Direct 3D Method for Performance Prediction of Linear ... - CiteSeerX

Direct 3D Method for Performance Prediction of Linear Moving Coil Actuator with Various Topologies J. C. Compter1 E.A. Lomonova

2

J. Makarovic2

Technical University of Eindhoven 1

Department of Mechanical Engineering, Group of Precision Engineering Postbus 513, 5600 MB Eindhoven, The Netherlands phone/fax: +31 40 247 3647/ 243 5330, [email protected]

2

Department of Electrical Engineering, Group of Electromechanics and Power Electronics Postbus 513, 5600 MB Eindhoven, The Netherlands phone/fax: +31 40 247 3710/ 243 4364, [email protected]

Abstract: The direct 3-dimensional (3D) method represents a mathematical assembling of the magnetic field of separate cuboidal magnets and vectors of a coil current density of a linear moving coil actuator within the global space. The mathematical description allows a variation of design parameters of the actuator and position of the coils with respect to the magnet array. The presented method describes the complete analytical derivation of the 3D magnetic field and coil current. Then a numerical integration is applied to obtain the forces and torques appearing in 6 degrees of freedom (DoF) actuator as the functions of the relative position of the coils with respect to the magnets in 3D space. The method and its governing equations are implemented as a rapid-design computational tool in Mathcad, and further the simulation results are experimentally verified.

1.

INTRODUCTION

A complex mathematical description of physical phenomena and great number of design parameters, as, for example, magnetic and electric circuit dimensions of electromechanical actuators, need to be handled during a design, performance prediction or shape optimization of actuators. The mathematical equations of the phenomena can be linear or nonlinear, and this determines a complexity of mathematical apparatus used to solve the design problem and an amount of time needed for a computation. The primary strategy during the linear moving coil actuator (LMCA) design [1] is to find a balance among costs, volume and performance that are evaluated in high accuracy actuators by losses at a certain force level; position dependency of the motor constant; forces and torques in other directions, as needed under the rising demands in the semi-conductor industry. Thus, all the forces F and torques T of the actuator along and around x-, y- and z-axes, respectively, should be investigated as the functions of the x-, y- and z-positions of the coils with respect to the magnet array and this means the full 6 DoF performance prediction. Previous design analyses done by an analytical equivalent electromagnetic circuit approach [2, 3, 4] show that the method can be easily programmed, and results can be quickly obtained, but still a high number of assumptions are the inherent sources of inaccuracies, and, therefore, the limited accuracy of the results is reached. Another so called combined methods that use both equivalent electromagnetic circuit and 2D finite element method (FEM) [5, 6, 7] reduce the number of assumptions, which results in higher accuracy, but increasing the computational time. These analyses involve a change in the model geometry and a new mesh generation. Further, the computational time depends on the number of the mesh elements and computational power of a processor. The design analyses are also carried out by 3D FEM [6]. In this case the highest accuracy could be achieved while the longest computational time is needed. Additionally, the quantities desired to be extracted, such as Fx, Fy and Fz and a force distribution do not emerge directly from the finite element computation, and a post-processing of node potentials is required. Although the three mentioned approaches are widely used and can be chosen appropriately for an electromagnetic study, they do not offer the unique combination of attributes as the presented 6 DoF behavior investigation, its simplicity, rapid extraction of parameters and flexibility for an investigation of different actuator topologies at the same time.

1

The purpose of the paper is to present a new theoretical and experimental investigation of LMCA (static approach) based on the general electromagnetic theory. Particular attention is paid to the development of general relations between forces (along x-, y- and z-axes), torques (around x-, y- and z-axes) and the coil positions (in x-, y- and z-directions) with respect to magnet array. Furthermore, an analysis of the magnetic circuit in the actuator yields the relationship between forces (torques) and coil currents while taking into account the magnet and coil dimensions. A more extended form might even allow an investigation directed to the influence of the magnetization direction and the rotation of the coils over one or more axes on the production of forces. This proposed direct method is further implemented in the form of rapid-design tool in a way that optimally uses Mathcad programming language, keeps the clear structure of the program for simple adjustments of any parameters and functions utilized in program and could be easily modified for the different actuator topologies. With the aim to demonstrate the features of the created approach the LMCA for linear drive systems with nanometer accuracy (see Figure 1), consisting of two current carrying coils, six magnets attached to the iron yoke, is selected. The approach is based on the proper orientation and placement of the local coordinate systems of individual parts of the actuator (permanent magnets, windings) in the global space, where the total forces and torques are calculated. The core of the method is the Lorentz force equation: r r r dF = B × J ⋅ dv ,

