Modeling and Realization of a Bearingless Flux-Switching Slice Motor

Abstract

This work introduces a novel bearingless slice motor design: the bearingless flux-switching slice motor. In contrast to state-of-the-art bearingless slice motors, the rotor in this new design does not include any permanent rotor magnets. This offers advantages for disposable devices, such as those used in the medical industry, and extends the range of bearingless slice motors toward high-temperature applications. In this study, our focus is on the analytical modeling of the suspension force torque generation of a single coil and the bearingless motor. We assessed motor performance in relation to motor topology by applying performance factors. A prototype motor was optimized, designed, and manufactured. We also presented the state-of-the-art nonlinear feedback control scheme used. The motor was operated, and both static and dynamic measurements were taken on a test bench, thus successfully demonstrating the functionality and applicability of the novel bearingless slice motor concept.

1. Introduction

A disk-shaped rotor inside a permanent magnetic biased air gap field allows three degrees of freedom (two tilting directions and axial displacement) to be stabilized passively by reluctance forces. Active control of the remaining three degrees of freedom (radial movement and rotation) is achieved by the stator coils and leads to a very compact, fully magnetically levitated drive. Such systems are called bearingless slice motors, and were first introduced in 1994 [1]. Continuous development led to their industrial application, mainly as pumps in the medical and semiconductor industries [2,3]. Other fields of application for bearingless slice motors are also being investigated. These include hermetically sealed process chambers for wafer processing [4], mixers and bioreactors for ultrapure fluid treatment [5], high speed drives for micro- turbines and compressors, [6] and blowers or fans for transporting special gases [7]. Further laboratory prototypes have been implemented, such as the viscometer [8], flywheel [9], and small scale wind turbine [10]. Apart from the typical advantages of magnetically suspended systems, such as contact-free operation, high cleanliness, lubrication-free operation, high lifetime with low maintenance, low vibration, and high speed range, bearingless slice motors additionally feature a relatively simple mechanical setup.

However, in all state-of-the-art bearingless slice motors, the permanent magnets that excite the air gap bias flux are located in the rotor system [4]. Thus, the rotor cross-section of bearingless slice motors bears a strong similarity to rotors of brushless permanent magnet synchronous machines. Especially in high-temperature or high-speed applications, and in disposable devices (where the rotor must be replaced frequently), permanent magnet-free rotors offer advantages. Using a magnetless rotor reduces manufacturing costs (rare-earth magnetic materials have increased significantly in price in recent years) and enhances thermal and mechanical robustness. Initial studies on bearingless slice motors without magnetic material in the rotor, also called bearingless reluctance slice motors, have already been published [11]. Using permanent magnets in the stator inherently demands a rotor with salient poles to influence the linked permanent magnetic flux of the stator coils for torque generation. Thereby the permanent magnetic air gap flux can either be homopolar or heteropolar. Only two homopolar bearingless slice motor designs without permanent magnets in the rotor have reportedly been built [12,13]. Results showed that force generation was comparable with standard slice motor systems, but that torque generation was reduced by principle, as the permanent magnetic field and flux linkage only varies but does not change its direction. Heteropolar drives do not feature this drawback in torque generation. There are three possible design variants of this kind: the flux reversal, doubly salient, and flux-switching permanent magnet machine [14]. This paper focuses on the bearingless flux-switching motor, as it appears to be a promising approach for development of slice motor systems.

This article is loosely based on a conference paper [15], but has been extended significantly throughout and includes new research results. It is organized as follows: Section 2 describes the structural design of the bearingless flux-switching motor. In Section 3 an analytical model for the single-phase torque characteristic is deduced. The overall performance in terms of suspension force and torque generation is estimated with the help of performance factors in Section 4. Optimization and design of the manufactured prototype are outlined in Section 5. Measurements are compared with the expected analytic and finite element simulation results in Section 6. Section 3 and Section 4 describe new results, and Section 6 has been extended with additional measurements from [16].

2. Bearingless Flux-Switching Slice Motor

The main structural characteristic of the flux-switching motor, shown in Figure 1, is the placement of the permanent magnets. They are located between the stator teeth, and separated electromagnetically [17]. The flux lines of each permanent magnet close mainly over the neighboring stator teeth, and thus do not penetrate the whole air gap, but especially influence the air gap region close to the permanent magnet. However, there is also a considerable amount of fringing flux closing over the outer side of the stator. To keep stray flux low, a small air gap length (compared to the magnet width) is favorable and necessary.

