IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014
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Electromagnetic Actuator Design Analysis
Using a Two-Stage Optimization Method With
Coarse–Fine Model Output Space Mapping
Jonathan 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 devices that suffer from inefficiencies. The conversion losses generate
internal heat, which is undesirable, as it leads to thermal loading
on the device. Temperature rise should be limited to enhance the
reliability, minimize thermal disturbance, and improve the output
performance of the device. This paper presents the application of
an optimization method to determine the geometric configuration
of a flexure-based linear electromagnetic actuator that maximizes
output force per unit of heat generated. A two-stage optimization
method is used to search for a global solution, followed by a
feasible solution locally using a branch and bound method. The
finite element magnetic (fine) model is replaced by an analytical
(coarse) model during optimization using an output space mapping technique. An 80% reduction in computation time is achieved
by the application of such an approximation technique. The measured output from the new prototype based on the optimal design
shows a 45% increase in air gap magnetic flux density, a 40%
increase in output force, and a 26% reduction in heat generation
when compared with the initial design before application of the
optimization method.
Index Terms—Electromagnetic devices, electromechanical effect, genetic algorithm (GA), optimization methods, quadratic
programming, reduced order systems.
I. I NTRODUCTION
T
HE CURRENT approach toward designing electromagnetic actuators is largely based on direct methods making
use of commercial software to generate designs iteratively. This
is a time-consuming process and will not necessarily yield the
optimal design [1], [2]. Moreover, the design process is now
complicated with increasing requirements such as consideration
Manuscript received March 8, 2013; revised September 13, 2013; accepted
November 25, 2013. Date of publication January 21, 2014; date of current
version 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 Engineering, Imperial College, London, SW7 2AZ, U.K. (e-mail: hhh09@imperial.
ac.uk; [email protected]).
T. J. Teo is with the Mechatronics Group, Singapore Institute of Manufacturing Technology, Singapore 638075 (e-mail: [email protected]).
V. P. Bui is with the Institute of High Performance Computing, Agency for
Science, Technology, and Research, Singapore 138632 (e-mail: buivp@ihpc.
a-star.edu.sg).
G. Yang is with the Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo 315201, China (e-mail: glyang@
nimte.ac.cn).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIE.2014.2301727
for multiphysics interaction. The device performance in terms
of power output and efficiency is closely linked to the electromagnetic and thermal interaction [3]. The power output of a
device is limited by its thermal efficiency, which is dependent
on the design [4]. An optimal design offers a balance between
the two competing requirements. Direct methods are no longer
intuitive due to the complex interactions, and optimization is a
useful mathematical tool to solve the inverse design problem
[5]. This ensures that the optimal design is selected within
the bounds defined by the constraints in terms of the output
performance and physical dimensions.
Recent applications of optimization methods have made use
of finite element (FE) models to simulate the device response
[6]. However, the increasing design requirements nowadays
mean that the application of fine models such as FE models
to the optimization process is still challenging as computation
power will be a limit. Space mapping (SM) is a technique that
makes use of a surrogate model in place of the fine model in the
optimization process [7]. Such a model can be derived from an
analytical or a coarse model through a mapping function. The
inaccuracies associated with the coarse model can be reduced
by using a mapping function with iterative parameter extraction
(PE) during optimization. The surrogate model therefore offers
a good balance between accuracy and computation effort.
Electromagnetic actuators are designed based on the application requirement. For example, the design optimization
of a transverse flux linear motor for railway traction [8] is
targeted at maximizing the thrust force while minimizing the
weight. The design analysis of a switch reluctance motor for
vehicle propulsion [9] is geared toward high torque density
and efficiency as measured by the torque per unit copper loss.
In this analysis, the test device is a linear electromagnetic
actuator with stationary permanent magnets (PMs) and moving
coil supported by flexure bearings. The actuator stroke length
is 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]. Thermal
loading due to conversion losses is undesirable as it is the major
source of positioning error [12]. Therefore, the design objective
is to improve the efficiency of the device to reduce thermal
loading.
The design analysis is aimed at identifying the geometric
configuration that maximizes the output force per unit heat generated. A single-objective mixed-variable optimization problem
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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 variables; thus, only feasible solutions can be accepted. An integrated approach dealing with the discrete variables would
facilitate the design process [13]. A two-stage optimization
method is proposed by first using genetic algorithm (GA) for
the global search. This is followed by a branch and bound (BB)
method for local minimization to determine the optimal feasible
solution. SM is implemented in the optimization process to
reduce the computation time. The current adaptation introduces
an iterative PE process that is decoupled from the main optimization process using a deterministic recursive least square
algorithm.
II. M ATHEMATICAL M ODELING
The basic configuration of the moving coil and stationary
magnet linear actuator is shown in Fig. 1. Such a linear actuator
can be adapted for precision applications such as positioning
and alignment. For instance, a flexure mechanism was used to
support the moving coil in [14], which enhanced the positioning
accuracy of the actuator. The actuator is able to deliver submicrometer positioning resolution with high actuating speed over
a few millimeters stroke. Due to the linear current–force relationship, they can also deliver direct and precise force control
capabilities, which are essential for the microscale/nanoscale
imprinting processes [15]. The design of such a device is
central around the conducting wire and its interaction with the
magnetic field generated by the PMs. Lorentz force is generated
due to the interaction of the conducting wire and the magnetic
field, which is the basis for actuation. At the same time,
internal heat generation is inevitable due to energy conversion
losses.
