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Abstract

Active magnetic bearings (AMBs) are widely used in high-speed turbines, precision machining, and energy storage systems due to their non-contact operation, high rotational speeds, and low maintenance. The 8-pole radial magnetic bearing (RMB), commonly adopted in AMB systems, faces limitations in load-carrying capacity and space utilization under heavy loads. To address this, a 12-pole RMB with E-type magnetic poles is proposed, enhancing stator space utilization and electromagnetic force generation. However, optimizing the 12-pole RMB is challenging due to the trade-off between maximizing electromagnetic force and minimizing stator volume, with limited research on its multi-objective optimization. This study fills this gap by systematically investigating the structural design and optimization of the 12-pole RMB. A mathematical model using an equivalent magnetic circuit is developed, validated by Finite Element Method (FEM) simulations, showing strong agreement. FEM further assesses magnetic flux density, dynamic performance, and structural parameter effects. Using the NSGA-II algorithm, a multi-objective optimization framework optimizes the conflicting objectives, selecting six key variables based on their Pareto effect. A prototype of the optimized 12-pole RMB is fabricated and tested, demonstrating stable levitation and confirming the design’s effectiveness. This study offers a systematic approach for designing 12-pole RMBs, providing valuable insights for industrial applications.

Keywords

heteropolar radial magnetic bearing preliminary design structure optimization design multi-objective optimization magnetic bearing

Introduction

Active magnetic bearings (AMBs), originally designed to address the limitations of conventional journal or ball bearings, demonstrate several advantages, including the ability to operate in vacuum environments without lubrication or contamination, as well as the capability to achieve higher rotational speeds.¹ These systems utilize passive or controllable electromagnetic forces to achieve non-contact rotor levitation, offering key benefits such as frictionless operation, reduced maintenance requirements, low vibration levels, higher rotational speeds, and enhanced operational reliability. A notable feature of AMBs is their ability to adjust damping and stiffness during operation, which allows for customization to meet various operational needs.^2,3 As a result, AMBs have found widespread applications in fields such as industrial turbine machinery, compressors, aerospace systems, energy storage devices, milling spindles, and medical equipment.^4–7

In an active magnetic bearing (AMB) system, stable rotor levitation is achieved through the coordinated operation of displacement sensors, microcontrollers, control algorithms, power amplifiers, and actuators.^8,9 Among these components, actuators play a critical role, as they receive control currents from the power amplifier and generate electromagnetic forces that maintain rotor levitation at each computational step.^10–12 Typically, actuators are classified into radial magnetic bearings (RMBs), which provide stabilization in two degrees of freedom (DOF), and axial magnetic bearings, which stabilize the rotor in one DOF. A standard AMB configuration consists of two RMBs and one axial bearing.¹³ Among these, RMBs have attracted significant research interest due to their essential role in supporting multi-DOF stabilization within magnetic bearing systems.^14,15

Depending on their structural configurations, radial magnetic bearings (RMBs) can be classified as either homopolar or heteropolar types. A homopolar RMB has poles of the same polarity within the same rotation plane, while a heteropolar RMB alternates its polarity perpendicularly. In practice, heteropolar RMBs are more commonly used in industry due to their simplicity and lower cost.¹⁶ Various heteropolar configurations are employed in applications requiring three, four, eight, twelve, or sixteen poles. Among these, the 8-pole RMB has attracted considerable research interest and is widely adopted in industrial applications. Numerous studies have focused on the design,^17–19 power loss,^20,21 and optimization^22,23 of 8-pole RMBs.

However, the limitations of the 8-pole RMB become evident as the demand for higher load capacity increases, particularly with larger rotor diameters. In an 8-pole RMB, the large gaps between stator poles result in inefficient space utilization, thereby restricting the generated electromagnetic force.^24,25 Additionally, the magnetic flux in the poles of an 8-pole RMB tends to saturate rapidly under heavy loads.²⁶ To overcome these challenges, an E-type magnetic pole configuration was developed by dividing each C-type magnetic pole into two sub-poles, increasing the pole count from eight to twelve. This modification optimizes space utilization between pole pairs, accommodating additional magnetic poles and windings. For the same stator diameter, the 12-pole RMB demonstrates superior load capacity and enhanced controllability compared to the 8-pole configuration.²⁷ Moreover, the reduced magnetic field coupling between poles makes the 12-pole RMB more suitable for applications requiring high load capacity in constrained spaces.^28,29 Comparative studies have quantified the improvements of the 12-pole RMB over the 8-pole configuration. Dimond et al. reported a 10% increase in load capacity when transitioning from 8 to 12 poles,³⁰ while Zhan and Zhu analyzed the relationships between the outer-to-inner stator diameter and losses with respect to frequency for both 8-pole C-type and 12-pole E-type RMBs, as shown in Figure 1. Additionally, the 12-pole RMB exhibits a reduced axial length compared to the 8-pole RMB, enhancing its compactness.³¹ Owing to these advantages, the E-type 12-pole configuration has been widely adopted by commercial Active Magnetic Bearing (AMB) manufacturers.³²

Figure 1.