(1)

r r where the first term B is the vector magnetic field density of the actuator magnets, J is the vector of current density in the coil, and dv is the volume element of the wire in the magnetic field. Thus, the modeling procedure starts with the description of the analytical equations of the magnetic field around a cuboidal magnet in the local coordinate system. Secondly, a mathematical composition of the separate permanent magnets with different dimensions into the global coordinate system is described analytically, and a treatment of magnets in vicinity of a ferromagnetic material is explained, yielding the total magnetic flux density of the magnet array in the area of interest (air gap). The second term of the Lorentz’s equation, the coil current density in all parts of the winding, is analytically described in the global space as a piecewise continuous function. Since the first two terms of equation (1) are defined, the forces and torques of the actuator along and around x-, y- and z-axes, respectively, are obtained by an integrating of the forces and torques over the volume of coils. Finally, the simulation results are presented and compared with experimental results.

2.

ANALITICAL DESCRIPTION OF CUBOIDAL PERMANENT MAGNET IN LOCAL SPACE

The modeling procedure starts with the analytical expressions of the cuboidal magnet in the air magnetized in zL direction. The following assumptions are made: - the B-H curve of the permanent magnet material in the second quadrant is linear, - the relative magnetic permeability of the magnet equals 1. The mathematical description of a cuboidal magnet is based on the surface magnetic charge model of a rectangular surface with uniformly distributed magnetic charges [8]: r σ 1 B x,y,z ( xL , yL ,z L ) = ∑ ( − 1)i+ j εx,y,z (Si ,T j ,Ri,j ) , 4π i,j =0 r

r

σ = Br ⋅ n ,

(2)

(3)

where

σ - density of the magnetic charge,

r n - unity vector normal to the magnet pole surface,

r B r - vector of the residual magnet remanence,

ε x = ln( Ri , j − T j ) - for Bx, ε y = ln( Ri , j − Si ) - for By,

(4)

2

 Si ⋅ T j  - for B , z  R ⋅ z  i , j L  

ε z = arctg 

Ri,j = Si2 + T j2 + z L2 , S i = xL − a ⋅ ( −1 )i ,

(5)

T j = yL − b ⋅ ( −1 ) , j

with a - 1/2 width of the pole surface, b - 1/2 length of the pole surface, c - 1/2 height of the magnet, xL, yL, zL - coordinates of the local space. The magnetic flux density components around the cuboidal magnet in the local space, which are derived from equation (2), are equal to (see Appendix for the symbols notation): BxL =

σ  ( β + ρ ) ⋅ (τ − ζ ) ⋅ ( β + υ ) ⋅ (φ − ζ )  , ln   4 ⋅ π  ( β + χ ) ⋅ (ψ − ζ ) ⋅ ( β + ω ) ⋅ (Θ − ζ ) 

(6)

ByL =

 (α + ρ ) ⋅ (α + τ ) ⋅ (υ − ε ) ⋅ (φ − ε )  , σ ln   4 ⋅ π  (α + χ ) ⋅ (α + ψ ) ⋅ (ω − ε ) ⋅ (Θ − ε ) 

(7)

BzL

 −1  α ⋅ β  −1  α ⋅ β  −1  ζ ⋅ α   tg   − tg   + tg    µ ρ η χ ⋅ ⋅      µ⋅ρ     .(8) σ  −1  α ⋅ ζ  −1  ε ⋅ β  −1  β ⋅ ε   tg tg = ⋅ −tg  + −      4 ⋅π   η ⋅τ   µ ⋅ω   η ⋅υ      ε ⋅ζ  −1  ε ⋅ ζ  +tg −1    − tg    µ ⋅φ   η ⋅Θ   

