Magnetic Field Equivalent Current Analysis-based Radial Force Control for Bearingless Permanent Magnet Synchronous Motors

Bearingless permanent magnet synchronous motors (BPMSMs), with all advantages of permanent magnet motors (PMSMs) and magnetic bearings, have become an important research direction in the bearingless motor field. To realize a stable suspension for the BPMSM, accurate decoupling control between the electromagnetic torque and radial suspension force is indispensable. In this paper, a concise and reliable analysis method based on a magnetic field equivalent current is presented. By this analysis method, the operation principle is analyzed theoretically, and the necessary conditions to produce a stable radial suspension force are confirmed. In addition, mathematical models of the torque and radial suspension force are established which is verified by the finite element analysis (FEA) software ANSYS. Finally, an experimental prototype of a 2-4 poles surface-mounted BPMSM is tested with the customized control strategy. The simulation and experimental results have shown that the motor has good rotation and suspension performance, and validated the accuracy of the proposed analysis method and the feasibility of the control strategy.


Introduction
Bearingless motors combine the characteristics of conventional motors and magnetic bearings.Compared with magnetic bearings, bearingless motors possess the advantages of shorter shaft length, higher critical speed, and lower cost [1].Among different types of bearingless motors, bearingless permanent magnet synchronous motors (BPMSMs) are highly valued around the world due to their virtues of small size, light weight, non-contact, no wear, no lubrication, high efficiency, high power factor, and so on [2].They have broad application prospects in centrifugal machines, aeronautics and astronautics, life science, etc. Especially due to their non-contact and no lubrication characteristics, BPMSMs are employed successfully in high purity environments, such as the semiconductor, pharmaceutical and medical industries.
Until now, the advanced technologies of the BPMSMs, including mathematical models, motor topologies and control strategies, have been investigated and discussed extensively by researchers.The complete mathematical models of the surface mounted BPMSMs are studied in [3] by using the Maxwell stress tensor method, which can effectively compensate for the effect of rotor eccentricity and Lorentz force.In [4], a new modeling method based on mechanical to electrical coordinate transformation is put forward, which proves an effective modeling method.For the motor topologies, Ooshima proposed a rotor design method considering demagnetization for a 2-DOF BPMSM [5].Besides, a 5-DOF BPMSM, consisting of a 3-DOF axial-radial magnetic bearing and a 2-DOF BPMSM, is presented in [6].At the same time, Kascak provided another novel 5-DOF BPMSM, only consisting of two conical BPMSMs to realize the 5-DOF active control [7].For the control strategies, the decoupling control algorithm based on the rotor flux orientation control (FOC) has been widely used in recent decades [4].The direct radial displacement control strategy, based on the relationship between the rotor radial displacement and radial suspension force, is pointed out in [8].In addition, many artificial-intelligence-based control strategies to effectively handle the nonlinearities of the BPMSM system are also presented, such as sliding mode variable structure control [9], artificial neural network inverse control [10] and genetic algorithm control.
The double-winding BPMSM has two sets of coupling windings in the stator slot, namely torque windings (pole pair is PM) and suspension force windings (pole pair is PB).Thus, the precise decoupling control between them is indispensable to realize the suspension operation.In [11], by marking the distribution situation of the torque winding flux linkage and suspension force winding flux linkage, the generation principle of radial suspension forces is studied.Then, the corresponding performance of the induction type bearingless motor can be verified by the 2D finite element analysis (FEA) model.Based on these methods, analyses of the bearingless motor suspension characteristics have become simple, direct and operable.However, as the calculated coefficients are hard to get, the mathematical models are inaccurate.In [12], the mathematical model of the surface-mounted BPMSM is established based on the principle of virtual displacement.The derived idea is as follows: firstly, according to the equivalent magnetic circuit principle, the inductance matrixes are deduced.Then the expressions of electromagnetic energy based on inductance matrixes are derived.Finally, the mathematical expressions of radial suspension forces can be deduced by partial derivation with respect to the electromagnetic energy expressions.This method makes the deduction of radial suspension forces easier and more visual.The inductance parameters, however, are difficult to obtain precisely through experimental measurements, therefore, the precise calculation of radial suspension force also cannot be achieved for the BPMSM.
In this paper, an accurate and reliable analysis method based on the magnetic field equivalent current is presented.Based on this analysis method, the practical motor windings are replaced by equivalent windings existing in different material spaces.Distributions of these equivalent windings only depend on air gap magnetic fields, and they are not subject to the mechanical structure features.Then, the interactions among the air gap magnetic fields can be converted to that of the carrying-current equivalent windings.Thus, some complex analyses of magnetic fields can be avoided and the difficulty of system analysis can be effectively reduced, which provides a systemic theoretical foundation for the control of radial suspension forces.