The comparison of C type and E-type magnetic pole with regard to stator diameter (a) and losses (b) by Zhan and Zhu.³¹

Compared to the extensive research on the design, modeling, analysis, and optimization of 8-pole radial magnetic bearings (RMBs), the 12-pole RMB has received relatively limited attention. Wang et al.³³ modeled and analyzed a hybrid permanent-magnet-biased E-type RMB, demonstrating advantages such as high stiffness and low channel coupling. Saeed et al.³⁴ investigated the nonlinear dynamics and bifurcation behavior of a 12-pole RMB, and experimentally evaluated its vibration response under a designed controller. Hamad et al.³⁵ optimized the AMB design by reducing the number of poles and air gap size. Kumar et al.²⁵ explored the use of AC excitation in 12-pole RMBs to counteract rotor unbalance. Ren and Ma³⁶ focused on the dynamic stability and vibration characteristics of 12-pole RMBs under PD control.

Despite the increasing importance of 12-pole radial magnetic bearings in industries, there remains a significant gap in the literature concerning their systematic design and optimization. And the design of magnetic bearings requires significant time, cost, and advanced knowledge in mechanical and electrical engineering, making the optimization of compact, efficient, reliable magnetic bearing systems a key research focus in the field. Existing studies often lack a unified framework that addresses both the preliminary design phase and the optimization process. To bridge this gap, this paper proposes a comprehensive and structured methodology for the design and multi-objective optimization of 12-pole RMBs, incorporating critical engineering constraints. This approach not only enhances the design efficiency but also ensures practical applicability for heavy load magnetic bearing systems. The proposed method represents a novel contribution by offering detailed guidance for 12-pole RMB development, serving as a valuable reference for both industrial applications and academic research.

This paper presents a comprehensive framework for the modeling, design, and multi-objective optimization of the 12-pole RMB. The remainder of the paper is structured as follows: Section “Modeling of 12-pole RMB” introduces the fundamental operating principles and establishes the nominal mathematical model for the bearing. Section “Preliminary design and analysis of 12-pole RMB” describes the preliminary structural design process and includes performance simulations of the initial design. Section “Optimization design of 12-pole RMB” outlines the cost functions, design variables, and constraints that are integral to the optimization process, along with an analysis of the performance of the optimized structure. Section “Experiment test” presents the levitation performance data collected during the experiment, and Section “Conclusion and future work” concludes with key insights and directions for future research.

Modeling of 12-pole RMB

Working principle of 12-pole RMB

The basic operating principle of a 12-pole radial magnetic bearing is illustrated in Figure 2. The single-degree-of-freedom (SDOF) system comprises several key components, such as the rotor, an E-type stator, displacement sensors, power amplifier, and windings arranged in differential control method. This configuration enables the electromagnets to achieve faster response times. Stable levitation of the rotor is maintained through a classic feedback control process: the position of the rotor is detected by non-contact sensors, and the sensor’s output voltage is sent to the controller, which calculates the necessary control signal. This control signal is then converted into a control current for the differential coils through power amplifiers. Finally, the resulting differential electromagnetic force attracts the rotor, bringing it back to its equilibrium position. This feedback control process occurs in microseconds and is repeated continuously, allowing for the levitation of the rotor.

Figure 2.

(a) Illustration of structure and (b) working principle of SDOF 12-pole RMB.

Figure 3(a) illustrates the basic structure of a 12-pole radial magnetic bearing, which consists of 4 central magnetic poles with larger pole areas and 8 sub-magnetic poles with smaller areas.³³ The distribution of the E-type magnetic poles is typically arranged in an “NSN” or “SNS” layout to minimize the potential cross-coupling of magnetic circuits between different pole groups. In general, the magnetic pole area and the number of coils turns in the central poles are twice those of the sub-magnetic poles, which simplifies the design and analysis. Two sets of single-degree-of-freedom (SDOF) differential control coils, aligned along the x-axis and y-axis, generate cooperative forces to levitate the rotor and counteract external disturbances.

Figure 3.

(a) Basic structure of 12-pole RMB and (b) its simplified equivalent magnetic circuit.

In the design and analysis of magnetic bearings, the equivalent magnetic circuit method is widely used due to its straightforward representation, which is achieved through appropriate simplifications. The magnetic circuit of 12-pole RMB is shown in Figure 3(b). A set of E-type magnetic pole group (e.g. magnetic pole 1, 2, and 3) consists of electromotive force (Ni), the magnetoresistance of air gap, stator and rotor (R), as well as magnetic flux ( $ϕ$ ). According to the Ohm’s law, the magnetic circuit of central magnetic pole and sub-magnetic pole all meets the relationship of $Ni = R ϕ$ . Regarding to the number of coils turns in sub-magnetic pole is half of the central magnetic pole; the electromotive force should meet $N i_{2} = 2 N i_{1} = 2 N i_{3}$ . By dividing the 12-pole RMB into 4 E-type magnetic circuit groups, the simplified calculation model and electromagnetic force can be deduced by Kirchhoff’s law.