Figure 2 shows the designation of the magnet dimensions, the direction of magnetization vector, and it determines the orientation of the magnet with respect to the local coordinate system, whose origin is fixed to the geometrical center of the magnet. The equation (2) for the internal part of the cuboidal magnet differs only in the z-component BzL where the remnant magnetic flux density is added:  −1  α ⋅ β  −1  α ⋅ β  −1  ζ ⋅ α   tg   − tg   + tg    ⋅ ⋅ µ ρ η χ      µ⋅ρ    .(9)  σ  −1  α ⋅ ζ  −1  ε ⋅ β  −1  β ⋅ ε   ⋅ −tg  BzL = Br +  + tg   − tg   4 ⋅π   η ⋅τ   µ ⋅ω   η ⋅υ      ε ⋅ζ  −1  ε ⋅ ζ  +tg −1    − tg    µ ⋅φ   η ⋅Θ   

3.

MATHEMATICAL COMPOSITION OF PERMANENT MAGNETS INTO GLOBAL SPACE

With the help of the equations (6) – (9), which describe the field of the cuboidal magnet magnetized in zLdirection, any desired magnetic array can be built in the global space by a shifting and/or turning the magnets in the global space where the magnetic field and the performance of the complete actuator has to be investigated.

3.1

Translation of permanent magnets in global space

The translation of the individual magnets in the global space is defined with a help of three position vectors that relate the origins of the local and global spaces, as shown in Figure 3. 3

From the origin to the center of the magnet holds: uuur 0L0 G = [ x0

T z0 ] .

y0

(10)

The position of point P in the local coordinate system is: uuur

( P0 )

= [ xL

L

L

yL

T zL ] .

(11)

And the position of P in the global space can be expressed by the two vectors above as: uuur

uuur

( P0 ) = ( P0 ) G

L

G

L

uuur + 0L0 G = [ xL + x0

yL + y0

T z L + z0 ] .

(12) Thus, when the middle point of the magnet is shifted from the origin of the global space by origin’s uuur vector 0L0 G (x0, y0, z0), the magnetic field of the magnet at the point P of the global space is calculated by uuur equations (6) – (9) whose coordinates are shifted by the vector 0L0 G : r r B xG ,yG ,z G ( xG , yG ,zG ) = B xL ,y L ,z L ( xL + x0 , yL + y0 ,z L + z0 ) . (13)

3.2

Rotation of permanent magnets in global space

The application of “Halbach” magnet arrays [9] in actuators is not exceptional nowadays. The major advantage of the addition of 90 degrees rotated magnets between the magnets indicated in Figure 1 is an improvement of the actuator performance. The rotation of individual magnets in the global space is defined by so-called rotation matrix. It is done by multiplying the rotational matrix MR by the vector of the magnetic field density at each point of the local space: r r BG = M R ⋅ B L .

(14)

This operation allows performing any rotation of the magnet and its magnetic field in the global space. The matrix consists of three unity vectors in the x-, y- and z-directions: r T u x = [1 0 0 ] , r T u y = [ 0 1 0] , r T u z = [ 0 0 1] .

(15)

The vectors are arranged in the single column matrix MR in a way that provides a certain rotation. For example, the rotation of the local coordinates around the x-axis by 90° counterclockwise means that xLaxis of the local space stays identical with XG of the global space and yL- and zL-axes are rotated. Mathematically it is represented by the exchange of the rotated axes’ unity vectors and multiplying the vectors by the sign of the global axes, which the local axis is rotated to (Figure 4): r    BLx  u xT r r  rT  . BG = M R ⋅ B L =  u z ⋅ sign ( YG )  ⋅  BLy    r u Ty ⋅ sign ( Z G )   BLz   

3.3

(16)

Replacement of permanent magnet and ferromagnetic material by the permanent magnets and its image

The primary goal of this method is to investigate the static actuator performance (force and torque prediction), thus, the eddy currents caused by transient effects (e.g. high-speed motion) are out of the scope of this method. For this reason, it is not necessary to analyze the magnetic field in the