The losses arise because of the following reasons: 1) resistive
heating of the conducting wire; 2) eddy current generated
in metallic parts; 3) hysteresis behavior in the magnet; and
4) mechanical losses at contacting surfaces. Eddy current and
hysteresis losses are generated in the presence of a changing
electric field. The changing field can be caused by an alternating
current or when there is relative motion between the electric
field and the magnets or metallic parts. Mechanical losses arise
because of the friction between moving surfaces in contact.
Finally, resistive heating is caused by the electrical resistivity
of the conducting wire.
In a static operation using direct current, resistive heating
is the only source of heat generation as there is no motion
and 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 wire
in a magnetic field with a flux density, i.e., B, that is directly
proportional to the current is given by (2). The expression
holds for a constant magnetic field directed perpendicular to
the direction of current flow. Thus
Q=
I 2 ρr lc
ac
F = BIlc .
(1)
(2)
Both the output force and the heat generation are functions
of the input current I, which would be kept constant at 1 A for
this design analysis. Equations (1) and (2) show that the device
output is central around the overall length lc of the conducting
wire 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 is
assumed, where the number of turns, i.e., nx and the number of
layers, i.e., ny , are defined to describe the wire configuration.
The external wire diameter (d = d0 + dt ) is adjusted for the
finite thickness of insulation material, i.e., dt , around the wire
diameter, i.e., d0 . The effective conductive cross-sectional area
is 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 in
Fig. 2. The coil with an inner radius, i.e., R, is suspended in the
magnetic field by a bobbin attached to a flexure bearing, which
is not shown. A small gap tolerance defined by l0 and g0 exists
between the coil and the magnet. The total length of the wire
for an idealized coil configuration is expressed in terms of the
design 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
2πnx {R + [(i − 1) + 0.5] d}
i=1
= πnx ny (2R + ny d)
(4)
Rext = ny d + 2g0 + 18
(5)
Lext = nx d + 2l0 + 10.
(6)
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A. Magnetic Model
1) FE Model: The analysis of the magnetic flux density
in the air gap is restricted to a sector of the device because
of the axis-symmetric design. The problem is reduced to a
magnetostatic analysis without the presence of the conducting
wire. In a current-free region, the problem is described by
Gauss’ law. Thus
∇ · B = 0.
(7)
Fig. 3. FE model with illustrated air gap magnetic flux density.
In the presence of PMs, there exist the following constitutive
material relationships in the magnetic fields:
B = μ(H + M)
(8)
where M is the magnetization vector within the magnets, and
μ denotes the material permeability. The magnetic field H can
be expressed in terms of the magnetic scalar potential such that
H = −∇ϕm
(9)
where ϕm is analogous to the definition of the electric potential
in electrostatics. Making use of the preceding expressions to
make appropriate substitutions into (7) gives the following
expression for the scalar potential in the region in and around
the PM:
∇2 ϕm = ∇ · M.
(10)
Equation (10) can be numerically solved by using an FE
method (FEM). Boundary conditions are imposed on the outer
surface of the computational domain to ensure the existence
and uniqueness of the solution. Then, the field strength and
the radial flux density in the air gap can be derived from (8)
and (9) by numerical differentiation. A 3-D FE model of the
device is constructed in the software package CST using the
magnetostatic solver. The simulation is performed within
the bounding box enclosing the device structure that defines
the computational domain. The FE modeling requires that the
whole domain be subdivided into discrete elementary volumes,
which provide the numerical approximation of the solution. The
solution space is meshed with a total of 205 000 tetrahedral
elements. In addition, information such as external sources
and material properties needs to be provided for a complete
FEM model. The core is constructed from mild steel, which
has an estimated relative permeability of 25. The intrinsic
magnetization of the N48H PMs is M = 690 kA/m, which is
used in the FE model as sources.
An output plot of the magnetic flux in the computation
domain is shown in Fig. 3. The magnetic flux density B used
for estimation of the force generation in (2) is derived from the
radial component of the flux density on the plane of interest.
The average across Nz × Nr points on the plane of interest can
be calculated using (11) by taking the mean of the integral sum.
The flux density is denoted by Bf for clarity in the subsequent
discussion. Thus
Bf =
Nr
Nz 1 B(ri , zj ).
Nz Nr j=1 i=1
(11)
Fig. 4. Two-dimensional magnetic circuit model.
2) Analytical Model: An analytical magnetic circuit model
is developed to estimate the air gap magnetic flux density. This
model is less accurate compared with a FE model due to model
simplifications. However, it requires little computational effort
and is ideal for implementation in an optimization process. The
mismatch in the model output between the analytical (coarse)
and FE (fine) models needs to be addressed in order to determine the optimal solution. An output mapping is introduced in
Section IV to account for this mismatch.
The coarse model is derived from an equivalent magnetic
circuit of a 2-D slice of the FE model, as illustrated in Fig. 4.
The C-shaped dual magnet configuration is represented by a
magnetic circuit with two parallel paths. The difference in the
two paths lies in the leakage path present in path 2, which takes
the place of the core in path 1. The magnetic flux in the leakage
path is confined to an imaginary path as though a core is present.
