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Electromagnetic actuator design analysis using a twostage optimization method with coarse-fine model

output space mappingJonathan Hey, Tat Joo Teo, Viet Phuong Bui, Guilin Yang, Ricardo Martinez-Botas

Abstract—Electromagnetic actuators are energy conversiondevices that suffers from inefficiencies. The conversion lossesgenerate internal heat which is undesirable as it leads to thermalloading on the device. Temperature rise should be limitedto enhance the reliability, minimize thermal disturbance andimprove output performance of the device. This paper presentsapplication of an optimization method to determine the geometricconfiguration of a flexure based linear electromagnetic actuatorthat maximizes output force per unit heat generated. A twostage optimization method is used to search for a global solutionfollowed by a feasible solution locally using a branch and boundmethod. The Finite Element magnetic (fine) model is replaced byan analytical (coarse) model during optimization using an outputspace mapping technique. An 80% reduction in computation timeis achieved by application of such an approximation technique.The measured output from the new prototype based on theoptimal design shows a 45% increase in air gap magnetic fluxdensity, a 40% increase in output force and a 26% reductionin heat generation when compared to the initial design beforeapplication of the optimization method.

Keywords—Optimization methods, Reduced order systems, Elec-tromechanical effect, Genetic algorithm, Quadratic programming,Electromagnetic devices

I. INTRODUCTION

THE current approach towards designing electromagneticactuators is largely based on direct methods making use

of commercial software to generate designs iteratively. This isa time consuming process and will not necessarily yield theoptimal design [1],[2]. Moreover, the design process is nowcomplicated with increasing requirements such as consider-ation for multi-physics interaction. The device performancein terms of power output and efficiency are closely linked tothe electromagnetic and thermal interaction [3]. In fact thepower output of a device is limited by its thermal efficiency

Copyright (c) 2013 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Manuscript received xxxxx, 2013. This work is supported by the Agencyfor Science, Technology and Research (A*STAR) of Singapore.

J. Hey and R. Martinez-Botas are with the Mechanical Engineering Depart-ment, Imperial College, Exhibition Road, London, SW7 2AZ, UK (e-mail:hhh09@imperial.ac.uk, r.botas@imperial.ac.uk)

T.J. Teo and G.L. Yang are with Mechatronics group, Singapore Instituteof Manufacturing Technology, 71 Nanyang Drive, Singapore, 638075 (e-mail:tjteo@simtech.a-star.edu.sg, glyang@simtech.a-star.edu.sg)

V.P. Bui is with the Institute of High Performance Computing (IHPC),1 Fusionopolis Way, 16-16 Connexis North, Singapore 138632 (e-mail:buivp@ihpc.a-star.edu.sg)

which is dependent on the design [4]. An optimal design offersa balance between the two competing requirements. Directmethods of design are no longer intuitive due to the complexinteractions and optimization is a useful design tool by solvingthe problem as an inverse mathematical problem [5]. Thisensures that the optimal design is selected within the boundsdefined by the constraints in terms of the output performanceand physical dimensions.

Recent applications of optimization methods make use ofFinite Element (FE) models to simulate the device response[6]. However, the increasing design requirements nowadaysmean that the application of fine models such as FE modelsto the optimization process is still challenging as computationpower will be a limit. Space mapping is a technique that makesuse of a surrogate model in place of the fine model in theoptimization process [7]. Such a model can be derived froman analytical or coarse model through a mapping function. Theinaccuracies associated with the coarse model can be reducedby using a mapping function with iterative parameter extractionduring optimization. The surrogate model therefore offers agood balance between accuracy and computation effort.

Electromagnetic actuators are designed based on the ap-plication requirement. For example, the design optimizationof a transverse flux linear motor for railway traction [8] istargeted at maximizing the thrust force while minimizing theweight. The design analysis of a switch reluctance motor forvehicle propulsion [9] is geared towards high torque densityand efficiency as measured by the torque per unit copper loss.In this analysis, the test device is a linear electromagneticactuator with stationary permanent magnets and moving coilsupported by flexure bearings. The actuator stroke length isup to 500µm and has an actuation speed of 100mm/s. Suchdevices are designed for use in precision applications [10] withpositioning accuracy of up to ±20nm [11]. Thermal loadingdue to conversion losses is undesirable as it is the major sourceof positioning error [12]. Therefore, the design objective is toimprove the efficiency of the device to reduce thermal loading.

The design analysis is aimed at identifying the geometricconfiguration that maximizes the output force per unit heatgenerated. A single objective mixed variable optimizationproblem is formulated due to the discrete nature of somedesign variables so only feasible solutions can be accepted. Anintegrated approach dealing with the discrete variables wouldfacilitate the design process [13]. A two stage optimizationmethod is proposed by first using Genetic Algorithm (GA)for the global search. This is followed by a Branch and

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Fig. 1. Moving coil and stationary magnet linear actuator

Bound (BB) method for local minimization to determine theoptimal feasible solution. Space mapping is implemented inthe optimization process to reduce the computation time. Thecurrent adaptation introduces an iterative parameter extractionprocess that is decoupled from the main optimization processusing a deterministic recursive least square algorithm.

II. MATHEMATICAL MODELING

The basic configuration of the moving coil and stationarymagnet linear actuator is shown in Fig. 1. Such a linearactuator can be adapted for precision applications such aspositioning and alignment. For instance, a flexure mechanismwas used to support the moving coil in [14] which enhancedthe positioning accuracy of the actuator. The actuator is able todeliver sub micron positioning resolution with high actuatingspeed over a few millimetres stroke. Due to the linear current-force relationship, they can also deliver direct and preciseforce control capabilities that are essentials for the micro/nano-scale imprinting processes [15]. The design of such a deviceis central around the conducting wire and its interactionwith the magnetic field generated by the permanent magnets(PM). Lorentz force is generated due to the interaction of theconducting wire and the magnetic field which is the basisfor actuation. At the same time, internal heat generation isinevitable due to energy conversion losses.

