Comparison and Optimization of a Magnetic Lead Screw Applied in Wave Energy Conversions

Abstract

In order to improve the power density of the wave power generation system, a magnetic lead screw (MLS) is introduced in this paper as a speed-increasing device in a wave energy converter (WEC), which converts low-speed linear wave motion to high-speed rotational motion. Then, several types of MLSs with different topologies, which were optimized, are described. Finite element analysis (FEA) was used to investigate and evaluate their electromagnetic performances, such as the thrust force, torque, air gap magnetic density, etc. The optimized MLS prototype was manufactured, and the measurements verify the FEA results. Finally, a magnetic lead screw hybrid generator (MLSHG) for application to the WEC is proposed. This MLS significantly improved the power density under different wave conditions. When the moving speed v = 0.15 m/s, the output power of the MLSHG and outer-PM linear tubular generator were 923 W and 87 W, respectively, when the load resistance was 5 Ω. The output power of the MLSHG was more than 10 times compared to that of the outer-PM linear tubular generator in a fair comparison. Here, it is shown that the power density and output power of were MLSHG increased greatly.

1. Introduction

The oceans cover 75% of the earth and have an estimated 2.5 TW of exploitable wave energy worldwide. The potential for harvesting energy from ocean waves is immense [1]. A large number of different wave energy conversion devices have been proposed and applied in different countries [2,3]. At present, wave energy converters (WEC) can be divided into four categories: hydraulic, pneumatic, gearbox, and direct-drive wave energy converters (DD-WEC), as shown in

Table 1. Different types of wave energy conversion systems.

Type	Device	Description
Hydraulic	Oyster [8] Waveroller [9]	Hydraulic WEC uses wave energy to drive the hydraulic device to act on the hydraulic turbine, which in turn drives the traditional motor to generate electricity.
Pneumatic	Limpet [10] Spar buoy [11]	Pneumatic WEC use air as a medium for conversion. Under the action of the wave heave, the wave energy is converted into the pressure energy and kinetic energy of the air.
Gearbox	Columbia Power WEC [12] Wave Dragon [13]	Gearbox WEC uses a sealed drive shaft to convert the energy of wave ups and downs into mechanical energy in a rotating motor through a gearbox, thereby generating electrical energy.
Direct-drive	AWS [14] TE 5 [15]	Direct-drive WEC generally use linear generators to directly convert the mechanical energy of waves into electrical energy.

. Compared with other types of WEC devices, the DD-WEC generally uses a linear generator to convert wave energy without medium devices, so that the generation system is simplified [4,5,6]. It is considered to be one of the most promising offshore power generation methods in the wave energy power generation field today [7].

The average ocean wave speed is generally slow, which leads to the disadvantages of the large weight and low-power density observed in DD-WECs. Thus, in order to improve the power generation performance and reduce the size, a tubular primary permanent magnet linear generator (TPPMLG) has been proposed for wave energy extraction [16]. The TPPMLG has unique advantages, such as a reduced number of PMs, no transverse edge effect, and a higher voltage obtained under low-speed conditions. The tubular linear generator with external PMs, which is introduced in [17], can clearly improve the air gap magnetic flux and power density.

As another means of achieving a high power density, the mechanical screw was introduced to WECs, as it can effectively transfer low-speed linear motion to high-speed rotational motion. However, due to the mechanical wear of and regular maintenance required for the mechanical screw, it cannot adapt to complex ocean wave conditions. Inspiration can be drawn from the mechanical screw, and the magnetic lead screw (MLS) has been proposed as a device that maintains the capacity for transferring low-linear motion to high-rotational motion [18,19,20]. In addition, the MLS has the advantages of being maintenance-free and possessing overload protection, a simple structure, low vibration, and a higher thrust density than traditional linear PM generators [21,22,23]. At present, with the development of PM material and topologies, many new types of MLSs have appeared [24,25]. In order to simplify the manufacturing process of spiral magnets, an MLS approximates spiral magnetic poles by using semicircular rings [26]. Furthermore, a new reluctance-based magnetic lead screw with a simple and robust structure was proposed, which uses discretized PMs and ferromagnetic semi-rings to form an approximate helically shaped structure [27].

In this paper, the working principle and magnetic field of the MLS applied in the WEC are analyzed and calculated. Meanwhile, the electromagnetic performance, thrust force, torque, and air gap magnetic density of a surface-mount magnetic lead screw (SMMLS), triangular tooth magnetic lead screw (TTMLS), and Halbach array magnetic lead screw (QHAMLS) are evaluated and compared. Maxwell, a business finite element analysis (FEA) software, was used to optimize these topologies through 2D and 3D analyses. Then, a prototype was manufactured and tested. Finally, a wave energy generator with QHAMLS is proposed in this paper and compared with the cylindrical linear generator at different speeds and load conditions. In this paper, we propose a suitable topology of MLS that was selected for the wave energy converter. Based on this topology, a magnetic screw compound linear generator was created and designed as the core device for the DD-WEC. The output power of the new generator is more than 10 times that of the general linear generator under certain operating conditions, which can significantly improve the wave energy conversion efficiency.

