Application of the Finite Element Method in Design and Analysis of Permanent-Magnet Motors

15 Pages·2007·4.18 MB·English

FINITE ELEMENTS published by WSEAS Press 138
Application of the Finite Element Method in Design and Analysis of
Permanent-Magnet Motors

ARASH KIYOUMARSI1, PAYMAN MOALLEM1, MOHAMMADREZA HASSANZADEH2 and
MEHDI MOALLEM3

1Department of Electronic Engineering, Faculty of Engineering, University of Isfahan, Isfahan
2Faculty of Electrical Engineering, Abhar Islamic Azad University, Abhar, Ghazwin
3Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan
IRAN

Abstract- In this research, the results of approximate analytical methods and Finite Element Method (FEM),
those are used for prediction of airgap flux density distribution, are compared. In this comparison, permanent-
magnet direct-current (PMDC) motors and brushless permanent magnet motors are considered. In addition, a
coupled magnetic field, electrical circuit, and mechanical system program by which the FEM analysis is
accomplished, is briefly discussed. Then, time stepping finite element method is used for the magnetic field
analysis. At last, an example of shape design optimization, i.e., optimal shape design of an interior permanent-
magnet (IPM) synchronous motor, is considered.

Key-Words: The Finite Element Method, Brushless Permanent Magnet Motors, DC Motors.

1 Introduction magnets with radial magnetization are suitable for the
BDCM [8-9].
Prior to the development of reliable high-power
Permanent-magnet synchronous motors (PMSM)
solid-state switching devices, the DC motor was the
have higher torque to weight ratio as compared to
dominant electric machine for all variable-speed
other AC motors. There are different rotor topologies
motor drive applications. The DC motor turns out to
that divide into two basic types, i.e., Exterior
be the most economical choice in the automotive
Permanent-Magnet motors (EPM) and Interior
industries for cranking motors, wind shield wiper
Permanent-Magnet motors (IPM). Surface–mounted
motors, blower motors and power window motors
permanent-magnet synchronous motor (SPMSM)
[1]. In this paper, first, the magnetic flux density of
and inset permanent-magnet motor, belong to
both a six-pole, 29-slot PMDC motor and a series-
former and buried or interior–type permanent-
exited four-pole, 21-slot field-winding DC motor are
magnet synchronous motor (IPMSM) and flux
obtained based on an iterative analytical method.
concentration or spoke–type permanent-magnet
Then, the magnetic field is modeled based on a two-
synchronous motor, belong to the latter. Because of
dimensional field analysis method considering the
the mechanical and electromagnetic properties of
effect of rotor slots [1-3]. The results of calculations
each type, different topologies have different
are compared with finite element method results and
advantages and disadvantages used in high-speed
predicted output characteristics of both motors are
applications [10, 11]. They have different control
also compared with those obtained by real output
measurements [4-7]. strategies and there is usually torques, speed, angular
position and current-control loops in the control
Brushless permanent magnet motors can be
system.
divided into the PM synchronous AC motor (PMSM)
Interior permanent-magnet synchronous motor,
and PM brushless DC motor (PMBDCM). The
has many advantages over other permanent-magnet
former has sinusoidal airgap flux and back EMF, thus
synchronous motors. It has usually larger quadrature
has to be supplied with sinusoidal current to produce
axis magnetizing reactance than direct axis
constant torque. The PMBDCM has the trapezoidal
magnetizing reactance. These unequal inductances in
back EMF, so the rectangular current waveform in its
different axes, enable the motor to have both the
armature winding is required to obtain the low ripple
properties of SPMSM and synchronous reluctance
torque. Generally, the magnets with parallel
motor (SynchRel) [12]. The total resultant
magnetization are used in the PMSM while the
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instantaneous torque of a brushless permanent- Then, the magnetic field is modeled based on a two-
magnet motor has two components, a constant or dimensional field analysis method considering the
useful average torque and a pulsating torque which effect of rotor slots [1-3]. The results of calculations
causes torque ripple. There are three sources of are compared with finite element method results and
torque pulsations. The first is field harmonic torque predicted output characteristics of both motors are
due to non-ideal distribution of flux density in the also compared with those obtained by real output
airgap, i.e., non-sinusoidal in the PMSM or non- measurements.
trapezoidal in the PM brushless DC motor. The
second is due to the cogging torque or detent torque
caused by the slotted structure of the armature and 2.1 Analytical Method
the rotor permanent-magnet flux. The third is Figs. 1-a and 1-b show the frame assembly and
reluctance torque, produced due to unequal armature of two ideal motors. Fig. 1-c shows
permeances of the d- and q-axis. This torque is armature and field current densities applied to the
produced by the self-inductance variations of phase FWDC motor in this study. Having determined the
windings when the magnetic circuits of direct- and structure of the magnetic circuit (Fig. 2) and electric
quadrature-axis are unbalanced [13]. circuit of both motors, the node permeance matrix
In IPM synchronous motor, the effective airgap and the node magnetic flux source vector can be
length on the d-axis is large so the variation of the d- found and interactive calculation is used to solve the
axis magnetizing inductance, Lmd, due to magnetic equation. The permeability of segment i, for the
saturation, is minimal. For the q-axis, there is an (k+1)th iteration, considering magnetic saturation, is
inverse condition, i.e., the effective airgap length on then corrected and determined by the following
the q-axis is small and therefore the saturation effects expression [5]:
are significant [14-16].
µk =µk−1+λk {µk −µk−1} (1)
i i i i i