The losses arise for the following reasons; (i) resistiveheating of the conducting wire, (ii) eddy current generatedin metallic parts, (iii) hysteresis behaviour in the magnet and(iv) mechanical losses at contacting surfaces. Eddy currentand hysteresis losses are generated in the presence of achanging electric field. The changing field can be caused byan alternating current or when there is relative motion betweenthe electric field and the magnets or metallic parts. Mechanicallosses arise due to friction between moving surfaces in contact.Finally, resistive heating is caused by the electrical resistivityof the conducting wire.

In a static operation using direct current, resistive heatingis the only source of heat generation as there is no motionand changing electric field [16]. The heat generation (Q) isdependent on the current (I), wire resistivity (ρr), length (lc)and cross sectional area (ac) as shown by (1). The Lorentzforce (F) acting on the conducting wire in a magnetic field witha flux density (B) that is directly proportional to the currentgiven by (2). The expression holds for a constant magneticfield directed perpendicular to the direction of current flow.

Q =I2ρrlcac

(1)

Fig. 2. Cross sectional view of magnet, core, air gap and coil

F = BIlc (2)

Both the output force and heat generation are functions ofthe input current (I) which would be kept constant at 1A forthis design analysis. Equation (1) and (3) shows that the deviceoutput is central around the overall length (lc) of the con-ducting wire used. In order to perform the subsequent designanalysis, the following assumptions and parameterization iscarried out using Fig. 2 for illustration. A regular packing ofthe wire is assumed where the number of turns (nx) and layers(ny) is defined to describe the wire configuration. The externalwire diameter (d = d0 +dt) is adjusted for the finite thicknessof insulation material (dt) around the wire of diameter, d0. Theeffective conductive cross sectional area is given by (3).

ac =π (d− dt)2

4(3)

nx,ny and d0 are taken as the design variables while theother geometric parameters are kept constant as defined inFig. 2. The coil with an inner radius, R, is suspended in themagnetic field by a bobbin attached to a flexure bearing whichis not shown. A small gap tolerance defined by l0 and g0 existsbetween the coil and magnet. The total length of wire for anidealized coil configuration is expressed in terms of the designvariables given by (4). The external radius (Rext) and length(Lext) in [mm] is given by (5) and (6).

lc =

ny∑i=1

2πnx {R+ [(i− 1) + 0.5] d} = πnxny (2R+ nyd)

(4)

Rext = nyd+ 2g0 + 18 (5)

Lext = nxd+ 2l0 + 10 (6)

A. Magnetic model1) Finite Element model: The analysis of the magnetic

flux density in the air gap is restricted to a sector of thedevice because of the axis-symmetric design. The problem isreduced to a magneto-static analysis without the presence ofthe conducting wire. In a current-free region, the problem isdescribed by Gauss law:

∇ ·B = 0 (7)

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In the presence of permanent magnets, there exist the followingconstitutive material relationships in the magnetic fields:

B = µ (H + M) (8)

where M is the magnetization vector within the magnets, µ0

denotes the material permeability. The magnetic field H canbe expressed in terms the magnetic scalar potential such that

H = −∇ϕm (9)

where ϕm is analogous to the definition of the electric potentialin electrostatics. Making use of the above expressions tomake appropriate substitutions into (7) gives the followingexpression for the scalar potential in the region in and aroundthe permanent magnet:

∇2ϕm = ∇ ·M (10)

Equation (10) can be solved numerically by using a finiteelement method (FEM). Boundary conditions are imposed onthe outer surface of the computational domain to ensure the ex-istence and uniqueness of the solution. Then, the field strengthand radial flux density in the air gap can be derived from (9),(8) by numerical differentiation. A 3D Finite Element (FE)model of the device is constructed in the software package CSTusing the magneto-static solver. The simulation is performedwithin the bounding box enclosing the device structure thatdefines the computational domain. The FE modelling requiresthat the whole domain be subdivided into discrete elementaryvolumes which provide the numerical approximation of thesolution. The solution space is meshed with a total of 205,000tetrahedral elements. In addition, information such as externalsources and material properties needs to be provided for acomplete FEM model. The core is constructed from mildsteel which has an estimated relative permeability of 25. Theintrinsic magnetization of the N48H permanent magnets isM = 690 kA/m which are used in the FE model as sources.

An output plot of the magnetic flux in the computationdomain is shown in Fig. 3. The magnetic flux density (B)used for estimation of the force generation in (2) is derivedfrom the radial component of the flux density on the plane ofinterest. The average across Nz × Nr points on the plane ofinterest can be calculated using (11) by taking the mean of theintegral sum. The flux density is denoted by Bf for clarity inthe subsequent discussion.

Bf =1

NzNr

Nz∑j=1

Nr∑i=1

B (ri, zj) (11)

2) Analytical model: An analytical magnetic circuit modelis developed to estimate the air gap magnetic flux density. Thismodel is less accurate compared to a Finite Element model dueto model simplifications. But it requires little computationaleffort and is ideal for implementation in an optimization pro-cess. The mismatch in the model output between the analytical(coarse) and Finite Element (fine) model needs to be addressedin order to determine the optimal solution. An output mappingis introduced in section IV to account for this mismatch.

Fig. 3. Finite Element model with illustrated air gap magnetic flux density

Fig. 4. 2D Magnetic circuit model

Fig. 5. Equivalent magnetic circuit

The coarse model is derived from an equivalent magneticcircuit of a 2D slice of the Finite Element model as illustratedin Fig. 4. The C-shape dual magnet configuration is representedby a magnetic circuit with two parallel paths. The differencein the two paths lies in the leakage path present in path 2 thattakes the place of the core in path 1. The magnetic flux inthe leakage path is confined to an imaginary path as though acore is present. This idealization will lead to the model outputinaccuracies. The magnetic flux (Φ), motive force (υmg) andreluctance (R) are analogous to the electrical equivalent ofcurrent, voltage and resistance respectively. R is calculatedfrom the magnetic path length (l), permeability (µ) and crosssectional area (A) as shown in (12). The cross sectional areais simply a product of the path width (W ) and characteristicdimension (D). The magnets are modelled as an ideal sourcewith an internal reluctance in series. The magnetic motiveforce is given by (13) where Br is the remanent magnetic fluxdensity, µr is the recoil permeability and lmg is the length(polarization direction) of the magnet.

