A Theoretical Method for Designing Thin Wobble Motor Using an Electromagnetic Force and an Electropermanent Magnet for Application in Portable Electric Equipment

applied sciences

Article A Theoretical Method for Designing Thin Wobble Motor Using an Electromagnetic Force and an Electropermanent Magnet for Application in Portable Electric Equipment

Sang Yong Park, Buchun Song and Yoon Su Baek *

Department of Mechanical Engineering, Yonsei University, Seoul 03722, Korea; [email protected] (S.Y.P.); [email protected] (B.S.) * Correspondence: [email protected]

Abstract: The thin wobble motors that are required to hold rating shafts employ an electropermanent magnet. This turns the holding force on and off by applying a momentary electrical pulse. To design the magnet devices without the need for finite element analyses, a theoretical force model is necessary for predicting the attractive force. In this paper, first, a force model is derived by estimating the permeance around the air gap. A magnetic circuit is constructed, employing a relatively simple method to build the model in clouding leakage flux. Thus, the basic structure and driving principle are also presented. Next, an analytical force model is constructed on the basis of distribution parameter analysis between the stator and the rotating shaft. The design of the electromagnet core and the control method are presented. Finally, a prototype model of the motor that is 30 mm in diameter and 7 mm in thick is fabricated. The two models are verified by comparing the results of FEM with the results of the experiments. They can properly predict the attractive force, so the thin

wobble motor with holding force can be applied in portable electric equipment. Keywords: wobble motor; permeance; magnetic circuit; leakage flux; electropermanent magnet; Citation: Park, S.Y.; Song, B.; Baek, Y.S. A Theoretical Method for force model Designing Thin Wobble Motor Using an Electromagnetic Force and an Electropermanent Magnet for Application in Portable Electric 1. Introduction Equipment. Appl. Sci. 2021, 11, 881. Electromagnetic actuators, which have many advantages, including their small size, https://doi.org/10.3390/app11020881 light weight, high precision, high accuracy, and high efficiency, have been widely used as rotary driving motors in various fields such as industrial machinery, medicine, human care Received: 25 December 2020 device, electronic parts assembly, military equipment and mobile robots. These motors Accepted: 14 January 2021 are manufactured depending on the target application in various ways. For example, new Published: 19 January 2021 motors shapes have been developed using physical and chemical phenomena such as electromagnetic force, smart polymers, shape-memory alloy (SMA), electrostatic force, etc. Publisher’s Note: MDPI stays neutral Some of the materials used to drive these motors are difficult to control, and, therefore, with regard to jurisdictional claims in there are limitations to their use. For this reason, we chose electromagnetically driven published maps and institutional affil- motors that are easy to manufacture and control and can easily calculate the generated iations. force. However, these motors require a reduction gear device to increase their driving torque. The added reduction gear equipment increases the total volume of the driving motor, making it difficult to drive or install. In addition, it may cause mechanical backlash, added friction, or assembly errors, which may further occur to malfunctions when motor is Copyright: © 2021 by the authors. operated. To solve these problems, there have been a number of research and development Licensee MDPI, Basel, Switzerland. efforts aiming to increase torque through the application of various types of reduction gear This article is an open access article device [1,2]. Several new types of stepping motors have been developed, such as a wobble distributed under the terms and motor [3–6]. In this research, we selected a wobble motor because it is simple structure, conditions of the Creative Commons with an eccentric structure, and the torque generated per unit volume is very high due to Attribution (CC BY) license (https:// the gear ratio while it also has the advantage of a precise control [7–9]. However, existing creativecommons.org/licenses/by/ wobble motors have the main disadvantage that they cannot fix a rotating shaft while 4.0/).

Appl. Sci. 2021, 11, 881. https://doi.org/10.3390/app11020881 https://www.mdpi.com/journal/applsci Appl. Sci. 2021, 11, 881 2 of 14

the shaft is halted. To fix the shaft, we introduce a new type of wobble actuator with an electropermanent magnet (EPM). It comprises semi-hard (Alnico 5) and neodymium (NdFeB) magnets, and it can be used to fix the rotating shaft. In this paper, we introduce the two theoretical force models for application with portable electric devices and fabricate a thin electromagnetic wobble motor that is 50 mm in diameter and 7 mm thick including the EPM device. First, it is necessary to predict the magnetic force generated by the magnets of the EPM device. The most accurate method predicting force is used by commercial finite element method software. FEM is one of the ways to solve problems using numerical approximation for problems that are difficult to obtain an accurate theoretical interpretation. In addition, FEM has the advantage of being compatible with Auto CAD or other design programs, and accuracy of the model analysis and user convenience are very high, and high-quality analysis is possible even if there is a lack of professional knowledge about FEM. On the other hand, it will take several hours to obtain the precise analysis of the complex model and variables in the model. However, this comes with the problem of having to perform numerous iterations in the initial design step. On the other hand, a more systematic method is magnetic circuit analysis. An EPM device with two magnets is expressed as a model in which a magnetic flux source and an internal reluctance are connected in parallel. The magnetic flux paths, including the reluctance of the core of the magnetic material and the reluctance in the air gap, are shown as an electric circuit. The method for calculating the magnetic force in a magnetic circuit is similar to calculate the current in an electric circuit and is very simple. Since the fringing effect and the leakage flux around the air gap are not considered, the results of magnetic circuit analysis led to a considerably large degree of error. This is because it is difficult to accurately include them in magnetic circuits. Therefore, we introduce a theoretical force method into the initial design for use in the magnetic circuit. In addition, we design an EPM device including magnets with a flat plate and a constant air gap, and propose a simple 3D model for the leakage flux and fringing effect around the gap. The paths of magnetic flux are modeled using a simplified method consisting of several arcs, straight lines, and a mean flux line. Additionally, we introduce the schematic of the EPM device to validate the proposed theoretical force model, and the basic principle and structure, are presented in a straightforward manner. Its effectiveness is confirmed on the basis of a comparison with the results of FEM analysis, and experiments carried out using a prototype. Second, in order to analyze the characteristics existing between the electromagnet core and the rotating shaft, it is necessary to predict the magnetic flux density and magnetic force when changing the air gap. To this end, we introduce a distribution parameter method in order to propose a theoretical force model. This method was used to calculate the gap by mapping the air gap, which does not have a constant structure, onto the Cartesian coordinates. We also present a schematic of a thin wobble motor in order to validate the proposed force model, as well as the optimization of core shape, basic driving principle, basic structure and control method of the motor. Finally, the experimental results are presented for the EPM device and a thin wobble motor.

2. Driving Principle and Structure 2.1. Driving Principle and Structure of the Wobble Motor Figure1 shows the basic working principle and structure of the wobble motor, which comprises of cylindrical stator A and columnar rotating shaft rotor B. The rolling point of Figure1a is in contact with the stator A and the rotor B. Rs, RR, Rc and Sc denote in the radius of stator A, the radius of a rotor B, the central point of rotor B, and central point of stator A, respectively. The primary feature of the wobble motor is that rotor B rolls along stator A’s inner surface as shown in Figure1b. If the input current is supplied in a clockwise direction in a continuous manner, the force will be generated from the input current. The attractive force pulls the rotor, and the rotor rolls along the inner of the stator via the generated attractive force. The driving direction of stator A is in the opposite direction of rolling direction of rotor B [10–15]. Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of 15

Appl. Sci. 2021, 11, 881 generated attractive force. The driving direction of stator A is in the opposite direction3 of of 14 rolling direction of rotor B [10–15].

