This application is a continuation of international application no. PCT/CN2024/082548 filed March 20, 2024, which claims priority to Chinese patent application NO. 202310832430.9, filed July 07, 2023, and the entire disclosures of the above-identified applications is incorporated herein by reference.

This application relates to the technical field of vibrators, specifically to a moving-magnet vibrator with nonlinear term cancellation.

For the designed vibrators in bone conduction headphones and/or haptic feedback actuators, the moving-magnet design has many advantages. For example, the coil has good heat dissipation, and the movable assembly acting as a load is not heated; the coil adopts a hollow shaft with the magnet provided inside, enabling a compact overall structure. In addition, since the coil is stationary, it avoids the vulnerability of coil connection wires to damage. Moreover, the moving-magnet design enables a high peak force and a high ratio of peak force to moving mass, thereby achieving a high acceleration G-value.

The existing designed moving-magnet vibrator often suffer from relatively high nonlinear terms, due to certain deficiencies in the combination of magnets and coils. That is, the force or acceleration value of the movable assembly have relatively high distortion in a low-frequency band or a high-frequency band, which is also referred to as the total harmonic distortion (THD). FIG. 22 is a THD test chart of the existing moving-magnet vibrator. It can be seen that the distortion reaches 99% around 25 Hz, and the distortion reaches 46% around 100 Hz. Such a large distortion indicates that, near low frequencies, the distortion of the audio signal or haptic feedback signal causes the perceived sound quality or haptic feedback to deviate significantly from the actual case. Generally, when the distortion is greater than 10%, it is unacceptable according to audio standards.

One objective of the disclosure is to provide a moving-magnet vibrator with nonlinear term cancellation.

The moving-magnet vibrator with nonlinear term cancellation includes moving-magnet vibrator body. The moving-magnet vibrator body includes an outer cylinder, a vibration transmission plate, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure, and the movable assembly includes a magnet combination structure. The stator assembly is fixed inside the outer cylinder, the vibration transmission plate is fixed on the outer cylinder, and the movable assembly is fixedly connected to the vibration transmission plate through at least one contact point. The movable assembly moves while the stator assembly remains stationary, and the movable assembly is referred to as a moving component.

The movable assembly is configured to be simultaneously subjected to paired electromagnetic forces of push and pull, thereby presenting push-pull structural characterizes.

The disclosure is further explained with reference to the accompanying drawings and embodiments.

• FIG. 1 is a diagram illustrating a cross-sectional view of Embodiments 1 and 2 of the disclosure.
• FIG. 2 is a diagram illustrating closed magnetic flux curves of a coil and a permanent magnet in Embodiments 1 and 2 of the disclosure.
• FIG. 3 is a magnetic domain analysis diagram of Embodiments 1 and 2 of the disclosure.
• FIG. 4 is a diagram illustrating a relationship between magnetic domains and a stator assembly in Embodiments 1 and 2 of the disclosure.
• FIG. 5 is a force analysis diagram of magnetic domains and a movable assembly in Embodiments 1 and 2 of the disclosure.
• FIG. 6 is a force analysis diagram of the movable assembly in Embodiments 1 and 2 of the disclosure.
• FIG. 7 is a diagram illustrating a cross-sectional view of Embodiments 3 and 4 of the disclosure.
• FIG. 8 is a diagram illustrating closed magnetic flux curves of a coil and a permanent magnet in Embodiments 3 and 4 of the disclosure.
• FIG. 9 is a magnetic domain analysis diagram of Embodiments 3 and 4 of the disclosure.
• FIG. 10 is a force analysis diagram of magnetic domains, a movable assembly, and a stator assembly in Embodiments 3 and 4 of the disclosure.
• FIG. 11 is a force analysis diagram of the magnetic domains and the movable assembly in Embodiments 3 and 4 of the disclosure.
• FIG. 12 is a diagram illustrating a cross-sectional view of Embodiments 5 and 6 of the disclosure.
• FIG. 13 is a diagram illustrating closed magnetic flux curves of a coil and a permanent magnet in Embodiments 5 and 6 of the disclosure.
• FIG. 14 is a magnetic domain analysis diagram of Embodiments 5 and 6 of the disclosure.
• FIG. 15 is a force analysis diagram of magnetic domains, a stator assembly, and a movable assembly in Embodiments 5 and 6 of the disclosure.
• FIG. 16 is a force analysis diagram of the magnetic domains and the movable assembly in Embodiments 5 and 6 of the disclosure.
• FIG. 17 is a diagram illustrating a cross-sectional view of Embodiments 7 and 8 of the disclosure.
• FIG. 18 is a diagram illustrating closed magnetic flux curves of a coil and a permanent magnet in Embodiments 7 and 8 of the disclosure.
• FIG. 19 is a magnetic domain analysis diagram of Embodiments 7 and 8 of the disclosure.
• FIG. 20 is a force analysis diagram of magnetic domains, a stator assembly, and a movable assembly in Embodiments 7 and 8 of the disclosure.
• FIG. 21 is a force analysis diagram of the movable assembly in Embodiments 7 and 8 of the disclosure.
• FIG. 22 is a diagram illustrating THD test of a moving-magnet vibrator in the related art.
• FIG. 23 is a diagram illustrating THD test of the moving-magnet vibrator in Embodiments 1 and 2 of the disclosure.
• FIGS. 24 to 40a are schematic diagrams of the magnet component in the disclosure.
• FIGS. 41 to 53 are schematic diagrams of the coil component in the disclosure.
• FIGS. 54 to 59 are schematic diagrams of the magnetic domains in the disclosure.

The disclosure will be described in detail below, and the technical solutions in the embodiments of the disclosure will be clearly and completely described. Apparently, the described embodiments are only a part of the embodiments of the disclosure, not all of them. Based on the embodiments of the disclosure, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the disclosure.

For the design of nonlinear term cancellation, there are 2N magnetic domains within a vibrator, in which the magnetic domains are in pairs, defined as magnetic domains D_1,i and D_2,i, where i=1, 2, 3, ..., N. Closed main magnetic flux lines of a coil and closed main magnetic flux lines of a permanent magnet each pass through the magnetic action domains D_1,i and D_2,i· In the magnetic domain D_1,i, a direction of the magnetic flux lines of the coil is the same as a direction of the magnetic flux lines of the permanent magnet, and in the magnetic domain D_2,i, a direction of the magnetic flux lines of the coil is opposite to a direction of the magnetic flux lines of the permanent magnet. Alternatively, in the magnetic domain D_1,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet; and in the magnetic domain D_2,i, the direction of the magnetic flux line of the coil is the same as the direction of the magnetic flux lines of the permanent magnet.

When the direction of the magnetic flux lines of the coil passing through a certain magnetic domain is the same as the direction of the magnetic flux lines of the permanent magnet passing through the magnetic domain, a total magnetic flux is equal to a sum of a magnetic flux generated by the coil and a magnetic flux generated by the permanent magnet. When the direction of the magnetic flux lines of the coil passing through a certain magnetic domain is opposite to the direction of the magnetic flux lines of the permanent magnet passing through the magnetic domain, the total magnetic flux is equal to a difference between the magnetic flux generated by the coil and the magnetic flux generated by the permanent magnet.

There is at least one magnetic action domain in the vibrator. The magnetic action domain refers to a spatial region where there is a certain electromagnetic field or multiple electromagnetic fields, which causes mutual forces between various components surrounding the magnetic action domain. Such region is defined as the magnetic action domain, or the magnetic field for short.

The magnetic domain is a spatial region filled with electromagnetic energy, is generally composed of air or a medium with a low magnetic permeability (e.g., a relative magnetic permeability < 1000), and includes a region where a magnet material is located. The moving-coil vibrator with nonlinear term cancellation of the disclosure includes at least one magnetic action domain. The magnetic action domain refers to a spatial region where a certain electromagnetic field or multiple electromagnetic fields exist, which causes mutual forces between various components surrounding the magnetic action domain. Such region is defined as the magnetic action domain, referred to as the magnetic domain for short. The magnetic domain is a spatial region where magnetic forces interact with each other, such as a spatial region between permanent magnets (generating attractive or repulsive interactions), or a spatial region enclosed by a permanent magnet and a magnetic conductor (generating attractive interactions), or a spatial region enclosed by magnetic conductors (yoke iron) magnetized by a permanent magnet, or a spatial region where magnetic force interactions occurs in a permanent magnet (a magnetic permeability of a hard magnetic material forming the permanent magnet is close to the magnetic permeability of air).

The magnetic domains have various types as follows.

1. 1) A space between permanent magnets is filled with medium (air, with a relative magnetic permeability slightly greater than 1).

The above medium may be replaced with paramagnetic materials (paramagnetic substances), diamagnetic materials (diamagnetic substances), or ferromagnetic materials (ferromagnetic substances) with the relative magnetic permeability less than 1000. Examples are given as follows.

1. a. Paramagnetic substances: The relative magnetic permeability of the paramagnetic substances is slightly greater than 1. Examples include air, oxygen, tin, aluminum, and lead. When the paramagnetic substances are placed in a magnetic field, the magnetic flux density B increases slightly.
2. b. Diamagnetic substances: The relative magnetic permeability of the diamagnetic substances is slightly less than 1. Examples include hydrogen, copper, graphite, silver, and zinc. Diamagnetic substances are also known as antimagnetic substances. When the diamagnetic substances are placed in a magnetic field, the magnetic induction intensity B decreases slightly.
3. c. Ferromagnetic substances: The relative magnetic permeability of the ferromagnetic substances is much greater than 1 but less than 1000. Examples include iron, steel, cast iron, nickel, cobalt and other substances are ferromagnetic substances. The ferromagnetic substances with the relative magnetic permeability less than 1000 include, for example, cobalt, unannealed cast iron, annealed cast iron, etc. For example, magnetorheological fluid has a relative magnetic permeability below 10.

As illustrated in FIG. 54, there are permanent magnet 1 and permanent magnet 2 surrounded by air. The permanent magnets attract each other.

Magnetic domain D1: The spatial region enclosed by the air medium between permanent magnet 1 and permanent magnet 2.

Magnetic domain D2: The spatial region enclosed by part of permanent magnet 2 and the air medium around part of permanent magnet 2.

Magnetic domain D3: The spatial region enclosed by the entire permanent magnet 1 and the air medium near permanent magnet 1.

Magnetic domain D4: The spatial region enclosed by the entire permanent magnet 1, the entire permanent magnet 2, and the air medium near permanent magnet 1 and permanent magnet 2.

Magnetic domain D5: The spatial region enclosed by the air medium on the side of permanent magnet 2 away from permanent magnet 1.

Magnetic domain D6: The spatial region enclosed by the permanent magnet material medium surrounding part of permanent magnet 1.

As illustrated in FIG. 55, there are permanent magnet 1 and permanent magnet 2 surrounded by air. The permanent magnets attract each other. D1-D6 may be defined in a similar way.

2) A space between a permanent magnet and a magnetic conductor is filled with a medium (air, with a relative magnetic permeability close to 1).

3) A space between magnetic conductors is filled with a medium (air, with a relative magnetic permeability close to 1), as illustrated in FIGS. 56-57.

Magnetic domain D1: The spatial region enclosed by the air medium between magnetic conductor 1 and magnetic conductor 2.

Magnetic domain D2: The spatial region enclosed by part of the permanent magnet, part of magnetic conductor 2, and the surrounding air medium.

Magnetic domain D3: The spatial region enclosed by the entire magnetic conductor 1, part of the permanent magnet, and the air medium near magnetic conductor 1.

Magnetic domain D4: The spatial region enclosed by the entire magnetic conductor 1, the entire magnetic conductor 2, the entire permanent magnet, and the air medium near them.

Magnetic domain D5: The spatial region enclosed by the air medium on the side of magnetic conductor 2 away from magnetic conductor 1.

Magnetic domain D6: The spatial region enclosed by the permanent magnet material medium surrounding part of the permanent magnet.

4) A space between magnetic conductors is filled with a medium (magnetorheological fluid, with a relative magnetic permeability between 5 and 9), as illustrated in FIG. 58.

5) The spatial region inside a permanent magnet is filled with a medium (permanent magnet material, with a relative magnetic permeability<1000).

As illustrated in FIG. 59, magnetic domain D6 in the previous examples contains a permanent magnet material as the medium. For example, the magnetic permeability of sintered ferrite, samarium-cobalt and neodymium-iron-boron is approximately 1.05, the magnetic permeability of bonded ferrite is approximately 1.05, and the magnetic permeability of bonded neodymium magnets approximately ranges from 1.1 to 1.7.

There are two types of magnetic action domains: the first type is the magnetic action domain enclosed inside the movable assembly or the stator assembly, and the second type is the magnetic action domain enclosed between the movable assembly and the stator assembly. We are more interested in the second type of magnetic action domain. Therefore, through the analysis of the second type of magnetic action domain, a force analysis of the movable assembly may be obtained, thereby obtaining a resultant force of the movable assembly of the vibrator system, and further obtaining the vibration equation.

Referring to FIGS. 1 to 5, a design method of a moving-magnet vibrator with nonlinear term cancellation includes conditions as follows.

1. (1) A moving-magnet vibrator body 11 is provided. The moving-magnet vibrator body 11 includes an outer cylinder 1, a vibration transmission plate 7, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure, the movable assembly includes a magnet combination structure, the coil combination structure includes a coil 3 and a first magnetic conductor 4, and the magnet combination structure includes a permanent magnet 6 and a second magnetic conductor 5. The stator assembly is fixed inside the outer cylinder 1, the vibration transmission plate 7 is fixed on the outer cylinder 1, and the movable assembly is fixedly connected to the vibration transmission plate 7 through at least one contact point. The movable assembly moves while the stator assembly remains stationary, and the movable assembly is referred to as a moving component.
2. (2) The movable assembly is simultaneously subjected to paired electromagnetic forces of push and pull, presenting push-pull structural characteristics.