4

ferromagnetic yoke, but it is essential to take the presence of the ferromagnetic material in vicinity of the permanent magnet into account. Therefore, the magnet and the ferromagnetic material are replaced by the original magnet with its image. To ensure that this replacement is equivalent the following assumptions are made: - the magnetic permeability goes to infinity - the effect of finite dimensions of the yoke can be neglected. Under these assumptions the permanent magnet and the yoke can be replaced by the magnet and its image (Figure 5). In a case a finite value of the magnetic permeability of the iron yoke has to be used, the image magnet has to have a lower value of the remanence magnetic density Br than the original magnet. The new Br value of the image magnet can be obtained with help of B. Hague treatment ([10], pp. 106-107). From the assumption made in this subsection and section 2 follow that the model is based on linear behavior of the magnets and ferromagnetic materials. Therefore the saturation of magnetic parts, which can appear during the period of high acceleration (extreme values of current densities), is not considered. These nonlinearities should be investigated by means of FE analysis. Sections 2 and 3 show how to describe analytically the total magnetic flux density of the magnetic array starting from the equations of the cuboidal magnet, and then the assembling of different cuboidal magnets in to the one magnetic array with ferromagnetic material. The following section explains a structure of the piecewise continuous function of the coil current density that can be found in equation (1) as the second term .

4.

PIECEWISE CONTINUOS FUNCTION OF COIL CURRENT DENSITY

The vector of the coil current density is defined in this case in terms of sine and cosine functions or constants in eight parts of the coil (Figure 6): r - part 1: J =  J x J y J z  = J ⋅ [ −1 0 0] ,   r - part 2: J = J ⋅  − cos (θ ) sin (θ ) 0  , r - part 3: J = J ⋅ [ 0 1 0] , r - part 4: J = J ⋅ sin (θ ) cos (θ ) 0  . r - part 5: J = J ⋅ [1 0 0] , r - part 6: J = J ⋅  cos (θ ) − sin (θ ) 0  , r - part 7: J = J ⋅ [ 0 −1 0] , r - part 8: J = J ⋅  − sin (θ ) − cos (θ ) 0  . The vector of the current density is defined by the piecewise continuous function. It is attractive to define a parameter m (0..1), which value is coupled with the actual position on a wire loop (Figure 7) and can be used to refer to one of the current density functions mentioned above. The current in the second coil of LMCA has the same value but flows counterclockwise. For this reason the current vector is multiplied by –1 in all parts of the second coil. When the vectors of the magnetic field density (section 2) and current density (section 4) are defined the forces and torques produced by the coils can be calculated.

5.

FORCE AND TORQUE OF THE ACTUATOR

In this analytical treatment the reluctance forces between the current carrying coils and iron yokes are neglected. Therefore only the Lorentz’s formula is used. Then, the total force and torque of the linear moving coil actuator are found as the volume integrals: ur r r F = ∫∫∫ J × BG ⋅ dV ,

(17)

ur r r r T = ∫∫∫ r × J × BG ⋅ dV ,

(18)

coil

coil

5

r r where J is the vector of the current density, BG is the vector of the magnetic field density in the global

r

space and r is the distance of the volume element dV from an observation point. The integrals are calculated numerically. Therefore, the three dimensional coil regions are subdivided into points Pi that lay on the loops Li. The points Pi and loops Li are defined by three parameters (Figure 7): - k goes from the inside to outside circumference of the coil, - l determines the height of the coil, - m goes around the coil. These parameters are local, therefore, the relation between them and the global coordinates should be established. Further, the parameters k and l determine the integration loops equally spaced from each other for the next step of integration. The total force and torque of the actuator are calculated in two steps. First, all forces and torques are summed separately on each loop. Afterwards, the forces and torques on the adjacent loops are integrated by using the following formula with the nine-point approximation [11]:

∫∫

n

f ( x, y ) ⋅ dx ⋅ dy = b ⋅ h∑ wi ⋅ f ( xi , yi ) + R .

(19)

i

S

with h as the height, b as the width of the coil sides. The position of the integration lines and the weight functions wi are given by Abramowitz [11]. The function f(xi,yi) is in this case the force or torque on the summation loop Li, and R is the remainder. 6.