Ri =li

µiAiwhere Ai = WDi (12)

υmg =Brlmgµr

(13)

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TABLE I. SUMMARY OF MODEL PARAMETERS

Magnet properties Expression UnitCoercivity Hmg = 690 kA/mRemanance Br = 0.87 TMagnet length T = 4.5 mmGeometricconstantsEnd gap 1 L0 = 5 mmEnd gap 2 G0 = 18 mmPermeabilityVacuum µ0 = 4π × 10−7

Magnet µr = 1.003µ0 Tm/ACore µcr,1 = µcr,2 = 25µ0 Tm/AGap µg = µ0 Tm/ALeakage path µlk = µ0 Tm/APath length (li)Magnet lmg = T mmCore1 lcr,1 = L+L0+G+G0 mmCore2 lcr,2 = L+ L0 mmGap lg = G mmLeakage path llk = G+G0 mmCharacteristicdimension (Di)Magnet Dmg = L mmCore Dcr,1 = Dcr,2 = 5 mmGap Dg = L mmLeakage path Dlk = 5 mm

The equivalent magnetic circuit is shown in Fig. 5. Theconservation of magnetic flux in path 1 and path 2 leads tothe expression (14). The magnetic flux density in the air gapis simply the flux density per unit area as shown in (15). Theselection of the width size (W ) is inconsequential since itwould be eliminated in the calculation. The material properties,geometric variables and constants used for calculation of theflux density are illustrated in Fig. 4 and listed in TABLE I.

Φ =2υ

RT + 2Rm +Rgwhere RT =

(Rlk +Rc2) (Rc1)

Rlk +Rc2 +Rc1(14)

Bc = Φ/Ag (15)

III. EXPERIMENTAL TESTING AND MODEL VALIDATION

Fig. 6 shows the experimental set up for measuring the radialcomponent of the magnetic flux density at the midpoint of theplane of interest along the z-direction (due to finite size of theprobe). A unidirectional hall probe coupled with a Lakeshore460 gaussmeter is used for measuring the flux density withan accuracy of 0.001T. It is positioned along z-axis with amotorized planar stage. This ensures that there is a consistentstep size (1mm) along the measurement path while maintaininga level position. The output force is measured using a SchaevitzFC2231 load cell as shown in Fig. 6. The load cell is pre-calibrated for direct measurement of the output force withan accuracy of ±0.05N. In this test, the device is poweredby a linear amplifier using a close loop current controller tomaintain the desired input current level. The heat generationis equal to the supplied electrical power given by the productof the supply voltage and current.

The measured air gap magnetic flux density is shown inFig. 7. A comparison between the measurement and modeloutput is made for validation of the Finite Element (FE) model.

Fig. 6. Experimental set up for measuring magnetic flux density and deviceinput-output characteristics

Fig. 7. Magnetic flux density (r component) along the z-direction

There is good agreement between the FE model estimationand measured data as shown by the close fit of the plots.The RMS error between the measured and estimated data is0.022T representing an averaged error of less than 7.3% of themeasurement range (0.3T). The measured air gap flux densityhas an average value of 0.146T while the model estimatesan average flux density of 0.150T. The average air gap fluxdensity is calculated using (11) with the radial component ofthe magnetic flux density shown in Fig. 7.

Fig. 8 shows the input output relationship of the device.There is a linear relationship between the output force andinput current as shown in Fig. 8(a). This agrees well withthe analytical model given by (2) which also predicts a linearrelationship based on an averaged magnetic flux density (B).However, a slight difference in the slope of the lines shows theminor deviation between the model estimated and measuredflux density as discussed earlier. As for the heat generation,there is a quadratic relationship with the input current asdescribed by (1). The deviation between the model estimateand measurement is more pronounced at higher levels of inputcurrent as shown in Fig. 8(b). This is caused by the changein material properties as the device temperature changes.Electrical resistivity increases with temperature and it directlyaffects the amount of heat generation based on (1). As such,it is expected that some discrepancy will arise since this effectis not explicitly accounted for in the model.

The input is an independent parameter in the optimization

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Fig. 8. Input output characteristic; (a) force and (b) heat generation

TABLE II. MODEL OUTPUT DEVIATION AT DESIGN POINT

Output parameter Measured(a)

Model estimated(b)

%deviation= b−a

a ×100%Magnetic flux density [T] 0.146 0.150 +2.7Force [N] 5.28 5.32 +0.6Heat generation [W] 4.11 3.74 -8.9

and it is kept constant at a nominal value of 1A. Thus,the models are validated against the measurement at thisdesign point. The force as measured using the load cell is5.28N while the heat generation is derived from the suppliedpower as 4.11W. The force is overestimated by 0.6% whilethe heat generation is underestimated by 8.9% for reasonsdiscussed earlier. Nevertheless, the model estimates do notdeviate beyond 10% from the measurement at the design point(1A input current) as summarized in TABLE II.

IV. OUTPUT SPACE MAPPING

Equation (16) describes a general optimization problem withinput variables x and model response R (x) subjected to asuitable objective function U with the associated constraints.x∗ is the optimal solution to this optimization problem.

x∗ = arg minx

U (R (x)) (16)

The underlying concept of Space Mapping (SM) is to matchthe optimal solution derived from a coarse model to the actualsolution from a fine model during the optimization process.The coarse model is typically less accurate due to modelsimplification. However, the coarse model is simple to evaluateand computationally efficient. The aim is to make use of thecoarse model to replace the fine model during the optimizationprocess. However, the mismatch between the coarse and finemodel would not result in an optimal solution.

