**Performance Enhancement of a Magnetic System in a Ultra Compact 5-DOF-Controlled Self-Bearing Motor for a Rotary Pediatric Ventricular-Assist Device to Diminish Energy Input** †

## **Masahiro Osa 1,\*, Toru Masuzawa 1, Ryoga Orihara 1 and Eisuke Tatsumi 2**

Received: 15 February 2019; Accepted: 13 April 2019; Published: 15 April 2019

**Abstract:** Research interests of compact magnetically levitated motors have been strongly increased in development of durable and biocompatible mechanical circulatory support (MCS) devices for pediatric heart disease patients. In this study, an ultra-compact axial gap type self-bearing motor with 5-degrees of freedom (DOF) active control for use in pediatric MCS devices has been developed. The motor consists of two identical motor stators and a centrifugal levitated rotor. This paper investigated a design improvement of the magnetic circuit for the self-bearing motor undergoing development in order to diminish energy input by enhancing magnetic suspension and rotation performances. Geometries of the motor were refined based on numerical calculation and three-dimensional (3D) magnetic field analysis. The modified motor can achieve higher suspension force and torque characteristics than that of a previously developed prototype motor. Oscillation of the levitated rotor was significantly suppressed by 5-DOF control over rotating speeds up to 7000 rpm with lower energy input, indicating efficacy of the design refinement of the motor.

**Keywords:** axial gap; self-bearing motor; double stator structure; 5-degrees of freedom active control; design refinement; ventricular assist device; pediatric

## **1. Introduction**

Mechanical circulatory support (MCS) is widely used for heart disease therapy. However, clinically available and implantable MCS devices are not applicable for pediatric patients, which have small body surface area (BSA < 0.7 mm2) due to anatomical limitations [1,2]. Almost all pediatric patients have to rely on using EXCOR pediatric ventricular assist device (VAD), which is a extracorporeal pulsatile flow pump developed by Berlin Heart Inc. in Germany [3]. Pulsatile devices with diaphragm and valve configurations potentially limit a device's lifetime and have the risk of thrombosis. Currently, there have been increasing research interests in pediatric heart treatment with implantable rotary MCS devices specifically designed for pediatric circulatory support [4]. In 2010, MCS device development for pediatric patients was supported as a national project named PumpKIN (Pump for kids, infants and neonates) in the United States (US) [1,2,4]. A tiny rotary MCS device (Jarvik2015) for pediatric

circulation is being developed by Jarvik Heart Inc. in the US [5–7]. However, the Jarvik device is now facing technical difficulties such as deterioration of mechanical durability, blood clotting and blood destruction, due to a mechanically contacting bearing to suspend a spinning rotor impeller. Hence, the development of next-generation MCS devices that can completely levitate a rotating impeller are in demand due to its high durability and better blood compatibility.

A magnetic suspension system is one of the strongest candidates to suspend the rotating impeller without mechanical contact. The biggest advantages of maglev technology are its high-speed motor drive, less heat generation due to no material wear, zero friction, less blood trauma, anti-thrombogenicity and increased mechanical reliability. In one of the earliest studies, 5-DOF-controlled maglev motors were developed [8,9]. The maglev system indicated high suspension stability, however, the device needed to be bigger due to the larger number of actuators: two radial magnetic bearings and two axial magnetic bearings. Miniaturization of the magnetic suspension system has a significant role in success of pediatric MCS device development. Reduction of actively controlled positions is a general strategy to miniaturize the magnetic suspension systems. For example, single DOF-controlled maglev systems, which are utilizing passive stability using magnetic coupling force or repulsive PM magnetic bearings to suspend the non-controlled DOF, were developed to achieve a simple structure and compact device size [10–16]. However, a larger number of passively stabilized axes potentially causes significant deterioration of suspension stability of the magnetic system. Hence, 2-degrees of freedom (DOF)-controlled radial maglev motors and 3-DOF-controlled double stator axial maglev motors have often been developed in blood pump applications [17–24]. These miniaturized maglev motors are successfully applied to implantable and extracorporeal MCS devices for adult patients, whereas further miniaturization of the maglev motors is required for use in rotary pediatric MCS devices. In extremely small magnetic suspension systems, there is a limitation of passive stabilization in multiple axes because the system cannot have sufficient capacity of passive stiffness and suspension force due to few spaces to have enough permanent magnet volume and turn number of control windings. Hence, a technical breakthrough is needed to achieve an ultra-compact magnetic system.

This study developed a compact pediatric MCS device with a novel self-bearing motor utilizing a 5-DOF-control concept that was newly developed in our laboratory, and the developed device demonstrated noncontact suspension and sufficient pump performance [25–28]. However, further improvement of magnetic suspension stability is necessary to achieve higher mechanical reliability and energy conservation system required for clinically applicable MCS devices. In this paper, design improvement of magnetic circuit for the 5-DOF-controlled self-bearing motor was investigated to enhance the magnetic suspension performance and energy efficiency by using theoretical calculation and three-dimensional (3D) finite element method (FEM) analysis. Static force and torque characteristics [28], dynamic suspension performance and energy consumption of the improved maglev motor with 5-DOF control are evaluated.

## **2. Materials and Methods**

## *2.1. 5-DOF-Controlled Self-Bearing Motor for Pediatric Ventricular Assist Device (VAD)*

## 2.1.1. Over View of Maglev Pediatric VAD with 5-DOF-Controlled Self-Bearing Motor

The 5-DOF-controlled self-bearing motor is driven as an axial gap type surface permanent magnet synchronous motor, which has 6-slot and 4-pole structure. Figure 1 shows a schematic of the self-bearing motor which is consists of two identical motor stators and a levitated impeller. The levitated impeller is axially suspended with the both stators. Integrated windings for suspension and rotation control are wound on each stator tooth. A motor torque and a suspension force are produced with double stator mechanism which can enhance motor torque and radial passive stability.

An axial position (*z*) and rotating speed (ω*z*) are actively regulated with a 4-pole control magnetic field. Radial positions (x and y) and tilting angles (θ*x* and θ*x*) are actively regulated with a 2-pole control magnetic field. 5-DOF of impeller postures are independently regulated by overlapping the different control magnetic fields in the magnetic gap [27]. A developed centrifugal blood pump for pediatric patients can regulate flow rate from 0.5 to 2.5 L/min against head pressure of around 100 mmHg at rotating speeds of 4500–5500 rpm.

**Figure 1.** Structure of pediatric ventricular assist device with the axial gap type double stator 5-degrees of freedom (5-DOF)-controlled self-bearing motor.

## 2.1.2. Characterization of Suspension Force and Torque

The motor produces axial suspension force and rotating torque with a single rotating magnetic field based on vector control algorithm. An axial position (*z*) of the levitated impeller is actively regulated by field strengthening and field weakening as shown in Figure 2. A rotating speed (ω*z*) of the rotor is controlled by conventional q-axis current regulation. The axial suspension force and the rotating torque are linearly produced with d-axis current id and q-axis current *iq*.

$$F\_z = k\_{Fz}(i\_d - i'\_d) \tag{1}$$

$$T\_{Oz} = k\_{T\ell z}(i\_{\emptyset} + i'\_{\neq}) \tag{2}$$

Inclination angles (θ*x* and θ*y*) and radial positions (*x* and *y*) of the levitated rotor can be controlled with p ± 2 pole algorithm. The control magnetic field can simultaneously produce an inclination torque and a radial suspension force. Inclination torque around the *y*-axis and the radial suspension force in *x* direction are produced with the double stator mechanism as shown in Figure 3. The magnitude and the direction of the inclination torque and the radial suspension force can be linearly regulated with respect to excitation current supplied to the top stator and the bottom stator as following equations.

$$T\_{\partial \mathbf{x}/y} = k\_{T\partial \mathbf{x}/y} \left( i\_{\text{top}} + i\_{\text{bottom}} \right) \tag{3}$$

$$F\_{x/y} = k\_{\rm Fx/y} (i\_{\rm top} - i\_{\rm botton}) \tag{4}$$

**Figure 2.** Axial position control by utilizing field strengthening and field weakening.

**Figure 3.** Inclination and radial position control with p ±2 pole rotating magnetic field.

*2.2. Suspension Force and Torque Enhancement with Modified Magnetic Circuit of the Maglev Motor*

## 2.2.1. Design Strategy of Suspension Performance Enhancement

The motor uses magnetic flux density produced by the permanent magnet as a main flux density to produce the suspension force and the rotating torque. Enhancement of the permanent magnet flux contributes to higher suspension force production. However, in miniaturized motor, there is difficulty to have sufficiently large cross-sectional area of the magnetic flux path, and it has possibility of deterioration of magnetic suspension force due to the magnetic saturation. Furthermore, excessively increased negative stiffness due to the high magnetic intensity potentially deteriorate controllability of the magnetic system. Hence, well design of the magnetic circuit which can keep a good balance between magnetic flux density produced by the permanent magnet and electromagnet is required to achieve sufficient magnetic suspension stability. In this study, design goal is to enhance the suspension force produced by the electromagnet without change of non-excited axial attractive force for avoiding instability caused due to the axial negative stiffness.

2.2.2. Design Refinement of Magnetic Circuit for the 5-DOF-Controlled Self-Bearing Motor to Enhance the Suspension Force, the Motor Torque and Reduce the Energy Input

Design improvement of a magnetic circuit for the 5-DOF-controlled self-bearing motor was performed based on following design strategy to enhance the magnetic suspension performance. (1) Keeping device size such as the outer diameter of 22 mm, the total height of 33 mm and the total volume of the previously developed prototype motor. (2) Maintaining the axial negative stiffness kz within ±10% of that produced by the previously developed prototype motor. (3) Maximizing the force coefficient in the axial direction ki defined as a slope of the suspension force to excitation current.

Geometries representatively characterizing the magnetic circuit of the self-bearing motor: pole height lp, pole cross sectional area Ap, magnetic gap length lg and permanent magnet thickness lm, were numerically designed with theoretical calculation and fixed by using 3-D FEM magnetic field analysis as shown in Figure 4. Height and cross-sectional area of the stator pole were determined as 9.3 mm and 17.0 mm2 based on the theoretical calculation. These geometries can increase in a turn numbers of coils and effectively maximize the force coefficient with slight change of the negative stiffness and the non-excited force. Parametric study in the magnetic gap length and the PM thickness were then performed. Variable parameters of the magnetic gap length of 1.3–1.7 mm and the PM thickness of 0.8–1.2 mm were chosen considering fabrication. Each combination of the magnetic gap length and PM thickness were simulated, and the non-excited axial negative stiffness kz and the force coefficient ki were estimated. A suspension index is defined as ratio of the force coefficient to the negative stiffness, which indicates rotor displacement with respect to excitation current of 1 A. The geometry which can achieve the biggest suspension index and satisfy the above design strategy was chosen as optimal design of the motor, which achieves well suspension performance with lower energy input.

**Figure 4.** Variable geometries of magnetic circuit for the self-bearing motor in design improvement using 3-D finite element method (FEM) magnetic field simulation.

The numerically estimated force coefficient and non-excited attractive force of the self-bearing motor with different geometries in the magnetic gap length and the PM thickness are shown in Figure 5. The suspension index in each motor geometry is listed in Table 1. Red and yellow colored cells indicate satisfying design requirements in the negative stiffness and the force coefficient. The optimal geometry in the FEM simulation to maximize the force coefficient (ki < 1.2 N/A) and maintain the axial negative stiffness (15.2 N/mm < kz < 18.6 N/mm) is the shortest magnetic gap length of 1.3 mm and the thinnest PM of 0.8 mm as shown in red colored cell in Table 1. The force coefficient and the negative stiffness of the optimally designed motor are 2.0 N/A and 17.1 N/mm. Deterioration of the magnetic flux density with reduction of the PM thickness can be compensated by reducing the magnetic gap length. The shorter magnetic gap length and the thinner PM thickness can reduce magnetic resistance for the electromagnet and effectively enhance the magnetic suspension force production with excitation current. The geometries of the previously developed prototype motor and the improved motor with the final design are summarized in Table 2.

**Figure 5.** Force coefficient and negative stiffness of the motor with different magnetic gap length and permanent magnet thickness.

**Table 1.** Estimated results of suspension index with different permanent magnet thickness and magnetic gap length.

**Table 2.** Motor geometries of the prototype motor and the improved motor.

*2.3. Developed System of 5-DOF-Controlled Maglev Motor with Modified Magnetic Circuit*

## 2.3.1. Fabrication of 5-DOF-Controlled Maglev Motor

A 5-DOF-controlled self-bearing motor for pediatric VAD shown in Figure 6 was developed referring to motor geometries determined by using 3D FEM magnetic field analysis. The outer diameter and the total height are 22 mm and 33 mm. The length of magnetic gap of the developed motor is set to 1.3 mm. The material used for magnetic core of the motor stator and the rotor back iron is soft magnetic iron (SUY-1). The permanent magnets of 0.8 mm thickness are made of Nd- Fe-B, that has coercivity and residual flux density of 907 kA/m and 1.36 T, respectively. Concentrated cupper windings of 105 turns are wound on each stator tooth. Pump clearance between the pump casing and levitated rotor in the axial and radial direction are 0.3 mm and 0.5 mm.

**Figure 6.** Developed 5-DOF-controlled maglev motor with modified magnetic circuit. Pump casing with sensor holder, motor stator and rotor with permanent magnets.

