A Linear Magnetic Gear

ΦAbstract – Linear magnetic gears, when coupled with a linear permanent magnet machine, are a high-efficiency, high-reliability and force-dense alternative to traditional linear actuation solutions. It is shown that the force density of a linear magnetic gear is affected by the required output stroke and gear ratio. A technique based on lumped parameter magnetic circuits is presented. It is employed for prediction of the flux density waveforms and the thrust force of the magnetic gear, which are in good agreement with those predicted by magnetostatic finite element analysis.

A prototype 3.25:1 linear magnetic gear is described. It is shown that the force capability of the linear magnetic gear can be very sensitive to manufacturing tolerances, in particular those related to the spacing between the pole-piece rings.

Index Terms—Linear machine, linear magnetic gear, lumped parameter modeling, permanent magnets

I. INTRODUCTION HE INCREASING demand for high force density linear actuators is currently being met almost exclusively by employing a hydraulic actuator or an

electromechanical actuator with a lead/ball/rollerscrew and nut to transform rotary motion to linear motion. However, in addition to the poor hydraulic power transmission density, the reliability and maintenance requirements of both actuation systems as well the higher risk of jamming of electromechanical actuators can be significant issues.

Therefore, despite their relatively poor thrust force density, linear electromagnetic actuators are increasingly being considered for applications spanning from industrial automation to automotive active suspension [1], [2]. An alternative approach to increase the force transmission capability of electromagnetic actuators is to employ a linear magnetic gear with a linear brushless permanent magnet machine [3]-[5].

Further, since a linear magnetic gear exhibits inherent overload protection, viz. when the output armature is subjected to a load force which is larger than the pull-out force it will slip harmlessly, thereby preventing physical damage to the magnetic gear and the systems connected to it. This feature is very important for safety-critical applications.

This paper discusses the force density of linear magnetic gears, and systems comprising a magnetic gear coupled to a linear permanent magnet machine. It is shown that the volumetric system force density is maximised for magnetic designs with large airgap diameters and smaller output strokes.

The paper also describes a lumped parameter magnetic circuit model of a linear magnetic gear. Results from the model are compared with finite element analysis and

Φ This work was supported in part by the UK Engineering and Physical

Science Research Council, EPSRC, through the award of a studentship. R. C. Holehouse, K. Atallah and J. Wang are with the Department of

Electrical Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK (e-mail: [email protected]).

experiments on a prototype 3.25:1 linear magnetic gear. Finally, it is shown that manufacturing tolerances may have significant effects on the force transmission capability of the magnetic gear.

II. LINEAR MAGNETIC GEAR PRINCIPLE OF OPERATION The linear magnetic gear, as presented and analyzed in

[3]-[5] consists of three concentric tubular armatures with the outer and inner carrying arrays of permanent magnets having different numbers of poles, and the intermediate having a set of annular ferromagnetic pole-pieces, Fig. 1, which modulate the magnetic fields produced by the magnet arrays.

The numbers of poles of the permanent magnet armatures, and the number of ferromagnetic pole-pieces, are related by,

!

ns = ph + pl (1) where

!

ns is the number of pole-pieces per active length,

and

!

ph and

!

pl are the number of pole-pairs on the high- and low-speed permanent magnet armatures, respectively.

It may be shown [3] that when the low-speed permanent magnet armature is held stationary the gear ratio between high-speed permanent magnet armature and the pole-piece armature is given by,

!

Gr =ns

ph (2)

This arrangement will be the preferred one in many practical applications, since the mechanical design is simplified.

III. FORCE DENSITY In [3] it was shown that an active force density of

2 MN/m3 could be achieved. However, unlike rotary gears, not all magnetic components simultaneously contribute to force transmission. Therefore, the volumetric force density,

!

VG

, considering all the magnetically active components of the gear, depends on the output stroke and the gear ratio.

If

!

"A

is the active force density, considering only the volume

!

VA

of the magnetically active components at any instant of time, the transmitted force is given by:

AA VF != (3) For a gear ratio

!

Gr, an output stroke

!

S and a high-speed armature of active length

!

LA

, the total length of the gear will be:

SGLL rAG += (4) and,

( ) !!"