The rest of this paper is organized as follows. The working principle and topology of the MLS are described in Section 2. Different topologies of MLSs and a comparison of their performances are described in Section 3. The manufacturing and testing of the prototype of the MLS are described in Section 4. A generator applied in the WEC with MLS is proposed in Section 5. Finally, there is a conclusion in Section 6.

2. Principles of the MLS

The magnetic lead screw is mainly composed of several components: a mover (screw), rotor (nut), and a pair of spiral permanent magnets with opposite radial magnetization directions placed outside the mover and inside the rotor. Compared with the mechanical screw, the magnetic screw functions on the basis of helical magnetic coupling transmission. There is no mechanical contact between the rotor and the mover, and the mutual conversion of the “thrust-torque” and “displacement-angle” is realized. Figure 1 is a two-dimensional equivalent diagram of the MLS. In order to analyze and explain the working principle of the MLS, the thrust of the mover is analytically predicted by the following two assumptions: (1) we assume that the mover of the MLS and the permanent magnet of the mover part of the axial length are infinite, so that the magnetic field distribution generated by the permanent magnet has axial symmetry and periodicity in the Z direction; (2) the magnetic permeability of the back iron of the mover and the rotor is infinite [28].

Figure 1 shows the two-dimensional (2-D) equivalent diagram of the MLS, where τ_p is the pole pitch, L_m is the width of the PM, H_m is the thickness of PM, L_t is the length of the rotor, g is the air gap length, R_r is the inner diameter of the mover, R_s is the outer diameter of the rotor, R_m is the outer diameter of the mover, and Z_d is the axial displacement. The magnetic field analysis is confined to two regions. The permeability of the airspace region is μ₀, and the permeability of the PM region is μ₀μ_r. Therefore:

$$B={\mu }_{0}M+{\mu }_0{\mu }_{r}H\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{In} \mathrm{the} \mathrm{PM}$$

(1)

$$B={\mu }_{0}H\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{In} \mathrm{the} \mathrm{air} \mathrm{space}$$

(2)

where M is the remanent magnetization, and M is related to the remanence B_rem by $M=B_{rem}/{\mu }_0$. In this system, M is given by:

$$M=M_r{e}_r+M_z{e}_z$$

(3)

Because the PMs are magnetized radially, the residual magnetization M has only a radial component M_r, and the axial component M_z = 0, $M_r$ is expressed into a Fourier series expansion, given by:

$$M_r={\displaystyle \sum _{n=0,1,2\cdots }^{\infty }4\left(\frac{B_{rem}}{{\mu }_0}\right)}\frac{\sin \left[\left(2n+1\right)\cdot \frac{\pi }{2}\cdot \frac{L_m}{{\tau }_p}\right]}{\left(2n+1\right)\pi }{\widehat{\alpha }}_r$$

(4)

where ${\widehat{\alpha }}_r$ is the magnetic flux vector in the radial direction, and the radial and axial components of the flux density in the airspace region $B_{rI}$ and $B_{zI}$ can be expressed as:

$$B_{rI}(r,z)=-{\displaystyle \sum _{n=0,1,2\cdots }^{\infty }{a}_{In}{I}_1(m_{n}r)+b_{In}{K}_1(m_{n}r)\cos (m_{n}z)}$$

(5)

$$B_{zI}(r,z)={\displaystyle \sum _{n=0,1,2\cdots }^{\infty }{a}_{In}{I}_0(m_{n}r)-b_{In}{K}_0(m_{n}r)\sin (m_{n}z)}$$

(6)

in which I₀ and K₀ are modified Bessel functions of the first kind. I₁ and K₁ are modified Bessel functions of the second kind of the order of 0 and 1, respectively. m_n is defined as $m_n=\left(2n+1\right)\pi /{\tau }_p$. The radial and axial components of the flux density in the PM region B_rII and B_zII can be expressed as:

$$B_{rII}(r,z)=-{\displaystyle \sum _{n=0,1,2\cdots }^{\infty }\left[\begin{array}{l}\left(F_{An}(m_{n}r)+a_{IIn}\right)I_1(m_{n}r)-\\ \left(F_{Bn}(m_{n}r)+b_{IIn}\right)K_1(m_{n}r)\end{array}\right]}\cos (m_{n}z)$$

(7)

$$B_{zII}(r,z)=-{\displaystyle \sum _{n=0,1,2\cdots }^{\infty }\left[\begin{array}{l}\left(F_{An}(m_{n}r)+a_{IIn}\right)I_0(m_{n}r)+\\ \left(F_{Bn}(m_{n}r)-b_{IIn}\right)K_0(m_{n}r)\end{array}\right]}\sin (m_{n}z)$$

(8)

In order to obtain the unknown coefficients in (5)–(8), some parameters are defined as follows:

$$P_n=\frac{4B_r}{{\tau }_p}\sin \left[\left(2n+1\right)\frac{\pi }{2}\cdot \frac{L_m}{{\tau }_p}\right]$$