2N magnetic domains D_1,i and D_2,i which are designed as N symmetrical pairs, are provided in the moving-magnet vibrator body 11, where N is 1, 2, 3, ..., 100, and i=1, 2, 3, ....

The number of a permanent magnet in the magnet combination structure and the number of a coil in the coil combination structure are limited, in such a manner that N_magnet>N_coil or N_magnet < N_coil , where N_magnet represents the number of the permanent magnet, and N_coil represents the number of the coil, N_magnet=1, 2, 3, ..., 100, and N_coil=1, 2, 3, ..., 100.

In the push-pull structural characteristics, linear terms of the electromagnetic force acting on the movable assembly are superimposed so as to be increased, and nonlinear terms of the electromagnetic force acting on the movable assembly are partially or completely canceled so as to be decreased.

The closed main magnetic flux lines of the coil in the coil combination structure and the closed main magnetic flux lines of the permanent magnet in the magnet combination structure each pass through the magnetic domains D_1,i and D_2,i. The magnetic domain is a spatial region filled with electromagnetic energy, is generally composed of air or a medium with low magnetic permeability (e.g., a relative magnetic permeability<1000), and includes a region where the magnet material is located. In the magnetic domain D_1,i, a direction of magnetic flux lines of the coil is the same as a direction of magnetic flux lines of the permanent magnet, while in the magnetic domain D_2,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet. Alternatively, in the magnetic domain D_1,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet, while in the magnetic domain D_2,i the direction of the magnetic flux lines of the coil is the same as the direction of the magnetic flux lines of the permanent magnet.

Specifically, the vibration transmission plate 7 may be a rectangular, circular, racetrack-shaped or three-dimensional structures according to different application scenarios, and may be used in various combinations according to different application scenarios. The vibration transmission plate 7 is usually fixed on a top surface, bottom surface or middle of the outer cylinder 1.

The stator assembly is fixed inside the outer cylinder 1, and may be arranged on an inner side wall, top surface or bottom surface of the outer cylinder 1.

The movable assembly is fixedly connected to the vibration transmission plate 7 through at least one contact point, in which the contact point includes point contact and surface contact. There may be one contact point, two contact points or multiple contact points.

The number of the permanent magnet 6 and the number of the coil 3 are limited, in which there are one permanent magnet, and two coils.

Two symmetrically paired magnetic domains D_1,1 and D_2,1 are provided in the moving-magnet vibrator body 11. The closed main magnetic flux lines of the coil 3 and the closed main magnetic flux lines of the permanent magnet 6 each pass through the magnetic domains D_1,1 and D_2,1. In the magnetic domain D_1,1, the direction of the magnetic flux lines of the coil 3 is the same as the direction of the magnetic flux lines of the permanent magnet 6, while in the magnetic domain D_2,1, the direction of the magnetic flux lines of the coil 3 is opposite to the direction of the magnetic flux lines of the permanent magnet 6.

The moving component is subjected to 2 forces. Each component force includes two parts: one part is a linear term of an excitation current i, and the other part is a nonlinear term of the excitation current i: $${\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n}}\left(\mathrm{i}\right)={\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},\mathrm{linear}}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},\mathrm{nonlinear}}\left(\mathrm{i}\right),$$ where n = 1, 2, 3, ..., 2N - 1, 2N.

Accordingly, a resultant force on the moving component also includes two parts: one part is a linear term of the current i, and the other part is a nonlinear term of the current i, which is expressed as: $$\begin{array}{l}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force}}\left(\mathrm{i}\right)\\ ={\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{linear}}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{nonlinear}}\left(\mathrm{i}\right)\end{array}$$ where: $$\begin{array}{l}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force}}\left(\mathrm{i}\right)\\ ={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n}}\left(\mathrm{i}\right)}={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},2}\left(\mathrm{i}\right)\right)}\end{array}$$ $$\begin{array}{l}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{linear}}\left(\mathrm{i}\right)={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},\mathrm{linear}}\left(\mathrm{i}\right)}\\ ={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{linear},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{linear},2}\left(\mathrm{i}\right)\right)}\end{array}$$ $$\begin{array}{l}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{nonlinear}}\left(\mathrm{i}\right)={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},\mathrm{nonlinear}}\left(\mathrm{i}\right)}\\ ={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{nonlinear},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{nonlinear},2}\left(\mathrm{i}\right)\right)}\end{array}$$

That is, the nonlinear terms in each component force are partially or completely canceled. In the final total resultant force ∑_i(F_1,i + F_2,i), the nonlinear terms of the total resultant force relative to the current are partially or completely canceled, and the linear terms are superimposed so as to be increased, thereby obtaining the moving-magnet vibrator with nonlinear term cancellation.

The movable assembly and the stator assembly are in concave-convex shapes and arranged in an interleaved engagement manner, and the closed main magnetic flux lines of the coil 3 and the closed main magnetic flux lines of the permanent magnet 6 alternately pass through the movable assembly and the stator assembly.

The design method further includes conditions as follows.

(3.1) Looking outward from a center, the permanent magnet 6 is located inside and the coil 3 is located outside.

(3.2) N_coil=2, n=1.

(3.3) When N_coil>1, the directions of the currents in the adjacent coils 3 are opposite; and for two adjacent coils 3, the polarities of the electromagnetic fields at the adjacent end faces are the same.

The magnetic conductor is arranged at a position of the outer cylinder 1 close to the coil 3, so as to minimize the magnetic resistance of the magnetic circuit of an electromagnet generated by the coil 3. The permanent magnet 6 in the magnet assembly is isolated by the magnetic conductor. A yoke iron is arranged around the coils 3 and the permanent magnet 6, or alternatively, for the coil combination structure, a part of the outer cylinder 1 close to the coil 3 is magnetically conductive.

Referring to FIGS. 1 to 5, a moving-magnet vibrator device with nonlinear term cancellation designed by the method of Embodiment 1 includes a moving-magnet vibrator body 11. The moving-magnet vibrator body 11 includes an outer cylinder 1, a vibration transmission plate 7, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure. The movable assembly includes a magnet combination structure. The coil combination structure includes a coil 3 and a first magnetic conductor 4. The magnet combination structure includes a permanent magnet 6 and a second magnetic conductor 5. The outer cylinder 1 may be magnetic or non-magnetic. Preferably, the outer cylinder is a magnetic conductive to reduce magnetic resistance. A cross-section of the outer cylinder may be circular, square, special-shaped, etc., and may be continuous or discontinuous, such as columnar connection or grid-like discontinuity.

The coil combination structure further includes a first magnetic conductive ring 2. Viewed from a center outward, the coil 3 is located outside and the permanent magnet 6 is located inside. There is one permanent magnet 6, and two coils 3. Directions of the currents in the adjacent coils 3 are opposite, and polarities of electromagnetic fields at adjacent end faces of the two adjacent coils 3 are the same. Two vibration transmission plates 7 are provided, and the two vibration transmission plates 7 are respectively fixed on a top surface and a bottom surface of the outer cylinder 1. The permanent magnet 6 is fixed in the second magnetic conductor 5. Two ends of the second magnetic conductor 5 are respectively fixed on the vibration transmission plates 7. The first magnetic conductor 4 is fixed in the middle of an inner side wall of the outer cylinder 1. The two coils 3 are respectively fixed on two sides of the first magnetic conductor 4. A first magnetic conductive ring 2 is fixedly arranged on an outer side of the coils 3. The coils 3 and the first magnetic conductive ring 2 are fixed on the inner side wall of the outer cylinder 1. The movable assembly and the stator assembly are in concave-convex shapes and arranged in an interleaved engagement manner (i.e., in a concave-convex interleaved engagement shape). Closed main magnetic flux lines of the coil 3 and closed main magnetic flux lines of the permanent magnet 6 alternately pass through the movable assembly and the stator assembly. Two magnetic domains D_1,1 and D_2,1 designed as a symmetrical pair are provided in the moving-magnet vibrator body. The closed main magnetic flux lines of the coil 3 and the closed main magnetic flux lines of the permanent magnet 6 each pass through the magnetic domains D_1,1 and D_2,1. In the magnetic domain D_1,1, the direction of the magnetic flux lines of the coil 3 is the same as the direction of the magnetic flux lines of the permanent magnet 6, and in the magnetic domain D_2,1, the direction of the magnetic flux lines of the coil 3 is opposite to the direction of the magnetic flux lines of the permanent magnet 6.

To further describe the design method of the moving-magnet vibrator with nonlinear term cancellation, referring to FIGS. 2 and 3, an air gap 1 constitutes a magnetic action domain D_1,1, and an air gap 2 constitutes a magnetic action domain D_2,1. In the magnetic action domain, a total magnetic flux/magnetic induction intensity generated by the superposition of the magnetic field generated by the permanent magnet 6 and the magnetic field generated by the electromagnet of the coil 3 causes mutual forces between the components around the magnetic domain.

The current passing through the coil C₁ and the current passing through the coil C₂ are represented as i, but the direction of the current in the coil C₁ is opposite to the direction of the current in the coil C₂. Assuming that the magnetic flux corresponding to the coil C₁ is Φ_i1, the magnetic flux corresponding to the coil C₂ is Φ_i2, and the magnetic flux corresponding to the permanent magnet is Φ_m. In the magnetic domain D₁ (magnetic action domain D₁), the direction of the magnetic flux lines corresponding to the coil C₁ is the same as the direction of the magnetic flux lines corresponding to the permanent magnet, so the total magnetic flux in the magnetic domain D₁ is a sum of Φ_i1 and Φ_m. In the magnetic domain D₂ (magnetic action domain D₂), the direction of the magnetic flux lines corresponding to the coil C₂ is opposite to the direction of the magnetic flux lines corresponding to the permanent magnet, so the total magnetic flux in the magnetic domain D₂ is a difference between Φ_i2 and Φ_m. Assuming that the direction of the magnetic flux lines of the permanent magnet 6 is a positive direction in each magnetic domain, we have: $${\mathrm{\Phi }}_{\mathrm{D}1}={\mathrm{\Phi }}_{\mathrm{m}}+{\mathrm{\Phi }}_{\mathrm{i}1}$$ and $${\mathrm{\Phi }}_{\mathrm{D}2}={\mathrm{\Phi }}_{\mathrm{m}}-{\mathrm{\Phi }}_{\mathrm{i}2}.$$

Assuming that the magnetic resistances of the magnetic circuits formed by the electromagnetic fields generated by the currents in the coil 1 and the coil 2 are Z_i1 and Z_i2 respectively, where N is the number of turns of the coil, and i is the current intensity, we have: $${\varnothing }_{\mathrm{i}1}\ast {\mathrm{Z}}_{\mathrm{i}1}=\mathrm{N}\ast \mathrm{i};$$ and $${\varnothing }_{\mathrm{i}2}\ast {\mathrm{Z}}_{\mathrm{i}2}=\mathrm{N}\ast \mathrm{i}.$$

Since the magnetic circuit structures of the coils C₁ and C₂ are symmetrically designed, Z_i1 = Z_i2 = Z_i , so Ø_i1 = Ø_i2 = Ø_i ; in addition, assuming that the magnetic conductance of the magnetic circuit formed by the electromagnetic field generated by the current is G₁, we have: $${\varnothing }_{\mathrm{i}}={\mathrm{G}}_{\mathrm{i}}\ast \mathrm{N}\ast \mathrm{i}.$$

The magnetic flux corresponding to the permanent magnet 6 may also be represented by the equation of magnetic induction intensity. Assuming that the magnetic induction intensity at the magnetic pole end face of the permanent magnet is B_m and the area of the magnetic pole end is S_m, we may obtain Ø_m = B_m * S_m.

Thus, we have: $$\left\{\begin{array}{c}{\varnothing }_{\mathrm{D}1}={\varnothing }_{\mathrm{m}}+{\varnothing }_{\mathrm{i}}={\varnothing }_{\mathrm{m}}+{\varnothing }_{\mathrm{i}}={\mathrm{B}}_{\mathrm{m}}{\mathrm{S}}_{\mathrm{m}}+{\mathrm{G}}_{\mathrm{i}}\ast \mathrm{N}\ast \mathrm{i}\\ {\varnothing }_{\mathrm{D}2}={\varnothing }_{\mathrm{m}}-{\varnothing }_{\mathrm{i}}={\varnothing }_{\mathrm{m}}-{\varnothing }_{\mathrm{i}}={\mathrm{B}}_{\mathrm{m}}{\mathrm{S}}_{\mathrm{m}}-{\mathrm{G}}_{\mathrm{i}}\ast \mathrm{N}\ast \mathrm{i}\end{array}\right..$$

Referring to FIG. 2, FIG. 2 illustrates the closed magnetic flux lines of the coil C1, the coil C2 and the permanent magnet. In FIG. 2, the closed magnetic flux lines generated by the coil C1 passes through the magnetic gap D1, the closed magnetic flux lines generated by the coil C2 passes through the magnetic gap D2, and the closed magnetic flux generated by the permanent magnet sequentially passes through the magnetic gap D1 and the magnetic gap D2.

