Electromagnetic Actuator Design Analysis Using a Two-Stage ... · Electromagnetic Actuator Design Analysis Using a Two-Stage Optimization Method With Coarse–Fine Model Output Space

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014 5453

Electromagnetic Actuator Design AnalysisUsing a Two-Stage Optimization Method With

Coarse–Fine Model Output Space MappingJonathan Hey, Tat Joo Teo, Member, IEEE, Viet Phuong Bui, Member, IEEE,

Guilin Yang, Member, IEEE, and Ricardo Martinez-Botas

Abstract—Electromagnetic actuators are energy conversion de-vices that suffer from inefficiencies. The conversion losses generateinternal heat, which is undesirable, as it leads to thermal loadingon the device. Temperature rise should be limited to enhance thereliability, minimize thermal disturbance, and improve the outputperformance of the device. This paper presents the application ofan optimization method to determine the geometric configurationof a flexure-based linear electromagnetic actuator that maximizesoutput force per unit of heat generated. A two-stage optimizationmethod is used to search for a global solution, followed by afeasible solution locally using a branch and bound method. Thefinite element magnetic (fine) model is replaced by an analytical(coarse) model during optimization using an output space map-ping technique. An 80% reduction in computation time is achievedby the application of such an approximation technique. The mea-sured output from the new prototype based on the optimal designshows a 45% increase in air gap magnetic flux density, a 40%increase in output force, and a 26% reduction in heat generationwhen compared with the initial design before application of theoptimization method.

Index Terms—Electromagnetic devices, electromechanical ef-fect, genetic algorithm (GA), optimization methods, quadraticprogramming, reduced order systems.

I. INTRODUCTION

THE CURRENT approach toward designing electromag-netic actuators is largely based on direct methods making

use of commercial software to generate designs iteratively. Thisis a time-consuming process and will not necessarily yield theoptimal design [1], [2]. Moreover, the design process is nowcomplicated with increasing requirements such as consideration

Manuscript received March 8, 2013; revised September 13, 2013; acceptedNovember 25, 2013. Date of publication January 21, 2014; date of currentversion May 2, 2014. This work was supported by the Agency for Science,Technology, and Research of Singapore.

J. Hey and R. Martinez-Botas are with the Department of Mechanical Engi-neering, Imperial College, London, SW7 2AZ, U.K. (e-mail: [email protected]; [email protected]).

T. J. Teo is with the Mechatronics Group, Singapore Institute of Manufactur-ing Technology, Singapore 638075 (e-mail: [email protected]).

V. P. Bui is with the Institute of High Performance Computing, Agency forScience, Technology, and Research, Singapore 138632 (e-mail: [email protected]).

G. Yang is with the Ningbo Institute of Materials Technology and Engineer-ing, Chinese Academy of Sciences, Ningbo 315201, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2014.2301727

for multiphysics interaction. The device performance in termsof power output and efficiency is closely linked to the electro-magnetic and thermal interaction [3]. The power output of adevice is limited by its thermal efficiency, which is dependenton the design [4]. An optimal design offers a balance betweenthe two competing requirements. Direct methods are no longerintuitive due to the complex interactions, and optimization is auseful mathematical tool to solve the inverse design problem[5]. This ensures that the optimal design is selected withinthe bounds defined by the constraints in terms of the outputperformance and physical dimensions.

Recent applications of optimization methods have made useof finite 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 (SM) is a technique thatmakes use of a surrogate model in place of the fine model in theoptimization process [7]. Such a model can be derived from ananalytical or a coarse model through a mapping function. Theinaccuracies associated with the coarse model can be reducedby using a mapping function with iterative parameter extraction(PE) during optimization. The surrogate model therefore offersa good 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 toward 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 (PMs) and movingcoil supported by flexure bearings. The actuator stroke lengthis up to 500 μm and has an actuation speed of 100 mm/s.Such a device is designed for use in precision applications[10] with positioning accuracy of up to ±20 nm [11]. Thermalloading due to conversion losses is undesirable as it is the majorsource of positioning error [12]. Therefore, the design objectiveis to improve the efficiency of the device to reduce thermalloading.

The design analysis is aimed at identifying the geometricconfiguration that maximizes the output force per unit heat gen-erated. A single-objective mixed-variable optimization problem

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5454 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014

Fig. 1. Moving coil and stationary magnet linear actuator.

is formulated due to the discrete nature of some design vari-ables; thus, only feasible solutions can be accepted. An in-tegrated approach dealing with the discrete variables wouldfacilitate the design process [13]. A two-stage optimizationmethod is proposed by first using genetic algorithm (GA) forthe global search. This is followed by a branch and bound (BB)method for local minimization to determine the optimal feasiblesolution. SM is implemented in the optimization process toreduce the computation time. The current adaptation introducesan iterative PE process that is decoupled from the main opti-mization process using a deterministic recursive least squarealgorithm.

