The effect of permanent magnet position on axial and radial load capability in axial hybrid magnetic bearing

Abstract

The axial and radial load capacity of axial hybrid magnetic bearing (HMB) is critical for the magnetic levitation system. In this paper, the effect of permanent magnet (PM) position on axial and radial magnetic force and stiffness in axial HMB is investigated. Six different configurations are considered and the equivalent magnetic circuits of HMBs for each configuration is built and studied based on the distribution of magnetic field and magnetic leakage. The dependence of axial and radial magnetic force and stiffness on the axial displacement, radial displacement and control current is calculated and investigated for different configurations. To validate the calculated results, the axial and radial magnetic forces for each configuration are simulated by finite element method. A good agreement between the calculated and simulated results validated the proposed magnetic circuit models.

Keywords

Hybrid magnetic bearing permanent magnet position load capability equivalent magnetic circuit finite element method

1. Introduction

Magnetic bearing (MB) is a mechatronic device that uses the electromagnetic force to support a rotating shaft without mechanical friction. MB has many advantages against the conventional mechanical bearing, such as no material wear, no wear particles, no lubrication requirements and less localized heat generation [1,2]. Rotors supported by magnetic bearings can not only rotate at high speed, but can also be used in vacuum. Therefore MB is capable for many special machines like turbo compressor, vacuum technology, energy storage flywheels, blood pump and so on [3–5].

In order to realize the noncontact magnetic levitation, the rotor must be levitated in five degree of freedom (DOF), including three translational movements and two tilting movements. The traditional 5-DOF active controlled magnetic levitation system (MLS) is achieved by two radial MBs and one axial thrust MB [1]. The rotor is actively stabilized in radial and tilting direction by two radial MBs and in axial direction by axial thrust MB. However, these MBs occupy more space of MLS and may cause more power consumption. In order to simplify the MLS structure and reduce the power consumption, the number of active controlled DOFs is necessary to be reduced in MLS designing. Magnetic levitation systems with less than 5 active controlled DOFs have been proposed and achieved [6–8]. Furthermore, one-DOF controlled magnetic bearings attract more attention, which minimizes the number of actively controlled DOFs to one [9,10]. A full MLS supported by two axial HMBs is developed and used for maglev blood pump [11,12]. Asama J. et al. proposed a novel one-DOF controlled bearingless motor system which combines a one-DOF MLS and a three phase motor [13]. Li et al. proposed a simple structure for a micro MB that can be stabilized using only one-axis control and have realized a micro MB for an 8-mm-diameter rotor [14]. Kuroki J. et al. proposed a simple structure of one DOF controlled MLS and the rotor could be levitated without mechanical contact and rotated at 39,000 rpm [11]. In one-DOF controlled MLS, only axial MBs are located at both ends of the rotor and provide both axial and radial levitation forces. Axial motion is actively controlled by the axial component of magnetic force. The radial and tilting motions are passively stabilized by the radial component of magnetic force. It is necessary to study the magnetic force and stiffness of axial MBs.

An additional motor element has to be installed in the drive system to rotate the shaft. This leads to a rather long axial length of the rotor shaft and decrease the critical speed of the rotor, when the rotor shape is cylindrical [15–17]. A bearingless motor concept has been proposed, which can generate both suspension force and torque in a single unit. To generate the suspension force, additional windings are generally wound in the stator. The bearingless motor combines a motor with magnetic bearings together, thus it possesses the advantages of compactness, a simple structure, and cost reduction in comparison to the conventional magnetic bearing drive.

MBs can be classified into three different types: passive magnetic bearing (PMB), active magnetic bearing (AMB) and hybrid magnetic bearing (HMB). PMBs achieve magnetic levitation by permanent magnetic attractive or repulsive forces without power loss [10]. AMBs have better control ability and high stiffness. Nonetheless, AMBs have high power consumption due to the bias current. In HMBs permanent magnet (PM) provides the bias magnetic flux substituted that from the bias current, which reduces the control current and energy loss. Therefore, HMB is widely used for one DOF actively controlled MLS [13,14]. In all these HMBs, the PM is located in the stator or the rotor to provide bias magnetic flux. The PM bias magnetic flux and the control magnetic flux across the gap and generate the magnetic force between the stator and the rotor. The position of PM influences the operating point of PM and the distribution of magnetic field, which will affect the axial and radial load capability of HMB. During HMB designing, the cross-sectional area and length of PM is commonly discussed and optimized [18,19]. However, the effect of permanent magnet position on axial and radial load capability of HMB is less discussed in literatures.

In this paper, the effect of PM positions on the axial and radial magnetic force in axial HMB was studied. The model of axial HMBs with different PM position is created and analyzed by using the equivalent magnetic circuit method. The magnetic circuits of PM bias magnetic flux and control magnetic flux are analyzed for each PM position based on the calculation of magnetic flux leakage and operating point of PM. Axial and radial magnetic forces of HMB are calculated by virtual displacement method. The dependence of magnetic force and stiffness on radial and axial displacement and control current is discussed for different PM positions. Finite element method (FEM) simulations are also carried out to verify the effectiveness of the proposed models and calculations.