Referring to FIG. 3, FIG. 3 is a diagram illustrating the relationship between the movable assembly, the magnetic domains D_1,1, D_2,2 and the stator assembly. In the magnetic domain D_1,1, the movable assembly is subjected to a rightward attractive force F₁ from the stator assembly, and in the magnetic domain D_2,1, the movable assembly is subjected to a leftward attractive force F₂ from the stator assembly. Taking the rightward direction as the positive direction, the resultant force of the stator assembly on the movable assembly is F₁ - F₂.

Referring to FIG. 5, FIG. 5 is a force analysis diagram of the isolated movable assembly. The movable assembly is subjected to forces from the stator assembly, that is, a rightward attractive force F₁ and a leftward attractive force F₂, and the resultant force is F₁ - F₂, expressed as: $${\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet}}={\mathrm{F}}_1-{\mathrm{F}}_2.$$

Further derive the equation for the electromagnetic force generated by each magnetic domain. The magnitude of the electromagnetic attractive force acting on a magnetized ferromagnetic object is proportional to the total area of the magnetic poles through which the magnetic flux lines passes and the square of the magnetic induction intensity. If the magnetic induction intensity B is uniformly distributed along the surface of the magnetic pole and the calculated air gap length is small, the equation for calculating the electromagnetic attractive force is calculated by the Maxwell formula, expressed as: $$\mathrm{F}=\mathrm{C}\frac{{\mathrm{B}}^2\cdot \mathrm{S}}{2{\mathrm{\mu }}_0}=\mathrm{C}\frac{{\varnothing }^2}{2{\mathrm{\mu }}_0\cdot \mathrm{S}}$$ where:

• F: Electromagnetic attractive force
• B: Magnetic flux density or magnetic induction intensity
• Ø: Magnetic flux passing through the medium
• S: Area of the magnetic pole through which the magnetic flux passes
• µ₀: Air permeability
• C: Coefficient related to the combination type and shape of the magnetic pole end face, with different values for different scenarios. When the force is generated between permanent magnets, it is denoted as C_m2m, usually taking a value of 1, and the accurate value is obtained through actual measurement in the actual design process. When the force is generated between a permanent magnet and a magnetic conductor (yoke iron), it is denoted as C_m2y, usually taking a value of 1/2, and the accurate value is obtained through actual measurement in the actual design process. When the force is generated between magnetic conductors (yoke iron), it is denoted as C_y2y, usually taking a value of 1/4, and the accurate value is obtained through actual measurement in the actual design process.

Applying the above equation to calculate the electromagnetic attractive forces in the above magnetic domain 1 and magnetic domain 2: $$\left\{\begin{array}{c}{F}_1=C_{y2y}\frac{{\left({\varnothing }_m+{\varnothing }_i\right)}^2}{2{\mu }_0\cdot S_{D1}}\\ F_2=C_{y2y}\frac{{\left({\varnothing }_m-{\varnothing }_i\right)}^2}{2{\mu }_0\cdot S_{D2}}\end{array}\right.$$ where S_D1 and S_D2 represent the areas of the annular end faces corresponding to magnetic domain 1 and magnetic domain 2 respectively, and S_D1 = S_D2 = S_D, thus: $$\left\{\begin{array}{c}{F}_1=C_{y2y}\frac{{\left({\varnothing }_m+{\varnothing }_i\right)}^2}{2{\mu }_0\cdot S_{D1}}=\frac{C_{y2y}}{2{\mu }_0\cdot S_D}{\left({\varnothing }_m+{\varnothing }_i\right)}^2\\ F_2=C_{y2y}\frac{{\left({\varnothing }_m-{\varnothing }_i\right)}^2}{2{\mu }_0\cdot S_{D2}}=\frac{C_{y2y}}{2{\mu }_0\cdot S_D}{\left({\varnothing }_m-{\varnothing }_i\right)}^2\end{array}\right.$$ $$\left\{\begin{array}{c}{F}_1=F_{1,\mathit{linear}}+F_{1,\mathit{nonlinear}}\\ F_2=F_{2,\mathit{linear}}+F_{2,\mathit{nonlinear}}\end{array}\right.$$ where: $$\left\{\begin{array}{c}{F}_{1,\mathit{linear}}=\frac{C_{y2y}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2+2{\varnothing }_m{\varnothing }_i\right)\\ F_{1,\mathit{nonlinear}}=\frac{C_{y2y}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2\end{array}\right.$$ $$\left\{\begin{array}{c}{F}_{2,\mathit{linear}}=\frac{C_{y2y}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2-2{\varnothing }_m{\varnothing }_i\right)\\ F_{2,\mathit{nonlinear}}=\frac{C_{y2y}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2\end{array}\right.$$ Since: $$F_{\mathit{moving}-\mathit{magnet}}=F_1-F_2$$ We obtain $$F_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=F_{\mathit{moving}-\mathit{magnet},\mathit{linear}}+F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}$$ $$\left\{\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=F_{1,\mathit{linear}}-F_{2,\mathit{linear}}\\ F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}=F_{1,\mathit{nonlinear}}-F_{2,\mathit{nonlinear}}\end{array}\right.$$ Substitute F_1,linear , F_2,linear , F_1,nonlinear , F_1,nonlinear into F_{moving-magnet,linear} and F_{moving-magnet,nonlinear} respectively, we obtain: $$\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=F_{1,\mathit{linear}}-F_{2,\mathit{linear}}\\ =\frac{C_{y2y}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2+2{\varnothing }_m{\varnothing }_i\right)-\frac{C_{y2y}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2-2{\varnothing }_m{\varnothing }_i\right)\\ =\frac{C_{y2y}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2+2{\varnothing }_m{\varnothing }_i-\left({\varnothing }_m^2-2{\varnothing }_m{\varnothing }_i\right)\right.\\ =\frac{C_{y2y}}{2{\mu }_0\cdot S_D}\left(4{\varnothing }_m{\varnothing }_i\right)\\ =\frac{2C_{y2y}{\varnothing }_m}{{\mu }_0\cdot S_D}{\varnothing }_i\end{array}$$ Since: $${\varnothing }_{\mathrm{m}}={\mathrm{B}}_{\mathrm{m}}{\mathrm{S}}_{\mathrm{m}}$$ $${\varnothing }_i=G_i\mathit{Ni}$$ It follows that: $$F_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=\frac{2C_{y2y}{\varnothing }_m}{{\mu }_0\cdot S_D}{\varnothing }_i=\frac{2C_{y2y}{B}_m{S}_m}{{\mu }_0\cdot S_D}{G}_i\mathit{Ni}=\frac{2C_{y2y}{B}_m{S}_m{G}_{i}N}{{\mu }_0\cdot S_D}\ast i$$ Calculate F_{moving-magnet,nonlinear} in a similar way as follows: $$\begin{array}{l}{F}_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}=F_{1,\mathit{nonlinear}}-F_{2,\mathit{nonlinear}}\\ =\frac{C_{y2y}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2-\frac{C_{y2y}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2=0\end{array}$$ Thus, the resultant force acting on the moving magnet, which serves as the moving member, is: $$\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet}}=F_{\mathit{moving}-\mathit{magnet},\mathit{linear}}+F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}\\ =\frac{2C_{y2y}{B}_m{S}_m{G}_{i}N}{{\mu }_0\cdot S_D}\ast i+0\\ =\frac{2C_{y2y}{B}_m{S}_m{G}_{i}N}{{\mu }_0\cdot S_D}\ast i\end{array}$$

From the above derivation process, the following characteristics may be seen:

1. 1) In the linear term of the resultant force F_{moving-magnet,linear}, the linear terms of the component forces F_1,linear and F_2,linear are superimposed, so that the linear term of the resultant force F_{moving-magnet,linear} and the coil current is larger.
2. 2) In the nonlinear term of the resultant force F_{moving-magnet,nonlinear}, the nonlinear terms of the component forces F_1,nonlinear and F_2,nonlinear cancel each other out, so that the nonlinear term of the resultant force F_{moving-magnet,nonlinear} is zero.

The above design method is called the design method of the moving-magnet vibrator with nonlinear term cancellation. This method can not only be used to design vibrators, but also be applied to design actuators. The moving-magnet vibrators or actuators obtained by the above method are called moving-magnet vibrator devices or actuators with nonlinear term cancellation.

Referring to FIG. 22, it can be seen from the FIG. 22 that the total harmonic distortion in the low-frequency band is greatly reduced, decreasing from the original peak value of 99% to below 15%, realizing a significant improvement.

The reduction of the distortion curve is equivalently interpreted as the reduction of the resonant frequency of the vibrator system, thereby achieving better sound quality. In addition, the reduction of the distortion curve may be further equivalently interpreted as the improvement of the sensitivity of the vibrator system and the reduction of power consumption.

Referring to FIGS. 7-11, a design method of a moving-magnet vibrator with nonlinear term cancellation includes conditions as follows.

1. (1) A moving-magnet vibrator body 11 is provided. The moving-magnet vibrator body 11 includes an outer cylinder 1, a vibration transmission plate 8, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure, the movable assembly includes a magnet combination structure. The coil combination structure includes a coil 3 and a first magnetic conductor 7, and the magnet combination structure includes a permanent magnet 6 and a second magnetic conductor 4. The stator assembly is fixed inside the outer cylinder 1, the vibration transmission plate 8 is fixed on the outer cylinder 1, and the movable assembly is fixedly connected to the vibration transmission plate 8 through at least one contact point. The movable assembly moves while the stator assembly remains stationary, and the movable assembly is referred to as a moving component.
2. (2) The movable assembly is simultaneously subjected to paired electromagnetic forces of push and pull, presenting push-pull structural characteristics.

2N magnetic domains D_1,i and D_2,i which are designed as N symmetrical pairs are provided in the moving-magnet vibrator body 11, where N is 1, 2, 3, ..., 100, and i=1, 2, 3, ....

The number of a permanent magnet in the magnet combination structure and the number of a coil in the coil combination structure are limited, in such a manner that N_magnet>N_coil or N_magnet < N_coil , where N_magnet represents the number of the permanent magnet, and N_coil represents the number of the coil, N_magnet=1, 2, 3, ..., 100, and N_coil=1, 2, 3, ..., 100.

In the push-pull structural characteristics, linear terms of an electromagnetic force acting on the movable assembly are superimposed so as to be increased, while nonlinear terms of the electromagnetic force acting on the movable assembly are partially or completely canceled so as to be decreased.

The closed main magnetic flux lines of the coil in the coil combination structure and the closed main magnetic flux lines of the permanent magnet in the magnet combination structure each pass through the magnetic domains D_1,i and D_2,i. The magnetic domain is a spatial region filled with electromagnetic energy, is generally composed of air or a medium with low magnetic permeability (e.g., a relative magnetic permeability<1000), and includes a region where the magnet material is located. In the magnetic domain D_1,i, a direction of magnetic flux lines of the coil is the same as a direction of magnetic flux lines of the permanent magnet, while in the magnetic domain D_2,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet. Alternatively, in the magnetic domain D_1,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet, while in the magnetic domain D_2,i the direction of the magnetic flux lines of the coil is the same as the direction of the magnetic flux lines of the permanent magnet.

The number of the permanent magnet 6 and the number of the coil 3 are limited, in which there are two permanent magnets, and one coil.

Two symmetrically paired magnetic domains D_1,1 and D_2,1 are provided in the moving-magnet vibrator body 11. The closed main magnetic flux lines of the coil 3 and the closed main magnetic flux lines of the permanent magnet 6 each pass through the magnetic domains D_1,1 and D_2,1. In the magnetic domain D_1,1, the direction of the magnetic flux lines of the coil 3 is opposite to the direction of the magnetic flux lines of the permanent magnet 6, while in the magnetic domain D_2,1, the direction of the magnetic flux lines of the coil 3 is the same as the direction of the magnetic flux lines of the permanent magnet 6.

The moving component is subjected to 2 forces. Each component force includes two parts: one part is a linear term of an excitation current i, and the other part is a nonlinear term of the excitation current i: $$\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet},n}\left(i\right)=F_{\mathit{moving}-\mathit{magnet},n,\mathit{linear}}\left(i\right)+F_{\mathit{moving}-\mathit{magnet},n,\mathit{nonlinear}}\left(i\right),\\ \mathrm{where}\mathrm{n}=1,2,3,\dots ,2\mathrm{N}-1,2\mathrm{N}.\end{array}$$ where n = 1,2 3, ..., 2N - 1,2N.

Accordingly, a resultant force on the moving component also includes two parts: one part is a linear term of the current i, and the other part is a nonlinear term of the current i, which is expressed as: $$\begin{array}{l}{F}_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force}}\left(i\right)\\ =F_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force},\mathit{linear}}\left(i\right)+F_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force},\mathit{nonlinear}}\left(i\right)\end{array}$$ where: $$\begin{array}{l}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force}}\left(\mathrm{i}\right)\\ ={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n}}\left(\mathrm{i}\right)}={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},2}\left(\mathrm{i}\right)\right)}\end{array}$$ $$\begin{array}{c}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{linear}}\left(\mathrm{i}\right)={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},\mathrm{linear}}\left(\mathrm{i}\right)}\\ ={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{linear},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{linear},2}\left(\mathrm{i}\right)\right)}\end{array}$$ $$\begin{array}{c}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{nonlinear}}\left(\mathrm{i}\right)={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},\mathrm{nonlinear}}\left(\mathrm{i}\right)}\\ ={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{nonlinear},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{nonlinear},2}\left(\mathrm{i}\right)\right)}\end{array}$$

That is, the nonlinear terms in each component force are partially or completely canceled. In the final total resultant force ∑_i(F_1,i + F_2,i), the nonlinear terms of the total resultant force relative to the current are partially or completely canceled, and the linear terms are superimposed to be increased, thereby obtaining the moving-magnet vibrator with nonlinear term cancellation.

