Torque Density Improvement of a Low-Speed High-Torque Swirling Actuator Driven by Electromagnetic Radial Force With Integrated Mechanical Gears

The torque density improvement of a low-speed high-torque swirling actuator is presented in this article. The swirling actuator is driven by the electromagnetic radial force and integrated mechanical gears. The electromagnetic radial force generates the circular motion of the swirler, which is converted into the low-speed rotor rotation by the mechanical gears. First, the dimensions of the swirler with 12-pole permanent magnets are optimized to enhance the electromagnetic radial force by analytical calculation and three-dimensional finite-element analysis. Second, an improved gear set with a reduced pressure angle and an increased transmission ratio is designed. The gear efficiency and torque are investigated analytically considering the friction loss. Two prototypes are designed, and the experimental results exhibit that the peak torque density is improved from 27 to 64 Nm/L with a small volume of 0.16 L.


I. INTRODUCTION
Low-speed high-torque electric machines are widely utilized in applications, such as wind turbines [1], traction motors [2], and industrial robots [3]. There are two main categories of low-speed high-torque electric machines. The first is the geared machine, which combines the high-speed electric machine and speed reducer. The second is the direct-drive electric machine.
Geared machines can achieve high-torque densities with high-transmission-ratio speed reducers. Typical mechanical speed reducers include spur gears, planetary gears, cycloidal drives, and strain wave gears. In [4], a bilateral-drive planetary gear with a high transmission ratio of up to 96.2 was proposed for robotic actuators. The two-stage speed reducer that combines the planetary gear and cycloidal drive can reach a transmission ratio of up to 204 [5]. The strain wave gear has a high single-stage transmission ratio with a compact size. For example, the actuators FHA-mini equipped with strain wave gears have peak torque densities of 36-63 Nm/L when the transmission ratio is 50, whereas the peak torque densities increase to 52-99 Nm/L when the transmission ratio is 100 [6].
Compared with geared machines, direct-drive electric machines need less regular maintenance. Moreover, the nobacklash characteristic makes them suitable for precise position control. For conventional permanent magnet (PM) machines, a large pole number is necessary to realize low speed and high torque. The 22-pole direct-drive PM motors in [7] and [8] have the peak torque densities of 7 and 17 Nm/L with the identical volume of 0.13 L, whereas the 40-pole direct-drive PM motor in [9] has a peak torque density of 21 Nm/L with a volume of 0.15 L. There have been some notable direct-drive machines aiming to improve the torque density. In [10], a claw-pole-stator PM motor was proposed. The stator has 3 claw-pole layers and the rotor has 20 poles of PMs. A peak torque density of 37.7 Nm/L was realized with a volume of 0.2 L in three-dimensional (3-D) finite-element analysis (FEA). In [11], an outer-rotor switched flux memory machine was presented. The hybrid stator has low-coerciveforce magnets and rare-earth PMs, both of which are in the spoke-type configuration. The optimized model reached a continuous torque density of 31.6 Nm/L with a volume of 0.16 L in 3-D FEA. In [12], a double-stator spoke-array PM vernier machine realized a continuous torque density of 66 Nm/L with a large volume of 30 L. In [13], a combined radial-axial flux PM vernier machine was developed. A set of toroidal windings were configured on the stator and the end-windings were utilized to couple with the axial rotors. With the enlarged rotor area that contributes to torque generation, the torque density was enhanced to 58 Nm/L in a 34-pole model with a volume of 5.4 L.