SM uses a function to map the input or response from thecoarse model onto the fine model. The mapping function isdenoted by P (x) as shown by (17) which can be linear ornonlinear. P (x) can be determined during problem formulation

or iteratively during the optimization process itself. The finemodel can be substituted by the coarse model using anapproximation shown in (18). The optimization process nowonly requires evaluation of the coarse model which would becomputationally less demanding. The optimal solution can thenbe estimated as xf through the inversion given in (20).

xc = P (xf ) (17)

Rc (P (xf )) ≈ Rf (xf ) (18)

x∗c = arg min

xc

U (Rc (xc)) (19)

x∗f = P−1(x∗

c) (20)

ε = ||Rf (xf )−Rc(xc)|| (21)

The mapping function is found by minimizing the ParameterExtraction (PE) error (21). In the original SM, Bandler et.al. uses a least square regression to determine the coefficientof a linear mapping function [17]. Such a method requiresa finite number of fine model evaluations prior to the opti-mization process to initiate the parameter extraction process.The number of evaluations required depends on the order anddimension of the mapping function which increases with themismatch. In the Aggressive Space Mapping (ASM) method[7], an iterative quasi newton method uses estimates of theJacobian matrix to direct the search towards a function thatminimizes (21). The parameter extraction is an intermediatestep of the main optimization process. It works in tandem withthe iterative search algorithm that is characteristic of gradientbased numerical methods. Thus, it limits the type of algorithmthat can be used for the main optimization.

If there are large differences in structure between underlyingmodels, the output misalignment between the coarse and finemodel response will be significant even after input mapping.Output mapping can overcome this deficiency by introducinga transformation of the coarse model response based onan output mapping function O (Rc(xc)) as defined in (22).The mapped output from the coarse model is known as thesurrogate model response, Rs(xc). An example of the outputmapping function is given by (23) where γ(i) is a weightingfactor on the response mismatch (∆R).

Rs(xc) = O (Rc(xc)) (22)

Rs(xc) = O (Rc(xc)) = Rc(xc) +

m∑i=1

γ(i)∆R(i) (23)

∆R(i) = Rf (xf )−Rc(x(i)c ) (24)

The weighting factor γ(i) in (24) is determined from asequence of m pairs of coarse and fine model responses ina parameter extraction process [18]. Updating the surrogatemodel is done iteratively by solving (25) for x∗

c and evaluatingthe fine model response at this point to generate a new pairof coarse fine model response. The process repeats until the

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response mismatch is less than a set limit. At the final iteration,the optimal solution (x∗

c ) is estimated as the final solution (x∗f ).

x∗c = arg min

xc

U (O (Rc (xc))) (25)

The analytical magnetic model presented in the precedingsection shows that the magnetic flux density in the air gap is afunction of the magnet length (L), gap size (G) and the othergeometric constants (l0, g0) which are defined in Fig. 2. Thus,the coarse model input (26) is mapped to the original designvariables (27) by the definition (28). This is an exact mappingas defined by the geometric relationship.

xc = [ L G d ] (26)

xf = [ nx ny d0 ] (27)

d = d0 + dt, L = nxd+ 2l0 and G = nyd+ 2g0 (28)

There is no need to estimate the input mapping functionP (xf ) since it is already known. Output mapping is used tomatch the response of the coarse model (14)-(15) and finemodel (11). The model response mismatch is defined as (29).

∆R = Bf −Bc (29)

The coarse model adopts some idealization for modelling theleakage flux at the open end of the magnetic circuit. Theleakage flux is dependent on relative reluctance of the magneticpaths. The reluctance is in turn dependent on the magnetic pathlength as described earlier. Thus, the magnet length (L) andgap size (G) are critical parameters that affects this additionalcomponent of flux leakage. It is reasonable to assume that thecoarse fine model response mismatch is due to the unaccountedcomponent of flux leakage. A linear function of (L, G) in theform shown in (30) is used to model the mismatch (∆R).Physical insight of the mismatch means that the mappingfunction is kept simple but still effective at improving theaccuracy of the surrogate model.

∆R = θy = [ 1 L G ]

[θ0θ1θ2

](30)

ε = ||∆R−∆R|| (31)

The Parameter Extraction error defined by (31) allows theparameter, θ, to be determined using a least square method.This process can be done iteratively during the main optimiza-tion process by using a recursive least square algorithm. Themajor steps of the algorithm is given by (32)-(34). Q is ameasure of the uncertainty in the estimated parameter θ. Qwill be reduced over time when more data points are madeavailable and hence the solution will converged to the optimalestimate of θ that minimizes (31). A large Q(0) should bechosen to move the estimate θ(0) towards its final solution θ.

Q(k) =

[Q(k−1) −

Q(k−1)yT(k)y(k)Q(k−1)

1 + y(k)Q(k−1)yT(k)

](32)

Fig. 9. Output mapping and parameter extraction (PE)

K(k) =Q(k−1)y

T(k)

1 + y(k)Q(k−1)yT(k)

(33)

θ(k) = θ(k−1) + K(k)

[∆R(k−1) − y(k)θ(k−1)

](34)

For y ∈ R1xn, there needs to be k ≥ n iterations toensure uniqueness of the estimated parameter (θ). Evaluationof the fine model response is necessary at each major iterations(k) of the optimization process to update this parameter. Theparameter extraction process is terminated once the averagedmapping error, 〈ε〉, defined by (35) is less than a predeterminedtarget level. Finally, the surrogate model output is givenby (36). The key steps of implementing an output mappingtogether with the optimization process is illustrated in Fig. 9.

〈ε〉 =

√∑ki

[ε(k)]2

k(35)

Bs = Bc + [ 1 L G ]

[θ0θ1θ2

](36)

This current adaptation of space mapping introduces aniterative parameter extraction process that is decoupled fromthe main optimization process but runs parallel to it. Thismeans that stochastic algorithms can also be used in the mainoptimization process which can be an advantage for searchspaces with multiple minima. The current application of spacemapping makes use of a recursive least square algorithm forthe parameter extraction process. It is a deterministic algorithmwhich involves only algebraic operations. Thus, convergence isguaranteed and a unique set of parameters will be determinedas long as the preconditions are met as discussed earlier.