Furthermore, the movable assembly and the stator assembly are in concave-convex shapes and arranged in an interleaved engagement manner, and the closed main magnetic flux lines of the coil 3 and the closed main magnetic flux lines of the permanent magnet 6 alternately pass through the movable assembly and the stator assembly.

The design method further includes conditions as follows.

(3.1) Looking outward from a center, the coil 3 is located inside and the permanent magnet 6 is located outside.

(3.2) N_magnet=2, where n is a natural number, and n=1.

(3.3) When N_magnet>1, the polarities of the two opposite end faces of the adjacent permanent magnets are the same.

The magnetic conductor is arranged at a position of the outer cylinder 1 close to the coil 3, so as to minimize the magnetic resistance of the magnetic circuit of an electromagnet 6 generated by the coil 3. The permanent magnet 6 in the magnet assembly is isolated by the magnetic conductor. A yoke iron is arranged around the coil 3 and the permanent magnet 6, or alternatively, for the coil combination structure, a part of the outer cylinder 1 close to the coil is magnetically conductive.

Referring to FIGS. 7 to 11, a moving-magnet vibrator device with nonlinear term cancellation designed by the method of Embodiment 3 includes a moving-magnet vibrator body 11. The moving-magnet vibrator body includes an outer cylinder 1, a vibration transmission plate 8, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure. The movable assembly includes a magnet combination structure. The coil combination structure includes a coil 3 and a first magnetic conductor 7. The magnet combination structure includes a permanent magnet 6 and a second magnetic conductor 4. The coil combination structure further includes a first magnetic conductive ring 5, and the magnet combination structure further includes a second magnetic conductive ring 2. Viewed from a center outward, the coil 3 is inside and the permanent magnet 6 is outside. There are two permanent magnets 6, and one coil. The two opposite end faces of adjacent permanent magnet 6 have the same polarity. There is one vibration transmission plate 8 which is fixed on a top surface of the outer cylinder 1. Preferably, the outer cylinder is a magnetic conductive to reduce magnetic resistance.

An end of the first magnetic conductor 7 is fixed on a bottom surface of the outer cylinder 1. The coil 3 is wound around and fixed on the first magnetic conductor 7, and the first magnetic conductive ring 5 is fixed at an end of the first magnetic conductor 7. An L-shaped vibration transmission bracket 9 is provided. A horizontal part of the vibration transmission bracket 9 is parallel to a vibration direction. The second magnetic conductor 4 is fixed on the horizontal part of the vibration transmission bracket 9. The two permanent magnets 6 are fixedly arranged on two sides of the second magnetic conductor 4, and the two permanent magnets 6 are fixed on the horizontal part of the vibration transmission bracket 9. The movable assembly and the stator assembly are in concave-convex shapes and arranged in an interleaved engagement manner. Closed magnetic flux lines of the coil 3 and closed magnetic flux lines of the permanent magnet 6 alternately pass through the movable assembly and the stator assembly. Two magnetic domains D_1,1 and D_2,1 which are designed as a symmetrical pair are provided in the moving-magnet vibrator body, the closed magnetic flux lines of the coil and the closed magnetic flux lines of the permanent magnets each pass through the magnetic domains D_1,1 and D_2,1. In the magnetic domain D_1,1, a direction of magnetic flux lines of the coil is opposite to a direction of magnetic flux lines of the permanent magnet. In the magnetic domain D_2,1, the direction of the magnetic flux lines of the coil is the same as the direction of the magnetic flux lines of the permanent magnet.

To further describe the design method of the moving-magnet vibrator with nonlinear term cancellation, referring to FIG 8, an air gap 1 constitutes a magnetic action domain D_1,1, and an air gap 2 constitutes a magnetic action domain D_2,1, thereby forming a magnetic domain pair D = (D_1,1, D_2,1). In the magnetic action domain, a total magnetic flux/magnetic induction intensity generated by the superposition of the magnetic field generated by the permanent magnet 6 and the magnetic field generated by the electromagnet of the coil 3 causes mutual forces between the components around the magnetic domain.

With regard to the magnetic domain pair D = (D_1,1, D_2,1), assuming that the current passing through the coil is i, the magnetic flux corresponding to the coil is Φ_i. The magnetic flux corresponding to the permanent magnet M₁ is Φ_m1, and the magnetic flux corresponding to the permanent magnet M₂ is Φ_m2. In the magnetic domain D_1,1 (magnetic action domain D_1,1), the direction of the magnetic flux lines corresponding to the coil C is opposite to that corresponding to the permanent magnet M₁, so the total magnetic flux in the magnetic domain D_1,1 is the difference between Φ_i and Φ_m1. In the magnetic domain D_2,1 (magnetic action domain D_2,1), the direction of the magnetic flux lines corresponding to the coil is to the same as that corresponding to the permanent magnet M₂, so the total magnetic flux in the magnetic domain D_2,1 is the sum of Φ_i and Φ_m2. Since the magnetic field formed by the permanent magnet is static, assuming that the direction of the magnetic flux lines of the permanent magnet is positive, the magnetic flux is also positive, then: $${\mathrm{\Phi }}_{\mathrm{D}1,1}={\mathrm{\Phi }}_{\mathrm{m}1}-{\mathrm{\Phi }}_{\mathrm{i}1};$$ and $${\mathrm{\Phi }}_{\mathrm{D}2,1}={\mathrm{\Phi }}_{\mathrm{m}2}+{\mathrm{\Phi }}_{\mathrm{i}2}.$$

Assuming that the magnetic resistance of the magnetic circuit formed by the electromagnetic field generated by the coil current i is Z_i, N is the number of turns of the coil, and i is the current intensity, we have: $${\varnothing }_i\ast {\mathrm{Z}}_i=N\ast i.$$

Assuming that the magnetic conductance of the magnetic circuit formed by the electromagnetic field generated by the current is G_i, we have: $${\varnothing }_i={\mathrm{G}}_i\ast N\ast i.$$

The magnetic flux corresponding to the permanent magnet may also be expressed by the equation of magnetic induction intensity. Assuming that the magnetic induction intensities at the magnetic pole end faces of the permanent magnet M₁ and the permanent magnet M₂ are B_m1 and B_m2 respectively, and B_m1 = B_m2 = B_m. In addition, assuming that the areas of the magnetic pole end faces of the permanent magnet 1 and the permanent magnet 2 are S_m1 and S_m2 respectively, and S_m1 = S_m2 = S_m, it follows that Ø_m1 = Ø_m2 = Ø_m = B_mS_m.

Thus, we have: $${\varnothing }_{D1}=-{\varnothing }_i+{\varnothing }_m=-{\mathrm{G}}_i\ast N\ast i+B_m{S}_m$$ $${\varnothing }_{D2}={\varnothing }_i+{\varnothing }_m={\mathrm{G}}_i\ast N\ast i+B_m{S}_m.$$

Referring to FIG. 8, FIG. 8 illustrates only the closed magnetic flux lines of the coil, the permanent magnet M₁ and the permanent magnet M₂. In FIG. 8, the closed magnetic flux lines generated by the permanent magnet M₁ passes through the magnetic gap D_1,1, the closed magnetic flux lines generated by the permanent magnet M₂ passes through the magnetic gap D_2,1, and the closed magnetic flux generated by the coil sequentially passes through the magnetic gap D₁ and the magnetic gap D₂.

Referring to FIG. 9, FIG. 9 is a diagram illustrating the relationship between the movable assembly, the magnetic domains D_1,1, D_2,2 and the stator assembly. In the magnetic domain D_1,1, the movable assembly is subjected to a leftward attractive force F₁ from the stator assembly, and in the magnetic domain D_2,1, the movable assembly is subjected to a rightward attractive force F₂ from the stator assembly. Taking the rightward direction as the positive direction, the resultant force of the stator assembly on the movable assembly is -F₁ + F₂.

Referring to FIG. 10, FIG. 10 is a force analysis diagram of the isolated movable assembly. The movable assembly is subjected to forces from the stator assembly, that is, a leftward attractive force F₁ and a rightward attractive force F₂, and the resultant force is -F₁ + F₂, expressed as: $${\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet}}=-{\mathrm{F}}_1+{\mathrm{F}}_2.$$

Further derive the equation for the electromagnetic force generated by each magnetic domain. The magnitude of the electromagnetic attractive force acting on a magnetized ferromagnetic object is proportional to the total area of the magnetic poles through which the magnetic flux lines passes and the square of the magnetic induction intensity. If the magnetic induction intensity B is uniformly distributed along the surface of the magnetic pole and the calculated air gap length is small, the equation for calculating the electromagnetic attractive force is calculated by the Maxwell formula, expressed as: $$\mathrm{F}=\mathrm{C}\frac{{\mathrm{B}}^2\cdot \mathrm{S}}{2{\mathrm{\mu }}_0}=\mathrm{C}\frac{{\varnothing }^2}{2{\mathrm{\mu }}_0\cdot \mathrm{S}}$$ where:

• F: Electromagnetic attractive force
• B: Magnetic flux density or magnetic induction intensity
• Ø: Magnetic flux passing through the medium
• S: Area of the magnetic pole through which the magnetic flux passes
• µ₀: Air permeability
• C: Coefficient related to the combination type and shape of the magnetic pole end faces, with different values for different scenarios. When the force is generated between permanent magnets, it is denoted as C_m2m, usually taking a value of 1, and the accurate value is obtained through actual measurement in the actual design process. When the force is generated between a permanent magnet and a magnetic conductor (yoke iron), it is denoted as C_m2y, usually taking a value of 1/2, and the accurate value is obtained through actual measurement in the actual design process. When the force is generated between magnetic conductors (yoke iron), it is denoted as C_y2y, usually taking a value of 1/4, and the accurate value is obtained through actual measurement in the actual design process.

Applying the above equation to calculate the electromagnetic attractive forces in the above magnetic domain D₁ and magnetic domain D₂: $$\left\{\begin{array}{c}{F}_1={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left(-{\varnothing }_i+{\varnothing }_{\mathrm{m}}\right)}^2}{2{\mu }_0\cdot S_{D1}}\\ F_2={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_i+{\varnothing }_{\mathrm{m}}\right)}^2}{2{\mu }_0\cdot S_{D2}}\end{array}\right.$$ where S_D1 and S_D2 represent the areas of the annular end faces corresponding to magnetic domain D_1,1 and magnetic domain D_2,1 respectively, and S_D1 = S_D2 = S_D, thus: $$\left\{\begin{array}{c}{F}_1={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{m}}-{\varnothing }_i\right)}^2}{2{\mu }_0\cdot S_{D1}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{\left({\varnothing }_i-{\varnothing }_{\mathrm{m}}\right)}^2\\ F_2={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{m}}+{\varnothing }_i\right)}^2}{2{\mu }_0\cdot S_{D2}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{\left({\varnothing }_i+{\varnothing }_{\mathrm{m}}\right)}^2\end{array}\right.$$ $$\left\{\begin{array}{c}{F}_1=F_{1,\mathit{linear}}+F_{1,\mathit{nonlinear}}\\ F_2=F_{2,\mathit{linear}}+F_{2,\mathit{nonlinear}}\end{array}\right.$$ where: $$\left\{\begin{array}{c}{F}_{1,\mathit{linear}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2-2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\\ F_{1,\mathit{nonlinear}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2\end{array}\right.$$ $$\left\{\begin{array}{c}{F}_{2,\mathit{linear}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2+2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\\ F_{2,\mathit{nonlinear}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2\end{array}\right.$$ Since: $$F_{\mathit{moving}-\mathit{magnet}}=-F_1+F_2$$ We obtain: $$\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=F_{\mathit{moving}-\mathit{magnet},\mathit{linear}}+F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}\\ \left\{\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=F_{1,\mathit{linear}}-F_{2,\mathit{linear}}\\ F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}=F_{1,\mathit{nonlinear}}-F_{2,\mathit{nonlinear}}\end{array}\right.\end{array}$$ Substitute F_1,linear , F_2,linear , F_1,nonlinear , F_2,nonlinear into F_{moving-magnet,linear} and F_{moving-magnet,nonlinear} respectively, we obtain: $$\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=-F_{1,\mathit{linear}}+F_{2,\mathit{linear}}\\ =-\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2-2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)+\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2+2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\\ =-\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2-2{\varnothing }_{\mathrm{m}}{\varnothing }_i-\left({\varnothing }_m^2+2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\right.\\ =\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left(4{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\\ =\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{\varnothing }_{\mathrm{m}}}{{\mu }_0\cdot S_D}{\varnothing }_i\end{array}$$ Since: $${\varnothing }_{\mathrm{m}}=B_m{S}_m$$ $${\varnothing }_i={\mathrm{G}}_i\mathit{Ni}$$ It follows that: $$F_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{\varnothing }_{\mathrm{m}}}{{\mu }_0\cdot S_D}{\varnothing }_i=\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m}{{\mu }_0\cdot S_D}{\mathrm{G}}_i\mathit{Ni}=\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m{\mathrm{G}}_{i}N}{{\mu }_0\cdot S_D}\ast i$$ Calculate F_{moving-magnet,nonlinear} in a similar way as follows: $$\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}=-F_{1,\mathit{nonlinear}}+F_{2,\mathit{nonlinear}}\\ =-\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2-\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2=0\end{array}$$ Thus, the resultant force acting on the moving magnet, which serves as the moving component, is: $$\begin{array}{c}{F}_{\mathit{moving}-\mathit{magnet}}=F_{\mathrm{moving}-\mathrm{magnet},\mathit{linear}}+F_{\mathrm{moving}-\mathrm{magnet},\mathit{nonlinear}}\\ =\frac{{2\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m{\mathrm{G}}_{i}N}{{\mu }_0\cdot S_D}\ast i+0\\ =\frac{{2\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m{\mathrm{G}}_{i}N}{{\mu }_0\cdot S_D}\ast i\end{array}$$

From the above derivation process, the following characteristics may be seen:

1. 1) In the linear term of the resultant force F_{moving-magnet,linear}, the linear terms of the component forces F_1,linear and F_2,linear are superimposed, so that the linear term of the resultant force F_{moving-magnet,linear} and the coil current is larger.
2. 2) In the nonlinear term of the resultant force F_{moving-magnet,nonlinear}, the nonlinear terms of the component forces F_1,nonlinear and F_2,nonlinear cancel each other out, so that the nonlinear term of the resultant force F_{moving-magnet,nonlinear} is zero.