(a)

(b)

FigureFigure 1. Driving 1. Driving principle principle of ofa wobble a wobble motor: motor: (a ()a Schematic) Schematic of of a a wobble wobble motor; motor; ( (b)) Driving principle.principle. 2.2. Driving Principle and Structure of the EPM 2.2. DrivingThe EPMPrinciple composes and Structure of two of magnetic the EPM materials, NdFeB magnet and another Alnico5 magnet;The EPM a coil composes is wrapped of two around magnetic these, materi as shownals, inNdFeB Figure magnet2. The basicand another principle Alnico5 driving magnet;of the a EPM coil is was wrapped showed around using these, the MagNet as shown 7.0 in electromagnetic Figure 2. The basic analysis principle software driving from of Mentor.the EPM If was the magnetizationshowed using directionsthe MagNet of the7.0 twoelectromagnetic magnets are oppositelyanalysis software combined, from the Mentor.magnetic If the flux magnetization only circulates directions inside connected of the two iron magnets as shown are in oppositely Figure2a. After combined, a switching the magneticcurrent flux pulse only of thecirculates opposite inside polarity connect applieded iron to as Alinico5, shown in the Figure magnetization 2a. After a directionsswitch- ingof current the two pulse magnets of the were opposite thesame, polarity and applied the EPM to Alinico5, generated the an magnetization external magnetic directions flux as of shownthe two in magnets Figure2 wereb. The the magnetization same, and the direction EPM generated of the NdFeBan external material magnetic is unchanged flux as shownbecause in Figure it has 2b. high The coercivity. magnetization In ferromagnetic direction of materials, the NdFeB the material phenomenon is unchanged of magnetic be- causehysteresis it has high occurs. coercivity. J.A. Ewing In ferromagneti coined the termc materials, “hysteresis” the phenom to describeenon his of observationmagnetic hys- that teresisthe magnetic occurs. J.A. induction Ewing coined in a metal the term behind “hys theteresis” applied to current,describe and his observation he also demonstrated that the magneticthat the induction molecules in of a a metal ferromagnetic behind the material applied acted current, as reversible and he also permanent demonstrated magnets that [16 ]. theThe molecules normal of coercivity a ferromagnetic of Alnico material 5 is approximately acted as reversible 50 kA/m, permanent and the remnant magnets flux [16]. density The normalis approximately coercivity of 1.2 Alnico T. The 5 normalis approximately coercivity of50 NdFeBkA/m, and is approximately the remnant flux 1000 density kA/m, andis approximatelythe remnant flux1.2 T. density The normal is approximately coercivity of 1.2~1.3 NdFeB T [17is ].approximately 1000 kA/m, and the remnantWhen flux a switching density is current approximately is applied 1.2~1.3 to the coilT [17]. wound, the magnetization direction of the Alnico 5 magnet changes, and the external magnetic force generates attractive force. Appl. Sci.Appl. 2021, Sci. 11, 2021x FOR, 11 PEER, 881 REVIEW 4 of 15 4 of 14

Figure 2.Figure Basic principle 2. Basic principledriving an driving electropermanent an electropermanent magnet: (a)magnet: In the no (currenta) In the state, no currentthe two state, the magnetstwo are magnetsopposite arely polarized; oppositely (b polarized;) In the switching (b) In the current switching state, current the two state, magnetic the two magnets magnetic are magnets polarizedare in polarized the same indirection. the same direction.

When3. Derivation a switching of current Theoretical is applied Force Modelto the coil wound, the magnetization direction of the Alnico3.1. Theoretical 5 magnet Force changes, Analysis and ofthe the extern EPMal magnetic force generates attractive force. Figure3 shows a full schematic of the EPM device for holding the rotating shaft 3. Derivationof the wobbleof Theoretical motor. Force Alnico Model 5 and NdFeB magnets are described as square magnets in 3.1. Theoreticalthe schematic, Force Analysis but the of two the EPM magnets used in the theoretical analysis and the experiment were replaced with cylindrical magnets. To predict the attractive force, first, the structure Figure 3 shows a full schematic of the EPM device for holding the rotating shaft of is assumed to be a simple structure with a constant gap, and a core that contains two the wobble motor. Alnico 5 and NdFeB magnets are described as square magnets in the permanent magnets. Additionally, we assume that the rotating shaft is replaced with the schematic, but the two magnets used in the theoretical analysis and the experiment were plate. Second, in order to simplify the calculations for magnetic flux density, including the replaced with cylindrical magnets. To predict the attractive force, first, the structure is leakage flux, we simplified the path through which the leakage flux flows. Additionally, assumed to be a simple structure with a constant gap, and a core that contains two per- the fringing flux was assumed to be the leakage flux, and the path of flux was consisted manent magnets. Additionally, we assume that the rotating shaft is replaced with the as a combination of arcs and straight line, as shown in Figure4a [ 18]. The fringing fluxes plate. Second, in order to simplify the calculations for magnetic flux density, including the were considered a kind of leakage flux. It was also assumed that the paths detouring on leakage flux, we simplified the path through which the leakage flux flows. Additionally, the top outside of the surrounding air gap were connected in the quadrant with a straight the fringing flux was assumed to be the leakage flux, and the path of flux was consisted line and the length of the air gap [19]. Figure4b shows a consideration of the mean leakage as a combinationflux line. The of arcs specifications and straight of line, the electromagnetic as shown in Figure elements 4a [18]. in The use fringing and their fluxes parameters were consideredare summarized a kind inof Tableleakage1. flux. It was also assumed that the paths detouring on the top outside of the surrounding air gap were connected in the quadrant with a straight For the leakage flux paths of Pdx, as shown in Figure4a, the length lx of the magnetic line andflux the path length is equalof the toairg gap+ π [19].x, which Figure is the4b shows sum of a theconsideration lengths of two of the quadrant mean leakage arcs of radius flux line.x and Thethe specifications length of the of gap. the electrom The permeanceagnetic inelements the path in is use as follows:and their parameters are summarized in Table 1. 1 µ0dAx µ0bdx Pdx = = = (1)

The permeance of the total leakage ﬂux can be obtained by integrating Equation (1). Appl. Sci. 2021, 11, 881 5 of 14 Appl. Sci. 2021, 11, x FOR PEER REVIEW 5 of 15

Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 15

Figure 3. Schematic of the switchable eletropermanent magnet construction. Figure 3. Schematic of the switchable eletropermanent magnet construction.

Table 1. Parameters of the electropermanent magnet device.