V. OPTIMIZATION

A. Design objectiveThe optimization process is defined by the objective function

which is subjected to appropriate physical constraints. Anobjective function of the form shown in (37) is chosen whereRs(xc) is defined by (38). The aim is to determine the optimal

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design configuration that maximizes the output force (F ) perunit heat generated (Q).

x∗c = arg min

xc

U (Rs (xc)) (37)

Rs(xc) =F

Qwhere F =

F

F0, Q =

Q

Q0(38)

F0 and Q0 are the norm of the output force and heat gen-eration respectively. Normalization of the outputs is essentialas they often represent different quantities which can carryvalues of varying orders of magnitude. This would affectthe optimization process since the algorithms are dependenton the sensitivity of the objective function against the input.Normalization creates non dimensional quantities which aremore suited for optimization. A natural selection for the normis the median of the expected minimum and maximum valueof the output. However, in this instance, output measurementsare available from the initial design as discussed in section III.Thus, they are taken as estimates of the norm. F0 and Q0 havevalues of 5.4 N and 4.1 W respectively.

Rs (xc) = −πQ0Bs (d− dt)2

F0ρrI(39)

Rs (xc) = −πQ0 (d− dt)2

F0ρrI

term 1︷︸︸︷Bc +

term 2︷ ︸︸ ︷θ0 + θ1L+ θ2G

(40)

Using expressions (1)-(3) to substitute for F and Q in theresponse function, Rs(xc), leads to the compact form shown in(39). It can be expanded by replacing Bs with (36) leading to afinal expression shown in (40). In the process, term 2 is addedto the response function but it does not change the convexityof the original coarse model output Bc. It is a monotonicallyincreasing/decreasing function (order of less than 2). However,it is not within the scope of this paper to provide mathematicalprove of the convexity of the response function (40). A generalapproach is adopted to ensure a global optimal solution isobtained based on the careful selection of the optimizationalgorithm and appropriate set up procedure. The details willbe discussed in the next subsection.

Mathematical constraints need to be defined to reflect thephysical constraints dictated by the application requirementand availability of components. For example, the copper wirestypically come in diameters based on the AWG standard.In this analysis, the wire diameter is chosen from a non-exhaustive set of wire diameters between the sizes AWG 30 to14. The minimum magnet length considered in this analysis islimited to 10mm while the maximum length is set at 100mm.This reflects the typical length of magnets that are readilyavailable. A constraint is also imposed on the air gap size forpractical reasons such as access for instrumentations to takemeasurements. The constraints are summarized by (41)-(44).

d[mm] ∈ {0.28, · · · , 1.692} (41)

10 ≤ L[mm] ≤ 110 (42)

3.5 ≤ G[mm] ≤ 8 (43)

F [N] ≥ 1.3F0, Q[W] ≤ 0.7Q0 (44)

Output constraints can also be defined as required by the ap-plication. In this case, a minimum output force and maximumheat generation target is benchmarked against the performanceof the initial design with a 30% improvement as shown by(44). However, output constraints need to be well defined asselection of extreme values could lead to non-convergence.Normalization of the constraints is also carried out for the samereason discussed earlier. The constraints are normalized againstthe upper or lower limit depending on where it is minimumor maximum constraint as illustrated in (45)-(47).

xmin ≤ x ≤ xmax (45)

x =x− x0x0

for x0 = xmin || xmax (46)

x ≥ 0 minimum || x ≤ 0 maximum (47)

The objective function defined in (37) is a continuousfunction of xc. Although the function is continuous for therange of xc, the original design variables (xf ) itself onlyaccept discrete values. The number of turns (nx) and number oflayers (ny) are integer values while the wire diameter (d0) onlyaccepts discrete values based on the AWG standard. In orderto address this issue, the proposed optimization methodologyinvolves two stages. Firstly, a global search using GeneticAlgorithm (GA) is performed to find the optimum solutionto the continuous problem. This is followed by a Branch andBound (BB) method to search feasible solutions locally for theoptimal. The local minimization uses a Sequential QuadraticProgramming (SQP) algorithm.

B. Optimization algorithm1) Global search: Genetic Algorithm (GA) is chosen for the

global search over gradient based method because of the latterslimitations in returning a global optimal solution especiallyso for objective functions which have multiple local minima.GA uses an evolutionary approach to search for the fittestindividual within the solution space which is defined here bythe fitness function (39). The GA implemented here is basedon the GA solver available in the Global Optimization Toolboxin Matlab which gives the user the flexibility of selecting theGA parameters making it a useful tool for such applications[19]. The software redefines the problem in terms of originalfitness function and nonlinear constraints using the Lagrangianrepresentation in what is known as the Augmented LagrangianGenetic Algorithm. The GA parameters used in this analysisis tabulated in TABLE III.

The output mapping described in section IV is incorporatedinto the optimization process during the global search. TheGA parameters are set to be less restrictive to allow for awider search of the solution space for the global optimum.It also aids the Parameter Extraction (PE) process since aninitial divergence of the solution allows for a more even outputmapping. Evaluation of the coarse and fine model responseis carried out at each major iterations of the GA and PE isdone using (32)-(34). The PE is terminated once the set target

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TABLE III. GENETIC ALGORITHM PARAMETER SETTING

GAparameter

Value/Type Description

Population 50 Size of the populationCreationfunction

Uniform Random initial population with a uniform distributionwithin the bounds of the linear constraints

Crossoverfraction

0.5 The fraction of the population created by crossover(the remaining individual undergo mutation)

Crossoverfunction

Intermediate Function used to create new individuals (childrencreated from the weighted average of parents)

Mutation Adaptivefeasible

Generate mutation directions that are adaptive de-pending on the relative success of each generation

Penaltyfactor

2 Penalty factor for violating the constraints to reset theAugmented Lagrangian subproblem

Constrainttolerance

0.0001 Determine solution feasibility with respect to nonlin-ear constraints

level of averaged mapping error is met. For this applicationthe target is fixed at 0.0075T which represents 2.5% of themeasured range. With sequential PE, the surrogate modelwould give better estimates of the magnetic flux density as theoptimization process progresses towards the optimum point.