2 Characteristics of a PMDC and a
FWDC Starter Motor
The influence of magnetic saturation on
electromagnetic field distribution in both permanent-
magnet direct-current (PMDC) and field-winding
(wound-field) direct-current (FWDC) motors with
the same output mechanical power, have been
Fig. 1-a Fig. 1-b
studied. An approximate analytical method and FEM
are used for prediction of airgap flux density
distribution. No-load and rotor-lucked conditions,
according to experimental measurements, and the
FEM and analytical method studies of the motor,
have been studied. A sensitivity analysis has also
been done on the major design parameters that affect
motor performance. At last, these two DC motors are

compared, in spite of their differences, on the basis of
Fig. 1-c
measured output characteristics.
Fig. 1. Frame of ideal DC motor and applied
Boules developed a two-dimensional field
current densities to the prototype motor
analysis technique by which the magnet and armature

fields of a surface–mounted brushless synchronous
Fig. 3 shows results of this method for FWDC
machine can be predicted [1,2]. In a comprehensive
motor magnetization curve. Results of design
proposed model, Zhu et al. presented an analytical
sensitivity analysis are also evident on this figure for
solution for predicting the resultant instantaneous
changing the number of turn of armature windings
magnetic field in the radially-magnetized BDCM and
[1-3].
PMDC under any load conditions and commutation

strategies [3]. In this research, first, the magnetic flux
density of both a six-pole, 29-slot PMDC motor and
a series-exited four-pole, 21-slot FWDC motor are
obtained based on an iterative analytical method.
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The no-load flux distribution is also shown in Fig.
7 in details with magnifications. The direction and
magnitude of the flux density distribution vector is
also included in the Fig. 7.

Fig. 2. Magnetic equivalent circuit of the motors
(a) (b)
Fig. 7. (a) Flux lines, and (b) colored vector
plot of vector field: no-load conditions; FWDC
motor

Figs. 8 and 9 show the flux distribution and radial
and tangential components of flux density curve at

no-load without and with the magnetic holder, for the
Fig. 3. Flux per pole of series-exited DC motor
PMDC motor. Finally, Fig. 10 shows the flux
vs. line current
distribution at full-load with the magnetic holder

[6,7].
2.2 Finite Element Method

Figs. 4 and 5 show the flux distribution and radial
and tangential components of flux density curves at
no-load and rated load, respectively.

(a) (b)
Fig. 4. Flux lines: no-load conditions; FWDC Fig. 8. (a), Frame of the prototype PMDC motor
motor and rotor windings, (b), Equipotential lines for
magnetic vector potential: no-load conditions;
PMDC motor

Fig. 5. Flux lines: rated-load conditions; FWDC
motor
Fig. 6 shows the equi-potential lines for magnetic
vector potential and their color map by considering
the effect of screw and bolt on the stator frame, for
the FWDC motor.
(a) (b)

Fig. 6. (a) Flux lines and (b) colored contour
Fig. 9. (a) Equipotential lines for magnetic vector
plot, effect of stator frame screw: no-load
potential: no-load conditions, PMDC motor,
conditions; FWDC motor
magnetic holder considered.
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solutions and also winding distribution in the stator
slots.