The air gap flux density in flux-switching machines is often higher than in comparable electric machines, which leads to an increased torque capacity. This is due to the flux concentration capability of this design. The surfaces of the stator teeth normal to the magnetization direction collect permanent magnetic flux, and concentrate it towards the much smaller area of the stator teeth adjoining the air gap.

Conventional flux-switching motors with mechanical bearings represent a relatively recent electrical drive topology introduced in 1955 [18]. Regarding the bearingless version, initial concept studies were published in 2011 [19], and the first prototype went into operation in 2013 [20]. Starting in 2014, Chinese research groups also published their theoretical results regarding bearingless flux-switching motors [21,22]. In 2016 a two-level bearingless flux-switching motor prototype was also reported [23]. Theoretical considerations regarding a linear bearingless flux-switching motor were presented in [24]. These works show the feasibility of this basic concept. Force and torque generation is comparable to standard bearingless slice motors with rotor permanent magnets. The main difference lies in the high rotor pole number of the flux-switching motor, which makes it unfavorable for high speeds. Hence, low speed, high torque direct-drive systems seem to be the most favorable field of application. Due to the saliency of the rotor, the permanent magnetic flux linked with the stator coils changes its direction, resulting in an induced stator back-electromotive force. This means that the energized coil also generates torque [17]. Additionally, the superposition of the stator field and permanent magnetic fields, leads to the creation of bearing forces, due to field strengthening and weakening [20]. Figure 2 shows the principle of force and torque generation schematically. Combining both concepts enables bearingless motor operation for this kind of machine.

The general force generation of one coil is nonlinearly correlated with the corresponding coil current according to [25]:

$${\left(\begin{array}{ccc}{F}_x\hfill & F_y\hfill & T_z\hfill \end{array}\right)}^T=M_Q(\phi )\cdot i_s{}^2+M_L(\phi )\cdot i_s+M_C(\phi )$$

(1)

where F_x and F_y are the forces towards and perpendicular to the coils axis, i_s is the coil current, ϕ stands for the rotor angle, and the vectors M_Q, M_L and M_C describe the force and torque components that have a quadratic, a linear and no relationship with the coil currents, respectively. This nonlinear relationship complicates the decoupling of torque, x-, and y-force that are necessary for stable bearingless operation. Hence, in the next section, a model of the coil force and torque characteristic is derived to gain a better understanding of this characteristic, and the importance of each component in Equation (1).

3. Modeling of the Single-Phase Characteristic

Suspension forces were computed from the magnetic field distribution in the air gap B_δ, which consists of the permanent magnetic and the stator current excited fields. The relations are:

$$F_x=\frac{lr}{2{\mu }_0}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{B}_{\delta }{\left(\gamma ,\phi \right)}^2}\hspace{0.17em}\cos \left(\gamma \right)d\gamma$$

(2)

and:

$$F_y=\frac{lr}{2{\mu }_0}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{B}_{\delta }{\left(\gamma ,\phi \right)}^2}\hspace{0.17em}\sin \left(\gamma \right)d\gamma$$

(3)

where l represents the axial rotor and stator stack length, r stands for the mean air gap radius below the salient poles, and µ₀ describes the permeability of air. The torque can be computed from several flux linkages by:

$$T_z=\frac{1}{2}\frac{d{\Psi }_{i,s}}{d\phi }{i}_s+\frac{d{\Psi }_{PM,s}}{d\phi }{i}_s+\frac{1}{2}\frac{d{\Psi }_{PM}}{d\phi }{I}_{PM}$$

(4)

Here, Ψ_i,s denotes the flux created by the stator coil, Ψ_PM the net flux created by the permanent magnet (PM) and Ψ_PM the permanent magnetic flux that is linked with the stator coil. All of these fluxes can be computed from the magnetic air gap field distribution B_δ, which results from the superposition of B_δ,i (created by the energized stator coil) and B_δ,PM (excited by the permanent magnets). The permanent magnet was replaced by the current source I_PM and a corresponding inductivity.