## 2.3.2. Control System for Magnetic Levitation and Rotation with Digital PID Controllers

Digital PID controllers are implemented on a microprocessor board MicrolabBox (dSPACE GmbH, Paderborn Germany) with MATLAB/Simlink for 5-DOF active control. Figure 7 shows a schematic diagram of a 5-DOF control system. An axial position and inclination angles around the x and y axes of the levitated rotor are measured by three eddy current sensors (PU-03A, Applied Electronics Corporation). Radial positions of the levitated rotor in x and y direction are measured with other two eddy current sensors. A rotating angle of the levitated rotor is determined by outputs of three Hall effect sensors (Asahi KASEI Corporation) with a sensitivity of a 30-degree electrical angle. The rotating speed is calculated by time derivative of the rotating angle. Required current to produce the control magnetic flux density integrating three phase two-pole field and three-phase four-pole field synchronized with rotating PM field is calculated with PID controllers and is independently supplied to each coil by power amplifier (PA12A, Apex Microtechnology Corporation). Sampling and control frequency is 10 kHz. Control gains of the digital PID controllers for magnetic suspension and rotation were determined based on the previously measured motor suspension force and torque characteristics, and then, manually tuned in dynamic performance evaluation.

**Figure 7.** Schematic diagram of control system for 5-DOF-controlled maglev motor with position sensor, PID controller implemented in the microprocessor and power amplifier.

A block diagram for axial position and rotation control, inclination angle and radial position control are shown in Figures 8 and 9, respectively. Positive and negative d-axis current are determined by a PID feedback loop to produce an axial suspension force. q-axis current of both stators is regulated with a PI feedback loop for a conventional rotating speed control. Required current for inclination angle and radial position control are calculated by the other four PID feedback loops to determine amplitude and phase angle of two-pole rotating magnetic field produced by the top and bottom stators. PID gains for the position control and PI gains for the rotating speed control were set using a limit sensitivity method, and then manually tuned as shown in Table 3. Control gains of PID/PI controller of the previously developed prototype motor are also listed in Table 3.

**Figure 9.** Block diagram of feedback loop for the inclination and the radial position control based on P +/- 2 pole algorithm.

**Table 3.** Control gains for impeller positioning and rotating speed regulation.

## *2.4. Magnetic Suspension Performance Evaluation of the Newly Developed Maglev Motor*

2.4.1. Magnetic Flux Distribution Measurement and Static Magnetic Suspension Force and Torque Characteristics Measurement

Magnetic flux distribution produced by the rotor permanent magnets in the magnetic gap was measured without excitation. Four/two pole magnetic flux density produced by the electromagnet at excitation current of 1 A were then measured without the rotor permanent magnets. After that, static magnetic suspension force and torque characteristics: an axial negative stiffness kz, a radial stiffness kr, and suspension force of the designed motor was evaluated at excitation current of 1 A and magnetic gap length of 1.3 mm. The axial and radial suspension force were measured with load cell, and the inclination torque was calculated as a product of the measured force and rotor radius.

## 2.4.2. Dynamic Characteristics of Developed 5-DOF-controlled Maglev Motor

The rotor was magnetically levitated in water medium with 5-DOF control. The water flow was shut off by closed outlet port of the centrifugal pump to evaluate basic magnetic suspension characteristics by minimizing hydraulic force disturbance. Magnetic suspension performance with respect to increase in the rotating speed of the rotor was evaluated. The rotating speed was increased from 1000 rpm to 7000 rpm. Oscillation amplitude in axial direction and radial direction, maximum inclination angle around x and y axes, and power consumption of the motor during magnetic levitation and rotation were evaluated. The maximum oscillation amplitude was defined as half of the peak-to-peak value of rotor vibration.

## **3. Results**

The measured magnetic flux density is shown in Figure 10. The magnetic flux density produced by the rotor permanent magnets did not significant difference between the prototype motor and the improved motor. In contrast, the magnetic flux density produced by the electromagnet of the improved motor significantly increased. The peak of the four/two pole magnetic flux density produced by the improved motor increased by 61% and 76% compare to the prototype motor.

Static suspension characteristics: stiffness, suspension force and torque, of the developed maglev motor which has the modified magnetic circuit and the previously developed maglev motor are shown together in Figure 11. The axial negative stiffness of the improved motor decreased by 17%, however, the radial stiffness was not significantly changed. The deterioration of the radial passive stability did not occur. The axial suspension force increased by 50 %, and the radial suspension force slightly decreased. Both the inclination torque and the rotating torque increased by 84% and 34%, respectively.

**Figure 10.** Magnetic flux density distribution in the magnetic gap. (**a**) Four pole magnetic flux density produced by the rotor magnet. (**b**) Four pole magnetic flux density for the axial position and the rotation control. (**c**) Two pole magnetic flux density for the radial position and the inclination control.

**Figure 11.** Static suspension characteristics of the developed maglev motor. (**a**,**b**) Axial and radial stiffness. (**c**,**d**) Magnetic suspension force in axial and radial direction at excitation current of 1 A. (**e**,**f**) Torque characteristics at excitation current of 1A.

The improved motor successfully achieves non-contact levitation and rotation up to the rotating speed of 7000 rpm. The maximum axial and radial oscillation amplitude and the maximum inclination angle around x and y axes with respect to the increase in the rotating speed of the levitated rotor are shown in Figures 12–14. In the lower speed range of 1000–3000 rpm, the oscillation amplitude of the levitated rotor was slightly increased in the prototype motor. In contrast, the oscillation amplitude in axial and radial direction, and the inclination angle of the improved motor were significantly suppressed around 20 μm, 100 μm and 0.4 degrees over every operational speed by enhancement of the suspension characteristics. The power consumption of the developed motor during magnetic levitation and rotation was shown in Figure 15, and it was in the range of 1–6 W at the rotating speeds of 1000–7000 rpm.

**Figure 12.** Maximum oscillation amplitude of the levitated rotor in axial direction with respect to increase in the rotating speed.

**Figure 13.** Maximum oscillation amplitude of the levitated rotor in radial direction with respect to increase in the rotating speed.

**Figure 14.** Maximum inclination angle of the levitated and rotated rotor around x and y axes with respect to increase in the rotating speed.

**Figure 15.** Power consumption of the developed maglev motor with respect to the increase in the rotating speed.

## **4. Discussion**

Impeller suspension technique using magnetic suspension much contributes to enhance device durability and blood compatibility of the rotary MCS devices. In an ultra-compact maglev motor, optimization of the magnetic circuit for suspension system plays a significant role in the next generation rotary pediatric VADs development.

The lower negative stiffness of the magnetic system can be effective to reduce the suspension index, whereas the decreased stiffness will cause the deterioration of the passive stability in rotor radial direction and the motor torque. The negative stiffness was strategically adjusted to be maintained by keeping peak level of the magnetic flux density produced by the rotor permanent magnets in order to avoid the motor deterioration above mentioned in this study. Decrease in the pole surface area, magnetic gap length and permanent magnet thickness successfully played a significant role in enhancement of the magnetic flux density produced by the electromagnet due to increase in the turn numbers of coils and reduction of the magnetic resistance of the magnetic circuit. Magnetic saturation in the rotor iron could not be occurred because the magnetic flux density ratio of the improved motor and the prototype motor almost uniform in arbitrary angle in the air-gap.

The developed maglev motor successfully achieves the much higher suspension force coefficient ki maintaining the negative stiffness kz in the axial suspension characteristics. The suspension index of ki/kz = 0.47 mm/A is significantly higher than that of the previously developed motor (ki/kz = 0.27 mm/A). The axial negative stiffness was lower than estimated result. One of the causes of the above may have been deteriorated permanent magnet flux caused by reduced magnet volume due to coating thickness. Although the radial suspension characteristics slightly decreased, the deterioration of the total magnetic suspension performance will not occur because the magnitude of the radial suspension force is absolutely small. Even the radial suspension force produced by the newly developed motor is much effective to suppress the resonance and disturbance. The grossly increased inclination torque and the rotating torque due to the increase in the control magnetic flux density will contribute to achieve better suspension stability and low energy consumption.

The axial and radial oscillation amplitude and the inclination angle of the levitated rotor were small enough for rotary blood pump operation to prevent blood trauma. The improved motor demonstrated better magnetic suspension performance with the lower PID control gains than that of the previously developed prototype motor, that is indicating efficacy of the magnetic circuit design refinement to achieve higher mechanical reliability and lower energy input. First, resonance around 50 Hz, which is calculated from the measured radial stiffness and mass of the rotor implies that the resonance will not influence actual pump operation due to lower frequency than the operational frequency of the pediatric pump. The oscillation of the impeller was well suppressed, and resonance peak was not found at any rotating speed. Safeness against the resonance was experimentally verified. Frequency response measurement in all actively controlled axes should be required to investigate advanced dynamic characteristics as a next step.

The power consumption at the operating speed range (4000–5500 rpm) of the pediatric VAD was 2.4–4.9 W. The improved motor with modified magnetic circuit achieved more than 50% reduction of the power consumption compare with that of the previously developed motor. The decreased power consumption should be small enough for pediatric pump operation. The input power increased as increase in the rotating speed. This is due to increase in the copper loss caused by increased suspension current at higher speed rotation and the iron loss. Material change of the stator magnetic core will effectively reduce power consumption at higher rotating speed. In actual pump operation, required rotating torque increases to produce hydraulic output. In contrast, power consumption for suspension may become smaller due to reduced rotor oscillation by higher viscus damping of blood. The blood is filled in the pump cavity, however, magnetic properties of the blood do not affect to the magnetic performance of the self-bearing motor. Total energy input will be evaluated during circulation in future.

## **5. Conclusions**

The ultra-compact 5-DOF-controlled self-bearing motor has been developed for pediatric MCS devices. Shortened magnetic gap and PM thickness effectively increased the control magnetic flux density due to the reduction of magnetic resistance maintaining PM magnetic flux density. In addition, increase in the turn number of the control windings almost double also played a significant role in enhancement of the control flux density production. As a result, the static magnetic suspension characteristics were successfully increased by up to 34–84% by refining the magnetic circuit of the motor. The dynamic magnetic suspension performance and further stable magnetic suspension with high speed rotation is successfully indicated due to the improved force and torque capacity with respect to the excitation current. Energy input was drastically reduced by less than half (1–5 W) because of the reduction of copper loss with low input current. The results indicate the efficacy of the magnetic circuit refinement of the proposed 5-DOF-controlled self-bearing motor. As a next step, dynamic suspension characteristics during pumping in actual circulation condition and pump performance of pediatric rotary VAD will be investigated.

**Author Contributions:** M.O. wrote the paper; T.M. and E.T. contributed in designing the experimental devices; M.O. performed 3D FEM analysis; M.O. and R.O. conducted the experiments; M.O. analyzed the numeric simulation and experimental results.

**Funding:** This work was supported by Japanese Society for the Promotion of Science (JSPS) KAKENHI Grant-in-Aid for Young Scientists (B) Grant Number 16K18036.

**Conflicts of Interest:** The authors declare no conflicts. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

## **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Optimal Magnetic Spring for Compliant Actuation—Validated Torque Density Benchmark †**

## **Branimir Mrak 1,2,\*, Bert Lenaerts 2, Walter Driesen 2 and Wim Desmet 1,3**

Received: 18 January 2019; Accepted: 18 February 2019; Published: 22 February 2019

**Abstract:** Magnetic springs are a fatigue-free alternative to mechanical springs that could enable compliant actuation concepts in highly dynamic industrial applications. The goals of this article are: (1) to develop and validate a methodology for the optimal design of a magnetic spring and (2) to benchmark the magnetic springs at the component level against conventional solutions, namely, mechanical springs and highly dynamic servo motors. We present an extensive exploration of the magnetic spring design space both with respect to topology and geometry sizing, using a 2D finite element magnetostatics software combined with a multi-objective genetic algorithm, as a part of a MagOpt design environment. The resulting Pareto-optima are used for benchmarking rotational magnetic springs back-to-back with classical industrial solutions. The design methodology has been extensively validated using a combination of one physical prototype and multiple virtual designs. The findings show that magnetic springs possess an energy density 50% higher than that of state-of-the-art reported mechanical springs for the gigacycle regime and accordingly a torque density significantly higher than that of state-of-the-practice permanently magnetic synchronous motors.

**Keywords:** magnetic spring; optimal design; component benchmarking; compliant actuation; parallel elastic actuators (PEA); series elastic actuators (SEA)

## **1. Introduction**

The principles of elastic actuation, first introduced by Alexander et al. [1], whether in series [2] or in parallel [3] elastic actuators have been consistently proven to improve actuator performance in service robotics. These systems rely on the high torque and force density of mechanical springs to reduce peak power requirements and to improve the actuator's energy efficiency. For example, in work done by Mettin et al. [4], the energy consumption is reduced by 55%. The goal of this paper is to offer a robust spring solution, in the form of magnetic springs, that can extend the use of elastic actuation from service robotics to widespread industrial robots but also a much broader family of highly dynamic industrial motion systems.

A mechanical spring stores energy as the potential energy of elastic deformation. Spring design for highly dynamic loads in industrial use is typically limited by the long lifetime requirements and often leads to suboptimal designs for the purposes of elastic actuation. Conventionally, it was considered that for some metals there is a stress level called the fatigue limit, that can be sustained with an infinite lifetime [5]. Nowadays, this value is still often used in the design together with the stochastic design methods. However, the existence of a fatigue limit has been disputed even in the lab environment due to inclusions in the crystal lattice [6] of steel. Local stresses can lead to fatigue in any kind of metallic springs [5–8] and industrial environments impose additional risks (i.e., corrosive environment, temperature variations, mechanical handling, manufacturing limitations etc.). Often, high safety factors are employed to guarantee a robust design for a full product line, leading to heavy springs with high inertia.