II. MATHEMATICAL MODELING

The basic configuration of the moving coil and stationarymagnet linear actuator is shown in Fig. 1. Such a linear actuatorcan be adapted for precision applications such as positioningand alignment. For instance, a flexure mechanism was used tosupport the moving coil in [14], which enhanced the positioningaccuracy of the actuator. The actuator is able to deliver submi-crometer positioning resolution with high actuating speed overa few millimeters stroke. Due to the linear current–force rela-tionship, they can also deliver direct and precise force controlcapabilities, which are essential for the microscale/nanoscaleimprinting processes [15]. The design of such a device iscentral around the conducting wire and its interaction with themagnetic field generated by the PMs. Lorentz force is generateddue to the interaction of the conducting wire and the magneticfield, which is the basis for actuation. At the same time,internal heat generation is inevitable due to energy conversionlosses.

The losses arise because of the following reasons: 1) resistiveheating of the conducting wire; 2) eddy current generatedin metallic parts; 3) hysteresis behavior in the magnet; and4) mechanical losses at contacting surfaces. Eddy current andhysteresis losses are generated in the presence of a changingelectric field. The changing field can be caused by an alternatingcurrent or when there is relative motion between the electricfield and the magnets or metallic parts. Mechanical losses arisebecause of the 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, i.e., Q,is dependent on the current, i.e., I , wire resistivity, i.e., ρr,

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

length, i.e., lc, and cross-sectional area, i.e., ac, as shown by(1). The Lorentz force, i.e., F , acting on the conducting wirein a magnetic field with a flux density, i.e., B, that is directlyproportional to the current is given by (2). The expressionholds for a constant magnetic field directed perpendicular tothe direction of current flow. Thus

Q =I2ρrlcac

(1)

F =BIlc. (2)

Both the output force and the heat generation are functionsof the input current I , which would be kept constant at 1 A forthis design analysis. Equations (1) and (2) show that the deviceoutput is central around the overall length lc of the conductingwire used. In order to perform the subsequent design analysis,the following assumptions and parameterization are carried out,using Fig. 2 for illustration. A regular packing of the wire isassumed, where the number of turns, i.e., nx and the number oflayers, i.e., ny , are defined to describe the wire configuration.The external wire diameter (d = d0 + dt) is adjusted for thefinite thickness of insulation material, i.e., dt, around the wirediameter, i.e., d0. The effective conductive cross-sectional areais given by

ac =π(d− dt)

2

4. (3)

nx, ny , and d0 are taken as the design variables, whereas

the other geometric parameters are kept constant as defined inFig. 2. The coil with an inner radius, i.e., 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 the magnet. The total length of the wirefor an idealized coil configuration is expressed in terms of thedesign variables given by (4). The external radius, i.e., Rext,and length, i.e., Lext, in millimeters, are given by (5) and (6).Thus

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)

HEY et al.: ELECTROMAGNETIC ACTUATOR DESIGN ANALYSIS USING A TWO-STAGE OPTIMIZATION METHOD 5455

A. Magnetic Model

1) FE Model: The analysis of the magnetic flux densityin the air gap is restricted to a sector of the device becauseof the axis-symmetric design. The problem is reduced to amagnetostatic analysis without the presence of the conductingwire. In a current-free region, the problem is described byGauss’ law. Thus

∇ ·B = 0. (7)

In the presence of PMs, there exist the following constitutivematerial relationships in the magnetic fields:

B = μ(H+M) (8)

where M is the magnetization vector within the magnets, andμ denotes the material permeability. The magnetic field H canbe expressed in terms of 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 preceding expressions tomake appropriate substitutions into (7) gives the followingexpression for the scalar potential in the region in and aroundthe PM:

∇2ϕm = ∇ ·M. (10)

Equation (10) can be numerically solved by using an FEmethod (FEM). Boundary conditions are imposed on the outersurface of the computational domain to ensure the existenceand uniqueness of the solution. Then, the field strength andthe radial flux density in the air gap can be derived from (8)and (9) by numerical differentiation. A 3-D FE model of thedevice is constructed in the software package CST using themagnetostatic solver. The simulation is performed withinthe bounding box enclosing the device structure that definesthe computational domain. The FE modeling requires that thewhole domain be subdivided into discrete elementary volumes,which provide the numerical approximation of the solution. Thesolution space is meshed with a total of 205 000 tetrahedralelements. In addition, information such as external sourcesand material properties needs to be provided for a completeFEM model. The core is constructed from mild steel, whichhas an estimated relative permeability of 25. The intrinsicmagnetization of the N48H PMs is M = 690 kA/m, which isused 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 usedfor estimation of the force generation in (2) is derived from theradial component of the flux density on the plane of interest.The average across Nz ×Nr points on the plane of interest canbe calculated using (11) by taking the mean of the integral sum.The flux density is denoted by Bf for clarity in the subsequentdiscussion. Thus

Bf =1

NzNr

Nz∑j=1

Nr∑i=1

B(ri, zj). (11)

Fig. 3. FE model with illustrated air gap magnetic flux density.