(9)

$$c_{1n}=I_0(m_n{R}_s)\hspace{1em}\hspace{1em}\hspace{1em}{c}_{2n}=K_0(m_n{R}_s)$$

$$\begin{gathered} c_{3n}=I_0(m_n{R}_r)\hspace{1em}\hspace{1em}\hspace{1em}{c}_{4n}=K_0(m_n{R}_r) \\ {c}_{5n}=I_0(m_n{R}_m)\hspace{1em}\hspace{1em}\hspace{1em}{c}_{6n}=K_0(m_n{R}_m) \\ {c}_{7n}=I_1(m_n{R}_m)\hspace{1em}\hspace{1em}\hspace{1em}{c}_{8n}=K_1(m_n{R}_m) \end{gathered}$$

(10)

where the coefficients $F_{An}(m_{n}r)$ and $F_{Bn}(m_{n}r)$ are as follows:

$$F_{An}\left(m_{n}r\right)=\frac{P_n}{m_n}{\displaystyle {\int }_{m_n{R}_r}^{m_{n}r}\frac{K_1\left(x\right)dx}{I_1\left(x\right)K_0\left(x\right)+K_1\left(x\right)I_0\left(x\right)}}$$

(11)

$$F_{Bn}\left(m_{n}r\right)=\frac{P_n}{m_n}{\displaystyle {\int }_{m_n{R}_r}^{m_{n}r}\frac{I_1\left(x\right)dx}{I_1\left(x\right)K_0\left(x\right)+K_1\left(x\right)I_0\left(x\right)}}$$

(12)

The coefficients a_In, b_In, a_IIn, b_IIn can be obtained through (13)–(15):

$$\left[\begin{array}{cc}{\mu }_r\left(\frac{c_{5n}}{c_{6n}}-\frac{c_{1n}}{c_{2n}}\right)& -\left(\frac{c_{5n}}{c_{6n}}-\frac{c_{3n}}{c_{4n}}\right)\\ \left(\frac{c_{7n}}{c_{8n}}+\frac{c_{1n}}{c_{2n}}\right)& -\left(\frac{c_{7n}}{c_{8n}}+\frac{c_{3n}}{c_{4n}}\right)\end{array}\right]\left[\begin{array}{c}{a}_{In}\\ a_{IIn}\end{array}\right]=\left[\begin{array}{c}{F}_{An}\left(m_n{R}_m\right)\left(\frac{c_{5n}}{c_{6n}}\right)+F_{Bn}\left(m_n{R}_m\right)\\ F_{An}\left(m_n{R}_m\right)\left(\frac{c_{7n}}{c_{8n}}\right)-F_{Bn}\left(m_n{R}_m\right)\end{array}\right]$$

(13)

$$b_{In}=\frac{c_{1n}}{c_{2n}}{a}_{In}$$

(14)

$$b_{IIn}=\frac{c_{3n}}{c_{4n}}{a}_{II}{}_n$$

(15)

Using Lorentz’s force law to calculate the force exerted on the mover, the magnetomotive force generated by the PM, whose demagnetization curve is close to a straight line, can be replaced by the magnetomotive force generated by the corresponding magnet equivalent current-carrying coils:

$$F_{PM}=H_c\times h_m$$

(16)

where H_c is the coercive force of the PM, h_m is the thickness of the PM, assuming B_r is the residual magnetic flux density of the PM material, μ₁ is the vacuum permeability, and μ₂ is the recovery permeability of the PM material, so that H_c = B_r/μ₁μ₂. After applying Lorentz’s force law to integrate the surface S of each current-carrying coil, the force received by each current-carrying coil can be obtained as:

$$\overrightarrow{F}={\displaystyle \int \left(\overrightarrow{H_c}\times \overrightarrow{B_I}\right)}dS$$

(17)

where $\overrightarrow{H_c}$ is a vector of magnitude H_c, and its direction is consistent with the direction of the current flowing in the equivalent current-carrying coil. $\overrightarrow{B_I}$ is the magnetic induction intensity vector in the air gap region, and the integral of the area at the same time dS = 2πrdr. The radial component of the magnetic flux density causes the mover to produce an axial thrust F_t, whose value is:

$$F_t=\frac{4\pi B_r}{{\mu }_1{\mu }_2}{\displaystyle \sum _{n=0,1,2\cdots }^{\infty }{K}_{rn}\sin \left(\frac{m_n{L}_m}{2}\right)}\sin \left(m_n{Z}_d\right)$$

(18)

where L_m is the axial length of the PM and Z_d is the axial displacement, as shown in Figure 1. The coefficient K_rn is given by:

$$K_{rn}={\displaystyle {\int }_{R_s+g}^{R_s}\left[a_{In}{I}_1\left(m_{n}r\right)+b_{In}{K}_1\left(m_{n}r\right)\right]}rdr$$

(19)

According to the working principle of MLS, if energy loss in the process of magnetic force transmission is not taken into account in the calculation, the rotor power P_t is equal to the mover power P_r. The power transfer of MLS can be expressed as:

$$P_r=T_n\times \omega =F_t\times v=P_t$$

(20)

where T_n is the rotor torque and F_t is the mover thrust force. A gear ratio may be defined as the ratio of the rotor angle speed $\omega$ (rad/s) to the mover linear speed v $(\mathrm{m}\cdot \mathrm{s})$.