Magnetic gears utilize the flux modulation effect to realize noncontact speed reduction and power transmission [14]. Among magnetic gears, coaxial magnetic gears normally have low transmission ratios, whereas cycloidal magnetic gears can have higher transmission ratios [15]. Many magnetic-geared machines integrating the coaxial magnetic gears have been reported. In [16], the coaxial magnetic gear was connected to a PM machine in the axial direction. In [17], an outer-rotor PM motor was inserted inside the high-speed rotor of a coaxial magnetic gear to build a magnetic-geared motor. The torque density was 40 Nm/L with a volume of 3.5 L. In [18], a stator with nonoverlapping windings was combined with the outer rotor of the coaxial magnetic gear. The resultant machine is known as the pseudo direct drive. A continuous torque density of over 60 Nm/L with a volume of 1.9 L was reported. The magnetic-geared machine in [19] has an inner stator, a modulation ring, and an outer rotor. Both the modulation ring and outer rotor have consequent-pole PMs. Thus, the machine was named as the dual-layer magnetic-geared PM machine. The peak torque density reached 89 Nm/L with a volume of 5.7 L in FEA. In addition, Halbach-layer PMs were adopted in the dual-layer magnetic-geared PM machine in [20]. The authors realized the torque density of 77.3 Nm/L with a volume of 4.9 L at the current density of 8 A/mm 2 and stated that the torque density could exceed 150 Nm/L with a higher current density. In [21], an axial-flux focusing magnetic gear was integrated with an outer stator for an ocean generator application. The machine volume was 5 L and the calculated and measured peak torque densities were 114 and 94.4 Nm/L, respectively.
The proposed swirling actuator is driven by the electromagnetic radial force, which leads to the circular motion of the swirler [22]. The identical motion can be found in several special cycloidal motors. In [23], the cyclic expansion and contraction of the shape-memory-alloy wires generate the circular motion. In [24], the external rotating magnetic fields generate the circular motion. Involute-type mechanical gears were integrated in [22], [23], and [24] to convert the circular motion into low-speed rotation. The gear mechanism is identical to that of cycloidal drives. A high transmission ratio can be achieved by a small gear tooth number difference. The principle of the proposed swirling actuator was verified in our previous article [22]; however, we could only achieve a peak torque density of 27 Nm/L with a small volume of 0.16 L. In contrast, the direct-drive PM machines with similar volumes of 0.1-0.2 L have the peak torque densities up to 37.7 Nm/L [7], [8], [9], [10], [11]. In this article, the peak torque density of the swirling actuator is improved to 64 Nm/L with the identical volume of 0.16 L. The target applications of the swirling actuator are robotic joints and wearable rehabilitation devices, which operate at very low speeds and require high-torque densities with compact sizes. Conventional geared machines have electric motors and mechanical gears connected in the axial direction, whereas the swirling actuator integrates the electromagnetic part and mechanical gears in the radial direction. The flat structure of the swirling actuator is attractive for applications, such as the wearable exoskeleton [25].
Section II introduces the basic structure and operation principle of the swirling actuator. In Section III, the electromagnetic radial force is enhanced by the optimization of swirler dimensions. Section IV investigates the design of mechanical gears. Experimental results of two prototypes are presented in Section V to evaluate the torque density improvement. Section VI provides the conclusions. Fig. 1 shows the basic structure of the swirling actuator. Three-phase double-layer nonoverlapping windings are wound around the teeth of the inner stator. The swirler surrounds the stator and has 12-pole PMs mounted on the inner surface. The swirler has a circular motion with a nominal radius of r 0 and an angular speed of ω s . The pins through the swirler holes prevent the rotation and allow only the radial movement of the swirler. There are Z sw gear teeth on the outer surface of the swirler and Z r gear teeth on the inner surface of the rotor. The gears convert the circular motion of the swirler into the rotor rotation at an angular speed of ω r . The transmission ratio G between ω s and ω r is given by the gear tooth numbers as