Fundamental Theory of the Magnetic Field Equivalent Current Analysis Method
Assuming that the motor windings are designed as spatially symmetric distributions, the permeability in the air gap is equal everywhere, the leakage inductances of permanent magnets and magnetic saturation are ignored.The air gap magnetic fields are thought to be produced by a series of carrying-current equivalent windings.The pole pairs of equivalent windings are equal to that of the original air gap magnetic fields.The distributions of the equivalent windings are flexibly designed according to the actual air gap magnetic fields.The equivalent full pitch windings are employed in this paper and they can rotate with the change of the air gap magnetic fields.The current in equivalent windings is considered as direct current.Then, the interaction relationships between torque winding air gap magnetic field and suspension force winding air gap magnetic field can be obtained indirectly from that of the corresponding equivalent currents.Figure 1 shows the schematics of magnetic field equivalent current analysis method.In this paper, the analysis method based on carrying-current equivalent windings is defined as the magnetic field equivalent current analysis method.The equivalent windings produced by the torque winding air gap magnetic field are defined as the equivalent torque windings, and the equivalent windings generated by the suspension force winding air gap magnetic field are defined as the equivalent suspension force windings.In addition, on the basis of field distributions, many structural forms can be composed of equivalent torque windings and equivalent suspension force windings.Among them, intensifying factor and softening factor are two key structural forms, as shown in Figure 2. The upper layer of windings corresponds to equivalent suspension force windings, while the lower layer of windings corresponds to equivalent torque windings.Intensifying factor and softening factor are formed by two sets of equivalent windings with their centre line overlapped.The enhanced resultant air gap magnetic fields can be produced by the intensifying factor, and the maximum magnetic motive force (MMF) can be formed.The reduced resultant air gap magnetic fields can be produced by the softening factor, and the minimum MMF can be generated.If the centre lines of the two sets of equivalent windings are no longer overlapped, but there exists an inclusion relation in space, the magnetic field superposition effect between windings is the same as that of intensifying factor and softening factor.Then, these structure forms are defined as partial-intensifying factor and partial-softening factor.

The Principle of Radial Suspension Force Generation for the BPMSM
Figure 3 shows the stator configuration and the 2D FEA model of a surface-mounted BPMSM.To orientate and start easily, it is usual to bring the A phase winding axes of the torque windings in line with that of the suspension force windings.The suspension force windings are embedded in the outer layer of the stator slot, and the phase sequence arrangement is NBA→NBB→NBC, PB = 2.The torque windings are embedded in the inner layer of the stator slot, and the phase sequence arrangement is NMA→NMB→NMC, PM = 1.The motor structure diagram of the 2-4 poles BPMSM in the rotational coordinates is illustrated in Figure 4. Due to the fact that the pole pair of the permanent magnet air gap magnetic field is the same as that of the torque windings, their resultant magnetic field can similarly be considered as the products of currents in the torque windings [13].The diagrams of the motor structure and operation principle for the 2-4 poles BPMSM, based on the conventional analysis method and on the magnetic field equivalent current analysis method, are depicted in Figure 4a,b, respectively.The windings NM stand for equivalent torque windings, and windings NB represent equivalent suspension force windings.Then the principles of radial suspension force generation are compared below.Figure 4a illustrates the principle based on the conventional method for the BPMSM [14,15].The symmetrical 2-pole excitation flux ψMd is generated by the d-axis component NMd of the torque windings.When a suspension force winding current IBd is provided, the symmetrical 4-pole flux ψBd is generated in the air gap.The directions of the torque winding fluxes ψMd and the torque winding fluxes ψBd are same in the air gap 1, and thus, the resultant air gap flux density is increased.In the air gap 3, their flux directions are opposite, so the resultant air gap flux density is decreased.Then, according to Maxwell's stress tensor, the radial suspension force F is produced and points to the positive x-axis.The opposite-direction radial suspension force in the x-axis can be produced with a negative current IBd.Moreover, the vertical radial suspension force F in the y-axis can be produced by providing the suspension current IBq in the suspension winding NBq.In short, the directions of the radial suspension forces are affected by the phase of the 4-pole suspension force winding current.
Figure 4b depicts the principle based on the proposed magnetic field equivalent current analysis method.In the air gap 1, an intensifying factor is formed and a softening factor is formed at the position of air gap 3. The direction of the composite MMF is the positive x-axis.Thus, a radial suspension force F is generated which is oriented in the positive x-axis.The analysis result of the proposed method agrees with that of conventional method.In the same way, if the equivalent suspension force windings NB rotate a 90° electrical angle in the counter-clockwise direction, the vertical radial suspension force F in the positive y-axis is generated.The radial suspension force F in the negative x-axis can be produced through rotating NB by 180° electrical angle in the counter-clockwise direction.In conclusion, the direction of radial suspension force rotation has relations with that of equivalent suspension force winding rotation, and the rotating angular velocity is PB times to that of equivalent suspension force windings, as can be seen in [16].The description of the principle for the 2-4 poles BPMSM also applies to other pole pair compositions and various types of bearingless motors.