Mathematical modeling of 12-pole RMB

Considering a E-type magnetic pole group shown in Figure 4(a), the two magnetic circuits within a E-type magnetic pole group can be expressed by Ampere loop theorem by proper simplification as:

\frac{3}{2} N_{central} I = l_{fe} H_{fe} + 2 δ_{0} H_{air}

(1)

Where, N_central represents the number of coils of the central magnetic pole; I represent the excitation current in the E-type magnetic pole. The symbols H_fe and H_air represents the magnetic field strength of the core and air gap, respectively. l_fe is the circuit length in the core, and δ₀ is the displacement of the air gap.

Figure 4.

(a) Structure and ideal magnetic paths of a group of E-type magnetic pole and (b) its detailed structure descriptions.

By introducing $H_{fe} = \frac{B_{central}}{μ_{0} μ_{r}}$ and $H_{air} = \frac{B_{central}}{μ_{0}}$ into (1), the magnetic flux density of central magnetic pole $B_{central}$ can be expressed as:

B_{central} = \frac{3 N_{central} I μ_{0}}{4 δ_{0}}

(2)

Where, $μ_{0}$ an $μ_{r}$ d represents the permeability of air and relative permeability, respectively.

According to MAXWELL electromagnetic force formula $F = \frac{B^{2} A}{2 μ_{0}}$ (A is the pole face area of magnetic pole), the electromagnetic force generated by the central magnetic pole is:

F_{central} = \frac{9 μ_{0} N_{central}^{2} I^{2} A_{central}}{32 δ_{0}^{2}}

(3)

Where, $A_{central}$ is the pole face area of central magnetic pole.

Given the relationship of $B_{central} = B_{side}$ and $A_{central} = 2 A_{side}$ , the electromagnetic force of sub-magnetic pole is half of that of the central magnetic pole. And its vertical component of the electromagnetic force $F_{side}$ is expressed as:

F_{side} = \frac{9 μ_{0} N_{central}^{2} I^{2} A_{central}}{64 δ_{0}^{2}} \cos α

(4)

Where, $α$ is the angle between the central line of sub-magnetic pole and central magnetic pole (Figure 4(b)).

Therefore, the electromagnetic force generated by a E-type magnetic pole can be deduced as:

F = F_{center} + 2 F_{side} = ϵ \frac{9 μ_{0} N_{central}^{2} I^{2} A_{central}}{32 δ_{0}^{2}} (1 + \cos α)

(5)

Considering the differential control of two sets of E-type magnetic pole shown in Figure 2(b), the resultant force excited to rotor is the sum of $F_{1}$ , and $F_{2}$ . By assuming the bias current, control current, and displacement of the rotor away from the balancing point as $I_{0}$ , $Δ I$ , and $Δ δ_{0}$ respectively, the differential electromagnetic force can be expressed as:

\begin{array}{l} F = \frac{9 μ_{0} N_{central}^{2} A}{32} [(\frac{{(I_{0} + Δ I)}^{2}}{{(δ_{0} + Δ δ)}^{2}} - \frac{{(I_{0} - Δ I)}^{2}}{{(δ_{0} - Δ δ)}^{2}}) \\ + (\frac{{(I_{0} + Δ I)}^{2}}{{(δ_{0} + Δ δ c o s α)}^{2}} - \frac{{(I_{0} - Δ I)}^{2}}{{(δ_{0} - Δ δ c o s α)}^{2}}) c o s α] \end{array}

(6)

By taking Tylor’s expansion at the set equilibrium point ( $δ_{0}$ , $I_{0}$ ) of (6) and omitting the higher order terms, the linearized differential electromagnetic force can be expressed as:

F = k_{i} Δ I + k_{δ} Δ δ

(7)

Where, $k_{i} = \frac{9 μ_{0} N_{central}^{2} A I_{0}}{8 δ_{0}^{2}} (1 + \cos α)$ represents the current stiffness, and $k_{δ} = - \frac{9 μ_{0} N_{central}^{2} {AI}_{0}^{2}}{8 δ_{0}^{3}} (1 + \cos^{2} α)$ is displacement stiffness.

Preliminary design and analysis of 12-pole RMB

Preliminary design flow of 12-pole RMB

The primary objective in designing radial magnetic bearings is to achieve maximum load capacity alongside optimal dynamic performance. The detailed methodology for the preliminary design of a 12-pole radial magnetic bearing is systematically outlined in Figure 5.

Figure 5.

Preliminary design procedure of 12-pole RMB.