2. Configuration and principle of axial HMB

Figure 1 shows the schematic geometry of traditional axial HMB, which includes a rotor and a stator. The stator consists of stator iron core, PM and control coils. The PM is magnetized along the axial direction and produces the PM bias magnetic flux. The control coils is wound inside the stator iron core and produces the control magnetic flux. The rotor is made of soft magnetic iron and has the same magnetic pole shape to the stator. There is air gap between the stator and rotor in order to ensure the rotor can suspend and rotate freely. The PM bias flux and control magnetic flux share the same paths and go through PM, stator iron core and rotor iron core. The magnetic force between the rotor and stator is generated by the PM bias flux and the control flux. The magnetic force can be adjusted by changing the control current and balance the applied load and disturbance force.

Fig. 1.

(a) Configurations of axial HMB and (b) magnetic flux path of axial HMB.

In the magnetic levitation system with less than five controlled DOFs, the rotor is suspended in the axial direction and the radial direction. The axial HMBs locate at each end of the rotor and provide both axial and radial levitation forces. During the working process, the magnetic force is dependent on rotor displacement and control current. The rotor displacement includes the axial and radial displacement. The relationships between the magnetic force and the rotor displacement and control current are nonlinear. After linearizing over small current and displacement, the magnetic force can be expressed as a function of rotor displacement and control current. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F_{z}=K_{iz}i+K_{z}z\end{eqnarray}$ (1) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle F_{r}=K_{ir}i+K_{r}{\Delta}r.\end{eqnarray}$ (2) Where K _z and K _r are the axial and radial stiffness, respectively. K _iz and K _ir are the axial and radial current stiffness, respectively. i is the current of HMB. z and Δr are the axial and radial displacement, respectively. In this paper, influence of PM position on the axial and radial magnetic force and stiffness will be investigated.

3. Magnetic circuit analysis of axial HMB

In order to study the effect of PM position on the magnetic force characteristics, the HMBs with six different PM positions are analyzed, as listed in Table 1. The volume of PM for each configuration was designed to be the same. Since the cross-sectional areas of inner pole and outer pole are equal, the axial length of PM in magnetization direction l _m is the same in different configurations.

In order to simplify the analysis of magnetic circuit of axial HMB, some assumptions were made as follows:

1. The permeability of iron core of the stator and rotor is infinite.

2. The finite coercivity of iron core is ignored.

3. In the calculation of the magnetic permeance of air gap, the magnetic field in gap is approximately divided into magnetic flux tubes with different shapes.

3.1. Equivalent magnetic circuit of PM bias magnetic flux

PM bias magnetic flux distribution in axial HMBs for each configuration is shown in Fig. 2. The PM is placed in the inner pole for configuration (I, III, V) and in the outer pole for configuration (II, IV, VI). For configuration I and II, the PM is place away from the gap. For configuration III and IV, the PM is place on one side of the gap. For configuration V and VI, the PM is placed on the both side of the gap.

Table 1
Configurations for different PM position in axial HMBs

Inner pole Outer pole

Non-connected to gap I II

On one side of gap III IV

On both side of gap V VI

	Inner pole	Outer pole
Non-connected to gap	I	II
On one side of gap	III	IV
On both side of gap	V	VI

Fig. 2.

Bias flux paths of axial HMB for six configurations with different PM position.

Since the magnetic leakage distribution is different for different configuration. The leakage paths for PM bias flux are classified into three categories:

1. The magnetic leakages between the inner poles of rotor and stator, including $P_{1}^{\text{I}}$ , $P_{1}^{\text{II}}$ , $P_{1}^{\text{III}}$ , $P_{1}^{\text{IV}}$ , $P_{1}^{\text{V}}$ and $P_{1}^{\text{VI}}$ ;

2. The magnetic leakages between the outer poles of rotor and stator, including $P_{2}^{\text{I}}$ , $P_{2}^{\text{II}}$ , $P_{2}^{\text{III}}$ , $P_{2}^{\text{IV}}$ , $P_{2}^{\text{V}}$ , $P_{2}^{\text{VI}}$ , $P_{3}^{\text{I}}$ , $P_{3}^{\text{II}}$ , $P_{3}^{\text{III}}$ , $P_{3}^{\text{IV}}$ , $P_{3}^{\text{V}}$ and $P_{3}^{\text{VI}}$ ;

3. The magnetic leakages from PM to itself, including $P_{4}^{\text{I}}$ , $P_{4}^{\text{II}}$ , $P_{4}^{\text{III}}$ , $P_{4}^{\text{IV}}$ , $P_{4}^{\text{V}}$ , $P_{4}^{\text{VI}}$ .

Because the PM is divided into two parts in configuration V and VI, the leakage paths from PM to itself $P_{4}^{\text{V}}$ and $P_{4}^{\text{VI}}$ include two parts.

Moreover, magnetic leakages from the outer surface of inner poles to the inner surface of outer poles, $P_{5}^{\text{I}}$ and $P_{5}^{\text{II}}$ will be considered in configuration I and II in which the PM is not close to the air gap.

Fig. 3.

Equivalent magnetic circuit of the bias flux in axial HMB with different PM positions.

