Experimental Verification of Passive Axial Electrodynamic Suspension in a Bearingless Motor

This article investigates a bearingless motor with passive electrodynamic axial suspension. The axial suspension force is generated by a specific coil configuration called a figure-eight coil. Radial directions and tilting angles are stabilized by passive permanent magnet bearings. Since axial electrodynamic force increases with rotational speed, it must overcome a certain minimum threshold speed to compensate for the rotor weight and the unstable axial force caused by the permanent magnet bearing. Theoretical equations are derived for the braking torque caused by the suspension current and for the steady-state axial equilibrium position at constant rotational speed. A method based on the braking torque equation is proposed for correcting the mismatch between the magnetic center of the bearingless motor and the middle point of the axial clearance. This method sets the middle point between upper and lower touchdown positions in the same place where the motor current is minimum during passive axial suspension. Axial suspension is confirmed in the experiment with a noncontact laser sensor.


I. INTRODUCTION
Some applications of electric machines, such as blood pumps and high-speed drives, require durable and reliable devices. Usually, rotating electric machines are supported by mechanical ball bearings. Those bearings wear due to friction and demand regular preventive maintenance. It is possible to reduce friction by substituting the mechanical ball bearings by magnetic forces.
Either active control or passive stabilization can be used to achieve noncontact magnetic suspension. If the shaft position is completely regulated by active magnetic forces, it is said that 5 degrees of freedom (5DOF) are actively controlled [1]. The radial direction, tilting angle, and axial position are regulated by a feedback control system. The 5DOF suspension strategy requires several position sensors and inverters, adding significant costs and complexity due to suspension control. By combining passive techniques with active position control, it is possible to achieve full noncontact suspension with fewer actively regulated DOF. There are several examples of 2DOF [2], [3], [4], [5] and 1DOF [6], [7], [8], [9] actively positioned bearingless motors in the literature. According to the Earnshaw theorem, it is not possible to achieve full passive suspension only through permanent magnet bearings in static state [10]. However, it is possible to combine permanent magnet bearings with other passive suspension techniques, such as hydrodynamic bearings [10], superconductor levitation [11], gyroscopic effects [12], diamagnetic materials at room temperature [13], and electrodynamic forces generated by eddy currents. Among these passive suspension techniques, electrodynamic suspension was chosen for this project, because the motor coils that generate torque are also used for passive stabilization, they provide stronger stiffness than that of room temperature diamagnetic levitation, and they do not require superconductors with a dedicated cooling system. The electrodynamic suspension principle was already employed in several electrodynamic bearings [14], [15], [16], [17], [18], [19], [20], [21], [22] and bearingless motors [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. Full passive suspension was confirmed in a bearingless motor that combines axial electrodynamic suspension and radial permanent magnet bearings [29], [30], [31]. However, the rotor remains in axial touchdown position until a certain minimum rotational speed is surpassed. In case full suspension is required for all rotational speeds, it is possible to operate these bearingless motors as a single-drive 1DOF machine up to the threshold speed of axial suspension [30], [35].
This article presents a surface-mounted permanent magnet bearingless motor with axial electrodynamic suspension [32], [33], [34], [35]. The radial directions and tilting angles are stabilized by radial permanent magnet bearings. The axial electrodynamic force is proportional to the axial displacement and increases with rotational speed. This principle is based on the lift and guidance coils applied in the Japan railway high-speed maglev train [36], [37], [38], [39]. By arranging the stator coils in a pattern that resembles the number eight (8), a stabilizing suspension current is induced when the rotor is axially displaced. This principle was previously adapted for bearingless motors [23], [28], but, owing to the lack of superconducting electromagnets, flux density and electrodynamic forces are considerably lower than that of the maglev train for high-speed transportation. The machine covered in this article has a double rotor structure with three layers of permanent magnets [33], [34], [35] for improved radial flux density and passive axial electrodynamic force. This radial air-gap rotor shape offers the possibility of axial suspension with 1DOF control at low rotational speed. The rotor is made axially asymmetric by removing one of the outer layers of permanent magnets [35]. This asymmetric topology allows simple commutation between the active and passive suspension modes by turning OFFthe suspension control loop without any physical switching.