Although the functionality of the magnetic spring (Figure 1) can be compared to that of a mechanical spring, the underlying physical principles are utterly different. Magnetic springs store potential energy in the magnetic field of permanent magnets (PM), where no fatigue failure mechanism is involved and thus have a virtually infinite lifetime [9], assuming the device is properly designed. This allows the use of compliant actuation concepts [10] in highly dynamic industrial applications with stringent lifetime demands.

**Figure 1.** Conceptual drawing of (**a**) a rotational motion (torsional) magnetic spring and (**b**) linear motion (translational) and figure indicating torque and force generated due to the displacement.

With elastic actuators, it is possible to deliver more mechanically reactive power to the system, under the assumption of a higher torque density of springs compared to motors. Considering the evident benefit of using mechanical springs in service robotics in improving dynamic behavior, it is necessary to prove that magnetic springs have the same or higher energy density than conventional solutions with mechanical springs, in order to showcase their potential for the design of industrial motion systems.

Some of the target applications are torque oscillation, compensation in continuous rotation in internal combustion engines and windmills, reciprocating and intermittent motion in weaving looms [11], fast switching valves [12] (valvetrains in internal combustion engines), reciprocating pumps and compressors [13] and other tools and machines with a highly dynamic reciprocating motion. Additionally, magnetic springs have been reported for use in vibration reduction and vibration isolation [14] as well as for static load compensation [15]. It is worth mentioning that magnetic springs are topologically identical to passive magnetic bearings (PMB) and magnetic clutches. The main difference is the magnetic load point of the permanent magnets: in a magnetic spring the magnets are loaded over the entire B-H curve in each loading cycle, while for PMB and clutches the operating point remains constant for a constant mechanical load.

Unlike the previous efforts on the topic, where effort was focused on a specific use, this paper studies the optimal design of a magnetic spring in more detail and demonstrates systematically the impact of a magnetic spring on the performance of highly dynamic industrial actuators. This article is based on conference paper [16] where the optimal component design methodology was presented. That methodology was extended with a more elaborate, reproducible validation campaign including dynamic validation data, and multiple virtual optimal design points in requirement space. Additionally, for the purposes of benchmarking, more stress was put on the experimental validation and virtual validation using models of differing complexity. For the same reason, the mechanical and magnetic spring, including temperature effects on both the magnetic and mechanical springs, is considered. The closed form magnetic spring scaling model with model limitations is presented, as opposed to the intuitive yet incomplete model in the conference paper, making the experimental validation fully

reproducible. Finally, in the discussion section, there is a significant amount of new benchmarking data for torque density comparison of magnetic springs and permanent magnet synchronous motors (PMSM). In addition, the exact data points and the designs will be made available via a link or in the addendum.

## **2. Materials and Methods**

Within this article, the focus is primarily on the component design cycle but we will also present its complementarity with the system design cycle (Figure 2). The main subject of this study is rotational magnetic springs, although some of the considerations regarding energy density can also be translated to linear magnetic springs. Regarding the environment where the magnetic springs can be used, we consider that due to the limitations of permanent magnetic materials, environments where PMSM can operate are considered to be suitable for magnetic springs.

**Figure 2.** The co-dependent nature of the system design and component design cycles through linked modelling approaches.

Although the FE model of a detailed design geometry is an indispensable tool for component design, in system optimization the computational cost of finite elements (FE) can be prohibitively expensive. On the other hand, a scalable 1D dynamic model of a magnetic spring is the ideal model for the sizing of different drivetrain components and system optimization. Therefore, we define a 1D scalable model based on first principles, where cost and inertia of a magnetic spring are calculated directly from required reactive energy. This model can be iteratively updated based on the FE model results, as a result of the virtual validation where a 1D model is compared to optimal component designs coming from component optimization design.

A standard way to compare energy-storing devices is a Ragone chart [17]. It typically shows the tradeoff between energy density and power density, i.e., some energy storage components should be used when a high energy density is required (e.g., Li-ion batteries) and others when high instantaneous power is required (supercapacitors, flywheels). The bottleneck of such a static approach when it comes to highly dynamic drivetrains is the disregard for lifetime and system dynamics. In the highly dynamic applications targeted within this study, the mechanical power delivered to the system is

significantly limited by the torque density of the actuator i.e., the ratio of torque limitation and inertia of the said actuator.

Therefore, it is necessary to know the inertia of the spring alongside the torque characteristics. Phenomenologically we can analyze the energy density of a spring. For elastic springs this will be the surface under the stress-strain curve (for the linear elastic model where the relation of stress and strain is linearly described by Young's modulus). Equivalently, for an idealized magnetic spring, the energy density is equivalent to the surface under the BH curve (Figure 3), which can be calculated as

$$E\_{\text{max}} = \int\_{H\_{\text{c}}}^{0} \mathcal{B}(H) dH \approx 2BH\_{\text{max}} \tag{1}$$

where *B* is the magnetic flux density and *H* is magnetic field strength.

**Figure 3.** The potential energy of springs; permanent magnet energy density calculated from B-H characteristic is a measure of the maximum theoretical energy density of a magnetic spring.

Two assumptions about the magnetic spring have to be made in order to create a 1D scalable model. First of all, equal distribution of magnets between stator and rotor, resulting in perfect canceling of the magnetic field in the magnets in the case where maximum potential energy is stored within the magnet. Secondly, a fixed form factor of the rotor, following the 1st assumption and optimal rotor diameter achieved from FE simulation. It is important to note that variation of a permanent magnetic energy density of 30.79% for a 100 ◦C difference can result in a significant stiffness variation with temperature, while mechanical springs will normally have less than 5% [18] for the same region. Keep in mind that the magnetic springs are not expected to generate significant heat, yet, for an expected environmental variation of ±20 ◦C, the magnetic spring will have a variation of ±5%. Although the SmCo material has a higher Curie temperature and can allow for a slightly higher operational temperature, this large stiffness variation and the possibility of demagnetization limits the magnetic spring from operating in a high-temperature environment where mechanical springs face less severe limitations.

Furthermore, realistic designs of a magnetic spring will always have a lower energy density than the maximum theoretical limit, due to effects like fringing and flux leakage. Therefore, we can define the design efficiency as an energy density ratio of a realistic magnetic spring and an ideal magnetic spring

$$
\eta\_{mat} = \frac{E\_{FE}}{E\_{max}}\tag{2}
$$

and use it for 1D model correction based on FE results.

For the realistic embodiment of the magnetic spring concept, there is a range of feasible variants, both continuous (geometry sizing) and discrete (topological). By permutation of the discrete variants such as the PM materials in Table 1, or rotor and stator topologies shown in Figure 4, we can generate a number of topologies (Table 2), of which a number can be pruned out early in the design.

**Table 1.** Overview of considered permanent magnet materials.

1 sintered Neodymium Iron Boron (NdFeB); 2 sintered Samarium Cobalt (SmCo), both anistropic material with limited magnetization; 3 plasto bonded NdFeB, isotropic material with free magnetization.

**Figure 4.** Overview of parametrized PM rotor topologies used in design optimization; (**a**) arc surface mounted magnets; (**b**) rectangular surface mounted magnets; (**c**) ring magnet—special case of arc surface mounted magnet; (**d**) buried arc magnets; (**e**) buried rectangular magnets; (**f**) internal magnets.

For instance, surface mounted topologies are most suitable for achieving high torque density, and so are high energy density PM materials. However, in the case of PM materials, sintered NdFeB offer only limited magnetizations and are, as such, limiting in design options. The possibility to have more varied and better-suited magnetization is also why bonded rare-earth magnet solutions were studied. Of the listed topologies, the most promising were optimized and studied in more detail using MagOpt software [19].

When setting up the design specifications, it important to note that magnetic spring will not necessarily have a linear characteristic. In fact, except for small strokes around equilibrium positions, it is more likely to produce a quasi-sinusoidal characteristic. The above mentioned linear region can be extended by specific geometries of the magnet and back-iron. However, it has been noted that this can lead to lower design efficiency. Additionally, it is not a given fact that a linear characteristic is the most suitable solution for a given application case. An example of utilizing nonlinear spring can be found in Reference [20] where stable and unstable equilibria of magnetic spring can be used instead of a locking mechanism. Under this consideration, we need an alternative to spring stiffness to translate the system design specifications into component design specifications.

Specifying stroke and potential energy of a spring is adequate since it does not over-constrain the optimization problem by imposing a desired torque characteristic. The magnetic spring potential energy can be evaluated from torque characteristic and stroke as

$$E = \int\_{\theta\_1}^{\theta\_2} T(\theta)d\theta. \tag{3}$$

In order to evaluate each design variant, a 2D magnetostatics model of the geometry is calculated (Figure 5), for a range of *θ* sufficient to capture the desired rotational orders. Normally, odd higher orders, (3rd and 5th harmonic) are present for symmetric sine distortion. Therefore, in this analysis anywhere from 11 up to 21 *θ* points were used for a single design evaluation, with the lower numbers proven to be sufficient. For long rotors with the aspect ratio of length to diameter of more than two, the 2D approach should be sufficient, as cap effects can be disregarded. For disc geometries, on the other hand, it is necessary to use a 3D FEM. Since we are interested in high bandwidth actuators, it makes sense to focus on low inertia, long shaft solutions.

**Figure 5.** Overview of parametrized PM rotor topologies used in design optimization with surface mounted topologies a–c being most suitable for high torque density; (**a**) anti-aligned magnets resulting in unstable equilibrium with maximum energy stored in PM; (**b**) aligned magnets resulting in stable equilibrium; no energy stored in PM.

For each FEM evaluation, a list of metrics of interest can be calculated, either by pre-processing the specifications and the geometry or by post-processing the FEM solution. The considered design metrics are:

	- -Stored energy
	- -Stroke
	- -Higher harmonic content (Fourier decomposition/THD)

The main objective of the design is to make a spring that fits the described energy and stroke specifications while minimizing inertia and cost. Within this article, the discussed cost of magnetic spring is merely the bulk material cost and is as such most useful for comparing different topological variations of magnetic springs, but also to get a first, rough idea of the magnetic spring cost in an industrial motion system. Although, in the latter case, other cost components, such as development, manufacturing and installation costs should be considered. All of these factors are heavily influenced by the volume of production and other economic factors. The cost comparison of different magnetic spring topological variations is considered valid, under the assumption that all of the considered sintered magnet geometries use the same manufacturing technology, especially with respect to magnetization i.e., only straight or diametrical magnetizations are considered. For a thorough design optimization, the MagOpt package [19] was used together with an opensource 2D FE solver for magnetostatic problems [21]. Other listed metrics were monitored for reasons of design safety (demagnetization) and possible unwanted dynamic effects (higher harmonic content). So far, loss models have not been considered, assuming that the efficiency of a magnetic spring is very high compared to a servo-drive since ohmic losses and the drive losses are completely avoided [22] in magnetic springs.

In order to validate the above described modeling approach, a prototype of a magnetic spring has been built (Figure 6). A magnetic spring consists of two diametrically magnetized ring NdFeB, N42H magnets, one on the stator and one on the rotor, with soft magnetic back iron to prevent flux leakage. For testing modularity, the spring has a built-in deep groove ball bearing. The bearing adds to the losses in the magnetic spring, which should be avoided in future designs, with a higher level of spring integration into the existing drivetrain.

**Figure 6.** Explosion view of the prototyped magnetic spring design.

Experimental validation of the following modeling approach was conducted with a test rig, in Figure 7, that consists of a position controlled PMSM motor (1), driving an inertial wheel (3) with the assistance of spring (4). The torque sensors (2) are installed between the motor and the spring, and spring and the flywheel, using bellow couplings to avoid alignment issues or over-constrained rotation axes. Both dynamic and static experiments were conducted using the same setup. Note that here below couplings are adding serious elasticity in the system between the PMSM rotor and magnetic spring rotor but also the magnetic spring rotor and flywheel. This stiffness of the below couplings is, however, several orders of magnitude higher than that of the used magnetic spring and as such is not relevant for primary dynamics due to the reciprocating motion. For monitoring of power flows, the sensors (encoders and torque sensors), described in Table 3, are used together with fully observable controller inputs.

**Figure 7.** Experimental test rig consisting of a (1) position controlled PMSM, (2) torque sensors, (3) flywheel—load and the developed prototype of a magnetic spring (4).

**Table 3.** Experimental setup sensor specifications.

## **3. Results**

## *3.1. Component Design Experimental Validation*

The measurement results (Figure 8) show a good qualitative and quantitative fit of the static measurement and a good qualitative fit with respect to the low loss hypothesis. In Figure 8a, a slight skewing of the sinusoidal curve is visible. This phenomenon is related to the eccentricity of the magnetic center of design and the mechanical rotation axis due to the manufacturing tolerances and it can be captured in static stiffness modelled as a skewed sine due with single order eccentricity

$$T\_{st}(\theta) = A \sin\left(\frac{2\pi\theta}{T} + \Delta T \sin\frac{\theta}{N}\right) \tag{4}$$

where *Tst* is the static torque of the magnetic spring as a function of angle *θ*. *A* is the torque amplitude in Nm. *T* is the period of the spring torque characteristic in radians, and depends on the pole pair number of the magnetic spring. Δ*T*, in radians, represents the skewing of the characteristic.