Assuming linear material behavior without any saturation effects, and ignoring tangential flux components, the magnetic field distribution in the air gap was modeled by the rotor-angle-dependent magnetic permeance of the air gap, which can be computed as the product of the rotor permeance P_r(γ,ϕ) and the stator tooth factor P_s(γ), where γ describes the angle of the point under consideration on the circumference in the middle of the air gap. The stator tooth factor depends on the stator dimensions, and can be expressed by a series of rectangular functions that are defined as:

$$\Pi \left(\gamma \right)=\{\begin{array}{cc}\hfill 0\hfill & \hfill if\hspace{0.17em}\left|\gamma \right|<0.5,\hfill \\ \hfill 0.5\hfill & \hfill if\hspace{0.17em}\left|\gamma \right|=0.5,\hfill \\ \hfill 1\hfill & \hfill if\hspace{0.17em}\left|\gamma \right|>0.5,\hfill \end{array}$$

(5)

via:

$$P_s\left(\gamma \right)={\displaystyle \sum _{i=0}^{N_s-1}\Pi \left(\frac{\gamma -\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$N_s$}\right.i+{\zeta }_s}{{\zeta }_s}\right)}+{\displaystyle \sum _{i=1}^{N_s}\Pi \left(\frac{\gamma -\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$N_s$}\right.i-{\zeta }_s}{{\zeta }_s}\right)}$$

(6)

with:

$${\zeta }_s={\sin }^{-1}\left(\raisebox{1ex}{$t{w}_s$}\!\left/ \!\raisebox{-1ex}{$\left(d{o}_r+\delta \right)$}\right.\right)$$

(7)

where N_s denotes the number of stator teeth, tw_s the stator tooth width, do_r the rotor outer diameter and δ the air gap length. A stator tooth factor P_s(γ) = 1 indicates a magnetically conductive stator part at the circumferential angle γ, whereas P_s(γ) = 0 represents the locations of the stator slots. In a similar manner, the expression for the rotor permeance can be formulated as:

$$P_r\left(\gamma ,{\phi }_r\right)=P_{r\hspace{0.17em}\min }+\left(P_{r\hspace{0.17em}\max }-P_{r\hspace{0.17em}\min }\right){\displaystyle \sum _{i=0}^{N_r-1}\Pi }\left(\frac{\gamma -\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$N_r$}\right.i+\phi }{2{\sin }^{-1}\left(\raisebox{1ex}{$t{w}_r$}\!\left/ \!\raisebox{-1ex}{$\left(d{o}_r+\delta \right)$}\right.\right)}\right)$$

(8)

with the number of rotor teeth N_r, and tw_r denoting the rotor tooth width. Here, P_{r max} represents the permeance of the smaller outer air gap, whereas P_{r min} stands for the corresponding permeance of the lager outer air gap of the rotor due to the saliency. The air gap flux density can then be calculated using the expression:

$$B_{\delta }\left(\gamma ,\phi \right)=B_{\delta ,s}\left(\gamma ,\phi \right)+B_{\delta ,PM}\left(\gamma ,\phi \right)=\frac{2}{l\cdot \left(d{o}_r+\delta \right)}{P}_r\left(\gamma ,\phi \right)\cdot P_s\left(\gamma \right)\cdot \left({\theta }_s\left(\gamma ,\phi \right)+{\theta }_{PM}\right)$$

(9)

The product P_s(γ)∙P_r(γ,ϕ) can be interpreted as an addition of the rotor and stator permeances, which sum up in parallel. The last unknown parameters to be evaluated in Equation (9). were the magneto-motive forces. Figure 3 shows the magnetic circuits with the process by which they were calculated.

For a single energized phase, the magneto-motive force distribution can be computed as:

$${\theta }_s\left(\gamma ,\phi \right)=\{\begin{array}{cc}\hfill -\frac{P_{et1}\left(\phi \right)}{P_{rt1}\left(\phi \right)+P_{et1}\left(\phi \right)}\hspace{0.17em}N\cdot i_s\hfill & for\hspace{0.17em}2\pi -9{\zeta }_s<\gamma <2\pi -7{\zeta }_s,\hfill \\ \hfill \frac{P_{rt1}\left(\phi \right)}{P_{rt1}\left(\phi \right)+P_{et1}\left(\phi \right)}\hspace{0.17em}N\cdot i_s\hfill & for\hspace{0.17em}2\pi -3{\zeta }_s<\gamma <2\pi -{\zeta }_s,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \\ \hfill -\frac{P_{et2}\left(\phi \right)}{P_{rt2}\left(\phi \right)+P_{et2}\left(\phi \right)}\hspace{0.17em}N\cdot i_s\hfill & for\hspace{0.17em}7{\zeta }_s<\gamma <9{\zeta }_s,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \\ \hfill \frac{P_{rt2}\left(\phi \right)}{P_{rt2}\left(\phi \right)+P_{et2}\left(\phi \right)}\hspace{0.17em}N\cdot i_s\hfill & for\hspace{0.17em}{\zeta }_s<\gamma <3\zeta \hspace{0.17em},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \\ \hfill 0\hfill & else,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \end{array}$$