Parameter Symbol Value Unit Thickness b 2 mm Length L 22 mm Pole length n 4 mm Pole length q 11.6 mm Pole length h 5.5 mm Pole height p 2 mm Pole width a 2 mm (a) (b) Magnet diameter d 2 mm FigureFigure 4. Paths 4. Paths modelMagnet model for leakage length for leakage flux: flux:(a) Paths (a) Paths for the Lm for leaked the leaked flux; flux;(b) Mean 8 ( b) Mean length length of flux of mmpath. flux path. Air gap g 0.1 mm TableIf the 1. coreParameters andRotor the diameter of plate the electropermanent have a constant magnetair gap, r device. as shown in 5 Figure 5a, it is assumed mm that the leakage flux paths diverting to the outside of the gap are connected to two quad- Stator diameter R 5.1 mm rants of the same Parametermagnetic flux in a straight Symbolline along the length Value of the air gap. Unit Each Number of turns N 120 leakage flux path Thicknessindicated by a line has a reluctance, b and all of 2them in the magnetic mm Angle θ 45 deg circuit are connectedLength in parallel, expressing it as shown L in Figure 5b. 22 Therefore, it is mm pos- Applied larger than current i 5 A sible to use a permeancePole length that has an inverse relationship n with a reluctance. 4 mm DiameterPole length of the coil dc q 0.18 11.6 mm mm Pole lengthμ h 5.5−7 mm Permeability of the free space 0 410π × H/m Pole heightμ p 2 mm Relative permeability of the magnet r 1.05 H/m Pole width a 2 mm A 2 Cross-sectionalMagnet diameterarea of the air gap g d5.26 2 mm mm 2 Cross-sectionalMagnet area length of two magnets A Lmm 6.28 8 mm mm Air gap g 0.1 mm For theRotor leakage diameter flux paths of P , as shown r in Figure 4a, the 5 length l of the mm magnetic Stator diameterdx R 5.1x mm +π flux path isNumber equal to of turnsg x , which is the sum N of the lengths of 120two quadrant arcs of radius x and the lengthAngle of the gap. The permeance θin the path is as follows:45 deg Applied larger than current i 5 A 1 μμdA bdx Diameter of the coilP == dc00x = 0.18 mm dx ℜ+π × −7 (1) Permeability of the free space dxµ0 lgx x 4π 10 H/m Relative permeability of the magnet µr 1.05 H/m 2 Cross-sectionalThe permeance area of of the the airtotal gap leakage fluxA cang be obtained 5.26by integrating Equationmm (1). 2 Cross-sectional area of two magnets Am 6.28 mm (a) (b) Figure 5. Analysis model consisting of an electropermanent magnet: (a) Flux leakage of the electropermanent magnet; (b) Magnetic circuit.

ℜ ℜ In Figure 5b, the reluctance c and o in the iron are ignored. Therefore, when the

permeances of the leakage flux path are Pgoglmomi,,PP , Pand Pg , respectively, they can be ex- ℜℜℜℜ ℜ ℜ Φ Φ pressed as reciprocal reluctances goglmomi,, , and g . m . r and m are the source of the magnet, the magnetic flux of the magnetic flux passing through the air gap. To more accurately consider the leakage flux generated in the gap, it is regarded as a geometry as shown in Figure 6. The permeance of each of the flux paths through the gap can be ex- =+ + + =++ pressed as permeance Pgo PPPP12342( ) 4 and Pgl 2(PP56 ) 4 P 7. Permeance P1 is μ 0 ab/ g . The mean length of the flux path of permeance P2 can be considered to be equal to the length of a line drawn midway between the radius and a quarter of circumference

as shown in Figure 6b. Permeance P2 and P3 with a quadrant of the cylindrical volume μ μ are 0.528 0b and 0.528 0 a , respectively. Appl. Sci. 2021, 11, x FOR PEER REVIEW 6 of 15

Appl. Sci. 2021, 11, 881 (a) (b) 6 of 14 Figure 4. Paths model for leakage flux: (a) Paths for the leaked flux; (b) Mean length of flux path.

If the core and theIf the plate core have and a the constant plate have air gap, a constant as shown air in gap, Figure as shown 5a, it inis assumed Figure5a, it is assumed that the leakagethat flux the paths leakage diverting flux paths to the diverting outside of to the the gap outside are connected of the gap to are two connected quad- to two quad- rants of the samerants magnetic of the same flux in magnetic a straight flux line in aalong straight the linelength along of the the air length gap. ofEach the air gap. Each leakage flux pathleakage indicated flux path by a indicated line has bya reluctance, a line has a reluctance,and all of andthem all in of the them magnetic in the magnetic circuit circuit are connectedare connected in parallel, in parallel,expressing expressing it as shown it as in shown Figure in 5b. Figure Therefore,5b. Therefore, it is pos- it is possible to sible to use a permeanceuse a permeance that has that an inverse has an inverserelationship relationship with a reluctance. with a reluctance.

(a) (b)

Figure 5. AnalysisFigure 5. model Analysis consisting model ofconsisting an electropermanent of an electropermanent magnet: (a )magnet: Flux leakage (a) Flux of theleakage electropermanent of the elec- magnet; (b) Magnetic circuit.tropermanent magnet; (b) Magnetic circuit.

ℜ ℜ In Figure 5b, theIn reluctance Figure5b, thec reluctanceand o in the

permeances of permeancesthe leakage flux of the path leakage are Pgoglmomi ﬂux,,PP path , Pand are PPgog ,, respectively,Pgl, Pmo, Pmi and theyP cang, respectively, be ex- they can be expressed as reciprocalℜℜℜℜ reluctances <ℜgo, <ℜgl, <Φmo,

to the length ofof a line circumference drawn midway as shown between in Figure the radius6b. Permeance and a quarterP2 ofand circumferenceP3 with a quadrant of the

as shown in Figurecylindrical 6b. Permeance volume areP2 0.528and µP03b withand 0.528a quadrantµ0a, respectively. of the cylindrical volume μ μ are 0.528 0b and 0.528 0 a , respectively.

(a) (b)

Figure 6. PermeanceFigure 6. Permeance of ﬂux paths of throughflux paths the through air gap: the (a )air Permeance gap: (a) Permeance of parallel of to parallel end of to the end core; of the (b) core; Simple-shaped volumes of permeance.(b) Simple-shaped volumes of permeance.