The above design method is called the design method of the moving-magnet vibrator with nonlinear term cancellation. This method can not only be used to design vibrators, but also be applied to design actuators. The moving-magnet vibrators or actuators obtained by the above method are called moving-magnet vibrator devices or actuators with nonlinear term cancellation.

Referring to FIGS. 12 to 16, a design method of a moving-magnet vibrator with nonlinear term cancellation includes conditions as follows.

1. (1) A moving-magnet vibrator body 11 is provided. The moving-magnet vibrator body 11 includes an outer cylinder 1, a vibration transmission plate 10, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure, the movable assembly includes a magnet combination structure. The coil combination structure includes a coil 7 and a first magnetic conductor 5, and the magnet combination structure includes a permanent magnet 3 and a second magnetic conductor 2. The coil combination structure is fixed inside the outer cylinder 1, the vibration transmission plate 10 is fixed on the outer cylinder 1, and the movable assembly is fixedly connected to the vibration transmission plate 10 through at least one contact point. The movable assembly moves while the stator assembly remains stationary, and the movable assembly is referred to as a moving component.
2. (2) The movable assembly is simultaneously subjected to paired electromagnetic forces of push and pull, presenting push-pull structural characteristics.

2N magnetic domains D_1,i and D_2,i which are designed as N symmetrical pairs are provided in the moving-magnet vibrator body 11, where N is 1, 2, 3, ..., 100, and i=1, 2, 3, ....

The number of a permanent magnet in the magnet combination structure and the number of a coil in the coil combination structure are limited, in such a manner that N_magnet>N_coil or N_magnet < N_coil , where N_magnet represents the number of the permanent magnet, and N_coil represents the number of the coil, N_magnet=1, 2, 3, ..., 100, and N_coil=1, 2, 3, ..., 100.

In the push-pull structural characteristics, linear terms of an electromagnetic force acting on the movable assembly are superimposed so as to be increased, while nonlinear terms of the electromagnetic force acting on the movable assembly are partially or completely canceled so as to be decreased.

The closed main magnetic flux lines of the coil in the coil combination structure and the closed main magnetic flux lines of the permanent magnet in the magnet combination structure each pass through the magnetic domains D_1,i and D_2,i. The magnetic domain is a spatial region filled with electromagnetic energy, is generally composed of air or a medium with low magnetic permeability (e.g., a relative magnetic permeability<1000), and includes a region where the magnet material is located. In the magnetic domain D_1,i, a direction of magnetic flux lines of the coil is the same as a direction of magnetic flux lines of the permanent magnet, while in the magnetic domain D_2,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet. Alternatively, in the magnetic domain D_1,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet, while in the magnetic domain D_2,i, the direction of the magnetic flux lines of the coil is the same as the direction of the magnetic flux lines of the permanent magnet.

The number of the permanent magnet 3 and the number of the coil 7 are limited, in which there are one permanent magnet 3, and two coils.

Four symmetrically paired magnetic domains D_1,1 , D_2,1 , D_1,2 and D_2,2 are provided in the moving-magnet vibrator body 11. The closed main magnetic flux lines of the coil 7 and the closed main magnetic flux lines of the permanent magnet 3 each pass through the magnetic domains D_1,1, D_2,1, D_1,2and D_2,2. In the magnetic domain D_1,1, the direction of the magnetic flux lines of the coil 7 is the opposite to the direction of the magnetic flux lines of the permanent magnet 3, while in the magnetic domain D_2,1, the direction of the magnetic flux lines of the coil 7 is the same as the direction of the magnetic flux lines of the permanent magnet 3.

The moving component is subjected to four forces. Each component force includes two parts: one part is a linear term of an excitation current i, and the other part is a nonlinear term of the excitation current i: $$F_{\mathit{moving}-\mathit{magnet},n}\left(i\right)=F_{\mathit{moving}-\mathit{magnet},n,\mathit{linear}}\left(i\right)+F_{\mathit{moving}-\mathit{magnet},n,\mathit{nonlinear}}\left(i\right),$$ where n = 1, 2, 3, ..., 2N - 1, 2N.

Accordingly, a resultant force on the moving component also includes two parts: one part is a linear term of the current i, and the other part is a nonlinear term of the current i, which is expressed as: $$\begin{array}{l}{F}_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force}}\left(i\right)\\ =F_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force},\mathit{linear}}\left(i\right)+F_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force},\mathit{nonlinear}}\left(i\right)\end{array}$$ where: $$\begin{array}{l}{F}_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force}}\left(i\right)\\ ={\displaystyle \sum _{n=1}^{2N}{F}_{\mathit{moving}-\mathit{magnet},n}\left(i\right)}={\displaystyle \sum _{n=1}^N\left(F_{\mathit{moving}-\mathit{magnet},n,1}\left(i\right)+F_{\mathit{moving}-\mathit{magnet},n,2}\left(i\right)\right)}\end{array}$$ $$\begin{array}{l}{F}_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force},\mathit{linear}}\left(i\right)={\displaystyle \sum _{n=1}^{2N}{F}_{\mathit{moving}-\mathit{magnet},n,\mathit{linear}}\left(i\right)}\\ ={\displaystyle \sum _{n=1}^N\left(F_{\mathit{moving}-\mathit{magnet},\mathit{linear},1}\left(i\right)+F_{\mathit{moving}-\mathit{magnet},\mathit{linear},2}\left(i\right)\right)}\end{array}$$ $$\begin{array}{l}{F}_{\mathit{moving}-\mathit{magnet},\mathit{resultant}\mathit{force},\mathit{nonlinear}}\left(i\right)={\displaystyle \sum _{n=1}^{2N}{F}_{\mathit{moving}-\mathit{magnet},n,\mathit{nonlinear}}\left(i\right)}\\ ={\displaystyle \sum _{n=1}^N\left(F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear},1}\left(i\right)+F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear},2}\left(i\right)\right)}\end{array}$$

That is, the nonlinear terms in each component force are partially or completely canceled. In the final total resultant force ∑_i(F_1,i + F_2,i), the nonlinear terms of the total resultant force relative to the current are partially or completely canceled, and the linear terms are superimposed to be increased, thereby obtaining the moving-magnet vibrator with nonlinear term cancellation.

The movable assembly and the stator assembly are in concave-convex shapes and arranged in an interleaved engagement manner, and the closed main magnetic flux lines of the coil 7 and the closed main magnetic flux lines of the permanent magnet 3 alternately pass through the movable assembly and the stator assembly.

The design method further includes conditions as follows.

(3.1) Looking outward from a center, the coil 7 is located inside and the permanent magnet 3 is located outside.

(3.2) N_coil=2, where n is a natural number, and n=1.

(3.3) When N_coil>1, the currents in adjacent coils have opposite directions, and the polarities of the electromagnetic fields on the two end faces of two adjacent coils are the same.

The magnetic conductor is arranged at a position of the outer cylinder 1 close to the coil 7, so as to minimize the magnetic resistance of the magnetic circuit of an electromagnet 3 generated by the coil 7. The permanent magnet 3 in the magnet assembly is isolated by the magnetic conductor. A yoke iron is arranged around the coil 7 and the permanent magnet 3, or alternatively, for the coil combination structure, a part of the outer cylinder 1 close to the coil is magnetically conductive.

Referring to FIGS. 12 to 16, a moving-magnet vibrator device with nonlinear term cancellation designed by the method of Embodiment 5 includes a moving-magnet vibrator body 11. The moving-magnet vibrator body 11 includes an outer cylinder 1, a vibration transmission plate 10, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure. The movable assembly includes a magnet combination structure. The coil combination structure includes a coil 7 and a first magnetic conductor 9. The magnet combination structure includes a permanent magnet 3 and a second magnetic conductor 2. The coil combination structure further includes a first magnetic conductive ring 4 and a second magnetic conductive ring 5. Viewed from a center outward, the coil 7 is inside and the permanent magnet 3 is outside. There are one permanent magnet 3, and two coils 7. The currents in adjacent coils have opposite directions. For two adjacent coils, the polarities of the electromagnetic fields on the adjacent faces are the same. There is one vibration transmission plate 10 fixed on a top surface of the outer cylinder 1. An end of the first magnetic conductor 9 is fixed on a bottom surface of the outer cylinder 1, and the two coils 7 are wound around and fixed on the first magnetic conductor 9. The second magnetic conductive ring 5 is fixed at an end of the first magnetic conductor 9, and the first magnetic conductive ring 4 is wound around and fixed on a middle part of the first magnetic conductor 9 and located between the two coils 7. An L-shaped vibration transmission bracket 8 is provided, a horizontal part of the vibration transmission bracket 8 is parallel to a vibration direction. The permanent magnet 3 is fixed in middle of the horizontal part of the vibration transmission bracket 8, and two second magnetic conductors 2 are located on two sides of the permanent magnet 3 and fixed on the horizontal part of the vibration transmission bracket 8. The movable assembly and the stator assembly are in concave-convex shapes and arranged in an interleaved engagement manner. Closed magnetic flux lines of the coils 7 and closed magnetic flux lines of the permanent magnet 3 alternately pass through the movable assembly and the stator assembly. Four magnetic domains D_1,1, D_2,1, D_1,2and D_2,2 which are designed as two symmetrical pairs are provided in the moving-magnet vibrator body, where D_1,1 is symmetric to D_2,1, and D_1,2 is symmetric to D_2,2. The closed magnetic flux lines of the coils 7 and the closed magnetic flux lines of the permanent magnet 3 each pass through the magnetic domains D_1,1, D_2,1, D_1,2 and D_2,2 respectively. In the magnetic domain D_1,1, a direction of magnetic flux lines of the coil 7 is opposite to a direction of magnetic flux lines of the permanent magnet 3. In the magnetic domain D_2,1, the direction of the magnetic flux lines of the coil 7 is the same as the direction of the magnetic flux lines of the permanent magnet 3.

To further illustrate the design method of the moving-magnet vibrator with nonlinear term cancellation, referring to FIGS. 12 to 14, there are 4 air gaps forming magnetic action domains D_1,1, D_2,1, D_1,2 and D_2,2 respectively. In the magnetic action domain, the total magnetic flux/magnetic induction intensity generated by the superposition of the magnetic field generated by the permanent magnet 3 and the magnetic field generated by the electromagnet of the coils 7 causes mutual forces between the components around the magnetic domain. The above magnetic domains D_1,1, D_2,1, D_1,2 and D_2,2 are enclosed by the stator assembly and the movable assembly. Therefore, in these magnetic domains, mutual component forces are generated between the stator assembly and the movable assembly.

The current passing through the coil C₁ is i₁, the current passing through the coil C₂ is i₂ , and the corresponding magnetic fluxes of the coils are Φ_i1 and Φ_i2 respectively. The magnetic flux corresponding to the permanent magnet M₁ is Φ_M1·

The magnetic domains D_1,1, D_2,1, D_1,2 and D_2,2 may be paired into magnetic domain pairs D_j = (D_1,j, D_2,j) according to symmetry, where j=1, 2. The domain pairs include D₁ = (D_1,1, D_2,1) and D₂ = (D_1,2, D_2,2).

1. 1) For the magnetic flux of the magnetic domain pair D_j = (D_1,j, D_2,j) when j = 1, that is, the magnetic flux of the magnetic domain pair D₁ = (D_1,1, D_2,1)·
   In the magnetic domain D_1,1, the direction of the magnetic flux lines corresponding to the coil C₁ is opposite to that corresponding to the permanent magnet M₁. Therefore, the total magnetic flux in the magnetic domain D_1,1 is the difference between Φ_i1 and Φ_M1 = Φ_m. In the magnetic domain D_2,1, the direction of the magnetic flux lines corresponding to the coil C₂ is the same as that corresponding to the permanent magnet M₁. Therefore, the total magnetic flux in the magnetic domain D_2,1 is the sum of Φ_i2 and Φ_M1 = Φ_m.
   Assuming i₁ = i₂ = i, Φ_i1 = Φ_i2 = Φ_i, and the direction of the magnetic flux lines of the magnet M₁ is positive (the magnetic flux is also positive), then: $${\mathrm{\Phi }}_{\mathrm{D}1,1}={\mathrm{\Phi }}_{\mathrm{M}1}-{\mathrm{\Phi }}_{\mathrm{i}1}={\mathrm{\Phi }}_{\mathrm{m}}-{\mathrm{\Phi }}_{\mathrm{i}}$$ $${\mathrm{\Phi }}_{\mathrm{D}2,1}={\mathrm{\Phi }}_{\mathrm{M}1}+{\mathrm{\Phi }}_{\mathrm{i}2}={\mathrm{\Phi }}_{\mathrm{m}}+{\mathrm{\Phi }}_{\mathrm{i}}$$
2. 2) For the magnetic flux of the magnetic domain pair D_i = (D_1,j, D_2,j) when j = 2, that is, the magnetic flux of the magnetic domain pair D₂ = (D_1,2, D_2,2)·

In the magnetic domain D_1,2, only the magnetic flux lines corresponding to the coil C₁ pass through, so the total magnetic flux is only Φ_i1 = Φ_i. In the magnetic domain D_2,2, only the magnetic flux lines corresponding to the coil C₂ pass through, so the total magnetic flux is only Φ_i2 = Φ_i.