SIMULATION RESULTS OF LINEAR MOVING COIL ACTUATOR

The presented method is used to analyze the air coil actuator with topology shown in Figure 1 (the dimensions of the permanent magnets and coils of the actuator are specified in Tables 1 and 2). The governing equations and whole procedure are implemented in CAS software Mathcad. The Mathcad program is performed on a PC with AMD Athlon 650MHz processor, 128 MB RAM. The coils, shifted in z-plane by 1.45 mm downwards from the middle position of the coils in the air gap, are carrying a current of 120 mA. The current density level in the coils is far below 20 A/mm2. FEA can prove that the reluctance forces between the coils and iron yokes can be neglected at this current level. Each of the nine integration loop in the coil has 200 equally distributed points, and all forces and torques, produced by the actuator, are calculated over an x-y plane consisting of 15×8 coil positions. The y- and z-components of the magnetic field densities in the air gap are shown in Figure 8 and 9, respectively. They are calculated on the y-z plane, in the middle of the magnets in x-direction (in accordance with Figure 1), divided by 21×21 points that builds up a grid with the size of 4.5 mm in ydirection and of 0.345 mm in z-direction. The middle position of the air gap corresponds to the point with coordinates (11, 11) along the y- and z-axes. For a better comparison of the simulation and measurement results the absolute values of the forces along y- (main force) and z-directions are computed (see Figure 10 and 11, respectively). They are calculated on the x-y plane, shifted downwards by 1.45 mm from the middle of the air gap in z-direction (in accordance with Figure 1), divided by 8×15 positions that builds up a grid with the size of 0.5 mm in both x- and ydirections. The calculation time of 12 minutes is to obtain the magnetic field density in the air gap, and all forces and torques produced by the actuator at the 8×15 positions. The calculation time may be reduced up to 2 minutes and 40 seconds if the number of the points on one integration loop is reduced to 40, but this reduction deteriorates the accuracy of the results. For only an approximate comparison of the proposed method and FE method computational time PC with Intel IV 2.4 GHz, 1MB RAM and 3D FE program that creates a mesh of the air gap (magnets air and coil of the actuator) with about 100 000 tetrahedral elements is used [12]. In this case only the calculation of the magnetic field and the three forces in x-, y- and z-directions for only one position of the coils respect to the magnets takes about 25 minutes.

6

7.

EXPERIMENTAL RESULTS

The experimental results are obtained by measuring the forces of the actuator with a 4-component piezoelectric dynamometer KISTLER 9272A, capable to measure simultaneously three perpendicular forces and one torque. The actuator with the sensor is attached to a mechanical setup with 3 translational DoF movements and a resolution of 0.01 mm in each direction. The outputs of the sensor are the charge signals converted to voltage output signals by a KISTLER charge amplifier. Because the output signals fall down with a certain time constant of the amplifier, the coils of the actuator are supplied by a 14 Hz bipolar square current. The fast rise and fall down times of the current minimize the influence of the time constant of the amplifier. Then the rms values of the output signals, which are proportional to forces, are measured by rms meters and an oscilloscope. Because of that the measured forces have always the positive values. The measured forces Fy and Fz are shown in the Figures 12, 14 and 13, 15, respectively. They are measured with coils fed with the rectangular current waveform with an amplitude of 120 mA. The other force Fx and torque Tz components are also given by the force sensor, but they are so small that it is not feasible to obtain reliable data-readings. The shapes of the force surfaces are identical in comparison with the simulation results except for the Fz force component that does not reach a zero force level, and whose minimum does not coincide with the middle position of the measurement range in x-direction. These differences can be caused by: - the threshold of the force sensor that is less than 20 mN, - the crosstalk between Fy and Fz forces of the sensor that is less than 2%, - the fact that the magnetization of the permanent magnets differs from the ideal one assumed during the simulation (average and local spatial strength and direction), - the position shift (approximately 0.5 mm) of the coil center from the magnet center in x direction. The Figures 16, 17, 18 and 19 present the relative errors in the measurement of Fy and Fz. The relative error is defined as: Er =

a − a% a

(20)

where a is the calculated value, and a% is the measured value. The maximum error of the Fy component measured by the oscilloscope is equal to 7.5%, and of 3.7% for the force measured by rms meter. The error of the parasitic force Fz is rising with the decreasing force, which is caused by the reasons explained in the previous paragraph. The error, of the measurement provided with the oscilloscope, is 3.8 % at the force level of 340 mN and rises up to 10% at 200 mN. The experimental error, of the measurement with the rms meter, is 2.2% at the force value of 340 mN and rises up to 10.7% at 200 mN.

8.