**Figure 8.** Component validation (**a**) static measurement of magnetic spring torque characteristic (**b**) dynamic measurement for identification of spring inertia and losses.

More complex aberration of the center of rotation can be captured with more addends in the sine argument, written as a Fourier decomposition, although, the most common issue of static alignment of mechanical and magnetic rotational axes results in a synchronous rotational order where *N* = 1. In Table 4, the identified parameters show that, apart from skewing, the peak torque value has less than 1% error compared to the FE model.

**Table 4.** Component design validation results.

1 CAD drawing of spring prototype; 2 based on bearing lubricant viscosity; 3 based on sliding torque; both from SKF model for W 61902-2Z bearing under C load.

For dynamic component identification (Figure 8b), torque and position measurements are filtered using 4th order 0-phase low pass filters with cutoff frequency at 200 Hz. The model parameters are fitted using Opti toolbox [Opti] non-linear least squares. The model of the simple magnetic spring in a direct dynamic form can be written as

$$T\_{dyn} = T\_{sl}(\theta) - J\ddot{\theta} - \mathcal{c}\_{\mathbb{P}}\dot{\theta} - T\_{\mathbb{C}}\left(\dot{\theta}\right),\tag{5}$$

where *TC*, is dynamic Coulomb friction with hysteresis effect. The principal intrinsic losses of the magnetic spring are expected to be caused by the velocity proportional eddy currents in the permanent magnets. This way these can be set apart from the bearing friction that is dominated by a sliding and rolling friction that is constant above a certain speed (breakaway torque) and modeled as *TC*. Additionally, viscous friction due to the lubricant viscosity, also contributes to bearing losses. The results of the dynamic parameter identification show that the losses are primarily dominated by the bearing friction *TC*. Moreover, the speed proportional component is lower than the anticipated lubrication viscosity, proposed by the a priori bearing model [23]. Therefore, based on this experiment, it is impossible to discern between bearing losses and intrinsic magnetic spring losses. Nevertheless, a clear conclusion is that magnetic spring losses are negligible compared to the other energy sinks in the highly dynamic drivetrain.

The magnetic spring assisted drivetrain shown in Figure 7 can be operated between the unstable equilibria, similar to a parallel elastic actuator with a locking mechanism [3] or an inverted pendulum. The system is operating as follows (Figure 9). At *t* = −0 s, the load is held in a stable equilibrium. Initially, an FF torque pulse is applied together with a negative damping controller in order to excite the natural resonance of the system (phase 1. Start-up). Due to the negative damping, the load is slowly brought in the neighborhood of the unstable equilibrium where a stable PID controller is switched on in order to hold the load in position with 0-torque control (phase 2. 0-torque wait).

Once a reciprocating motion is required, the controller starts to operate in a catch-release fashion with a small FF torque pulse initiating the motion and pushing the load towards the next unstable equilibrium. Due to the magnetic spring torque, the load is accelerated until reaching the middle point, where the spring starts to decelerate it. Upon reaching the surroundings of the next unstable equilibrium, the motor is activated again, with a feedback controller, in order to stabilize the load in the endpoint. In this fashion, the motor is delivering only the bare minimum of the required torque.

The same motor operating without a magnetic spring while driving the same load (Figure 9), requires a peak torque of 25 Nm while in case of the magnetic spring assisted setup it is only 8 Nm. Therefore, the required peak torque is approximately three times lower in a case where a magnetic spring is used. The significant reduction can also be observed in energy consumption per cycle of

reciprocating motion. The energy required for operation of magnetic spring assisted drivetrain is reduced from 29.07 J per cycle to 5.05 J per cycle, signifying an almost six-fold energy reduction. Here, the energy consumption is calculated as a sum of the measured mechanical power (torque sensors, encoders) at the motor output shaft and the ohmic losses calculated from the torque reference and motor datasheet parameters (phase resistance and torque constant).

It is visible that initially, during the start-up, the energy required to initialize the spring assisted setup is higher. This is, however, not a serious downside of the spring assisted actuator, considering that in the industrial application cases the drivetrain is only seldomly initiated, before long hours of operation, making the start-up energy consumption a negligible segment of the total energy consumption. For this reason, and for the convenience of tracking the energy consumption during the operational behavior (Figure 9, phase 3, reciprocating motion) the plotted energy is reset in the middle of the experiment (Figure 9, Phase 2, 0-torque wait. Alternatively, it is also possible to run the spring assisted system at a much higher torque in order to achieve a faster transient than it is possible with the motor only. In that case, a bang-bang controller can be used to accelerate the load as quickly as possible between two end positions.

**Figure 9.** Proof-of-concept. Comparison of dynamic operational data for a magnetic spring with minimum motor torque vs. no spring setup with peak torque operation; controller structure and tuning have an impact on the exact values.

## *3.2. Model Based-Optimal Component Design*

Detailed design optimization of the selected five most interesting topologies was done. The resulting Pareto fronts of different magnetic spring topologies can be compared for a fixed energy requirement and stroke. In Figure 10 it can be seen that sintered NdFeB is preferred over bonded magnets for reasons of both lower cost inertia. The added effect of using isotropic material (bonded NdFeB) to achieve a wider variety of magnetization is smaller than the added cost and inertia that results from lower flux densities in these materials. Interesting enough, low inertia levels can be achieved for each topology, irrelevant of the magnet geometry. However, the amount of material required to do so results in the lowest cost design with surface mounted arc magnets.

**Figure 10.** Optimization results plotted as Pareto-fronts for five stator and rotor topologies selected after design space pruning for different energy and stroke specifications.

Additional conclusions regarding design rules can be drawn from optimization results through Pareto optimal parameters. In Figure 11 normalized histograms (i.e., non- dimensional value on the y-axis) of the pareto optimal designs are plotted for each of the five selected topologies, showing the parameter distribution for the optimal designs that lie on the Pareto front.

Further analysis, shows that pole pitch in quasi Halbach arrays is optimally fully pitched with pitch factor values (i.e., the ratio of magnet coverage and pole pitch) approaching 1 (Figure 11e,f), which results in closest possible design to a real Halbach magnetization. On the other hand, standard multipole array values optimally have short pitchpoles with pitch factor values between 0.75 and 0.85 in order to prevent short-circuiting of the permanent magnet flux. The specific value of pitch factor, in this case, depends on the magnetic air gap between stator and rotor magnets as this represents the magnetic resistance of the parallel flux path. Another difference between Halbach and standard multipole arrays is in the thickness of the magnets (Figure 11c,d).

Finally, the scalable 1D model of magnetic springs can be validated using both the experimental validation and the detailed FEM of the designs presented here. Note that the two designs have different requirements as well as geometry sizing, and pole pair number. The single experimental design maps into one point, while the pareto front shows a dispersion of the possible designs. Therefore, the 1D model visible in Figure 12 Should be considered as a line partitioning the feasible component space from the infeasible.

stator

**Figure 11.** Optimal parameters histograms for five selected stator and rotor topologies selected after design space pruning.

**Figure 12.** Validation of 1D scaling model for (**a**) magnetic spring inertia and (**b**) magnetic spring bulk material cost using experimental data and virtual validation data.

## **4. Discussion**

Following the optimization results, the impact of magnetic spring on system performance can be analyzed from different perspectives. To compare magnetic springs to mechanical springs side-to-side phenomenologically, maximum theoretical energy density based on first principles is considered alongside with the realistically feasible energy density following from the optimization result. Since desired lifetime has a direct influence on stress level in mechanical springs and therefore also on energy density, we can plot energy density vs. required lifetime for mechanical and magnetic springs (Figure 13).

**Figure 13.** Benchmarking magnetic springs vs. mechanical springs; magnetic springs have increasingly higher energy density for high life cycle numbers.

The maximum theoretical energy density of the magnetic spring of *Emax*52*S* = 828 kJ/m3 is already higher than that of the Murakami model based gigacycle energy density of steel springs at *EMurakami* = 506 kJ/m3. The difference between feasible energy density achieved with the feasible designs is even more dramatic. With NdFeB 42H grade and arc magnets, we are able to design a magnetic spring with an energy density of *Emodel*42*H* = 404 kJ/m3, while a specific mechanical spring described in Reference [9] possesses an energy density of *Emech* = 210 kJ/m3 with possible fatigue failure already at megacycles. However, it is difficult to generalize on feasible gigacycle mechanical designs for all the designs as the range of safety factors used within these applications is usually in quite a large range. Nevertheless, while being conservative we can say that the resulting increase in energy density is at least 50%, considering that the minimax design efficiencies of 0.6 achieved in this paper are larger than those in mechanical spring designs where safety factors are usually moderately higher than 2. Consequently, magnetic springs are specifically relevant for highly dynamic drivetrains in manufacturing machines e.g., a weaving loom operating shedding frames at 10 Hz reaches into megacycles after only 27.8 h and reaches well into a gigacycle regime in its standard operational age.

The benchmarking against PMSM is performed using a combination of real-life data from PMSM datasheets and model comparison using the developed magnetic spring modelling toolchain. Several types of servomotors are considered, with a preference for highly dynamic ones with high torque density. Extrapolation from the datasheet points can be carried out using the relation.

$$J = nT\_{peak} \, ^{5/3} \, \tag{6}$$

which is valid for both springs and motors, assuming a fixed rotor aspect ratio (diameter/length). To reduce the cost and size of an electric drive solution, a reducer with transmission ratio *n* may be employed. However, the reflected inertia with a geared solution is always higher, given that

$$J = n^2 J\_{motor} \tag{7}$$

and **2** > **5**/**3**. In Figure 14a only peak torques are considered, which are limited by the magnetic design of motor and spring. This results in a misleading image of rather "smaller" PMSMs (Maxon) having a higher torque density than magnetic springs with one or even two pole pairs. Note that for these "smaller" motors with natural cooling the difference between nominal torque and the peak torque is also greater. Figure 14b presents a more relevant image for highly dynamic industrial applications since here the nominal torque provided by the motor is compared to the spring peak torque.

**Figure 14.** Benchmarking torque density of magnetic springs (1D scaling model) with pole pair numbers Npp = 1–5 vs. off-the-shelf highly dynamic PMSM (**a**) peak torque vs. inertia and (**b**) nominal torque vs. inertia.

Nominal torque depends on the thermal design of the motor and cooling circuit, and in this analysis, all of the conventional air and liquid cooling methods were considered. The allowed dynamic peak can be higher than this limit, depending on the overload potential and the dynamic nature of the load, however, for exact quantification of this effect, a more detailed system dynamics analysis, outside of the scope of this article, should be considered. Additionally, for magnetic springs no thermal limitation is considered since the losses associated with generating torque are non-existent and the dynamic losses due to the eddy currents have been shown to be negligible. In conclusion, as a result of high torque density of magnetic springs the actuator bandwidth can be systematically improved for predetermined reciprocating profiles. This effect is more significant for small air-cooled motors, and less pronounced for larger designs. For exact quantification, a more detailed study on system optimization of magnetic spring assisted drivetrains is needed, considering optimal control strategy, relative sizing of spring and motor, sizing and selection of other system components (e.g., gearbox, motion conversion mechanism), and capturing motion requirements.

## **5. Conclusions**

A detailed component design methodology has been developed and validated. A theoretical energy density was established based on physical insight in energy stored in permanent magnets. Detailed design optimization results show that design up to 60% of material efficiency are manufacturable. Best results are achieved with surface mounted arc sintered NdFeB magnets. The modeling approach is validated for a manufactured prototype, by static and dynamic component characterization on the experimental test rig. Following these results, based on 1D scalable models of magnetic springs, the energy density of a mechanical and magnetic spring can be compared for a long lifetime. Magnetic springs have at least a 50% higher power/energy density than mechanical springs with the with the added benefit of no fatigue failure. Additionally, using 1D scalable models of

magnetic springs, a comparison between torque density of magnetic springs and PMSM off-the-shelf motors shows the added value of magnetic springs for preplanned reciprocating motion systems. The added benefit is specifically dramatic for partial strokes when magnetic springs with two and more pole pairs are employed when drivetrain peak acceleration is increased by 33% for the worst case scenario.

Conceptually, also the impact of magnetic spring on system behavior is experimentally demonstrated. The results show a six times lower energy consumption, and three times lower peak torque for a magnetic spring assisted drivetrain. Future studies should, however, consider a detailed analysis of system level design of highly dynamic drivetrains and quantify the associated cost reduction resulting from possible motor downsizing and improvement in bandwidth and energy efficiency in a more systematic manner.

Based on the dynamic measurement performed on the prototype, magnetic spring losses do not seem to be a relevant issue for the design of spring assisted reciprocating drivetrains, where due to the low pole pair number, the frequency of the magnetic field is rather low. Nevertheless, the demagnetization *Hdemag* field is directly influenced by the temperature, and the rise in temperature is directly caused by losses. Therefore, it is important to notice that for using magnetic spring with higher pole pair number, than considered here, for e.g., torque ripple reduction [24], demagnetization can still be a possible issue. For such cases, a better understanding of thermal behavior and losses might lead to savings related to the selection of lower temperature grade magnets related to lower Dysprosium content.