Analysis on Producing Stable Radial Suspension Force for the BPMSM
Based on the magnetic field equivalent current analysis method, the necessary condition to produce a stable radial suspension force in a certain direction can be derived.In this section, the counter-clockwise rotation direction is defined as the rotation positive direction.It is supposed that torque winding air gap magnetic field rotates in the positive direction with mechanical angular velocity ωM.Then, to obtain a stable radial suspension force in a certain direction, the absolute angular velocity of the radial suspension force should be set as 0 rad/s, and its direction should be fixed.Treating the torque winding air gap magnetic field as a reference, the relative angular velocity ωF of radial suspension force is −ωM.The relative angular velocity ω of suspension force winding air gap magnetic field can be obtained as [17]: The absolute angular velocity ωB of suspension force winding air gap magnetic field can be written as: Likewise, when the angular velocity of the torque winding air gap magnetic field is −ωM, (that is, rotating in the negative direction), the absolute angular velocity of suspension force winding air gap magnetic field can be expressed as −PMωM/PB.
Therefore, the necessary condition to generate a stable radial suspension force is that the angular velocity ωB of suspension force winding air gap magnetic field should be PM/PB times that of the torque winding air gap magnetic field, and their directions should remain the same.In other words, the suspension force winding air gap magnetic field and the torque air gap winding magnetic field have the same rotation electric angular velocity and rotation direction: The verification of the necessary condition by means of the magnetic field equivalent current analysis method is as follows: the relative motion analysis situations are shown in Figures 5 and 6.
Among them, large dimension and small dimension of windings represent equivalent suspension force windings and equivalent torque windings, respectively.
If PB = PM + 1 (that is PB = 2, PM = 1), when the equivalent torque windings rotate π/2 rad mechanical angle in positive direction, according to the necessary condition, the equivalent suspension force windings must rotate θB = PMθM/PB = π/4 rad mechanical angle in the same direction, as can be seen from Figure 5b.There is an intensifying factor at α = 0 rad and a softening factor at α = π rad.Thereby, radial suspension force is produced towards the direction of 0 rad.In this situation, the rotation electrical angle of the equivalent suspension force windings PB·θB = 2·π/4 = π/2 rad is the same as that of the equivalent torque windings PM·θM = 1·π/2 = π/2 rad.When the equivalent torque windings rotate π rad mechanical angle in positive direction, the equivalent suspension force windings must rotate θB = PMθM /PB = π/2 rad mechanical angle in positive direction, as shown in Figure 5c.There also is an intensifying factor at α = 0 rad and a softening factor at α = π rad, so the direction of the radial suspension force remains 0 rad.The rotation electrical angle of the equivalent suspension force windings PB·θB = 2·π/2 = π rad is equal to that of the equivalent torque windings PM·θM = 1·π = π rad.In Figure 5d, when the equivalent torque windings rotate 3π/2 rad mechanical angle in the positive direction, the same conclusion could be obtained.In consequence, when PB = PM + 1, to generate a stable and reliable radial suspension force, the rotation electrical angular of the equivalent suspension force windings PB·θB is equal to that of equivalent torque windings PM·θM.Besides, the rotation direction of equivalent suspension force windings is the same as that of equivalent torque windings, and the converse is also true.If PB = PM − 1 (that is PB = 1, PM = 2), when the equivalent torque windings rotate π/4 rad mechanical angle in the positive direction, the equivalent suspension force windings must rotate θB = PMθM /PB = π/2 rad mechanical angle in the same direction to hold the radial suspension force in a stationary direction, as plotted in Figure 6b.The intensifying factor is at α = 0 rad and a softening factor is at α = π rad.Thereby, the radial suspension force points to the direction of 0 rad.Then, the rotation electrical angle of the equivalent suspension force windings PB·θB = 1·π/2 = π/2 rad is equal to that of the equivalent torque windings PM·θM = 2·π/2 = π rad.When the equivalent torque windings rotate π/2 rad mechanical angle in positive direction, the equivalent suspension force windings must rotate π rad mechanical angle in positive direction, as shown in Figure 6c.The rotation electrical angle of the equivalent suspension force windings PB·θB = 1·π = π rad is also equal to that of equivalent torque windings PM·θM = 2·π/2 = π rad.In Figure 6(d), when the equivalent torque windings rotate 3π/4 rad mechanical angle in positive direction, the same conclusion could be obtained.Thus, when PB = PM−1, the conclusion is consistent with that of the condition of PB = PM + 1.In brief, the electrical angular velocity of torque winding air gap magnetic field PM·ωM is in complete agreement with that of suspension force winding air gap magnetic field PB·ωB.The current frequency of suspension force windings and torque windings can be expressed as: Therefore, the necessary condition to generate a stable and reliable radial suspension force is that the suspension force winding air gap magnetic field rotates in the same direction as the torque winding air gap magnetic field with PM/PB times to the angular velocity ωM of the torque winding air gap magnetic field.