CONCLUSIONS

The method and governing equations are developed to predict the force (torque) distribution in the linear moving coil actuator. From the simulation and measurement results it can be generally concluded that the relative errors are in the range of 4% up to 10% (for high and low force levels respectively), and the proposed method sufficiently captures the complexity of the physical phenomena. Besides, a higher accuracy can be expected when the more precise measurement setup is used or when higher force levels are applied. The proposed mathematical model has all attributes to be used as a standard tool for designer of high precision moving coil actuators where a prediction of parasitic forces is quite important, and it could be implemented as a basic algorithm within the rapid-design software. The novel method is significantly reducing a development time of the air coils actuators, it makes an evaluation of new actuator topologies very fast, and, consequently, gives more space for creativity during research. It can also be used for an optimization, investigation of an influence of tolerances (direction of magnetization, dimensions, position deviations etc.) for all types of linear air coil actuators and other design related studies. Moreover, the application of the developed method for other actuator topologies is the issue for the foregoing publications that are currently under preparation.

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REFERENCES [1] COMPTER, J. C., S. A. J. HOL and E. A. LOMONOVA: ‘Lorentz actuator for nanometer accuracy’, EUSPEN International Conference, May 2002, Twente, The Netherlands, pp. 49-52 [2] GIERAS, J. F.: ‘Linear induction drives’, (Clarendon Press, Oxford, 1994) [3] GIERAS, J. F., and Z. J. PIECH: ‘Linear synchronous motors, Transportation and automation systems’, (CRC Press, London 2000) [4] JUNG, S. Y., J. S. CHUN and H. K. JUNG: ‘Performance evaluation of slotless permanent magnet linear synchronous motor energized by partially exited primary current’, IEEE Transactions on Magnetic, September 2001, vol. 37, no. 5, pp. 3757-3761 [5] MAKAROVIC, J., A. J. A. VANDENPUT, E. A. LOMONOVA, H. KRIJT, S. STIKKELORUM: ‘Simple method of linear induction motor redesign’, IFAC Conference on Mechatronic Systems, December 2002, Berkeley, CA, USA, pp.55-60 [6] REECE, A. B. J., and T. W. PRESTON: ‘Finite element methods in electrical power engineering’, (Clarendon Press, Oxford, 2000) [7] RIZOV, P., A. IVANOV and O. NEJKOV: ‘A finite element method application for obtaining certain parameters of the power tool motors’, CWIEME Conference proceedings, June 2000, Berlin, Germany, pp.51-56 [8] YONNET, J. P. and G. AKOUN: ‘3D Analytical calculation of the forces exerted between two cuboidal magnets’, IEEE Transaction on magnetics, 1984, vol. Mag-20, no. 5, pp. 1962-1964 [9] ZHU, Z. Q. and D. HOWE: ‘Halbach permanent magnet machines and applications: a review’, IEE Proc.-Electric Applications, July 2001, vol 148, no. 4, pp. 299-308 [10] HAGUE, B.: ‘The principles of electromagnetism, applied to electrical machines’, (Dover publications, New York, 1962) [11] ABRAMOWITZ, M. and I. A. STEGUN: ‘Handbook of mathematical functions with formulas, graphs, and mathematical tables’, (Dover Publications, New York, 1965) [12] JANSEN, J. W., E. A. LOMONOVA, A. J. A. VANDENPUT, J. C. COMPTER.: ‘Design tool for 6-DoF planar motor with moving PM and stand still coils’, LDIA, September 2003, Birmingham, UK (to be published)

APPENDIX The following analytical expressions are used in the equations (5), (6), (7) and (8): α=a−x, β = b − y, ε=a+x, ζ = b+y, η=c+z, µ = z − c,

(21) (22) (23) (24) (25) (26)

ρ = α 2 + β2 + µ 2 ,

(27)

τ = α 2 + ζ 2 + η2 ,

(28)

υ = ε 2 + β2 + η2 ,

(29)

φ = ε2 + ζ 2 + µ 2 ,

(30)

χ = α 2 + β2 + η2 ,

(31)

ψ = α2 + ζ 2 + µ2 ,

(32)

ω = ε 2 + β2 + µ 2 ,

(33)

Θ = ε 2 + ζ 2 + η2 .