(10)

with the two permeances below the directly energized stator tooth sections:

$$\begin{array}{l}{P}_{et1}\left(\phi \right)={\displaystyle \underset{2\pi -{\zeta }_s}{\overset{2\pi -3{\zeta }_s}{\int }}{P}_r\left(\gamma ,\phi \right)\cdot P_s\left(\gamma \right)\hspace{0.17em}}d\gamma ,\hfill \\ P_{et2}\left(\phi \right)={\displaystyle \underset{3{\zeta }_s}{\overset{{\zeta }_s}{\int }}{P}_r\left(\gamma ,\phi \right)\cdot P_s\left(\gamma \right)\hspace{0.17em}}d\gamma ,\hfill \end{array}$$

(11)

and the two permeances below the two legs of the return flux path:

$$\begin{array}{l}{P}_{rt1}\left(\phi \right)={\displaystyle \underset{2\pi -7\zeta }{\overset{2\pi -9{\zeta }_s}{\int }}{P}_r\left(\gamma ,\phi \right)\cdot P_s\left(\gamma \right)\hspace{0.17em}}d\gamma ,\hfill \\ P_{rt2}\left(\phi \right)={\displaystyle \underset{9{\zeta }_s}{\overset{7{\zeta }_s}{\int }}{P}_r\left(\gamma ,\phi \right)\cdot P_s\left(\gamma \right)\hspace{0.17em}}d\gamma .\hfill \end{array}$$

(12)

The magneto-motive force resulting from the permanent magnets was obtained by superposition of the magneto-motive forces of the individual permanent magnets between the teeth by:

$${\theta }_{PM}\left(\gamma ,\phi \right)={\displaystyle \sum _{i=0}^{N_s-1}{\left(-1\right)}^i{\theta }_{PM,seg}\left(\gamma -\frac{2\pi }{N_s}i,\phi -2\pi \cdot \frac{N_r-N_s}{N_s\cdot N_r}i\right)}$$

(13)

The term (−1)ⁱ accounts for the two alternating directions of magnetization in the stator permanent magnets. The shift in the circumferential air gap angle and the rotor angle considers the different locations of the stator teeth holding the permanent magnets. The flux paths of the permanent magnetic field of a single magnet are shown in the right-hand image of Figure 3. With this knowledge, the magneto-motive force distribution can be derived, yielding:

$${\theta }_{PM,seg}\left(\gamma ,{\phi }_r\right)=\{\begin{array}{cc}\hfill -\frac{P_{t2}\left(\phi \right)}{P_{t1}\left(\phi \right)+P_{t2}\left(\phi \right)}\cdot {\theta }_{PM,op}\left({\phi }_r\right)\hfill & for\hspace{0.17em}2\pi -9{\zeta }_s<\gamma <2\pi -{\zeta }_s,\hfill \\ \hfill \frac{P_{t1}\left(\phi \right)}{P_{t1}\left(\phi \right)+P_{t2}\left(\phi \right)}\cdot {\theta }_{PM,op}\left(\phi \right)\hfill & for\hspace{0.17em}{\zeta }_s<\gamma <9{\zeta }_s,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \\ \hfill 0\hfill & else,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hfill \end{array}$$

(14)

with:

$$\begin{array}{l}{P}_{t1}\left(\phi \right)=P_{et1}\left(\phi \right)+P_{rt1}\left(\phi \right),\hfill \\ P_{t2}\left(\phi \right)=P_{et2}\left(\phi \right)+P_{rt2}\left(\phi \right).\hfill \end{array}$$

(15)

The permanent magnetic source voltage in the operating point was calculated by:

$${\theta }_{PM,op}\left(\phi \right)=H_c\cdot l_m-R_{PM}\cdot \frac{H_c\cdot l_m}{R_{PM}+\left(R_{os}\left|\right|\left(R_{t1}\left(\phi \right)+R_{t2}\left(\phi \right)\right)\right)}$$

(16)

where reluctance R_t1(ϕ) = 1/P_t1(ϕ), R_t2(ϕ) = 1/P_t2(ϕ), and the constant outer stray flux reluctance R_os can be assumed to be a semi-circular flux tube similar to that defined in [26].