According to the series-parallel connection of each magnetic reluctance, the equivalent magnetic circuits for axial HMBs with different PM position are shown in Fig. 3. F _PM is the magnetomotive force (MMF) of PM and given by $\begin{eqnarray}\displaystyle F_{pm}=H_{m}l_{m} & & \displaystyle\end{eqnarray}$ (3) Where H _m is the magnetic field strength at operating point for PM, and lm is the axial length of PM.

The magnetic permeances for each PM bias magnetic flux leakage are calculated as follows [20]. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{1}^{\text{I}}=P_{1}^{\text{II}}=P_{1}^{\text{IV}}=P_{1}^{\text{VI}}=\frac{0.5{\mu}_{0}(R_{2}-R_{1})R_{1}}{0.22g+0.2(R_{2}-R_{1})}\end{eqnarray}$ (4) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{1}^{\text{III}}=P_{1}^{\text{V}}=\frac{0.5{\mu}_{0}(R_{2}-R_{1})R_{1}}{0.22(g+l_{m})+0.2(R_{2}-R_{1})}\end{eqnarray}$ (5) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{2}^{\text{I}}=P_{2}^{\text{II}}=P_{2}^{\text{III}}=P_{2}^{\text{V}}=\frac{{\mu}_{0}l_{m}R_{2}}{0.22g+0.4l_{m}}\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{2}^{\text{IV}}=P_{2}^{\text{VI}}={\displaystyle \frac{{\mu}_{0}l_{m}R_{2}}{0.22(g+l_{m})+0.4l_{m}}}\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{3}^{\text{I}}=P_{3}^{\text{III}}=P_{3}^{\text{V}}=\frac{{\mu}_{0}R_{3}^{2}}{0.22g+0.4l_{m}}\end{eqnarray}$ (8) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{3}^{\text{IV}}=P_{3}^{\text{VI}}={\displaystyle \frac{{\mu}_{0}R_{3}^{2}}{0.22(g+l_{m})+0.4R_{3}}}\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{3}^{\text{II}}={\displaystyle \frac{{\mu}_{0}R_{3}^{2}}{0.22(g+2l_{m})+0.4R_{3}}}\end{eqnarray}$ (10) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{4}^{\text{I}}=P_{4}^{\text{III}}=P_{4}^{\text{V}}=0.6{\mu}_{0}{\pi}R_{1}\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{4}^{\text{II}}=P_{4}^{\text{IV}}=P_{4}^{\text{VI}}=0.42{\mu}_{0}{\pi}R_{2}+0.6{\mu}_{0}{\pi}R_{3}\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{5}^{\text{I}}=P_{5}^{\text{II}}={\displaystyle \frac{2{\pi}{\mu}_{0}l_{m}}{\ln \left({\displaystyle \frac{R_{2}}{R_{1}}}\right)}}\end{eqnarray}$ (13) Where R ₁ is diameter of inner pole, R ₂ and R ₃ are internal and external diameter of outer pole, g is the axial length of air gap and Δr is the radial displacement.

Fig. 4.

Demagnetization curve for PM.

The magnetic permeance of inner gap P _gi and that of outer gap P _go are dependent on the structure parameters, radial and axial displacement. P _gi and P _go can be written as [18,19] $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{gi}={\mu}_{0}\left(\frac{{\pi}R_{1}^{2}-4R_{1}{\Delta}r}{g}+{\displaystyle \frac{8R_{1}{\Delta}r}{e+2g}}\right)\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{go}={\mu}_{0}(R_{3}+R_{2})\left({\displaystyle \frac{{\pi}(R_{3}-R_{2})-4{\Delta}r}{g}}+{\displaystyle \frac{8{\Delta}r}{{\Delta}r+2g}}\right).\end{eqnarray}$ (15)

P _gi and P _go are the same for each configuration.

The total magnetic permeance for each configuration is calculated according to equivalent magnetic circuit models. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{Z}^{\text{I}}=P_{4}^{\text{I}}+P_{5}^{\text{I}}+{\displaystyle \frac{1}{{\displaystyle \frac{1}{P_{gi}+P_{1}^{\text{I}}}}+{\displaystyle \frac{1}{P_{go}+P_{2}^{\text{I}}+P_{3}^{\text{I}}}}}}\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{Z}^{\text{II}}=P_{4}^{\text{II}}+P_{5}^{\text{II}}+{\displaystyle \frac{1}{{\displaystyle \frac{1}{P_{gi}+P_{1}^{\text{II}}+P_{1}^{\text{III}}}}+{\displaystyle \frac{1}{P_{go}+P_{2}^{\text{II}}}}}}\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{Z}^{\text{III}}=P_{4}^{\text{III}}+{\displaystyle \frac{1}{{\displaystyle \frac{1}{P_{gi}}}+{\displaystyle \frac{1}{P_{go}+P_{1}^{\text{III}}+P_{2}^{\text{III}}+P_{3}^{\text{III}}}}}}\end{eqnarray}$ (18) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{Z}^{\text{IV}}=P_{4}^{\text{IV}}+{\displaystyle \frac{1}{{\displaystyle \frac{1}{P_{gi}+P_{1}^{\text{IV}}+P_{2}^{\text{IV}}+P_{3}^{\text{IV}}}}+{\displaystyle \frac{1}{P_{go}}}}}\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{Z}^{\text{V}}=2P_{4}^{\text{V}}+P_{3}^{\text{V}}+{\displaystyle \frac{1}{{\displaystyle \frac{1}{P_{gi}}}+{\displaystyle \frac{1}{P_{go}+P_{1}^{\text{V}}+P_{2}^{\text{V}}+P_{3}^{\text{V}}}}}}\end{eqnarray}$ (20) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle P_{Z}^{\text{VI}}=2P_{4}^{\text{VI}}+P_{3}^{\text{VI}}+{\displaystyle \frac{1}{{\displaystyle \frac{1}{P_{gi}+P_{1}^{\text{VI}}+P_{2}^{\text{VI}}+P_{3}^{\text{VI}}}}+{\displaystyle \frac{1}{P_{go}}}}}.\end{eqnarray}$ (21)