Step 0: Design inputs

The mass of rotor $m$ and the rotor diameter $D_{2}$ of the permanent magnet synchronous motor are provided as input parameters, along with the maximum rotational speed $n$ and dynamic balancing level requirement. The preliminary design parameters are listed in Table 1.

Table 1.

Preliminary parameter list of 12-pole rmb.

Parameters	Symbol	Descriptions	Unit
Design input	$m$	Rotor’s mass	40 kg
	$n$	Maximum rotation speed	25,000 rpm
	$D_{1}$	Outer diameter of motor’s rotor	96 mm
	$G$	Dynamic balance level	—
Initial parameters	$δ_{0}$	Air gap	0.4 mm
	$B_{s}$	Saturation magnetic flux density	1.5 T
	$I_{m}$	Maximum current	5 A
	$J$	Maximum current density	5 A/mm²
Experience coefficients	$P_{1}$	Magnetic flux density coefficient	0.8
	$P_{2}$	Magnet pole shoe coefficient	0.15
	$P_{3}$	Laminated silicon steel coefficient	0.9
	$P_{4}$	Anti-interference factor	0.9
	$λ$	Winding space factor	0.2
	$λ_{2}$	Winding filling factor	0.6
	$t_{h}$	Arc length ratio coefficient	1

Step 1: Initial parameters

Several parameters have to be selected at the beginning of the design process. The air gap $δ_{0}$ between the rotor and stator of RMB varies according to specific design requirements. Generally, the air gap is chosen within a range of 0.4–0.6 mm, by balancing trade-offs. The maximum current is limited by the power amplifier hardware, while the current density is constrained by thermal conditions, typically ranging from 3 to 6 A/mm². The saturation magnetic flux density $B_{s}$ is influenced by the material type of silicon steel. For this design, the selected silicon steel sheets are of the DW310-35 brand whose saturation magnetic flux density is around 1.5 T.

Step 2: Loading capacity calculation

Determining the maximum load capacity is the primary objective in the design of RMB. The target loading force of a group of E-type magnetic poles can be calculated by multiplying the static load $F_{sta}$ and dynamic load $F_{dyn}$ by dynamic factor $D_{f}$ as:

F_{m} = \frac{D_{f} (F_{sta} + F_{dyn})}{\sqrt{2}}

(8)

Where, $D_{f} = 3.5$ is set to guarantee the loading redundancy of the designed RMB.

The static load capacity $F_{sta}$ provides the electromagnetic force to levitate the rotor stationary in the direction of gravity. The value of $F_{sta}$ is half of the rotor’s weight considering two RMBs will support the rotor together. The dynamic load capacity $F_{dyn}$ will overcome the unbalancing force of rotor caused by unbalance mass of rotor, and its value can be calculated according to ISO 1940 standard as:

F_{dyn} = m ω^{2} 9549 \frac{G}{n}

(9)

Where, $ω$ is the maximum angular speed of rotor, and G is the required dynamic balance level.

By taking values into (9), the demanded maximum load capacity $F_{m}$ of a group of E-type magnetic pole is 598 N.

Step 3: Determining stator and winding parameters

As is shown in Figure 4(a), the inner diameter of the stator can be expressed as $D_{2} = D_{1} + 2 δ_{0}$ once the outer diameter of motor’s rotor and air gap are settled at first. In the following design flow, the experience coefficients listed in Table 1 will be used.

The maximum allowable magnetic flux density $B_{m}$ should be less than the saturation magnetic flux density of the used silicon steel. Accordingly, $B_{m}$ can be expressed as $B_{m} = P_{1} B_{s}$ . Once $B_{m}$ is settled, the cross-sectional area of central magnetic pole $A_{central}$ can be calculated as:

A_{central} = \frac{2 μ_{0} F_{m}}{ϵ {B_{m}}^{2} (1 + \cos α)}

(10)

The width of central magnetic pole $t$ can be calculated in (11) by distributing the arc length at the radius of $D_{2}$ . The coefficient $t_{h}$ has to be chosen carefully because it may cause difficulty in assembling the windings.

t = \frac{t_{h} π D_{2}}{8 (1 + t_{h})}

(11)

In common configurations of 12-pole RMB, the width of sub-magnetic pole is half of that value of $t$ . And the inner diameter of rotor can be designed initially as $D_{0} = D_{1} - 2 c_{1}$ , where $c_{1}$ equals $t$ to guarantee that both rotor and stator can achieve maximum magnetic flux density simultaneously.