The mean lengthThe of mean the flux length path of for the permeance ﬂux path for P4 permeancewith one eighthP4 with of onethe spherical eighth of the spherical volume can bevolume approximated can be approximatedby considering bythe considering mean flux line the to mean be situated ﬂux line 0.65 to of be the situated 0.65 of way between the center of the sphere and the circumference, as shown in Figure 5b. There- μ fore, P4 with one-eighth of the spherical volume is 0.308 0 g . P5 and P6 with one-half of μπ()+ μπ()+ the half annulus are 2/ln1/0 atg and 2/ln1/0btg, respectively. P7 with one- μ half of the quadrant of the spherical shell is 0.5 0t . Permeance Pmo and Pmi detouring μ ππ()+ μ into the air without going through the plate are 0t /ln1 and 0t /2, respectively. ℜ The reciprocal of permeance is defined as reluctance. t is the total reluctance; thus, all reluctance in the magnetic circuit can be calculated. To calculate the magnetic field in consideration of the leakage flux, we used Gauss’ law and a magnetic force circuit, which can be expressed as follows

ℜ=11 = t ++ + (2) PPtmomiglgo P0.5( PP )

π 2 +=+Φ dB()/8ar B Bab g leak

Φ= − ℜ leak()/Ni H m L m leak (3)

()μπab/2 g+ℜ 1/ Ni − d2 ( B + B )/8 H = 0 leak r a m πμ2 ++ℜ() μ dabgL00/8 /2 1/ leak m

where Hm is the axial magnetic field intensity when combining two magnets, Bg is the Φ magnetic field density in the air gap, and leak is the pole-to-pole leakage flux. The axial

magnetic flux densities of the Alnico 5 and NdFeb magnets are Ba and Br , respectively. Finally, the generated attractive force can be obtained using the Maxwell stress tensor, which can be written as

2 Ni− H L = μ mm Fab0  (4) 2g

3.2. Theoretical Force Analysis of Motor The side view used for theoretical analysis is showed in Figure 7. An idealized model was assumed in which the end effects were neglected. The idealized model is thus similar to an idealized rotating machine. A distribution equation of the electromagnetic system can be expressed using Maxwell’s equation and Ohm’s law. The magnetic vector potential Appl. Sci. 2021, 11, 881 7 of 14

the way between the center of the sphere and the circumference, as shown in Figure5b. Therefore, P4 with one-eighth of the spherical volume is 0.308µ0g. P5 and P6 with one-half of the half annulus are 2µ0a/π ln(1 + t/g) and 2µ0b/π ln(1 + t/g), respectively. P7 with one-half of the quadrant of the spherical shell is 0.5µ0t. Permeance Pmo and Pmi detouring into the air without going through the plate are µ0t/π ln(1 + π) and µ0t/2, respectively. The reciprocal of permeance is deﬁned as reluctance.

1 1

2 πd (Ba + Br)/8 = Bgab + Φleak

Φleak = (Ni − Hm Lm)/

2 (µ0ab/2g+1/

where Hm is the axial magnetic field intensity when combining two magnets, Bg is the magnetic field density in the air gap, and Φleak is the pole-to-pole leakage flux. The axial magnetic flux densities of the Alnico 5 and NdFeb magnets are Ba and Br, respectively. Finally, the generated attractive force can be obtained using the Maxwell stress tensor, which can be written as Ni − H L 2 F = µ ab m m (4) 0 2g

3.2. Theoretical Force Analysis of Motor

Appl. Sci. 2021, 11, x FOR PEER REVIEW The side view used for theoretical analysis is showed in Figure7. An idealized8 modelof 15 was assumed in which the end effects were neglected. The idealized model is thus similar to an idealized rotating machine. A distribution equation of the electromagnetic system can be expressed using Maxwell’s equation and Ohm’s law. The magnetic vector potential A A is isdefined defined as as B=B ∇×=∇ A ×. TheirA. Their solutions solutions have have previously previously been been studied studied in related in related research research areasareas [20–23]. [20–23 Therefore,]. Therefore, we we briefly briefly describe describe the the theoretical theoretical analysis analysis of ofthe the flux flux density density of of thethe air airgap. gap. To Toestablish establish analytical analytical solutions solutions as asa function a function of ofthe the air air gap, gap, variations variations in inthe the r-direction are ignored, and ∇⋅A0 = can also be assumed, and six terms of g, h, t, p,μ r-direction are ignored, and ∇ · A = 0 can also be assumed, and six terms of g, h, t, p,0 µ0 μ andand rµ r areare air air gap, gap, the the pole pole width, width, thickness thickness of of the the stator, pole pitch, thethe permeabilitypermeabilityof ofair airgap gap andand thethe permeabilitypermeability ofofthe thematerial, material, respectively. respectively.

FigureFigure 7. Geometric 7. Geometric model model for fortheoretical theoretical analysis. analysis.

TheThe governing governing equation equation of ofthe the electromagnetic electromagnetic phenomena phenomena in inthe the wobble wobble motor motor can can be calculated as be calculated as ∂A ∇2A = µσ − v × (∇ × A) (5) 2 ∂∂tA ∇=AvAμσ  −×∇×() (5) ∂t

μπ=×−72 where 0 410/NA is the permeability of the free space, v is the speed of the mov- − ing rotor (m/s), and σ is electrical conductivity ( ()Ω⋅m 1 ). B is expressed from the Fou- rier series as

∞  = jt()ωβ− x B  Bem j (6) n=1,3,5

= π β β = π β ω where BBndm 4/ sin(/2) and (/)np. and are the wave number and angular velocity, respectively. The y-direction flux density in the air gap can be calculated using the following equation: ∂∂AA =∇×yy − = + BA=ijBiBjxj ∂∂yx ∞ ∂A λλωβ− ==II ()λλ12yy + jt() x BCeDeexII,12 II II (7) ∂y n=1,3,5, ∞ ∂A λλωβ− =−II =βλ()12yy + λ j()tx BjCeDeeyII,12 II II ∂y n=1,3,5,   where i and j indicate x-and y-axis unit vectors, respectively, and II represents the layer of the air gap. λγγα=±22 + αβμσβγμσ 22 = + = where 1,2( 4/2,) jsv 0s , 0 vx , and s and vs indicate slip and a synchronous speed, respectively. The boundary conditions are as follows: BBBeBBBBBBA,;,and==jt()ωβ− x μμ=;,0 = IIyygy==0 II,,,,, y m II II g III g II x r III x III =∞ III (8)

The coefficients in Equation (8) form boundary conditions that can be solved, and the coefficients is as follows: Appl. Sci. 2021, 11, 881 8 of 14

−7 2 where µ0 = 4π × 10 N/A is the permeability of the free space, v is the speed of the − moving rotor (m/s), and σ is electrical conductivity ((Ω · m) 1). B is expressed from the Fourier series as ∞ _ j(ωt−βx) B = ∑ Bme j (6) n=1,3,5···

where Bm = 4B/nπ sin(βd/2) and β = (nπ/p). β and ω are the wave number and angular velocity, respectively. The y-direction ﬂux density in the air gap can be calculated using the following equation:

_ _ _ _ ∂Ay ∂Ay B = ∇ × A = ∂y i − ∂x j = Bx i + Bj j ∞ ∂A = II = λ1y + λ2y j(ωt−βx) Bx,II ∂y ∑ λ1CII e λ2DII e e (7) n=1,3,5,··· ∞ ∂AII λ1y λ2y j(ωt−βx) By,II = − ∂y = ∑ jβ λ1CII e + λ2DII e e n=1,3,5,···

_ _ where i and j indicate x-and y-axis unit vectors, respectively, and II represents the layer of the air gap. p 2 2 2 2 where λ1,2 = γ ± γ + 4α /2 , α = β + jµ0σsvsβ , γ = µ0σvx , and s and vs indicate slip and a synchronous speed, respectively. The boundary conditions are as follows: Appl. Sci. 2021, 11, x FOR PEER REVIEW 9 of 15

j(ωt−βx) BII |y = 0 , BII,y = Bme ; BII |y =g , BII,g = BIII,g and µBII,x = µr BIII,x; BIII |y = ∞, AIII = 0 (8)