3.2. Calculation of PM operating point

The magnetomotive force of PM F _pm can be obtained by calculating the operating point of PM. Figure 4 shows the demagnetization curve of the NdFeB rare-earth PM which is the hysteresis loop in the second quadrant and represents the region of interest in the operation of PM. The sign of magnetic field intensity H is reversed. For NdFeB rare-earth PM, the demagnetization curve magnet is nearly a straight line. The operating point of PM can be determined by the intersection between the demagnetization curve and the load line. The slope of load line tanθ is referred to as the permeance coefficient. $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{m}=B_{r}+{\displaystyle \frac{B_{r}}{H_{c}}}H_{m}\end{eqnarray}$ (22) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \tan {\theta}={\displaystyle \frac{B_{m}}{H_{m}}}={\displaystyle \frac{B_{m}S}{H_{m}S}}={\displaystyle \frac{{\phi}_{m}}{H_{m}S}}={\displaystyle \frac{H_{m}l_{m}P_{z}}{H_{m}S}}={\displaystyle \frac{l_{m}P_{z}}{S}}\end{eqnarray}$ (23) Substituting (23) into (22), the operating point (H _m, B _m) of PM can be expressed as $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B_{m}={\displaystyle \frac{B_{r}\tan {\theta}}{\tan {\theta}-{\displaystyle \frac{B_{r}}{H_{c}}}}}={\displaystyle \frac{B_{r}{\displaystyle \frac{l_{m}P_{z}}{S}}}{{\displaystyle \frac{l_{m}P_{z}}{S}}-{\displaystyle \frac{B_{r}}{H_{c}}}}}\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H_{m}={\displaystyle \frac{B_{r}}{\tan {\theta}-{\displaystyle \frac{B_{r}}{H_{c}}}}}={\displaystyle \frac{B_{r}{\displaystyle \frac{l_{m}P_{z}}{S}}}{{\displaystyle \frac{l_{m}P_{z}}{S}}-{\displaystyle \frac{B_{r}}{H_{c}}}}}\end{eqnarray}$ (25) Where P _z is the total external PM circuit permeance which was calculated for each configuration in (16)–(21).

Fig. 5.

Magnetic control flux of axial HMBs with different PM positions. (a) for configuration I, III, V; (b) for configuration II, IV, VI.

3.3. Equivalent magnetic circuit of magnetic control flux

When the HMB works, the control current is applied to generate the control magnetic flux and magnetic force. The magnetic control flux distribution and leakages for each configuration are shown in Fig. 5. Figure 5(a) shows the magnetic flux distribution for configurations I, III, IV, in which PM is in the inner pole. Figure 5(b) shows the magnetic flux distribution for configurations II, IV, VI, in which PM is in the outer pole. The equivalent magnetic circuits for different configurations are shown in Fig. 6.

In the equivalent magnetic circuits, the magnetic control flux shares the same path with the PM bias magnetic flux. The PM is considered as the magnetic reluctance in the equivalent magnetic circuits of magnetic control flux. The magnetic permeance of PM is expressed as $\begin{eqnarray}\displaystyle P_{m}^{e}={\displaystyle \frac{{\mu}_{0}{\mu}_{r}S}{l_{m}}} & & \displaystyle\end{eqnarray}$ (26) where μ_r is the relative permeability of PM. The magnetic permeance of other parts are same to that in the equivalent magnetic circuits of PM bias flux.