Analytical Model of Radial Suspension Force for the BPMSM
According to Maxwell's stress tensor method, the radial suspension force vector F as shown in Figure 5b can be expressed as [3,18]: with μ0 as the air permeability, Bn and Bt being the radial and tangential components of air gap flux density for the BPMSM, respectively, A being the stator inner surface area, r as the stator inner diameter, l representing the motor length and θ being the space angle.
Taking only the fundamental components of current into account, the air gap flux density B1 caused by a permanent magnet and torque winding current loading A1, and the air gap flux density B2 excited by a suspension force winding current loading A2 can be described as: while the current loadings of torque windings and suspension force windings A1, A2 can be expressed as follows: where: . δ0= hm + g is the equivalent air gap length without rotor eccentricity, hm is the permanent magnet thickness, g is the mechanical air gap length, gM and gB correspond to the fundamental winding factors of torque windings and suspension force windings, respectively, PM and PB are their pair poles.I1m = IM + If is the torque winding resultant current magnitude, IM and If represent as the magnitude of the torque winding current and the equivalent current of the permanent magnets, respectively.I2m is the suspension force winding current value.N1, N2 are the turn numbers of torque windings and suspension force windings, respectively, ω1 and ω2 correspond to their electrical angular frequencies, λ1 and λ2 are their initial phase angle.
The radial flux density component Bn and tangential flux density component Bt in the air gap are given by: Supposing PM = PB ± 1 and ω1 = ω2, the radial suspension force components in the x-and y-direction are obtained as: where and m is the phase number for the BPMSM.
The first part is the Maxwell controllable suspension forces influenced by the air gap flux density B1 and B2, whereas the second part is the radial component of the Lorentz forces generated by the interaction of the air gap flux density B1 with the current loading A2 and the interaction of the air gap flux density B2 with the current loading A1.It can be seen from ( 9) that the amplitude of the controllable radial suspension forces is linear with the suspension force winding current value I2m.And the direction depends only on phase difference between the torque winding air gap flux density B1 and suspension force winding air gap flux density B2, given by λ1-λ2.Due to r >> δ0 for the BPMSM, kl is much less than km, thus the radial component of Lorentz forces can be ignored.
Therefore, under dq synchronous rotating reference frame, the radial suspension force Fd and Fq can be expressed as: with the IMd and IBd as the d-axis components of torque and suspension force winding currents, the IMq and IBq corresponding to their q-axis components, respectively.Except for currents in the suspension force windings, the rotor eccentricity e will also cause a radial suspension force, namely, unilateral magnetic force.When rotor eccentric displacement occurs, the motor air gap length δ will change which can be approximately expressed as: with x = e cos φ, y = e sin φ, and φ being the deviation angle between the eccentricity orientation and x-axis.Thus, the radial suspension force can be expressed by [19,20]: Here, In order to support the stable suspension, it is necessary to compensate the unilateral magnetic forces by a reasonable adjustment of the suspension force winding currents.

