Research on direct-drive motor for wave energy conversion based on Halbach array

Abstract

In this paper, an improved Halbach array structure of a direct-drive linear motor for wave energy conversion is proposed, and its magnetic circuit structure is modeled and the induced electromotive force (EMF) generated by vertical wave energy conversion is deduced. Then, using the internal and external parameters of the wave energy conversion direct-drive motor, that is, four internal parameters of the magnetic circuit topology and two external parameters of the mover motion, the influence of these parameters on the induced EMF of the direct-drive motor in the wave energy conversion is analyzed. Since the theoretical analysis results of the model are consistent with the results of the two kinds of permanent magnet materials analyzed by the finite element method (FEM), the model and analysis method of the direct drive motor are correct. The investigation show that the modeling method proposed in this paper can be used to optimize the energy conversion of the direct-drive motor from two aspects: the internal parameters of the magnetic circuit topology of the direct-drive motor and the external parameters of the mover motion.

Keywords

Halbach array structure magnetic circuit permanent magnets (PM)wave energy direct-drive linear motor

1. Introduction

With the further development of wave energy, direct-driven linear generators [1,2], such as linear synchronous PM generators, flux-switching PM linear generators, and linear magnetic gear generators, stand out from many high-efficiency conversion schemes for wave power generation. Without gears and hydraulic systems, direct-drive linear motors have become a key point for improving power generation conversion. From the perspective of improving the conversion efficiency of direct-drive linear motors, the current literature mostly focuses on two aspects. One is the control method of the power generation conversion system, such as damping control [3–7], power control [8,9], power factor control [10], and other control methods [11,12]; the other is the design and optimization of the generator structure, especially the optimization of the electromagnetic topology. In terms of electromagnetic topology optimization, the optimization of permanent magnet structure has become an important point [13–21], such as magnet segmentation [13–15], trapezoidal magnet pole [16,17], creating magnet pole with different magnet materials [18,19], Halbach structure [20,21], etc. Due to the low energy density of ocean waves, the speed of ocean waves changes with time, and linear generators must operate at low speed and instantaneous speed change [21,22]. Therefore, the optimization of the electromagnetic structure of linear motors has become a key point of ocean wave energy conversion [23]. Traditional direct-drive motors for wave energy conversion rarely use Halbach magnetic pole structure, and even less use the improved Halbach. This paper mainly focuses on the improved Halbach topology based on the Halbach array, which is used for wave energy harvesting.

Combined with the reciprocating operation characteristics of the direct-drive linear energy conversion device, this paper proposes an improved Halbach array for wave energy conversion in low-speed motion, which is based on the improvement of the Halbach array and uses a magnetic circuit topology with different heights of radially magnetized permanent magnets and axially magnetized permanent magnets. It not only has the characteristics of unilateral self-shielding effect of Halbach array [14,20,24], but also reduces the cost of permanent magnets without significantly reducing the air gap flux density. In Section 2, the magnetic circuit model of the improved Halbach array is first established. Then, based on the magnetic circuit, the no-load electromotive force generated by the motor armature driven by the monochromatic wave is deduced. Finally, two kinds of permanent magnets are used to verify the structure magnetic circuit and no-load induced electromotive force by the finite element method. The proposed wave energy conversion model is correct. In Section 3, the influence of the main parameters of the Halbach array structure on wave energy conversion is discussed. The conversion of waves into electromotive force is analyzed from six parameters of array structure parameters and mover motion parameters.

The contributions of this paper are (1) a direct-drive motor using Halbach array that can reduce the volume of permanent magnets is theoretically modeled and verified by the finite element method, and (2) the internal parameters and external parameters of the wave energy conversion motor, that is, the four parameters of the array structure and the two parameters of the mover motion, are used to jointly analyze the conversion motor, and use it for the design and optimization of the motor.

2. Design and analysis

2.1. Magnetic circuit modeling

Due to the regularity of its magnet arrangement structure, the magnetic circuit model is used to analyze the improved Halbach structure used in this paper, which is an effective method and can achieve a certain engineering accuracy [25]. Figure 1 shows the no-load magnetic circuit model of the improved Halbach magnet array, in which the red arrows indicate radial magnetization magnets, and the blue arrows indicate axial magnetization magnets. The heights of the axial magnets and the radial magnets are unequal. The heights of the axial magnets and the radial magnets are unequal. The no-load magnetic flux path and the equivalent magnetic circuit within a pair of poles are drawn with lines in Fig. 1.

Fig. 1.

Schematic diagram of the no-load magnetic circuit model of the improved Halbach structure.