Table 1. Comparison of FEM and analytical method
for the FWDC motor
FEM Analytical
Analysis Method
No-load
ω 5828.84 5263
m [RPM]
T 0.033 0.030
Fig. 10. (a) Equipotential lines for magnetic vector e[Nm]
potential: full-load conditions, PMDC motor, Φ 96.57e-5 100.2e-5
P[Wb]
magnetic holder considered.
B
mid−aigap[T] 0.7 0.7025
Table 1 includes the results of comparison of the B 1.5 1.4
FEM analysis of the FWDC motor and the two- Yoke[T]
dimensional field distribution analytical method. In I =100A
a
this analysis, the average value of the rotor angular
ω 1985.5 1966.1
speed, rotor output shaft torque, flux per pole, mid- m [RPM]
airgap flux density distribution waveform and yoke T 0.3368 0.33065
e[Nm]
flux density distribution waveform are considered.
Magnetic holders are devised to prevent the ΦP[Wb] 105.84e-5 103.88e-5
demagnetization of the PMs during the influence of a B
1.064 1.0421
mid−aigap[T]
strong armature reaction field on the stator field. The
FEM results have completely validated the operation B 1.622 1.5
Yoke[T]
of the holders during different load conditions.
I =150A
These two prototype direct-current motors, i.e., a a
PMDC and a FWDC are compared from the point of ω 1747.9 1768
m [RPM]
view of output characteristics. Results of an
approximate analytical method, a previously- Te[Nm] 0.533 0.529396
developed two-dimensional field analysis technique Φ 111.648e-5 110.88e-5
P[Wb]
and finite element method are compared for
B
prediction of airgap flux density distribution and 1.13 1.1722
mid−aigap[T]
possible replacement of the FWDC motor with
B 1.727 1.6212
PMDC motor is briefly studied. The results of a few Yoke[T]
experimental measurements are also involved in this I =300A
a
study and the results of all methods are also
compared. ωm [RPM] 0 0
T 1.2644 1.28
e[Nm]

3 Brushless Permanent Magnet Motors ΦP[Wb] 132.432e-5 35.0e-5
In a comprehensive proposed model, Zhu et al. B
1.596 1.733
mid−aigap[T]
presented an analytical solution for predicting the
resultant instantaneous magnetic field in the radially- B 2.08 2.12
Yoke[T]
magnetized BDCM and PMDC under any load

condition and commutation strategy [3].
Cross section of the 5 HP, 1500 rpm, 4-pole
Recently, Zhu et al.[17] extended Rasmussen’s
surface-mounted brushless permanent magnet motor
model[9] to predict magnetic field due to the
[10], is shown in Figure 11. Figure 12 shows
armature reaction both in the three phase
comparison of flux lines in radial magnetization
overlapping and non-overlapping stator windings.
configuration. Figure 13 shows comparison of
Finally, open–circuit field distribution and load
analytical and numerical results. The results shows
condition field distribution can be expressed as
that the flux density curves of that new improved
relative permeance functions including the field
model follows the FE curves more closely, especially
at the corners of the stator slots. As a result, the
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analytical model is a powerful tool for torque the Magnetic Equivalent Circuit(MEC) model of
pulsation calculations, using magnetic flux density machine, Fig. 14, which takes into account the non–
distribution in the airgap. Also average torque can be linear characteristics of the iron in the machine and
obtained using useful flux per pole and unit the FEM method is also accomplished and carried
length of stator. The results show that the flux out.
density curves of improved model follow the FE
curves more closely, especially at the corners of the
stator slots.

Fig. 11. Cross section of the machine
Fig. 14. Magnetic Equivalent Circuit model

(Non-Linear elements are in black)

The fundamental component of flux density
distribution is obtained using MEC analysis
(B1:MEC), the two-dimensional Cartesian-based
coordinates method(B3:2DR)[8], the two-
dimensional polar-based coordinates method with
stator slot effects (B4:2DS) [9], and FEM analysis
(B5:FEM), are compared in Table 2. The figures
show the peak of fundamental component of flux
density distribution considering both the parallel and
radial magnetization. The good agreement of the

Fig. 12. Equipotential lines: flux distribution (open- analytical method with FE results has proved validity
circuit condition) at time t=0 (Radial Magnetization) of this method for fast calculation of back-EMF and
torque ripples.

Table 2. Comparison of different field analysis
methods
Parallel Radial
B1(MEC) -------- 0.8200
B3(2DP) 0.8262 0.8385
B4(2DS) 0.8138 0.8260
B5(FEM) 0.7818 0.7953
The flux densities are in Tesla.