The sum of R_t1(ϕ) and R_t2(ϕ) can be assumed to be nearly constant over the rotor angle, exhibiting only small variances compared to the overall mean value. In this case, θ_PM,op becomes independent of the rotor angle. This simplification was assumed in all of the following considerations. Using Equation (9)., the magnetic field strength in the air gap due to the energized coil and due to the stator permanent magnets can be computed.

Figure 4 shows the standardized rotor and stator permeances as well as the standardized radial magnetic field strengths for two different rotor angles. The chosen topology featured 12 stator teeth and 10 salient rotor poles. This standard topology for stator permanent magnet flux-switching drives [27] exhibits negligible reluctance forces and low cogging torque. Additionally, the rotor pole width tw_r was chosen to be 1.5 times the iron tooth width tw_s, to reduce higher harmonics in the back electromotive force, and to increase the torque [26,28]. As can be seen from the P_r characteristic in Figure 4, the ratio P_{r min}/P_{r max} was set at 10% in this example. The scale basis of the B-fields was given by the corresponding maximum absolute magnetic field strength over all rotor angles.

From the B-fields, the radial and tangential forces for one energized coil in relation to the coil axis were calculated using Equations (2) and (3), as illustrated in Figure 5. To obtain these results, the ratio between the maximum permanent magnetic field and the maximum of the field created by the coil current was set to be 2:1. Due to the symmetrical topology, no cogging forces would occur. The forces correlate linearly with the phase current and are approximately sinusoidal. Note that amplitude and offset of the tangential force depend on the ratio of P_{r min}/P_{r max}, but only to a limited extent, as illustrated in Figure 5. The force characteristic obtained from the analytical model agreed very well with the finite-element simulation results, and the results published in [23,29]. However, the reluctance forces could not be ignored for all rotor angles in the radial force direction. This was problematic because the state-of-the-art control schemes rely on linearity, which is not given for a significant reluctance part. The choice of iron tooth width tw_s (relative to the rotor tooth width tw_r) did not change the general characteristic and influenced only higher harmonic orders.

The torque components from Equation Equation (4). were evaluated with the help of:

$$\begin{array}{l}{\Psi }_{PM}=lr{\displaystyle \underset{0}{\overset{2\pi }{\int }}{B}_{\delta ,PM}\left(\gamma ,\phi \right)d\gamma },\hfill \\ {\Psi }_{PM,s}=lr{\displaystyle \underset{-\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2N_s$}\right.}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2N_s$}\right.}{\int }}{B}_{\delta ,PM}\left(\gamma ,\phi \right)d\gamma },\hfill \\ {\Psi }_{i,s}=lr{\displaystyle \underset{-\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2N_s$}\right.}{\overset{\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2N_s$}\right.}{\int }}{B}_{\delta ,s}\left(\gamma ,\phi \right)d\gamma },\hfill \end{array}$$

(17)

as shown in Figure 6. For the variant analyzed with twelve stator teeth and ten rotor teeth, both cogging torque and reluctance torque were negligible. The permanent magnetic stator coil flux, the induced voltage and therefore the linear-current-dependent torque were broadly sinusoidal. This torque is a function of P_{r min}/P_{r max}: If this ratio increases, the torque decreases. Again, different rotor tooth widths tw_s did not change the fundamental wave but influenced the higher harmonic values.

4. Overall Performance of the Bearingless Flux-Switching Motor

The analytical model and the finite-element simulations consistently showed that there was a high quadratic term in the generation of the single-phase x-force. This term arose from a significant armature reaction, which is typically small in systems where the current flux has to penetrate a permanent magnet. This case of a negligible armature reaction is given in nearly all state-of-the-art bearingless slice motors, but here the significant armature reaction resulted in the additional x-force component, with quadratic dependency on the stator current described. Such a quadratic term was incompatible with the proposed nonlinear control scheme, as it relied on a linear relationship. However, the quadratic term in the suspension force could be eliminated when two opposing coils featuring the same single-phase characteristic are connected in the same phase.

4.1. Linearization and Decoupling

The upper graphic in Figure 7 shows two identical opposing coils energized with the same current i_t. All forces cancel out, and only torque is created. In contrast, the force phase illustrated in the lower graphic in Figure 7 shows inverted currents in the opposing coils leading to compensation for the quadratic forces, which are independent of the current signs, and summation of the linear forces. An inverted winding direction instead of a negative current would lead to the same effect.

