Radial magnetic bearings: An overview

Accepted Manuscript Radial Magnetic Bearings: An Overview Weiyu Zhang, Huangqiu Zhu PII: DOI: Reference:

S2211-3797(17)30863-X http://dx.doi.org/10.1016/j.rinp.2017.08.043 RINP 893

To appear in:

Results in Physics

Received Date: Revised Date: Accepted Date:

20 May 2017 24 August 2017 24 August 2017

Please cite this article as: Zhang, W., Zhu, H., Radial Magnetic Bearings: An Overview, Results in Physics (2017), doi: http://dx.doi.org/10.1016/j.rinp.2017.08.043

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1

Radial Magnetic Bearings: An Overview Weiyu Zhang, Huangqiu Zhu School of Electrical and Information Engineering, Jiangsu University, 212013 Zhenjiang, China Abstract—Radial magnetic bearings (RMBs) are one of the most commonly used magnetic bearings. They are used widely in the

field of ultra-high speed and ultra-precise numerical control machine tools, bearingless motors, high speed flywheels, artificial heart pumps, and molecular pumps, and they are being strengthened and extended in various important areas. In this paper, a comprehensive overview is given of different bearing topologies of RMBs with different stator poles that differ in their construction, the driving mode of electromagnets, power consumption, cost, magnetic circuits, and symmetry. RMBs with different poles and couplings between the two bearing axes in the radial direction responsible for cross-coupling generation are compared. In addition, different shaped rotors are compared, as the performances of magnetic bearing-rotor systems are of great concern to rotor constructions. Furthermore, the parameter design methods, the mathematical models and control strategies of the RMBs are described in detail. From the comparison of topologies, models and control methods for RMBs, the advantages, disadvantages and utilizable perspectives are also analyzed. Moreover, several possible development trends of the RMBs are discussed. Index Terms—Radial magnetic bearings (RMBs), Topologies, Mathematical mode, Control strategies, Development trends

I.

INTRODUCTION

A

magnetic bearing is a machine element utilizing a magnetic suspension force to suspend a rotor without any mechanical contact [1]-[5]. Thus far, various magnetic bearings such as axial magnetic bearings (single degree freedom magnetic bearings) [6]-[7], radial magnetic bearings (two degree of freedom magnetic bearings) [8]-[17], radialaxial bearings (three degree of freedom magnetic bearings) [18]-[20], four degree of freedom magnetic bearings[21], and five degree of freedom magnetic bearings have been developed [22]. In these types of magnetic bearings, the radial magnetic bearings (RMBs) have been greeted mostly with enthusiasm, as they are used as the basic components of other degrees of freedom magnetic bearings (except for the single degree freedom magnetic bearings). As new bearing topologies are continually released, RMBs have become increasingly attractive. They have demonstrated potential applications in centrifugal machines, turbo molecular pumps, compressors, ultra-high speed and ultra-precise numerical control machine tools, bearingless motors, high speed flywheels, life sciences, and in many other fields [23]-[24]. II.

BEARING TOPOLOGIES

One key issue of a magnetic bearing-rotor system is to design reasonable topological structures (bearing and rotor), which affect the overall performance of the magnetic bearing–rotor system. A.

4-pole RMBs and 8-pole RMBs Most RMBs currently installed are based on four electromagnets, of which the simplest magnetic bearing is the 4-pole RMB, the structure and its magnetic circuit are shown in Fig. 1. However, the most commonly used RMB in many industrial applications is the heteropolar 8-pole RMB, as shown in Fig. 2. The 8-pole RMBs includes the heteropolar and homopolar 8-pole RMBs. As seen in Fig. 2(a) and (b), the two 8-pole structures will both form four independent

magnetic loops with four independent coils, which results in nearly uncoupled magnetic forces in four directions. Consequently, there are, in general, two bipolar amplifiers or occasionally four unipolar power amplifiers in an 8-pole RMB system. In heteropolar RMBs, all eight poles are distributed in the same plane, the N pole and S pole are arranged mutually crosswise in the circumferential direction. Although the stators and rotors of heteropolar RMBs generally adopt laminated structures, the eddy current and hysteresis loss are still larger, though on the surface the eddy current and hysteresis loss are still relatively large. Meanwhile, because the line of magnetic force of homopolar RMBs is parallel to the axis of rotor, the coupling effect and power loss of homopolar RMBs are smaller than heteropolar RMBs, and the dynamic performance of homopolar RMBs systems is better than heteropolar RMBs systems. For active magnetic bearing (AMB) systems, two or four amplifiers must be kept in normal working condition due to the bias current remaining, and using fewer power amplifiers is possible by adopting a differential winding scheme [25]. In other words, if one coil is designated a special coil for the bias current in all poles and it is independent of the coils for control currents, not all the amplifiers are constantly in supply, and the power consumption and cost can be greatly reduced. However, the disadvantage is that the additional coil for bias current will generate much more heat. Thus, an additional cooling system should be used, and the overall cost of the bearing system will increase consequently. In contrast, for the hybrid magnetic bearing (HMB) system, the consumption and cost of amplifiers are relatively less than the AMB system because the bias magnetic field is generated by the permanent and not the bias current to produce it. Taken together, the coupling effect and power loss of a homopolar HMB with 4-pole pairs shown in Fig. 2 (b) are even smaller than the heteropolar AMB shown in Fig. 2 (a). Furthermore, a feasible method for lowering the cost is to reduce the number of magnetic poles. The minimum pole