This idealization will lead to the model output inaccuracies. The
magnetic 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 the
magnetic path length l, permeability μ, and cross-sectional
area A, as shown in (12). The cross-sectional area is simply a
product of the path width, i.e., W , and characteristic dimension,
i.e., D. The magnets are modeled as an ideal source with an
internal reluctance in series. The magnetic motive force is given
by (13), where Br is the remanent magnetic flux density, μr
is the recoil permeability, and lmg is the length (polarization
direction) of the magnet. Thus
Ri =
υmg =
li
where Ai = W Di
μi Ai
(12)
Br lmg
.
μr
(13)
The equivalent magnetic circuit is shown in Fig. 5. The
conservation of magnetic flux in paths 1 and 2 leads to the
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 10, OCTOBER 2014
Fig. 5. Equivalent magnetic circuit.
TABLE I
S UMMARY OF M ODEL PARAMETERS
Fig. 6. Experimental setup for measuring magnetic flux density and device
input–output characteristics.
Fig. 7.
expression (14). The magnetic flux density in the air gap is
simply the flux density per unit area, as shown in (15). Thus
Φ=
2υ
(Rlk + Rc2 )(Rc1 )
where RT =
(14)
RT +2Rm +Rg
Rlk +Rc2 + Rc1
Bc = Φ/Ag .
(15)
The selection of the width size W is inconsequential since it
would be eliminated in the calculation. The material properties,
geometric variables, and constants used for calculation of the
flux density are illustrated in Fig. 4 and listed in Table I.
III. E XPERIMENTAL T ESTING AND M ODEL VALIDATION
Fig. 6 shows the experimental setup for measuring the radial
component of the magnetic flux density at the midpoint of
the plane of interest along the z-direction (due to the finite
size of the probe). A unidirectional hall probe coupled with
a Lakeshore 460 gaussmeter is used for measuring the flux
density with accuracy of 0.001 T. It is positioned along the
z-axis with a motorized planar stage. This ensures that there is
a consistent step size (1 mm) along the measurement path while
maintaining a level position. The output force is measured using
a Schaevitz FC2231 load cell, as shown in Fig. 6. The load
cell is precalibrated for direct measurement of the output force
with accuracy of ±0.05 N. In this test, the device is powered
Magnetic flux density (r component) along the z-direction.
by a linear amplifier using a closed-loop current controller to
maintain the desired input current level. The heat generation is
equal to the supplied electrical power given by the product of
the supply voltage and current.
The measured air gap magnetic flux density is shown in
Fig. 7. A comparison between the measurement and model
outputs is made for validation of the FE model. There is a
good agreement between the FE model estimation and measured data, as shown by the close fit of the plots. The rootmean-square error between the measured and estimated data is
0.022 T, representing an averaged error of less than 7.3% of the
measurement range (0.3 T). The measured air gap flux density
has an average value of 0.146 T, whereas the model estimates
an average flux density of 0.150 T. The average air gap flux
density is calculated using (11) with the radial component of
the 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 and
the input current, as shown in Fig. 8(a). This agrees well with
the analytical model given by (2), which also predicts a linear
relationship based on an averaged magnetic flux density B.
However, a slight difference in the slope of the lines shows
the minor deviation between the model estimated and measured
flux densities, as discussed earlier. As for the heat generation,
there is a quadratic relationship with the input current, as
described by (1). The deviation between the model estimate
and the measurement is more pronounced at higher levels
of input current, as shown in Fig. 8(b). This is caused by
the change in material properties as the device temperature
changes. Electrical resistivity increases with temperature, and
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mismatch between the coarse and fine models would not result
in an optimal solution.
SM uses a function to map the input or response from the
coarse model onto the fine model. The mapping function is
denoted by P (x), as shown by (17), which can be linear or
nonlinear. P (x) can be determined during problem formulation or iteratively during the optimization process itself. The
fine model can be substituted by the coarse model using an
approximation shown in (18). The optimization process now
only requires evaluation of the coarse model, which would
be computationally less demanding. The optimal solution can
be then estimated as xf through the inversion given in (20).
Thus
xc = P (xf )
(17)
Rc (P (xf )) ≈ Rf (xf )
Fig. 8.
Input–output characteristic. (a) Force. (b) Heat generation.
(18)
x∗c = arg min U (Rc (xc ))
(19)
x∗f = P −1 (x∗c )
(20)
xc
TABLE II
M ODEL O UTPUT D EVIATION AT D ESIGN P OINT
ε = Rf (xf ) − Rc (xc ).
it directly affects the amount of heat generation based on (1).
As such, it is expected that some discrepancy will arise since
this 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, the
models are validated against the measurement at this design
point. The force as measured using the load cell is 5.28 N,
whereas the heat generation is derived from the supplied power
as 4.11 W. The force is overestimated by 0.6%, whereas the heat
generation is underestimated by 8.9%, for reasons discussed
earlier. Nevertheless, the model estimates do not deviate beyond
10% from the measurement at the design point (1-A input
current), as summarized in Table II.