The number of turns of central magnetic pole can be derived by placing the value of $B_{m}$ and $I_{m}$ into (2) as:

N_{central} = \frac{4 B_{m} δ_{0}}{3 I_{m} μ_{0}}

(12)

The pole face area of central magnetic pole $A_{central}$ is expressed as $A_{central} = P_{3} tL$ , where $L$ is the axial length of the stator. By taking the expression of $A_{central}$ $F_{m}$ , and $I_{m}$ into (10), the axial length of the stator can be derived as:

L = \frac{32 F_{m} δ_{0}^{2}}{9 ϵ μ_{0} N_{center}^{2} I_{m}^{2} P_{3} t (1 + \cos α)}

(13)

Also, the diameter of winding coil $d_{w}$ , current density J, and maximum current $I_{m}$ should meet the relationship of $\frac{π d_{w}^{2}}{4} J = I_{m}$ . Therefore, $d_{w}$ is represented as:

d_{w} ⩾ \sqrt{\frac{4 I_{m}}{π J}}

(14)

What is worth mentioning is that in practical design, the value of $d_{w}$ should be selected and updated from related standard (such as AWG standardized diameter) after calculation.

The sector area $A_{cu}$ in Figure 4(b) between central and sub-magnetic pole can be derived from the relationship of $λ A_{cu} ⩾ \frac{3}{2} N_{center} \frac{π d_{w}^{2}}{4}$ as:

A_{cu} ⩾ \frac{3 π N_{center} d_{w}^{2}}{8 λ}

(15)

At the same time, the value of $A_{cu}$ can be expressed approximately by the difference of two sectors as:

A_{cu} = \frac{1}{2} α {(\frac{D_{2}}{2} + H)}^{2} - \frac{1}{2} α {(\frac{D_{2}}{2})}^{2} - \frac{3}{4} Ht

(16)

Where, $H$ represents the central magnetic pole leg length which consists of maximum coil cross-section height $h_{1}$ and safety redundancy height $h_{2}$ . The maximum coil cross-section height $h_{1}$ can be expressed from $H = h_{1} + h_{2}$ as:

h_{1} = \frac{\sqrt{2 α A_{cu} + {(\frac{2 α D_{2} - 3 t}{4})}^{2}} - \frac{2 α D_{2} - 3 t}{4}}{α (1 + P_{2})}

(17)

Then, the outer diameter of the RMB stator $D_{4}$ can be determined as:

D_{4} = D_{3} + 2 c_{2} + 2 w

(18)

Where, $D_{3} = 2 \sqrt{{(\frac{D_{2}}{2} \cos (a \sin (\frac{t}{D_{2}})) + H)}^{2} + {(\frac{t}{2})}^{2}}$ and $w$ is the depth of stator welding groove. $c_{2}$ is the stator yoke thickness and it can be set as $c_{2} = \frac{t}{2}$ initially.

Step 4: Determining winding dimensions

Unlike the 8-pole RMB, where the magnetic pole and winding sizes are equal, the coils of the central and sub-magnetic poles in 12-pole RMB may cause mechanical interference. To mitigate this, the maximum available coil width for the central magnetic pole $w_{1}$ must be constrained. As is illustrated in Figure 6(a), the maximum width space of central magnetic pole coil should meet the relationship of:

w_{1} ⩽ P_{4} ((\frac{D_{2}}{2} + h_{2}) \tan (\frac{2}{3} Alpha) - \frac{t}{2} - t_{b})

(19)

Where, the factor $P_{4}$ is used to avoid the mechanical interference by assuming the available width for winding is partly occupied.

Figure 6.

(a) Illustrations for maximum winding width and (b) cross-sectional area parameters.

Once the maximum winding width is determined, the number of layers and rows can be calculated according the value of $N_{central}$ . The same process applies for sub-magnetic pole as well.

The preliminary designed parameters for the 12-pole RMB, based on the requirements for mass, rotational speed, dynamic balance level, and motor diameter, are summarized in Table 2.

Table 2.

Preliminary designed parameters.

Symbol	Description	Value	Unit
$D_{0}$	Rotor inner diameter	76	mm
$D_{1}$	Rotor outer diameter	96	mm
$δ_{0}$	Air gap	0.4	mm
$D_{2}$	Stator inner diameter	96.8	mm
$α$	Angle between magnetic poles	33	Degree
$t$	Central magnetic pole width	20	mm
$H$	Central magnetic pole height	36	mm
$N_{central}$	Number of turns	104	—
$L$	Axial length of stator	36	mm

Characteristics of the 12-pole RMB

A finite element method model (FEM) is developed and analyzed by MAXWELL software based on the parameters listed in Table 2.

Figure 7(a) and (b) illustrates the maximum magnetic flux density distribution and vectors when maximum current $I_{m}$ is applied to all E-type magnetic pole groups. The results show that both the magnetic flux density and the field vectors are uniformly distributed. When excited by the maximum current, the magnetic flux density reaches approximately 1.3 T in the magnetic pole leg, which slightly exceeds the value of $B_{m}$ .

Figure 7.

(a) Magnetic flux density distribution and (b) vectors distribution.

Figure 8 illustrates the interdependencies among electromagnetic force, control current, and displacement within a differential control group. In the central operating range of both control current and displacement, the current stiffness and displacement stiffness exhibit strong linearity; however, this linearity degrades as the silicon steel material approaches saturation under maximum current conditions.

Figure 8.