However, for the flux-switching motor the proposed method cancels out all significant quadratic effects, leaving only a force and torque generation that correlates linearly with the stator phase currents. Additionally, separation into only force-creating and only torque-creating phases is achieved, thus implementing a separated and decoupled winding system.

4.2. Topology Evaluation

For a single-phase torque and suspension force characteristic with linear phase-current dependency, the force-current matrix T_m can be deduced [19]. It describes the force and torque generation of all phases and their corresponding currents by the simple relation:

$${\left(\begin{array}{ccc}{F}_x\hfill & F_y\hfill & T_z\hfill \end{array}\right)}^T=T_m(\phi )\cdot i_s$$

(18)

where i_s is the vector of all stator phase currents. For the control scheme, the inverse relation was more important, because the current-force matrix K_m helps to compute the phase currents necessary to generate specific forces and drive torque. A common way to calculate the inverse relation is given by:

$$i_s=K_m(\phi )\cdot {\left(\begin{array}{ccc}{F}_x\hfill & F_y\hfill & T_z\hfill \end{array}\right)}^T$$

(19)

$$K_m(\phi )=T_m{(\phi )}^T{\left(T_m(\phi )T_m{(\phi )}^T\right)}^{-1}$$

(20)

which minimizes the copper loss in the coils [25]. With the help of K_m, the overall performance of the motor can be estimated. In [20], two benchmark factors were defined to assess the operational characteristic of a bearingless motor. The first factor is called force factor and is defined by:

$$c_t=\frac{4}{N_s}\frac{T_{min,\hspace{0.17em}overall}}{T_{max,\hspace{0.17em}phase}}$$

(21)

This represents the overall worst-case torque generation capability T_{min, overall}, in relation to the single-phase torque maximum T_{min, phase}. In standard sinusoidal multi-phase synchronous machines, this factor is given by m/2, where m is the number of torque phases. This factor was used to normalize the torque factor to values between 0 and 1. A similar factor, called force factor or lift-off factor [30], was defined for the suspension force generation:

$$c_f=\frac{4}{N_s}\frac{F_{min,\hspace{0.17em}overall}}{F_{max,\hspace{0.17em}phase}}$$

(22)

These two factors were used to estimate the performance of the bearingless flux-switching motor. The standard topology with 12 stator teeth, featuring a double separated three-phase winding system, was used. Starting with the single-phase force and torque characteristics in Figure 8, the T_m-matrix, K_m-matrix and performance factors were computed as functions of the rotor tooth number. The tangential force of the single-phase characteristic depends on the geometry, but only to a very limited extent, as indicated in Figure 8. For the evaluation, a mean representative characteristic of the tangential force was used. As can be seen in Figure 9, there are three different feasible winding systems, all of which were considered in the evaluation.

Table 1. Performance factors of different bearingless flux-switching motors, featuring twelve stator teeth and a double three-phase winding system.

Winding Scheme (Corresponding to Figure 9)	Number of Stator Teeth Ns = 12
	Nr = 8		Nr = 10		Nr = 12		Nr = 14		Nr = 16
(a) double three-phase	cf = 0	ct = 1	cf = 0.5	ct = 1	cf = 0.2	ct = 0 *	cf = 0	ct = 1	cf = 0.5	ct = 1
	CF↓	CT−	CF↓	CT−	CF↓	CT↑	CF↓	CT−	CF↓	CT−
(b) double three-phase	cf = 0.2	ct = 1	cf = 0.1	ct = 1	cf = 0.2	ct = 0 *	cf = 0.4	ct = 1	cf = 0.3	ct = 1
	CF↓	CT−	CF↓	CT−	CF↓	CT↑	CF↓	CT−	CF↓	CT−
(c) double three-phase	cf = 0.2	ct = 0.6	cf = 0.3	ct = 0.6	cf = 0.2	ct = 0 *	cf = 0.3	ct = 0.6	cf = 0.2	ct = 0.6
	CF↓	CT−	CF↓	CT−	CF↓	CT↑	CF↓	CT−	CF↓	CT−

Dark and light red backgrounds indicate: very poor and poor respectively, green: good, and yellow: moderate performance. CF stands for cogging force and CT for cogging torque. The symbols ↓, − and ↑ indicate low, medium and high values, respectively. A * indicate drives which cannot create torque at certain rotor angles but might feature acceptable mean torque values considering one electrical period.

summarizes the resulting performance factors and estimations of cogging torque and cogging force. The best overall bearingless motor was the typical 12/10 topology with winding set (a), but the 12/14 variant with winding scheme (b) also appeared to be promising. The winding scheme (c) featured non-optimal torque generation because the three phases were not shifted by 120° each, and are therefore of minor importance. Based on these results, the common 12/10 bearingless flux-switching topology was chosen to be implemented in a prototype.