IV. O UTPUT SM
Equation (16) describes a general optimization problem with
input variables x and model response R(x) subjected to a
suitable objective function U with the associated constraints.
x∗ is the optimal solution to this optimization problem, i.e.,
x∗ = arg min U (R(x)) .
(16)
The mapping function is found by minimizing the PE error
(21). In the original SM, Bandler et al. used a least square
regression to determine the coefficient of a linear mapping
function [7]. Such a method requires a finite number of fine
model evaluations prior to the optimization process to initiate
the PE process. The number of evaluations required depends
on the order and dimension of the mapping function, which
increases with the mismatch. In the aggressive SM method
[7], the PE is an intermediate step of the main optimization
process. It uses an iterative quasi-Newton method to estimate
the Jacobian matrix to direct the search toward a function
that minimizes (21). However, it suffers from the limitations
common to numerical methods such as convergence difficulties
and sensitivity to the initial estimate.
If there are large differences in structure between underlying
models, the output misalignment between the coarse and fine
model responses will be significant even after input mapping.
Output mapping can overcome this deficiency by introducing a
transformation of the coarse model response based on an output
mapping function O(Rc (xc )) as defined in (22). The mapped
output from the coarse model is known as the surrogate model
response, i.e., Rs (xc ). An example of the output mapping
function is given by (23), where γ (i) is a weighting factor on
the response mismatch, i.e., ΔR. Thus
Rs (xc ) = O (Rc (xc ))
(22)
x
The underlying concept of SM is to match the optimal solution
derived from a coarse model to the actual solution from a fine
model during the optimization process. The coarse model is
typically less accurate due to model simplification. However,
the coarse model is simple to evaluate and computationally
efficient. The aim is to make use of the coarse model to replace
the fine model during the optimization process. However, the
(21)
Rs (xc ) = O (Rc (xc )) = Rc (xc ) +
m
γ (i) ΔR(i) (23)
i=1
.
ΔR(i) = Rf (xf ) − Rc x(i)
c
(24)
The weighting factor γ (i) in (24) is determined from a
sequence of m pairs of coarse and fine model responses in a PE
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process [17]. Updating the surrogate model is iteratively done
by solving (25) for x∗c and evaluating the fine model response
at this point to generate a new pair of coarse and fine model
responses. The process repeats until the response mismatch is
less 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 = arg min U (O (Rc (xc ))) .
(25)
xc
The analytical magnetic model presented in the preceding
section shows that the magnetic flux density in the air gap is
a function of the magnet length L, gap size G, and the other
geometric constants (l0 , g0 ), which are defined in Fig. 2. Thus,
the coarse model input (26) is mapped to the original design
variables (27) by the definition (28). This is an exact mapping
as defined by the geometric relationship. Thus
xc = [L
xf = [nx
G
ny
d]
d0 ]
d = d0 + dt, L = nx d + 2l0 , G = ny d + 2g0 .
(26)
(27)
(29)
The coarse model adopts some idealization for modeling the
leakage flux at the open end of the magnetic circuit. The leakage
flux is dependent on the relative reluctance of the magnetic
paths. The reluctance is, in turn, dependent on the magnetic
path length, as described earlier. Thus, the magnet length L and
the gap size G are critical parameters that affect this additional
component of flux leakage. It is reasonable to assume that
the coarse and fine model response mismatch is due to the
unaccounted component of flux leakage. A linear function of
(L, G) in the form shown in (30) is used to model the mismatch
ΔR. Physical insight of the mismatch means that the mapping
function is kept simple but still effective at improving the
accuracy of the surrogate model. Thus
⎡ ⎤
θ0
(30)
ΔR̂ = y∗ θ = [ 1 L G ] ⎣ θ1 ⎦
θ2
ε = ΔR − ΔR̂.
Output mapping and PE.
minimizes (31). A large Q(0) should be chosen to move the
estimate θ(0) toward its final solution θ. Thus
T
y(k) Q(k−1)
Q(k−1) y(k)
(32)
Q(k) = Q(k−1) −
T
1 + y(k) Q(k−1) y(k)
(28)
There is no need to estimate the input mapping function
P (xf ) since it is already known. Output mapping is used to
match the response of the coarse model (14) and (15) and fine
model (11). The model response mismatch is defined as
ΔR = Bf − Bc .
Fig. 9.
(31)
The PE error defined by (31) allows the parameter θ to
be determined using a least square method. This process can
be iteratively done during the main optimization process by
using a recursive least square algorithm. The major steps of
the algorithm is given by (32)–(34). Q is a measure of the
uncertainty in the estimated parameter θ. Q will be reduced
over time when more data points are made available, and hence,
the solution will converge to the optimal estimate of θ that
K(k) =
T
Q(k−1) y(k)
T
1 + y(k) Q(k−1) y(k)
θ(k) = θ(k−1) + K(k) ΔR(k−1) − y(k) θ(k−1) .
(33)
(34)
For y ∈ R1xn , there needs to be k ≥ n iterations to ensure
uniqueness of the estimated parameter θ. Evaluation of the fine
model response is necessary at each major iteration k of the
optimization process to update this parameter. The PE process
is terminated once the averaged mapping error, i.e., ε, defined
by (35) is less than a predetermined target level. Finally, the
surrogate model output is given by (36). Thus
2
k i ε(k)
(35)
ε =
k
⎡ ⎤
θ0
Bs = Bc + [ 1 L G ] ⎣ θ1 ⎦ .