Assuming the magnetic resistances of the magnetic circuits formed by the electromagnetic field generated by the current i in the coils C₁ and C₂ are both Z_i, N represents the number of turns of the coils C₁ and C₂, and i is the current intensity, we have: $${\varnothing }_i\ast {\mathrm{Z}}_i=N\ast i$$

Assuming that the magnetic conductance of the magnetic circuit formed by the electromagnetic field generated by the current is G_i, we have: $${\varnothing }_i={\mathrm{G}}_i\ast N\ast i.$$

The magnetic flux corresponding to the permanent magnet 3 may also be expressed by the equation of magnetic induction intensity. Assuming that the magnetic induction intensities at the magnetic pole end faces of the permanent magnet M₁ is B_m1, the area of the magnetic pole end face is S_m, it follows that Ø_m = B_mS_m.

Thus, we have: $$\left\{\begin{array}{c}{\varnothing }_{D1,1}={\varnothing }_m-{\varnothing }_i=B_m{S}_m-{\mathrm{G}}_i\ast N\ast i\\ {\varnothing }_{D2,1}={\varnothing }_m+{\varnothing }_i={\mathrm{G}}_i\ast N\ast i+B_m{S}_m\end{array}\right.$$ $$\left\{\begin{array}{c}{\varnothing }_{D1,2}=-{\varnothing }_i=-{\mathrm{G}}_i\ast N\ast i\\ {\varnothing }_{D2,2}={\varnothing }_i={\mathrm{G}}_i\ast N\ast i\end{array}\right.$$

Referring to FIGS. 13 and 14, FIGS. 13 and 14 show the closed magnetic flux lines of the coils C₁ and C₂, as well as the closed magnetic flux lines of the magnet M₁. In FIGS. 13 and 14, the closed magnetic flux lines generated by the coil C₁ pass through the magnetic gaps D_1,1, and D_2,1, the closed magnetic flux lines generated by the coil C₂ pass through the magnetic gaps D_2,1, and D_2,2 , and the closed magnetic flux lines generated by the magnet M₁ sequentially pass through the magnetic gaps D_1,1 and D_2,1.

Referring to FIG. 15, FIG. 15 is a diagram of the relationship between the movable assembly, the magnetic domains D_1,1, D_2,1, D_1,2 and D_2,2 and the stator assembly. In the magnetic domain D_1,1, the movable assembly is subjected to a rightward attractive force F_1,1 from the stator assembly. In the magnetic domain D_2,1, the movable assembly is subjected to a leftward attractive force F_2,1 from the stator assembly. In the magnetic domain D_1,2, the movable assembly is subjected to a leftward attractive force F_1,2 from the stator assembly; in the magnetic domain D_2,2, the movable assembly is subjected to a rightward attractive force F_2,2 from the stator assembly.

Assuming the resultant force corresponding to the magnetic domain pair D_j = (D_1,j, D_2,j) is F_j (its positive or negative indicates different directions of the force), and taking the rightward direction as positive, the resultant force exerted on the movable assembly by the stator assembly is: $$F_{\mathit{moving}-\mathit{magnet}}=F_1+F_2={\mathrm{F}}_{1,1}-{\mathrm{F}}_{2,1}-{\mathrm{F}}_{1,2}+{\mathrm{F}}_{2,2}$$ $$F_{\mathit{moving}-\mathit{magnet}}=F_1+F_2=\left({\mathrm{F}}_{1,1}-{\mathrm{F}}_{2,1}\right)+\left(-{\mathrm{F}}_{1,2}+{\mathrm{F}}_{2,2}\right)$$ where F_j is the resultant force of the component forces corresponding to the magnetic domain pair D_j = (D_1,j, D_2,j).

Referring to FIG. 15, FIG. 15 is a force analysis diagram of the isolated movable assembly. The movable assembly is subjected to component forces F_1,1, F_2,1, F_1,2 and F_2,2 from the stator assembly, and its resultant force is: $$F_{\mathit{moving}-\mathit{magnet}}=F_1+F_2=\left(F_{1,1}-F_{2,1}\right)+\left(-F_{1,2}+F_{2,2}\right)$$

The above equation may also be expressed by including the direction of the force through the sign of the component force as: $$F_{\mathit{moving}-\mathit{magnet}}={\displaystyle \sum _{j=1}^2{F}_j=\sum _{j=1}^2\left(F_{1,j}+F_{2,j}\right)}$$

The various component forces are grouped into two-by-two paired magnetic domain pairs. Each magnetic domain pair corresponds to the resultant force of a different magnetic domain pair D_j, such as F1 = F_1,1 - F_2,1 and F2 = -F_1,2 + F_2,2, then the total resultant force is calculated.

Further derive the equation for the electromagnetic force generated by each magnetic domain. The magnitude of the electromagnetic attractive force acting on a magnetized ferromagnetic object is proportional to the total area of the magnetic poles through which the magnetic flux lines passes and the square of the magnetic induction intensity. If the magnetic induction intensity B is uniformly distributed along the surface of the magnetic pole and the calculated air gap length is small, the equation for calculating the electromagnetic attractive force is calculated by the Maxwell formula, expressed as: $$\mathrm{F}=\mathrm{C}\frac{{\mathrm{B}}^2\cdot \mathrm{S}}{2{\mathrm{\mu }}_0}=\mathrm{C}\frac{{\varnothing }^2}{2{\mathrm{\mu }}_0\cdot \mathrm{S}}$$ where:

• F: Electromagnetic attractive force
• B: Magnetic flux density or magnetic induction intensity
• Ø: Magnetic flux passing through the medium
• S: Area of the magnetic pole through which the magnetic flux passes
• µ₀: Air permeability
• C: Coefficient related to the combination type and shape of the magnetic pole end faces, with different values for different scenarios. When the force is generated between permanent magnets, it is denoted as C_m2m, usually taking a value of 1, and the accurate value is obtained through actual measurement in the actual design process. When the force is generated between a permanent magnet and a magnetic conductor (yoke iron), it is denoted as C_m2y, usually taking a value of 1/2, and the accurate value is obtained through actual measurement in the actual design process. When the force is generated between magnetic conductors (yoke iron), it is denoted as C_y2y, usually taking a value of 1/4, and the accurate value is obtained through actual measurement in the actual design process.

1. 1) Calculate F_j when j = 1, corresponding to the magnetic domain pair D_j = (D_1,j, D_2,j) when j = 1.
   The resultant force of the component forces corresponding to the magnetic domain pair D₁ = (D_1,1, D_2,1) is F₁ = F_1,1 - F_2,1. Applying the above formula to calculate the electromagnetic attractive forces in the magnetic domains D_1,1 and D_2,1, we have: $$\left\{\begin{array}{c}{F}_{1,1}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{D}1,1}\right)}^2}{2{\mu }_0\cdot S_{D1,1}}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left(-{\varnothing }_i+{\varnothing }_{\mathrm{m}}\right)}^2}{2{\mu }_0\cdot S_{D1,1}}\\ F_{2,1}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{D}2,1}\right)}^2}{2{\mu }_0\cdot S_{D2,1}}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_i+{\varnothing }_{\mathrm{m}}\right)}^2}{2{\mu }_0\cdot S_{D2,1}}\end{array}\right.$$ where S_D1,1 and S_D2,1 represent the areas of the annular end faces corresponding to magnetic domain D_1,1 and magnetic domain D_2,1 respectively, and S_D1,1 = S_D2,1 = S_D, thus: $$\left\{\begin{array}{c}{F}_{1,1}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{m}}-{\varnothing }_i\right)}^2}{2{\mu }_0\cdot S_{D1,1}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{\left(-{\varnothing }_i+{\varnothing }_{\mathrm{m}}\right)}^2\\ F_{2,1}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{m}}+{\varnothing }_i\right)}^2}{2{\mu }_0\cdot S_{D2,1}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{\left({\varnothing }_i+{\varnothing }_{\mathrm{m}}\right)}^2\end{array}\right.$$ $$\left\{\begin{array}{c}{F}_{1,1}=F_{1,1,\mathit{linear}}+F_{1,1,\mathit{nonlinear}}\\ F_{2,1}=F_{2,1,\mathit{linear}}+F_{2,1,\mathit{nonlinear}}\end{array}\right.$$ where: $$\left\{\begin{array}{c}{F}_{1,1,\mathit{linear}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2-2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\\ F_{1,1,\mathit{nonlinear}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2\end{array}\right.$$ $$\left\{\begin{array}{c}{F}_{2,1,\mathit{linear}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2+2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\\ F_{2,1,\mathit{nonlinear}}=\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2\end{array}\right.$$ Since: $${\mathrm{F}}_1={\mathrm{F}}_{1,1}-{\mathrm{F}}_{2,1}$$ We obtain $$F_{1,\mathit{linear}}=F_{1,\mathit{linear}}+F_{1,\mathit{nonlinear}}$$ $$\left\{\begin{array}{c}{F}_{1,\mathit{linear}}=F_{1,1,\mathit{linear}}-F_{2,1,\mathit{linear}}\\ F_{1,\mathit{nonlinear}}=F_{1,1,\mathit{nonlinear}}-F_{2,1,\mathit{nonlinear}}\end{array}\right.$$ Substitute F_1,1,linear , F_2,1,linear , F_{1,1,nonlinear} , F_{2,1,nonlinear} into F_1,linear and F_1,nonlinear respectively, we obtain: $$\begin{array}{c}{F}_{1,\mathit{linear}}=F_{1,1,\mathit{linear}}-F_{2,1,\mathit{linear}}\\ =\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2-2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)-\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2+2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\\ =\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left({\varnothing }_m^2-2{\varnothing }_{\mathrm{m}}{\varnothing }_i-\left({\varnothing }_m^2+2{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\right.\\ =\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}\left(-4{\varnothing }_{\mathrm{m}}{\varnothing }_i\right)\\ =-\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{\varnothing }_{\mathrm{m}}}{{\mu }_0\cdot S_D}{\varnothing }_i\end{array}$$ Since: $${\varnothing }_{\mathrm{m}}=B_m{S}_m$$ $${\varnothing }_i={\mathrm{G}}_i\mathit{Ni}$$ It follows that: $$F_{1,\mathit{linear}}=-\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{\varnothing }_{\mathrm{m}}}{{\mu }_0\cdot S_D}{\varnothing }_i=-\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m}{{\mu }_0\cdot S_D}{\mathrm{G}}_i\mathit{Ni}=-\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m{\mathrm{G}}_{i}N}{{\mu }_0\cdot S_D}\ast i$$ Calculate F_1,nonlinear in a similar way as follows: $$\begin{array}{c}{F}_{1,\mathit{nonlinear}}=F_{1,1,\mathit{nonlinear}}-F_{2,1,\mathit{nonlinear}}\\ =\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2-\frac{{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}}{2{\mu }_0\cdot S_D}{{\varnothing }_i}^2=0\end{array}$$ Thus, the resultant force D₁ = (D_1,1, D_2,1) of component forces is: $$\begin{array}{c}{F}_1=F_{1,\mathit{linear}}+F_{1,\mathit{nonlinear}}\\ =-\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m{\mathrm{G}}_{i}N}{{\mu }_0\cdot S_D}\ast i+0\\ =-\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m{\mathrm{G}}_{i}N}{{\mu }_0\cdot S_D}\ast i\end{array}$$
2. 2) Calculate F_j when j = 2, corresponding to the magnetic domain pair D_j = (D_1,j, D_2,j) when j = 2.