II. BASIC STRUCTURE AND OPERATION PRINCIPLE
The electromagnetic radial force is generated between the stator and swirler. Let us define p s and p sw as the pole-pair numbers of the magnetomotive forces (MMFs) generated by the stator windings and swirler PMs, respectively. Based on the principle of bearingless motors [26], to generate the active radial force, p s and p sw should satisfy the following equation: The 12-pole PMs on the swirler mainly generate the six pole-pair MMF. Thus, the stator windings in Fig. 1 are designed to generate the five and seven pole-pair MMFs.
The resultant electromagnetic radial force can be decomposed into F d and F q . The F d is in the eccentric direction of the swirler and F q is perpendicular to F d . In Fig. 1, the swirler is eccentric in the x-direction. Thus, the d-and q-axes are on the x-and y-axes, respectively. The swirler has the velocity v sw0 in the q-axis with the magnitude of ω s r 0 . Thus, the mechanical power P sw0 of the swirler is given by The mechanical power of the rotor is given by where T is the rotor output torque. The gear efficiency η g is defined as By substituting (1), (3), and (5) into (4), the rotor output torque is derived as Equation (6) indicates that the rotor output torque can be enhanced by increasing the electromagnetic radial force F q , transmission ratio G, and gear efficiency η g . The nominal swirler eccentric radius r 0 should be equal to the nominal center distance between the rotor and swirler gears. The calculation of the gear center distance is introduced in the Appendix. In our previous article [22], the nominal gear center distance was 0.5 mm. Thus, r 0 is also set as 0.5 mm in this article. Note that r 0 should be shorter than the nominal air-gap length g 0 , which is the air-gap length when the swirler is concentric with the stator.  mechanical angular position θ s in the static coordinate system. The A pm (θ s ) and A i (θ s ) are the MMFs generated by the swirler PMs and stator currents, respectively. For k = 1, 2, 3, …, the 12-pole PMs generate [(2k − 1)6]th MMF components, whereas the 12-slot double-layer nonoverlapping windings can generate [(2k − 1)6 ± 1]th MMF components to satisfy (2) for active radial force generation. In Fig. 2, the dominant sixth component of A pm (θ s ) and fifth and seventh components of A i (θ s ) are shown. The P(θ s ) is the permeance function describing the stator slot effect and θ t is the span angle of the stator teeth. The g(θ s ) is the nonuniform air-gap length between the stator and swirler. Let us define θ d as the swirler eccentric direction angle and set the initial value of θ d as 0. Then, θ d is given by

III. ENHANCEMENT OF ELECTROMAGNETIC RADIAL FORCE
Using The radial air-gap flux density B(θ s ) is given as where μ 0 is the permeability of vacuum; and μ r and t m are the relative permeability and thickness of PMs, respectively. As shown in Fig. 2, the amplitude of B(θ s ) is high where the air-gap length is short. Using Maxwell's stress tensor method [27] and considering only the radial air-gap flux density [28], the electromagnetic radial forces on the swirler in the static coordinate system are derived as where r s is the stator outer radius and l s is the stack length. Using θ d , F x and F y are transformed into the rotational dq coordinate system as The stator phase currents i u , i v , and i w are also transformed into the d-and q-axis currents as (13) Note that the d-axis in a conventional PM motor is based on the rotational angular position of the rotor PMs, whereas the d-axis in the swirling actuator is based on the swirler eccentric direction.
To simplify the analytical derivations of F d and F q , only the dominant sixth component of A pm (θ s ) and the fifth and seventh components of A i (θ s ) are considered. Thus, A pm (θ s ) is given as where B r is the residual flux density of PMs. The A i (θ s ) is given as where θ i is the initial phase shift of the phase currents. The amplitudes A I5 and A I7 are derived as (16) where N S is the turn number per stator tooth and I s is the amplitude of the phase currents. Moreover, the following approximation is conducted: From (7) to (17), F d and F q are simplified into where k d and k i are the displacement force factor and current force factor, respectively. The k d and k i are derived as The partial derivative of k i with respect to t m is derived as For a certain value of g 0 , k i reaches the maximum when (21) is zero, i.e., the PM thickness t m = μ r g 0 . Fig. 3 shows the analytical results of F q with respect to t m . The i d is zero and i q is 3.3 A. There are five solid curves corresponding to different values of g 0 from 0.6 to 1.0 mm with a step of 0.1 mm. Moreover, the red dashed curve satisfying t m = μ r g 0 is also shown, which intersects with the other five curves at the peak points. The F q increases as g 0 becomes short. Thus, the design point should be on the curve satisfying t m = μ r g 0 , and has a g 0 as short as possible. As the relative permeability of PMs μ r is close to 1, the PMs are expected to be as thin as the air-gap length. However, in the real design, PMs thinner than 1 mm are avoided due to the fabrication difficulty and possible irreversible demagnetization. In [22], the design point was at t m = 2 mm and g 0 = 1 mm. In this article, g 0 = 0.6 mm and t m = 1 mm are selected.
The optimization is verified by 3-D FEA. Fig. 4 shows the FEA and calculated results of electromagnetic radial forces F d and F q with respect to the q-axis current i q . The calculated forces are obtained by numerical integrations using the original waveforms of A pm (θ s ), A i (θ s ), and g(θ s ). The d-axis current i d is zero. The F d does not change with i q , whereas F q is proportional to i q . The calculated results agree well with the FEA results. The current force factor k i is enhanced from 27.5 to 51.2 N/A in the improved electromagnetic part. The F d when i d = 0 is also enhanced from 94 to 173 N.