3.4. Calculation of axial and radial magnetic force and stiffness

The MMF at air gap is provided by the PM bias flux and magnetic control flux. The MMF at inner air gap and outer air gap can be calculated by: $\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}F_{mi}\\ F_{mo}\end{array}\right]=H_{m}l_{m}\left[\begin{array}{@{}c@{}}k_{i}^{j}\\ k_{o}^{j}\end{array}\right]+NI\left[\begin{array}{@{}c@{}}k_{ei}^{j}\\ k_{eo}^{j}\end{array}\right] & & \displaystyle\end{eqnarray}$ (27) where $k_{i}^{j}$ , $k_{o}^{j}$ are the bias flux MMF coefficient, $k_{ei}^{j}$ , $k_{eo}^{j}$ are the control flux MMF coefficient, which are given by $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left[\begin{array}{@{}cc@{}}k_{i}^{\text{I}} & k_{o}^{\text{I}}\\[6.0pt] k_{i}^{\text{II}} & k_{o}^{\text{II}}\\[6.0pt] k_{i}^{\text{III}} & k_{o}^{\text{III}}\\[6.0pt] k_{i}^{\text{IV}} & k_{o}^{\text{IV}}\\[6.0pt] k_{i}^{\text{V}} & k_{o}^{\text{V}}\\[6.0pt] k_{i}^{\text{VI}} & k_{o}^{\text{VI}}\end{array}\right]=\left[\begin{array}{@{}cc@{}}\displaystyle \frac{P_{go}+P_{2}^{\text{I}}+P_{3}^{\text{I}}}{P_{gi}+P_{go}+P_{1}^{\text{I}}+P_{2}^{\text{I}}+P_{3}^{\text{I}}} & \displaystyle {\displaystyle \frac{P_{gi}+P_{1}^{\text{I}}}{P_{gi}+P_{go}+P_{1}^{\text{I}}+P_{2}^{\text{I}}+P_{3}^{\text{I}}}}\\[12.0pt] \displaystyle {\displaystyle \frac{P_{go}+P_{2}^{\text{II}}}{P_{gi}+P_{go}+P_{1}^{\text{II}}+P_{2}^{\text{II}}+P_{3}^{\text{II}}}} & \displaystyle {\displaystyle \frac{P_{gi}+P_{1}^{\text{II}}+P_{3}^{\text{II}}}{P_{gi}+P_{go}+P_{1}^{\text{II}}+P_{2}^{\text{II}}+P_{3}^{\text{II}}}}\\[15.60004pt] \displaystyle {\displaystyle \frac{P_{go}+P_{1}^{\text{III}}+P_{2}^{\text{III}}+P_{3}^{\text{III}}}{P_{gi}+P_{go}+P_{1}^{\text{III}}+P_{2}^{\text{III}}+P_{3}^{\text{III}}}} & \displaystyle {\displaystyle \frac{P_{gi}}{P_{gi}+P_{go}+P_{1}^{\text{III}}+P_{2}^{\text{III}}+P_{3}^{\text{III}}}}\\[15.60004pt] \displaystyle \frac{P_{go}}{P_{gi}+P_{go}+P_{1}^{\text{IV}}+P_{2}^{\text{IV}}+P_{3}^{\text{IV}}} & \displaystyle {\displaystyle \frac{P_{gi}+P_{1}^{\text{IV}}+P_{2}^{\text{IV}}+P_{3}^{\text{IV}}}{P_{gi}+P_{go}+P_{1}^{\text{IV}}+P_{2}^{\text{IV}}+P_{3}^{\text{IV}}}}\\[15.60004pt] \displaystyle \frac{P_{go}+P_{1}^{\text{V}}+P_{2}^{\text{V}}+P_{3}^{\text{V}}}{P_{gi}+P_{go}+P_{1}^{\text{V}}+P_{2}^{\text{V}}+P_{3}^{\text{V}}} & \displaystyle \frac{P_{gi}}{P_{gi}+P_{go}+P_{1}^{\text{V}}+P_{2}^{\text{V}}+P_{3}^{\text{V}}}\\[15.60004pt] \displaystyle \frac{P_{go}}{P_{gi}+P_{go}+P_{1}^{\text{VI}}+P_{2}^{\text{VI}}+P_{3}^{\text{VI}}} & \displaystyle \frac{P_{gi}+P_{1}^{\text{VI}}+P_{2}^{\text{VI}}+P_{3}^{\text{VI}}}{P_{gi}+P_{go}+P_{1}^{\text{VI}}+P_{2}^{\text{VI}}+P_{3}^{\text{VI}}}\end{array}\right]\end{eqnarray}$ (28) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \left[\begin{array}{@{}cc@{}}k_{ei}^{\text{I}} & k_{eo}^{\text{I}}\\[6.0pt] k_{ei}^{\text{II}} & k_{eo}^{\text{II}}\\[6.0pt] k_{ei}^{\text{III}} & k_{eo}^{\text{III}}\\[6.0pt] k_{ei}^{\text{IV}} & k_{eo}^{\text{IV}}\\[6.0pt] k_{ei}^{\text{V}} & k_{eo}^{\text{V}}\\[6.0pt] k_{ei}^{\text{VI}} & k_{eo}^{\text{VI}}\end{array}\right]=\left[\begin{array}{@{}cc@{}}\displaystyle \frac{(P_{go}+P_{3}^{\text{IV}}){\displaystyle \frac{P_{m}}{P_{gi}+P_{m}}}}{\left(\frac{P_{gi}P_{m}}{P_{gi}+P_{m}}+P_{1}^{\text{III}}\right)+P_{go}+P_{3}^{\text{IV}}} & \displaystyle \frac{\frac{P_{gi}P_{m}}{P_{gi}+P_{m}}+P_{1}^{\text{III}}}{\left({\displaystyle \frac{P_{gi}P_{m}}{P_{gi}+P_{m}}}+P_{1}^{\text{III}}\right)+P_{go}+P_{3}^{\text{IV}}}\\[19.20007pt] \displaystyle \frac{{\displaystyle \frac{P_{go}P_{m}}{P_{go}+P_{m}}}+P_{3}^{\text{I}}}{P_{gi}+P_{1}^{\text{I}}+\left({\displaystyle \frac{P_{go}P_{m}}{P_{go}+P_{m}}}+P_{3}^{\text{I}}\right)} & \displaystyle \frac{(P_{gi}+P_{1}^{\text{I}})\frac{P_{m}}{P_{go}+P_{m}}}{P_{gi}+P_{1}^{\text{I}}+\left(\frac{P_{go}P_{m}}{P_{go}+P_{m}}+P_{3}^{\text{I}}\right)}\\[15.60004pt] \displaystyle \frac{(P_{go}+P_{3}^{\text{IV}}){\displaystyle \frac{P_{m}}{P_{gi}+P_{m}}}}{\left({\displaystyle \frac{P_{gi}P_{m}}{P_{gi}+P_{m}}}+P_{1}^{\text{III}}\right)+P_{go}+P_{3}^{\text{IV}}} & \displaystyle \frac{{\displaystyle \frac{P_{gi}P_{m}}{P_{gi}+P_{m}}}+P_{1}^{\text{III}}}{\left({\displaystyle \frac{P_{gi}P_{m}}{P_{gi}+P_{m}}}+P_{1}^{\text{III}}\right)+P_{go}+P_{3}^{\text{IV}}}\\[19.20007pt] \displaystyle \frac{{\displaystyle \frac{P_{go}P_{m}}{P_{go}+P_{m}}}+P_{3}^{\text{I}}}{P_{gi}+P_{1}^{\text{I}}+\left({\displaystyle \frac{P_{go}P_{m}}{P_{go}+P_{m}}}+P_{3}^{\text{I}}\right)} & \displaystyle \frac{(P_{gi}+P_{1}^{\text{I}}){\displaystyle \frac{P_{m}}{P_{go}+P_{m}}}}{P_{gi}+P_{1}^{\text{I}}+\left({\displaystyle \frac{P_{go}P_{m}}{P_{go}+P_{m}}}+P_{3}^{\text{I}}\right)}\\[19.20007pt] \displaystyle \frac{(P_{go}+P_{3}^{\text{IV}}){\displaystyle \frac{P_{m}}{P_{gi}+P_{m}}}}{\left({\displaystyle \frac{P_{gi}P_{m}}{P_{gi}+P_{m}}}+P_{1}^{\text{III}}\right)+P_{go}+P_{3}^{\text{IV}}} & \displaystyle \frac{{\displaystyle \frac{P_{gi}P_{m}}{P_{gi}+P_{m}}}+P_{1}^{\text{III}}}{\left(\frac{P_{gi}P_{m}}{P_{gi}+P_{m}}+P_{1}^{\text{III}}\right)+P_{go}+P_{3}^{\text{IV}}}\\[19.20007pt] \displaystyle \frac{{\displaystyle \frac{P_{go}P_{m}}{P_{go}+P_{m}}}+P_{3}^{\text{I}}}{P_{gi}+P_{1}^{\text{I}}+\left({\displaystyle \frac{P_{go}P_{m}}{P_{go}+P_{m}}}+P_{3}^{\text{I}}\right)} & \displaystyle \frac{\left(P_{gi}+P_{1}^{\text{I}}\right){\displaystyle \frac{P_{m}}{P_{go}+P_{m}}}}{P_{gi}+P_{1}^{\text{I}}+\left({\displaystyle \frac{P_{go}P_{m}}{P_{go}+P_{m}}}+P_{3}^{\text{I}}\right)}\end{array}\right]\end{eqnarray}$ (29) By using method of virtual displacement, the expression of axial and radial magnetic force of HMB can be calculated as $\begin{eqnarray}\displaystyle F_{z}\text{} & =\text{} & \displaystyle -{\displaystyle \frac{\partial W_{g1}}{\partial g}}-{\displaystyle \frac{\partial W_{g2}}{\partial g}}\approx -{\displaystyle \frac{1}{2}}F_{mi}^{2}{\displaystyle \frac{\partial G_{i}}{\partial g}}-{\displaystyle \frac{1}{2}}F_{mo}^{2}{\displaystyle \frac{\partial G_{o}}{\partial g}}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle {\displaystyle \frac{-F_{mi}^{2}{\mu}_{0}R_{1}}{2}}\left[{\displaystyle \frac{{\pi}R_{1}-4e}{g^{2}}}+{\displaystyle \frac{16e}{(e+2g)^{2}}}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -{\displaystyle \frac{F_{mo}^{2}{\mu}_{0}(R_{2}+R_{3})}{2}}\left[{\displaystyle \frac{{\pi}(R_{3}-R_{2})-4e}{g^{2}}}+{\displaystyle \frac{16e}{(e+2g)^{2}}}\right]\end{eqnarray}$ (30) $\begin{eqnarray}\displaystyle F_{r}\text{} & =\text{} & \displaystyle -\frac{\partial W_{g1}}{\partial g}-{\displaystyle \frac{\partial W_{g2}}{\partial g}}\approx -{\displaystyle \frac{1}{2}}F_{mi}^{2}{\displaystyle \frac{\partial G_{i}}{\partial e}}-{\displaystyle \frac{1}{2}}F_{mo}^{2}{\displaystyle \frac{\partial G_{o}}{\partial e}}\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle -\left[{\displaystyle \frac{F_{mi}^{2}{\mu}_{0}R_{1}}{2}}+{\displaystyle \frac{F_{mo}^{2}{\mu}_{0}(R_{2}+R_{3})}{2}}\right]\left[{\displaystyle \frac{4}{g}}-{\displaystyle \frac{16e}{(e+2g)^{2}}}\right].\end{eqnarray}$ (31)