2 number for an RMB to provide desired forces in the twodimensional configuration space is three. The 3-pole RMBs are usually considered the best magnetic bearings in the sense of bearing compactness when the journal diameters are smaller. Therefore, the 3-pole RMBs have been widely studied in recent decades. Compared with the 8-pole type, the 3-pole RMB (as shown in Fig. 3) has several advantages. ① At least one driver can be saved since it requires at most three power amplifiers. ② It has a smaller remagnetization frequency and, hence, includes less iron loss in the rotor. For the 8-pole RMB, Fig. 4 indicates that when a point on the rotor travels from P1 to P8 (i.e., one revolution), the associated magnetic fields (magnetic field strength H and magnetic flux density B) will vary two cycles [26]. In contrast, in Fig. 5, as a point on the rotor travels from Q1 to Q4 (i.e., one revolution), the associated magnetic fields will vary one cycle. The results in Fig. 5 indicate that the remagnetization frequency for the 3pole RMB is only half of that of the 8-pole type. ③ Reducing the pole number can leave more space for heat dissipation, coil winding, and sensor installation. All of the above advantages can lead to lower costs for the 3-pole RMB, which occurs solely from the number of magnetic poles. In addition, the operational performance, energy loss and system costs can be deeply affected by the connection method of windings and their driving modes [26]. y i1 Pole

Pole

N

N

S

Permanent magnet

S

N

S N

S

Control coils (b) Homopolar RMBs

Fig.2 Prototype and magnetic circuit of 8-pole RMBs

y

i1

i2 Q2

O

Q3

Q1

i0 Fig.3 Structure diagram and magnetic circuit of 3-pole RMBs P1,P5

x

O

i4 H

P2,P6

Stator

i2

Control coils

P4,P8

P3,P7

Fig.1 Prototype and magnetic circuit of 4-pole RMBs

i1 P3 i3

P4 P5

Fig.4 Magnetic characteristics of 8-pole RMBs

y P2

Stator B

P1 P8

O

Q1

i4 x

P7

H

Q2

P6 i2

x

Q4

B

i3

Stator

Q4

Control coils

Pole Rotor Q3

(a) Heteropolar RMBs

Fig.5 Magnetic characteristics.of 3-pole RMBs B.

3-pole RMBs According to the connection method of windings and the driving modes of 3-pole RMBs, they can be further classified

3 as 3-pole RMBs driven by three power amplifiers [26], 3-pole RMBs driven by two power amplifiers [27]-[28], and 3-pole RMBs driven by a 3-phase converter [29], as shown in Fig. 3, 6 and 7, respectively. In these topologies of 3-pole RMBs, the 3-pole RMBs driven by three power amplifiers are primarily studied. As opposed to the most common radial magnetic bearing with asymmetric structure (even number of poles), 3-pole RMBs with only three poles for each bearing can simplify the structure of the bearing and reduce the size of the bearing system greatly. In [26], a 3-pole RMBs is introduced, and its radial control is realized by three independent unipolar power amplifiers. Compared with 4-pole or 8-pole types driven by four or two power amplifiers, the proposed 3-pole RMBs driven by three power amplifiers can lead to lower cost, lower power consumption, less iron loss and better heat dissipation. However, the cost and power consumption of the 3-pole RMBs can also be further reduced; therefore, a more advanced type of 3-pole RMB has been developed, known as 3-pole RMBs driven by two power amplifiers [27]–[28]. Fig. 6 shows a 3-pole RMB driven by a two power amplifier system. In Fig. 6, the three poles are arranged in a radically symmetric “Y” structure for better decrease in the power consumption and cost. For the 3-pole RMB driven by two power amplifiers with a “Y” structure, there are two windings wrapped around the three poles. The upper two poles share the same set of windings i2, though in different directions. Another winding i1 is wrapped independently around the third pole. Given the existence of gravity in the negative ydirection, the goal for the “Y” type structure with two sets of windings is to support the rotor weight when the rotor is at the steady state. That is, there exists only a bias current for i2, whereas no bias current is needed for i1 in usually suspended cases for rotors. Consequently, only two power amplifiers are required under such a “Y” type structure and less copper loss in the steady-state will occur due to the peculiar pole arrangement, winding scheme and bias currents.

y

by a 3-phase inverter, the cost and complexity of the magnetic bearing might be significantly reduced [29]. Fig. 7 shows a 3-pole RMB driven by a 3-phase inverter system. However, for the 3-pole RMB, the main problem with the discussed arrangement is the fact that its asymmetry will lead to cross-coupling between the two bearing axes. For the RMBs with symmetrical arrangements (4-pole RMBs and 8pole RMBs, which can be called 4-phase bearings), there is no coupling effect and the force in the x-direction would be fully independent of changes in control current and the position in the y-direction. The reason for the strong crosscoupling is the wrong assumption of a linear dependence between the magnetic forces in the electromagnets and the related control current upon which our model and the 3-2phase transformation are based. For the control of 4-phase bearings, the same assumption is commonly made. However, because of the geometrical symmetry of the magnet arrangement, a nonlinearity does not lead to cross-coupling. A simple solution for the problem is to control the three separate electromagnets. However, such a control scheme will require independent control of all three winding currents and it cannot be compatible with a 3-phase system. In addition, some decoupling approach considered from a mathematical model perspective may work. Thus, the designed decoupling method of how to implement a digital controller is worth studying. This may seem daunting, but it is actually quite easy to achieve with the development of power electronics technology and control measures. In another words, a mathematical decoupling in a digital controller should be possible without the requirement of much calculation power by gain scheduling based on a lookup table. Here, the drawback would be the necessity of good knowledge of the system and a high sensitivity to parameter changes. y iB