IV. GEAR DESIGN INVESTIGATION
In our previous article [22], it was indicated that the gear design parameters may affect the actual eccentric radius of the swirler and gear efficiency. Moreover, (6) shows that increasing the transmission ratio is expected to enhance the output torque. In this section, the gear design parameters are investigated, and a new gear set with identical dimensions and a higher transmission ratio is designed. The gear efficiency is analyzed and the effects of gear design parameters on the output torque are illustrated. Fig. 5 shows two gear sets to explain the effect of the pressure angle. The swirler gear is eccentric and meshes with the rotor gear in the x-direction. At the meshing point, there is a pressure force F sw on the swirler perpendicular to the tooth surface. The F sw can be decomposed into the radial force F r and tangential force F t . The angle between F t and F sw is the pressure angle α. The F sw and F t can be written as the functions of F r

A. GEAR DESIGN PARAMETERS
The tangential force F t generates the rotor output torque. If the friction is ignored, the rotor output torque is given by where r r is the rotor pitch circle radius. By substituting (23) into (24), the rotor output torque is given by  The gear set in Fig. 5(a) has a large pressure angle α of 42 • , which is identical to the previous prototype in [22]. In contrast, the gear set in Fig. 5(b) has a smaller α of 20 • . From (25), with the identical F r , the gear set in Fig. 5(b) can generate a higher torque because of the smaller α. Fig. 6 shows the gear design candidates. The rotor tooth number is given by where m is the module describing the size of gear teeth. The values of m in Fig. 6 are 0.4, 0.5, 0.6, 0.8, and 1 mm, which are standard values listed in JISB1702-1. As m decreases, the gear tooth becomes smaller and the tooth numbers increase. The rotor pitch circle radius r r is about 45 mm and should lead to integer tooth numbers. The tooth number difference (Z r − Z sw ) is set as 1 to maximize the transmission ratio G according to (1). For involute-type gears, such a small tooth number difference may cause undesirable tooth interference. The method to check the interference is introduced in Chapter 2.9 of [29]. The selected gears should have no interference. Gear 1 with α = 42 • and G = 113 was used in the previous prototype in [22]. According to (6) and (25), a small α and a large G are desirable to enhance the output torque. Thus, in the prototypes in this article, Gear 2 with α = 20 • and G = 150 is adopted. Fig. 7 shows the forces on the swirler and the velocities of the swirler and rotor. The F pin is the pressure force from the pins. The friction f on the tooth surface is perpendicular to F sw . Let us define the actual eccentric radius of the swirler as r m . It will be shown in Section V that r m is shorter than the nominal eccentric radius r 0 and depends on the operation points. The β is the angle between the swirler eccentric direction and meshing angular position. From (47) in the Appendix, β is derived as