Finally, we would like to comment un utility of magnetic springs. In this article, magnetic springs are primarily intended for enabling elastic actuation in industrial applications, where this was not feasible so far, due to the catastrophic failures that can result not only in significant down times, but also damage to the machine, processed goods and operator (e.g., weaving loom or punching tool failure). Alternatively, it might be possible for magnetic spring to replace mechanical springs in applications where the use of mechanical springs is established, for reasons of reduced downtime. However, the cost of magnetic springs is still expected to be higher than that of highly commoditized mechanical springs, meaning that a trade-off study, done from a perspective of specific industrial application will be necessary in order to determine when to use the magnetic springs. The results presented in this article provide a head start in such a study.

**Author Contributions:** Conceptualization, B.M., W.D. (Walter Driesen) and W.D. (Wim Desmet); Data curation, B.M.; Formal analysis, B.M.; Funding acquisition, W.D. (Walter Driesen) and W.D. (Wim Desmet); Investigation, B.M.; Methodology, B.M., B.L., W.D. (Walter Driesen) and W.D. (Wim Desmet); Supervision, W.D. (Walter Driesen) and W.D. (Wim Desmet); Validation, B.M.; Visualization., B.M.; Writing – original draft, B.M.; Writing – review & editing, B.M., B.L., W.D. (Walter Driesen) and W.D. (Wim Desmet).

**Funding:** The research of B. Mrak as an Early Stage Researcher was funded by a grant within the European Project EMVeM Marie Curie Initial Training Network (GA 315967). This research was partially supported by Flanders Make, the strategic research centre for the manufacturing industry within Flanders Make project Profensto\_icon.

**Acknowledgments:** The authors would like to thank Linz Center of Mechatronics GmbH and Siegfried Silber for their support through the use of MagOpt software and the associated infrastructure.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Condition Monitoring of Active Magnetic Bearings on the Internet of Things †**

## **Alexander H. Pesch \* and Peter N. Scavelli**

Department of Engineering, Hofstra University, 104 Weed Hall, Hempstead, NY 11549, USA<EMAIL_ADDRESS>

Received: 20 January 2019; Accepted: 14 February 2019; Published: 20 February 2019

**Abstract:** A magnetic bearing is an industrial device that supports a rotating shaft with a magnetic field. Magnetic bearings have advantages such as high efficiency, low maintenance, and no lubrication. Active magnetic bearings (AMBs) use electromagnets with actively controlled coil currents based on rotor position monitored by sensors integral to the AMB. AMBs are apt to the Internet of Things (IoT) due to their inherent sensors and actuators. The IoT is the interconnection of physical devices that enables them to send and receive data over the Internet. IoT technology has recently rapidly increased and is being applied to industrial devices. This study developed a method for the condition monitoring of AMB systems online using off-the-shelf IoT technology. Because off-the-shelf IoT solutions were utilized, the developed method is cost-effective and can be implemented on existing AMB systems. In this study, a MBC500 AMB test rig was outfitted with a Raspberry Pi single board computer. The Raspberry Pi monitors the AMB's position sensors and current sensors via an analog-to-digital converter. Several loading cases were imposed on the experimental test rig and diagnosed remotely using virtual network computing. It was found that remote AMB condition monitoring is feasible for less than USD 100.

**Keywords:** Active Magnetic Bearing; AMB; Internet of Things; IoT; Condition Monitoring

## **1. Introduction**

Active Magnetic Bearings (AMBs) support a rotor with a magnetic field such that the rotor is levitated [1]. AMBs are an alternative to other types of bearings such as rolling element bearings and fluid film bearings [2]. AMBs have advantages over other types of bearings such as the potential for higher efficiency and rotational speeds. In addition, AMBs do not need lubrication, which is advantageous in clean rooms, and food or medical processing (e.g., [3]). AMBs do not need service and are useful in subsea [4] and space applications [5]. The control in the airgap clearance can be exploited, for example, for machining tool positioning [6] and active balancing [7].

Passive magnetic bearings use permanent magnet stators to repel a rotor with permanent magnets with opposing polarization [8]. The system is naturally stable, tending toward the equilibrium in the center of the bearing. Passive magnetic bearings tend to be more efficient than AMBs with electromagnets, because there is no current consumption. However, they lack the load capacity and performance of the actuated AMBs. Permanent-magnet-biased AMBs have permanent magnets to provide initial pulling force on a ferromagnetic rotor. Permanent-magnet-biased AMBs serve as a way to achieve some of the advantages of passive magnetic bearings and AMBs and still utilize integral sensors and actuators [9]. Zero-bias and low-bias AMBs also use integral sensors and high-efficiency actuators and require sophisticated control laws (e.g., [10–16]) to take advantage of the nonlinear flux.

AMBs use electromagnetic actuators to generate an attractive force on a ferromagnetic rotor. The magnitude of the attractive force increases as the rotor moves closer to the stator. Therefore, the setup is naturally unstable and requires stabilizing feedback control in order to function. An AMB inherently includes some form of position sensing. The rotor position signal is used to calculate required coil current to control rotor position and achieve stable levitation. A typical type of sensor is the noncontact eddy current position probe. AMB position sensors have conveniently been utilized for online health monitoring [17]. In addition, real-time knowledge of rotor position and coil current can be used to determine bearing forces that can be used to indirectly monitor rotor loading [18].

AMBs are designed, built, and implemented by engineers and technicians with expert knowledge. For example, AMB controller design must be customized to rotor geometry because of inertial and gyroscopic cross-coupling and to avoid excitation of flexible modes. Once commissioned on-site, the AMB can be used by the end user with relatively little training [19]. However, the end user may not be able to effectively troubleshoot complications with the AMB system that may arise after the commissioning process, because the end users are not trained to recognize or diagnose these complications. In such cases, a field service technician from the AMB's Original Equipment Manufacturer (OEM) must go on-site to perform service. This results in down time for the system supported by the AMB and increased expense to the customer. A solution is to make AMBs part of the Internet of Things (IoT). This would allow for increased productivity and decreased costs.

The exact definition and scope of the IoT is still being developed. However, the IoT basically enables the interconnection of physical devices. This interconnection allows the devices to send and receive data over the Internet. This enables value creation beyond the mere sum of the "thing-based function" and "IT-based service" [20]. By putting AMBs on the IoT, a remote user such as an off-site OEM technician could access the AMB, and diagnose malfunctions without the need for on-site examination. Therefore, (to reduce or eliminate equipment down time) the OEM technician can recommend corrective action immediately, or even preemptively.

Early cases of what would become known as IoT were in the area of radio-frequency identification (RFID) tags (e.g., [21]). Since then, there has been much development of IoT because of the significant impact on people's everyday lives [22]. There are several instances of industry beginning to take advantage of IoT technology [23]. More recently, IoT has been applied towards structural health monitoring [24]. For example, IoT has been used for monitoring the position of steel in a continuous casting process [25]. In addition, IoT has been used for the monitoring of vibrations in electric motors [26]. This suggests the potential of AMBs when coupled with the IoT.

There has been some previous work in the area of IoT tools used for AMBs. These are mostly proprietary industrial systems used toward facilitating automated commissioning. However, there has not been enough work in the application of off-the-shelf IoT hardware, which is low-cost and readily available. An early work in remote operation of AMBs is found in [27], where a local area network (LAN) is used to facilitate communication with real-time AMB controllers in a laboratory environment. The utilized LAN is a direct connection between the remote computer used for system interfacing and computers on the AMBs with dedicated hardware for real-time control and ethernet communications. This method is successful at interfacing with the AMBs for conducting experiments at a safe distance but was hardware intensive. In [28], a remote computer is used to communicate with a server computer via Transmission Control Protocol/Internet Protocol (TCP/IP). The server passes data via an RS-232 connection with a digital signal processor, which in turn controls an AMB system. The setup is used to remotely tune the AMB controller gains. Jayawant and Davies [29] developed an automated commissioning scheme capable of remote commissioning AMBs via TCP/IP and a Simple Object Access Protocol (SOAP) interface. SOAP sends packaged datasets between computers on a high level. Because the data are compiled and packaged before being transmitted via SOAP, the method can facilitate data transfer between different types of systems, e.g., differing operating systems. With SOAP, the actual data vector from the AMB is passed between the local and remote computers. Data processing can take place on either or both computers. In [30], SOAP-based remote

commissioning is applied to a fluid film bearing AMB test rig and an industrial turbo-machine. Similar remote commissioning methods are utilized in [31] for a 3.3 MW motor-driven compressor and in [32] for a high-temperature gas-cooled reactor.

In the present study, an AMB test rig was augmented with an off-the-shelf IoT gateway that is low-cost and readily available. The local device was programmed to read the AMB's sensors and perform data processing for condition monitoring. A remote user could then log into the device through Virtual Network Computing (VNC) via TCP/IP to observe the sensor signals. This approach differs from the SOAP approach in that the AMB data vectors are not transmitted to the remote computer. In the current approach, all data processing is done on the local IoT gateway and the resulting frames are used remotely for condition monitoring. The usefulness of the developed system for condition monitoring of AMBs was demonstrated by operating the test rig under different conditions, and presenting the remote user interface, illustrating how the condition of the system was evaluated. Preliminary results for this study are presented in [33].

The next section explains the concept of using the IoT for condition monitoring of AMBs. Then, the experimental system used to demonstrate the proposed method is detailed, including cost information. Next, the experimental results are presented. The practical issue of sampling time when utilizing the IoT gateway device is then discussed. Finally, the paper is ended with concluding remarks.

## **2. Materials and Methods**

This section covers the proposed method and the materials used to implement the experimental demonstration. Specifically, an introduction to AMBs and the proposed method for condition monitoring of AMBs on the IoT is discussed in the next subsection. Then, the AMB test rig for the experimental demonstration is presented. Finally, subsections for the hardware and software for IoT implementation are presented.

## *2.1. AMB and Condition Monitoring on the IoT*

An AMB uses a magnetic field to support a rotating shaft. The magnetic field is generated by an array of electromagnets around the rotor. The electromagnetic force induced on the ferromagnetic rotor is inherently unstable. The position of the shaft is measured in real time by a noncontact position sensor. The position data are used (by a controller) to calculate how much current is needed in the electromagnetic coils to maintain a stable levitation. A typical control setup for AMBs is shown in Figure 1 [34]. Figure 1a shows a common biasing scheme for a single AMB axis. Figure 1b shows a control block diagram for a generic AMB-rotor system, which is Multiple-Input-Multiple-Output (MIMO) because a rotor may have multiple AMBs. Therefore, an AMB, by its nature, includes sensors and actuators. It is suitable for condition monitoring for traditional rotordynamic faults [35]. It is also highly apt to be extended to the IoT.

**Figure 1.** Typical AMB control scheme: (**a**) one control axis biasing; and (**b**) MIMO Control block diagram.

By accessing the information available in an AMB, a remote service technician can monitor the condition of the AMB system. Many types of information may be available in a AMB controller, such as rotor speed, hours in operation, temperature, flux, etc., but the signals used for the experimental demonstration in this work were position and current. Figure 2 illustrates the overall concept for the proposed method of AMB condition monitoring via the IoT. In the proposed scheme, the rotor position and the coil current, available from the AMB, are accessed for the IoT. Therefore, a service technician can log in remotely to the gateway and troubleshoot the AMB system. The remote technician is granted the ability to diagnose a variety of equipment malfunctions. The technician might be able to recommend corrective or even preventative action to the AMB end users without the need for an on-site service visit.

**Figure 2.** Condition monitoring of AMB system via the IoT concept.

For example, if the rotor position is low and/or the coil current is high, shaft overloading may be indicated. The technician can recommend checking the application, or "spec out" a larger AMB. Another scenario could involve the remote technician observing an overly large orbit indicating large unbalance. The technician could recommend rotor balancing before continuing operation.

## *2.2. Experimental System*

The experimental system used to develop the proposed IoT condition monitoring solution consists of a stand-alone AMB test rig coupled with an IoT gateway and other hardware. The AMB test rig is model MBC500 by LaunchPoint Technologies, Inc. It has sensor signals that are readily available via a front breakout panel. The IoT gateway selected is the Raspberry Pi 3 Model B single board computer. The experimental system is shown in Figure 3. Figure 3 shows the overall system with: the AMB test rig, added sensor amplifiers and ADC unit on a solderless breadboard, and Raspberry Pi with Ethernet cable to connect to the Internet. The levitated shaft of the test rig has metal collars, which are movable, removable, made of differing materials for differing weights, and may have an adjustable unbalance screw added.

Additional hardware is required for interfacing the Raspberry Pi with the analog signals of the AMB rig. Specifically, an analog-to-digital converter (ADC) board and amplifiers were used to condition the sensor signals to the appropriate range. The following subsections detail the AMB test rig configuration, electronic hardware for IoT implementation, and corresponding software.

## 2.2.1. AMB Test Rig

The MBC500 AMB test rig consists of a single shaft supported by two radial AMBs at either end. Shaft position was monitored by Hall effect sensors to the immediate outside of the magnetic coils. The shaft is stainless steel and 12.5 mm in diameter. Rotation is driven by an air turbine near the inboard AMB.

**Figure 3.** Experimental AMB test rig with attached Raspberry Pi single board computer, ADC, and conditioning circuitry.

Movable collars were added to the shaft to create reconfigurable weight and unbalance loads. The collars are approximately 10.5 mm wide and 28 mm in diameter. Two collars (one aluminum and one stainless steel) were used for the present test. The aluminum collar is 15 g and the stainless steel collar is 36 g. The position sensors are 2.8 mm from the ends of the shaft. The locations of the AMBs and the collars on the shaft are shown in Figure 4.