(a) The relationship between electromagnetic force, control current, and displacement and (b) the relationship between electromagnetic force and control current under different displacements.

To assess the impact of design parameters on levitation forces, a set of E-type magnetic pole groups is analyzed by simulation. As shown in Figure 9(a), the electromagnetic force increases with the thickness of the rotor $c_{1}$ and stator $c_{2}$ , peaking at approximately 10 mm. Figure 9(b) demonstrates that reducing the magnetic pole height $H$ results in an increased electromagnetic force, indicating that the magnetic pole height should be minimized, as long as design requirements are met. Additionally, increases in parameters $t$ and $L$ contribute to higher electromagnetic force, as illustrated in Figure 9(c). However, a larger value of $t$ may complicate the assembly of windings. Figure 9(d) depicts the relationship among inductance, the number of turns, and displacement. The inductance of the winding is directly influenced by the number of turns $N$ , with a smaller $N$ generally being preferable, as higher inductance makes it more challenging to achieve the required current.

Figure 9.

(a) The influence of c₁ and c₂ to electromagnetic force, (b) the influence of h₁ and h₂ to electromagnetic force, (c) the influence of t and L to electromagnetic force, and (d) the influence of N and $Δ δ$ to inductance.

Optimization design of 12-pole RMB

Based on the preliminary design results presented in Table 2, a multi-objective optimization is conducted using the Non-dominated Sorting Genetic Algorithm II (NSGA-II)³⁷ in ISIGHT, which has been widely applied to various engineering design problems and is often regarded as a benchmark against which other optimization algorithms.³⁸ The cost functions, design parameters, and constraints are described in the following section. The results of the optimization are subsequently presented and analyzed.

Objectives

In this optimization design, two conflicting objectives were selected to guide the performance improvement of the 12-pole Radial Magnetic Bearing (RMB): maximizing the magnetic force (equation (20)) while minimizing the stator volume (equation (21)). These objectives are critical for enhancing the bearing’s load-carrying capacity and ensuring compactness, which are essential for practical applications in high-speed rotating machinery. These two objectives are inherently conflicting, as increasing the magnetic force typically requires larger magnetic poles or higher currents, which in turn increases the stator volume, while minimizing the stator volume may compromise the magnetic circuit ability to generate sufficient force. This trade-off necessitates a multi-objective optimization approach, which was addressed using the NSGA-II algorithm to identify the Pareto front solutions, as discussed in the following sections.

max (F) = max (k_{i} Δ I + k_{δ} Δ δ)

(20)

min (V) = min [L (\frac{π (D_{4}^{2} - D_{3}^{2})}{4} + 4 Ht + \frac{8 Ht}{2})]

(21)

Selection of variables

The variables involved in the optimization process must be carefully selected, as they directly influence the efficiency, convergence behavior, and overall quality of the optimization outcomes. For the 12-pole RMB structure, seven variables were identified for analysis: magnetic pole angle (α), pole height (H), maximum control current (I), axial length (L), coil number of the central pole (N), magnetic pole width (t), and thickness of the stator yoke (c₂). To evaluate their impact on the optimization objectives, the optimal Latin Hypercube method^39,40 was applied with 200 sampling points, and the resulting Pareto effect plot is presented in Figure 10.

Figure 10.

Pareto response plot to magnetic force (in blue) and volume (in green).

The analysis reveals that the variable α has significantly less influence on both objectives compared to the other six variables. Consequently, the remaining six variables were selected for further optimization analysis.

Constraints

Constraints on magnetomotive force

The maximum magnetomotive force of central magnetic pole generated by $N_{central}$ and $I_{m}$ should be restricted as a function of maximum flux density $B_{m}$ , levitation gap $δ_{0}$ , and permeability of air $μ_{0}$ .

N_{central} I_{m} - \frac{4 B_{m} δ_{0}}{3 μ_{0}} ⩽ 0

(22)

Also, the magnetic flux density in the stator yoke and poles should be restricted below the saturation flux density $B_{s}$ , and they are expressed as:

\frac{3 N_{central} I_{m}}{2 (R_{central} + R_{side}) c_{2} L} - B_{s} ⩽ 0

(23)

\frac{3 N_{central} I_{m}}{(R_{central} + R_{side}) tL} - B_{s} ⩽ 0

(24)

Where, $R_{central} = \frac{δ_{0}}{μ_{0} A_{central}}$ and $R_{sdie} = \frac{2 δ_{0}}{μ_{0} A_{central}}$ are the air gap reluctance of the central magnetic pole and sub-magnetic pole.

