rA Right forces in the A direction (N) F rB Right forces in the B direction (N) F rC Right forces in the C direction (N) k fd Axial force-displacement stiffness (N/m) k ia Axial force-current stiffness (N/m) F lp Magnetic forces of the magnets in the left radial air gaps at the equilibrium position (N) F rp Magnetic forces of the magnets in the right radial air gaps at the equilibrium position (N) k lir Left radial force-current stiffness (N/m) k rir Right radial force-current stiffness (N/m) F lx Left resultant forces in the X directions (N) F ly Left resultant forces in the Y direction (N) F rx Right resultant forces in the X direction (N) F ry Right resultant forces in the Y direction (N) F m Magnetomotive force of the left and right permanent magnets (

Table 1 .
Main design parameters.

Table 1 .
Main design parameters.