B. ANALYSIS OF GEAR EFFICIENCY
When the actuator is in the steady state, the resultant force of the swirler is the centripetal force towards the stator center. The forces on the swirler in the radial and tangential directions satisfy the following equations: where m sw is the swirler mass. The friction f is given by where μ is the coefficient of friction. The swirler velocity v sw is in the q-axis, whereas the rotor velocity v r at the meshing point is in the tangential direction. The angle between v sw and v r is β. When β is not zero, there is a relative motion between the swirler and rotor at the meshing point. The friction loss P f is calculated as Then, the mechanical power of the rotor considering the friction loss is given by By substituting (3) and (33) into (5), the gear efficiency η g is given by Finally, by (1), (6), (22), (23), and (28)-(34), the rotor output torque is derived as Fig. 8 shows the calculated torque with respect to G. The eccentric radius r m , q-axis force F q , and pressure angle α are set as 0.5 mm, 100 N, and 20 • , respectively. The coefficient of friction μ depends on the gear materials, lubrication, fabrication accuracy, and surface roughness. It is difficult to calculate or measure μ at this stage. Here, μ is assumed to be 0.3, 0.4, and 0.5. Fig. 8 indicates that the torque increases as G increases, but on the other hand, it decreases when μ increases. Gear 2 increases the transmission ratio from 113 to 150 compared with Gear 1. This time, even higher transmission ratios are not attempted because the small gear teeth may have a problem in mechanical fabrication precision. Fig. 9(a) shows the calculated torques of Gear 1 and Gear 2 with respect to r m when F q = 100 N and μ = 0.4. Fig. 9(b) shows the calculated η g and β with respect to r m of Gear 1 and Gear 2. It is observed that the gear efficiency η g increases as the eccentric radius r m increases. The reason is that the rotor mechanical power P r increases as r m increases as shown in (34). Moreover, the maximum efficiency of Gear 2 is 72%, whereas the maximum efficiency of Gear 1 is 88%. As shown in Fig. 7, a large β indicates that the relative velocity between the swirler and rotor is high. As a result, the friction loss P f is high. The gear efficiency of Gear 2 is lower than that of Gear 1 due to the larger β of Gear 2. However, Gear 2 has a higher torque than Gear 1 owing to the high transmission ratio.

V. EXPERIMENTS
Two prototypes were built to verify the torque density improvement by 1) the electromagnetic optimization and 2) the improved gear design. Table 1 lists the specifications of Prototype 1, Prototype 2, and the previous prototype in [22]. Compared with the previous prototype, Prototype 1 reduces the nominal air-gap length and PM thickness to enhance the electromagnetic radial force. Prototype 2 further adopts the new Gear 2 with a reduced pressure angle and an enlarged transmission ratio. Fig. 10(a)-(c) shows the fabricated gears  of Prototype 1, Prototype 2, and the previous prototype in [22]. Fig. 10(d) shows the complete prototype including the mechanical structure. Fig. 11 shows the front sectional view of the 3-D model of the prototypes. The pins go through the frames and swirler to prevent the rotation and restrict the eccentric radius of the swirler. The shaft is fixed with the rotor gear and supported by two mechanical bearings. As shown in Fig. 7, the unbalanced radial force on the rotor is