It includes the improvements of a bearingless motor with axial electrodynamic suspension. Experimental results up to 8065 r/min are shown, instead of the 5000 r/min limit considered in [34], making it possible to confirm axial suspension without additional support structures for weight relief. A method based on the relation between the braking torque and the axial position is proposed to minimize mismatches between the magnetic center of the permanent magnets and the center of axial mechanical clearance. This method finds the origin of the axial displacement z by second-degree fitting of rms motor currents measured with an oscilloscope. Axial suspension was confirmed in the experiment by a noncontact laser sensor.
Throughout this article, finite element analysis (FEA) and theoretical equations are shown. They are used to estimate the braking torque due to suspension current and the minimum rotational speed that is required to achieve axial electrodynamic suspension. For stable axial suspension, the axial electrodynamic stiffness must be higher than the unstable axial stiffness generated by the radial permanent magnet bearings. Moreover, to lift the rotor from touchdown condition, the electrodynamic force must be higher than the sum of the rotor weight and the repulsive axial force by the permanent magnet bearings. This repulsive force is proportional to axial displacement. Thus, the minimum force required for suspension also depends on the axial touchdown position. Fig. 1 shows a simplified 3-D model of the bearingless motor and the radial permanent magnet bearings. A front view and a section cut of the machine are shown side by side. This machine is a permanent magnet synchronous motor with concentrated rectangular coreless coils. The stator coils are divided into two layers. Coils in the upper layer are connected in parallel association with the coil directly below. Each of these pairs is called a figure-eight coil set, owing to their shape that resembles the number eight. The rotor has three layers of four-pole permanent magnets. The permanent magnets in the upper and lower layers are polarized in an opposite direction to that of the central permanent magnet layer. The rotor has an outer and inner parts connected by an aluminum frame; however, for better visualization, it is omitted in Fig. 1. Permanent magnets in the outer part have their magnetic poles aligned with those in the inner part [33].
There are two pairs of permanent magnet rings positioned at both ends of the rotor shaft. In each pair, one permanent magnet is attached to the rotor shaft and the other one is mounted on the stator side. Owing to the magnetic pole configuration of these pairs, there are repulsive forces between the rotor and stator. These repulsive forces centralize the shaft in the radial direction and prevent tilting. On the other hand, they also repel the rotor shaft along the axial direction, also known as thrust direction, causing axial instability. To achieve stable axial suspension, the unstable axial stiffness by the radial permanent magnet bearings must be compensated for by the axial electrodynamic stiffness.  Fig. 2(a) shows a planar representation of the figure-eight coil A 1 connected to the phase A terminal. It consists of the upper and lower coils A u1 and A l1 , respectively, connected in parallel. It is assumed that the magnetic flux density distribution is sinusoidal along the angular direction. The magnetic field generated by the rotor permanent magnets is represented by the red vectors, coming out of the sheet plane, and the blue vectors, entering the sheet plane. When the rotor is displaced from the central axial position z = 0, an imbalance between back-electromotive forces (back-EMF) in the upper and lower coils generates a circulating current i za1 . When i za1 flows in the counter-clockwise direction in the upper coil, it flows in the clockwise direction in the lower coil, and vice versa. Interaction between i za1 and the magnetic field crossing the horizontal coil segments results in Lorentz forces along the axial direction. Those forces push the rotor and stator back to the central axial position. Fig. 2(b) shows the path of an external motor current i a supplied by an inverter. A Lorentz force is generated along the vertical segments of the figure-eight coil set. The motor torque is, thus, actively regulated by the q-axis component of i a . The figure-eight coil set A 1 is connected in series with the coil set A 2 in the opposite side of the stator, as shown in Fig. 3. Assuming that the upper and lower coils have the same impedance value, the current i a divides itself equally between the upper and lower paths. It should be noted that the external motor current i a /2 flows in the same direction in the upper and lower coils. It is the opposite of what happens to the suspension current i za1 . Since forces generated along the horizontal coil segments cancel each other, it is not possible to regulate the axial suspension force through the direct axis component of i a .