The magnetic field of the Halbach array is enhanced on one side, and weakened on the other side, even to zero, so the magnetic flux leakage of the array in the axial direction can be ignored. In the derivation of this paper, an assumption is made, that is, the iron core works in an unsaturated state, the core magnetic resistance R _c is ignored, and other leakage magnetic fluxes such as slot leakage magnetic flux and inter-pole magnetic flux leakage are also ignored. According to Kirchhoff’s law of magnetic circuit, in one magnetic pole, Eq. ((1)) can be obtained. $\begin{eqnarray}\displaystyle H_{cr}h+H_{ca}{\tau}_{ma}/2=(R_{mr}+R_{ma}/2+R_{g}){\phi}_{g} & & \displaystyle\end{eqnarray}$ (1) where H _cr and R _mr represent the magnetic field strength and reluctance of the radially magnetized magnet respectively, H _ca and R _ma represent the magnetic field strength and reluctance of the axially magnetized magnet, h represents the height of the radially magnetized magnet, and τ_ma represents the axially magnetized magnet. The length of the magnet, R _g represents the reluctance along the length of the air gap of the motor, and 𝜙_g represents the magnetic flux of the magnetic circuit. According to the structure as shown, R _mr, R _ma andR _g can be represented by Eqs ((2)), ((3)) and ((4)), respectively. $\begin{eqnarray}\displaystyle R_{mr}\text{} & =\text{} & \displaystyle \frac{h}{{\mu}_{rcr}{\mu}_{0}L_{ly}({\tau}-{\tau}_{ma})/2}\end{eqnarray}$ (2) $\begin{eqnarray}\displaystyle R_{ma}\text{} & =\text{} & \displaystyle \frac{{\tau}_{ma}}{{\mu}_{rca}{\mu}_{0}L_{ly}h_{m}}\end{eqnarray}$ (3) $\begin{eqnarray}\displaystyle R_{g}\text{} & =\text{} & \displaystyle \frac{{\delta}}{{\mu}_{0}L_{ly}({\tau}-{\tau}_{ma})/2}\end{eqnarray}$ (4) where, μ_rca is the relative permeability of the axial magnetized permanent magnet, μ_rcr is the relative permeability of the radial magnetized permanent magnet, μ₀ is the vacuum permeability, δ is the air gap length of the motor, τ is the pole pitch of the Halbach structure, and L _ly is the width of the motor in the y direction.

Substituting Eqs ((2)), ((3)) and ((4)) into Eq. ((1)), the magnetic flux of the magnetic circuit can be expressed as ((5)). $\begin{eqnarray}\displaystyle {\phi}_{g}=\frac{H_{cr}h+H_{ca}{\tau}_{ma}/2}{\frac{2h}{{\mu}_{rcr}{\mu}_{0}L_{ly}({\tau}-{\tau}_{ma})}+\frac{1}{2}\frac{{\tau}_{ma}}{{\mu}_{rca}{\mu}_{0}L_{ly}(r_{ma}-r_{mi})}+\frac{2{\delta}}{{\mu}_{0}L_{ly}({\tau}-{\tau}_{ma})}}. & & \displaystyle\end{eqnarray}$ (5) Divide the magnetic flux by the air gap area to get the air gap flux density ((6)). $\begin{eqnarray}\displaystyle B_{g}=\frac{{\phi}_{g}}{A_{g}}=\frac{H_{cr}h+H_{ca}{\tau}_{ma}/2}{\frac{2h}{{\mu}_{rcr}}+\frac{{\tau}_{ma}({\tau}-{\tau}_{ma})}{2{\mu}_{rca}h_{m}}+2{\delta}}{\mu}_{0}. & & \displaystyle\end{eqnarray}$ (6) Considering the influence of stator slotting on the air-gap magnetic density, the Carter coefficient is used to solve this problem. Set the Carter coefficient k _c = 1.3 [26], the air-gap flux density can be rewritten as Eq. ((7)). $\begin{eqnarray}\displaystyle B_{g}=\frac{H_{cr}h+H_{ca}{\tau}_{ma}/2}{\frac{2h}{{\mu}_{rcr}}+\frac{{\tau}_{ma}({\tau}-{\tau}_{ma})}{2{\mu}_{rca}h_{m}}+2k_{c}{\delta}}{\mu}_{0}. & & \displaystyle\end{eqnarray}$ (7)