Fig. 4. Two-dimensional magnetic circuit model.

2) Analytical Model: An analytical magnetic circuit modelis developed to estimate the air gap magnetic flux density. Thismodel is less accurate compared with a FE model due to modelsimplifications. However, it requires little computational effortand is ideal for implementation in an optimization process. Themismatch in the model output between the analytical (coarse)and FE (fine) models needs to be addressed in order to deter-mine the optimal solution. An output mapping is introduced inSection IV to account for this mismatch.

The coarse model is derived from an equivalent magneticcircuit of a 2-D slice of the FE model, as illustrated in Fig. 4.The C-shaped dual magnet configuration is represented by amagnetic circuit with two parallel paths. The difference in thetwo paths lies in the leakage path present in path 2, which takesthe place of the core in path 1. The magnetic flux in the leakagepath is confined to an imaginary path as though a core is present.This idealization will lead to the model output inaccuracies. Themagnetic flux, i.e., Φ, motive force, i.e., υmg, and reluctance,i.e., R, are analogous to the electrical equivalent of current,voltage, and resistance, respectively. R is calculated from themagnetic path length l, permeability μ, and cross-sectionalarea A, as shown in (12). The cross-sectional area is simply aproduct of the path width, i.e., W , and characteristic dimension,i.e., D. The magnets are modeled as an ideal source with aninternal reluctance in series. The magnetic motive force is givenby (13), where Br is the remanent magnetic flux density, μr

is the recoil permeability, and lmg is the length (polarizationdirection) of the magnet. Thus

Ri =li

μiAiwhere Ai = WDi (12)

υmg =Brlmg

μr. (13)

The equivalent magnetic circuit is shown in Fig. 5. Theconservation of magnetic flux in paths 1 and 2 leads to the


Fig. 5. Equivalent magnetic circuit.

TABLE ISUMMARY OF MODEL PARAMETERS

expression (14). The magnetic flux density in the air gap issimply the flux density per unit area, as shown in (15). Thus

Φ=2υ

RT +2Rm+Rgwhere RT =

(Rlk +Rc2)(Rc1)

Rlk+Rc2 +Rc1(14)

Bc=Φ/Ag. (15)

The selection 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.

III. EXPERIMENTAL TESTING AND MODEL VALIDATION

Fig. 6 shows the experimental setup for measuring the radialcomponent of the magnetic flux density at the midpoint ofthe plane of interest along the z-direction (due to the finitesize of the probe). A unidirectional hall probe coupled witha Lakeshore 460 gaussmeter is used for measuring the fluxdensity with accuracy of 0.001 T. It is positioned along thez-axis with a motorized planar stage. This ensures that there isa consistent step size (1 mm) along the measurement path whilemaintaining a level position. The output force is measured usinga Schaevitz FC2231 load cell, as shown in Fig. 6. The loadcell is precalibrated for direct measurement of the output forcewith accuracy of ±0.05 N. In this test, the device is powered

Fig. 6. Experimental setup for measuring magnetic flux density and deviceinput–output characteristics.

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

by a linear amplifier using a closed-loop current controller tomaintain the desired input current level. The heat generation isequal to the supplied electrical power given by the product ofthe supply voltage and current.

The measured air gap magnetic flux density is shown inFig. 7. A comparison between the measurement and modeloutputs is made for validation of the FE model. There is agood agreement between the FE model estimation and mea-sured data, as shown by the close fit of the plots. The root-mean-square error between the measured and estimated data is0.022 T, representing an averaged error of less than 7.3% of themeasurement range (0.3 T). The measured air gap flux densityhas an average value of 0.146 T, whereas the model estimatesan average flux density of 0.150 T. 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 andthe input 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 showsthe minor deviation between the model estimated and measuredflux densities, 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 the measurement is more pronounced at higher levelsof input current, as shown in Fig. 8(b). This is caused bythe change in material properties as the device temperaturechanges. Electrical resistivity increases with temperature, and


Fig. 8. Input–output characteristic. (a) Force. (b) Heat generation.

TABLE IIMODEL OUTPUT DEVIATION AT DESIGN POINT

it directly affects the amount of heat generation based on (1).As such, it is expected that some discrepancy will arise sincethis effect is not explicitly accounted for in the model.