Analytical Model of Electromagnetic Torque for the BPMSM
Because the d-and q-axis components of the torque winding inductances are approximately equal (Ld = Lq) for the surface-mount BPMSM, the mathematical formula of electromagnetic torque has the same form as the conventional PMSM in a dq synchronous rotating reference frame which can be given by [21]: where, ψ1d = ψMd + ψfd, ψ1q = ψMq + ψfq, ψ1d and ψ1q are the d-and q-axis components of the equivalent torque winding air gap flux linkage, respectively.ψMd and ψMq are d-and q-axis components of the torque winding air gap flux linkage, respectively.ψfd and ψfq are d-and q-axis components of the air gap flux linkage excited by the permanent magnet.

FEA Validation of the Model
To verify radial suspension force generation, the 2D FEA simulation model is developed by means of the software-ANSYS (ANSYS Inc., Cecil Township, PA, USA) for the BPMSM, as shown in Figure 3b.The main specifications of the BPMSM simulation model are listed in Table 1.In Figures 7-9, the FEA results of the radial suspension force are compared with the mathematical model calculation values.This leads to the following conclusions: the amplitudes of the radial suspension force increase with the suspension force winding current I2m, as illustrated in Figure 7a.When the suspension force winding current I2 is small (less than 3 A), the radial suspension forces show a linearly increasing trend.The FEA results are in good agreement with the mathematical model calculations and the error between them is less than 3 percent, as depicted in Table 2.When the current has reached 3 A, the motor core enters a saturated state.Thus the FEA results are significantly less than the values of the mathematical model calculation, and the error between them is more than 10 percent, so the rated suspension force current is set at 2 A in this paper, and the maximum radial suspension force available in this case is 244.65 N. Besides, from Figure 7b, it can be seen that the direction of the radial suspension force is proportional to the initial phase angle λ2 of the suspension force winding current.It is almost unaffected by saturation of the magnetic circuit which is completely consistent with the mathematical model calculation values.In conclusion, when the motor operates in a linear state, the amplitude and direction of the radial suspension force change with the magnitude and phase angle of the suspension force winding current, respectively.On the basis of this rule, the orientation control of radial suspension force can be realized, which can build a foundation for the BPMSM.In Figure 8, the radial suspension force component in the x-direction changes in cosines with initial phase angle λ2 of suspension force winding current, and the y-direction component is dominated by the sines rule with initial phase angle λ2.The variation tendencies of the FEA results and mathematical model calculations are essentially identical.When the suspension force winding current is zero, the related FEA for a unilateral magnetic force is carried out, as shown in Figure 9.   It's concluded that the FEA results basically agree well with the mathematical model calculation values, and the error between them is less than 2.7 percent (Table 3).Due to the assembly auxiliary bearing, the maximum rotor radial deviation ranges from −0.3 mm to 0.3 mm, and the amplitude of maximum unilateral magnetic force is 175 N.

Prototype Machine Structure
The 5-DOF suspension control for the BPMSM is an essential prerequisite to realize its high-speed and stable operation.The prototype BPMSM, consisting of a 3-DOF axial-radial magnetic bearing and a 2-DOF BPMSM, has been built, as shown in Figure 10.The 2-DOF on the right end of the high-speed machine and the rotor rotation are controlled by the BPMSM unit, while the other 2-DOF on the left end and the axial direction are controlled by the 3-DOF axial-radial magnetic bearing unit.Among them, the BPMSM unit is the key technology to realize the stable operation of the high-speed machine.Thus, this paper mainly focuses on the 2-DOF BPMSM unit.
The stator and rotor iron core are wire cut from DW465_50 silicon steel sheet and neodymium-iron-boron (NdFeB) is selected as the permanent magnet material.The rotor radial positions are measured by eddy-current displacement sensors and the rotation speed is measured by a photoelectric encoder.If there is no external disturbance or load imposed on the rotor, the radial suspension force just supports the rotor in the balance position.