Control coils iA

Pole

x O

i2

iC

O

x Fig.7 Prototype and magnetic circuit of 3-pole RMBs driving

i1 C. Fig.6 Structure diagram and magnetic circuit of 3-pole RMBs driving by two power amplifiers In these cases, to realize the radial control of one bearing, two independent unipolar power amplifiers are still needed. However, if one RMB can be driven by only a 3-phase power inverter, it would be increasingly attractive. Therefore, if a magnetic bearing could be designed to use the 3-phase power modules, which is the 3-pole RMB driven

6-pole RMBs A good way to solve the cross-coupling problem existing in 3-pole RMBs is to change the asymmetric structure (odd numbered poles) into a symmetric structure (even numbered poles). One way to achieve this is an arrangement consisting of six magnets around the rotor. In this case, two opposite electromagnets are connected to the same phase of the 3phase converter. The structure diagram and magnetic circuit of a 6-pole RMBs with two opposite magnets connected to

4 the same phase of the 3-phase converter is shown in Fig. 8. iC

Control coils

Pole

iB iA iA

Rotor

O

iB

iC

Fig.8 Structure diagram and magnetic circuit of 6-pole RMBs with two opposite magnets connected to the same phase of the 3-phase converter Reference [30] introduces another type of horseshoe 6-pole magnetic bearing (as shown in Fig. 9), which is an active magnetic bearing (AMB). The proposed construction includes three electromagnets as the necessary minimal number of actuators required for the rotor suspension in the bearing space. A smaller number of execution units and electronics circuits is allowed to minimize bearing size, production costs and energy consumption. The disadvantages of this solution correspond to the lower stability region and strong nonlinearities. The three coil radial AMB belongs to a heteropolar construction bearing, which consists of three pole-pairs located 120 degrees apart. These three electromagnets generate sufficient electromagnetic forces acting on the rotor for suspension and damping purposes. Three PWM power amplifiers are used to supply and measure coil current. For identification purposes, the system is also equipped with flux and temperature sensors. The power amplifiers are controlled by PWM waves with specified frequency and variable duty cycles. These control signals can be applied directly in the digital form or hardware generated using analogue reference signals. i1

y Pole Control coils

x O i2

Rotor

i3

Fig.9 Structure diagram and magnetic circuit of horseshoe 6pole RMBs The 6-pole RMBs mentioned above are AMBs, and another type of RMBs includes HMBs, as shown in [31]. The arrangement of the proposed magnetic bearing is shown in Fig. 10. Such magnetic bearings consist of an iron ring with three permanent magnet poles and three iron poles. The permanent magnet poles marked 1, 2, 3 are used to generate bias flux by each of the permanent magnets. The iron poles

marked a, b, and c are wounded with magnetic coils to produce control flux. In Fig. 10, the dashed lines indicate the bias flux, and the solid lines indicate the control flux. When the rotor is in the equilibrium position, flux densities generated by permanent magnets in all air gaps are equal due to the symmetry of structure, which means there are no magnetic forces between the stator and rotor in both the x and y directions. When a disturbance force in the negative bdirection is imposed, the rotor will drift away from the equilibrium position. Then, the position deviation will be detected by the position sensors and the controller will generate a magnetic suspension force in the positive ydirection based on the error detection. As a result, the flux density in the air gap between the iron pole b and rotor is increased, whereas the magnetic flux densities in the air gaps of the iron pole a and c are accordingly decreased. Finally, a magnetic force in the rotor along the positive b-direction is created to drag the rotor back to the equilibrium position. y b Permanent Contol iB 1 magnet coils iA 2

iC

a x

O

c

3 -b

Pole

Stator

Fig.10 Prototype and magnetic circuit of 6-pole RMBs(HMBs) In addition to the magnetic bearing having one piece of the stator, a new 6-pole RMB with two stator pieces is proposed in [32]. The bearing type is shown in Fig. 11. The bearing itself consists of two stators on both sides of the bearing with the coils attached to the stator and the corresponding coils of the bearing are connected together. The permanent magnets provide the bias flux for the bearing. As [29] reported, for 3pole magnetic bearings, linear equations are usually applied in a control system. However, the simplified equation is relatively valid only for small changes in the current and displacements, which considers the magnetic forces as being proportional to the current and displacements. In actuality, the magnetic forces are proportional to the power of two of the current and displacements, which is a nonlinear relationship, especially for large displacement of the rotor. To solve the cross-coupling problem for a 3-pole bearing with an asymmetric structure, [29] proposes a good method that involves turning one of the stators to create a 6-pole bearing (double pieces of stators) with a symmetrical structure. The modification is shown in Fig. 12. Such a bearing with double pieces of stators is perfectly symmetrical to displacements and shows virtually no cross-coupling effects. Further advantages of the described bearing in Fig. 11 and in Fig. 12 are the feasibility of large air gaps that are due to

5 the permanent magnetic biasing. To obtain the same magnetomotive force, which can be achieved already by small volumes of rare-earth magnets by electrical coils, an extremely high winding space would be necessary. Since the price of neodymium-iron-boron magnets reduced drastically in recent years, it is very economic to use permanent magnet biasing. Radial control coil Radial stator

Aluminum ring Shaft

Permanent magnet

Rotor

Fig.11 Structure diagram of 6-pole RMBs with two pieces of stator in a bearing Turning one of the stators

Fig.12 Structure diagram of 6-pole RMBs with two pieces of stator in a bearing (design with one ring turned ) D.