When the rotor rotates one circle, the mover also correspondingly moves in the axial direction through a lead displacement λ = $p\times {\tau }_p$, where p is the number of magnetic poles. Therefore, the relationship between the angular velocity $\omega$ of the rotor and the linear velocity v of the mover can be expressed as:

$$\omega =\frac{2\pi }{\lambda }v$$

(21)

The ratio of the rotor angular velocity $\omega$ of the MLS to the linear velocity v of the mover is defined as the transmission ratio G, and its expression is:

$$G=\frac{F_t}{T_n}=\frac{\omega }{v}=\frac{2\pi }{\lambda }$$

(22)

This means that a linear speed of 0.5 m/s can be converted into a rotational speed of 1500 rpm, so that low-speed wave motion can be converted into high-speed rotational motion, greatly improving the conversion efficiency and power density of the wave power generation device.

3. Magnetic Lead Screw Topology

For the application of wave power generation, the model of the surface-mounted magnetic screw (SMMLS), triangular tooth magnetic screw (TTMLS), and quasi-Halbach permanent magnet magnetic screw (QHAMLS) were established and optimized by FEA. In order to determine the most suitable topology for wave power generation, the electromagnetic performances of these magnetic screws, including the thrust, torque, and air gap magnetic density, were compared and evaluated.

3.1. Surface-Mount Magnetic Lead Screw

The SMMLS adopts a cylindrical structure, which mainly includes a linear mover and a rotor. The outer surface of the mover and the inner surface of the rotor are attached by a spiral permanent magnet ring and placed coaxially. The adjacent PM rings are radially magnetized in opposite directions. The topology and magnetic flux path of the SMMLS are as shown in Figure 2.

Since the spiral permanent magnet rings are placed on both the mover and the rotor back irons, the magnetic field distribution in the air gap of the MLS is in three dimensions. However, as the air gap length is much smaller than the pole pitch, its field distribution can be simplified as a 2D axisymmetric model, as depicted in Figure 3, which can be used for predicting the magnetic field distribution and thrust force by FEA. The magnetizing direction of the spiral permanent magnet ring is shown in Figure 3. It can be seen that the spiral permanent magnet ring is magnetized alternately with the N and S poles.

The thrust force for the minimum and maximum positions are shown in Figure 4, respectively. When the mover is at the position of Z_d = 0, there is only a radial force between the rotor and mover, and the axial thrust is the lowest, as shown in Figure 4a. When the mover is at the position of Z_d = τ_p/2, which is half of the pole pitch, the maximum thrust force can be achieved, as shown in Figure 4b.

3.2. Quasi-Halbach Array Magnetic Lead Screw

In order to improve the thrust and electromagnetic performance of the magnetic screw, the quasi-Halbach permanent magnet array was applied. The spiral permanent magnet rings with four magnetization directions were placed on the surface of the rotor and the mover. The three-dimensional structure and magnetic flux path are shown in Figure 5. Owing to the quasi-Halbach permanent magnet array, the magnetic flux leakage was significantly reduced, and the air gap magnetic field was effectively improved.

Figure 6 shows the arrangement of the axial and radial magnetizing PMs of the QHAMLS. The magnetic field distribution of the QHAMLS is depicted in Figure 7. Analyzing the magnetic flux lines between the mover and rotor, it can be seen that the lines mostly pass between the adjacent magnetic poles, while fewer lines pass through the back iron. Compared with the SMMLS, the quasi-Halbach permanent magnet array causes the magnetic field energy to be more concentrated. Therefore, it improves the air gap magnetic density and the power density of the QHAMLS. The positions of the minimum and maximum thrust forces are shown in Figure 7a,b, respectively.

3.3. Triangular Tooth Magnetic Lead Screw

The TTMLS embeds the spiral triangular permanent magnet rings in the back iron and enables the axially magnetized permanent magnets to be kept flush with the back iron, as shown in Figure 8. According to the principle of minimum reluctance, the magnetic field lines of the permanent magnets directly form a loop through the back iron and the air gap, which reduces reluctance and magnetic leakage between the magnetic poles. At the same time, it can produce a magnetic concentration effect, thereby improving the electromagnetic characteristics of the TTMLS. More importantly, this topology can withstand a greater wave impact force, because the permanent magnets are embedded in the back iron, preventing the permanent magnets from falling off and breaking.

Figure 9 shows the distribution of the axially magnetized PMs and back iron in the TTMLS. It can be seen from the figure that the permanent magnet is sandwiched between two spiral back irons, and the magnetization directions of the adjacent PMs are opposite. In addition, the thickness of the back iron is greater than that of the PMs so as to ensure that the permanent magnet does not cause permanent magnet displacement due to linear or rotational movement during the operation, and the mechanical strength of the magnetic screw and the stability of the movement process are improved.