The coefﬁcients in Equation (8) form boundaryλ conditionsg that can be solved, and the Be()βμ+ λ μ 2 = m 02 coefﬁcients is as follows: CII λλ βλλ()−+12ggλ −Δg jeBm(βµ120+ eλ2µ)e 2 CII = λ g λ g j −e 1 +e 2 λ g − (9) β( βμ+λ1 λ μ 1λ2 ∆) Bem ()01 λ g (9) D = Bm(βµ +λ µ)e 1 II λλgg0 1 DII =jeβλ()12λ g −+Δ eλ λg jβ(e 112λ1−e 2 λ2+∆)

λλggg g whereΔ=∆ ()/=ee12 (e −λ1 −βμμeλ2 )βµ /µ. where 0 . 0 Finally, in order to calculate the Equation (7) on the basis of the change in the air Finally, in order to calculate the Equation (7) on the basis of the change in the air gap gap [24], the air gap of the wobble motor can be simpliﬁed by representing it as a rectangular [24], the air gap of the wobble motor can be simplified by representing it as a rectangular coordinate system as shown in Figure8. coordinate system as shown in Figure 8.

FigureFigure 8. Simplified 8. Simpliﬁed geometry. geometry.

TheThe air air gap, gap, g, g,between between the the rotor rotor and and stator stator as a as function a function of θ of θis isexpressed expressed and and

simplified.simpliﬁed. R sR sandand RRrr are are thethe radius of stator stator and and roto rotor,r, respectively. respectively. In In addition, addition, the the distance of the air gap, g, between these points on the stator and the rotor was then calculated.

gmmnn=−+−()()22 sr sr (10) −±Δδδ ±Ψ2 θδ ×−±Ψ θδ × ±Δ =−+tan ( ) − tan ( )  22 2 2  secθθ sec sec θ sec θ 

Δ=+−Ψ=+−δθδ2222 δθδ 2222 where sec (RRsr ), sec ( ) . The magnetic flux density can be calculated by substituting the valid gap obtained from Equation (10) into Equation (9). The final magnetic force can be determined using Maxwell’s stress tensor, which can be written as

2 2 ⋅ ( BB− ) ()BBxy yx F ==dA, F dA (11) xyAAμμ 002

4. Validation of Theoretical Force Model 4.1. Modeling and Structure of Motor with EPM To validate the proposed two force models, we developed a prototype based on a thin wobble motor with EPM 30 mm in diameter and 7 mm in thickness, the schematic of which is as shown in Figure 9. It was composed of a stator with an electromagnet core and coil, rotating shaft, and an EPM device. First, we decided to consider the shape of the core, as this was important for increasing the generated force, and then the core was simply α β optimized. The parameters of the core were TT12,,, , as shown in Figure 10a. To increase Appl. Sci. 2021, 11, 881 9 of 14

distance of the air gap, g, between these points on the stator and the rotor was then calculated. q 2 2 g = (ms − mr) + (ns − nr)

r (10) 2 = −δ±∆ − δ±Ψ + tan θ×(−δ±Ψ) − tan θ×(δ±∆) sec2 θ sec2 θ sec2 θ sec2 θ

p 2 2 2 2 p 2 2 2 2 where ∆ = δ + sec θ(Rs − δ ), Ψ = δ + sec θ(Rr − δ ). The magnetic ﬂux density can be calculated by substituting the valid gap obtained from Equation (10) into Equation (9). The ﬁnal magnetic force can be determined using Maxwell’s stress tensor, which can be written as

2 2 I B · B I By − |Bx| = x y = Appl. Sci. 2021, 11, x FOR PEER REVIEW Fx dA , Fy dA 10(11) of 15 A µ0 A 2µ0 4. Validation of Theoretical Force Model 4.1.the Modeling magnetic and force Structure and to ofdecrease Motor with the EPMleak of magnetic flux, we also carried out electro- magneticTo validate field analysis, the proposed and the two original force models, material we of developedthe core was a prototypepure iron. basedIn this on FEM a thinanalysis, wobble the motor magnetic with flux EPM density 30 mm was in obtained diameter by and exciting 7 mm two in thickness, cores when the the schematic rotor was ofmomentarily which is as shownin contact in Figure with the9. It end was of composed the stator. ofThe a stator optimal with shape an electromagnet for the core in coreorder andto increase coil, rotating attractive shaft, force and anand EPM the magnetic device. First, flux wedensity decided distribution to consider of the shapecore were of thedetermined core, as this through was important FEM, as shown for increasing in Figure the 10b. generated force, and then the core was simplySecond, optimized. we propose The parameters a wobble ofwith the EPM core device were T so1, thatT2, α the, β ,rotating as shown shaft in Figurecan be 10fixeda. Towithout increase the the continuous magnetic use force of anda power to decrease supply. theTher leakefore, of magneticit is to be used flux, weto solve also carriedthe diffi- outculties electromagnetic presented by field existing analysis, wobble and motors, the original and the material schematic of theis represented core was pure in Figure iron. In11. this3D FEMFEM analysis,simulation the was magnetic conducted flux densityto estimate was the obtained flow of by the exciting magnetic two flux cores of when EPM, the but rotorthe results was momentarily of FEM was in expressed contact with in the2D endas shown of the stator.in Figure The 11 optimal to confirm shape the for change the core of inmagnetic order to flux. increase The attractivediameter forceand length and the of magneticthe two magnets flux density used distributionin FEM were of 2 the mm core and were8 mm, determined respectively. through FEM, as shown in Figure 10b.

FigureFigure 9. 9.Schematic Schematic of of the the thin thin wobble wobble motor motor with with an an Electropermanent Electropermanent magnet. magnet.

Second, we propose a wobble with EPM device so that the rotating shaft can be fixed without the continuous use of a power supply. Therefore, it is to be used to solve the difficulties presented by existing wobble motors, and the schematic is represented in Figure 11. 3D FEM simulation was conducted to estimate the flow of the magnetic flux of EPM, but the results of FEM was expressed in 2D as shown in Figure 11 to confirm the change of magnetic flux. The diameter and length of the two magnets used in FEM were 2 mm and 8 mm, respectively.

(a) (b) Figure 10. Shape of the core: (a) Parameters for optimization of the core; (b) optimized core and the magnetic flux density distribution.

When the magnetization directions of the two magnets are opposite, the magnetic flux flows only toward the inside as shown Figure 11a. It means that the rotating shaft in unaffected. When the current for changing the magnetization direction of the Alnico 5 magnet is applied, the magnetization direction is changed. However, when a current is applied to the coil wound, the magnetization direction of NdFeb does not change. There- fore, the magnetization direction of the two magnets is the same, and an external magnetic flux is generated as shown in Figure 11b. Appl. Sci. 2021, 11, x FOR PEER REVIEW 10 of 15

the magnetic force and to decrease the leak of magnetic flux, we also carried out electromagnetic field analysis, and the original material of the core was pure iron. In this FEM analysis, the magnetic flux density was obtained by exciting two cores when the rotor was momentarily in contact with the end of the stator. The optimal shape for the core in order to increase attractive force and the magnetic flux density distribution of the core were determined through FEM, as shown in Figure 10b. Second, we propose a wobble with EPM device so that the rotating shaft can be fixed without the continuous use of a power supply. Therefore, it is to be used to solve the difficulties presented by existing wobble motors, and the schematic is represented in Figure 11. 3D FEM simulation was conducted to estimate the flow of the magnetic flux of EPM, but the results of FEM was expressed in 2D as shown in Figure 11 to confirm the change of magnetic flux. The diameter and length of the two magnets used in FEM were 2 mm and 8 mm, respectively.