2.2. Induced potential analysis

Since the leakage flux is ignored, the working flux density is the air gap flux density, that is, B _m = B _g. The magnetic density obtained by the above formula (7) is the constant average magnetic density under the pole arc of the permanent magnet, and the amplitude in the pole arc of the positive half cycle is always positive B _m, and the amplitude in the pole arc of the negative half cycle is always negative B _m. Since B _m is an oddly symmetric waveform in the period, the Fourier trigonometric function expansion of this waveform can obtain that both the DC component and the cosine component are zero, that is, a ₀ = 0 and a _n = 0, and the sine component is the expression (8). $\begin{eqnarray}\displaystyle b_{n}=(-1)^{n+1}\frac{4B_{m}}{(2n-1){\pi}}\sin ((2n-1){\beta})\quad (n=1,2,3,\ldots ). & & \displaystyle\end{eqnarray}$ (8) Therefore, substituting a ₀, a _n, and b _n for the corresponding coefficient values of the Fourier formula (9), the Fourier decomposition of the magnetic flux density waveform is expressed as expression (10). $\begin{eqnarray}\displaystyle f_{m}\text{} & =\text{} & \displaystyle \frac{a_{0}}{2}+\sum _{n=1}^{\infty }(a_{n}\cos ((2n-1){\theta}_{ae})+b_{n}\sin ((2n-1){\theta}_{ae}))\end{eqnarray}$ (9) $\begin{eqnarray}\displaystyle f_{m}\text{} & =\text{} & \displaystyle \mathop{\sum }_{n=1}^{\infty }b_{n}\sin ((2n-1){\theta}_{ae})=\mathop{\sum }_{n=1}^{\infty }(-1)^{n+1}\frac{4B_{m}}{(2n-1){\pi}}\sin ((2n-1){\beta})\sin ((2n-1){\theta}_{ae}).\end{eqnarray}$ (10)

Therefore, taking the value of n in Expression (10) as 1, the fundamental wave Expression (11) of the magnetic flux density can be obtained. $\begin{eqnarray}\displaystyle f_{m1}=B_{m1}\sin {\theta}_{ae}=\frac{4}{{\pi}}B_{m}\sin {\beta}\sin {\theta}_{ae}. & & \displaystyle\end{eqnarray}$ (11) Where B _m1 is the fundamental amplitude of working flux density, 𝛽 is the half pole arc of permanent magnet, and θ_ae is the electrical angle of the motor. Then by multiplying the air-gap flux density by the motor pole area, the fundamental wave of the air-gap flux can be expressed by Eq. (12). $\begin{eqnarray}\displaystyle {\phi}_{m1}=\frac{B_{av}{\pi}DL}{p}\sin {\theta}_{ae}=\frac{DL}{(p/2)}\frac{4}{{\pi}}B_{m}\sin {\beta}\sin {\theta}_{ae}. & & \displaystyle\end{eqnarray}$ (12) Where, p is the number of poles of the motor, and B _av is the average magnetic density of the air gap. L is the width of linear motor in this paper, and Correspondingly, L is the shaft length of circular rotating motor. D is the linear motor virtual diameter of the linear motor in this paper, and correspondingly, D is the diameter of the circular rotating motor. The virtual diameter in this paper is related to the number of pole pairs and pole distance, that is, D = pτ∕π.

The direct-drive wave energy linear power generation belongs to the reciprocating power generation system. One of its key features is that the mover of the linear motor reciprocates in the vertical direction with the wave. Assuming that driven by an ideal monochromatic wave, the motor mover exhibits sinusoidal motion, and its reciprocating displacement can be expressed by Eq. (13). $\begin{eqnarray}\displaystyle x(t)=H\sin ({\omega}t). & & \displaystyle\end{eqnarray}$ (13) Where, x (t), or written as x, is the movement displacement of the motor mover changing with time, which is also the displacement in the vertical direction of the wave, H is the amplitude of the sinusoidal motion of the wave, and ω is the frequency of the wave motion.

Since the electrical angular displacement θ_ae of the linear motor is the product of the mechanical angular displacement θ_r and the number of pole pairs P, and the mechanical angular displacement is also related to the pole distance τ and the motion displacement x, the motion displacement can be converted into electrical angular displacement, that is, ${\theta}_{ae}=P{\theta}_{r}=(p/2)\frac{{\pi}}{{\tau}}x$ . Then, the electrical angular displacement is substituted into the Eq. (12), and the expression of the fundamental wave of the magnetic flux can be rewritten to obtain the Eq. (14). $\begin{eqnarray}\displaystyle {\phi}_{m1}=\frac{DL}{(P/2)}\frac{4}{{\pi}}B_{m}\sin {\beta}\sin (\frac{p}{2}\frac{{\pi}}{{\tau}}x(t)). & & \displaystyle\end{eqnarray}$ (14) Applying Faraday’s law to Eq. (14), the induced potential of direct-drive wave energy linear power generation can be obtained (15). $\begin{eqnarray}\displaystyle V(t)=N\frac{\text{d}{\phi}_{m1}}{\text{d}t}=k\cos \frac{p{\pi}x}{2{\tau}}x^{\prime }. & & \displaystyle\end{eqnarray}$ (15) Where, k is the coefficient related to the amplitude of the induced voltage, that is, $k=N\frac{DL}{(p/2)}\frac{4}{{\pi}}B_{m}\frac{p{\pi}}{2{\tau}}\sin {\beta}=\frac{4NDLB_{m}}{{\tau}}\sin {\beta}$ , N is the effective coil turns of the motor. Since x is the abbreviation of the displacement x (t) of the reciprocating motion of the motor mentioned above, x ^′ represents the derivation of the displacement x with respect to time, which is the speed of the reciprocating motion.