The input is an independent parameter in the optimization,and it is kept constant at a nominal value of 1 A. Thus, themodels are validated against the measurement at this designpoint. The force as measured using the load cell is 5.28 N,whereas the heat generation is derived from the supplied poweras 4.11 W. The force is overestimated by 0.6%, whereas the heatgeneration is underestimated by 8.9%, for reasons discussedearlier. Nevertheless, the model estimates do not deviate beyond10% from the measurement at the design point (1-A inputcurrent), as summarized in Table II.

IV. OUTPUT SM

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, i.e.,

x∗ = argminx

U (R(x)) . (16)

The underlying concept of SM is to match the optimal solutionderived from a coarse model to the actual solution from a finemodel during the optimization process. The coarse model istypically less accurate due to model simplification. However,the coarse model is simple to evaluate and computationallyefficient. The aim is to make use of the coarse model to replacethe fine model during the optimization process. However, the

mismatch between the coarse and fine models would not resultin 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 formula-tion or iteratively during the optimization process itself. Thefine model can be substituted by the coarse model using anapproximation shown in (18). The optimization process nowonly requires evaluation of the coarse model, which wouldbe computationally less demanding. The optimal solution canbe then estimated as xf through the inversion given in (20).Thus

xc =P (xf ) (17)

Rc (P (xf )) ≈Rf (xf ) (18)

x∗c = argmin

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 PE error(21). In the original SM, Bandler et al. used a least squareregression to determine the coefficient of a linear mappingfunction [7]. Such a method requires a finite number of finemodel evaluations prior to the optimization process to initiatethe PE process. The number of evaluations required dependson the order and dimension of the mapping function, whichincreases with the mismatch. In the aggressive SM method[7], the PE is an intermediate step of the main optimizationprocess. It uses an iterative quasi-Newton method to estimatethe Jacobian matrix to direct the search toward a functionthat minimizes (21). However, it suffers from the limitationscommon to numerical methods such as convergence difficultiesand sensitivity to the initial estimate.

If there are large differences in structure between underlyingmodels, the output misalignment between the coarse and finemodel responses will be significant even after input mapping.Output mapping can overcome this deficiency by introducing atransformation of the coarse model response based on an outputmapping function O(Rc(xc)) as defined in (22). The mappedoutput from the coarse model is known as the surrogate modelresponse, i.e., Rs(xc). An example of the output mappingfunction is given by (23), where γ(i) is a weighting factor onthe response mismatch, i.e., ΔR. Thus

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

Rs(xc) =O (Rc(xc)) = Rc(xc) +m∑i=1

γ(i)ΔR(i) (23)

ΔR(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 in a PE


process [17]. Updating the surrogate model is iteratively doneby solving (25) for x∗

c and evaluating the fine model responseat this point to generate a new pair of coarse and fine modelresponses. The process repeats until the response mismatch isless than a set limit. At the final iteration, the optimal solution,i.e., x∗

c, is estimated as the final solution, i.e., x∗f . Thus

x∗c = argmin

xc

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

The analytical magnetic model presented in the precedingsection shows that the magnetic flux density in the air gap isa function 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. Thus

xc = [L G d] (26)

xf = [nx ny d0] (27)

d = d0 + dt, L = nxd+ 2l0, 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) and (15) and finemodel (11). The model response mismatch is defined as

ΔR = Bf −Bc. (29)

The coarse model adopts some idealization for modeling theleakage flux at the open end of the magnetic circuit. The leakageflux is dependent on the relative reluctance of the magneticpaths. The reluctance is, in turn, dependent on the magneticpath length, as described earlier. Thus, the magnet length L andthe gap size G are critical parameters that affect this additionalcomponent of flux leakage. It is reasonable to assume thatthe coarse and fine model response mismatch is due to theunaccounted component of flux leakage. A linear function of(L,G) in the form shown in (30) is used to model the mismatchΔR. Physical insight of the mismatch means that the mappingfunction is kept simple but still effective at improving theaccuracy of the surrogate model. Thus

ΔR̂ =y∗θ = [ 1 L G ]

⎡⎣ θ0θ1θ2

⎤⎦ (30)

ε = ‖ΔR−ΔR̂‖. (31)

The PE error defined by (31) allows the parameter θ tobe determined using a least square method. This process canbe iteratively done during the main optimization process byusing a recursive least square algorithm. The major steps ofthe algorithm is given by (32)–(34). Q is a measure of theuncertainty in the estimated parameter θ. Q will be reducedover time when more data points are made available, and hence,the solution will converge to the optimal estimate of θ that

Fig. 9. Output mapping and PE.

minimizes (31). A large Q(0) should be chosen to move theestimate θ(0) toward its final solution θ. Thus

Q(k) =

[Q(k−1) −

Q(k−1)yT(k)y(k)Q(k−1)

1 + y(k)Q(k−1)yT(k)

](32)

K(k) =Q(k−1)y

T(k)

1 + y(k)Q(k−1)yT(k)

(33)