Control of Torque and Radial Suspension Force for the BPMSM
The air gap magnetic field of the BPMSM is composed of the suspension force winding air gap magnetic field, torque winding air gap magnetic field and permanent magnet air gap magnetic field.The torque production mechanism is the same as that of PMSM, namely, the product of the interactions of the permanent magnet and torque winding air gap magnetic field.In addition, because of its long effective air gap length for the surface-mounted BPMSM, the torque winding air gap magnetic field can be ignored compared with the permanent magnet air gap magnetic field.Then, the radial suspension force could be considered as the interactions between the suspension force winding air gap magnetic field and permanent magnet air gap magnetic field.Thus, the decoupling control between torque and radial suspension force can be realized by adoption of the traditional FOC [22].
Figure 11 shows the relative position between torque windings and suspension force windings.In order to facilitate the control of the radial suspension force, the xoy coordinate is set up along the displacement sensors.It is supposed that the angle between the x-axis and A phase axes of the torque windings (PM = 1) or the nearest A phase axes relative to the positive x-axis (PM > 1) is θM, and the angle between the x-axis and A phase axes of suspension force windings (PB = 1) or the nearest A phase axes relative to the positive x-axis (PB > 1) is θB.If θM ≠ θB, defining Δθ = θB − θM, (see Figure 11a).An advanced mechanical angle Δθ must be compensated by the suspension force winding currents.Based on the relative position relations between the suspension force winding air gap magnetic field and the torque winding air gap magnetic field, the result is equivalent to Δθ = 0°, (see Figure 11b).If PB = PM + 1, the advanced electrical angle θ1 must be compensated by the suspension force winding currents.If PB = PM − 1, the lag electrical angle θ1 must be compensated by the suspension force winding currents.Thus, the suspension force winding air gap magnetic field and torque winding air gap magnetic field have the same direction of rotation and electric angular velocity.In the control system, the suspension force winding current command iBd will directly control the component of the radial suspension force in the x-axis.Then, due to the current component iBq advancing iBd with mechanical angle 90°/PB, if PB = PM -1, the current component iBq will generate a radial suspension force lagging 90° with respect to the x-axis.If PB = PM + 1, the current component iBq will produce a radial suspension force advancing 90° with respect to the x-axis.Consequently, the 2-DOF radial suspension force control for the BPMSM can be realized.
When the BPMSM runs at load states, the torque winding current component iMq will produce an air gap magnetic field in the q-axis.Then, an advancing electrical angle of θ1 (< 90°) is produced by the torque winding air gap magnetic field.In other words, the suspension force winding air gap magnetic field lags a mechanical angle θ1/PM with respect to the torque winding air gap magnetic field.If PB = PM + 1, the angle of suspension force is θ1/PM + (-PBθ1/PM) = −θ1.To obtain a stable suspension force, an advancing mechanical angle θ1/PB must be compensated by the suspension force winding currents.If PB = PM − 1, the angle of radial suspension force is θ1/PM + PBθ1/PM = θ1.Similarly, an advancing mechanical angle θ1/PB must be compensated by the suspension force winding currents.The advanced angle θ1 of the torque winding current can be obtained by the proportional relation the air gap magnetic field of torque winding current component iq and the original torque winding air gap magnetic field.Based on the rotor flux-orientation control, it is concluded that: Thus, Equations ( 12) and ( 13) can be expressed respectively as: Hence, the radial suspension forces and torque are controlled by changing the suspension force winding current (IBd, IBq) and torque winding current (IMq), respectively.Moreover, compared with CRPWM, the space vector pulse width modulation (SVPWM) method can reduce the inverter switching frequency and decrease the harmonic content of the inverter output current.Hence, a control strategy based on the SVPWM method is proposed to realize the FOC in this section.Figure 12 shows the vector control block diagram, which consists of two basic schemes, namely, torque control and radial suspension force control.
In torque control, the rotational speed signal n is obtained by an optical-electricity encoder in the torque control system.The error between the speed command n* and measurement n is amplified in the proportional-integral (PI) controllers, and the torque winding current command iMq* can be generated by the torque control algorithm.Due to the adoption of field orientation control, the torque current commands iMd* is defined as zero.Then, the reference voltage vectors VMα* and VMβ* can be generated through the current PI controllers and PARK inverse transformation.Finally, through the SVPWM modulation module, six PWM signals are generated to drive the voltage source inverter.
In radial suspension force control, the radial displacement closed-loop control is adopted as well as the current closed-loop control.The radial positions x and y of the rotor are measured by eddy current sensors.The error between the displacement commands and measurements is amplified in the proportional-integral-differential (PID) controllers.Thus, radial suspension force commands  On the basis of the above analyses, the double closed control circuits are used both in the torque control and radial suspension force control systems.Thereinto, PID controller tuning is one of the key and more difficult points for stable suspension operation.In this paper, the engineering tuning method is adopted, due to its advantages of simplicity, valid and easy operation.Below, the specific setting steps are given.Firstly, set the integral gain ki and differential gain kd to zero, increase the proportional gain kp until the output of the loop starts to oscillate.Then the kp should be set as 0.6-0.7 times the current value.Increase the integral gain ki until the loop oscillates as well, and the ki should be set as 1.5-1.8times the current value.Then, keep ki unchanged and change the value of kp for tracking control.If the condition is improved, continue to adjust the kp until a satisfactory result is achieved.Otherwise increase the original kp a bit, adjust the ki to improve the control effect.Repetition goes on until the best coefficients of the integral gain ki and the proportional gain kp are achieved.Follow the same steps above, the appropriate differential gain kd can also be obtained.