A. MECHANICAL DESIGN OF PROTOTYPES
(36)  If the friction f is ignored, based on (25), F ur is given by Equation (37) indicates that reducing the pressure angle α can reduce the unbalanced radial force F ur . By changing α from 42 • to 20 • , F ur is reduced by 60%. Moreover, the load force on a single bearing is given by The peak torque of prototype 2 is 10 Nm. From (37) and (38), the peak value of F b is calculated to be 40 N, which is only 2% of the radial load capacity of 1920 N of the bearings (B6801ZZ). Thus, the unbalanced radial force has little effect on the lifetime of the bearings. Fig. 12(a) shows the top view of the 3-D model of the prototypes including the six pins. Fig. 12(b) shows the ideal pin configuration when r m = r 0 . The diameters of frame holes and swirler holes are both D h , whereas the diameter of the pins is D p . The pins are tangential with the frame holes and swirler holes. The center distance between the frame holes and swirler holes is the eccentric radius of the swirler. To restrict the maximum eccentric radius to r 0 , D h and D p need to satisfy (39) In the prototypes, D h and D p are set as 2.8 and 2.3 mm, respectively. To prevent the rotation of the swirler, at least three pins are necessary. Moreover, the diameter of pins should be smaller than the width of the swirler gear. By increasing the number of pins to six, the load force on a single pin is reduced and thin pins can be used. The peak torque of prototype 2 was 10 Nm and the equivalent force that the pins need to overcome is 238 N. Considering the worst condition in which the force is applied on a single pin, the FEA result of the maximum von Mises stress on the pin is 185 MPa. In contrast, the yield strength of the pin material (stainless steel 440C) is 450 MPa. Thus, the pins have adequate stiffness.
In the actual operation, r m is shorter than r 0 as shown in Fig. 12(c). It can be observed that the centers of the frame hole, swirler hole, and pin are no longer collinear when r m < r 0 . From Fig. 7, the force component imposed on the swirler tooth and opposite to F d is given by The swirler is pushed back towards the concentric position by F dn . As a result, the eccentric radius r m is reduced. By substituting (24) and (37) into (39), F dn is derived as The F dn increases as the torque increases. Consequently, the required F d to overcome F dn and keep r m close to r 0 also increases. Fig. 13 shows the control block of the swirling actuator. A three-phase voltage source inverter drives the swirling actuator. Two displacement sensors measure the displacements x and y of the swirler in the x-and y-directions. The eccentric direction angle θ d and eccentric radius r m are given by

B. TORQUE EVALUATION
The d-and q-axis currents i d and i q are regulated by proportional-integral controllers. Fig. 14 shows the picture of the experimental platform. A load geared machine (Fuji  Hensokuki VX02-025M) with the rated speed and torque of 60 r/min and 30 Nm is connected to the swirling actuator shaft through a torque meter (Onosokki SS100). The rotor angular speed ω r is regulated by the load machine. Moreover, the same lubrication grease is applied on Gear 1 and Gear 2. Fig. 15 shows the calculated torque, measured torque, and measured eccentric radius r m with respect to the d-axis current i d . The q-axis current is 1.5 A and the rotor speed is 5 r/min. The coefficient of friction μ is assumed to be 0.4, and the FEA results of F q are used for the calculation. It is noted that r m increases as i d increases, indicating that increasing the d-axis force F d can enlarge r m . The torque calculated by assuming r m = r 0 has a large discrepancy compared with the measured torque due to the discrepancy between r m and r 0 . If the measured value of r m is used for the calculation, the discrepancy becomes small. Prototype 2 has a higher torque than that of prototype 1. Fig. 16 shows the current operation points of prototype 1 and prototype 2 at the rotor speed of 5 r/min. For each value of i q , i d that generates the maximum torque is selected. The current amplitude is limited by the peak current density of 40 A/mm 2 . It is shown that the required i d increases as i q increases. Prototype 1 suffers from the insufficient i d when i q exceeds 3.5 A. In contrast, Prototype 2 needs less i d to generate the maximum torque. This indicates that the smaller pressure angle α of Gear 2 reduces the required F d . As a result, the required i d is reduced. Fig. 17 shows the calculated torque, measured torque, and measured eccentric radius r m with respect to the q-axis current i q . The measured torque of the previous prototype in [22] is also shown for comparison. It is noted that r m decreases as the torque increases. There is an obvious decrease of r m in prototype 1 when i q exceeds 3.5 A due to the limited i d as shown in Fig. 16. For both prototype 1 and prototype 2, the calculated torques with the measured r m agree with the measured torques when i q is lower than 3 A. When i q is larger, there are discrepancies between the calculated and measured torques. The possible reason is that μ is not constant and increases with the torque. The peak measured torques of prototype 1 and prototype 2 are 7.4 and 10.0 Nm, respectively. The corresponding peak torque densities are 47 and 64 Nm/L, which are significantly improved compared with the peak torque density of 27 Nm/L of the previous prototype in [22].