(36)
θ2
The key steps of implementing an output mapping together with
the optimization process is illustrated in Fig. 9.
This current adaptation of SM introduces an iterative PE process that is decoupled from the main optimization process but
runs parallel to it. A recursive least square algorithm is used for
the PE process. It is a deterministic algorithm, which involves
only algebraic operations. Thus, convergence is guaranteed, and
a unique set of parameters will be determined as long as the
preconditions are met, as discussed earlier.
V. O PTIMIZATION
A. Design Objective
The optimization process is defined by the objective function, which is subjected to appropriate physical constraints.
HEY et al.: ELECTROMAGNETIC ACTUATOR DESIGN ANALYSIS USING A TWO-STAGE OPTIMIZATION METHOD
TABLE III
GA PARAMETER S ETTING
An objective function of the form shown in (37) is chosen,
where Rs (xc ) is defined by (38). The aim is to determine the
optimal design configuration that maximizes the output force F
per unit heat generated Q. Thus
x∗c = arg min U (Rs (xc ))
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(37)
xc
Rs (xc ) =
F̄
F
Q
where F̄ =
, Q̄ =
.
F0
Q0
Q̄
(38)
F0 and Q0 are the norms of the output force and heat generation, respectively. Normalization of the outputs is essential as
they often represent different quantities that can carry values of
varying orders of magnitude. This would affect the optimization
process since the algorithms are dependent on the sensitivity of the objective function against the input. Normalization
creates nondimensional quantities, which are more suited for
optimization. A natural selection for the norm is the median
of the expected minimum and maximum values of the output.
However, in this instance, output measurements are available
from the initial design, as discussed in Section III. Thus, they
are taken as estimates of the norm. F0 and Q0 have values of
5.4 N and 4.1 W, respectively. Thus
πQ0 Bs (d−dt )2
F0 ρr I
⎡term 1
⎤
term 2
2
πQ0 (d − dt) ⎣
Rs (xc ) = −
Bc +θ0 +θ1 L+θ2 G⎦.
F0 ρr I
Rs (xc ) = −
(39)
(40)
take measurements. The constraints are summarized by the
following:
d[mm] ∈ {0.28, . . . , 1.692}
10 ≤ L[mm] ≤ 110
3.5 ≤ G[mm] ≤ 8
F [N] ≥ 1.3F0 , Q[W] ≤ 0.7Q0 .
Output constraints can be also defined as required by the
application. In this case, a minimum output force and maximum
heat generation target is benchmarked against the performance
of the initial design with a 30% improvement, as shown by (44).
However, output constraints need to be well defined as selection
of extreme values could lead to nonconvergence. Normalization
of the constraints is also carried out for the same reason
discussed earlier. The constraints are normalized against the
upper or the lower limit depending on whether it is minimum
or maximum constraint, as illustrated in the following:
x̄ =
Using expressions (1)–(3) to substitute for F and Q in the
response function Rs (xc ) leads to the compact form shown in
(39). It can be expanded by replacing Bs with (36), leading to a
final expression shown in (40). In the process, term 2 is added
to the response function, but it does not change the convexity
of the original coarse model output Bc . It is a monotonically
increasing/decreasing function (order of less than 2). However,
it is not within the scope of this paper to provide mathematical
proof of the convexity of the response function (40). A general
approach is adopted to ensure that a global optimal solution
is obtained based on the careful selection of the optimization
algorithm and appropriate setup procedure. The details will be
discussed in the next subsection.
Mathematical constraints need to be defined to reflect the
physical constraints dictated by the application requirement
and availability of components. For example, the copper wires
typically come in diameters based on the AWG standard. In
this analysis, the wire diameter is chosen from a nonexhaustive
set of wire diameters between the sizes AWG 30 and 14. The
minimum magnet length considered in this analysis is limited
to 10 mm, whereas the maximum length is set at 100 mm.
This reflects the typical length of magnets that are readily
available. A constraint is also imposed on the air gap size
for practical reasons such as access for instrumentations to
(41)
(42)
(43)
(44)
x − x0
for x0 = xmin xmax .
x0
(45)
The objective function defined in (37) is a continuous function of xc . Although the function is continuous for the range of
xc , the original design variables (xf ) themselves only accept
discrete values. The number of turns nx and the number of
layers ny are integer values, whereas the wire diameter d0 only
accepts discrete values based on the AWG standard. In order to
address this issue, the proposed optimization methodology involves two stages. First, a global search using GA is performed
to find the optimum solution to the continuous problem. This is
followed by a BB method to search feasible solutions locally for
the optimal. The local minimization uses a sequential quadratic
programming (SQP) algorithm.
B. Optimization Algorithm
1) Global Search: GA is chosen for the global search over
a gradient-based method because of the latter’s limitations in
returning a global optimal solution, particularly so for objective functions, which have multiple local minima. GA uses
an evolutionary approach to search for the fittest individual
within the solution space, which is defined here by the fitness
function (39). The GA implemented here is based on the
GA solver available in the Global Optimization Toolbox in
MATLAB, which gives the user the flexibility of selecting the
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Fig. 11.
Fig. 10. BB algorithm.