#$$%

&+=+=

A

r

A

rA

A

AG

L

SG

FSGL

L

VV 1

' (5)

A Linear Magnetic Gear

Robert C. Holehouse, Kais Atallah, and Jiabin Wang

T

978-1-4673-0142-8/12/$26.00 ©2012 IEEE 563

Therefore, the volumetric force density, considering the total volume of the magnetically active components, is given by:

!

"G

="A

1+Gr

S

LA

(6)

Fig. 2 shows the variation of the gear force density with the gear ratio. It can be seen that the highest force density is achieved when the output stroke is significantly shorter than the length of the high-speed armature. It can also be seen that the force density decreases when the gear ratio is increased.

Fig. 2. Variation of gear force density with gear ratio.

For many applications, with a specified output stroke

and force requirement, it is the overall system force density that is of most interest; i.e. the combined density of the gear and the high-speed low-force machine driving it. If a machine of force density

!

"m

is employed to drive the magnetic gear, its volume will be given by:

!

Vm

=F

Gr

1

"m

(7)

resulting in a total volume,

!!"

#$$%

&++=

AA

r

Amr

sL

SG

GFV

'''

1111 (8)

and a force density:

Ar

m

A

r

As

L

SG

G++

=

!

!

!!

11

(9)

Fig. 3 shows the force density of a system comprising a linear magnetic gear coupled to a naturally-cooled PM linear machine (force density

!

"m

= 300 kN /m3 ). It can be

seen that the system force density is again highest when the required output stroke is significantly shorter than the active length of the high-speed armature. It can also be seen that for any ratio of output stroke to active length there exists a gear ratio for which an optimum system force density exists.

Fig. 3. Variation of system force density with gear ratio.

IV. LUMPED PARAMETER MAGNETIC CIRCUIT OF A LINEAR MAGNETIC GEAR

Although finite element analysis is a relatively accurate gear design and analysis tool, it can be computationally intensive. Therefore, a lumped parameter circuit model may enable the accurate prediction of magnetic gear key performance indicators, such as force transmission.

Analytical models of both rotary and linear magnetic gears based on computation of the field distribution have been presented and very good agreement found with finite element predictions [6], [7]. However, derivation of the necessary field equations is not a trivial task.

A lumped parameter magnetic circuit model of a rotary magnetic gear was presented in [8], and very good agreement was found with finite element predictions. However, this model applies a correction factor to MMF source values to consider flux leakage at the magnet boundaries, and finite element analysis is necessary to tune the correction coefficients.

The model presented in this paper divides the gear into a number of rectangular blocks, Fig. 4. Each linear block within the model is modeled as a network of axial and radial reluctances, Fig. 5a. For the permanent magnet blocks two radial MMF sources are added, Fig. 5b. By considering axial leakage paths within each permanent magnet, it is not necessary to employ a correction factor.

Fig. 1. Cross-sectional schematic of the prototype linear magnetic gear.

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The total number of axial divisions is selected so that each pole-piece, and each space between pole-pieces, is divided into an integer number of blocks.

Analysis presented here considers both linear and non-linear isotropic steel for the pole-pieces, and infinitely permeable high-speed armature and stator yokes.

Each linear radial reluctance is calculated as

!

Rr=

l / 2

µ0µr2"r#w

(10)

where

!

l is the radial length of the element,

!

µ0 is the

permeability of free space,

!

µr is the relative permeability

of the linear material,

!

r is the radius of the machine at the midpoint of the section represented by the reluctance, and Δw the axial width of each block.

Similarly, each linear axial reluctance is calculated as

!

Ra=

"w / 2

µ0µr2#rl

(11)

The non-linear lumped parameter model considers the pole-pieces as being comprised of a network of two radial and two axial non-linear reluctances, Fig 5c.

It is assumed that the relationship between magnetic field,

!

H , and flux density,

!

B , within the non-linear pole-pieces may be presented as in Fig. 6, and given by

!

H ="1B +"

nBn (12)

where

!

"1 and

!

"n are coefficients, and the order

!

n is determined by the material nonlinearity [9].

Hence the MMF drop across the non-linear radial reluctances supporting flux

!

"may be given by

!

Fr

="1l / 2

2#r$w% +

"nl / 2

2#r$w( )n% n (13)

and similarly the MMF drop across non-linear axial reluctances as

!