Where W _g1 and W _g2 are the magnetic energy of inner and outer air gap. The axial and radial stiffness can be calculated by derivative of the suspension force with respect to displacement: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{z}={\displaystyle \frac{\partial F_{z}}{\partial g}}\end{eqnarray}$ (32) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle K_{r}={\displaystyle \frac{\partial F_{r}}{\partial e}}.\end{eqnarray}$ (33)

4. The effect of PM position on load capacity

In order to study the effect of PM positions on load capacity of HMB, the models of axial HMB with different configurations are investigated. The parameters of axial HMBs are summarized in Table 2.

Fig. 6.

Equivalent magnetic circuit of the bias flux in axial HMB for different configuration. (a) for configuration I, III, V; (b) for configuration II, IV, VI.

Table 2

Structural parameters of axial HMB

Parameter	Value	Parameter	Value
R ₁ (mm)	8	h _so (mm)	50
R ₂ (mm)	15	h _si (mm)	35
R ₃ (mm)	17	l _m of PM (mm)	5
h _ro (mm)	25	Remanence (T)	1.3
h _ri (mm)	15	Coercivity (kA/m)	990

4.1. Axial bias magnetic force and stiffness

According to (27)–(31), the relationship between the axial bias magnetic force and axial displacement for different PM positions are calculated and shown in Fig. 7. It can be seen that the axial magnetic force decreases with increasing axial displacement for all configurations, which is well-known in traditional axial HMBs. By comparing the calculation results, the six different configurations in decreasing order of the axial bias magnetic force are given as: V, VI, III, IV, I and II. The axial bias magnetic force is maximum when PM is on both sides of inner gap (V). When PM is on the same axial position, F _z for PM in the inner pole (I, III, V) is larger than F _z for PM in the outer pole (II, IV, VI), respectively. The reason is that the geometric shape of PM in the inner pole is cylinder and magnetic leakage from PM to itself occurs on the outer surface. And the PM in the outer pole is ring cylinder and the magnetic leakage from PM to itself occurs both on the outer surface and on the inner surface. The magnetic flux density in the gap is larger when PM is in the inner pole for configuration I, III, V. When PM is on the same radial positions, it is found that F _z decreases in the order: PM on both side of gap (V, VI), PM on one side of gap (III, IV), PM non-connected to gap (I, II). The reason is that the magnetic flux density will be enhanced when PM on the both side of gap. The axial stiffness of HMBs with different PM positions is shown in Fig. 9(b). The axial stiffness is negative and the value decreases with increasing the axial displacement. Therefore the rotor cannot be stabilized in axial direction only by the PM bias magnetic force. The active control is necessary for axial suspension.

Fig. 7.

Axial magnetic force and stiffness versus axial displacement.

Fig. 8.

Radial magnetic force and stiffness versus radial displacement.

4.2. Radial bias magnetic force and stiffness

Figure 8(a) shows the relationships between radial bias magnetic force and radial displacement for different PM positions. The radial magnetic force increases with increasing radial displacement for all configurations. When PM is positioned on the same axial position, F _r for PM in the outer pole (II, IV, VI) is larger than F _r for PM in the inner pole (I, III, V), respectively. This is opposite to the situation of axial bias magnetic force. The radial bias magnetic force can be enhanced by placing PM in the outer pole. When PM is on the same radial position, the radial bias magnetic force decreases in the order of on both side of gap (V, VI), on one side of gap (III, IV), non-connected to gap (I, II), which is similar to the situation of axial bias magnetic force.

Fig. 9.

Axial magnetic force and stiffness versus ampere turns in axial HMB with different PM positions.

Figure 8(b) shows the dependence of radial bias stiffness K _r on radial displacement Δr. The K _r is positive for all configurations, which means the radial bias magnetic force can be used for the rotor stabilization in radial direction. K _r decreases with increasing radial displacement in the considering region 0 < Δr < 0. 5 mm. When PM is on the same radial position, K _r for PM on both side of both side of gap is larger than Kr for PM on one side of gap and non-connected to gap. This means that K _r is enhanced when PM is adjoining to air gap. When PM is on the same radial position, K _r can be enhanced for PM in the outer pole.

Fig. 10.

Radial magnetic force and stiffness versus ampere turns in axial HMB with different PM positions.

4.3. Axial magnetic force and stiffness versus ampere-turns

Figure 9(a) shows the relationship between the axial magnetic force and ampere turns of control coils when g = 0. 5 mm and Δr = 0. When NI is positive, the control magnetic flux and PM bias magnetic flux are in the same direction and PM is in magnetizing state. Then F _z increases with increasing NI. When NI is negative, the control magnetic flux and PM bias magnetic flux is in the opposite direction and PM is in demagnetizing state. Then F _z decreases with increasing NI in the negative region. The force-current factor is nearly linear for configuration I, III, V, in which PM is in the inner pole. And the force-current factor become more nonlinear for configuration II, IV and VI, in which PM is in the outer pole. It is because that the operating point of PM is lower and the leakage from PM to itself is larger when PM is in the outer pole.