Appl. Sci. 2021, 11, 881 10 of 14 Figure 9. Schematic of the thin wobble motor with an Electropermanent magnet.

(a) (b) Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 15 Figure 10. ShapeFigure of the 10. core: Shape (a) Parametersof the core: (a for) Parameters optimization for optimization of the core; of (b the) optimized core; (b) optimized core and core the and magnetic ﬂux density the magnetic flux density distribution. distribution. When the magnetization directions of the two magnets are opposite, the magnetic flux flows only toward the inside as shown Figure 11a. It means that the rotating shaft in unaffected. When the current for changing the magnetization direction of the Alnico 5 magnet is applied, the magnetization direction is changed. However, when a current is applied to the coil wound, the magnetization direction of NdFeb does not change. There- fore, the magnetization direction of the two magnets is the same, and an external magnetic flux is generated as shown in Figure 11b.

Appl. Sci. 2021, 11, x FOR PEER REVIEW 11 of 15

(a) (b)

Figure 11.Figure Basic principle 11. Basic driving principle a proposed driving electropermanent a proposed electropermanent manent magnet: manent(a) Flux magnet:path (a) Flux path when when the magnetization direction of magnets is in the opposite direction; (b) Flux path through air gap betweenthe iron magnetization and shaft. direction of magnets is in the opposite direction; (b) Flux path through air gap between iron and shaft. 4.2. Experimental System and Driving Principle The drivingWhen method the of magnetization the motor and directionsthe experimental of the system two magnets of driving are motor opposite, are the magnetic as shownflux in Figure flows 12. only Current toward is applied the inside to two as electromagnets shown Figure simultaneously, 11a. It means and that the the rotating shaft rotationalin speed unaffected. of the wobble When motor the current is contro forlled changing by the frequency the magnetization of the controller. direction The of the Alnico 5 magnet is applied, the magnetization direction is changed. However, when a current U, V, W, and Z at four-phase from the signal controller matches with coil A, coil B, coil C, and coil Dis wrapped applied around to the the coil(a) four-pole wound, core. the The magnetization coil is operated direction as follows:(b) of (coil NdFeb A and does not change. coil B), (coilTherefore, B and coil the C), magnetization (coil C and coil D), direction and (coil of D the and two coil magnetsA). is the same, and an external Figure 11. Basic principle driving a proposed electropermanent manent magnet: (a) Flux path magneticwhen the magnetization flux is generated direction as of shown magnets in is Figurein the opposite 11b. direction; (b) Flux path through air gap between iron and shaft. 4.2. Experimental System and Driving Principle 4.2. ExperimentalThe driving System method and ofDriving the motor Principle and the experimental system of driving motor are as shownThe driving in Figure method 12. Currentof the motor is applied and the to experimental two electromagnets system of driving simultaneously, motor are and the rotationalas shown in speed Figure of 12. the Current wobble is motorapplied is to controlled two electromagnets by the frequency simultaneously, of the and controller. the The U,rotational V, W, and speed Z atof four-phasethe wobble motor from theis contro signallled controller by the frequency matches of withthe controller. coil A, coil The B, coil C, andU, V, coil W, Dand wrapped Z at four-phase around from the four-polethe signal controller core. The matches coil is operated with coil A, as coil follows: B, coil (coil C, A and coiland B),coil (coil D wrapped B and coilaround C), the (coil four-pole C and coilcore. D), The and coil (coil is operated D and as coil follows: A). (coil A and coil B), (coil B and coil C), (coil C and coil D), and (coil D and coil A). .

Figure 12. The driving method of the motor and total system of driving motor.

4.3. Experiment for Theoretical Model Validation To experimentally validate the proposed theoretical model, a prototype of a thin wobble motor with an EPM device was fabricated. Pictures of the prototype are shown in Figure 13. It has the same dimensions as the one used for FEM. The magnets used for the EPM were 2 mm in diameter, 8 mm in length, and around them was wrapped a coil with a diameter of 0.18mm for 120 turns, as shown in Figure 13b. Four electromagnetic arms of core were arranged around the wobble shaft and wrapped with a coil as shown in Figure . 13c. In the experiment, a core with a thickness of 1 mm, around which a coil with a diameter of 0.16 mm was wound for 150 turns, was used to create an attractive rotating shaft. FigureFigure 12. TheThe driving driving method method of the of themotor motor and andtotal totalsystem system of driving of driving motor. motor. To maximize the force and minimize the iron losses, the material was changed from pure iron to electrical4.3. Experiment steel. The for materialTheoretical selected Model Validationfor the electrical steel was 50PN350, charac- terized by Bsat = 1.62 T and an iron loss of 3.50 Watts per kilogram from Pohang Iron and To experimentally validate the proposed theoretical model, a prototype of a thin Steel Company (POSCO), and the core shape was then manufacturing by Electro Dis- wobble motor with an EPM device was fabricated. Pictures of the prototype are shown in charge Machining (EDM). Additionally, pure iron (SS400) was used as the material for the Figure 13. It has the same dimensions as the one used for FEM. The magnets used for the rotating shaft. This had the highest electrical conductivity, high permeability, low coercive EPM were 2 mm in diameter, 8 mm in length, and around them was wrapped a coil with a intensity, and was also easy to purchase. We fabricated a prototype of the EPM device diameter of 0.18mm for 120 turns, as shown in Figure 13b. Four electromagnetic arms of core were arranged around the wobble shaft and wrapped with a coil as shown in Figure 13c. In the experiment, a core with a thickness of 1 mm, around which a coil with a diameter of 0.16 mm was wound for 150 turns, was used to create an attractive rotating shaft. To maximize the force and minimize the iron losses, the material was changed from pure iron to electrical steel. The material selected for the electrical steel was 50PN350, charac- terized by Bsat = 1.62 T and an iron loss of 3.50 Watts per kilogram from Pohang Iron and Steel Company (POSCO), and the core shape was then manufacturing by Electro Dis- charge Machining (EDM). Additionally, pure iron (SS400) was used as the material for the rotating shaft. This had the highest electrical conductivity, high permeability, low coercive intensity, and was also easy to purchase. We fabricated a prototype of the EPM device Appl. Sci. 2021, 11, 881 11 of 14

Appl. Sci. 2021, 11, x FOR PEER REVIEWDischarge Machining (EDM). Additionally, pure iron (SS400) was used as the12 material of 15 for the rotating shaft. This had the highest electrical conductivity, high permeability, low coercive intensity, and was also easy to purchase. We fabricated a prototype of the EPM device using a Computer Numerical Control (CNC) machine. From the FEM results, it using a Computer Numerical Control (CNC) machine. From the FEM results, it was con- was conﬁrmed that a change occurs when the magnetization direction of the Alnico 5 firmed that a change occurs when the magnetization direction of the Alnico 5 momentarily momentarily larger than 5A. To change the magnetization direction of the Alnico 5 magnet larger than 5A. To change the magnetization direction of the Alnico 5 magnet in EPM in EPM device, we used the controller of an NI cRio-9022 from NATIONAL INSTRUMENTS device, we used the controller of an NI cRio-9022 from NATIONAL INSTRUMENTS and and a high-performance capacitor-type driver capable of momentarily applying a current a high-performance capacitor-type driver capable of momentarily applying a current of of larger than 5A. The signal controller was a TMS320F28335, and the driver was expressly larger than 5A. The signal controller was a TMS320F28335, and the driver was expressly mademade for use with four-pole four-pole cores. cores.