12-pole RMBs As [29] reported, the 12-pole arrangement shows virtually no cross-coupling effects. For large bearings, the 12 poles are not a disadvantage. However, for smaller bearings it would be an advantage to have fewer poles, as Fig. 13 demonstrates.

y iw

iv

U0

i0

Control coils Pole iu

O

iu

x

Rotor

iv iw Fig.13 Structure diagram and magnetic circuit of 12-pole RMBs E. Comparisons and Recommendations related to RMBs Table 1 shows the comparative analysis of parameters for the bearings presented above under certain conditions, and

therefore the advantages and challenges of each topology can be obviously determined. The specific conditions are as follows. (1) Thirteen types of RMBs are compared under the same given value of journal diameter. (2) The number of turns of coils is determined by magnetic circuit analysis to generate the worst case load capacity within limits of coil currents and flux densities. (3) The rated speeds for all thirteen are set the same to make a comparison. The detailed process for comparison of power consumption is as follows: when the power of a high frequency motor is shut off, the rotor speed is gradually reduced to zero under the combined influence of air friction, resistance loss, eddy current and hysteresis loss. The total kinetic energy is converted into the wind damages and power loss (copper loss, the hysteresis loss and the eddy current loss). In the process of natural slowdowns (where the rotor speed is reduced from 60,000 to 0 r/min), the energy (Ek) and power loss (Pk) of the rotor are as follows:  J p w2 E k   2  2 dE k dw w dJ p dw  Pk   J pw   J pw  dt dt 2dt dt 

(1)

where Jp is the moment of inertia, and w is the rotating angular velocity. Based on the Eq. (1), the total power loss including wind damages and power loss under different rotational speeds can be obtained according to the variation of rotor speed over time. In the experiment, the wind damages can be conceptually considered to be identical. Therefore, the differences among the measured power loss of the thirteen different RMBs can be as the benchmark for comparison. In Table 1, the symbol “√” represents “yes” and the symbol “ × ” represents “ no ” . From the column of power consumption (ranked from highest to lowest), it can be seen that the power consumption of RMBs driven by power amplifiers is higher than that by inverters. In addition, the stator geometry, number of poles, core materials, eddy current phenomenon, magnetic forces and bias current will affect the performances of RMBs and have special literatures for reference. (1) Changing the stator geometry of the RMBs to obtain the optimal arc width of the pole teeth will affect the current gain and position stiffness, and one can refer to [33] for further details. (2) Changing the number of stator poles and the coil controlling scheme will affect the specific load capacity, and one can refer to [24] for further details. (3) Changing the core materials will affect the structure and electromagnetic parameters of RMBs, and one can refer to [34] for further details. (4) Due to the eddy current phenomenon, the magnetic fluxes can generate an additional force vector in the radial direction. For further details regarding the eddy current influence on the parameters of RMBs, one can refer to [35].

6 (5) The magnetic force, which acts on the moving shaft and stiffness parameters of RMBs, is calculated under steady state conditions, and one can refer to [36] for further details.

(6) For active RMBs, the bias current will affect their specific load capacities and power consumption, and one can refer to [37] for further details.

TABLE 1 COMPARATIVE ANALYSIS OF PARAMETERS

Biasing method Legend Fi.g.1 Fi.g.2 (a) Fi.g.2 (b) Fi.g.3 Fi.g.6 Fi.g.7 Fi.g.8 Fi.g.9 Fi.g.10 Fi.g.11 Fi.g.12 Fi.g.13

√

√

√

4

Power consumption (rank from highest to lowest) 2

√

√

√

8

1

√

×

8

3

√

×

3 3 3 6 6 6 6 6 12

4 5 11 6 10 12 8 9 7

× × × √ × × × √ √

√ √ √ × √ √ √ × ×

Active

Hybrid

√ √ √ √ √ √

Driving model Power amplifier

Inverter

III.

Heteropolar

Homopolar

√

√

√ √

√ √ √

pole arrangement

√ √ √ √ √ √ √

√ √ √ √ √ √ √ √ √

Pole number

Symmetry

Coupling

√

×

bearings have the same size, and the axial length and maximum diameter of the rotor are also identical.

ROTOR TOPOLOGIES

In addition to the bearing topologies, the performances are of great concern to rotor construction forms of magnetic bearing-rotor systems. The most common forms are cylindrical rotors, but disturbance torque is inevitable when the rotor deviates from the equilibrium position. In view of the deficiency of existing cylindrical rotors of magnetic bearing-rotor systems, a new type of spherical rotors is equipped with spherical stators. When the rotor experiences some deflection or offset, the electromagnet will point to the rotor, and therefore the interference torque of the stator poles on the rotor will be reduced, improving the control precision of the magnetic bearing-rotor system. Furthermore, the control current and energy consumption will be obviously decreased. Taking a 6-pole RMB with two pieces of stator as an example, a double spherical rotor can be an effective and special design because there are two pieces of magnetic bearing in the bearing. The three different shapes of the rotor are compared in Fig. 14. The performance parameters are also compared, as shown in Table 2. In the contrast test, the three

Magnetic bearing

z y O double spherical rotor

spherical rotor

cylindrical rotor x

Fig.14 Comparison diagram of different rotor of RMBs

TABLE 2 COMPARATIVE ANALYSIS OF DIFFERENT ROTOR

The rotor offset in x-direction is 0.1mm Interference torque (mN.m) Suspended magnetic force(N) The rotor deflection around y-axis is 0.3° Interference torque (μN.m) Suspended magnetic force(N)