θ(k) = θ(k−1) +K(k)

[ΔR(k−1) − y(k)θ(k−1)

]. (34)

For y ∈ R1xn, there needs to be k ≥ n iterations to ensure

uniqueness of the estimated parameter θ. Evaluation of the finemodel response is necessary at each major iteration k of theoptimization process to update this parameter. The PE processis terminated once the averaged mapping error, i.e., 〈ε〉, definedby (35) is less than a predetermined target level. Finally, thesurrogate model output is given by (36). Thus

〈ε〉 =

√∑ki

[ε(k)

]2k

(35)

Bs =Bc + [ 1 L G ]

⎡⎣ θ0θ1θ2

⎤⎦ . (36)

The key steps of implementing an output mapping together withthe optimization process is illustrated in Fig. 9.

This current adaptation of SM introduces an iterative PE pro-cess that is decoupled from the main optimization process butruns parallel to it. A recursive least square algorithm is used forthe PE process. It is a deterministic algorithm, which involvesonly algebraic operations. Thus, convergence is guaranteed, anda unique set of parameters will be determined as long as thepreconditions are met, as discussed earlier.

V. OPTIMIZATION

A. Design Objective

The optimization process is defined by the objective func-tion, which is subjected to appropriate physical constraints.


An objective function of the form shown in (37) is chosen,where Rs(xc) is defined by (38). The aim is to determine theoptimal design configuration that maximizes the output force Fper unit heat generated Q. Thus

x∗c = argmin

xc

U (Rs(xc)) (37)

Rs(xc) =F̄

Q̄where F̄ =

F

F0, Q̄ =

Q

Q0. (38)

F0 and Q0 are the norms of the output force and heat gener-ation, respectively. Normalization of the outputs is essential asthey often represent different quantities that can carry values ofvarying orders of magnitude. This would affect the optimizationprocess since the algorithms are dependent on the sensitiv-ity of the objective function against the input. Normalizationcreates nondimensional quantities, which are more suited foroptimization. A natural selection for the norm is the medianof the expected minimum and maximum values of the output.However, in this instance, output measurements are availablefrom the initial design, as discussed in Section III. Thus, theyare taken as estimates of the norm. F0 and Q0 have values of5.4 N and 4.1 W, respectively. Thus

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 mathematicalproof of the convexity of the response function (40). A generalapproach is adopted to ensure that a global optimal solutionis obtained based on the careful selection of the optimizationalgorithm and appropriate setup procedure. The details will bediscussed 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. Inthis analysis, the wire diameter is chosen from a nonexhaustiveset of wire diameters between the sizes AWG 30 and 14. Theminimum magnet length considered in this analysis is limitedto 10 mm, whereas the maximum length is set at 100 mm.This reflects the typical length of magnets that are readilyavailable. A constraint is also imposed on the air gap sizefor practical reasons such as access for instrumentations to

TABLE IIIGA PARAMETER SETTING

take measurements. The constraints are summarized by thefollowing:

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 be also defined as required by theapplication. 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 as selectionof extreme values could lead to nonconvergence. Normalizationof the constraints is also carried out for the same reasondiscussed earlier. The constraints are normalized against theupper or the lower limit depending on whether it is minimumor maximum constraint, as illustrated in the following:

x̄ =x− x0

x0for x0 = xmin ‖ xmax. (45)

The objective function defined in (37) is a continuous func-tion of xc. Although the function is continuous for the range ofxc, the original design variables (xf ) themselves only acceptdiscrete values. The number of turns nx and the number oflayers ny are integer values, whereas the wire diameter d0 onlyaccepts discrete values based on the AWG standard. In order toaddress this issue, the proposed optimization methodology in-volves two stages. First, a global search using GA is performedto find the optimum solution to the continuous problem. This isfollowed by a BB method to search feasible solutions locally forthe optimal. The local minimization uses a sequential quadraticprogramming (SQP) algorithm.

B. Optimization Algorithm

1) Global Search: GA is chosen for the global search overa gradient-based method because of the latter’s limitations inreturning a global optimal solution, particularly so for objec-tive functions, which have multiple local minima. GA usesan evolutionary approach to search for the fittest individualwithin the solution space, which is defined here by the fitnessfunction (39). The GA implemented here is based on theGA solver available in the Global Optimization Toolbox inMATLAB, which gives the user the flexibility of selecting the


Fig. 10. BB algorithm.

GA parameters, making it a useful tool for such applications[18]. The software redefines the problem in terms of originalfitness function and nonlinear constraints using the Lagrangianrepresentation in what is known as the augmented LagrangianGA. The GA parameters used in this analysis are tabulatedin Table III.