B. AXIAL ELECTRODYNAMIC STIFFNESS
The equation for the axial electrodynamic stiffness was derived in the authors' previous article [33]. This section highlights the equations that are most relevant for this article.
The upper and lower coils, A u1 and A l1 , respectively, are assumed to have identical shape, resistance R, and selfinductance L. Mutual inductance between coils is neglected. This is assumed because the stator is coreless and the winding is concentrated. An FEA confirmed that the highest mutual inductance between neighbor coils corresponds to about 4% of their self-inductances. The rotor rotates at a constant mechanical rotational angular speed ω and is radially centralized. Magnetic flux linkages in the upper and lower coils, λ au1 and λ al1 , respectively, are derived as where K λ is the ratio of flux linkage variation in terms of axial position z. It is a constant that depends on the peak magnetic flux density, number of turns per coil, and coil shape [33]. The constant λ 0 is the flux amplitude when z = 0, p is the number of pole pairs, and t is the time variable.
The back-EMF in the upper and lower coils, ε au1 and ε al1 , respectively, are derived from (1) as Assuming that the upper and lower coils are connected in parallel association, the steady-state suspension current i za1 is derived in terms of rotational speed ω and axial position z as where θ c is the impedance angle of the coils, that is, the arcsin of the ratio from the inductive reactance to coil impedance. The angle θ c , in radians, is derived as If the rotor is axially centered, that is, z = 0, thus, i za1 is zero; otherwise, z = 0 results in a circulating suspension current i za1 .
Interaction between the suspension current i za1 and the magnetic field by the rotor permanent magnets results in Lorentz forces on the stator coils. Defining F zeT as the total force on the six figure-eight coil sets, the axial electrodynamic stiffness k ze is derived from the projection of F zeT on the rotor side as where c is the number of figure-eight coil sets. Table 1 gives the parameters of (5). The resistance R was directly measured on the coils. The average resistance of individual coils is 2.6 . The self-inductance L was calculated using a single-step 3-D FEA as 1.03 mH. It is numerically equal to the magnetic flux linked by the coil   equivalent to λ 0 + K λ z, was obtained through fast Fourier transform. The third harmonic component generates a small contribution to the axial stiffness [27]. The figure-eight coil connection is modeled by connecting the upper and lower coils in parallel association in the equivalent circuit of the FEA study. Rotational speed and axial position are constant for each simulation case. No linear approximations are assumed, including the back-iron yoke. The constant K λ was calculated through first degree fitting as 5.5 and 0.5 Wb/m for the fundamental and third harmonic components, respectively. The third harmonic contributes with 1.7% of the total theoretical axial stiffness at 10 000 r/min. Fig. 4 shows a comparison between the electrodynamic axial stiffness k ze calculated using a 3-D FEA along one electrical cycle and the results yielded by (5). Rotational speed and axial position are constant for each simulation case. The coil resistances in the FEA were set to 2.6 . Discrepancy of 3.1% is observed between the FEA results and the equation at 10 000 r/min. This small difference is probably caused by the mutual inductances that were neglected in the derivation process [33]. More accurate theoretical modeling might be possible considering the mutual inductances [26].

C. BRAKING TORQUE DUE TO SUSPENSION CURRENT
Assuming that the machine rotates at constant speed in a noload state, it is possible to estimate the braking torque due to the suspension currents by dividing its respective power losses by the mechanical rotational angular speed ω in rad/s [21].
As derived in (3), the amplitude of the circulating suspension current in the figure-eight coil is proportional to the axial displacement z and increases with rotational speeds when ω R/pL. The suspension current generates power losses in the figure-eight coil resistance. Defining the suspension rms current I za1 , neglecting the harmonic components, the total power loss P z in a stator with c figure-eight coil sets is derived from (3) as The braking torque is derived from (6) as It is concluded from (6) and (7) that both the power loss and the braking torque due to the axial suspension current are proportional to the square of axial displacement z. Fig. 5 shows a comparison between results yielded by (7) and FEA calculations of Lorentz torque in the coils. The markers represent the FEA results, whereas full lines represent the equation results. There is a good correspondence between both results.