The resultant force of the component forces corresponding to the magnetic domain pair D₂ = (D_1,2, D_2,2) is F₂ = -F_1,2 + F_2,2. Calculate the electromagnetic attractive forces in the magnetic domains D_1,2 and D_2,2, we have: $$\left\{\begin{array}{l}{F}_{1,2}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{D}1,2}\right)}^2}{2{\mu }_0\cdot S_{D1,2}}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{{\varnothing }_i}^2}{2{\mu }_0\cdot S_{D1,2}}\\ F_{2,2}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{D}2,2}\right)}^2}{2{\mu }_0\cdot S_{D2,2}}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{{\varnothing }_i}^2}{2{\mu }_0\cdot S_{D2,2}}\end{array}\right.$$ where S_D1,2 and S_D2,2 represent the areas of the annular end faces corresponding to magnetic domain D_1,2 and magnetic domain D_2,2 respectively, and S_D1,2 = S_D2,2 = S_D, thus: $$\left\{\begin{array}{l}{F}_{1,2}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{D}1,2}\right)}^2}{2{\mu }_0\cdot S_{D1,2}}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{{\varnothing }_i}^2}{2{\mu }_0\cdot S_D}\\ F_{2,2}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{\left({\varnothing }_{\mathrm{D}2,2}\right)}^2}{2{\mu }_0\cdot S_{D2,2}}={\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{{\varnothing }_i}^2}{2{\mu }_0\cdot S_D}\end{array}\right.$$ Consequently, $$F_2=F_{2,\mathit{linear}}+F_{2,\mathit{nonlinear}}=-F_{1,2}+F_{2,2}=-{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{{\varnothing }_i}^2}{2{\mu }_0\cdot S_D}+{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}\frac{{{\varnothing }_i}^2}{2{\mu }_0\cdot S_D}=0$$ We obtain: $$\left\{\begin{array}{c}{F}_{2,\mathit{linear}}=0\\ F_{2,\mathit{nonlinear}}=0\end{array}\right.$$ Since the resultant force on the movable assembly: $$F_{\mathit{moving}-\mathit{magnet}}=F_1+F_2$$ $$F_{\mathit{moving}-\mathit{magnet}}=F_{\mathit{moving}-\mathit{magnet},\mathit{linear}}+F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}$$ Thus: $$F_{\mathit{moving}-\mathit{magnet},\mathit{linear}}=F_{1,\mathit{linear}}+F_{2,\mathit{linear}}=-\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m{\mathrm{G}}_{i}N}{{\mu }_0\cdot S_D}\ast i+0=-\frac{2{\mathrm{C}}_{\mathrm{y}2\mathrm{y}}{B}_m{S}_m{\mathrm{G}}_{i}N}{{\mu }_0\cdot S_D}\ast i$$ $$F_{\mathit{moving}-\mathit{magnet},\mathit{nonlinear}}=F_{1,\mathit{nonlinear}}+F_{2,\mathit{nonlinear}}=0+0=0$$

From the above derivation process, the following characteristics may be seen:

1. 1) In the linear term of the resultant force F_{moving-magnet,linear}, the linear terms of the component forces F_1,linear and F_2,linear are superimposed, so that the linear term of the resultant force F_{moving-magnet,linear} still has a linear relationship with the coil current.
2. 2) In the nonlinear term of the resultant force F_{moving-magnet,nonlinear}, the nonlinear terms of the component forces F_1,nonlinear and F_2,nonlinear cancel each other out, so that the nonlinear term of the resultant force F_{moving-magnet,nonlinear} is zero.

The above design method is called the design method of the moving-magnet vibrator with nonlinear term cancellation. This method can not only be used to design vibrators, but also be applied to design actuators. The moving-magnet vibrators or actuators obtained by the above method are called moving-magnet vibrator devices or actuators with nonlinear term cancellation.

Referring to FIGS. 17 to 21, a design method of a moving-magnet vibrator with nonlinear term cancellation includes conditions as follows.

1. (1) A moving-magnet vibrator body 11 is provided. The moving-magnet vibrator body 11 includes an outer cylinder 1, a vibration transmission plate 9, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure, the movable assembly includes a magnet combination structure. The coil combination structure includes a coil 3 and a first magnetic conductor 4, and the magnet combination structure includes a permanent magnet 6 and a second magnetic conductor 5. The stator assembly is fixed inside the outer cylinder 1, the vibration transmission plate 9 is fixed on the outer cylinder 1, and the movable assembly is fixedly connected to the vibration transmission plate 9 through at least one contact point. The movable assembly moves while the stator assembly remains stationary, and the movable assembly is referred to as a moving component.
2. (2) The movable assembly is simultaneously subjected to paired electromagnetic forces of push and pull, presenting push-pull structural characteristics.

2N magnetic domains D_1,i and D_2,i which are designed as N symmetrical pairs are provided in the moving-magnet vibrator body 11, where N is 1, 2, 3, ..., 100, and i=1, 2, 3, ....

The number of a permanent magnet in the magnet combination structure and the number of a coil in the coil combination structure are limited, in such a manner that N_magnet>N_coil or N_magnet < N_coil , where N_magnet represents the number of the permanent magnet, and N_coil represents the number of the coil, N_magnet=1, 2, 3, ..., 100, and N_coil=1, 2, 3, ..., 100.

In the push-pull structural characteristics, linear terms of an electromagnetic force acting on the movable assembly are superimposed so as to be increased, while nonlinear terms of the electromagnetic force acting on the movable assembly are partially or completely canceled so as to be decreased.

The closed main magnetic flux lines of the coil in the coil combination structure and the closed main magnetic flux lines of the permanent magnet in the magnet combination structure each pass through the magnetic domains D_1,i and D_2,i. The magnetic domain is a spatial region filled with electromagnetic energy, is generally composed of air or a medium with low magnetic permeability (e.g., a relative magnetic permeability<1000), and includes a region where the magnet material is located. In the magnetic domain D_1,i, a direction of magnetic flux lines of the coil is the same as a direction of magnetic flux lines of the permanent magnet, while in the magnetic domain D_2,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet. Alternatively, in the magnetic domain D_1,i, the direction of the magnetic flux lines of the coil is opposite to the direction of the magnetic flux lines of the permanent magnet, while in the magnetic domain D_2,i the direction of the magnetic flux lines of the coil is the same as the direction of the magnetic flux lines of the permanent magnet.

The number of the permanent magnet 6 and the number of the coil 3 are limited, in which there are two permanent magnets, and three coils.

Six magnetic domains D_1,1, D_2,1, D_1,2, D_2,2, D_1,3, and D_2,3 which are designed as three symmetrical pairs are provided in the moving-magnet vibrator body, where D_1,1 is symmetric to D_2,1, D_1,2 is symmetric to D_2,2, and D_1,3 is symmetric to D_2,3 . The closed magnetic flux lines of the coils 3 and the closed magnetic flux lines of the permanent magnets 6 each pass through the magnetic domains D_1,1, D_2,1, D_1,2, D_2,2, D_1,3, and D_2,3. In the magnetic domain D_1,1, a direction of magnetic flux lines of the coil 3 is opposite to a direction of magnetic flux lines of the permanent magnet 6. In the magnetic domain D_2,1, the direction of the magnetic flux lines of the coil 3 is the same as the direction of the magnetic flux lines of the permanent magnet 6. In the magnetic domain D_1,2, the direction of the magnetic flux lines of the coil 3 is the same as the direction of the magnetic flux lines of the permanent magnet 6. In the magnetic domain D_2,2, the direction of the magnetic flux lines of the coil 3 is opposite to the direction of the magnetic flux lines of the permanent magnet 6.

The moving component is subjected to six forces. Each component force includes two parts: one part is a linear term of an excitation current i, and the other part is a nonlinear term of the excitation current i: $$F_{\mathit{moving}-\mathit{magnet},n}\left(i\right)=F_{\mathit{moving}-\mathit{magnet},n,\mathit{linear}}\left(i\right)+F_{\mathit{moving}-\mathit{magnet},n,\mathit{nonlinear}}\left(i\right),$$ where n = 1, 2, 3, ..., 2N - 1, 2N.

Accordingly, a resultant force on the moving component also includes two parts: one part is a linear term of the current i, and the other part is a nonlinear term of the current i, which is expressed as: $$\begin{array}{l}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force}}\left(\mathrm{i}\right)\\ ={\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{linear}}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{nonlinear}}\left(\mathrm{i}\right)\end{array}$$ where: $$\begin{array}{l}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force}}\left(\mathrm{i}\right)\\ ={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n}}\left(\mathrm{i}\right)}={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},2}\left(\mathrm{i}\right)\right)}\end{array}$$ $$\begin{array}{c}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{linear}}\left(\mathrm{i}\right)={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},\mathrm{linear}}\left(\mathrm{i}\right)}\\ ={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{linear},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{linear},2}\left(\mathrm{i}\right)\right)}\end{array}$$ $$\begin{array}{c}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{resultant}\mathrm{force},\mathrm{nonlinear}}\left(\mathrm{i}\right)={\displaystyle \sum _{\mathrm{n}=1}^{2\mathrm{N}}{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{n},\mathrm{nonlinear}}\left(\mathrm{i}\right)}\\ ={\displaystyle \sum _{\mathrm{n}=1}^{\mathrm{N}}\left({\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{nonlinear},1}\left(\mathrm{i}\right)+{\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet},\mathrm{nonlinear},2}\left(\mathrm{i}\right)\right)}\end{array}$$

That is, the nonlinear terms in each component force are partially or completely canceled. In the final total resultant force ∑_i(F_1,i + F_2,i), the nonlinear terms of the total resultant force relative to the current are partially or completely canceled, and the linear terms are superimposed to be increased, thereby obtaining the moving-magnet vibrator with nonlinear term cancellation.

The movable assembly and the stator assembly are in concave-convex shapes and arranged in an interleaved engagement manner. The closed main magnetic flux lines of the coils 3 and the closed main magnetic flux lines of the permanent magnet 6 alternately pass through the movable assembly and the stator assembly.

The design method further includes conditions as follows.

(3.1) Looking outward from a center, the coil 3 is located outside and the permanent magnet is located inside.

(3.2) N_coil=3, and N_magnet=2.

(3.3) When N_magnet > 1, the polarities of the two opposite end faces of the adjacent permanent magnets 6 are the same. When N_coil>1, the currents in adjacent coils 3 have opposite directions, and the polarities of the electromagnetic fields on the two end faces of two adjacent coils 3 are the same.

The magnetic conductor is arranged at a position of the outer cylinder 1 close to the coil 3, so as to minimize the magnetic resistance of the magnetic circuit of an electromagnet generated by the coil 3. A yoke iron is arranged around the coil 3 and the permanent magnet 6, or alternatively, for the coil combination structure, a part of outer cylinder a close to the coil is magnetically conductive.

Referring to FIGS. 17 to 21, a moving-magnet vibrator device with nonlinear term cancellation designed by the method of Embodiment 7 includes a moving-magnet vibrator body 11. The moving-magnet vibrator body 11 includes an outer cylinder 1, a vibration transmission plate 9, a stator assembly and a movable assembly. The stator assembly includes a coil combination structure, and the movable assembly includes a magnet combination structure. The coil combination structure includes a coil 3 and a first magnetic conductor 4, the magnet combination structure includes a permanent magnet 6 and a second magnetic conductor 8. The coil combination structure further includes a first magnetic conductor ring 2. When looking from a center outward, the coil 3 is located outside and the permanent magnet 6 is located inside. Two permanent magnets 6 are provided, and polarities of two opposite end faces of adjacent permanent magnets 6 are the same. Three coils 3 are provided, directions of currents in adjacent coils 3 are opposite, and polarities of electromagnetic fields at adjacent end faces of two adjacent coils 3 are the same. Two vibration transmission plates 9 are provided and are respectively fixed on a top surface and a bottom surface of the outer cylinder 1. Two permanent magnets 6 are fixed on two sides of the second magnetic conductor 8. The two permanent magnets are respectively fixed on magnetic conductive sleeves 5, and the magnetic conductive sleeves 5 are respectively fixed on the two vibration transmission plates 9. The three coils are sequentially fixed on an inner side wall of the outer cylinder 1. The first magnetic conductor 4 is fixedly arranged between adjacent coils 3. The first magnetic conductive ring 2 is fixedly arranged on an outer side of the outermost coils 3, and both the first magnetic conductor 4 and the first magnetic conductive ring 2 are fixed on the inner side wall of the outer cylinder 1. The movable assembly and the stator assembly are in concave-convex shapes and arranged in an interleaved engagement manner. Closed magnetic flux lines of the coils 3 and closed magnetic flux lines of the permanent magnets 6 alternately pass through the movable assembly and the stator assembly. Six magnetic domains D_1,1, D_2,1, D_1,2, D_2,2, D_1,3, and D_2,3 which are designed as three symmetrical pairs are provided in the moving-magnet vibrator body 1, where D_1,1 is symmetric to D_2,1, D_1,2 is symmetric to D_2,2, and D_1,3 is symmetric to D_2,3. The closed magnetic flux lines of the coils 3 and the closed magnetic flux lines of the permanent magnets 6 each pass through the magnetic domains D_1,1, D_2,1, D_1,2, D_2,2, D_1,3, and D_2,3. In the magnetic domain D_1,1, a direction of magnetic flux lines of the coil 3 is opposite to a direction of magnetic flux lines of the permanent magnet 6. In the magnetic domain D_2,1, the direction of the magnetic flux lines of the coil 3 is the same as the direction of the magnetic flux lines of the permanent magnet 6. In the magnetic domain D_1,2, the direction of the magnetic flux lines of the coil 3 is the same as the direction of the magnetic flux lines of the permanent magnet 6. In the magnetic domain D_2,2, the direction of the magnetic flux lines of the coil 3 is opposite to the direction of the magnetic flux lines of the permanent magnet 6.

Referring to FIG. 18, the magnetic action domains D_1,1, D_2,1, D_1,2, D_2,2, D_1,3, and D_2,3 are formed by 6 air gaps respectively. In the magnetic action domain, the total magnetic flux/magnetic induction intensity generated by the superposition of the magnetic field generated by the permanent magnets and the magnetic field generated by the electromagnet of the coils causes mutual forces between the components around the magnetic domain. The above magnetic domains D_1,1, D_2,1, D_1,2 , D_2,2 , D_1,3, and D_2,3 are enclosed by the stator assembly and the movable assembly. Therefore, in these magnetic domains, mutual forces are generated between the stator assembly and the movable assembly.

FIG. 19 illustrates the closed magnetic flux lines of the coils C₁, C₂, and C₃, as well as the closed magnetic flux lines of the magnets M₁ and M₂. The closed magnetic flux lines generated by the coil C₁ pass through the magnetic gaps D_1,2 and D_1,3, the closed magnetic flux lines generated by the coil C₂ pass through the magnetic gaps D_1,1 and D_2,1, and the closed magnetic flux lines generated by the coil C₃ pass through the magnetic gaps D_2,2 and D_2,3. The closed magnetic flux lines generated by the magnet M₁ sequentially pass through the magnetic gaps D_1,1 and D_1,2, and the closed magnetic flux lines generated by the magnet M₂ sequentially pass through the magnetic gaps D_2,1 and D_2,2.