Spherical rotor 0.8513 4.85 Spherical rotor -43.5 4.6

The calculating results show that the interference torque of the spherical magnetic bearing-rotor system is 12.99% and

Cylindrical rotor Double spherical rotor 6.55 5.35 113.35 79.8 Cylindrical rotor Double spherical rotor -133 -285.9 0.156 95.6 the suspended magnetic force is 4.27% of that of the cylindrical magnetic bearing-rotor system when rotor reaches the offset of 0.1 mm in the x-direction, showing that spherical

7 rotor relative to the cylindrical rotor has greatly improved anti-interference torque and suspended magnetic force. However, the performance parameters of the double spherical rotor lag far behind that of the spherical rotor. They are much closer to the cylindrical rotor. In addition, the calculating results show that the interference torque of the spherical magnetic bearing-rotor system is 32.71% and the suspended magnetic force is 29.49 times than that of the cylindrical magnetic bearing-rotor system when rotor deflects 0.3°around y-axis, showing that spherical rotor relative to the cylindrical rotor has greatly improved anti-interference torque and degenerative suspended magnetic force. However, the performance parameters of the double spherical rotor lag far behind that of the spherical rotor whether interference torque or magnetic force, which has the worst performance in terms of three RMBs . They are much closer to the cylindrical rotor. Meanwhile, the calculating results show that the cylindrical rotor just takes only a very tiny force when the rotor makes a deflection. IV.

PARAMETER DESIGN METHODS

To improve the performances of RMBs and better meet the engineering requirements, the design methods and optimization are especially necessary on the bearings. A.

Usual Parameter Design Steps The usual parameter design steps for the presented bearings are as follows: (1) Choice of Length of Air Gap and Permanent Magnet Usually, the air gap length of this presented RMB is set as 0.3-0.5mm. For safety, there is usually need to set some backup bearing, so the air gap length is limited as 0.150.25mm in the laboratory. If the designed bearing is hybrid magnetic bearing, the permanent magnet is the key techniques for design. The NdFe-B is used most commonly as the permanent magnet. The stator core and rotor are structured in stacked silicon steel sheets. (2) Calculations of key parameters in the core stator Once the core material has been fixed, the magnetization characteristic and saturation flux density has seemed bright. In the magnetization curve, the choice of working point for the AC magnetic bearing (as shown in Fig.1, Fig.2(a), Fig.2(b), Fig.3 and Fig.6) is different from the choice for DC magnetic bearing(as shown from Fig.7 to Fig.13). To make the most use of the core materials, the working point is usually selected at about half the demagnetization curve for the DC magnetic bearing [38]. Notes: The “ ac magnetic bearing ” is such a magnetic bearing that can be driven by an industrial three-phase power inverter. The “dc magnetic bearing” is usually driven either by four unipolar power amplifiers or two bipolar power amplifiers. (3) Calculations of magnetic pole area and ampere-turns In order to calculate the magnetic pole area and ampere-

turns more accurately, it is important to take the coefficient of magnetic leakage into consideration in calculation, which can guarantee the true value of magnetic flux flowing through the air gap. What’s more, setting the ratio of the minimum width of magnetic pole body and the arc length of pole shoe is another important factor, which can guarantee the magnetic saturation will not happen even at the minimum crosssectional in the magnetic circuit. Based on the above key parameter design steps, the rest of the parameter design results can be available in [8-17], therefore its calculative process is no longer listed here. B.

Optimization Design Method

The main optimization design methods for the presented bearings are as follows: (1) Parameter design method considering influence of eddy effect Because the movement state of rotor is significantly different from that of resting and suspended state due to the large additional magnetic field produced by eddy currents, the eddy current effect on the parameters design becomes particularly important. Firstly, the changes of the key parameters affected by the eddy current effect should be estimated. Then, by analyzing the parametric analysis results, an optimization scheme is proposed, which makes the parameters design accurate and close to the actual operation. The specific optimization scheme can reference to [39]. (2) Multiobjective optimization method The most common optimization design method of RMBs is optimized for single objective, such as smaller volume, high magnetic force, and lower power loss. The better results for single objective can be obtained by the single objective optimal process, while other important performances may be sacrificed. Therefore, the performances can be balanced by multiobjective optimization. The evolutionary algorithm (EA) has been applied in many industrial circles. The performances of the thrust magnetic bearing, such as volume, weight, and control performance, have been optimized by EA. Quantities of multiobjective optimization problems existing in industrial applications can be tackled by particle swarm optimization method. The specific optimization scheme can reference to [40], which adopts a multiobjective optimal design for a RMB with a particle swarm optimization method. To increase the optimization efficiency, reference [20] also proposed an integrated optimum methodology to realize the multiobjective optimization, which may be regarded as a reference case for RMBs. V.

MATHEMATICAL MODEL

Another key issue is the building of a comparatively accurate model of magnetic suspension force for a magnetic bearing-rotor system, which also affects the overall performance of the bearing system. For the sake of contrastive analysis, all the models listed below are based on the same type of bearing. Taking the 6-pole RMBs with two pieces of stator in each bearing as examples, their overall structures are shown in Fig. 11.

8

A. Modeling Process Based on Equivalent Magnetic Circuit Method The equivalent magnetic circuit method is the most classic modeling method of the magnetic suspension force for magnetic bearings, which are widely used in all kinds of magnetic bearings. To simplify the computation, the working air gap reluctance is only considered, and the magnetic flux leakage, the core reluctance, rotor reluctance and eddy current loss are ignored. The equivalent magnetic circuit diagram is shown in Fig. 15.