GA parameters, making it a useful tool for such applications
[18]. The software redefines the problem in terms of original
fitness function and nonlinear constraints using the Lagrangian
representation in what is known as the augmented Lagrangian
GA. The GA parameters used in this analysis are tabulated
in Table III.
The output mapping described in Section IV is incorporated
into the optimization process during the global search. The GA
parameters are set to be less restrictive to allow for a wider
search of the solution space for the global optimum. It also aids
the PE process since an initial divergence of the solution allows
for a more even output mapping. Evaluation of the coarse and
fine model responses is carried out at each major iterations of
the GA, and PE is done using (32)–(34). The PE is terminated
once the set target level of averaged mapping error is met. For
this application, the target is fixed at 0.0075 T, which represents
2.5% of the measured range. With sequential PE, the surrogate
model would give better estimates of the magnetic flux density
as the optimization process progresses toward the optimum
point.
2) BB With Local Minimization: The following is a description of the BB method illustrated in Fig. 10. First, if
the continuous global optimization solution is feasible, then
the process terminates. If not, then the solution space X ∈ R
is subdivided into smaller domains Z such that Z ⊆ X. BB is
based on dichotomy and exclusion principles [5]. The idea is to
discard subdomains that do not yield solutions that are better
than the current best and search the remaining subdomains
for the optimal solution in a systematic manner. Branching
is a process of creating these subdomains by subdividing the
solution space based on interval analysis. A common method is
to define the limits of the subdomain by the next higher (x+
i ) or
)
feasible
value
of
the
current
variable
(x
).
lower (x−
i
i
Process of branching and the solution tree.
At each step of branching, two new subproblems are created. The process is followed by solving the subproblem as
a continuous optimization problem with the new constraint
limits. The process of branching and solving the subproblem
is continued for all the variables until a feasible solution is
obtained. The objective function value, i.e., f , for this solution
then becomes the upper bound for the remaining subproblems
(bounding). Branches that have higher objective function values
are eliminated from further consideration. The upper bound
(fmin ) is updated when a feasible solution yields a lower
objective function value. The process will terminate once all
the 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 branches
in the solution tree, as illustrated in Fig. 11. Each branch in
the solution tree is a possible solution, which makes up the
solution set ζ. Not all branches are evaluated as the process
of bounding discards branches that would not yield the optimal
feasible solution. For example, if subproblem 3 has an objective
function value greater than the current best solution fmin , then
that branch will be fathomed from the solution set. This reduces
the total number of evaluations needed to search for the optimal
feasible solution.
SQP is used for local minimization during the branching step.
The SQP implemented is based on the “fmincon” solver available in the Optimization Toolbox of MATLAB. The method
uses a quadratic approximation of the Lagrangian function as a
replacement of the original objective and constraint functions.
It searches for the minimum point in an iterative manner
using the function and its derivative to determine a search
direction. The Hessian matrix or the second-order derivative
is approximated using a quasi-Newton updating method. The
method approximates the Hessian matrix from the first-order
derivatives, which leads to time saving as exact calculations can
be computationally costly.
VI. R ESULTS AND D ISCUSSION
The output force and the heat generated are functions of
the original design variables: the number of turns nx , the
number of layers ny of the coil, and the wire diameter d0 .
The design variables are linked to the external geometry by
HEY et al.: ELECTROMAGNETIC ACTUATOR DESIGN ANALYSIS USING A TWO-STAGE OPTIMIZATION METHOD
5461
Fig. 12. Force variation with geometry and wire diameter.
Fig. 13. Heat generation variation with geometry and wire diameter.
(5) and (6). The external length is directly dependent on the
magnet length, whereas the external radius is a direct offset of
the gap size. A parametric analysis is performed to investigate
how the output force and the heat generation are dependent on
the external device geometry and the choice of wire diameter.
This is done using the input–output relationship (1) and (2), the
parameterization of the coil length (4), and the estimate of the
magnetic flux density using the analytical magnetic model (14)
and (15).
Output force is plotted against the external length for selected
external radii, as shown in Fig. 12. The output force increases
with length initially for a given radius. However, the output
force reaches a maximum before decreasing, particularly so
for devices with larger radii. The reduction in force generated
is due to the decreasing trend of the magnetic flux density
with both external length and radii of the device. However,
the coil length is increased with the external dimensions as
well. The output force, being a product of the two terms, has
a characteristic maximum point, which reflects this underlying
relationship. The effect is less pronounced for smaller radii as
the decrease in field strength is less significant with magnet
length for smaller gap sizes.
Output force significantly changes with the choice of wire
diameter, as shown by the offset between each plot in
Fig. 12(a)–(c). It is possible to have a device with similar
external dimensions but different output force by adjusting the
wire diameter. However, the choice of wire diameter is a critical
factor in determining the amount of heat generated. The heat
generated increases with the overall dimensions of the device
due to the increase in coil length. However, increase in heat
generation is amplified for smaller wire diameter, as shown by
the steeper gradients of the (
) plots in Fig. 13. Heat generation
is an inverse function of wire diameter, and it can be minimized
by increasing the wire diameter.