Analysis on Experimental
To verify the validity of the proposed analysis method, relevant experiments are designed and conducted for the surface-mounted BPMSM.The basic parameters of the experimental prototype are as follows: rated power PN is 500 W, rotor outer diameter dro is 73 mm, permanent magnet thickness hm is 2 mm, air gap length g is 1 mm, torque winding resistance is 2.07 Ω, the d-and q-axis inductance of the torque windings are 8 mH, suspension force winding resistance is 1.13 Ω, the d-and q-axis inductance of the suspension force windings are 3.6 mH, the mutual inductance between torque and suspension force windings is 0.16 mH, air gap length g1 with the auxiliary bearing is 0.3 mm.
Figure 13 shows an experiment system which takes a TMS320F2812 digital signal processor as the BPMSM drive controller, and an intelligent power module (IPM) as its power driver [23,24].Besides, the range of analog input voltage is 0-3V in the digital converter of DSP board, but the output voltage of the displacement sensor QH8500 ranges from −2 V to −18 V and the output range of the CNSE151-100 Hall sensor is −24 mA-24 mA.Thus, the corresponding signals detected by the sensors must be transferred to the DSP board through the corresponding interface circuit.After that, when the rotor is located in the equilibrium position, the output voltage is set as 1.5 V.The output voltage should be 3 V when the rotor is located at the maximum displacement with the positive x-axis and y-axis, while the output voltage should be 0 V when the rotor is located at the minimum displacement of the negative x-axis and y-axis.The voltage signal is linear with the rotor displacement in the other position.Based on this method, the radial position of rotor can be detected accurately.Besides, the current interface circuit must be employed to transform the sensor output signal to the range of 0-3 V. Finally, the PID controller parameters can be achieved by the method above: in the torque control system, the current closed loop kp is 0.6, and ki is 0.04, the speed closed loop kp is 0.6, and ki is 0.00005; in the suspension force control system, the current closed loop kp is 0.90, and ki is 0.005, the displacement closed loop kp is 1.8, ki is 0.006 2, and kd is 0.85.Then, the suspension operation tests are carried out to verify the effectiveness of the proposed equivalent winding current analysis method.Figure 14 shows the radial displacement waveforms in the x-and y-direction with the start of suspension.When the radial suspension force control system is not active, the rotor stops on a random position of the auxiliary bearing due to the influence of gravity.From Figure 14, it can be found that the original position of the rotor is (−0.05 mm, −0.3 mm).When the radial suspension force control system is active, the rotor comes rapidly back to the equilibrium position (0 mm, 0 mm).At this time, the rise time in the y-direction is 0.3 s, the settling time is 1 s, and the maximum deviation is 67 μm, thus the maximum overshoot is less than 22.67%, which is much smaller than the air gap at the equilibrium point (g0 = 0.3 mm).In the x-direction, the vibration amplitude is about 20 μm, and the vibration amplitude is almost 32 μm in the y-direction.The deviations of radial displacements, which are mainly caused by mass unbalance of the rotor, are acceptable.The deviations can be reduced by more precise control schemes with an unbalance compensation circuit.The frequency of the radial displacement vibration is dependent on the rotating speed of the rotor.Figures 15-17 show the rotor position angle and rotation speed waveforms, the radial displacement waveforms in the x-and y-direction, and the waveform of relationship between them, respectively, when the rated currents are applied.In Figure 15, the rotor position angle changes periodically with time in the range of 0°-360°.The corresponding frequency is 50 Hz, which is in agreement with the rotational speed calculation formula, and the oscillation of the rotational speed is slight.Thereby, the reasonable operating characteristic of speed closed loop is verified at the speed of 3000 r/min.
From Figure 16 and Figure 17 the maximal displacement in the positive x-axis is about 16 µm and 12 µm in the negative x-axis.The maximal displacement in the positive y-axis is about 18 µm and 21 µm in the negative y-axis, when the rotor runs steadily at 3000 r/min.They are significantly less than the length of air gap at the equilibrium point.It is found that the steady suspension characteristics of proposed 2-4 BPMSM can be obtained at 3000 r/min.Due to the effects of gravity, the displacements in the y-direction are larger than that in the x-direction.Thus, the eccentric displacement track diagrams are nearly-circular or elliptical, as shown in Figure 17.   Figure 18 describes the responses of radical displacement in the x-and y-direction when the rotation speed is changed from 1200 r/min to 3000 r/min.It can be found that the rotational speed command n* is accelerated suddenly from 1200 r/min to 3000 r/min at 0.4 s.After a 0.2 s settling time, the stable suspension operation is realized and the overshoot is 0. The short working time of the proposed experimental prototype shows that the dynamic response of the BPMSM is good.At this point, there is a little vibration in the x-and y-direction when the speed has been changed.But the changes of vibration amplitudes for the radical displacements which may originate from the motor's unbalance mass are slight.The vibration peak-to-peak value is about 28 μm in the x-direction, and the peak-to-peak value is almost 39 μm in the y-direction.The radial displacement deviations are within acceptable ranges, which can be reduced further by more precise control schemes with an unbalance compensation circuit, so the independent control between radial displacement and rotating speed can be realized.The validity and feasibility of the control strategy is confirmed, which establishes a solid foundation for further deliberation and research, and has some theoretical and practical reference value.