GA1

GA2

GB1

GB2

GW1

GC2

m

Fm

Fig.15 The equivalent magnetic circuit diagram where GA1,GB1,GC1,GA2,GB2 and GC2 describe the radial airgap permeances, φmis the excitation flux, and Fm is the magnetomotive force provided to the outer circuit of the AC 2-DOF HMB. According to the Kirchhoff's Laws of magnetic circuit, the model of the radial suspension forces (Fx, Fy) can be written as follows:

  Fx  3 1 0   x  3  1 0   ix   kir     k xy       2 0 1  i y    Fy  2 0 1   y  (2)  0 N r Fm Sr 0 Fm 2 Sr  kir  2 2 , k xy  2 3  r r where ix, iy are the current components in the x-axis and y-axis, respectively, transformed from the three-phased currents by Clark coordinate transformation; kxy is the radial forcedisplacement coefficient; kir is the radial force-current coefficient; and μ0 is the permeability in vacuum. Sr is the face area of the radial magnetic pole; δr is the length of the uniform air-gap without rotor eccentricity; and Nr is the number of turns that the radial control coils. B. Modeling Process based on the Maxwell Tensor Method For more precise control of the ac RMB (The “ac RMB” is a magnetic bearing that can be driven by an industrial threephase power inverter), it is particularly important to build a correct mathematical model of the suspension forces. Due to the working principle of the ac RMB, it has similarities with the suspension subsystem of a bearingless motor, thus the modeling method for radial suspension forces of the

bearingless motor could be referenced for modeling the radial suspension forces of the ac RMB. A new modeling method based on the Maxwell tensor method for suspension forces of the ac RMB is proposed. Compared with existing methods, the invention method designed specifically for the ac magnetic bearings overcomes the inaccuracy and has the advantages of directness and universality. The Maxwell force acting on the ac RMB is defined by the control magnetic flux density generated by control coils and the biased magnetic flux density generated by magnet permanent. Thus, the two parts of the magnetic flux density must be analyzed, and we then add them all to obtain the resultant magnetic flux density. According to the Maxwell tensor method, the Maxwell force per unit area along a certain dimensional mechanical angle on the rotor surface can be expressed as B 2 ( , t )  dS B 2 ( , t ) (3) dF ( )    (lrd ) 2 0 2 0 where l is the equivalent length of the rotor, and r is radius of the rotor. dS is the per-unit area, and θ is the dimensional mechanical angle.

The model of the radial suspension forces (Fx, Fy) can be written as follows:   Fx  3 1 0   x  3  1 0   ix      k xy    y   2 kir 0 1  i  F 0 1 2  y       y (4)  2 2 3lrH m hm 0 N3r lrH m hm 0  ; k xy  kir  8 0 2 4 03  where Hm is themagnetic field intensity at the work point for the permanent magnet. hm is the length of the permanent magnet in the direction of magnetization. The verification experiment was designed and the encouraging experimental results showed that the model based on the Maxwell tensor method was closer to the test results compared with the equivalent magnetic circuit method. Furthermore, the Maxwell tensor method caused the modeling results on radial suspension forces of the ac magnetic bearing to be more accurate, universal, and direct.

VI.

CONTROL STRATEGIES

Another key issue is to establish the appropriate control strategies for a magnetic bearing-rotor system, which seriously affects the overall performance of a bearing system. A.

PID The proportional plus integral plus differential (PID) control scheme, which is a traditional control scheme, has been used widely in RMB systems due to its ease of use. However, the control performance of the RMBs system when using the conventional PID control scheme is not very satisfactory due to the unmeasured parameters variations and unavoidable external disturbances. Therefore, with the development of modern control theories, many types of advanced control methods have been recently proposed for

9 RMBs systems, such as fuzzy control [41], sliding mode control [42], model predictive control [43]-[44], and fault tolerant control [45]. These advanced control methods not only enrich the control theory of RMBs systems but also

improve their performance in different aspects. A comparison of the performances of different control strategies is shown in Table 3.

TABLE 3 PERFORMANCES COMPARISON OF DIFFERENT CONTROL STRATEGIES

PID

Advantages

Disadvantages

Simple to be realized

Not suitable for unknown or unfixed model

FC

Suitable for unknown or unfixed model

SMC

Fully self-adaptive against external disturbances and parameter variations

MPC

Strong robustness and can effectively overcome the process of uncertainty, nonlinear and shunt resistance, and can process controlled variable and control variables in various constraints.

FTC

Stability and reliability can be guaranteed after a failure occurs

Control precision of the system and the dynamic quality of variation may be reduces by simple fuzzy information processing Control precision and stability of system will be affected by the chattering, caused by sliding mode switch control.

Improvement approach Optimize the parameter s of PID controller to adapt the changing of model parameters.

Extended technique (Artificial Immune Bee Colony Optimization( AIBCO)

Unite other intelligent optimization algorithm to improve c ontrol performance

Fuzzy bang–bang relay control; Fuzzy-neural network control

Filtering method or Elimination of disturbance and uncertainty

Sliding Mode Control Based on Fuzzy Reaching Law; Discrete-time Fuzzy Sliding Mode Control

The model predictive control problem for nonlinear and time-varying system has not been solved yet.