The output mapping described in Section IV is incorporatedinto the optimization process during the global search. The GAparameters are set to be less restrictive to allow for a widersearch of the solution space for the global optimum. It also aidsthe PE process since an initial divergence of the solution allowsfor a more even output mapping. Evaluation of the coarse andfine model responses is carried out at each major iterations ofthe GA, and PE is done using (32)–(34). The PE is terminatedonce the set target level of averaged mapping error is met. Forthis application, the target is fixed at 0.0075 T, which represents2.5% of the measured range. With sequential PE, the surrogatemodel would give better estimates of the magnetic flux densityas the optimization process progresses toward the optimumpoint.

2) BB With Local Minimization: The following is a de-scription of the BB method illustrated in Fig. 10. First, ifthe continuous global optimization solution is feasible, thenthe process terminates. If not, then the solution space X ∈ R

is subdivided into smaller domains Z such that Z ⊆ X . BB isbased on dichotomy and exclusion principles [5]. The idea is todiscard subdomains that do not yield solutions that are betterthan the current best and search the remaining subdomainsfor the optimal solution in a systematic manner. Branchingis a process of creating these subdomains by subdividing thesolution space based on interval analysis. A common method isto define the limits of the subdomain by the next higher (x+

i ) orlower (x−

i ) feasible value of the current variable (xi).

Fig. 11. Process of branching and the solution tree.

At each step of branching, two new subproblems are cre-ated. The process is followed by solving the subproblem asa continuous optimization problem with the new constraintlimits. The process of branching and solving the subproblemis continued for all the variables until a feasible solution isobtained. The objective function value, i.e., f , for this solutionthen becomes the upper bound for the remaining subproblems(bounding). Branches that have higher objective function valuesare eliminated from further consideration. The upper bound(fmin) is updated when a feasible solution yields a lowerobjective function value. The process will terminate once allthe possible solutions in the solution set, i.e., ζ, are exhausted,and 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 that would not yield the optimalfeasible solution. For example, if subproblem 3 has an objectivefunction value greater than the current best solution fmin, thenthat branch will be fathomed from the solution set. This reducesthe total number of evaluations needed to search for the optimalfeasible solution.

SQP is used for local minimization during the branching step.The SQP implemented is based on the “fmincon” solver avail-able in the Optimization Toolbox of MATLAB. The methoduses a quadratic approximation of the Lagrangian function as areplacement of the original objective and constraint functions.It searches for the minimum point in an iterative mannerusing the function and its derivative to determine a searchdirection. The Hessian matrix or the second-order derivativeis approximated using a quasi-Newton updating method. Themethod approximates the Hessian matrix from the first-orderderivatives, which leads to time saving as exact calculations canbe computationally costly.

VI. RESULTS AND DISCUSSION

The output force and the heat generated are functions ofthe original design variables: the number of turns nx, thenumber of layers ny of the coil, and the wire diameter d0.The design variables are linked to the external geometry by


Fig. 12. Force variation with geometry and wire diameter.

(5) and (6). The external length is directly dependent on themagnet length, whereas the external radius is a direct offset ofthe gap size. A parametric analysis is performed to investigatehow the output force and the heat generation are dependent onthe external device geometry and the choice of wire diameter.This is done using the input–output relationship (1) and (2), theparameterization of the coil length (4), and the estimate of themagnetic flux density using the analytical magnetic model (14)and (15).

Output force is plotted against the external length for selectedexternal radii, as shown in Fig. 12. The output force increaseswith length initially for a given radius. However, the outputforce reaches a maximum before decreasing, particularly sofor devices 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 significantly changes with the choice of wirediameter, as shown by the offset between each plot inFig. 12(a)–(c). It is possible to have a device with similarexternal dimensions but different output force by adjusting thewire diameter. However, the choice of wire diameter is a criticalfactor in determining the amount of heat generated. The heatgenerated increases with the overall dimensions of the devicedue to the increase in coil length. However, increase in heatgeneration is amplified for smaller wire diameter, as shown bythe steeper gradients of the () plots in Fig. 13. Heat generationis an inverse function of wire diameter, and it can be minimizedby increasing the wire diameter.

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

Such parametric study serves as a guide for sizing the de-vice and selection of wire diameter. A suggested approach isoutlined below using Figs. 12 and 13 for illustration.

1) Set minimum output force requirement.2) Choose design with minimum external dimensions that

meet this output force requirement (see Fig. 12).3) Choose the design that has the largest wire diameter.4) Estimate heat generated for the selected design (see

Fig. 13).5) Reduce output force if heat generation is too high.6) Restart the design process with modified requirements.

This approach is an iterative design selection based on avail-able knowledge of the device behavior against the selectedvariables. The design objective in this case is to determinethe most compact design, which results in the least amount ofheat generation for a target level of output force. The manualapproach shown here can be tedious, and the result may besuboptimal. Thus, it is more effective by automating the processthrough the formulation of a mathematical problem that incor-porates the design objective and constraints in one objectivefunction. The problem can be solved using an optimizationmethodology such as the one presented earlier.