2) Branch and Bound (BB) with local minimization: Thefollowing is a description of the Branch and Bound (BB)method as illustrated in Fig. 10. Firstly, if the continuous globaloptimization solution is feasible then the process terminates. Ifnot then the solution space X ∈ R is subdivided into smallerdomains Z such that Z ⊆ X . Branch and bound is based ondichotomy and exclusion principles [5]. The idea is to discardsub domains that do not yield solution that are better thanthe current best and search the remaining subdomain for theoptimal solution in a systematic manner. Branching is a processof creating these subdomains by subdividing the solution spacebased on interval analysis. A common method is to define thelimits of the subdomain by the next higher (x+i ) or lower (x−i )feasible value of the current variable (xi).

At each step of branching, two new sub problems arecreated. The process is followed by solving the sub problemas a continuous optimization problem with the new constraintlimits. The process of branching and solving the sub problemis continued for all the variables until a feasible solution isobtained. The objective function value (f ) for this solutionthen becomes the upper bound for the remaining sub prob-lems (bounding). Branches that have higher objective functionvalues are eliminated from further consideration. The upperbound (fmin) is updated when a feasible solution yields alower objective function value. The process will terminate onceall the possible solutions in the solution set (ζ) is exhaustedand it will return the optimal feasible solution (x∗).

Branching is so termed because it generates new branchesin the solution tree as illustrated in Fig. 11. Each branch inthe solution tree is a possible solution which makes up thesolution set (ζ). Not all branches are evaluated as the processof bounding discards branches which would not yield theoptimal feasible solution. For example, if subproblem 3 has anobjective function value greater than the current best solution(fmin), then that branch will be fathomed from the solutionset. This reduces the total number of evaluations needed tosearch for the optimal feasible solution.

Sequential Quadratic Programming (SQP) is used for localminimization during the branching step. The SQP implemented

Fig. 10. Branch and Bound (BB) algorithm

Fig. 11. Process of branching and the solution tree

is based on the ‘fmincon’ solver available in the OptimizationToolbox of Matlab. The method uses a quadratic approximationof the Lagrangian function as a replacement of the originalobjective and constraint functions. It searches for the minimumpoint in an iterative manner using the function and its deriva-tive to determine a search direction. The Hessian matrix orsecond order derivative is approximated using a quasi-Newtonupdating method. The method approximates the Hessian matrixfrom the first order derivatives which leads to time saving asexact calculations can be computationally costly.

VI. RESULTS AND DISCUSION

The output force and heat generated are functions of theoriginal design variables; number of turns (nx), number oflayers (ny) of the coil and the wire diameter (d0). The designvariables are linked to the external geometry by (5) and (6).The external length is directly dependent on the magnet length

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while the external radius is a direct offset of the gap size. Aparametric analysis is performed to investigate how the outputforce and heat generation is dependent on the external devicegeometry and the choice of wire diameter. This is done usingthe input-output relationship (1)-(2), parameterization of thecoil length (4) and estimate of the magnetic flux density usingthe analytical magnetic model (14)-(15).

Output force is plotted against the external length forselected external radii as shown in Fig. 12. The output forceincreases with length initially for a given radius. However, theoutput force reaches a maximum before decreasing, especiallyso for device with larger radii. The reduction in force generatedis due to the decreasing trend of the magnetic flux densitywith both external length and radii of the device. However,the coil length is increased with the external dimensions aswell. The output force being a product of the two terms hasa characteristic maximum point which reflects this underlyingrelationship. The effect is less pronounced for smaller radii asthe decrease in field strength is less significant with magnetlength for smaller gap sizes.

Output force changes significantly with the choice of wirediameter as shown by the offset between each plot in Fig.12(a), (b) and (c). It is possible to have a device with similarexternal dimensions but different output force by adjustingthe wire diameter. However, the choice of wire diameter isa critical factor in determining the amount of heat generated.The heat generated increases with overall dimensions of thedevice due to the increase in coil length. However, increasein heat generation is amplified for smaller wire diameter asshown by the steeper gradients of the (B) plots in Fig. 13.Heat generation is an inverse function of wire diameter and itcan be minimized by increasing the wire diameter.

Such parametric study serves as a guide for sizing thedevice and selection of wire diameter. A suggested approachis outlined below using Fig. 12, 13 for illustration.

1) Set minimum output force requirement2) Choose design with minimum external dimensions that

meets this output force requirement (Fig. 12)3) Choose the design which has the largest wire diameter4) Estimate heat generated for selected design (Fig. 13)5) Reduce output force if heat generation is too high6) Restart the design process with modified requirementsThis approach is an iterative design selection based on avail-

able knowledge of the device behaviour against the selectedvariables. The design objective in this case is to determinethe most compact design which results in the least amountof heat generation for a target level of output force. Themanual approach shown here can be tedious and the resultmay be suboptimal. Thus, it is more effective by automatingthe process through the formulation of a mathematical prob-lem that incorporates the design objective and constraints inone objective function. The problem can be solved using anoptimization methodology such as the one presented earlier.