The magnetic line distribution of the TTMLS is shown in Figure 10. Figure 10a shows the magnetic field line distribution at the opposite axial magnetic pole position (Z_d = 0), and Figure 10b shows the relative displacement τ_p/2 (Z_d = τ_p/2). Analyzing the magnetic field loop between the rotor and mover, we can see that the magnetic flux of the axially magnetized permanent magnet is concentrated in the triangular back iron, and there is almost no magnetic leakage between the magnetic poles, which improves the magnetic flux density of the air gap to a certain extent.

4. Optimization of Different Topologies

For the magnetic screw, the thrust density F_density and the shear stress F_suface, as important parameters for characterizing the electromagnetic performance of the magnetic screw, can be regarded as standards for measuring the optimization results of the magnetic screw. The specific expression is:

$$F_{density}=\frac{F_t}{V}=\frac{F_t}{\pi \left(R_s^2-R_r^2\right)L_t}$$

(23)

$$F_{suface}=\frac{F_t}{S}=\frac{F_t}{2\pi \left(R_m+g/2\right)L_t}$$

(24)

where the effective working volume V is the volume of the cylinder surrounding the rotor and the mover, and the effective working area S is the area enclosed by the air gap between the rotor and the mover, and its expression is:

$$S=2\pi (R_m+g/2)$$

(25)

$$V=\pi \left(R_s^2-R_r^2\right)$$

(26)

In order to better optimize and compare the thrust characteristics and electromagnetic performances of the three different topological MLSs, it was necessary to maintain the consistency of the initial volume of the three MLSs—that is, to determine the design parameters shown in

Table 2. Parameter of MLS.

Symbol	Parameter	Value
$R_s$	Outer radius of the rotor (mm)	37
$R_m$	Outer radius of the mover (mm)	21
$R_r$	Inner radius of the mover (mm)	6
$L_t$	Rotor length (mm)	60

.

4.1. Optimization of the SMMLS

The PM is the only way to establish the magnetic field of the SMMLS. Its volume directly affects the magnetic field strength and electromagnetic performance of the magnetic screw. Therefore, the thickness and lead of the PM must be optimized first; that is, the geometric size of the PM must be determined. First, one must fix the total thickness of the PM and the back iron of the mover and the rotor according to the air gap length, so as to keep the values of the inner and outer diameters of the SMMLS unchanged. Then, the air gap magnetic field is optimized by changing the thickness and lead of the PM, and the optimal thrust density point is selected to determine the two key parameters of the PM thickness and lead that affect the thrust and electromagnetic performance of the SMMLS. The main parameters of the MLS that were initially selected are shown in

Table 3. Main parameters of the SMMLS.

Symbol	Parameter	Value
$\lambda$	Magnetic lead (mm)	30
τp	Pole pitch (mm)	15
Rs	Outer radius of the rotor (mm)	37
Rm	Outer radius of the mover (mm)	21
Rr	Inner radius of the mover (mm)	6
Lm	Width of the PM (mm)	15
Hm	Thickness of the PM (mm)	5
Hr	Back iron thickness (mm)	10
Lt	Rotor length (mm)	60
g	Length of the air gap (mm)	1
$B_r$	Residual magnetic field strength	1.23

.

According to the structural parameters of the SMMLS shown in Table 3, a finite element optimization model was established, the value range of the lead was determined to be 10 mm–60 mm, and the value range of the thickness of the permanent magnet was 2 mm–9 mm. The curve of the thrust density of the SMMLS during the process is shown in Figure 11. It can be seen from the figure that, when the lead was 10 mm, the thrust density increased with the thickness of the magnet, and the change was not obvious. This was caused by the more serious air gap magnetic saturation that occurred when the lead was small. When the lead was 20 mm–60 mm, the permanent magnet varied from 2 mm–7 mm, and the thrust density increased significantly as the thickness of the permanent magnet increased. However, when the thickness of the permanent magnet exceeded 7 mm, the air gap reached magnetic saturation, which caused the thrust density to change less. It was finally determined that the thickness and lead of the permanent magnet were 7 mm and 20 mm, respectively.

Figure 12 shows the curve of the shear stress changing with the outer radius of the mover under different leads. It can be seen that, when the lead was constant, the shear stress was not affected by the change in the outer radius of the mover. The fluctuation in the shear stress was actually caused by the magnetic leakage effect between the magnetic poles under different leads, so that the outer radius of the mover was fixed to 21 mm.

The air gap area between the rotor and the mover is a part of the magnetic flux circuit. The length of the air gap significantly affects the strength of the air gap magnetic field. Generally speaking, the larger the air gap is, the weaker the corresponding air gap magnetic field will be. We performed finite element simulation calculations on the magnetic screw under different air gap sizes, and the results are shown in Figure 13. It can be seen from the figure that, when the air gap length varied from 0.5 mm–5 mm and the other parameters remained unchanged, the thrust varied from 2.74 kN–0.85 kN, and the thrust decreased linearly with the increase in the air gap length; thus, the air gap length should be reduced as much as possible while considering the mechanical processing factors. The final air gap length was determined to be 1 mm.