III. RADIAL PERMANENT MAGNET BEARINGS AND REQUIREMENTS FOR AXIAL SUSPENSION A. STABILIZING PASSIVE RADIAL STIFFNESS
The electrodynamic suspension principle employed in this machine stabilizes the rotor shaft in the axial direction. Passive radial permanent magnet bearings are used to stabilize the radial direction and tilting angle of the shaft. There are two of those, one mounted on the upper end and another mounted on the lower end of the rotor shaft. Fig. 6 shows a radial permanent magnet bearing. The inner permanent magnet ring is mounted on the rotor shaft. The outer ring is mounted on the structure frame. The magnets are assembled in a repulsive configuration.
The repulsive force between the magnet rings stabilizes the radial direction, but it also destabilizes the axial direction. In this configuration, the unstable axial stiffness k zp is twice as high as the stable radial stiffness k r p [10], that is, A positive stiffness indicates a stabilizing force, whereas a negative stiffness indicates a destabilizing force. This relation is valid for static permanent magnets in the absence of soft magnetic materials and with small displacements. For the axial shaft position to become stable, the electrodynamic axial stiffness k ze by the figure-eight coils must compensate for the destabilizing axial stiffness k zp by the two permanent magnet bearings, satisfying the following condition: Equations (8) and (9) indicate that the minimum electrodynamic stiffness increases with the radial stiffness of the permanent magnet bearings. Therefore, the stabilizing radial stiffness k r p should be as low as possible. As for a permanent magnet machine with an iron core stator, there are magnetic pull forces between the rotor and stator that cause radial instability. In that case, the minimum k r p can be specified to compensate for that unstable radial stiffness. On the other hand, the machine presented in this article has coreless stator; therefore, there are no static magnetic pull forces in the radial direction. For this reason, the radial stiffness k r p did not have a minimum specification and was arbitrarily chosen.

B. CONDITIONS FOR AXIAL SUSPENSION
The minimum rotational speed to achieve axial suspension with a certain radial permanent magnet bearing can be estimated with the support of FEA. Fig. 6 shows an axial offset d o between the permanent magnets mounted on the rotor and stator. This is the rated displacement when the motor permanent magnets are aligned with the center of the figure-eight coils, that is, z = 0. Considering the offset d o , the axial force F zp , resultant from both radial permanent magnets, is derived as where F o = −2d o k zp . The force F o is adjusted to partially compensate for the rotor weight. By relieving part of the axial load on the shaft, the minimum electrodynamic stiffness requirement for achieving axial suspension is reduced as well.
Assuming a constant rotational speed, defining the rotor mass m and the acceleration of gravity g, at steady-state equilibrium condition, resultant axial force on the suspended rotor is zero, where k ze + 2k zp > 0 indicates the condition for a stable axial stiffness. The minimum electrodynamic stiffness for achieving axial suspension is derived by solving (11) at the lower axial touchdown position z = z t where k zp < 0, k ze > 0, and z t < 0. When the rotational speed increases, so does the electrodynamic stiffness k ze . When k ze overcomes the minimum value of condition (12), the steadystate axial equilibrium position z b of the suspended rotor is derived from (11) as The steady-state equilibrium position z b can be calculated in terms of the rotational speed ω by substituting (5) into (13) The limit steady-state equilibrium position z b for very high rotational speeds is derived from (14) as At steady-state full passive suspension, the higher the electrodynamic stiffness, the lower the absolute value of the axial displacement. However, z b approaches (15) as the rotational speed approaches infinity. If a steady-state equilibrium position close to z b = 0 is desired, the axial displacement between permanent magnets in the radial bearing must be carefully adjusted to have F o as close as possible to the weight force. Table 2 summarizes the dimensions and grades of the neodymium permanent magnets employed in the radial permanent magnet bearings. The unstable axial stiffness k zp was measured with a sixaxis force sensor. The outer permanent magnet was fixed to the sensor. The inner permanent magnet was radially centralized with the outer ring. The distance between both rings was adjusted during the experiment in steps of 0.2 mm. The axial force was measured for several axial displacements and k zp was estimated through first-degree fitting as k zp = −5.66 N/mm.