Fig. 14(a) shows that a total of 15 major iterations of the GAis required before it converges to a solution. The averaged map-ping error at each major iteration is shown in Fig. 14(c). Thereis a decreasing trend with the iteration due to the least square al-gorithm used for PE. As more coarse and fine model responsesare made 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 meanless number of iterations required (three in this case).


Fig. 14. GA parameters at each major iteration. (a) Objective function value.(b) Constraint violation. (c) Averaged mapping error.

An averaged mapping error of 0.006 T (representing an errorof 2% of the measurement range) is achieved after three majoriterations. Since the error is below the set target level, the PEprocess is terminated beyond the third iteration. A sharp dropin objective function value is seen from the third to fourth iter-ations, as shown in Fig. 14(a). The output mapping does affectthe solution as expected. The mapping function is kept simple toavoid adding unnecessary minima to the solution space. The PEprocess needs to be tied to a termination criterion, which allowsthe optimization solution to converge eventually. Fig. 14(b)shows that the near-optimal solution is close to the constraintsindicated by the higher constraint violation at the ninth toeleventh iterations. However, the GA setup is robust enough tomove it away to another point, which is selected as the optimalsolution since it has a lower objective function value.

Each evaluation of the fine model requires about 12 min and23 s on a workstation running at 3 GHz with 32 GB of randomaccess memory. The application of output mapping resulted inan 80% reduction in computation time as the fine model isreplaced by the surrogate model after the third iteration. Thetotal time required for the optimization process to completeis approximately 40 min with output mapping. A similar op-timization performed using the commercial software packageCST with a GA optimization toolbox required approximatelythree days to yield a solution. However, the problem formu-lation in terms of the objective function and design variablesdiffers from the current optimization due to the limitation ofthe software. For example, having the wire diameter as a designvariable is not an option in the software. Such limitationsresulted in a reduced dimension of the problem; however, thecomputation time required is still much more considerable thanthe proposed method.

TABLE IVCOMPARISON OF MAGNETIC FLUX DENSITY ESTIMATES

TABLE VRESULTS FROM GLOBAL SEARCH USING SQP STARTING AT INITIAL

POINT, GA, AND LOCAL SEARCH FOR OPTIMAL FEASIBLE SOLUTION

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 models. The coarse modelestimate is 20.8% more than the fine model at the optimumpoint. This would have led to a suboptimal 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 0.212 T,as opposed to 0.206 T 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 the finemodel with the surrogate one during the optimization processafter an initial stage of “model tuning” during the PE process.

The solution returned from the global search using GA iscompared against those obtained using an SQP algorithm withdifferent initial estimates, as shown in Table V. Gradient-basedsearch algorithms such as SQP guarantee an optimal solutionwhen the search space has a single minimum point. Otherwise,the algorithm could yield a local or a global minimum, depend-ing on 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 there aremultiple minima. Thus, the two-stage algorithm has the advan-tage of using different algorithms suited for each stage of theoptimization process. Using GA for the global search increasesthe probability of obtaining a global optimum, as shown bythe lower objective function value obtained. Thereafter, the BBmethod with local minimization would yield the optimal amongthe feasible solutions around this 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 performancein the selected output measures. The output force has beenincreased by 39.9%, whereas the heat generation is reducedby 26.2%. The performance improvements are based on theoutput measurements from the original device (initial design)and the newly fabricated prototype (optimal design). Measure-ments are made using the same experimental setup described


TABLE VIDESIGN ATTRIBUTES BEFORE AND AFTER OPTIMIZATION

TABLE VIICOMPARISON OF OUTPUT PERFORMANCE (MEASURED)

Fig. 15. Radial component of magnetic flux density profile.

in Section III. The measurements are tabulated in Table VII.The optimal design also resulted in significant reduction in theexternal dimensions and overall device volume, which leads toless material usage and cost savings.

Fig. 15 shows the magnetic flux distribution along thez-direction at the midpoint of the plane of interest for the initialand optimal designs. The mean line shows the average valueof each plot. There is a substantial increase in the average dueto the more compact design. The optimal design shows betteroverall magnetic distribution, although there is still a decreasingtrend toward 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 increased by45.2% 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 to the better magnetic performance, as discussedearlier. Fig. 16(b) shows the heat generation against the inputcurrent for both designs. There is a significant reduction in heatgeneration simply by the use of wires with larger diameter. Asboth outputs are functions of the coil configuration, there existsan optimum configuration that meets the design objective. Theoptimization process has returned a design that maximizes theair gap magnetic flux density by adjusting the gap geometry. Atthe same time, it maximized the coil length exposed to this fieldwhile selecting the ideal wire diameter to balance the amount of

Fig. 16. Input–output characteristic. (a) Force. (b) Heat generation.

force and heat generated. These parameters have some interde-pendence, which makes the problem difficult to solve manually.All these have to be done while ensuring that the constraints aremet. The two-stage optimization method shown in this paperovercomes some of these difficulties and challenges faced bydesigners. Moreover, the application of the methodology ledto significant improvements in the performance of the newprototype.