Since the three-phase windings of the linear motor are symmetrical in space, that is, each phase winding occupies one third of the cycle, so by rewriting the parameter θ_ae containing the spatial position information of the induced voltage Eq. (15), the three-phase induced electromotive force of the wave reciprocating motion is given by the following Eqs (16), (17) and (17), respectively. $\begin{eqnarray}\displaystyle V_{1}(t)\text{} & =\text{} & \displaystyle k\cos \left(\frac{p{\pi}}{2{\tau}}x(t)\right)x^{\prime }\end{eqnarray}$ (16) $\begin{eqnarray}\displaystyle V_{2}(t)\text{} & =\text{} & \displaystyle k\cos \left(\frac{p{\pi}}{2{\tau}}x(t)+\frac{2{\pi}}{3}\right)x^{\prime }\end{eqnarray}$ (17) $\begin{eqnarray}\displaystyle V_{3}(t)\text{} & =\text{} & \displaystyle k\cos \left(\frac{p{\pi}}{2{\tau}}x(t)-\frac{2{\pi}}{3}\right)x^{\prime }.\end{eqnarray}$ (18)

In order to extract maximum power from a reciprocating linear motor, the phase current of the motor must be in phase with its no-load induced phase electromotive force, and this condition can be achieved by PWM control of the current phase [8]. Therefore, the three-phase current expressions of the reciprocating linear motor can be expressed by Eqs (19), (20) and (21), respectively. $\begin{eqnarray}\displaystyle i_{1}(t)\text{} & =\text{} & \displaystyle I\cos \left(\frac{p{\pi}}{2{\tau}}x(t)+{\varphi}_{0}\right)x^{\prime }\end{eqnarray}$ (19) $\begin{eqnarray}\displaystyle i_{2}(t)\text{} & =\text{} & \displaystyle I\cos \left(\frac{p{\pi}}{2{\tau}}x(t)+{\varphi}_{0}+\frac{2{\pi}}{3}\right)x^{\prime }\end{eqnarray}$ (20) $\begin{eqnarray}\displaystyle i_{3}(t)\text{} & =\text{} & \displaystyle I\cos \left(\frac{p{\pi}}{2{\tau}}x(t)+{\varphi}_{0}-\frac{2{\pi}}{3}\right)x^{\prime }\end{eqnarray}$ (21) where, I is a coefficient related to the magnitude of the current, and φ₀ is the initial angle of the current. When the no-load induced potential and the current are in the same phase, φ₀ is equal to 0, and the motor can extract the maximum power under the control condition. When the induced potential of the reciprocating linear motor is in phase with the current, the total electromagnetic power of the motor can be obtained, which is the expression (22). From the expression, the electromagnetic power is proportional to the voltage amplitude coefficient, the current amplitude coefficient, and the square of the speed of the motor mover driven by the wave. $\begin{eqnarray}\displaystyle P_{em}\text{} & =\text{} & \displaystyle V_{1}(t)i_{1}(t)+V_{2}(t)i_{2}(t)+V_{3}(t)i_{3}(t)\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle k\cos \left(\frac{p{\pi}x}{2{\tau}}\right)x^{\prime }I\cos \left(\frac{p{\pi}x}{2{\tau}}\right)x^{\prime }+k\cos \left(\frac{p{\pi}x}{2{\tau}}+\frac{2{\pi}}{3}\right)x^{\prime }I\cos \left(\frac{p{\pi}x}{2{\tau}}+\frac{2{\pi}}{3}\right)x^{\prime }\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle +∼k\cos \left(\frac{p{\pi}x}{2{\tau}}-\frac{2{\pi}}{3}\right)x^{\prime }I\cos \left(\frac{p{\pi}x}{2{\tau}}-\frac{2{\pi}}{3}\right)x^{\prime }\nonumber\\ \displaystyle \text{} & =\text{} & \displaystyle \frac{3}{2}kI(x^{\prime })^{2}.\end{eqnarray}$ (22)

2.3. Simulation

In the previous part of the magnetic circuit structure model and the induced electromotive force, the induced electromotive force of the linear generator that reciprocates with the wave is deduced. This mainly analyzes the correctness of the induced electromotive force of the improved Halbach model with the wave motion. In this part, the correctness of the derivation in this paper is verified by simulating the two magnetic circuit parameters of the improved Halbach array model in commercial finite element analysis software. Due to the symmetry of the magnetic circuit, this paper uses the parameters in Table 1 to establish an analysis model of the number of pole pairs, and the magnetic flux line of the model is shown in Fig. 2, where the x-direction represents the motor axial direction, and the z-direction represents the motor radial direction. It can be found that the self-shielding effect on the left side of the Halbach permanent magnet is better.