Fa

="1#w / 2

2$rl% +

"n#w / 2

2$rl( )n% n (14)

The value of the MMF sources on each permanent magnet block is given by:

!

F = ±Hc

l

2 (15)

where

!

Hc is the coercive force of the magnet material. The

sign of the MMF being dependent on the direction of magnetization.

MMF sources in permanent magnet blocks shared with magnets with different magnetization directions are appropriately scaled to approximate any localized leakage, such that a source lying directly on a boundary will have zero MMF, increased linearly as the displacement between magnet boundary and MMF source increases.

a) Spacer, air-gap and linear pole-piece blocks.

b) Permanent magnet blocks.

c) Non-linear pole-piece blocks

Fig. 5. Block magnetic circuit for the lumped parameter model.

Fig. 6. Relationship between magnetic field and flux density for non-linear

pole-piece material Furthermore, in order to minimize complexity, end

effects are neglected and the linear magnetic gear is assumed to be infinitely long.

A rats-nest magnetic circuit of the gear is compiled with MATLAB®, and solved using WinSpice© circuit emulation program. A flow chart detailing the solving process at each time step is shown in Fig. 7.

Fig. 4. Division of the magnetic gear into blocks.

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The calculated flux within each MMF source is returned to MATLAB® for post processing, and the force on each permanent magnet armature is calculated as

!

F =1

"w#mj $ "fmj

j=1

N

% (16)

where

!

N is the total number of MMF sources in the permanent magnet armature, and

!

"mj and

!

fmj the flux and MMF of each source, respectively.

Fig. 7. Lumped parameter model solving process.

V. DEMONSTRATOR LINEAR MAGNETIC GEAR The lumped parameter magnetic circuit model has been

employed for the 3.25:1 linear magnetic gear presented in [5], the leading parameters of which are listed in table I. Predictions from the lumped parameter magnetic circuit models are compared with axisymmetric finite element analysis.

TABLE I LEADING DIMENSIONS OF THE MAGNETIC GEAR

Number of pole-pairs on high-speed armature 4 Number of active pole-pairs on stationary armature 9 Number of active ferromagnetic pole-pieces 13 Active length of high-speed armature 200 mm Active stator diameter 180 mm Airgap diameter 160 mm Airgap length 1 mm Radial pole-piece thickness 10 mm Magnet material NdFeB Br = 1.25 T

A. Air-gap Flux Density Figs. 8 and 9 show the radial flux densities in the air-

gaps adjacent to the stationary and high-speed armatures respectively, and their associated harmonic spectra. Figs. 10 and 11 show the radial flux densities and associated spectra within the two permanent magnet armatures. It can be seen for each there is good agreement between the finite element and lumped parameter magnetic circuit predictions. It may be worth mentioning that the larger discrepancies on the extremities are due to the infinitely long linear gear assumption adopted for the lumped parameter modeling.

The good agreement between the force-producing

harmonic on each spectral analysis suggests that the assumption of an infinitely long linear magnetic gear for the lumped parameter modeling is acceptable.

a) Flux density waveform.

b) Harmonic spectrum with pole-pieces.

Fig. 8. Radial flux density in the air-gap adjacent to the stationary armature due to high-speed armature magnets.



Fig. 9. Radial flux density in the air-gap adjacent to the high-speed armature due to stationary armature magnets.

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Fig. 10. Radial flux density within the stationary armature magnets due to high-speed armature magnets.



Fig. 11. Radial flux density within the high-speed armature magnets due to stationary armature magnets.

B. Force-displacement Characteristic When the pole-piece armature and the outer permanent

magnet armature are held stationary, Fig. 12 shows the variation of the force on the pole-piece armature with the position of the high-speed armature and Fig. 13 shows the force on the two permanent magnet armatures. It can be seen that there is very good agreement between lumped parameter magnetic circuit models and finite element analysis.

Fig. 12. Variation of predicted force on the pole-piece armature with displacement of the high-speed armature. Calculated with linear finite

element and lumped parameter methods.

Fig. 13. Variation of predicted force on the permanent magnet armatures

with displacement of the high-speed armature. Calculated with linear finite element and lumped parameter methods.