TABLE 3. Parameters Estimated for the Radial Permanent Magnet Bearing
Substituting k zp in (12), it is possible to estimate the minimum rotational speed required for achieving axial suspension. The rotor mass m is 2 kg and the lower touchdown position z t is -1 mm. Per mechanical design d o is set to 1 mm. The minimum k ze for axial suspension is, thus, estimated as 19.62 N/mm. Fig. 7 shows the plot of axial stiffness calculated through (5) along with the axial suspension threshold. From the figure, it is estimated that the rotor shaft will become axially suspended around 4200 r/min. The radial bearing parameters are given in Table 3.

A. DESCRIPTION OF THE EXPERIMENTAL SETUP
An experiment was conducted to verify the passive axial electrodynamic suspension in the bearingless motor. A 3-D model of the experimental setup is shown in Fig. 8. Besides the bearingless motor and the radial permanent magnet bearings described earlier, there are a few supporting structures included in the setup.
Two radial ball bearings are mounted on the frame. They act as touchdown bearings and restrict the maximum displacement of the rotor shaft. The maximum axial displacement is ±1 mm. The maximum radial displacement is 0.5 mm.
There is a laser sensor IL-030 by Keyence attached to the base. The sensor is used to assess the axial position and confirm the passive suspension. It is not used for any form of closed-loop control. The target of the laser sensor is the hexagonal head of the lower rotor bolt. The head of the bolt is flat, smooth, and reflective.
Around the upper end of the rotor shaft, there is a 3-Dprinted support that houses four Hall sensors. These Hall sensors face a two-pole permanent magnet ring attached to the upper end of the rotor shaft. The rotor angular position is calculated from the Hall sensor outputs and used in the feedback control of rotor rotational speed. There are two sensors facing opposite directions on both the x-and y-axes. The output  signal from these opposing sensors is averaged to compensate for the radial displacement. The output signals are previously calibrated and normalized to eliminate variations in amplitude and offset. The control algorithm also compensates for the angular displacement between the sensor permanent magnet and the magnetic rotor pole.
The machine is driven at constant rotational speed by a feedback control loop. Up to the threshold speed for axial suspension, the rotor sits on the lower touchdown ball bearing. Above the minimum speed, the rotor is expected to lift. Fig. 9 shows a picture of the rotor. It is composed of an inner and an outer parts that are mechanically connected by an aluminum frame. There are, in total, 24 neodymium N50 segmented permanent magnets, 12 on the inner part and 12 on the outer part. There are SS400 back-iron behind the permanent magnets on both the inner and outer rotor parts. The  inner permanent magnets are retained by a carbon fiber sleeve. Figs. 10 and 11 show, respectively, pictures of the stator coils and the experimental setup. The coreless coils are fixed to the mechanical support with epoxy resin.

B. FINDING THE AXIAL POSITION OF THE MAGNETIC CENTER
During the first experiment, it was noticed that there was a significant asymmetry between the motor current supplied by the inverter at lower and upper touchdown positions. Since the coil connections were correct, this asymmetry indicated a significant displacement between the magnetic center and the middle point between the upper and lower touchdown bearings.