It can be seen from Figure 14 that the optimized SMMLS had a peak thrust of 2.35 kN and a torque peak of 7.35 N·m. The thrust results obtained were placed into Equation (23) to obtain a thrust density of 9.3 MN/m². After that, the transmission ratio of the MLS could be obtained from the thrust and torque peaks as 319, and the lead parameter in Table 3 was factored into Equation (22) to calculate the transmission ratio as 314. Comparing the transmission ratios obtained by the two methods, it can be seen that the values of the transmission ratio calculated by the finite element analysis method and the formula (22) are in good agreement, thus verifying the accuracy of the thrust analysis results in the first section.

The air gap is an important site for the energy conversion of the MLS. The magnetic field strength of the air gap determines the power density of the MLS; thus, it must be analyzed. The air gap flux density of the SMMLS in one lead, when at the position of Z_d = τ_p/2, is shown in Figure 15. It can be seen from the figure that the axial and radial air gap magnetic density were 0.96 T and 1.03 T, respectively, and the air gap magnetic density was low in sine.

4.2. Optimization of the QHAMLS

The thrust and electromagnetic characteristics of the QHAMLS are also determined by the volume of the PM, but the PMs of the quasi-Halbach structure have two different magnetization methods, the axial and radial. Different axial and radial magnetization PM widths affect the thrust and has a greater impact on the electromagnetic performance. Since the radially magnetized PM guides the lines of the magnetic force in the air gap, it must be optimized. The relationship between the radial magnetization width h₀ of the PM and the axial magnetization width h₁ of the PM is as follows:

$${\tau }_p=h_1+h_0$$

(27)

The value range of the radial PM width was determined to be 1 mm–9 mm, and the variation curve between the radial PM width and the thrust was obtained by FEA, as shown in Figure 16. It can be seen from the figure that the peak thrust of the QHAMLS first increased and then decreased with the width of the radial PM. When the width was 4 mm, the peak thrust reached the highest value of 3.87 kN, and the thrust density was 15.4 MN/m².

The air gap flux density of the QHAMLS at the position of Z_d = τ_p/2 is shown in Figure 17. The axial and radial air gap flux density were 1.34 T and 1.1 T, respectively, and the sine degree was relatively high. Therefore, compared with the surface-mounted magnetic screw, the quasi-Halbach magnetization structure can simultaneously increase the amplitude and sine of the air gap magnetic density, thereby improving the transmission efficiency and reducing the mechanical vibration, which is more suitable for low-speed and high-thrust operations.

4.3. Optimization of the TTMLS

The TTMLS has the PMs embedded in the back iron, while the back iron occupies a larger space and reduces the volume of the PM, which affects the air gap magnetic density of the MLS to a certain extent. Therefore, it is necessary to find the optimal point of balance between the PM width l₀ and the triangular tooth width l₁ in order to maximize the thrust value. The relationship between the width of the triangle tooth and the width of the PM can be seen in the following formula:

$${\tau }_p=l_1+l_0$$

(28)

The value range of the triangular tooth width was determined by FEA to be 1 mm–9 mm, and the curve of the peak thrust changing with the tooth width was obtained, as shown in Figure 18. It can be seen that, as the width of the triangular teeth increased, the peak thrust first increased and then decreased. When the tooth width was 5 mm, the thrust reached the maximum value of 3.69 kN, and the thrust density was 14.7 MN/m² at this time.

When the magnetic pole position of the TTMLS was Z_d = τ_p/2, according to the air gap magnetic density shown in Figure 19, and the axial and radial air gap magnetic densities were 1.44 T and 2.01 T, respectively. Compared with the other two topological structures, although the triangular tooth structure increased the amplitude of the air gap magnetic density, the sine of the waveform was reduced, which caused the device to mechanically vibrate and made it unsuitable for low-speed operations.

4.4. Comparison and Evaluation

In this study, we first optimized the lead of the surface-mounted magnetic screw, the width and thickness of the permanent magnet, the outer radius of the mover, and the length of the air gap. After that, based on the optimization parameters of the SMMLS, the radial and axial magnetized PM widths and triangular tooth widths of the QHAMLS and TTMLS were optimized. Finally, the main parameters of the three topological structures of the MLSs were optimized, as shown in

Table 4. Parameter comparison.

Symbol	Parameter	SMMLS	QHAMLS	TTMLS
$\lambda$	Magnetic lead (mm)	20	20	20
τp	Pole pitch (mm)	10	10	10
Rs	Outer radius of the rotor (mm)	37	37	37
Rm	Outer radius of the mover (mm)	21	21	21
Rs	Inner radius of the mover (mm)	6	6	6
Lm	Width of the PM (mm)	10	10	10
h0	Width of the radial magnetized PM (mm)	—	4	—
hl	Width of the axial magnetized PM (mm)	—	6	—
l0	Permanent magnet tooth width (mm)	—	—	5
l1	Triangle tooth width (mm)	—	—	5
l2	Triangle tooth height (mm)	—	—	0.1
a	Triangle tooth angle (deg)	—	—	39.3
Hm	Thickness of the PM (mm)	7	7	7
Hr	Back iron thickness (mm)	8	8	8
Lt	Rotor length (mm)	60	60	60
g	Length of the air gap (mm)	1	1	1
$B_r$	Residual magnetic field strength (T)	1.23	1.23	1.23

.