Fig. 14 shows the ratio between the force on the pole-

piece armature and the high-speed armature. It is seen that there is generally good agreement between the two methods. It is also seen that the calculated ratio agrees well with the ratio given in (2).

Fig. 14. Variation of gear force ratio with displacement of the high-speed

armature. Calculated with linear finite element and lumped parameter methods.

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C. Effects of the Radial Thickness of the Pole-piece When the diameter of the air-gap adjacent to the

stationary armature is kept constant, and the volume of all permanent magnets is fixed, Fig. 15 shows the variation of the thrust force with the radial thickness of the pole-pieces, calculated with both linear and non-linear lumped parameter and finite element methods. It can be seen that there exists an optimum thickness to maximize the gear force capability.

It is also seen that for both linear and non-linear analysis there is very good agreement between the force predicted by the lumped parameter magnetic circuit modeling and finite element analysis. However, when the radial thickness of the pole-pieces is small, they exhibit higher saturation levels which explain the discrepancy between the linear and the non-linear analysis. Nevertheless, for manufacturing purposes, the selected designs should have larger radial thicknesses, therefore linear lumped parameter analysis is a valid design tool, with the benefit of notably lower computation time compared to non-linear lumped parameter analysis.

Fig. 15. Variation in force capability with pole-piece radial thickness.

Calculated with linear and non-linear finite element and lumped parameter methods.

D. Sensitivity to Number of Model Blocks Fig. 16 shows the variation of the force on the pole-piece

armature with number of axial blocks per pole-piece. It is seen that as the number of blocks is increased the calculated force tends towards the value predicted by finite element analysis. It can also be seen that for this particular gear, 5 to 10 axial blocks per pole-piece should be sufficient to achieve good accuracy.

Fig. 16. Variation in predicted gear force capability with number of axial blocks per pole-piece in linear lumped parameter model. Linear FE value

given for reference.

VI. EFFECTS OF MANUFACTURING TOLERANCES The 3.25:1 linear magnetic gear whose leading

dimensions are given in table I has been constructed. A cross-sectional schematic is shown in Fig. 1, and a more enlarged section is shown in Fig. 17. Fig. 18 shows the assembled pole-piece armature whilst Fig. 19 shows the linear magnetic gear on the test bed.

Fig. 17. Schematic of magnetic gear. (A close up view).

Fig. 18. Pole-piece armature with integrated high-speed armature.

Fig. 19. Magnetic gear on test-bed.

Fig. 20 compares the measured and predicted variations

of the transmitted force on the pole-piece armature with position when the high-speed armature is kept stationary. It can be seen that the measured force is markedly lower than predicted.

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Fig. 20. Variation of measured and predicted force on pole-piece armature

with position. (Locked high-speed armature).

This may be attributed to the mechanical construction of the pole-piece armature, where spacing between the pole-piece rings is maintained by polymer spacer rings, Fig. 17 and 18, whose axial dimensions could have exhibited large reductions when subjected to large and repeated compressive stresses applied by the tie-rods and the assembly nut.

In order to illustrate the sensitivity of the transmitted force to manufacturing tolerances, Fig. 21 shows the variation of the transmitted force with the relative change of the dimensions of the spacers from the design value. It can be seen that a 5% reduction in the axial dimension of the spacers, results in a 30% reduction in the transmitted force capability.

Fig. 21. Variation of transmitted force with the change of axial distance

between the pole-piece rings. It may be worth noting that the spacers employed in this

gear have a very low compressive modulus, therefore a 5% reduction in axial dimensions is highly probable due to the high levels of force employed to compress the pole-piece armature in order to ensure mechanical stiffness.

VII. CONCLUSIONS The paper presents the analysis and performance of a

linear magnetic gear. It is shown that lumped parameter magnetic circuit models can be employed to predict the key performance indicators, such as transmitted force. Measurements on a prototype linear magnetic gear have shown that manufacturing tolerances can have significant effects on the transmitted force capability. In particular, it has been shown that a 5% reduction in the axial dimensions of the non-magnetic spacers between the pole-pieces can result in a 30% reduction in the force transmission.

VIII. ACKNOWLEDGMENT The authors thank Mr. John Wilkinson for his support

and assistance in the design and construction of the prototype gear and associated test rig.