From (7), it is known that the braking torque due to the axial suspension current is proportional to the square of axial position z. Since the motor torque is proportional to the motor rms current, if the rotor is kept at constant rotational speed, it is possible to determine the central axial position by finding the axial position where the motor current is minimum. Fig. 12 shows the motor rms currents measured at upper and lower touchdown conditions at constant rotational speed 600 r/min. The inverter current i a is measured in the position indicated by current clamp 2 in Fig. 3. The points A and B in the plot correspond to the standard touchdown positions, 3.8 and 1.8 mm, respectively, as indicated by absolute measurements of the laser sensor. By inserting laminated steel sheets between the bearing support and the experiment frame, the touchdown was axially shifted and the current was measured at two additional positions C and D. Through second degree fitting, an equation for motor current in terms of the laser sensor output was derived. Differently from (7), zero current is not achieved at the estimated central axial position due to axial misalignment between the figure-eight coils. The axial misalignment between the coils is investigated in the back-EMF measurements of Section IV-C. Owing to this misalignment, the neutral axial position where circulating suspension current is zero is different among the figure-eight coil sets. However, the origin of the z-axis can be set for absolute minimum braking torque. In other words, it can be found by calculating the position where the derivative of the fitted motor current curve is equal to zero. The estimated position for the magnetic center is 3.7 mm as indicated by the laser sensor. It was 0.1 mm above the lower touchdown and 1.9 mm below the upper touchdown positions. In this condition, if the machine is started in lower touchdown, the axial displacement z is too small; thus, there is not enough axial force to lift the rotor and reach noncontact suspension. If the machine is started in upper touchdown, the initial displacement and, consequently, the braking torque are higher than that of the lower touchdown condition.
To adjust the axial displacement, the stator coil was shifted 0.9 mm upward with respect to the initial position. It was moved farther from the lower touchdown ball bearing and closer to the upper one. Fig. 13 shows the motor currents measured after the adjustment. The currents were measured both in upper and lower touchdown positions, A' and B', respectively, at constant rotational speed 1200 r/min. Additional measurements, C', D', E', and F', were carried out after shifting the touchdown bearings by inserting laminated steel sheets. The minimum motor current now occurs at 2.94 mm, as measured by the laser sensor. The upper and lower touchdowns are 0.97 mm away from the center, indicating that the mechanical adjustment was successful and the magnetic center of the bearingless motor is aligned with the middle point between upper and lower touchdown positions.

C. MEASUREMENT OF BACK-EMF
By cutting the parallel connection between the upper and lower coils, it is possible to drive this machine with only half winding while the other half is open, as shown in the circuit diagram in Fig. 14. This configuration permits measuring directly the back-EMF induced in the coils with open terminals. This connection is used only for assessing the back-EMF. It is not used during the other experiments.
An experiment was carried out to assess the back-EMF of each phase at constant 1200 r/min rotational speed at several fixed axial positions z. First, the upper coils were fed by the inverter while the lower coils had one open terminal, as shown in Fig. 14. All coils were in a star connection with common neutral point. The voltages V al , V bl , and V cl , measured between the open coil terminals and the neutral point, are equal to the sum of back-EMF in the lower coils for each phase. In a similar fashion, voltages V au , V bu , and V cu were measured by disconnecting the phase terminals from the upper coils and driving the rotor with the lower coils. Fig. 15(a) shows the back-EMF measured in the lower coils at lower touchdown  position z = −1 mm. A discrepancy is observed among the amplitudes of the back-EMF of the three phases, indicating a misalignment between the neutral magnetic center of the coils. Fig. 15(b) shows the first-degree polynomial fit of the amplitude of the fundamental harmonic component of back-EMF for the upper and lower coils at axial positions −1.4, −1.2, −1, 1, 1.2, and 1.4 mm. Displacements beyond the standard axial touchdown positions z = 1 mm and z = −1 mm were achieved by mechanically offsetting parts of the experimental setup using laminated steel sheets with 0.2 mm thickness. This procedure is similar to that of Section IV-B. The back-EMF is a first-degree function of axial position z, as derived in (2). Therefore, it is possible to calculate the constant K λ through first-degree fitting of the back-EMF curves. Moreover, the crossing points of the upper and lower back-EMF amplitude curves, P a , P b , and P c , are equivalent to the magnetic center of each phase [27]. P a , P b , and P c are positioned at z a = 0.16 mm, z b = −0.07 mm, and z c = −0.12 mm, respectively. This result explains why the minimum motor current in Figs. 12 and 13 is not zero.
The average ratio K λ for the whole winding is derived as The experimental value of K λ was calculated as 5.2 Wb/m, 5% lower than that estimated with FEA.