4.3. Control Scheme

With the help of Equation (20), a nonlinear feedback transformation was used to decouple the bearing forces and the drive torque in the control system. This method has been proven for bearingless slice motors [12,25]. Figure 10 illustrates this control scheme with outer proportional integral derivative (PID) position and proportional integral (PI) speed controller. Faster nested proportional (P) current controllers impress the necessary currents into the phases with the help of two standard three-phase half-bridge voltage inverters. Measuring the rotor position and the rotor angle is necessary for feedback to close the control loop.

5. Optimization and Design of the Prototype

A 3D finite element optimization of the geometry illustrated in Figure 1 was performed with the software package Maxwell 3D from Ansys. Previous work [26,31] has optimized the geometric parameters of standard (non-bearingless) flux-switching motors with respect to torque generation, from which favorable geometric parameters are known. For instance, choosing a rotor-to-stator diameter ratio between 0.55 and 0.6 was recommended. Additionally, it was favorable to have similar values for the stator tooth width, the magnet width and the stator slot width; a rotor tooth width of about 1.4 times greater than the stator tooth width was recommended.

In addition to the maximization of the motor torque, the increase in passive stabilizing stiffness values and the achievable bearing forces were criteria to be optimized. Finite-element simulations showed that saturation effects severely restricted the parameter choice, especially that of the stator geometry. Due to the limited design space, saturation effects occurred in the iron of the stator teeth (at higher current densities or torque creation). Hence, the geometry was adapted accordingly. The main geometric and electromagnetic characteristic parameters are shown in Figure 11. The corresponding geometric and electromagnetic values after the optimization process are summarized in

Table 2. Main geometric parameters of the prototype drive.

Variable	Description	Value	Unit
dos	stator iron outer diameter	266	mm
dor	rotor outer diameter	150	mm
hz	axial rotor and stator length	10	mm
δ	magnetic air gap	3	mm
mw	magnet width	8.5	mm
ml	magnet length	54	mm
laminated iron material		M330-35A
permanent magnet material		NdFeB N38

and

Table 3. Main electromagnetic parameters of the prototype drive.

Variable	Description	Value	Unit
kr	radial stiffness	−41.6	N/mm
kz	axial stiffness	7.3	N/mm
kϕ	tilt stiffness	21.4	Nm/rad
Jmax	maximum current density	6	A/mm2
Fmax	maximum bearing force	10	N
Tmax	maximum drive torque	1.5	Nm

. The air gap of 3 mm might seem very large when compared to standard electrical drives. However, bearingless slice motors are often used in applications with hermetically sealed rotors. Therefore, the rotor and stator are separated by a type of chamber wall. To simplify its manufacture and reduce production cost large air gaps of this kind are necessary. The whole optimization process for a bearingless flux-switching slice motor and the optimization criteria are outlined thoroughly in [32].

Figure 12 shows the manufactured prototype motor. The laminated stator iron stack is held by an aluminum frame to strengthen the structure. The permanent magnets are glued into pockets that separate the stator elements. The externally wound stator coils are easily slipped onto the stator teeth and connected. Hall and eddy current sensors are placed on the inside of the stator and between the stator teeth to measure the radial deflection and the rotor angle. An aluminum housing covers the rotor and serves as target for the eddy current sensors.

The power electronics inverter used is composed of two three-phase power modules. The rated phase voltage and current of the inverter are 300 V and 15 A, respectively. The nonlinear control scheme for proper decoupling of force and torque is implemented on a 32-bit digital signal processor, which is placed in the integrated control electronics of the inverter.

6. Measurement Results

6.1. Static Measurements

The prototype system was mounted on a test bench with a x-y-z table, load cell, torque sensor and load machine. This setup is universal and allows numerous values to be measured. Generally, the measurements showed good correlation with the finite-element simulation results.