**Figure 4.** Rotor configuration. Dimensions in mm.

The AMBs have eight poles and are wired differentially in the vertical and horizontal axes. Each AMB axis has a bias current of 0.5 A and a nominal gap of 400 μm. The AMB force constant, based on coil geometry, is 2.8 × 10−7 N·m2/A2. The current amplifier bandwidth is approximately 720 Hz. The AMB controller built into the MBC500 is local lead-lag type. The built-in controller was used for the IoT condition monitoring study (when the shaft collars were moved).

## 2.2.2. Hardware Added for IoT

The IoT gateway selected is the Raspberry Pi 3 B (~USD 40). The Raspberry Pi is a single board computer with a 1.2 GHz Broadcom BCM2837 Quad-Core CPU and 1 GB of RAM. It runs the Raspbian operating system, which is Debian based. The Raspberry Pi was selected for this study as it is one of the most readily available IoT gateways. It serves as a cost-effective solution to remote condition monitoring. It is widely obtainable and therefore available to be augmented to older AMB systems already commissioned. In addition, software developed for the Raspbian operating system can be relatively easily ported to other Linux-based systems. A limitation of this IoT gateway solution is it is relatively slow and nonreal-time sampling as a consequence of the operating system. Therefore, it is most useful for monitoring relatively slow rotors or systems with low frequency bearing modes, subharmonics, external excitations, and substructure modes. The issue of nonreal-time sampling is discussed further in Section 3.

The hardware interface of the Raspberry Pi is general purpose input–output (GPIO) digital pins. To read the analog signals from the AMBs, an ADC must be added. For the current study, a Texas Instruments ADS1115 4-channel 16-bit ADC was utilized (~USD 15 on Adafruit Industries, LLC breakout board). Communication between the Raspberry Pi and the ADC was implemented via standard I2C digital communication protocol. This protocol requires two wires, one for data transfer and one for a timing trigger. Figure 5a shows the basic scheme for experimental implementation of AMB condition monitoring via IoT.

**Figure 5.** Hardware added for IoT implementation: (**a**) overall scheme; and (**b**) op-amp wiring diagram.

The ADC input range is nondifferential, effectively 0–5 V. The AMB sensors' operation range is within ±4 V. The sensor signals were scaled and offset by an array of four summing amplifiers. Texas Instruments μA741cp general purpose operational amplifiers (each < USD 1) were used. The amplifiers were wired as shown in Figure 4b to achieve the usable signal voltage range of 0.5–4.5 V. *VI* is the input sensor signal from the AMB and *VO* is the output signal sent to the ADC.

A standard PC power supply (~USD 20) provides an economical source of +12 V and −12 V, as well as the 5 V needed to raise to sensor signals. The resistors were balanced to half the overall sensor signal. The exact value of the resistors was selected through trial-and-error for an acceptable impedance match.

One AMB of the fully levitated system was monitored to demonstrate the developed IoT condition monitoring system. The sensor signals monitored were shaft position in the horizontal and vertical directions (*x* and *y*, respectively) and the corresponding axis coil currents (*ix* and *iy*, respectively). The software loaded on the Raspberry Pi used these data to give insight to the operating conditions of the outboard AMB. The entire assembly of hardware added for IoT is less than USD 100.

## 2.2.3. IoT Condition Monitoring Software

IoT condition monitoring software was written to collect the output signals of the ADC unit, and to a display a visual representation of the data. This includes rotor position and coil current. The ADC samples each signal, and converts the sampled data from voltage to bits. The bit readout was communicated to the Raspberry Pi via I2C protocol. The bits were then scaled to recover the real-world signal values. This scaling involved applying an offset and a sensitivity to the vectors of collected data. The offset constants for the positions were obtained (during healthy levitation) by averaging the position vectors. These values were subtracted from all corresponding values in each corresponding vector. The sensitivity, found by comparing a known physical travel to the change in bits, was multiplied into each value in the respective vector. This yielded the calibrated position. A similar process was done for the current, but with the addition of offset to accurately represent the bias current applied.

Next, the vectors of position readings and current readings were plotted to display orbits (*x* vs. *y*) and position *y* with time and frequency. The frequency plot was generated by taking the Fast Fourier Transform (FFT) (via the NumPy library and the command *numpy.fft*) of a reconstructed position vector, as follows: the original sampled position vector had inconsistent time steps because of the nature of the Raspbian operating system. Since the processor was running both the operating system and the monitoring program, the loop running the code may be suspended to maintain the operating system's processes. (The exact sampling rate is discussed in Section 4.) To perform the FFT, which requires a consistent sampling rate, the original data vector and corresponding known time vector were resampled via linear interpolation. Using a resampled vector with 512 elements from the original approximately 400 over 5 s time history was selected. This number of virtual samples was selected to optimize the FFT algorithm while being similar to the actual number of data taken (to maintain fidelity).

The plotting of position and current in the time domain and in the frequency domain gave insight into the AMB system's operating condition. The software placed these figures, as well as the average value of each of the considered parameters after each instance, into a graphical user interface (GUI). The GUI was generated using the matplotlib library and the command *matplotlib.pyplot.plot*. This allowed taking the pre-allocated position and current vectors and expressing them visually. From the Raspberry Pi, a remote monitoring technician could observe the motions of the shaft within the bearing, and alert on-site operating technicians of a possible malfunction.

To connect remotely to the IoT AMB, Secure Shell (SSH), a cryptographic network protocol was enabled to allow a connection from an outside source. The Raspberry Pi hosted a server using the commercially available program *VNC Server*. The technician uses the corresponding program *VNC Viewer* (client) (both by RealVNC Ltd.) to facilitate the connection with a remote framebuffer (RFB) protocol. Similar IoT solutions utilizing VNC Server have been implemented in [36–38]. In general, an RFB protocol transmits screen pixels from one computer (over a network) to another and can also send control events, (e.g., mouse, keyboard, touch screen, etc.) in return [39]. For the current study, the RFB allowed the remote user to activate the IoT program as well as observe the operation of the bearing through the GUI. Therefore, not all data vectors for position, current, etc., need to be transmitted. The program performed data collection and displayed results for a set 5 s time interval. Future editions may allow the program to display latency, constantly update values and automatically replot figures.

## **3. Results**

Six trials were conducted to demonstrate the condition monitoring capabilities of the developed AMB IoT system. For each trial, the shaft collars were adjusted to create varying load conditions. Two trials were conducted with the shaft levitated, but not rotating. Two trials were conducted with the shaft rotating. Two trials were conducted with the shaft rotating with added unbalance. For each case, the GUI used by the remote service technician is presented to illustrate how the condition of the AMB system is monitored.

## *3.1. Nonrotating Tests*

Figures 6 and 7 show the GUI that a remote service technician would see when executing the IoT AMB monitoring software. The software was executed by calling VNC viewer on the local machine, connecting to the specific Raspberry Pi IoT gateway, and remotely calling the condition monitoring GUI program on the Raspberry Pi. The top of the GUI is a header that displays basic information. (The time period over which data are collected is displayed, in this case 5 s.) There is a blank expansion field for latency to be displayed by a later version of the software. The average values for position and current in the vertical and horizontal AMB axes are also displayed.

The two left plots are orbits, plotting data from vertical vs. horizontal AMB axes. The first plot displays rotor position and the second displays top coil current calculated from the recorded control current. The default scale for the position orbit is half the nominal AMB airgap of ±200 μm. The default scale serves as a limit for safe operation predetermined by expert users. Therefore, the technician can easily determine if the rotor is near the limit of safe operation if it nears one of the axes. The default scale for the current orbit is 0–1 A. A non-levitated rotor would sit at (0, −400) μm in the position orbit and (0,0) A in the current orbit.

The center right plot displays the time response of the rotor vertical position over the entire 5 s time history. The right plot displays the corresponding frequency spectrum found with an FFT of the resampled position data.

Figure 6 shows the results for the nonrotating shaft without added collars. The shaft levitated steadily near (0,0) μm, which is the center of the AMB. The coil current was near the bias current, 0.5 A. The calibration of the IoT condition monitoring system can differ from that of the AMB controller.

**Figure 6.** AMB IoT condition monitoring GUI display for non-rotating bare shaft.

**Figure 7.** AMB IoT condition monitoring GUI display for non-rotating shaft with two collars.

The frequency spectrum depicted in the figure shows only a 0 Hz component for static offset. Note that the AMB controller has no integral action (as in a common PID controller). Figure 7 is for the nonrotating shaft with the two added collars. The remote service technician can infer the static loading condition of the levitated rotor by noting the lower levitated position and increased static current. In the event that the static deflection was too low or the static current was too high, the remote service technician can diagnose rotor over loading and recommend proper corrective action to on-site personnel without the need for an in-person inspection.

## *3.2. Balanced Rotating Tests*

The rotor was rotated at 1200 RPM with and without the shaft collars. Figure 8 shows the IoT condition monitoring GUI for the case without the collars and Figure 9 for the case with the collars. The remote service technician can observe the orbit of the rotor inside the AMB air gap caused by the rotation. For both cases, the orbit was consistent and stable. The added weight of the shaft collars caused a static deflection downwards, and a corresponding increase of current (as with the nonrotating cases). The increase of gravity preloading also caused a slight bearing stiffness anisotropy, which led to vertical elongation of the orbit, which was observable by the remote service technician.

The rotation condition of the rotor was further observable in the time plot that displays a consistent harmonic wave. (The frequency spectrum had a peak at approximately 20 Hz, indicating the running speed.) The case with the shaft collars had a slightly higher peak at the running speed because of residual imbalance of the collars. The remote service technician can inspect the frequency spectrum for other components. For example, rotation off of the bearing centerline led to the appearance of a 2× rotation component at 40 Hz, as shown in Figure 9.

**Figure 8.** AMB IoT condition monitoring GUI display for rotating bare shaft.

**Figure 9.** AMB IoT condition monitoring GUI display for rotating shaft with two balanced collars.

## *3.3. Unbalanced Rotating Tests*

To induce a rotordynamic malfunction, unbalance masses were added to each shaft collar (in the form of a machine screw with exposed head). The resulting unbalance was approximately 6.5 g-mm per collar. Two unbalance tests were conducted. The first had both unbalance screws in the same direction on the rotor to create a static unbalance. The second had the unbalance screws in the opposite directions on the rotor to create a dynamic unbalance. Again, the shaft was rotated at 1200 RPM.

Figure 10 presents the IoT condition monitoring GUI for the static unbalance test and Figure 11 for the dynamic unbalance test. Observing the GUI in Figure 10, a remote service technician can diagnose the rotor unbalance from the slightly increased level of vibrations. This was seen in orbit size, vibration amplitude in time, and 1× frequency peak. The slight increase in level of vibration can alert the remote service technician to the added unbalance.

The dynamic unbalance test shows that the unbalance increased in the aluminum (inboard) collar but the added mass of the stainless steel (outboard) collar countered its own residual unbalance. Therefore, the remote service technician would observe a healthier orbit size, albeit lower, in the bearing gap. The ability to remotely access these data enables the remote service technician to recommend rotor balancing to the end user.

**Figure 10.** AMB IoT condition monitoring GUI display for rotating shaft with two collars and static unbalance.

**Figure 11.** AMB IoT condition monitoring GUI display for rotating shaft with two collars and dynamic unbalance.

## **4. Discussion**

A limitation of the developed IoT condition monitoring solution is the inconsistent sampling rate that stems from the operating system of the IoT gateway device. This is different from, for example, a dedicated microcontroller that has no operating system and no related background activities. The developed IoT program executed in Raspbian achieved a typical sampling rate of 100 Hz. However, it suffered from periodic delays created as the operating system performs background processes. These are the functions maintaining the operating system and functionality of peripherals and other programs

run by the remote user. Figure 12a shows the time stepping history of a characteristic 5 s condition monitoring run.

**Figure 12.** Characteristic sampling times for 5 s of IoT data: (**a**) time history; and (**b**) histogram.

The nominal sampling time of 0.01 s is presented as the baseline level in the figure. Frequently, the time was delayed to around 0.012 s. In addition, the data collection was pseudo-periodically delayed even further for several time steps, about every 1 s. This led to a time step as high as 0.018 s.

Figure 12b shows a histogram of the sampling time for this 5 s run. The histogram confirmed that the nominal sampling time was dominant, but interrupted by occasional delays. The current study overcame this limitation by visual inspection of orbits (which are not highly time dependent), and resampling to achieve a practical frequency spectrum. (However, this issue is important in further development of IoT for AMBs, i.e., execution of active control online.) A possible solution might be implementation of a real-time operating system. Another solution might be implementation of a programmable real-time unit on a single board computer.

## **5. Conclusions**

This study addressed the problem of remote condition monitoring of AMBs. A solution was proposed to use off-the-shelf IoT hardware and custom software to tie into an AMB's position and current signals. This allowed an OEM technician to observe the signals remotely. The proposed strategy was demonstrated on an AMB test rig. A Raspberry Pi gateway and VNC Server software were used to implement IoT connection. The IoT gateway and other associated hardware cost less than USD 100. Static loading and static and dynamic unbalances were imposed on the experimental rotor. For each case, the conditions of the AMB system were successfully monitored remotely.

Therefore, it was concluded that off-the-shelf IoT hardware and custom software is economical and effective for remote AMB condition monitoring. AMB OEMs can implement similar methods to remotely monitor their products, which are operating on-site for their clients, the end users. This ability will alleviate the need for on-site service calls, and prevent AMB down time.