D. AXIAL SUSPENSION AND MOTOR CURRENT RESULTS
Currents and axial positions were measured for constant rotational speeds up to 8065 r/min. Total coil current i al1 in coil A l1 and the motor current i a were measured with current clamp probes, as indicated in Fig. 3. The axial position was monitored by a laser sensor. Fig. 16(a) shows the axial position measured with the laser sensor at constant speeds 0 and 8065 r/min. The plotted axial position was calculated as the displacement from the magnetic center characterized in Fig. 13. With a static rotor sitting on the lower touchdown bearing, the measured axial position was −0.99 mm. At 8065 r/min, the average measured axial position was z = −0.28 mm with 15.4% ripple. Fig. 16(b) shows the currents measured at 8065 r/min. The experimental suspension current was not directly measured. Its instantaneous value was calculated as i za1 = i a /2 − i al1 . There is a current peak variation observed every two mechanical cycles. This variation may be caused by radial displacement or precession of the rotor shaft. Fig. 17(a) shows a comparison between the experimental and theoretical results for steady state equilibrium axial position z at several rotational speeds. The markers in the measured position plot are the average measurements of the laser sensor outputs. The error bars are equivalent to the peak-to-peak values measured by the oscilloscope. There is a reasonable correspondence. The theoretical values of steadystate axial position z were calculated with K λ = 5.2˜Wb/m, obtained from the back-EMF experiment. The rotor was expected to reach axial suspension around 4450 r/min; however,  in the experiment, suspension was only confirmed above 5319 r/min. This difference is attributed to uncertainty of parameters in the radial permanent magnet bearings, such as assembly error in the axial offset d o . There is a tilting angle between the rotor and the vertical direction, because a precession movement was observed. This inclination slightly decreases the resultant axial electrodynamic force. Assuming that d o is smaller than the design specification, the static axial load mg − F o is increased, resulting in a higher rotational speed requirement for axial suspension, even if the mechanical error is of the order of 0.1 mm. Fig. 17(b) shows the measured motor and suspension rms values of the fundamental current components. The currents were measured at rotational speeds up to 8065 r/min. The plot also shows the respective theoretical currents for each speed. The theoretical suspension current was calculated by dividing the amplitude of (3) by √ 2, whereas the rms motor current was estimated by dividing (7) by the torque-current factor k T = 0.0796 Nm/A. There are error bars plotted for the experimental suspension current values, because the suspension current amplitude varies every few mechanical cycles, as shown in Fig. 16(b). Up to 5319 r/min, the motor rms current increases because of the braking torque caused by the suspension current. The motor current reaches a peak during the transition to axial suspension and falls from about 1.15 A to 0.62 A. The measured current did not get as close to zero as the theoretical value because of the difference observed in the axial equilibrium position depicted in Fig. 17(a). There may also be unaccounted losses, such as those from the third harmonic component of the suspension current, pulsewidth modulation harmonics, proximity effect, and eddy currents on the cross section of the coil conductors and surface of the permanent magnets.

V. CONCLUSION
This article covered a surface-mounted permanent magnet bearingless motor with passive axial electrodynamic suspension and passive radial permanent magnet bearings.
The electrodynamic suspension principle was described. Equations for the braking torque and steady-state axial equilibrium position were derived. The electrodynamic stiffness increases with rotational speed; therefore, it must overcome a minimum speed threshold to achieve axial suspension. This threshold depends on parameters, such as the rotor weight, touchdown position, and the stiffness of the radial permanent magnet bearings. Both the suspension current and braking torque are proportional to the square of the axial displacement z. The passive radial permanent magnet bearing was described, and the minimum rotational speed for achieving passive axial suspension was estimated A method was proposed to compensate for the misalignment between the middle point between upper and lower touchdowns and the magnetic center where suspension current is minimum.
Axial suspension was confirmed for rotational speeds above 5319 r/min. The minimum speed for levitation observed during the experiment was higher than the preliminary estimation of 4200 r/min, calculated through (5) and (12) with a K λ = 5.5 Wb/m obtained through FEA. It was also higher than 4450 r/min, and the minimum speed estimated with the corrected K λ = 5.2 Wb/m was obtained from the back-EMF experiment. This difference is attributed to the uncertainty of the offset axial force F o in the radial permanent magnet bearings and a precession movement observed during suspension.