Table 1
Parameters of the modeled motor

Symbol (Unit) Parameter Value

r _mi (mm) Back iron height 10

h (mm) Main magnetic pole permanent magnet height 10

h _m (mm) Auxiliary magnetic pole permanent magnet height 9

τ (mm) Pole pitch 21

τ_ma (mm) Auxiliary magnetic pole permanent magnet length 12.5

δ (mm) Air gap length 2

D (mm) Linear motor virtual diameter 26.7

L (mm) Linear motor width 125.7

B _rca (T) Residual flux density of axially magnetized PM 1.14

B _rcr (T) Residual flux density of radially magnetized PM 1.14

μ_rca Relative permeability of axially magnetized PM 1.1

μ_rcr Relative permeability of radially magnetized PM 1.1

N Effective number of turns of the coil 100

Symbol (Unit)	Parameter	Value
r _mi (mm)	Back iron height	10
h (mm)	Main magnetic pole permanent magnet height	10
h _m (mm)	Auxiliary magnetic pole permanent magnet height	9
τ (mm)	Pole pitch	21
τ_ma (mm)	Auxiliary magnetic pole permanent magnet length	12.5
δ (mm)	Air gap length	2
D (mm)	Linear motor virtual diameter	26.7
L (mm)	Linear motor width	125.7
B _rca (T)	Residual flux density of axially magnetized PM	1.14
B _rcr (T)	Residual flux density of radially magnetized PM	1.14
μ_rca	Relative permeability of axially magnetized PM	1.1
μ_rcr	Relative permeability of radially magnetized PM	1.1
N	Effective number of turns of the coil	100

Fig. 2.

The magnetic flux lines of a pair of poles of the motor model.

Firstly, the parameters of the first magnetic circuit structure model used in the simulation are shown in Table 1, and the simulation results of the single-phase induced electromotive force are shown in Fig. 3. Assuming that the monochromatic wave drives the linear motor mover to do reciprocating motion, there is no energy loss, that is, the linear motor mover moves with the wave, and the movement frequency of the mover is the frequency of the wave. The main parameters of the monochromatic wave used in this paper are a height of 0.2 meters and a period of 2 s.

Fig. 3.

Analytical, finite element method no-load induced electromotive force for Magnetic Circuits in NdFeB, and their errors at extreme value.

Figure 3 express the single-phase induced electromotive force of analytical analysis and finite element analysis under the same motor model parameters and monochromatic wave parameters, and clearly show the frequency and time-varying amplitude obtained by these two methods.

The single-phase no-load electromotive force in this figure is obtained in the sinusoidal motion period of 2 s, and the sinusoidal envelope frequency of the electromotive force is equal to the frequency of the reciprocating motion of the direct drive motor, that is, the period is also 2 s. At the same time, it can be clearly observed from the figure that the induced electromotive force obtained by FEM is basically the same as the waveform trend of the electromotive force obtained by analytical calculation, that is, their frequency, phase and amplitude are basically the same.

It can also be seen from Fig. 3 that the error between the FEM electromotive force and the analytically calculated electromotive force is mainly concentrated at the extreme value of the induced electromotive force. The maximum error occurs at 0.0672 s, the error value is 0.2030 V, and the proportion of the error in the voltage amplitude is 1.52%.

Secondly, in order to further verify the analytical calculation, the simulation uses the second type of magnetic circuit structure model parameters, and the simulation results of the single-phase induced electromotive force are shown in Fig. 3. In order to have better comparability, the difference in the structural parameters of the two magnetic circuits is only the difference in the permanent magnetic material of the Halbach array. That is, the permanent magnet materials of axial magnetization and radial magnetization of the motor magnetic circuit in Fig. 3 are all NdFeB permanent magnet materials with residual magnetic density of 1.14T, and the permanent magnet materials magnetized in both directions in Fig. 4 are all ferrites with a residual magnetic density of 0.4T.

Fig. 4.

Analytical, finite element method no-load induced electromotive force for Magnetic Circuits in ferrite, and their errors at extreme value.