Constraints on available winding space

The maximum winding space available between central and sub-magnetic pole should be constrained by assuming a winding space filling factor $λ$ as:

N_{central} \frac{3 π d_{w}^{2}}{8} - λ (\frac{1}{2} α {(\frac{D_{2}}{2} + H)}^{2} - \frac{1}{2} α {(\frac{D_{2}}{2})}^{2} - \frac{3}{4} Ht) ⩽ 0

(25)

Similarly, the winding space between two sub-magnetic poles should be constrained as:

\frac{π d_{w}^{2} N_{central}}{4} - λ (\frac{1}{2} β {(\frac{D_{2}}{2} + H)}^{2} - \frac{1}{2} β {(\frac{D_{2}}{2})}^{2} - \frac{Ht}{2}) ⩽ 0

(26)

Where, $β = \frac{π}{2} - 2 α$ .

Constraints on avoiding coil interface

The windings of the central and sub-magnetic poles may interfere with each other if the design is not properly optimized. To mitigate this interference, an anti-interference factor P₄, is introduced, based on the assumption that the winding width does not exceed 90% of the maximum available width w₁ (Figure 6(a)). This relationship can be expressed as:

\frac{π d_{w}^{2} N_{central}}{4 λ_{2} (h_{1} - 2 t_{b})} - P_{4} ((\frac{D_{2}}{2} + h_{2}) \tan (\frac{2}{3} Alpha) - \frac{t}{2} - t_{b}) ⩽ 0

(27)

Similarly, for sub-magnetic pole there is:

\frac{π d_{w}^{2} N_{central}}{8 λ_{2} (h_{1} - 2 t_{b})} - P_{4} ((\frac{D_{2}}{2} + h_{2}) \tan (\frac{1}{3} Alpha) - \frac{t}{4} - t_{b}) ⩽ 0

(28)

To avoid the conflict of two sub-magnetic poles, their dimensions have to be restricted as:

\begin{matrix} P_{4} ((\frac{D_{2}}{2} + h_{2}) \tan (\frac{1}{3} Alpha) - \frac{t}{4} - t_{b}) \\ - ((\frac{D_{2}}{2} + h_{2}) \tan (\frac{β}{2}) - \frac{t}{4} - t_{b}) ⩽ 0 \end{matrix}

(29)

Constraints on the winding feasibility

Additionally, constraints on the feasibility of the winding’s cross-sectional dimensions (width and height) must be incorporated, assuming that the available winding space is fully utilized (Figure 6(b)).

\frac{P_{4} ((\frac{D_{2}}{2} + h_{2}) \tan (Alpha) - \frac{t}{2} - t_{b}) (h_{1} - 2 t_{b})}{d_{w}^{2}} - N_{central} ⩽ 0

(30)

For windings arranged at sub-magnetic pole, this constraint can be expressed as:

\frac{P_{4} ((\frac{D_{2}}{2} + h_{2}) \tan (\frac{1}{3} Alpha) - \frac{t}{4} - t_{b}) (h_{1} - 2 t_{b})}{d_{w}^{2}} - \frac{N_{central}}{2} ⩽ 0

(31)

Optimization results

To address the two conflicting objectives of maximizing magnetic force and minimizing stator volume, the NSGA-II algorithm was employed to identify the Pareto front solutions, where no objective can be improved without compromising the other. The NSGA-II parameters used in the optimization process included a population size of 200, 50 generations, a crossover distribution index of 10, and a mutation distribution index of 20. As shown in Figure 11(a), the Pareto front illustrates the necessary trade-off between the objectives, with an increase in magnetic force inevitably leading to a larger stator volume. Since all solutions on the Pareto front satisfy the minimum magnetic force requirement of 598 N, the solution with the smallest stator volume was selected as the optimal design. Additionally, the iteration history plots for both objectives, presented in Figure 11(b) and (c), indicate that the first feasible solution was identified after 5306 iterations.

Figure 11.

(a) Graph of Pareto front solutions, (b) iteration history of force, and (c) stator volume.

The initial values, optimal values, and boundaries of the design variables are summarized in Table 3. The initial values were derived from the preliminary design process, with their lower and upper boundaries set to 0.1 and 2 times the initial values, respectively, to allow sufficient exploration during optimization. However, the upper boundary of the control current (I) was constrained by hardware limitations. The optimization results indicate that four variables—namely, the coil number of the central pole (N_central), magnetic pole width (t), pole height (H), and control current (I)—decreased after optimization, while the axial length (L) increased.

Table 3.

Design variables and boundaries.