Fig. 14(a) shows that a total of 15 major iterations of theGenetic Algorithm (GA) is required before it converges to asolution. The averaged mapping error at each major iterationis shown in Fig. 14(c). There is a decreasing trend with theiteration due to the least square algorithm used for Parameter

Fig. 12. Force variation with geometry and wire diameter

Fig. 13. Heat generation variation with geometry and wire diameter

Extraction (PE). As more coarse and fine model response aremade available at each iteration, the output mapping can befine tuned such that the mapping error is minimized. There isa minimum number of iteration required for the PE process toreturn meaningful parameters as discussed in section IV. Thechoice of a simple linear mapping function (30) would mean

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Fig. 14. Genetic algorithm parameters at each major iteration; (a) objectivefunction value, (b) constraint violation and (c) averaged mapping error

less number of iterations required (3 in this case).An averaged mapping error of 0.006T (representing an error

of 2% of the measurement range) is achieved after 3 majoriterations. Since the error is below the set target level, the PEprocess is terminated beyond the 3rd iteration. A sharp dropin objective function value is seen from the 3rd to 4th iterationas shown in Fig. 14(a). The output mapping does affect thesolution as expected. The mapping function is kept simple toavoid adding unnecessary minima to the solution space. ThePE process needs to be tied to a termination criterion whichallows the optimization solution to converge eventually. Fig.14(b) shows the near-optimal solution is close to the constraintsindicated by the higher constraint violation at the 9th to 11thiteration. However, the GA set up is robust enough to move itaway to another point which is selected as the optimal solutionsince it has a lower objective function value.

Each evaluation of the fine model requires about 12 minutesand 23 seconds on a workstation running at 3GHz with 32GBof RAM. The application of output mapping resulted in a80% reduction in computation time as the fine model isreplaced by the surrogate model after the 3rd iteration. Thetotal time required for the optimization process to completeis approximately 40 minutes with output mapping. A sim-ilar optimization performed using the commercial softwarepackage CST with a genetic algorithm optimization toolboxrequired approximately 3 days to yield a solution. However,the problem formulation in terms of the objective functionand design variables differs from the current optimization dueto the limitation of the software. For example, having thewire diameter as a design variable is not an option in thesoftware. Such limitations resulted in a reduced dimension

TABLE IV. COMPARISON OF MAGNETIC FLUX DENSITY ESTIMATES

Model Magnetic flux density [T] Deviation [T] % deviationFine model 0.202 - -Coarse model 0.244 0.042 +20.8%Surrogate model 0.206 0.004 +2.0%

TABLE V. RESULTS FROM GLOBAL SEARCH USING (A) SEQUENTIALQUADRATIC PROGRAMMING (SQP) STARTING AT (INITIAL POINT), (B)

GENETIC ALGORITHM (GA) AND (C) LOCAL SEARCH FOR OPTIMALFEASIBLE SOLUTION

nx ny d0 Objectivefunctionvalue

124.77 (125) 10.30 (10) 0.621 (0.320) -0.899(a) SQP 81.94 (82) 13.27 (6) 0.867 (0.485) -1.532

37.86 (35) 11.00 (3) 0.545 (1.151) -1.831(b) GA 39.77 10.84 0.553 -1.865(c) Optimal feasible 38 11 0.535 -1.826

of the problem but yet the computation time required is stillconsiderable much more than the proposed method.

For the current purpose of output comparison, the fine modelresponse is evaluated at the optimal point. TABLE IV showsa comparison between the magnetic flux density estimatesfrom the fine, coarse and surrogate model. The coarse modelestimate is 20.8% more than the fine model at the optimumpoint. This would have led to a sub optimal solution due toan overestimate of the output from modeling inaccuracies.However, with output mapping the deviation from the finemodel is reduced to 2%. The average air gap magnetic fluxis calculated using (11) with measurement taken off the newprototype. The average flux density is measured to be 0.212Tas opposed to 0.206T estimated from the surrogate model. Theestimate from the surrogate model is in fact 2.8% less than themeasured value. Output mapping is an effective tool in outputcorrection. It leads to eventual time saving by replacing thefine model with the surrogate during the optimization processafter an initial stage of ‘model tuning’ during the PE process.

The solution returned from the global search using GeneticAlgorithm (GA) is compared against those obtained usinga Sequential Quadratic Programming (SQP) algorithm withdifferent initial estimates as shown in Table V. Gradient basedsearch algorithms such as SQP guarantees an optimal solutionwhen the search space has a single minimum point. Otherwise,the algorithm could yield a local or global minimum dependingon the initial point. The result shows that the search spacedoes indeed have several minimum points. Stochastic methodssuch as GA are better suited for a global search when thereare multiple minima. Thus, the two stage algorithm has theadvantage of using different algorithms suited for each stageof the optimization process. Using GA for the global searchincreases the probability of obtaining a global optimum asshown by the lower objective function value obtained. There-after, the Branch and Bound method with local minimizationwould yield the optimal amongst the feasible solutions aroundthis global solution.

The optimal feasible solution obtained from the design op-timization translates into a device with attributes summarizedin TABLE VI. The optimal design shows a better performance

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TABLE VI. DESIGN ATTRIBUTES BEFORE AND AFTER OPTIMIZATION

Design parameter Initial design Optimal DesignNo. of turns 82 38No. of layers 6 11Wire diameter [mm] 0.485 0.535External length [mm] 51.8 32.7External radius [mm] 22.9 25.8Device volume [mm3] 85,340 68,381

TABLE VII. COMPARISON OF OUTPUT PERFORMANCE (MEASURED)

Measured quantity Initialdesign (a)

OptimalDesign (b)

% change =b−aa ×100%

Force generation [N] 5.28 7.39 +39.9Heat generation [W] 4.11 3.03 -26.2Averaged magnetic flux density [T] 0.146 0.212 +45.2

Fig. 15. Radial component of magnetic flux density profile

in the selected output measures. The output force has beenincreased by 39.9% while the heat generation is reduced by26.2%. The performance improvements are based on the outputmeasurements from the original device (initial design) and thenewly fabricated prototype (optimal design). Measurementsare made using the same experimental set up described insection III. The measurements are tabulated in TABLE VII.The optimal design also resulted in significant reduction of theexternal dimensions and overall device volume which leads toless material usage and cost savings.