Assuming that the currents passing through the coils C₁, C₂, and C₃ are i₁, i₂, and i₃ respectively, and i₁ = i₂ = i₃ = i. The corresponding magnetic fluxes of the coils are Φ_i1, Φ_i2 and Φ_i3 respectively. For the simplicity of derivation, assuming that Φ_i1 = Φ_i2 = Φ_i3 = Φ_i (another case is N₁ = N₃ ≠ N₂, where N₁, N₂, N₃ are the number of turns of the coins C₁, C₂, and C₃, respectively; or the magnetic resistance of C₂ is different from that of C₁ and C₃ due to the magnetic circuit structures of C₁, C₂, and C₃, thus Φ_i1 = O_i3 ≠ Φ_i2. For this case, because of the overall symmetrical design, the conclusion that the nonlinear terms in the resultant force of the paired forces cancel each other still holds). The magnetic fluxes corresponding to the permanent magnets M₁ and M₂ are Φ_M1 = Φ_M2 = Φ_m.

The magnetic domains D_1,1, D_2,1, D_1,2, D_2,2, D_1,3, and D_2,3 may be grouped in pairs based on the symmetry. D_1,1 and D_2,1 are the first pair of symmetrically arranged magnetic domains, D_1,2 and D_2,2 are the second pair of symmetrically arranged magnetic domains, and D_1,3 and D_2,3 are the third pair of symmetrically arranged magnetic domains.

1. 1) Magnetic domain pair D_j = (D_1,j, D_2,j) when j=1.
   In the magnetic domain D_1,1, the direction of the magnetic flux lines corresponding to the coil C₂ is opposite to that corresponding to the permanent magnet M₁. Therefore, the total magnetic flux in the magnetic domain D_1,1 is the difference between Φ_i1 and Φ_M1 = Φ_m. In the magnetic domain D_2,1, the direction of the magnetic flux lines corresponding to the coil C₂ is the same as that corresponding to the permanent magnet M₂. Therefore, the total magnetic flux in the magnetic domain D_2,1 is the sum of Φ_i2 and Φ_M2 = Φ_m.
   Assuming that i₁ = i₂ = i₃ = i, Φ_i1 = Φ_i3 ≠ Φ_i2, and in the multiple magnetic domains, the direction of the magnetic flux lines of the permanent magnet is positive, and the magnetic flux is also positive, then: $${\mathrm{\Phi }}_{\mathrm{D}\mathrm{1,1}}={\mathrm{\Phi }}_{\mathrm{M}1}-{\mathrm{\Phi }}_{\mathrm{i}2}={\mathrm{\Phi }}_{\mathrm{m}}-{\mathrm{\Phi }}_{\mathrm{i}2}$$ $${\mathrm{\Phi }}_{\mathrm{D}\mathrm{2,1}}={\mathrm{\Phi }}_{\mathrm{M}1}+{\mathrm{\Phi }}_{\mathrm{i}2}={\mathrm{\Phi }}_{\mathrm{m}}+{\mathrm{\Phi }}_{\mathrm{i}2}$$
2. 2) Magnetic domain pair D_j = (D_1,j, D_2,j) when j=2.
   In the magnetic domain D_1,2, the direction of the magnetic flux lines corresponding to the coil C₁ is the same as that corresponding to the permanent magnet M₁. Therefore, the total magnetic flux in the magnetic domain D_1,2 is the sum of Φ₁₁ = Φ_i3 and Φ_M1 = Φ_m. In the magnetic domain D_2,2, the direction of the magnetic flux lines corresponding to the coil C₃ is opposite to that corresponding to the permanent magnet M₂. Therefore, the total magnetic flux in the magnetic domain D_2,2 is the difference of Φ_i3 = Φ_i1 and Φ_M2 = Φ_m.
   Assuming that i₁ = i₂ = i₃ = i, Φ_i1 = Φ_i3 ≠ Φ_i2, and in the multiple magnetic domains, the direction of the magnetic flux lines of the permanent magnet is positive, and the magnetic flux is also positive, then: $${\mathrm{\Phi }}_{\mathrm{D}\mathrm{1,2}}={\mathrm{\Phi }}_{\mathrm{M}1}+{\mathrm{\Phi }}_{\mathrm{i}1}={\mathrm{\Phi }}_{\mathrm{m}}+{\mathrm{\Phi }}_{\mathrm{i}1}$$ $${\mathrm{\Phi }}_{\mathrm{D}2,1}={\mathrm{\Phi }}_{\mathrm{M}1}-{\mathrm{\Phi }}_{\mathrm{i}3}={\mathrm{\Phi }}_{\mathrm{m}}-{\mathrm{\Phi }}_{\mathrm{i}1}$$
3. 3) Magnetic domain pair D_j = (D_1,j, D_2,j) when j=3.

In the magnetic domain D_1,3, only the magnetic flux lines corresponding to the coil C₁ pass through, so the total magnetic flux is only Φ_i1. In the magnetic domain D_2,3, only the magnetic flux lines corresponding to the coil C₃ pass through, so the total magnetic flux is only Φ_i3 = Φ_i1.

Assuming the magnetic resistances of the magnetic circuits formed by the electromagnetic field generated by the current i in the coils C₁, C₂ and C₃ are Z_i, N represents the number of turns of the coils C₁, C₂ and C₃, and i is the current intensity, we have: $${\varnothing }_i\ast {\mathrm{Z}}_i=N\ast i$$

Assuming that the magnetic conductance of the magnetic circuit formed by the electromagnetic field generated by the current is G_i, we have: $${\varnothing }_i={\mathrm{G}}_i\ast N\ast i.$$

Another case is: N₁ = N₃ ≠ N₂, where N₁, N₂, N₃ are the number of turns of the coins C₁, C₂, and C₃, respectively, or the magnetic resistance of C₂ is different from the magnetic resistance of C₁ and C₃ due to the magnetic circuit structures of C₁, C₂, and C₃, i.e., G_i1 = G_i3 ≠ G_i2, thus Φ_i1 = Φ_i3 ≠ Φ_i2. For this case, due to the overall symmetrical design, it still holds the conclusion that the nonlinear terms in the resultant force of the paired forces cancel each other. Thus: $${\varnothing }_{i,1}={\varnothing }_{i,3}={\mathrm{G}}_{i,1}\ast {\mathrm{N}}_{i,1}\ast i$$ $${\varnothing }_{i,2}={\mathrm{G}}_{i,2}\ast {\mathrm{N}}_{i,2}\ast i$$

The magnetic flux corresponding to the permanent magnet may also be expressed by the equation of magnetic induction intensity. Assuming that the magnetic induction intensities at the magnetic pole end faces of the permanent magnets M₁ and M₂ are both B_m1, the area of the magnetic pole end face is S_m, it follows that Ø_m = B_mS_m.

Thus, we have: $$\left\{\begin{array}{c}{\varnothing }_{D1,1}={\varnothing }_m-{\varnothing }_i=B_m{S}_m-{\mathrm{G}}_i\ast N\ast i\\ {\varnothing }_{D2,1}={\varnothing }_m+{\varnothing }_i=B_m{S}_m+{\mathrm{G}}_i\ast N\ast i\end{array}\right.$$ $$\left\{\begin{array}{c}{\varnothing }_{D1,2}={\varnothing }_m+{\varnothing }_i=B_m{S}_m+{\mathrm{G}}_i\ast N\ast i\\ {\varnothing }_{D2,2}={\varnothing }_m-{\varnothing }_i=B_m{S}_m-{\mathrm{G}}_i\ast N\ast i\end{array}\right.$$ $$\left\{\begin{array}{c}{\varnothing }_{D1,3}={\varnothing }_i={\mathrm{G}}_i\ast N\ast i\\ {\varnothing }_{D2,3}={\varnothing }_i={\mathrm{G}}_i\ast N\ast i\end{array}\right.$$

For another case: the number of turns N₁, N₂, N₃ of the coils C₁, C₂, and C₃ satisfy N₁ = N₃ ≠ N₂, or the magnetic resistance of C₂ is different from the magnetic resistance of C₁ and C₃ due to the magnetic circuit structures of C₁, C₂, and C₃, G_i1 = G_i3 ≠ G_i2, thus Φ_i1 = Φ_i3 ≠ Φ_i2. The above formulas become: $$\left\{\begin{array}{c}{\varnothing }_{D1,1}={\varnothing }_m-{\varnothing }_i=B_m{S}_m-{\mathrm{G}}_{i,2}\ast N_{i,2}\ast i\\ {\varnothing }_{D2,1}={\varnothing }_m+{\varnothing }_i=B_m{S}_m+{\mathrm{G}}_{i,2}\ast N_{i,2}\ast i\end{array}\right.$$ $$\left\{\begin{array}{c}{\varnothing }_{D1,2}={\varnothing }_m+{\varnothing }_i=B_m{S}_m+{\mathrm{G}}_{i,1}\ast N_{i,1}\ast i\\ {\varnothing }_{D2,2}={\varnothing }_m-{\varnothing }_i=B_m{S}_m-{\mathrm{G}}_{i,3}\ast N_{i,3}\ast i\end{array}\right.$$ $$\left\{\begin{array}{c}{\varnothing }_{D1,3}={\varnothing }_i={\mathrm{G}}_{i,1}\ast {\mathrm{N}}_{i,1}\ast i\\ {\varnothing }_{D2,3}={\varnothing }_i={\mathrm{G}}_{i,3}\ast {\mathrm{N}}_{i,3}\ast i\end{array}\right.$$

It can be seen from the above formulas that when N₁ = N₃ and G_i1 = G_i3, the magnetic fluxes in the magnetic domain pair D₂ = (D_1,2, D_2,2), as well as the magnetic domain pairs D₁ = (D_1,1, D_2,1) and D₃ = (D_1,3, D_2,3) still have the property that the nonlinear terms relative to the current in the corresponding component forces can be canceled.

Referring to FIG. 20, FIG. 20 is a diagram of the relationship between the movable assembly, the magnetic domains D_1,1, D_2,1, D_1,2, D_2,2, D_1,3, and D_2,3 and the stator assembly. In the magnetic domain D_1,1, the movable assembly is subjected to a leftward attractive force F_1,1 from the stator assembly. In the magnetic domain D_2,1, the movable assembly is subjected to a rightward attractive force F_2,1 from the stator assembly. In the magnetic domain D_1,2, the movable assembly is subjected to a rightward attractive force F_1,2 from the stator assembly; in the magnetic domain D_2,2, the movable assembly is subjected to a leftward attractive force F_2,2 from the stator assembly. In the magnetic domain D_1,3, the movable assembly is subjected to a leftward attractive force F_1,3 from the stator assembly; and in the magnetic domain D_2,3, the movable assembly is subjected to a rightward attractive force F_2,3 from the stator assembly. The resultant force corresponding to the magnetic domain pair D_j = (D_1,j, D_2,j) is defined as the resultant force F_j = (F_1,j, F_2,j) corresponding to the magnetic domain pair D_j. Assuming the rightward direction is positive, the sign of F_1,j and F_2,j is positive when the direction of F_1,j and F_2,j is rightward, and the sign of F_1,j and F_2,j is negative when the direction of F_1,j and F_2,j is leftward. Then, the resultant force exerted on the movable assembly by the stator assembly is: $${\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet}}={\mathrm{F}}_1+{\mathrm{F}}_2+{\mathrm{F}}_3=-{\mathrm{F}}_{1,1}+{\mathrm{F}}_{2,1}+{\mathrm{F}}_{1,2}-{\mathrm{F}}_{2,2}-{\mathrm{F}}_{1,3}+{\mathrm{F}}_{2,3}$$ $${\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet}}={\mathrm{F}}_1+{\mathrm{F}}_2+{\mathrm{F}}_3=\left(-{\mathrm{F}}_{1,1}+{\mathrm{F}}_{2,1}\right)+\left({\mathrm{F}}_{1,2}-{\mathrm{F}}_{2,2}\right)+\left(-{\mathrm{F}}_{1,3}+{\mathrm{F}}_{2,3}\right)$$ where F_j is the resultant force of the component forces generated on the movable assembly in the magnetic domain pair D_i = (D_1,j, D_2,j).

The above is a force analysis diagram of the isolated movable assembly. The movable assembly is subjected to component forces F_1,1, F_2,1, F_1,2, F_2,2, F_1,3, F_2,3 from the stator assembly. When calculating, first calculate F₁ according to the magnetic domain pair D_j = (D_1,j, D_2,j), then calculate the resultant force of the movable assembly as: $${\mathrm{F}}_{\mathrm{moving}-\mathrm{magnet}}=F_1+F_2+F_3=\left(-F_{1,1}+F_{2,1}\right)+\left(F_{1,2}-F_{2,2}\right)+\left(-F_{1,3}+F_{2,3}\right)$$

The above can also be expressed by including the direction of the force through the sign of the component force: $$F_{\mathrm{moving}-\mathrm{magnet}}={\displaystyle \sum _{i=1}^3{F}_i}={\displaystyle \sum _{i=1}^3\left(F_{1,i}+F_{2,i}\right)}$$

First calculate the resultant force of the component forces generated in the two-by-two paired magnetic domain pairs, i.e., calculate the resultant force F₁ = -F_1,1 + F_2,1 in the magnetic domain pair D₁ = (D_1,1, D_2,1), the resultant force F₂ = F_1,2 - F_2,2 in the magnetic domain pair D₂ = (D_1,2, D_2,2), and the resultant force F₂ = -F_1,3 + F_2,3 in the magnetic domain pair D₃ = (D_1,3, D_2,3), then calculate the total resultant force.