Symbol	Description	Initial value	Lower bound	Upper bound	Optimal value
N _central	Turns in central magnetic pole	104	0.1 N_central	2 N_central	87
t	Central magnetic pole width	20 mm	0.1 t	2 t	19.6 mm
H	Central magnetic pole height	36 mm	0.1 H	2 H	28.89 mm
c ₂	Stator yoke thickness	10 mm	0.1 c₂	2 c₂	10.02 mm
L	Axial length	36 mm	0.1 L	2 L	52.41 mm
I	Maximum control current	2.5 A	0.1 I	2.5 A	2.39 A

To validate the performance of the optimized 12-pole RMB, FEM simulations were conducted using the optimal structural parameters obtained from the NSGA-II optimization. As illustrated in Figure 12(a), when the y-axis is excited with the maximum control current, the magnetic flux densities in the stator yoke, central pole, and sub-pole of the upper side of the E-type winding all remain below the saturation magnetic flux density B_s, ensuring the linearity of the generated electromagnetic force. Additionally, to confirm the accuracy of the simplified mathematical model in equation (7), the current stiffness K_i was evaluated through both FEM and mathematical simulations, with results compared in Figure 12(b). The comparison demonstrates strong agreement between the mathematical expression of K_i and the FEM simulation results for control currents ranging from 0 to 1.6 A, with errors increasing slightly beyond this range due to the saturation effects of the silicon steel. However, this error has minimal impact on the designed structure’s performance, as the maximum control current scenario is rarely encountered due to the electromagnetic force redundancy considerations in the preliminary design phase. Overall, the simulation results confirm that the optimized model meets the design requirements and is suitable for fabrication and subsequent testing.

Figure 12.

(a) Plot of magnetic flux density of the optimized structure parameters and (b) current stiffness comparison of FEM and mathematical calculation.

Experiment test

Based on the multi-objective optimization results, a prototype of the 12-pole RMB was fabricated, as shown in Figure 13. Two sets of the optimized 12-pole RMBs (RMB1 and RMB2) were integrated into a magnetic bearing motor to provide levitation and counteract gravitational forces. The motor operates at a rated power of 75 kW and a rotational speed of 25,000 rpm. Displacement and control current signals were measured using an eddy current sensor and a current sensor, respectively, to monitor the rotor’s position and the bearing’s response. A basic PID (Proportional-Integral-Derivative) controller was implemented, achieving stable levitation from the initial position on the backup bearing and demonstrating the effectiveness of the optimized design.

Figure 13.

The fabricated 12-pole RMB prototype and motor test rig.

As shown in Figure 14, when a step levitation command is applied at approximately 40 ms, the displacement signals of RMB1 in the X and Y directions change from −125 and −12 µm to the reference center position (0 µm) within 60 ms. The displacement in RMB2 takes longer (80 ms) to settle to the target position. Regarding the control currents during levitation (Figure 15), the X+ and Y+ currents of RMB1 increase and stabilize at approximately 2.04 and 2.52 A, respectively. Due to the differential control mode, the X− and Y− currents settle at 0.7 and 1.16 A within ∼60 ms. A similar trend is observed in RMB2, where the X+ and Y+ currents stabilize at 2.90 and 2.48 A, respectively. The successful levitation demonstrates the effectiveness of the optimized 12-pole RMB.

Figure 14.

Levitation performance of the two designed 12-pole RMBs: (a) for RMB 1 and (b) for RMB 2.

Figure 15.

History of control current of the two designed 12-pole RMBs: (a) for RMB 1 and (b) for RMB 2.

Conclusion and future work

In this paper, a framework for the design and optimization of a 12-pole RMB is presented to address the research gap in 12-pole RMB studies. Using an equivalent magnetic circuit method, the electromagnetic force mathematical model for the 12-pole RMB is established, and its effectiveness is verified by comparison with FEM results. The detailed preliminary design process is presented, and the influence of structural parameters on performance metrics, such as electromagnetic force and winding inductance, is analyzed using the FEM method. To solve the conflicting optimization problem of maximizing electromagnetic force while minimizing stator volume, the NSGA-II optimization algorithm is utilized to identify the Pareto front solutions. The optimization results demonstrate that the optimized structure meets the design requirements. Finally, the optimized 12-pole RMB is fabricated and tested in a magnetic levitation motor, where successful levitation confirms the effectiveness of the design and optimization process. This study offers a structured framework for both industry and academia, establishing a foundation for further research on 12-pole RMBs.

Future work will focus on developing a more complex model by considering the eddy current effect and optimizing structural parameters with losses, vibration, and stability margins as constraints, in accordance with ISO and API standards.

Footnotes

Handling Editor: Carmen Bujoreanu

ORCID iD

Zigang Deng

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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Structure design,analysis,and optimization of 12-pole radial magnetic bearing

Abstract

Keywords

Introduction

Modeling of 12-pole RMB

Working principle of 12-pole RMB

Mathematical modeling of 12-pole RMB

Preliminary design and analysis of 12-pole RMB

Preliminary design flow of 12-pole RMB

Step 0: Design inputs

Step 1: Initial parameters

Step 2: Loading capacity calculation

Step 3: Determining stator and winding parameters

Step 4: Determining winding dimensions

Characteristics of the 12-pole RMB

Optimization design of 12-pole RMB

Objectives

Selection of variables

Constraints

Constraints on magnetomotive force

Constraints on available winding space

Constraints on avoiding coil interface

Constraints on the winding feasibility

Optimization results

Experiment test

Conclusion and future work

Footnotes

ORCID iD

Funding

Declaration of conflicting interests

References