(34)

TABLES Table 1: Permanent magnet parameters Magnet

8

material (NdFeB) magnetized along ZG axis residual magnet remanence, [T] coercitivity, [kA/m] magnetic permeability, [-] maximum energy product, [kJ/m3] half-width, [mm] half-length of side magnets, [mm] half – height, [mm] vertical gap between magnets, [mm] horizontal gap between magnets, [mm]

Table 2: Coil parameters Coils Material height, [mm] length, [mm] width, [mm] eye, [mm] width of a coil leg, [mm] coil pitch, [mm] diameter of wire, [mm] number of turns, [-]

Neolit Q3F Br=1.23 HcB=915 µr=1.07 BHmax=320 a=30 b=5 c =3.3 g=7 hg=10

copper hcu=3.8 ltot=61 btot=27.6 e=12 0.5(btot-e) 4b+hg 0.38 185

CAPTIONS Figure 1: Linear moving coil actuator. Figure 2: Designation and orientation of permanent magnet in local space. Figure 3: Translation of magnet in global space. Figure 4: Rotation of magnet around x-axis by 90°. Figure 5: Permanent magnet, ferromagnetic material and their equivalent replacements. Figure 6: Partition and current of the coil. Figure 7: Integration loops in the coil. Figure 8: Magnetic flux density in the y-direction (middle of air gap). Figure 9: Magnetic flux density in the z-direction (middle of air gap). Figure 10: Calculated force in the y-direction (main force). Figure 11: Calculated force in the z-direction (parasitic force). Figure 12: Measured force in the y-direction (by an oscilloscope). Figure 13: Measured force in the z-direction (by an oscilloscope). Figure 14: Measured force in the y-direction (by an rms meter). Figure 15: Measured force in the z-direction (by an rms meter). Figure 16: Relative error of Fy (by an oscilloscope). Figure 17: Relative error of Fy (by an rms meter). Figure 18: Relative error of Fz (by an oscilloscope). Figure 19: Relative error of Fz (by an rms meter).

ILLUSTRATIONS

9

2a 2c g XG

hg

4b

ZG Br

2b

Br

iron yoke

Br

YG

Br

Br

coil

permanent magnet

Figure 1: Linear moving coil actuator.

zL 2a

Br 0L

2c

yL

2b

xL

Figure 2: Designation and orientation of permanent magnet in local space.

ZG

zL

P Br 0L

yL

0G

YG

xL

XG

Figure 3: Translation of magnet in global space.

ZG yL

zL -YG

YG xL

0G= 0L

XG Figure 4: Rotation of magnet around x-axis by 90°.

Br Ferromagnetic material

Br Ferromagnetic material

Original magnet

Br

Original magnet

Br

Image magnet

Br

Image magnet

Br

a

b

10

Figure 5: Permanent magnet, ferromagnetic material and their equivalent replacements.

btot

ZG ltot

-Jx

1. 8.

XG

Jy

2.

7.

3.

5.

-Jy hcu e

J 6.

4.

Jx

YG

θ

Jy Jx

Figure 6: Partition and current of the coil.

ZG m=0..1

k=0..1

YG Pi(ki,li,mi)

XG

l=0..1

Li(ki,li,m=0..1)

magnetic flux density By, T

Figure 7: Integration loops in the coil.

position y, points

position z, points

Figure 8: Magnetic flux density in the y-direction (middle of air gap).

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magnetic flux density Bz, T position y, points

position z, points

force Fy, N

Figure 9: Magnetic flux density in the z-direction (middle of air gap).

position y, points

position x, points

force Fz, N

Figure 10: Calculated force in the y-direction (main force).

position x, points

position y, points

Figure 11: Calculated force in the z-direction (parasitic force).

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force Fy, N position y, points

position x, points

force Fz, N

Figure 12: Measured force in the y-direction (by an oscilloscope).

position y, points

position x, points

force Fy, N

Figure 13: Measured force in the z-direction (by an oscilloscope).

position y, points

position x, points

Figure 14: Measured force in the y-direction (by an rms meter).

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force Fz, N position y, points

position x, points

position x, points

Figure 15: Measured force in the z-direction (by an rms meter).

position y, points

position x, points

Figure 16: Relative error of Fy (by an oscilloscope).

position y, points Figure 17: Relative error of Fy (by an rms meter).

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position x, points

position y, points

position x, points

Figure 18: Relative error of Fz (by an oscilloscope).

position y, points Figure 19: Relative error of Fz (by an rms meter).

15