It can be observed from Fig. 4 that the error between the FEM electromotive force and the analytically calculated electromotive force is mainly concentrated at the extreme value of the induced electromotive force. The maximum error occurs at 0.0672 s, the error value is 0.0569 V, and the proportion of the error in the voltage amplitude is 1.18%.

Finally, based on the finite element simulation analysis of the previous two magnetic circuit structural parameters, the simulation results of the single-phase induced electromotive force have a good match with the analytical calculation results, and the envelope period of the electromotive force is consistent with the monochromatic wave period, so the simulation results show that the analytical calculation deduced in this paper is correct and can be used for subsequent research and analysis.

3. Results

In this part, two aspects are discussed on the influence of induced electromotive force, namely, the magnetic circuit topology parameters and the mover motion parameters.

First, the influence of the parameter changes of the improved Halbach topology on the no-load electromotive force of wave conversion is analyzed, and four variables τ, 𝛼, 𝛾 and 𝜅 are set. where, 𝛼 is the ratio of the radially magnetized permanent magnet width to a pole pitch in the magnetic circuit structure, and the expression is (23); 𝛾 is the proportional relationship between the number of motor poles and the pole pitch, and the expression is (24); and 𝜅 is the ratio of the height of the axially magnetized permanent magnet to the radially magnetized permanent magnet in the magnetic circuit structure, and the expression is (25). $\begin{eqnarray}\displaystyle {\alpha}\text{} & =\text{} & \displaystyle \frac{{\tau}-{\tau}_{ma}}{{\tau}}\end{eqnarray}$ (23) $\begin{eqnarray}\displaystyle {\gamma}\text{} & =\text{} & \displaystyle \frac{p}{{\tau}}\end{eqnarray}$ (24) $\begin{eqnarray}\displaystyle {\kappa}\text{} & =\text{} & \displaystyle \frac{h_{m}}{h}.\end{eqnarray}$ (25)

Figure 5 shows the single-phase induced electromotive force with different proportions 𝛼 of the width of the radially magnetized permanent magnet in a magnetic pole. Without changing the τ value, the amplitude of the sinusoidal envelope of the single-phase induced electromotive force has a downward trend with the increase of the radially magnetized magnet width ratio 𝛼. But the single-phase induced electromotive force waveform does not change the period and the number of extreme points with the increase of 𝛼.

Fig. 5.

Half-cycle single-phase induced electromotive force with different proportions 𝛼 of radially magnetized permanent magnet width in the magnetic pole.

Figure 6 depicts the single-phase induced electromotive force of the ratio 𝜅 of the radial magnetization to the axial magnetization width. From the waveform in Fig. 6, it can be found that the period, the number of extreme points, and the change trend of the potential waveform with the 𝜅 value are consistent with Fig. 5, that is, the parameters 𝜅 and 𝛼 have the same influence on the single-phase induced electromotive force. It can also be observed from Fig. 6 that when the array structure parameter 𝜅 decreases from 1 to about 0.7, the single-phase potential amplitude of the direct-drive motor changes little, and when 𝜅 drops over 0.7, the single-phase potential amplitude of the direct-drive motor changes greatly. This feature can be used to optimize the amount of permanent magnets used, while the single-phase potential amplitude changes little. That is, the usage amount of the permanent magnets is reduced due to the unequal heights in the axial direction and the radial direction, thereby reducing the usage cost of the permanent magnets.

Fig. 6.

Half-cycle single-phase induced electromotive force with different values of 𝜅.

Figure 7 shows that, in the case of fixed τ, the amplitude of the sinusoidal envelope of the single-phase induced electromotive force has a tendency to increase with the increase of 𝛾, the polarity change of the induced electromotive force in half a wave cycle is accelerated, and the number of the extreme points increases.

Fig. 7.

Half-cycle single-phase induced electromotive force with different values of 𝛾.

Figure 8 shows that the amplitude of the sinusoidal envelope of the single-phase induced electromotive force has a tendency to increase with the increase of τ, the polarity change of the induced electromotive force within half a wave cycle is weakened, and the number of extreme points is reduced.

Fig. 8.

Half-cycle single-phase induced electromotive force with different values of τ.

In addition, the influence of the reciprocating motion parameters of the mover on the no-load electromotive force is also analyzed, and two variables are set, namely the period T and the input drive signal x superimposed by the three components.

As the external wave frequency that drives the motor movement changes, that is, the wave period changes, the induced potential of the motor changes as shown in Fig. 9. It can be seen from the figure that as the wave period increases, the amplitude of the sinusoidal envelope of the single-phase induced electromotive force decreases, and the period of the potential waveform increases. And the period of the potential waveform is consistent with the wave period that drives the mover to move.