Fig. 15 shows the magnetic flux distribution along the z-direction at the midpoint of the plane of interest for the initialand optimal design. The mean line shows the average valueof each plot. There is substantial increase in the average dueto the more compact design. The optimal design shows betteroverall magnetic distribution though there is still a decreasingtrend towards the open end of the magnetic circuit. This pointsto the fact that there is substantial flux leakage at the openend. However, the flux leakage in such C-shaped dual magnetconfiguration is reduced by the selection of more compactmagnets. The averaged magnetic flux density has increasedby 45.2% on average for the optimal design.

Fig. 16(a) shows the comparison of the output force fromboth designs. It is clearly shown by the steeper slope for theoptimal design that it has a higher force constant. This increaseis due in part because of the better magnetic performanceas discussed earlier. Fig. 16(b) shows the heat generationagainst the input current for both designs. There is significantreduction in heat generation simply by the use of wires withlarger diameter. As both outputs are functions of the coil

Fig. 16. Input output characteristic; (a) force and (b) heat generation

configuration, there exists an optimum configuration that meetsthe design objective. The optimization process has returned adesign that maximizes the air gap magnetic flux density byadjusting the gap geometry. At the same time, it maximized thecoil length exposed to this field while selecting the ideal wirediameter to balance the amount of force and heat generated.These parameters have some interdependence which makesthe problem difficult to solve manually. All these has to bedone while ensuring that the constraints are met. The twostage optimization method shown in this work overcomessome of these difficulties and challenges faced by designers.Moreover, the application of the methodology led to significantimprovements in the performance of the new prototype.

VII. CONCLUSION

This paper presented a two stage optimization methodologyfor the design analysis of a linear electromagnetic actuator.Solving the design problem as an inverse mathematical prob-lem resulted in a design that yielded significant performanceimprovement over an initial design. An increase in air gapmagnetic flux density by 45.2% is achieved leading to anincrease in output force by 39.9% and reduction in heatgeneration by 26.2%. Moreover, the overall device size hasbeen reduced which means a more compact design and costsavings in terms of material usage. The application of GeneticAlgorithm for the global search followed by a Branch andBound method is a systematic way of searching for the optimalfeasible solutions. This makes the methodology suitable forapplication to a range of engineering design problems whichhave mixed variables.

Time spent during the design stage is very much reduced dueto the automation resulting from the integrated optimizationprocedure. Time savings is achieved by replacing the FiniteElement magnetic (fine) model with an analytical (coarse)model during the process. Model output mismatch is reducedwith output mapping. Mapping parameter extraction is doneusing a deterministic least square algorithm. This resulted

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in better estimates of the magnetic flux density using thesurrogate model with adjusted output. At the optimal solution,the surrogate model estimate is within a 2.0% deviation oforiginal Finite Element model output and 2.8% deviationfrom the measured value. The application of an output spacemapping technique resulted in a time saving of 80% withminimal sacrifice of the modelling accuracy.

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Jonathan Hey received the B.Eng (Hons) Degree in Mechanical Engineeringfrom Nanyang Technological University (NTU), Singapore, in 2008. He is arecipient of the A*STAR graduate scholarship. He is currently a candidatefor the Ph.D. degree at Imperial College London, UK. His research interest ison thermal management of electromechanical devices focused on disturbancemodel identification, compensation methods and thermal design analysis.

Tat Joo TEO received the B.S. degree in mechatronics engineering fromQueensland University of Technology, Australia, in 2003 and the Ph.D.degree in mechanical and aerospace engineering from Nanyang TechnologicalUniversity, Singapore, in 2009. In 2009, he joined Singapore Institute ofManufacturing Technology as a research scientist in the mechatronics group.His research interest is in the area of ultra-precision system, compliantmechanism theory, parallel kinematics, electromagnetism, electro-mechanicalsystem, thermal modeling and analysis, energy-efficient machine, and topo-logical optimization.

Viet Phuong BUI (M09) received the Master and Ph.D. degrees in electricalengineering from the Grenoble Institute of Technology (INPG), France, in2004 and 2007, respectively. Since 2007, he has been with the Institute of HighPerformance Computing (IHPC), A*STAR, Singapore, where he is currentlyworking as a research scientist with the Electronics and Photonics Department.He has been involving in electromagnetics research from fundamentals todevices and applications since 2003. His research areas of interest alsoinclude computational electromagnetics, and optimization methods applied inelectromagnetics.

Guilin Yang received the B. Eng degree and M. Eng degree from JilinUniversity, China, in 1985 and 1988 respectively, and Ph.D. degree fromNanyang Technological University in 1999. From 1988 to 1995, he had beenwith the School of Mechanical Engineering, Shijiazhuang Tiedao University,China, as a lecturer, a division head, and then the vice dean of the school.Since 1998, he has been with Singapore Institute of Manufacturing Technology(SIMTech), Singapore. Currently, he is a senior scientist and the manager ofthe Mechatronics Group. His research interests include precision mechanisms,electromagnetic actuators, parallel-kinematics machines, modular robots, in-dustrial robots, and rehabilitation devices. He has published over 190 technicalpapers in referred journals and conference proceedings, 8 book chapters, and2 books. He has also filed 12 patents.

Ricardo Martinez-Botas is a Professor of Turbomachinery at ImperialCollege London. He has an MEng (Hons) Degree in Aeronautical Engineeringfrom Imperial College London and a doctoral degree from the Universityof Oxford. He has developed the area of unsteady flow aerodynamics ofsmall turbines, with particular application to the turbocharger industry. Thecontributions to this area centre on the application of unsteady fluid mechanics,instrumentation development and computational methods. He has expandedhis research in the area heat transfer of electrical machines and batteries. Hehas published extensively in journals and peer reviewed conferences. He iscurrently Associate Editor of the Journal of Turbomachinery and is a memberof the editorial board of two other international journals. He is currently theTheme Leader for Hybrid and Electric Vehicles of the Energy Futures Lab atImperial College.