IX. REFERENCES

[1] J. Wang, W. Wang, K. Atallah, and D. Howe, “Demagnetisation assessment for three-phase tubular permanent magnet brushless machines,” IEEE Trans. Magn., vol. 44, no. 9, pp. 2195–2203, 2008.

[2] J. Wang, K. Atallah, and D. Howe, “Design considerations of tubular flux-switching permanent magnet machines,” IEEE Trans. Magn., vol. 44, no. 11, pp. 4026–4032, 2008.

[3] K. Atallah, J. Wang, S. Mezani, and D. Howe, “A novel high-performance linear magnetic gear,” Inst. Elect. Eng. J. Trans. Ind. Appl., vol. 126, no. 10, 2006.

[4] W. Li, K. T. Chau, and J. Li, “Simulation of a tubular linear magnetic gear using HTS bulks for field modulation,” IEEE Trans. Appl. Supercond., vol. 21, no. 3, pp.1167-1170 2011.

[5] R. C. Holehouse, K. Atallah, and J. Wang, "Design and Realisation of a Linear Magnetic Gear," IEEE Trans. Magn., vol. 47, pp. 4171-4174, October 2011.

[6] T. Lubin, S. Mezani, and A. Rezzoug, "Analytical Computation of the Magnetic Field Distribution in a Magnetic Gear," IEEE Trans. Magn., vol. 46, pp. 2611-2621, 2010.

[7] W. Li and K. T. Chau, "Analytical Field Calculation for Linear Tubular Magnetic Gears Using Equivalent Anisotropic Magnetic Permeability," Progress In Electromagnetics Research, vol. 127, pp. 155-171, 2012.

[8] M. Fukuoka, K. Nakamura, and O. Ichinokura, "Dynamic Analysis of Planetary-Type Magnetic Gear based on Reluctance Network Analysis," IEEE Trans. Magn., vol. 47, pp. 2414-2417, 2011.

[9] K. Nakamura and O. Ichinokura, “Dynamic Simulation of PM Motor Drive System based on Reluctance Network Analysis,” EPE-PEMC 2008, pp. 758-762, 2008

X. BIOGRAPHIES Robert C. Holehouse was born in Leeds, U.K. in 1984. He received

the M.Eng. degree (with honours) in electrical engineering from the University of Sheffield, Sheffield, U.K. in 2006.

In 2006 he started studying for the Ph.D. degree in the department of Electrical and Electronic Engineering, University of Sheffield, where he is currently working as a Research Associate. His research interests include linear magnetic gearing, and drive trains for tidal turbines.

Kais Atallah received the degree of Ingenieur d’Etat in Electrical

Engineering from Ecole Nationale Polytechnique, Algeria, 1988, and the PhD from the University of Sheffield, England, in 1993.

He is currently a Reader in Electrical Engineering at the University of Sheffield. From 1993 to 2000 he was a Postdoctoral Research Associate in the Department of Electronic & Electrical Engineering at the University of Sheffield. In 2006, he co-founded Magnomatics Ltd, where he was Director until July 2008. His research interests embrace fault-tolerant permanent magnet drives for safety-critical applications, magnetic gearing and ‘pseudo’ direct drive electrical machines, and drive-trains for wind/tidal turbines and electric/hybrid vehicles.

Jiabin Wang received the B.Eng. and M.Eng. degrees from Jiangsu University of Science and Technology, Zhengjiang, China, in 1982 and 1986, respectively, and the Ph.D. degree from the University of East London, London, U.K., in 1996, all in electrical and electronic engineering. Currently, he is a Professor in Electrical Engineering at the University of Sheffield, Sheffield, U.K. From 1986 to 1991, he was with the Department of Electrical Engineering at Jiangsu University of Science and Technology, where he was appointed a Lecturer in 1987 and an Associated Professor in 1990. He was a Postdoctoral Research Associate at the University of Sheffield, Sheffield, U.K., from 1996 to 1997, and a Senior Lecturer at the University of East London from 1998 to 2001. His research interests range from motion control to electromagnetic devices and their associated drives for applications in automotive, renewable energy, household appliances and aerospace sectors.

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A Linear Magnetic Gear