Conclusions
This paper proposes an analysis method based on equivalent winding currents.On that basis, the validity of the necessary conditions to produce a stable radial suspension force for bearingless motors is proved.The simulation model is established by the ANSYS simulation software and the simulation results verify the theoretical analysis.In addition, good rotation and suspension performance are obviously shown by the simulation and experimental results, as well as the accuracy of the proposed analysis method and the feasibility of the control strategy.Besides, the decoupling control between electromagnetic torque and radial suspension force is also successfully achieved.Thus, the new analysis method of the BPMSM has some theoretical significance and practical reference value for further research compared with other methods.

Figure 1 .
Figure 1.Schematic diagrams of magnetic field equivalent current analysis method.(a) 2-pole air gap magnetic field; (b) Equivalent winding currents of 2-pole air gap magnetic field; (c) 4-pole air gap magnetic field; (d) Equivalent winding currents of 4-pole air gap magnetic field.

Figure 4 .
Figure 4. Structure and operation principle diagram of the 2-4 poles BPMSM.(a) Based on conventional analysis method.(b) Based on the magnetic field equivalent current analysis method.

Figure 5 .
Figure 5. Schematic diagram of stable radial suspension force in a single direction (PB = PM + 1).(a) Initial state.(b) Rotation of equivalent torque windings with the mechanical angular 90°.(c) Rotation in the positive direction with the mechanical angular 180°.(d) Rotation in the positive direction at the mechanical angular 270°.

Figure 6 .
Figure 6.Schematic diagram of a stable radial suspension force in a single direction (PB = PM − 1).(a) Initial state.(b) Rotation of equivalent torque windings with the mechanical angular rotation of 45°.(c) Rotation in the positive direction with the mechanical angular rotation of 90°.(d) Rotation in the positive direction at the mechanical angular rotation of 135°.

Figure 11 .
Figure 11.Relative positions of torque windings and suspension force windings.
Fx* and Fy* are generated.Through the radial suspension force control algorithm, the Fx* and Fy* are transformed into the current commands iBd* and iBq* under the d-q rotating coordinate system.Then, the reference voltage vectors VBα* and VBβ* can be obtained through the current PI controllers and PARK inverse transformation.Finally, through the SVPWM modulation module, 6 PWM signals are produced to drive the voltage source inverter.

Figure 12 .
Figure 12.A vector control block diagram of BPMSM.

Figure 13 .
Figure 13.Experimental system of the prototype motor.DCdirect current; DSP-Digital Signal Processor.

Figure 14 .
Figure 14.Radial displacement waveforms in the x-and y-direction when the start of levitation.

Figure 15 .
Figure 15.Waveform of the rotor position angle and rotation speed when the stable suspension operation is realized.

Figure 16 .
Figure 16.Radial displacement waveforms in the x-and y-direction when the start of levitation.

Figure 17 .
Figure 17.The relationships between radical displacements in the x-and y-direction for BPMSM.

Figure 18 .
Figure 18.The responses of speed and radical displacement in the x-and y-direction when the rotation speed is changed.

Table 1 .
Specification of the prototype machine.

Table 2 .
Comparison of FEA and mathematical models in controllable radial suspension forces.

Table 3 .
Comparison of FEA and mathematical model in unilateral magnetic force.