Need continuous creation in predictive control strategy

Multi-objective modelpredictive control (MOMPC); Real-time weighted multiobjective model-predictive control(RW-MPC)

Fault detection and identification of nonlinear system is difficulty and key

Reduce failure source and dig into FTC algorithm in more depth

FTC of RMB based on coordinate transformation; FTC of RMB based on the self-sensing technology.

C. B.

FC Fuzzy logic is widely used in machine control. The term "fuzzy" refers to the fact that the logic involved can address concepts that cannot be expressed as the "true" or "false" but rather as "partially true". Due to the nonlinear properties and knowledge-based rules, fuzzy control (FC) has advantages in applications to magnetic suspension systems, which itself is a highly nonlinear electromechanical plant. FC is a nonlinear control strategy based on input-output membership functions and rule bases. It describes a control in a qualitative and intuitive way that emulates the heuristic rule-of-thumb strategies of experts. Fuzzy logic can be applied to diagnosis of magnetic bearing systems with knowledge-based expert systems. FC is employed to improve the performance of RMB or to overcome the position dependent non-linearity of magnetic bearings. Fault detection and supervision can apply fuzzy logic control concepts to abnormal operating conditions.

SMC In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamics of a nonlinear system by application of a discontinuous control signal that forces the system to "slide" along a cross-section of the system's normal behavior. The state-feedback control law is not a continuous length of time. Instead, it can switch from one continuous structure to another based on the current position in the state space. Hence, SMC is a variable structure control method, which is usually used in the field of magnetic bearings due to its associated robustness. Although SMC is fully self-adaptive against external disturbances and parameter variations, the control precision and stability of the system will be affected by the chattering. Therefore, the filtering method or elimination of disturbances and uncertainties can solve the problem. D.

MPC The capability of controlling the coils current with a wide

10 bandwidth is a crucial requirement, as it directly impacts the dynamic performances of the RMB. For such systems, Model Predictive Control (MPC) represents an attractive solution, due to its inherently fast dynamic response, lack of modulation requirement, easy inclusion of nonlinearities and constraints of the system, possibility of incorporating nested control loops in only one loop and the flexibility to include other system requirements in the controller. MPC considers a model of the system in order to predict its future behavior over a specific time horizon. Furthermore, due to the complex driving conditions during the transitional state, the traditional model predictive control algorithm with constant weight matrix cannot meet the requirement of improvement in the RMB system. Therefore, a real-time weight tuning strategy can solve time-varying multi-objective control problems, where the weight of each objective can be adjusted with respect to different operating conditions. E.

FTC The rotor displacements should be accurately measured by sensors as the feedback signals for the RMB controller. For the issue of safety, a position sensor fault may cause erroneous feedback signals and system instability. An existing fault tolerant control solution is to incorporate sensor redundancy, but the collocation of redundant sensors is expensive and sometimes difficult due to space limitations. However, the self-sensing technology of the RMB, in which rotor displacements are directly estimated from the coil voltages and currents, provides a new and effective method for achieving the sensor fault tolerant control of the RMB. Because the sensor-fault detection is only determined by the displacement output of the sensor, this fault tolerant control method will not introduce extra system faults when the parameter estimator itself fails. This causes the system to be more reliable than the method of using a simple comparator for fault detection.

Rotor

Conical-shaped stator

Fig.16 A conical-shaped magnetic bearing In addition to the traditional structure of RMBs that are only responsible for rotor position in the radial direction, some new topology structures are responsible for tilt control of rotors and have become a significant new trend. Therefore, the definitions of RMBs are no longer narrow. Furthermore, at high rotational speeds for magnetic bearings systems, the gyroscopic effect is a major factor influencing rotor stability. Some new topology structures can be invented to solve the gyroscopic effect problems. Fig. 17 shows such a new topology structure, which can effectively compensate gyroscopic effects and realize four degree of freedom control due to the large air gap and disc structure with double stator. Fig. 18 shows a ball joint type magnetic bearing, where the RMB can realize translational and rotational control simultaneously. The gyroscopic effects can be inhibited effectively because of the shaftless structure. Top stator Rotor position and rotation control coils Rotor Large air gap Tilt control coil Bottom stator

Fig.17 Double stator self-bearing with large air gap RMBs VII.

FUTURE TRENDS OF OTHER KEY TECHNOLOGIES

A.

Topology Structure If the RMBs can further reduce the overall cost of their construction, amplifiers and control system, there will be broader application prospects. Therefore, the more-reasonable topology structure of the RMBs becomes a timeless topic. Standard configuration of a magnetic bearing involves a pair of radial bearings and an axial bearing. It is also possible to give the two RMBs a conical shape and to provide axial suspending forces as well. Then, the axial bearing can be omitted, and the magnetic bearing will have a smaller axial size and simpler constructions than a standard magnetic bearing. As shown in Fig. 16, for the conical magnetic bearings, the rotor can be suspended by only a pair of conical-shaped RMBs.

Fig.18 Ball joint type magnetic bearing B.

Optimized Design Based on the reasonable topology structures, some optimized design methods of the RMBs are still important. The size of the stator and rotor, the number of poles, the structural style of the stators and rotor, and the resulting decoupling or coupling effect, driving mode of the control coils, among others, must all be optimally designed according to the special structure and operation requirement of the RMBs.