VII. CONCLUSION

This paper has presented a two-stage optimization methodol-ogy for 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 a 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 GA forthe global search followed by a BB method is a systematic wayof searching for the optimal feasible solutions. This makes themethodology suitable for application to a range of engineeringdesign problems, which have mixed variables.

Time spent during the design stage is very much reduced dueto the automation resulting from the integrated optimizationprocedure. Time savings are achieved by replacing the FEmagnetic (fine) model with an analytical (coarse) model duringthe process. Model output mismatch is reduced with output SM.PE is done using a deterministic least square algorithm. Thisresulted in better estimates of the magnetic flux density usingthe surrogate model with adjusted output. At the optimal solu-tion, the surrogate model estimate is within a 2.0% deviationof the original FE model output and 2.8% deviation from themeasured value. The application of an output SM techniqueresulted in a time saving of 80%, with minimal sacrifice of themodeling accuracy.


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Jonathan Hey received the B.Eng. (Hons.) degree inmechanical engineering from Nanyang Technologi-cal University, Singapore, in 2008. He is currentlyworking toward the Ph.D. degree at Imperial CollegeLondon, London, U.K.

His research interests include thermal manage-ment of electromechanical devices, with a focuson disturbance model identification, compensationmethods, and thermal design analysis.

Mr. Hey is a recipient of the Agency for Science,Technology, and Research of Singapore Graduate

Scholarship.

Tat Joo Teo (M’10) received the B.S. degreein mechatronics engineering from the QueenslandUniversity of Technology, Kelvin Grove, QLD,Australia, in 2003, and the Ph.D. degree in me-chanical and aerospace engineering from NanyangTechnological University, Singapore, in 2009.

In 2009, he joined the Singapore Institute of Man-ufacturing Technology, Singapore, as a ResearchScientist with the Mechatronics Group. His researchinterests include ultraprecision systems, compliantmechanism theory, parallel kinematics, electromag-

netism, electromechanical systems, thermal modeling and analysis, energy-efficient machines, and topological optimization.

Viet Phuong Bui (M’09) received the Master’s andPh.D. degrees in electrical engineering from theGrenoble Institute of Technology (INPG), Grenoble,France, in 2004 and 2007, respectively.

Since 2007, he has been with the Institute ofHigh Performance Computing, Agency for Science,Technology, and Research, Singapore, where he iscurrently a Research Scientist with the Electronicsand Photonics Department. Since 2003, he has beeninvolved in electromagnetics research from funda-mentals to devices and applications. His research

interests include computational electromagnetics and optimization methodsapplied in electromagnetics.

Guilin Yang (M’02) received the B.Eng. and M.Eng.degrees from Jilin University, Jilin, China, in 1985and 1988, respectively, and the Ph.D. degree fromNanyang Technological University, Singapore, in1999, all in mechanical engineering.

From 1998 to 2013, he was with the SingaporeInstitute of Manufacturing Technology, Singapore.He is currently a Senior Research Fellow with theNingbo Institute of Materials Technology and En-gineering, Chinese Academy of Sciences, Ningbo,China. He has authored/coauthored over 200 tech-

nical papers in refereed journals and conference proceedings, eight bookchapters, and two books. He has also filed 12 patents. His research interestsinclude precision mechanisms, electromagnetic actuators, parallel kinematicsmachines, modular robots, industrial robots, and rehabilitation devices.

Mr. Yang is currently an Editorial Board Member of Frontiers of MechanicalEngineering and the International Journal of Mechanisms and Robotic Systems,and is an Associate Editor of IEEE Access.

Ricardo Martinez-Botas received the M.Eng.(Hons.) degree from the Imperial College of London,London, U.K., and the Doctoral degree from the Uni-versity of Oxford, Oxford, U.K., both in aeronauticalengineering.

He is currently a Professor of Turbomachineryand the Theme Leader for Hybrid and Electric Ve-hicles of the Energy Futures Lab, Imperial Collegeof London. He has published extensively in journalsand peer-reviewed conferences. His research inter-ests include heat transfer of electrical machines and

batteries.Dr. Martinez-Botas is currently an Associate Editor of the Journal of Turbo-

machinery and is a Member of the editorial board of two other internationaljournals.

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Electromagnetic Actuator Design Analysis Using a Two-Stage ... · Electromagnetic Actuator Design Analysis Using a Two-Stage Optimization Method With Coarse–Fine Model Output Space