Fig. 9.

Single-phase induced electromotive force for wave period T.

Fig. 10.

Single-phase induced electromotive force of drive signal x superimposed by three components.

Fig. 11.

Single-phase induced electromotive force of drive signal x combined by three-section wave.

The driving signal x of the complex wave moving in the vertical direction can be decomposed into the superposition of the fundamental wave and multiple harmonic motions through Fourier decomposition. Assuming that the motion in the vertical direction is decomposed into the red waveform in Fig. 10, the waveform expression is (26). Driven by this red motion waveform, the single-phase induced electromotive force of the motor is shown as the black waveform in Fig. 10. In the figure, it can be observed that when the motion waveform is at the extreme value, the obtained single-phase potential is 0, mainly because the velocity is 0 at the extreme value of the displacement. The greater the difference between the two extremes, the greater the amplitude of the sinusoidal envelope of the single-phase induced potential. The period of the potential is equal to the period of the fundamental wave of the drive signal x, and the maximum value of the potential also occurs at the derivative of the fundamental wave of the signal x. $\begin{eqnarray}\displaystyle x=H\sin ({\omega}t)+0.8H\sin (2{\omega}t)+0.6H\sin (3{\omega}t). & & \displaystyle\end{eqnarray}$ (26)

On the other hand, a three-section combined wave on the basis of the three-component narrowband wave (26) is added as an input signal, and drive the motor. The input waveform is as in formula (27), and the output voltage waveform is as in Fig. 11. It can be observed from Fig. 11 that when the monochromatic wave is input for 0–2 s, the single-phase voltage waveform envelope is sinusoidal, and the envelope period is consistent with the input wave period. The two intervals of 2–4 s and 4–6 s are input waves for multiple component combinations. The single-phase potential (10) obtained by the input wave (27) driving the motor is consistent with the characteristics of the single-phase potential (9) of the input wave (26). $\begin{eqnarray}\displaystyle x=\left\{\begin{array}{@{}ll@{}}0.6H\sin ({\omega}t) & 0\leq t<2\\ 0.3H\sin (2{\omega}t)+0.2H\sin (3{\omega}t) & 2\leq t<4\\ 0.25H\sin (2{\omega}t)+0.15H\sin (4{\omega}t)+0.1H\sin (6{\omega}t) & 4\leq t\leq 6.\end{array}\right. & & \displaystyle\end{eqnarray}$ (27)

4. Conclusion

This paper introduces the magnetic circuit modeling of an improved Halbach permanent magnet linear motor for ocean wave energy extraction, and deduces the no-load electromotive force formula generated in the energy conversion under sine wave direct drive. This can be used to optimize the Halbach structure topology and analyze the effect on no-load EMF. The Halbach permanent magnet structure uses high-coercivity NdFeB permanent magnets as the main and auxiliary magnetic poles of unequal heights, which reduces the usage of the permanent magnets without significantly reducing the air-gap magnetic flux density. Firstly, the magnetic circuit model of the direct-drive linear motor for wave energy extraction is established, and then the electromotive force generated by the monochromatic wave drive is analyzed, and the finite element method is used to verify it. Finally, the influence of the magnetic circuit topology and mover motion on the single-phase no-load induced EMF is analyzed when the direct-drive motor reciprocates with the waves. However, since the conversion of wave energy into motor no-load EMF and power is highly dependent on the improved Halbach topology of the direct-drive linear motor, additional studies using the analytical methods presented in this paper are required.

Footnotes

Acknowledgements

This work was funded by a project that partially funded by National Science Foundation of China (51675265), the Advantage Discipline Construction Project Funding of University in Jiangsu Province (PAPD), and the Independent Research Funding of the State Key Laboratory of Mechanics and Control of Mechanical Structures (0515K01). The authors gratefully acknowledge this support.

References

Colli

V.D.

Cancelliere

Marignetti

Di Stefano

and Scarano

, A tubular-generator drive for wave energy conversion, IEEE Transactions on Industrial Electronics 53 (2006), 1152–1159.

Feng

Zhao

Zhang

and Hou

, Magnetic field-modulated linear permanent-magnet generator for direct-drive wave energy conversion, IET Electric Power Applications 14 (2020), 742–750.

Y.J.

Park

J.S.

Hyon

B.J.

and Lee

, Novel control strategy of wave energy converter using linear permanent magnet synchronous generator, IEEE Transactions on Applied Superconductivity 28 (2018), 5204705.

Bodrov

Shuttleworth

and Iacchetti

M.F.

, Modulation and feed-forward based damping matching strategy for linear generators, IEEE Transactions on Industry Applications 57 (2021), 5296–5305.