There are several promising directions for further development of AMBs and IoT. First, cybersecurity should be considered. In other words, mechanisms need to be developed to ensure only intended users can log in and access AMB data. In addition, a mobile application can be developed with which AMB users can check on the condition of the system from arbitrary locations. More complicated is the potential improvement of the IoT scheme for real-time use. Two possible solutions are a real-time operating system for the IoT gateway and using a real-time programmable unit, which would increase hardware cost. Real-time execution will lead to the next stage of development, a cyber physical system (i.e., the feedback control for the AMB will be done through the IoT, making a system in which the real-world dynamics of the system are dependent on the cyberworld). Then, an OEM service technician would not only be able to monitor the condition of an AMB system and diagnose problems, but also might be able to fix problems by changing the control law.

**Author Contributions:** The overall AMB condition monitoring scheme was devised by A.H.P. the coding for IoT implementation of the digital hardware was conducted by P.N.S. The authors worked together to perform the experiment.

**Conflicts of Interest:** The authors declare no conflicts of interest.

## **References**

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Stability and Performance Analysis of Electrodynamic Thrust Bearings†**

## **Joachim Van Verdeghem \*, Virginie Kluyskens and Bruno Dehez**

Department of Mechatronic, Electrical energy and Dynamic systems (MEED), Institute of Mechanics, Materials and Civil Engineering (IMMC), Université catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium<EMAIL_ADDRESS>(V.K<EMAIL_ADDRESS>(B.D.)

Received: 30 December 2018; Accepted: 29 January 2019; Published: 1 February 2019

**Abstract:** Electrodynamic thrust bearings (EDTBs) provide contactless rotor axial suspension through electromagnetic forces solely leaning on passive phenomena. Lately, linear state-space equations representing their quasi-static and dynamic behaviours have been developed and validated experimentally. However, to date, the exploitation of these models has been restricted to basic investigations regarding the stiffness and the rotational losses as well as qualitative stability analyses, thus not allowing us to objectively compare the intrinsic qualities of EDTBs. In this context, the present paper introduces four performance criteria directly related to the axial stiffness, the bearing energy efficiency and the minimal amount of external damping required to stabilise the thrust bearing. In addition, the stability is thoroughly examined via analytical developments based on these dynamical models. This notably leads to static and dynamic conditions that ensure the stability at a specific rotor spin speed. The resulting stable speed ranges are studied and their dependence to the axial external stiffness as well as the external non-rotating damping are analysed. Finally, a case study comparing three topologies through these performance criteria underlines that back irons fixed to the windings are not advantageous due to the significant detent force.

**Keywords:** performance criteria; damping; electrodynamic; energy efficiency; stability; stiffness; thrust bearing

## **1. Introduction**

Nowadays, magnetic bearings constitute a convincing alternative to classical solutions such as ball or journal bearings by ensuring contactless guiding of rotors, thereby reducing losses and removing mechanical wear and friction. These compelling bearing can be either active or passive. The former are based on current-controlled electromagnets exerting an attractive force on a ferromagnetic rotor, whereas the latter only rely on passive phenomena.

Electrodynamic bearings (EDBs) belong to passive magnetic bearings (PMBs) as they lean on electromagnetic forces generated by the appearance of induced currents in short-circuited conductors in relative motion with respect to a magnetic field produced by permanent magnets (PMs). Although their stiffness is quite low in comparison with active magnetic bearings (AMBs), these bearings are attractive as they require neither sensors nor power and control electronics, thereby being intrinsically more reliable, compact and energy-efficient [1]. EDBs can be of two types: radial or axial bearings. The former allows guiding the radial degrees of freedom of the rotor, whereas the latter provides the axial levitation.

Numerous models describing radial EDBs in quasi-static conditions [2,3], i.e., assuming constant spin speed and eccentricity, as well as in dynamic conditions were developed [4,5]. Although they have never been defined as such, several criteria allowing us to compare these EDBs came up along with these models.

Obviously, the stiffness induced by the electrodynamic effects is of primary interest given that it directly relates to the bearing stability and eccentricity. This stiffness is an increasing function of the rotor spin speed and can be characterised through two coefficients, namely the maximal stiffness and the electrical pole of the *R*-*L* equivalent circuit [6]. Several sensitivity analyses were performed on these two coefficients, thus yielding a first insight of the geometrical [7], electrical [8] and magnetic parameters [9] that strongly influence them.

In addition to the stiffness, attention is paid to the rotational losses required to provide the levitation force. Indeed, these losses are dissipated as heat and should therefore be limited to avoid significant temperature rises as well as to increase the energy efficiency. To this end, the null-flux concept was transposed to heteropolar EDBs, allowing us to conceive new topologies whose flux linkage is null when there is no rotor eccentricity [10]. In this way, there is no induced currents and therefore no losses in this position. Similarly, the null-E concept was then developed for homopolar bearings [11]. Simultaneously, analytical formulas were derived to evaluate these rotational losses [12,13].

The dynamic behaviour of radial EDBs constitutes a major issue as these bearings are always unstable in the absence of non-rotating damping, i.e., damping that does not depend on the rotor rotation [5,14]. Considering the difficulty of adding damping in a contactless way, thus being consistent with the magnetic bearing approach, this external damping should be minimised. To this end, analytical expressions were developed on the basis of quasi-static models to determine the minimal damping required to ensure the stability at a particular spin speed [2,9,15].

Despite their promising stability properties, electrodynamic thrust bearings (EDTBs) have focused much less research efforts. A bearing energy efficiency, defined as the ratio between the electrodynamic levitation force and the corresponding power losses, has been introduced as a performance criterion, even though external stiffnesses, such as the detent one, cannot be taken into account [16]. Recently, models describing both the axial quasi-static and dynamic behaviours of EDTBs have been derived and validated experimentally, allowing us to study their stiffness and rotational losses [17–20]. By contrast, although the beneficial effect of the external damping has been theoretically demonstrated, there is still no formula allowing us to determine the additional damping required to ensure the stability. Similarly, the spin speed ranges within which the EDTB is stable can still not be determined analytically.

In this context, the present paper introduces four performance criteria related to the bearing axial stiffness, the energy efficiency and the stability, allowing us to compare objectively EDTB topologies in terms of their intrinsic qualities. Analytical expressions of these criteria are derived on the basis of the dynamic models proposed in [17,18,21], thus being suitable for a wide variety of thrust bearing. In addition, static and dynamic stabilities are analysed analytically, providing conditions that ensure that the EDTB is stable at a particular spin speed and therefore allowing us to determine the stable spin speed ranges.

The paper is structured as follows. Section 2 depicts the thrust bearing topologies under study. Following on from this, the electromechanical model, comprising the electromagnetic and the rotor mechanical models, is described in Section 3. The stability of the system is then analysed in Sections 4 and 5. Section 6 defines the four performance criteria for EDTBs. The last section is devoted to a case study analysing three topologies through these criteria.

## **2. Bearing Description**

The thrust bearing being analysed is constituted of two independent subassemblies, namely the PM arrangements and the armature winding, in rotary motion relative to each other, as illustrated in Figure 1. Each of them can be attached either to the stator or to the rotor.

**Figure 1.** Bearing topologies with only one phase represented: (**a**) PMs are internal and the *p* coils of each set are connected in series, the two resulting sets being connected together in series; and (**b**) PMs are external and the *p* upper and the *p* lower coils are independently connected together in opposition.

The first subassembly comprises two PM arrangements, each producing an identical axial magnetic field with *p* pole pairs. These arrangements can:

The armature winding comprises *N* identical and evenly distributed phase windings. The latter are each constituted of two identical sets of *p* identical and evenly distributed coils, each set being predominantly magnetically linked to one PM arrangement, and can be of two types:

Besides, as illustrated in Figure 1a,b, respectively, both upper and lower sets of coils can be shifted by an angle equal to *π*/*p* or zero and can be connected either in series or in opposition. This connection is chosen on the basis of the angular shift that separates the upper and lower sets as well as the attractive or repulsive mode of the PM arrangements so as to ensure that the flux linked by the armature winding is null when the rotor is axially centred with respect to the stator, thereby respecting the null-flux principle.

## **3. Electromechanical Model**

Under the assumption of small rotor axial, radial and angular displacements and neglecting the inductance coefficient variations with these displacements, the axial dynamics of the system constituted of the rotor and the ETDB is decoupled from the radial and angular ones [22]. Assuming in addition that the rotor spin speed varies slowly compared to the axial dynamics, the latter can be described through a linear state-space representation as extensively derived in [18]:

$$
\begin{bmatrix}
\dot{z} \\
\dot{z} \\
\dot{F} \\
\left(\frac{\dot{T}}{z}\right)
\end{bmatrix} = \mathbf{A} \begin{bmatrix}
\dot{z} \\
\dot{z} \\
\dot{F} \\
\left(\frac{T}{z}\right)
\end{bmatrix} + \mathbf{B} \cdot \mathbf{F}\_{\ell\prime} \tag{1}
$$

with:

$$\mathbf{A} = \begin{bmatrix} -\frac{\mathbb{C}}{M} & -\frac{k\_c}{M} & \frac{1}{M} & 0\\ 1 & 0 & 0 & 0\\ -\frac{K\_\Phi^2 N}{2L\_c} & 0 & -\frac{R}{L\_c} & \omega\\ 0 & -\omega p^2 \left(\frac{K\_\Phi^2 N}{2L\_c}\right) & -\omega p^2 & -\frac{R}{L\_c} \end{bmatrix},\tag{2}$$

$$\mathbf{B} = \frac{1}{M} \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}^T.\tag{3}$$

where *z* and *z*˙ are, respectively, the rotor axial position and velocity; *F* and *T* are, respectively, the electrodynamic force and torque; *Fe* is the external axial force acting on the rotor; *C* is the external non-rotating damping; *M* is the rotor mass; *R* is the phase winding resistance; *Lc* is the cyclic inductance, thus taking into account the self and mutual inductance coefficients of the *N* phases constituting the armature winding; *K*Φ is the proportionality factor between the amplitude of the flux linked by the phase windings due to the PMs and the axial position; and *ke* is the external axial stiffness. The latter could, for example, arise from detent effects or be related to the axial stiffness induced by centring PM bearings added to the system so as to ensure the rotor radial and angular guidance. Hence, this stiffness is generally negative, as it is assumed hereafter.

Assuming quasi-static conditions, i.e., *z*˙ = 0, the axial electrodynamic stiffness *k*(*ω*) as well as the associated braking torque *T*(*ω*) can be retrieved from this dynamic model, yielding [18]:

$$\begin{split} k(\omega) &= -\frac{F(\omega)}{z} = \frac{K\_{\Phi}^{2}N}{2L\_{c}} \frac{\omega^{2}}{\omega^{2} + \left(\frac{1}{p}\frac{R}{L\_{c}}\right)^{2}} \\ T(\omega) &= -z^{2}\frac{K\_{\Phi}^{2}N}{2L\_{c}}\frac{R}{L\_{c}}\frac{\omega}{\omega^{2} + \left(\frac{1}{p}\frac{R}{L\_{c}}\right)^{2}} \end{split} \tag{4}$$

As depicted in Figure 2, illustrating the evolution of the stiffness, the latter increases with the spin speed and can be characterised through two coefficients, namely the rotor spin speed *ωe* = *R*/(*pLc*) related to the electrical pole and the asymptotic stiffness *k*∞, defined as:

$$k\_{\infty} = \frac{K\_{\Phi}^{2}N}{2L\_{c}}\tag{5}$$

The latter therefore corresponds to the maximal axial stiffness that can be generated by the EDTB. Let us point out that this stiffness appears explicitly in the state matrix **A**, given in Equation (2). On the contrary, as shown in Figure 2, the braking torque *T*(*ω*) reaches its maximal value when the speed is equal to *ωe* and then decreases asymptotically to zero.

**Figure 2.** Evolution of the electrodynamic stiffness *k* (solid line) and braking torque *T* (dashed line) with the spin speed *ω*.

## **4. Stability Analysis**

The behaviour of EDTBs is strongly dependent on the rotor spin speed and so is their stability. Hereinafter, general considerations about the stability of an EDTB coupled to the rotor are first derived. On this basis, the static and dynamic stability are then analysed, leading to conditions ensuring a stable behaviour at a specific rotor spin speed.

The following developments can be greatly simplified by considering the electrical pole as being much greater than the maximal natural frequency of the equivalent spring–mass system constituted of the rotor and the EDTB:

$$
\frac{R}{L\_c} \gg \sqrt{\frac{k\_{\infty}}{M}}.\tag{6}
$$

In this way, the electrical phenomena are much faster than the mechanical ones and thus do not have a significant impact on the rotor axial dynamics. Observing that the electromechanical model, given in Equation (1), depends on the stiffness as well as the rotor mass and not their square roots, Equation (6) can be expressed in a more convenient manner as:

$$\left(\frac{R}{L\_c}\right)^2 \gg \frac{k\_{\infty}}{M}.\tag{7}$$

To the authors' best knowledge, the latter hypothesis is verified in the vast majority of the experimental and numerical studies of EDTBs, including the case study in Section 7. In addition, let us assume a priori that the external damping satisfies:

$$2\left(\frac{\text{C}}{M}\right) \ll \frac{R}{L\_{\text{c}}},\tag{8}$$

This assumption is verified a posteriori in Section 4.3.