Fig. 11.

Calculated and simulated axial magnetic force variation with axial displacement.

Fig. 12.

Calculated and simulated radial magnetic force variation with radial displacement.

Fig. 13.

Calculated and simulated results of axial-force-current relationships.

Figure 9(b) shows the axial stiffness K _z for each configuration with NI = 0, ±1000 A. It can be seen that the value of K _z is enhanced for positive NI and is reduced for negative NI. When NI is positive (NI = 1000 A), the maximum K _z is −1680 N/mm for configuration II and the minimum K _z is −710 N/mm for configuration V. And K _z for PM in the outer pole (configuration II, IV, VI) is larger than K _z for PM in the inner pole (configuration I, III, V). When NI is negative (NI = 1000 A), the maximum K _z is −627 N/mm for configuration I and minimum K _z is −332 N/mm for configuration VI. And K _z for PM in the outer pole is larger than K _z for PM in the inner pole. The effect of PM position on the axial magnetic force and stiffness become more obvious with increasing positive NI in axial HMBs.

4.4. Radial magnetic force and stiffness versus ampere-turn

Figure 10(a) shows the relationship between the radial magnetic force F _r and ampere turns NI of the control coils when g = 0. 5 mm and Δr = 0. 25 mm. The dependence of F _r on NI for each configuration is similar. And F _r is enhanced by the positive NI and reduced by the negative NI, which is similar to the situation of F _z. Figure 10(b) shows the radial stiffness for different PM positions and NI. It can be seen that K _r is also enhanced by the positive NI and reduced by the negative NI, especially for configuration V and VI. In configuration VI, K _r can increase over four times by changing NI from −1000 A to 1000 A.

5. Finite Element analysis of axial HMBs

In order to verify the proposed model, FEM analysis is carried out. Ansoft Maxwell is used for 3-D FEM analysis of the axial HMB. By using the structure parameters in Table 2, six 3-D FEM models of axial HMBs with different PM positions are created in the software. The total number of mesh is 142610 for each model. The axial and radial magnetic force and stiffness are calculated and obtained for each configuration.

Figure 11 shows the axial magnetic force variation with axial displacement for different PM positions. Figure 12 shows the radial magnetic force variation with radial displacement for different PM positions. Figure 13 shows the axial magnetic force variation with NI. The calculated and simulated results are shown for comparison. There is a good agreement between the calculated and simulated results, which validates the proposed model.

6. Conclusion

In this paper, the effect of permanent magnet position on the axial and radial magnetic force properties of axial HMB is studied. Based on the spatial distribution of magnetic field and magnetic flux leakage, equivalent magnetic circuit model was proposed for axial HMBs with different PM positions. By using magnetic field division method, the magnetic permeance for each magnetic flux tube is calculated. The operating point of PM is obtained by the calculation of magnetic circuit. The dependence of the axial and radial magnetic force on axial and radial displacement and control current is calculated and analyzed. Some conclusions are as follows:

(1) In the proposed model, the detailed magnetic leakages of axial HMB is considered and the magnetic permeances for each part are calculated analytically, including the magnetic leakages between the inner poles of rotor and stator, the magnetic leakages between the outer poles of rotor and stator and the magnetic leakages from PM to itself. The dependence of axial and radial magnetic force and stiffness on the axial and radial displacement is obtained, which can be widely used for HMB design and analysis.

(2) The effect of PM position on axial and radial fore and stiffness is studied thoroughly for axial HMB. Six different configurations with different PM position are considered and the magnetic circuit model for each configuration is analyzed. The effect of PM position on load capability of HMB is less discussed in the literatures.

(3) In the configurations of axial HMB with different PM position, axial bias magnetic force is enhanced when PM is on both sides of air gap and in the inner pole. Axial stiffness does not change significantly for different PM positions in particular when axial displacement is large. It is validated by the calculation and simulation results, which can be applied in design of axial HMB.

(4) Axial magnetic force and stiffness is enhanced by positive NI and reduced by negative NI, which is more significant for PM in the outer pole. Radial magnetic force and stiffness is also enhanced by positive NI and reduced by negative NI, which is more significant for PM on the both side of air gap.

The axial and radial magnetic force is studied for each configuration by finite element method. A good agreement between the calculated and simulated results validated the proposed magnetic circuit models of axial HMBs for different PM positions. These conclusions can be used in the design of axial HMBs.

Footnotes

Acknowledgements

This work was supported by the National Key R&D Program of China [2018YFB200100].

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The effect of permanent magnet position on axial and radial load capability in axial hybrid magnetic bearing

Abstract

Keywords

1. Introduction

2. Configuration and principle of axial HMB

3.1. Equivalent magnetic circuit of PM bias magnetic flux

Table 1 Configurations for different PM position in axial HMBs Inner pole Outer pole Non-connected to gap I II On one side of gap III IV On both side of gap V VI

5. Finite Element analysis of axial HMBs

6. Conclusion

Footnotes

Acknowledgements

References

Table 1
Configurations for different PM position in axial HMBs

Inner pole Outer pole

Non-connected to gap I II

On one side of gap III IV

On both side of gap V VI