Such parametric study serves as a guide for sizing the device and selection of wire diameter. A suggested approach is
outlined 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 available knowledge of the device behavior against the selected
variables. The design objective in this case is to determine
the most compact design, which results in the least amount of
heat generation for a target level of output force. The manual
approach shown here can be tedious, and the result may be
suboptimal. Thus, it is more effective by automating the process
through the formulation of a mathematical problem that incorporates the design objective and constraints in one objective
function. The problem can be solved using an optimization
methodology such as the one presented earlier.
Fig. 14(a) shows that a total of 15 major iterations of the GA
is required before it converges to a solution. The averaged mapping error at each major iteration is shown in Fig. 14(c). There
is a decreasing trend with the iteration due to the least square algorithm used for PE. As more coarse and fine model responses
are made available at each iteration, the output mapping can be
fine-tuned such that the mapping error is minimized. There is
a minimum number of iteration required for the PE process to
return meaningful parameters, as discussed in Section IV. The
choice of a simple linear mapping function (30) would mean
less number of iterations required (three in this case).
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TABLE IV
C OMPARISON OF M AGNETIC F LUX D ENSITY E STIMATES
TABLE V
R ESULTS F ROM G LOBAL S EARCH U SING SQP S TARTING AT I NITIAL
P OINT, GA, AND L OCAL S EARCH FOR O PTIMAL F EASIBLE S OLUTION
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 error
of 2% of the measurement range) is achieved after three major
iterations. Since the error is below the set target level, the PE
process is terminated beyond the third iteration. A sharp drop
in objective function value is seen from the third to fourth iterations, as shown in Fig. 14(a). The output mapping does affect
the solution as expected. The mapping function is kept simple to
avoid adding unnecessary minima to the solution space. The PE
process needs to be tied to a termination criterion, which allows
the optimization solution to converge eventually. Fig. 14(b)
shows that the near-optimal solution is close to the constraints
indicated by the higher constraint violation at the ninth to
eleventh iterations. However, the GA setup is robust enough to
move it away to another point, which is selected as the optimal
solution since it has a lower objective function value.
Each evaluation of the fine model requires about 12 min and
23 s on a workstation running at 3 GHz with 32 GB of random
access memory. The application of output mapping resulted in
an 80% reduction in computation time as the fine model is
replaced by the surrogate model after the third iteration. The
total time required for the optimization process to complete
is approximately 40 min with output mapping. A similar optimization performed using the commercial software package
CST with a GA optimization toolbox required approximately
three days to yield a solution. However, the problem formulation in terms of the objective function and design variables
differs from the current optimization due to the limitation of
the software. For example, having the wire diameter as a design
variable is not an option in the software. Such limitations
resulted in a reduced dimension of the problem; however, the
computation time required is still much more considerable than
the proposed method.
For the current purpose of output comparison, the fine model
response is evaluated at the optimal point. Table IV shows
a comparison between the magnetic flux density estimates
from the fine, coarse, and surrogate models. The coarse model
estimate is 20.8% more than the fine model at the optimum
point. This would have led to a suboptimal solution due to
an overestimate of the output from modeling inaccuracies.
However, with output mapping, the deviation from the fine
model is reduced to 2%. The average air gap magnetic flux
is calculated using (11), with measurement taken off the new
prototype. The average flux density is measured to be 0.212 T,
as opposed to 0.206 T estimated from the surrogate model. The
estimate from the surrogate model is, in fact, 2.8% less than the
measured value. Output mapping is an effective tool in output
correction. It leads to eventual time saving by replacing the fine
model with the surrogate one during the optimization process
after an initial stage of “model tuning” during the PE process.
The solution returned from the global search using GA is
compared against those obtained using an SQP algorithm with
different initial estimates, as shown in Table V. Gradient-based
search algorithms such as SQP guarantee an optimal solution
when the search space has a single minimum point. Otherwise,
the algorithm could yield a local or a global minimum, depending on the initial point. The result shows that the search space
does indeed have several minimum points. Stochastic methods
such as GA are better suited for a global search when there are
multiple minima. Thus, the two-stage algorithm has the advantage of using different algorithms suited for each stage of the
optimization process. Using GA for the global search increases
the probability of obtaining a global optimum, as shown by
the lower objective function value obtained. Thereafter, the BB
method with local minimization would yield the optimal among
the feasible solutions around this global solution.
The optimal feasible solution obtained from the design optimization translates into a device with attributes summarized
in Table VI. The optimal design shows a better performance
in the selected output measures. The output force has been
increased by 39.9%, whereas the heat generation is reduced
by 26.2%. The performance improvements are based on the
output measurements from the original device (initial design)
and the newly fabricated prototype (optimal design). Measurements are made using the same experimental setup described
HEY et al.: ELECTROMAGNETIC ACTUATOR DESIGN ANALYSIS USING A TWO-STAGE OPTIMIZATION METHOD
5463
TABLE VI
D ESIGN ATTRIBUTES B EFORE AND A FTER O PTIMIZATION
TABLE VII
C OMPARISON OF O UTPUT P ERFORMANCE (M EASURED )
Fig. 16. Input–output characteristic. (a) Force. (b) Heat generation.