Figure 20a shows the comparison of the torques of the three topological magnetic screws. The peak torques of the SMMLS, QHAMLS, and TTMLS were 7.35 $\mathrm{N}\cdot \mathrm{m}$, 12.48 $\mathrm{N}\cdot \mathrm{m}$, and 10.59 $\mathrm{N}\cdot \mathrm{m}$, and it can be seen that the peak torque of the QHAMLS was 69.79% higher than that of the SMMLS and 44% higher than that of TTMLS of the same size. Figure 20b shows the comparison of the thrusts of the three topological MLS, in which the peak thrust of the SMMLS, QHAMLS, and TTMLS are 2.35 kN, 3.85 kN, and 3.46 kN. It can be seen that the peak thrust of the QHAMLS was 63.82% higher than that of the SMMLS and 47.23% higher than that of the TTMLS of the same size. By comparing the thrusts and torques of the three MLS topologies, it can be seen that the QHAMLS is more suitable for low-speed and high-thrust applications of wave power generation.

5. Application of the MLS in Wave Energy

Figure 21 is a typical point-suction direct-drive wave energy power generation device. The device is composed of an outer-PM linear tubular generator, a damping plate, a pontoon, a connecting bracket, and an anchor chain, among which is the wave energy conversion device (WEC). It is an outer-PM linear tubular generator. In this study, the QHAMLS was applied in the field of wave power generation, and the MLSHG was designed to replace the outer-PM linear tubular generator in order to complete the wave energy conversion, and the two types were compared and analyzed in different sea conditions. The output power of the energy conversion device was evaluated.

Due to the different hydrodynamic parameters of the inner and outer pontoons of the point-suction direct-drive wave power generation device, their vertical speeds are also different under the action of wave forces, and the inner pontoon basically remains stationary due to the presence of the damping plate. The outer pontoon moves up and down with the waves, and the relative speed between the inner and outer pontoons drives the mover of the MLSHG to produce a low-speed linear motion. Then, the low-speed linear motion of the mover is converted into the high-speed rotating motion of the rotor through the spiral magnetic coupling transmission of the MLS. Then, the PM synchronous generator rotor is driven to rotate at a high speed, and the output power of the wave power generation device is improved.

Figure 22a is a schematic diagram of the MLSHG. The structure is composed of a QHAMLS and an 8-pole, 12-slot PM synchronous rotating machine. The QHAMLS rotor and the rotating motor rotor share a rotor back iron. The MLS moves and the rotor, adopting the quasi-Halbach structure, realizes the decoupling between the two rotors and reduces the complexity of the assembly. The MLSHG was analyzed and compared with the 9-pole, 10-slot outer-PM linear tubular generator shown in Figure 22b. To ensure a fair comparison, the two energy conversion core devices had the same volume, and the specific structure and operating parameters are shown in

Table 5. Comparison of the power generation device structures and operating parameters.

Symbol	Parameter	Linear Generator	MLS Hybrid Generator
Rr	Inner radius of the mover (mm)	—	6
Rm	Outer radius of the mover (mm)	—	21
Rs	Outer radius of the rotor (mm)	—	37
h0	Width of the radial magnetized PM (mm)	—	4
hl	Width of the axial Magnetized PM (mm)	—	6
Din	Inner diameter (mm)	37	37
Dout	Outer diameter (mm)	77	77
Lt	Axial length (mm)	100	100
Nr	Number of rotor poles	9	8
Ns	Number of stator slots	10	12
Np	Number of motor phases	3	3
Hm	Thickness of the PM (mm)	7	7
Hr	Back iron thickness (mm)	8	8
τp	Pole pitch (mm)	9.5	10
g	Length of the air gap (mm)	1	1
v	Speed	0.2 m/s	0.2 m/s

.

5.1. Comparison under Constant Speed Conditions

The linear generator’s primary movement speed v = 0.2 m/s. From Equation (22), the rotating generator speed transformed by the magnetic screw conversion device $\omega$ = 62.8 rad/s.

Table 6. Output power comparison.

Speed (m/s)	Load (ohm)	MLS Hybrid Generator		Linear Generator
		Input Power (W)	Output Power (W)	Input Power (W)	Output Power (W)
0.1	5	1225	811	52	39
	10	1477	1158	31	26
0.15	5	1367	923	118	87
	10	1932	1534	68	58
0.2	5	1445	967	205	153
	10	2136	1722	124	104
025	5	1492	991	314	234
	10	2278	1826	189	161
0.3	5	1508	1003	440	330
	10	2356	1889	268	230

shows the output power under different load conditions. When the moving speed v = 0.25 m/s, the output power of the MLSHG and outer-PM linear tubular generator were 991 W and 234 W, respectively, at the load resistance of 5 Ω. Compared with the outer-PM linear tubular generator, the output power of the MLSHG was nearly up to 10 times higher. According to the data analysis shown in the table, as the movement speed increased, the output power of the MLSHG and outer-PM linear tubular generator gradually increased. However, under the same load and moving speed, the output power of the MLSHG was much higher than that of the outer-PM linear tubular generator. In particular, under the condition of a lower moving speed, its output power demonstrably increased.