C. LOSS AND THERMAL EVALUATIONS
To evaluate the power and losses of the prototypes, a digital power analyzer (Yokogawa WT1803E) is used to measure the input electric power P in and currents. The output power P r is  calculated by the measured rotor speed and torque from the torque meter. Then, the actuator efficiency is given by The losses consist of copper, iron, and mechanical losses. The iron loss is obtained by 3-D FEA, which is dominantly due to the eddy current loss in the swirler iron core that is not laminated but solid instead. Table 2 lists the powers and losses of the two prototypes at 5 and 30 r/min. The currents are i d = 1 A and i q = 1.5 A, corresponding to the rated current density of 10 A/mm 2 . The  = 1 A, i q = 1.5 A)

TABLE 3. Measured Temperatures of Windings and PMs
copper loss is dominant since the maximum speed of the prototypes is as low as 30 r/min. When the rotor speed increases, the torque decreases and the mechanical and iron losses increase. Due to the higher transmission ratio, prototype 2 has a higher current frequency at the same rotor speed. Thus, the iron loss of prototype 2 is slightly higher. At 30 r/min and 10 A/mm 2 , the actuator efficiencies of the two prototypes are 26.8% and 28.2%, respectively.
The thermal characteristics are investigated by measuring the winding and PM temperatures using thermocouples. Table 3 lists the measured temperatures at 10, 20, and 40 A/mm 2 . It is noteworthy that the prototype is air-cooled with a room temperature of 20 • C. After a 30-min operation at the rated current density of 10 A/mm 2 , the winding and PM temperatures reach the steady-state values of 70.8 • C and 45.1 • C, respectively. When the prototype is overloaded at 20 A/mm 2 for 1 min, the maximum winding and PM temperatures are 74.1 • C and 34.5 • C, respectively. Furthermore, the maximum winding and PM temperatures are 76.5 • C and 36.3 • C, respectively, after an operation of 10 s at the peak current density of 40 A/mm 2 . The temperature limit of the AIW copper wires is 220 • C, which is high enough for the aforementioned operation conditions. Meanwhile, the temperature limit of the PMs depends on the demagnetization condition. In the two prototypes, the PMs are made of N48H and have a thickness of 1 mm. Fig. 18 shows the FEA results of the irreversible demagnetization ratio on the PM at the current density of 40 A/mm 2 . The demagnetization ratio is defined as the amount of variation in the residual flux density of the PM. The irreversible demagnetization occurs around one corner of the PM when the PM temperature is 60 • C. The operating conditions shown in Table 3 are permissible since the PM temperatures do not go beyond 60 • C. However, if the actuator needs to be periodically overloaded, the overloaded operation time should be limited.

VI. CONCLUSION
This article presented the torque density improvement of the swirling actuator, which is driven by the electromagnetic radial force and high-transmission-ratio mechanical gears. By reducing the nominal air-gap length and PM thickness, the active electromagnetic radial force was enhanced by 86% compared with the previous model. An improved gear design with a higher transmission ratio of 150 and a reduced pressure angel of 20 • was also designed. Two prototypes were built and the torques, losses, and thermal characteristics were investigated. It was shown that the reduced pressure angle helped decrease the required d-axis current. The peak torque density was improved from 27 to 47 Nm/L with the enhanced electromagnetic radial force and was further improved to 64 Nm/L with the improved gear design. Fig. 19 shows the schematic for calculating the gear center distance r m . Point O 1 and O 2 are the centers of the rotor gear and swirler gear, respectively. The red and blue solid circles are the base circles of the rotor gear and swirler gear with the radii of r r and r sw , respectively. The red and blue dashed circles are the base circles of the rotor gear and swirler gear with the radii of r b1 and r b2 , respectively. The r b1 and r b2 satisfy the following equations: r b1 = r r cos α, r b2 = r sw cos α.

APPENDIX
The segment AD is the common tangent of the two base circles. Let us draw an auxiliary circle that is concentric with the rotor gear and has a radius of (r b1 − r b2 ). The segment O 2 E is the tangent of the auxiliary circle and is parallel with the segment AD. Thus, r m = r b1 − r b2 cos(α + β ) .
Moreover, the pitch circle radii are given by By substituting (44) and (46) into (45), r m is derived as