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 the
external dimensions and overall device volume, which leads to
less 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 initial
and optimal designs. The mean line shows the average value
of each plot. There is a substantial increase in the average due
to the more compact design. The optimal design shows better
overall magnetic distribution, although there is still a decreasing
trend toward the open end of the magnetic circuit. This points
to the fact that there is substantial flux leakage at the open
end. However, the flux leakage in such C-shaped dual magnet
configuration is reduced by the selection of more compact
magnets. The averaged magnetic flux density has increased by
45.2% on average for the optimal design.
Fig. 16(a) shows the comparison of the output force from
both designs. It is clearly shown by the steeper slope for the
optimal design that it has a higher force constant. This increase
is due in part to the better magnetic performance, as discussed
earlier. Fig. 16(b) shows the heat generation against the input
current for both designs. There is a significant reduction in heat
generation simply by the use of wires with larger diameter. As
both outputs are functions of the coil configuration, there exists
an optimum configuration that meets the design objective. The
optimization process has returned a design that maximizes the
air gap magnetic flux density by adjusting the gap geometry. At
the same time, it maximized the coil length exposed to this field
while selecting the ideal wire diameter to balance the amount of
force and heat generated. These parameters have some interdependence, which makes the problem difficult to solve manually.
All these have to be done while ensuring that the constraints are
met. The two-stage optimization method shown in this paper
overcomes some of these difficulties and challenges faced by
designers. Moreover, the application of the methodology led
to significant improvements in the performance of the new
prototype.
VII. C ONCLUSION
This paper has presented a two-stage optimization methodology for the design analysis of a linear electromagnetic actuator.
Solving the design problem as an inverse mathematical problem resulted in a design that yielded significant performance
improvement over an initial design. An increase in air gap
magnetic flux density by 45.2% is achieved, leading to an
increase in output force by 39.9% and a reduction in heat
generation by 26.2%. Moreover, the overall device size has
been reduced, which means a more compact design and cost
savings in terms of material usage. The application of GA for
the global search followed by a BB method is a systematic way
of searching for the optimal feasible solutions. This makes the
methodology suitable for application to a range of engineering
design problems, which have mixed variables.
Time spent during the design stage is very much reduced due
to the automation resulting from the integrated optimization
procedure. Time savings are achieved by replacing the FE
magnetic (fine) model with an analytical (coarse) model during
the process. Model output mismatch is reduced with output SM.
PE is done using a deterministic least square algorithm. This
resulted in better estimates of the magnetic flux density using
the surrogate model with adjusted output. At the optimal solution, the surrogate model estimate is within a 2.0% deviation
of the original FE model output and 2.8% deviation from the
measured value. The application of an output SM technique
resulted in a time saving of 80%, with minimal sacrifice of the
modeling accuracy.
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Jonathan Hey received the B.Eng. (Hons.) degree in
mechanical engineering from Nanyang Technological University, Singapore, in 2008. He is currently
working toward the Ph.D. degree at Imperial College
London, London, U.K.
His research interests include thermal management of electromechanical devices, with a focus
on disturbance model identification, compensation
methods, 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. degree
in mechatronics engineering from the Queensland
University of Technology, Kelvin Grove, QLD,
Australia, in 2003, and the Ph.D. degree in mechanical and aerospace engineering from Nanyang
Technological University, Singapore, in 2009.
In 2009, he joined the Singapore Institute of Manufacturing Technology, Singapore, as a Research
Scientist with the Mechatronics Group. His research
interests include ultraprecision systems, compliant
mechanism theory, parallel kinematics, electromagnetism, electromechanical systems, thermal modeling and analysis, energyefficient machines, and topological optimization.
Viet Phuong Bui (M’09) received the Master’s and
Ph.D. degrees in electrical engineering from the
Grenoble Institute of Technology (INPG), Grenoble,
France, in 2004 and 2007, respectively.
Since 2007, he has been with the Institute of
High Performance Computing, Agency for Science,
Technology, and Research, Singapore, where he is
currently a Research Scientist with the Electronics
and Photonics Department. Since 2003, he has been
involved in electromagnetics research from fundamentals to devices and applications. His research
interests include computational electromagnetics and optimization methods
applied in electromagnetics.
Guilin Yang (M’02) received the B.Eng. and M.Eng.
degrees from Jilin University, Jilin, China, in 1985
and 1988, respectively, and the Ph.D. degree from
Nanyang Technological University, Singapore, in
1999, all in mechanical engineering.
From 1998 to 2013, he was with the Singapore
Institute of Manufacturing Technology, Singapore.
He is currently a Senior Research Fellow with the
Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo,
China. He has authored/coauthored over 200 technical papers in refereed journals and conference proceedings, eight book
chapters, and two books. He has also filed 12 patents. His research interests
include precision mechanisms, electromagnetic actuators, parallel kinematics
machines, modular robots, industrial robots, and rehabilitation devices.
Mr. Yang is currently an Editorial Board Member of Frontiers of Mechanical
Engineering 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 University of Oxford, Oxford, U.K., both in aeronautical
engineering.
He is currently a Professor of Turbomachinery
and the Theme Leader for Hybrid and Electric Vehicles of the Energy Futures Lab, Imperial College
of London. He has published extensively in journals
and peer-reviewed conferences. His research interests include heat transfer of electrical machines and
batteries.
Dr. Martinez-Botas is currently an Associate Editor of the Journal of Turbomachinery and is a Member of the editorial board of two other international
journals.