FigureFigure 13. Pictures of of a a thin thin wobble wobble motor motor with with EPM. EPM. (a) (Partsa) Parts a thin a thin wobble wobble motor; motor; (b) Fabricated (b) Fabricated EPM device; (c) Core with the coil wound. EPM device; (c) Core with the coil wound.

4.4.4.4. Validation Results Results Figure 14 showsshows the the predictions predictions of of the the magnetic magnetic circuit circuit model model in three in three cases, cases, as well as well asas thethe experimental results results with with respect respect to to va variationriation in inthe the air air gap gap of the of the EPM EPM device. device. The The blackblack curve shows the the neglecting neglecting leakage leakage permeance; permeance; the the red red curve curve shows shows the the2D leakage 2D leakage permeance in consideration of 2× P and 2× P , as shown Figure 6; the blue curve shows permeance in consideration of 2 × 2P2 and 2 ×6 P6, as shown Figure6; the blue curve shows thethe 3D3D leakageleakage permeance; permeance; and and the the green green curv curvee shows shows the the 3D 3D FEM FEM results. results. When When the theair air gapgap waswas setset at 0, 0.1 mm and 0.2mm, the the meas measuredured forces forces were were 5.853 5.853 N, N, 1.762 1.762 N N and and 0.926 0.926 N, respectively.N, respectively. When When the the air air gap gap was was set set at at 0.1 0.1 mm, mm, the the results results without without considering considering magneticmag- fluxnetic and flux the and FEM the results FEM results showed showed considerable considerable error. However,error. However, the error the between error between the results inthe which results the in magnetic which the leakage magnetic flux leakage was 3D flux and was the 3D FEM and results the FEM was results the 20 percent.was the 20 Based onpercent. the plot, Based it can on bethe seen plot, that it can pole-to-pole be seen that leakage pole-to-pole did not changeleakage thedidholding not change force the when theholding air gap force was, when but thethe slopeair gap of thewas, force but vs.the theslope distance of the curve force changedvs. the distance significantly curve with changed significantly with the zero air gap. This is an important effect to consider when designing EPM devices, even those which operate over relatively small air gaps. Appl. Sci. 2021, 11, 881 12 of 14

Appl. Sci. 2021, 11, x FOR PEER REVIEW 13 of 15 the zero air gap. This is an important effect to consider when designing EPM devices, even those which operate over relatively small air gaps.

Figure 14. Experiment results, and comparisons of the force vs. air, calculated using the mantic Figure 14. Experiment results, and comparisons of the force vs. air, calculated using the mantic circuit with and without leakage flux. circuit with and without leakage flux. Table2 summarizes the results of FEM, the experimental results, and the theoretical modelTable results 2 summarizes according the to theresults applied of FEM, current. the Anexperimental experimental results, test wasand conductedthe theoretical using modelthe method results according shown in to Figure the applied 12. The current theoretically. An experimental calculated test stall was torque conducted and the using stall thetorque method generated shown fromin Figure FEM comprise12. The theoretically the moment calculated when the rotatingstall torque shaft and moves. the Itstall was torqueconfirmed generated thatthe from error FEM between comprise the the theoretical moment method when the and rotating the FEM shaft analysis moves. results It was was confirmed26 percent that on the average, error between and between the theoreti the experimentcal method and the FEM FEM analysis analysis results results was was 2612 percent percent on average. average, As and the between input current the expe to theriment electromagnet and the FEM core analysis was increased, results was the stall12 percenttorque average. of the wobble As the motor input also current increased. to the electromagnet core was increased, the stall torque of the wobble motor also increased. Table 2. Comparison of stall torque according to current variation. Table 2. Comparison of stall torque according to current variation. By Theoretical By Experiment Current (A) By FEM (mNm) Current (A) By TheoreticalMethod Method (mNm) (mNm) By FEM (mNm) By Experiment(mNm) (mNm) 0.1 0.10.166 0.166 0.134 0.134 0.118 0.118 0.2 0.20.672 0.672 0.523 0.523 0.456 0.456 0.3 0.31.488 1.488 1.173 1.173 1.034 1.034 0.4 0.42.64 2.64 2.088 2.088 1.843 1.843 0.5 4.128 3.264 2.857 0.5 4.128 3.264 2.857 5. Conclusions 5. Conclusions This paper proposed the two force models in order to theoretically calculate the This paper proposed the two force models in order to theoretically calculate the gen- generated force in a thin wobble motor with the EPM device. First, the EPM can change the erated force in a thin wobble motor with the EPM device. First, the EPM can change the holding force on and off by applying a momentary electrical pulse, and the structure and holding force on and off by applying a momentary electrical pulse, and the structure and driving principle are presented. It is difficult to establish an analysis model that accurately driving principle are presented. It is difficult to establish an analysis model that accurately presents the magnetic leakage fluxes, but by assuming their paths to consists of straight lines presentsand arcs, the the magnetic mean length leakage of the fluxes, flux pathbut by could assuming be obtained. their paths Therefore, to consists in order of to straight increase linesthe and accuracy arcs, the of the mean theoretical length of force the modelflux pa ofth thecould EPM be device,obtained. a method Therefore, considering in order to the increaseleakage the fluxes accuracy was proposed.of the theoretical Comparing force the model FEM of results the EPM and device, the force a method calculated consid- using eringthe proposedthe leakage method, fluxes was it was proposed. confirmed Comp thataring there the was FEM a reductionresults and in the the force error calcu- of the latedforce using generated the proposed when considering method, it was leakage confirmed fluxes that were there considered. was a reduction However, in therethe error were ofdifferences the force generated between thewhen theoretical considering and theleakage experimental fluxes were results, considered. because theHowever, loss of frictionthere wereand differences assembly error between of the the magnets theoretical and theand iron the influencedexperimental the results, experimental because measurements. the loss of frictionDespite and the assembly assembly error error, of the it successfully magnets and generated the iron influenced holding force the experimental at the rotating meas- shaft. urements. Despite the assembly error, it successfully generated holding force at the rotating shaft. These results can not only be used in selecting the initial design parameters Appl. Sci. 2021, 11, 881 13 of 14

These results can not only be used in selecting the initial design parameters necessary for FEM analysis for the design of EPM devices, but also indicate that the proposed magnetic circuit analysis results can be used as the basis for its design. Secondly, we simply presented the optimal design of the electromagnets with respect to various parameters, as well as a partial differential equation on the basis of the distribution parameters as a function of the variation of the air gap between the stator and the rotating shaft. The theoretical force model for the generated force of the electromagnetic ﬁeld using the distribution parameters was calculated by approximating the model using a linear coordinate system. We conducted experiments using a prototype motor with respect to the changing current. As shown in Table2, it was conﬁrmed that the theoretical method results and the FEM results were similar. Differences were observed in the experimental results owing to the loss of friction and error occurring when manufacturing the rotating shaft. However, our study shows advantages in that this method is less time consuming and systematic, because a large number of iterative analyses are not required during the initial design step. As a result, the proposed a thin motor with EPM device has great potential for commercialization as a driving part of small robots, in the medical industry, and especially in thin portable equipment in the future.