As an example, the force orbit and the torque characteristic at constant magneto-motive force are plotted in Figure 13. The force orbit was unique due to its offset in the tangential direction. Both the force and the torque curves differed only by about 10% from the expected values. However, there was a small asymmetry in the positive and negative half cycle. This fact is well known [33], and results from the nature of the associated permanent magnetic flux paths as the rotor moves. This asymmetry was also visible in the derived model by the characteristic curve of T_z,l(ϕ) in Figure 6.

6.2. Dynamic Suspension Measurements

Measurements were conducted under bearingless operation to check the functionality of the control scheme and its performance. The reaction of the bearingless motor for a step change of the reference position in the x-direction is illustrated in the left-hand image of Figure 14. It was observed that the new position was reached stably after about 80 ms. Due to the controller parameters used, an overshoot and oscillation were observed. Minimal cross coupling between the x- and y-axes was present because there was also a reaction in the y-position signal. This measurement was taken at zero speed.

Furthermore, a disturbance feed-forward x-force step was implemented, again at zero speed. The reaction of the bearingless motor is visible in the right-hand image of Figure 14. The error was compensated for, and the original position signal was reached after 100 ms. However, a cross coupling between the x and y axis was visible. This coupling was compensated by the outer position controllers, as long as it did not become too large. The reason of the coupling lies in the incomplete linearization of the model by the K_m(ϕ)-matrix. A decentered rotor, tilting effects, asymmetries in rotor and stator compositions, nonlinearities in the power electronics, higher harmonics in force and torque, as well as saturation effects, were not accounted for, as the control effort would have increased dramatically.

6.3. Motor Measurements

For measuring the performance of the torque generation, the bearingless drive was mechanically suspended and strained by means of a load machine. Several points of operation were evaluated. The efficiency and the load factor, which are plotted in relation to rotor speed and different torque values in Figure 15, were of major interest here.

As can be seen from the measurements, efficiencies of nearly 80% were reached. Compared to standard drives, the efficiency did not seem very high. This was caused by several issues. First, the air gap of this motor with 3 mm was very large. Therefore, relatively high magneto-motive forces needed to be applied, which also led to increased copper losses. Second, due to the slice motor composition, the axial length was only 10 cm and therefore very short. Therefore, the end winding area of the coils was much larger than the active axial coil area. Third, the rotor iron passed through the full hysteresis cycle. This is not the case in standard PM-excited rotors, where the rotor PM flux is not alternating. The efficiency of bearingless slice motors is seldom presented in literature, but examples can be found in Grabner [34], Karutz [35], and Steinert [36]. In these works the efficiency also typically ranged from 60% to 80%, depending on the speed and load. Higher speed usually increases the efficiency, as the mechanical output power increases at the same load and current. However, above a certain speed, iron losses become significant and diminish efficiency.

The power factor is defined as the ratio of the active power to the apparent power in the drive phases. As the inverter had to be dimensioned for apparent power, a high power factor was desirable to keep converter power and therefore costs low. Unfortunately, the stator inductivities in reluctance rotors are typically much higher as in permanent magnet excited rotors. Hence, the power factor is generally smaller. This was also true for the bearingless flux-switching motor. Starting from one at zero-drive torque, the power factor significantly decreased with higher torque (and therefore also higher current) at constant speed. This resulted from the fact that the induced voltage remains constant, while the inductive component rose with the current.

7. Discussion

We have presented a new motor topology, the bearingless flux-switching slice motor. After introducing the concept, we focused on modeling the suspension forces and the drive torque, using an analytical permeance model to examine the force and torque generated by a single coil. Linearization and decoupling of force and torque generation were introduced by connecting opposing coils in series; state-of-the-art nonlinear feedback control methods can thus be employed to stabilize the rotor and generate torque. Based on the single-phase characteristic, performance factors for force and torque were used to describe the performance of the overall bearingless motor. These factors were evaluated for bearingless slice motors featuring 12 stator teeth and a double three-phase winding system. A favorable topology was chosen for realization, optimized by 3D finite-element simulation and manufactured. This bearingless flux-switching slice motor was successfully put into operation, and the feasibility of the design approach demonstrated by measurements.

Examining further bearingless flux-switching motor topologies that feature stator tooth counts other than 12 and other winding sets seems promising for future research. A study aiming to increase the torque by adapting the geometry may also yield useful results. Finally, a novel control scheme that can deal with quadratic current dependencies on suspension forces and drive torque would remove the need for serial connection of opposing coils, thus greatly enhancing the freedom of design and the utilization of the stator coils.