## *4.1. General Considerations*

The model in Equation (1) being linear, the stability analysis can be performed through the study of the real part of the four eigenvalues of the state matrix **A** as a function of the spin speed. To this end, the characteristic polynomial can be easily derived, yielding:

$$\begin{split} P(s) &= s^4 + s^3 \left( \frac{\mathbb{C}}{M} + \frac{2R}{L\_{\mathfrak{c}}} \right) + s^2 \left( \frac{\mathbb{C}}{M} \frac{2R}{L\_{\mathfrak{c}}} + \frac{R^2}{L\_{\mathfrak{c}}^2} + \omega^2 p^2 + \frac{k\_{\infty} + k\_{\mathfrak{c}}}{M} \right) \\ &+ s \left( \frac{\mathbb{C}}{M} \frac{R^2}{L\_{\mathfrak{c}}^2} + \omega^2 p^2 \frac{\mathbb{C}}{M} + \frac{R}{L\_{\mathfrak{c}}} \frac{k\_{\infty} + 2k\_{\mathfrak{c}}}{M} \right) + \omega^2 p^2 \frac{k\_{\infty} + k\_{\mathfrak{c}}}{M} + \frac{R^2}{L\_{\mathfrak{c}}^2} \frac{k\_{\mathfrak{c}}}{M} . \end{split} \tag{9}$$

Under the hypothesis expressed in Equation (7) and assuming Equation (8) as verified, the polynomial in Equation (9) can be simplified as follows:

$$P(s) = s^4 + s^3 \frac{2R}{L\_\varepsilon} + s^2 \left(\frac{R^2}{L\_\varepsilon^2} + \omega^2 p^2\right) + s \left(\frac{C}{M} \frac{R^2}{L\_\varepsilon^2} + \omega^2 p^2 \frac{C}{M} + \frac{R}{L\_\varepsilon} \frac{k\_{\rm co} + 2k\_\varepsilon}{M}\right) + \omega^2 p^2 \frac{k\_{\rm co} + k\_\varepsilon}{M} + \frac{R^2}{L\_\varepsilon^2} \frac{k\_\varepsilon}{M}.\tag{10}$$

The root locus of the four eigenvalues can thus be obtained by finding the roots of Equation (10) for different spin speeds. However, when it comes to stability analyses, only the speeds at which the eigenvalues cross the imaginary axis are relevant as they define the spin speed ranges within which the bearing is stable. Figure 3a,b illustrates, respectively, the impact of the external damping and stiffness on the root locus. Only the two relevant eigenvalues, related to the mechanical behaviour, are represented, the remaining two, related to the electrical behaviour, being located far in the left half plane. The additional damping allows us to shift the complex conjugates parts of the root locus to the left by an amount equal to *C*/(2*M*), whereas the external stiffness strongly modifies their shape.

**Figure 3.** Root locus of both relevant eigenvalues: (**a**) evolution with the damping *C* = {0, 0.5, 1} Ns/m for *ke* = 0 N/m; and (**b**) evolution with the external stiffness |*ke*| = {0, 1 4 , 1 2 , 3 4 , 9 10 } *k*∞ N/m for *C* = 0 Ns/m.

As a result, there are at most three spin speeds, *ω*1, *ω*2 and *ω*3, defined in Figure 4, corresponding to intersections with the imaginary axis. More precisely, as shown in Figure 3, when the external damping approaches zero, the spin speed *ω*3 tends to infinity and therefore no longer exists. By contrast, increasing the damping allows us to move the spin speeds *ω*2 and *ω*3 towards each other until they are equal, when the damping reaches a specific value, denoted by *Cm* hereinafter. Beyond the latter damping, these two speeds do not exist anymore. Besides, as illustrated in Figure 3b, the presence of the speed *ω*2 strongly depends on the external stiffness.

**Figure 4.** Root locus: Spin speeds corresponding to intersections with the imaginary axis.

For determining these speeds, let us assume that *s* = j*h*, implying that the eigenvalue lies on the imaginary axis. In this case, Equation (10) can be separated into real and imaginary parts as follows:

$$\begin{cases} 0 = h^4 + \omega^2 p^2 \frac{k\_{\infty} + k\_{\varepsilon}}{M} + \frac{R^2}{L\_c^2} \frac{k\_{\varepsilon}}{M} - h^2 \left(\frac{R^2}{L\_c^2} + \omega^2 p^2\right) \end{cases} \tag{11}$$

$$\left\{ 0 = h \left( \frac{\mathbb{C}}{M} \frac{R^2}{L\_c^2} + \omega^2 p^2 \frac{\mathbb{C}}{M} + \frac{R}{L\_c} \frac{k\_{\infty} + 2k\_{\varepsilon}}{M} \right) - h^3 \frac{2R}{L\_c} \right\} \tag{12}$$

Solving Equation (12) for *h* yields three solutions. As demonstrated hereinafter, one solution is related to a static instability, whereas the other two are linked to a dynamic one.

## *4.2. Static Stability*

The trivial solution of Equation (12), i.e., *h* = 0, corresponds to the first intersection of the eigenvalues with the imaginary axis. Substituting this solution into Equation (11) and isolating *ω* leads to:

$$
\omega\_1 = \frac{1}{p} \frac{R}{L\_c} \sqrt{-\frac{k\_\varepsilon}{k\_\varepsilon + k\_\infty}}.\tag{13}
$$

This corresponds to the spin speed at which the stiffness induced by the electrodynamic effects exactly compensates for the external stiffness, i.e., *k*(*ω*1) = |*ke*|, as can be verified through Equation (4). Below this specific spin speed, the thrust bearing suffers from an instability as the external stiffness, whose effect is destabilising due to its negative value, is larger than the electrodynamic one. This instability can be qualified as static as it does not depend on the damping. The static stability condition can thus be stated as:

$$k(\omega) \ge |k\_t| \iff \omega \ge \omega\_1 \tag{14}$$

Two limiting cases can be studied. On the one hand, when there is no external stiffness, the speed *ω*1 is equal to zero and the static stability condition does not introduce any restriction on the rotor spin speed. On the other hand, when the external stiffness is larger, in absolute value, than the maximal electrodynamic stiffness, i.e., |*ke*| > *k*∞, the speed *ω*1 tends to infinity and the bearing is unstable regardless of the rotor spin speed.

## *4.3. Dynamic Stability*

Both remaining solutions of Equation (12) are linked to a dynamic instability as they depend on the damping. They can be calculated as follows:

$$h = \pm \sqrt{\frac{1}{2} \frac{\mathcal{C}}{M} \frac{R}{L\_c} + \omega^2 p^2 \frac{1}{2} \frac{\mathcal{C}}{M} \frac{L\_c}{R} + \frac{k\_{\infty} + 2k\_{\varepsilon}}{2M}}.\tag{15}$$

Substituting Equation (15) into Equation (11) and multiplying by 4*R*2/*L*2 *c* yields:

$$
\omega^4 f\_1 + \omega^2 f\_2 + f\_3 = 0,\tag{16}
$$

where:

$$\begin{split} f\_1 &= p^4 \frac{\mathbb{C}}{M} \left[ \frac{\mathbb{C}}{M} - 2\frac{R}{L\_c} \right] \\ f\_2 &= 2p^2 \frac{R}{L\_c} \left[ \frac{R}{L\_c} \frac{k\_{\infty}}{M} + \frac{R}{L\_c} \left( \frac{\mathbb{C}}{M} \right)^2 + \frac{k\_{\infty} + 2k\_c}{M} \frac{\mathbb{C}}{M} - 2\left( \frac{R}{L\_c} \right)^2 \frac{\mathbb{C}}{M} \right] \\ f\_3 &= \left( \frac{R}{L\_c} \right)^2 \left[ \left( \frac{R}{L\_c} \frac{\mathbb{C}}{M} \right)^2 + \left( \frac{k\_{\infty} + 2k\_c}{M} \right)^2 - 2\left( \frac{R}{L\_c} \right)^3 \frac{\mathbb{C}}{M} - 2\left( \frac{R}{L\_c} \right)^2 \frac{k\_{\infty}}{M} + 2\frac{R}{L\_c} \frac{\mathbb{C}}{M} \frac{k\_{\infty} + 2k\_c}{M} \right]. \end{split} \tag{17}$$

The polynomial in Equation (16) has at most two positive roots, thereby confirming that both eigenvalues related to the electrical behaviour never cross the imaginary axis. Under the hypothesis expressed in Equation (7) and still assuming that the damping satisfies Equation (8), the coefficients in Equation (17) can be greatly simplified, leading to:

$$f\_1 = -2p^4 \frac{C}{M} \frac{R}{L\_c} \tag{18}$$

$$f\_2 = 2p^2 \left(\frac{R}{L\_c}\right)^2 \left[\frac{k\_{\infty}}{M} - 2\frac{R}{L\_c}\frac{\mathcal{C}}{M}\right] \tag{19}$$

$$f\_3 = -2\left(\frac{R}{L\_c}\right)^4 \left[\frac{R}{L\_c}\frac{\mathcal{C}}{M} + \frac{k\_\infty}{M}\right].\tag{20}$$

Solving Equation (16) with these reduced coefficients allows us to determine both spin speeds *ω*2 and *ω*3 at which the relevant eigenvalues cross the imaginary axis, as shown in Figure 4:

$$\begin{cases} \omega\_{2,3}|\_{C \neq 0} = \frac{1}{p} \sqrt{\frac{1}{2} \frac{R}{L\_c} \frac{M}{C} \left[ \frac{k\_{\infty}}{M} - 2 \frac{\mathbb{C}}{M} \frac{R}{L\_c} \mp \sqrt{\Delta} \right]} \\\\ \Delta = \left( \frac{k\_{\infty}}{M} \right)^2 - 8 \frac{R}{L\_c} \frac{\mathbb{C}}{M} \frac{k\_{\infty}}{M} \end{cases} . \tag{21}$$

The value of these speeds is independent from the external stiffness *ke*, signifying that the intersections of the eigenvalues with the imaginary axis occur at the same spin speeds even when the shape of the root locus is modified by this stiffness, as shown in Figure 3b. By contrast, as mentioned in Section 4.1, the existence of these intersections strongly depends on the external damping and stiffness.

Figure 5 shows the evolution of both speeds *ω*2 and *ω*3 with the external damping. As expected, when the latter is equal to zero, the speed *ω*3 tends to infinity and therefore no longer exists, whereas the speed *ω*2 can be easily calculated by observing that the coefficient *f*1 in Equation (18) is equal to zero, implying that Equation (16) has only one positive solution:

$$\left.\omega\_2\right|\_{\mathbb{C}=0} = \frac{1}{p}\frac{R}{L\_{\mathbb{C}}} = \omega\_{\mathfrak{c}}.\tag{22}$$

This speed thus corresponds to spin speed *ωe* related to the electrical pole. Let us point out that spin speeds smaller than this particular speed can never lie on the imaginary axis and are therefore stable, from a dynamic point of view, regardless of the damping. As stated in Section 4.1, adding external damping enables moving the speeds *ω*2 and *ω*3 towards each other until they intersect, when the damping reaches *Cm*. Cancelling the coefficient Δ in Equation (21) allows us to determine both the damping *Cm* such that these two speeds are equal and the corresponding speed, denoted by *ωm*:

$$\int \mathcal{C}\_{m} = \frac{k\_{\infty}}{8} \frac{L\_{\text{c}}}{R} = \frac{K\_{\Phi}^{2} N}{16R} \tag{23}$$

$$\left\{\omega\_m = \frac{\sqrt{3}}{p} \frac{R}{L\_c}\right\} \tag{24}$$

Below this damping, the speeds *ω*2 and *ω*3 are distinct and the EDTB is unstable, from a dynamic point of view, when the spin speed belongs to the interval [*ω*2 ; *ω*3], as shown in Figure 4. By contrast, when the damping is larger than *Cm*, the eigenvalues only cross the imaginary axis at the speed *ω*1 and the EDTB is stable beyond the latter speed. Consequently, unlike their static counterparts, dynamic instabilities can be removed through additional non-rotating damping.

Finally, substituting the maximal damping given in Equation (23) into Equation (8) and considering that the assumption in Equation (7) is verified allows us to validate the relation in Equation (8) a posteriori, highlighting that the latter is not, as such, a hypothesis.

**Figure 5.** Evolution of the speeds *ω*2 and *ω*3 with the external damping *C*.

## *4.4. Stability Conditions*

In summary, the stability can be analysed on the basis of:

More precisely, when the maximal electrodynamic stiffness is larger than the external one, i.e., *k*∞ > |*ke*|, the stability is ensured at the spin speed *ω* provided that:

$$\begin{cases} \omega \le \omega\_2(\mathbb{C}) \text{ or } \omega \ge \omega\_3(\mathbb{C}) & \text{if } \mathbb{C} \in [0; \mathbb{C}\_m] \\ \omega \ge \omega\_1 \end{cases} \tag{25}$$

Finally, let us point out that the state-space representations in [17,21] yield an identical characteristic polynomial to Equation (9), thus widening the scope of the previous developments to these models.

## **5. Stable Speed Analysis**

Assuming that the eight parameters describing the dynamic behaviour of the system are identified, the speed ranges within which the EDTB is stable can be easily determined through the conditions defined in Equation (25). However, let us go one step further by analysing three different cases, depending on the relative importance of the external stiffness in comparison to the electrodynamic one.

#### *5.1.* |*ke*| ∈ [0 ; *k*∞ 2 ]