Author Contributions: FEM analysis, designed and manufactured of the motor, controller, and wrote the paper, S.Y.P.; edited and revision, analyzed the date, B.S.; investigation and validation S.Y.P. and B.S.; supervision, Y.S.B. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: This paper is no data. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Abbreviation

EPM Electropermanent Magnet

References 1. Viviani, A. Experimental and Theoretical Study of Hypocycloidal Motors with Two-Harmonic Field Windings. IEEE Trans. Power Appar. Syst. 1980, 1, 292–300. 2. Hayashi, I.; Iwatsuki, N.; Shibata, J.; Matsunaga, H.; Wada, S. An Electromagnetic Cycloid Motor. J. Jpn. Soc. Precis. Eng. 1995, 61, 95–99. [CrossRef] 3. Nagata, T.; Suzumori, K.; Kanda, T.; Uzuka, K.; Enomoto, I. Electro Direct-Drive Stepping Motor for Robots. In Proceedings of the IEEE International Conference on Robotics and Automation, New Orleans, LA, USA, 26 April–1 May 2004; pp. 4493–4498. 4. Kimura, H.; Hirose, S.; Nakaya, K. Development of the Crown Motor. In Proceedings of the 2001 ICRA, IEEE International Conference on Robotics and Automation, Seoul, Korea, 21–26 May 2001; pp. 2442–2447. 5. Okamoto, K.; Suzumori, K.; Kanda, T.; Yamada, Y. Development of Three-Chamber Micro Pneumatic Wobble Motor. Proc. Mach. Des. Tribol. Div. 2006, 295–297. [CrossRef] 6. De Cristofaro, S.; Stefanini, C.; Ng Pak, N.; Susilo, E.; Carrozza, M.C.; Dario, P. Electromagnetic wobble micromotor for microrobots actuation. Sens. Actuators A Phys. 2010, 161, 234–244. [CrossRef] 7. Jacobsen, S.C.; Price, R.H.; Wood, J.E.; Rytting, T.H.; Rafaelof, M. The Wobble Motor: Design, Fabrication and Testing of an Eccentric-Motion Electrostatic Microactuator. In Proceedings of the IEEE International Conference on Robotics and Automation, Scottsdale, AZ, USA, 14–19 May 1989; pp. 1536–1546. 8. Suzumori, K.; Hori, K. Micro Electrostatic Wobble Motor with Toothed Electrodes. In Proceedings of the IEEE Tenth Annual International Workshop, Nagoya, Japan, 26–30 January 1997; pp. 227–232. 9. Miyake, M.; Suzumori, K.; Uzuka, K. Development of a Thin Electromagnetic Wobble Motor. In Proceedings of the 2011 IEEE International Conference on Robotics and Biomimetics, Karon Beach, Phuket, Thailand, 7–11 December 2011; pp. 2523–2528. 10. Jacobsen, S.C.; Price, R.H.; Wood, J.E.; Rytting, T.H.; Rafaelof, M. The Wobble Motor: An Electrostatic, Planetary-Armature, Microactuator. In Proceedings of the IEEE Micro Electro Mechanical Systems, An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots, Salt Lake City, UT, USA, 20–22 February 1989. Appl. Sci. 2021, 11, 881 14 of 14

11. Jungreis, A.M.; Kelley, A.W. The axial air gap wobble motor—An appropriate topology for magnetic micromotors. In Proceedings of the Conference Record of the 1995 IEEE Industry Applications Conference Thirtieth IAS Annual Meeting, Orlando, FL, USA, 8–12 October 1995. 12. Mehergancy, M.; Nagarkar, P.; Senturia, S.D.; Lang, J.H. Operation of Microfabricated Harmonic and Ordinary Side-Drive Motors. In Proceedings of the IEEE Proceedings on Micro Electro Mechanical Systems, An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots, Napa Valley, CA, USA, 11–14 February 1990. 13. Dhuler, V.R.; Mehregany, M.; Phillips, S.M. An experimental technique and a model for studying the operation of harmonic side-drive micromotors. IEEE Trans. Electron Devices 1993, 40, 1977–1984. [CrossRef] 14. Deng, K.; Meheregany, M. Outer-Rotor Polysilicon Wobble Micromotors. In Proceedings of the IEEE Workshop on Micro Electromechanical Systems, Oiso, Japan, 25–28 January 1994. 15. Furuhata, T.; Hirano, T.; Lane, L.H.; Fontana, R.E.; Fan, L.S.; Fujita, H. Outer Rotor Surface-Micromachined Wobble Micromotor. In Proceedings of the IEEE Micro Electro Mechanical Systems, Fort Lauderdale, FL, USA, 10 February 1993. 16. Ewing, J.A.; Klaassen, H.G. Magnetic qualities of Iron. Philos. Trans. R. Soc. A 1893, 184, 985–1039. 17. Campbell, P.; Al-Murshid, S. A model of anisotropic Alnico magnets for ﬁeld Computation Magnetics. IEEE Trans. Magn. 1982, 18, 898–904. [CrossRef] 18. Hanselman, D.C. Brushless Permanent Magnet Motor Design; McGraw-Hill: Cranston, RI, USA, 1994. 19. Roters, H.C. Electromagnetic Devices, 1st ed.; John Wiley & Sons: New York, NY, USA, 1941. 20. Yamamura, S. Theory of Linear Induction Motors, 2nd ed.; Wiley: New York, NY, USA, 1979. 21. Wangsness, R.K. Electromagnetic Fields, 2nd ed.; Wiley: New York, NY, USA, 1986. 22. Grifﬁths, D.J. Introduction to Electrodynamics, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 1989. 23. Nasar, S.A. Linear Motion Electric Machines; John Wiley & Sons: New York, NY, USA, 1976. 24. Trimmer, W.; Jebens, R. An Operational Harmonic Electrostatic Motor. In Proceedings of the IEEE Micro Electro Mechanical Systems, An Investigation of Micro Structures, Sensors, Actuators, Machines and Robots, Salt Lake City, UT, USA, 20–22 February 1989; pp. 13–16.