11 A new parameter design method based on two sets of magnetic field systems (the stator core magnetic field system and the air gap field system) is a good method for calculating the structure and electromagnetism parameters of the RMBs. System performance comparison results show the control system based on the new parameter design method has stronger anti-disturbance characteristics than that based on the traditional parameter design method. The key steps are as follows: First, it is important to take the coefficient of magnetic leakage σ into consideration in the calculation, which can guarantee the true value of magnetic flux flowing through the air gap. Then, setting the ratio of the minimum width of the magnetic pole body and the arc length of pole shoe k is another important factor, which can guarantee the magnetic saturation will not occur even at the minimum cross-sectional in the magnetic circuit. Thus, we can also establish the functional relationships between the stator core magnetic flux density and air gap flux density by the coefficient k. Lastly, for the remainder of the structure parameters, the traditional parameter design method for magnetic bearings can be adopted for easier and higher efficiency parameters design, which can still ensure the accuracy of the design results based on the above achievements. C. Optimized Design Accurate Mathematic Model and Nonlinear Decoupled Control For more precise control of RMBs based on a reasonable topology design, it is particularly important to build an accurate model of radial suspension forces. For the RMBs having even numbered poles, such as 4-pole, 8-pole and 16pole RMBs, the equivalent magnetic circuit is the most commonly used method. For RMBs having even numbered poles, such as 6-pole, 12-pole, along with 3-pole RMBs, which are multiples of 3, the radial suspension force modeling method specifically for AC magnetic bearings based on the Maxwell tensor method is most recommended and can make the modeling process more accurate, direct and universal compared with the conventional modeling method. In addition, the mathematic model of radial suspension forces for RMBs is heavily influenced by magnetic coupling, force coupling, magnetic saturation, magnetic flux leakage, temperature rise, rotor eccentricity, eddy current, rotational speed and the change of load parameters. Therefore, the analysis of the results of the above by the static test and ANSYS simulation results of a prototype are quite essential. Then, based on the analysis results, the operation area is divided into several small sub-regions depending on the state of the working condition to establish an error correction model. For RMBs having strong nonlinear, strong coupling and external interference problems, especially for the 3-pole RMBs, various effective decoupling strategies should be proposed to realize decoupling control of the RMB system and to enhance its performance. In many decoupling algorithms, a dynamic decoupled control based on the dynamic model influenced by magnetic coupling, force

coupling, magnetic saturation, magnetic flux leakage, temperature rise, rotor eccentricity, eddy current, rotational speed and the change of load parameters may still be the best method for the RMB systems to achieve favorable static and dynamic steady performance. D.

Self-sensing RMBs and Estimation Methods The displacement sensor is a most common method in closed loop feedback links to realize precise location of rotors in RMB systems, and the most common of these are the eddy current sensors. However, because the eddy current sensor is operated based on the electromagnetic induction principle, it will couple with the magnetic field generated by the control coils, which leads to the precision of the sensor possibly being affected. In addition, the price of the sensor is high, which causes the overall price of the magnetic bearing to be high and limits the widespread application of RMBs. Meanwhile, a self-sensing RMB can be a perfect substitution for traditional RMBs with sensors in that the rotor position information can be deduced from the electromagnetic interaction between the stator and rotor. It is possible to design the bearing system without position sensors just by the measured current in the electromagnets. This results in a significant advantage because of a considerable reduction in manufacturing costs and the complexity of the system, as well as the elimination of the failure modes associated with the sensors. There are many detecting and estimation methods to obtain the radial displacement and velocity of the rotor for selfRMBs. Thus, it is necessary to compare various methods in order to take full advantage of the characteristics of each method according to the different structure and principle of the self-RMBs. VIII.

CONCLUSIONS

In this paper, different RMBs topologies are compared, which mainly differ in the cost, power consumption, symmetry and coupling between the two bearing axes in the radial direction responsible for cross-coupling generation. From the comparison of the different types of RMBs, the advantages and challenges of each topology was determined. Furthermore, rotors of different shapes were compared because the performances of magnetic bearing-rotor systems are of great concern to rotor construction form. Additionally, the parameter design methods, the mathematical models and control strategies are compared in detail. In addition to being important for bearing bodies, models and control strategies, future work for the RMB system may include new topology structures, optimized designs, accurate mathematic models, nonlinear decoupled control, self-sensing RMBs and estimation methods for self-RMBs. With the aid of this overview, the best fit topology for a certain application can be selected appropriately. The most important benefit of the contrastive analysis is that different constructions of the RMBs are worth attempting.

12 ACKNOWLEDGMENT This work is sponsored by Natural Science Foundation of Jiangsu Province (BK20150524), National Natural Science Foundation of China (51607080, 51675244), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (2014). REFERENCES [1]

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14 

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   

Radial magnetic bearings (RMBs) are one of the most commonly used magnetic bearings. They are used widely in the field of ultra-high speed and ultra-precise numerical control machine tools, bearingless motors, high speed flywheels, artificial heart pumps, and molecular pumps, and they are being strengthened and extended in various important areas. In this paper, a comprehensive overview is given of different bearing topologies of RMBs with different stator poles that differ in their construction, the driving mode of electromagnets, power consumption, cost, magnetic circuits, and symmetry. RMBs with different poles and couplings between the two bearing axes in the radial direction responsible for cross-coupling generation are compared. In addition, different shaped rotors are compared, as the performances of magnetic bearing-rotor systems are of great concern to rotor constructions. Furthermore, the parameter design methods, the mathematical models and control strategies of the RMBs are described in detail. From the comparison of topologies, models and control methods for RMBs, the advantages, disadvantages and utilizable perspectives are also analyzed. Moreover, several possible development trends of the RMBs are discussed. The overview is special and comprehensive in the field, and more importantly, there are no other references by far to specifically summarize the topologies of RMBs.