5.2. Comparison under Wave Speed Conditions

Assuming that the wave velocity is sinusoidal, it is v = $0.2\sin (2\pi t/T)$, where T is the wave period, which is defined as 2 s. Ignoring the wave diffraction and other factors, the non-load-induced electromotive forces generated by the MLSHG and the outer-PM linear tubular generator in one cycle are shown in Figure 23.

It can be seen from Figure 23 that the induced electromotive force was a wave curve, and the non-load-induced electromotive force amplitudes of the outer-PM linear tubular generator and the MLSHG were 36.8 V and 219.5 V, respectively. The comparison shows that the MLSHG used the principle of magnetic transmission to realize the speed-increasing operation of the device, and its non-load-induced electromotive force amplitude was about 6 times that of the outer-PM linear tubular generator. Figure 24 shows the instantaneous power waveforms of the two wave power generation devices in one cycle, when the wave speed was v = $0.2\sin (2\pi t/T)$ and the loads were 5 Ω and 10 Ω. Figure 24a shows that, when the loads were 5 Ω and 10 Ω, the instantaneous peak power of the outer-PM linear tubular generator was 212 W and 150 W, and the average power was 95.57 W and 66.68 W, respectively. Figure 24b shows that, when the loads were 5 Ω and 10 Ω, the peak instantaneous power of the MLSHG was 1150 W and 1650 W, and the average power was 820.89 W and 1063.83 W, respectively. It can be seen from the comparison of the results that the output power of the MLSHG was greatly improved compared with that of the outer-PM linear tubular generator in the case of low-speed wave motion.

6. Experiment and Discussion

From the above theoretical analysis and calculation, it can be seen that the MLSHG has a higher power density than the outer-PM linear tubular generator, can extract more wave energy, and is more suitable for low-speed and high-thrust wave power generation.

According to the design parameters in Table 4, and considering the actual processing conditions, the assembled MLS is shown in Figure 25. Among these parameters, the PM adopts N38 neodymium iron boron, a rare earth PM material, and the moving and rotor back iron adopts DW470-50, an iron core material. In order to measure the thrust characteristics of the magnetic screw, a rotating crankshaft platform, as shown in Figure 26. was used for the thrust testing. Among this apparatus, one end of the crankshaft connecting rod was connected with the driving motor through two plum blossom couplings, and the other end was connected with the mover of the magnetic screw through a tension sensor. When working, first the rotor is fixed, and the drive motor drives the crankshaft connecting rod to generate axial tension on the mover of the magnetic screw. As the mover moves, the distance between the magnetic poles of the PMs on the rotor increases, and the tension sensor senses the change of force, and it is converted into an electrical signal output and displayed on the oscilloscope.

Figure 27 compares the predicted value and the actual measured value of the thrust versus axial displacement. The results show that the peak thrust of the two-dimensional finite element analysis was 2.38 kN, the peak thrust obtained from the three-dimensional finite element analysis was 2.35 kN, and the measured peak thrust was 2.29 kN. It can be seen that the measured value was 2.6% lower than the three-dimensional finite element analysis result, and the results were more consistent, which further proves the accuracy of the above simulation results.

In order to further verify that the MLS, as a speed-increasing device, can convert low-speed linear motion into high-speed rotational motion, using the test platform shown in Figure 25, we demonstrated the MLS rotor was driven by the drive motor to rotate at a speed of 120 rpm, and the rotor drove the mover to move linearly on account of magnetic coupling, and the speed–time curve shown in Figure 28 was obtained. The curve, as can be seen from the figure, shows that when the rotor performed a rotating motion of $\omega$ = 120 rpm, the linear velocity of the mover fluctuated at around v = 0.04 m/s, which verifies the accuracy of the simulation results.

7. Conclusions

Here, the MLSHG was proposed for application in the DD-WEC. The key problems relating to the parameter selection in the design and manufacturing of the MLS were studied, the structural parameters were optimized, and a prototype was manufactured. Through FEA, the output powers of the MLSHG and outer-PM linear tubular generator under different wave speeds were compared. It can be concluded that, in the case of the same volume, the lower the wave velocity is, the higher the power of the MLSHG is compared to that of the outer-PM linear tubular generator. At a constant speed of 0.15 m/s, the output power of the MLSHG and outer-PM linear tubular generator were 923 W and 87 W, respectively, when the load was 5 Ω. Assuming that the wave velocity was 0.2sin (πt), which is a sinusoidal velocity, the average output power of the MLSHG and outer-PM linear tubular generator were 1063.83 W and 66.7 W at the load of 10 Ω. This shows that the MLSHG is more suitable for low-speed waves than the outer-PM linear tubular generator. In our follow-up research, we will continue to explore breakthroughs in the structure of the MLSHG in order to make it more suitable for the wave environment and maintain a stable power generation with changing wave speeds.