Mendonca

and Martinez

, A resistance emulation approach to optimize the wave energy harvesting for a direct drive point absorber, IEEE Transactions on Sustainable Energy 7 (2016), 3–11.

Zhang

Bodrov

Shuttleworth

and FIacchetti

, Current-modulation-based on-line resonance tuning strategy for linear generator drives, IEEE Transactions on Industrial Electronics 68 (2021), 2812–2822.

Huang

and Xiao

, High efficiency power extraction control method for direct-drive linear wave energy converters based on forced resonance, IET Electric Power Applications 14 (2020), 2624–2630.

Brooking

and Mueller

M.A.

, Power conditioning of the output from a linear vernier hybrid permanent magnet generator for use in direct drive wave energy converters, IET Proceedings – Generation Transmission and Distribution 152 (2005), 673–681.

Zhang

Qin

Peng

and Huang

, Modeling, control strategy, and power conditioning for direct-drive wave energy conversion to operate with power grid, Proceedings of the IEEE (2013), 925–941.

10.

Ching

T.W.

Chau

K.T.

and Li

, Power factor improvement of a linear vernier permanent-magnet machine using auxiliary DC field excitation, IEEE Transactions on Magnetics 52 (2016), 8204804.

11.

Shek

J.K.

Macpherson

D.E.

and Mueller

M.A.

, Reaction force control of linear electrical generators for direct drive wave energy conversion, IET Renewable Power Generation 1 (2006), 17–24.

12.

Fusco

and Ringwood

J.V.

, A simple and effective real-time controller for wave energy converters, IEEE Transactions on Sustainable Energy 4 (2013), 21–30.

13.

Dai

Cao

Long

Liang

and Wang

, Analytical modeling of an eddy-current adjustable-speed coupling system with a three-segment Halbach magnet array, Electric Power Components and Systems 43 (2015), 1891–1901.

14.

Xia

Guo

and Wang

, Modeling and analyzing of magnetic field of segmented Halbach array permanent magnet machine considering gap between segments, IEEE Transactions on Magnetics 50 (2014), 8106009.

15.

Guo

Wang

Shi

and Xia

, Modeling and analyzing of magnetic field in unequal segmented Halbach array permanent magnet machine, International Journal of Applied Electromagnetics and Mechanics 65 (2021), 333–353.

16.

Zhang

and Li

, Analytical model of magnetic field of a permanent magnet synchronous motor with a trapezoidal Halbach permanent magnet array, IEEE Transactions on Magnetics 55 (2019), 8105205.

17.

Duan

Xiao

Zhang

and Jing

, A novel 3-D analytical modeling method of trapezoidal shape permanent magnet Halbach array for multi-objective optimization, Journal of Electrical Engineering and Technology 14 (2019), 635–643.

18.

Isfahani

A.H.

Vaez-Zadeh

and Rahman

M.A.

, Performance improvement of permanent magnet machines by modular poles, IET Electric Power Applications 3 (2009), 343–351.

19.

Isfahani

A.H.

Vaez-Zadeh

and Rahman

M.A.

, Using modular poles for shape optimization of flux density distribution in permanent-magnet machines, IEEE Transactions on Magnetics 44 (2008), 2009–2015.

20.

Huang

Jing

and Li

, Analysis and application of discrete Halbach magnet array with unequal arc lengths and unequally changed magnetization directions, IEEE Transactions on Applied Superconductivity 28 (2018), 5201305.

21.

Jing

Lei

and Liu

, Design and experimental analysis of AC linear generator with Halbach PM arrays for direct-drive wave energy conversion, IEEE Transactions on Applied Superconductivity 24 (2014), 0502704.

22.

Zhang

Huang

and Xia

, Research on a PM slotless linear generator based on magnet field analysis model for wave energy conversion, IEEE Transactions on Magnetics 53 (2017), 8208404.

23.

Eigebaly

A.E.

and El-Nemr

M.K.

, Optimal design of slotless PM halbach array linear generator for wave energy converters at maximum power transfer condition, in: 2018 Twentieth International Middle East Power Systems Conference, MEPCON, 2018, pp. 271–276.

24.

Jin

Yuan

and Lin

, General analytical method for magnetic field analysis of Halbach magnet arrays based on magnetic scalar potential, Journal of Magnetics 18 (2013), 95–104.

25.

Huang

Zhou

Dong

Lin

Yang

and Cheng

, Magnetic equivalent circuit modeling of yokeless axial flux permanent magnet machine with segmented armature, IEEE Transactions on Magnetics 50 (2014), 8104204.

26.

Krishnan

, Permanent Magnet Synchronous and Brushless DC Motor Drives, CRC Press/Taylor & Francis, 2010.