4 Interior Permanent Magnet

Synchronous Motor
Fig. 13. Flux density distribution: radial
Kim et al. [18,19], in a Recent comprehensive
magnetization
proposed method, have presented a shape design
A comparison between lumped–parameter and optimization method to reduce cogging torque of an
distributed-parameter flux calculations , i.e., using IPM synchronous motor using the continuum
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sensitivity analysis combined with FEM. They used
the finite element nodes on the rotor outer surface as
control points and used the B-spline curves to relate
design variables vector and control points vector to
each other. There is only a slight difference between
initial shape and final shape of rotor outer surface.
They also presented an optimal shape design method
for reducing the higher back-EMF harmonics
generated in IPM synchronous motor; however, they
did not evaluate the optimal design at different load
conditions.
The proposed optimal design method in this
part is industrially applicable to the rotor and motor

drive operation is considered too. The optimal shape Fig. 15. Operating limits for prototype IPM
is evaluated at different load conditions which shows synchronous motor: family of maximum torque-per-
good improvement in all loading conditions. ampere curves, constant torque curves, rated stator
Electrical circuits’ equations of different stator current curve and voltage-limit ellipses
windings and rotor mechanical motion equations are
coupled to magnetic field equations, to obtain a
comprehensive model for the IPM motor drive.
4.2 Modeling of Motor-Drive System
The optimal shape design is obtained based on
4.2.1 Magnetic Field Model
addition of three circled-type holes that are drilled in
Rotor mesh and stator mesh are coupled together
the rotor iron. Motor-drive operation is also
at a slip interface to allow for rotation which is a
discussed.
cylindrical slip surface in 3D and a circular slip path

in two-dimension in the middle of the air gap. So,

there is a weak boundary condition enforced on this
4.1 Steady-State Operating Curves of IPM interface. Using Lagrange multiplier and Kth rotor
Synchronous Motors variable, A as magnetic vector potential at node k on
rk
The main torque control strategies for the speeds rotor, rotor can be coupled to the corresponding
lower than base speed operation are zero d-axis stator variables A , as:
si
current, maximum torque per unit current (MTPC), 4
maximum efficiency, unity power factor and constant A = ∑A N (k) (1)
rk si si
mutual flux linkage. The main control strategies for i=1
the speeds higher than base speed operation are
constant back-EMF and six-step voltage. The MTPC The local virtual work method is used for torque
control strategy provides maximum torque for a calculation in FEM method. In the rotor PM, the
given current. This minimizes copper losses for a magnetic field equation can be expressed as:
given torque; however, it does not optimize the total
losses. Maximum torque per current curve is shown ∇×ν∇× A=ν∇×M (2)
by curve 6 in Fig.15. The operating point, A, shown In the stator conductors, the magnetic field
in this figure is for below base speed. The path equation is given by:
ABCD, shows the above speed operation. Shape
∂A
design optimization of the motor-drive has been done ∇×ν∇×A= −J +σ (3)
s ∂t
in these five operating points, considering field
And in the rotor and stator iron regions is
weakening on path ABCD [15]. These are shown in
expressed as:
Fig. 15 by curves 8 to 10. According to motor-drive
∂A
limits given by these curves, two basic operating
∇×ν∇×A=σ (4)
conditions can be identified: infinite-speed operation ∂t

and finite-speed operation. Below and above the
rated speed, there are three control modes, current
limited, current-voltage and voltage limited regions. 4.2.2 The Electrical Circuit Model [20-22]
The motor-drive here is finite maximum speed drive The voltage equations of the phase stator
because the infinite speed operating point lies outside windings are given by:
the current-limited circle.
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d results (contour plots of the resultant absolute value
[V ]=[ (λ −λ)]+[R ]I −[R ]I (5)
ab dt a b a a b b of flux density distribution) of the time stepping
FEM for this IPM synchronous motor.
d
[V ]=[ (λ −λ)]+[R ]I −[R ]I (6)
bc dt b c b b c c 4.3 Shape Optimization Method

To reduce the pulsation torque, which consists of
4.2.3 Mechanical System Model
cogging torque and torque ripples, it is necessary to
Two equations of the rotation of rotor are given as
redistribute the flux in rotor. For flux pattern
follows,
optimization, it needs to change the iron and air
dω
combination in a practical approach. In this research,
J ( r)+Bω =T −T (7)
eq dt r e L small holes have been drilled in the flux path at rotor

dδ surface (Fig. 18). The place and radius of these holes
=ω −ω (8) are found to minimize the torque pulsations [22-40].
dt e r

The above-mentioned equations are coupled in the
two-dimensional circuit-field-torque coupled time
stepping finite element method. This model is used to
optimize the shape of the rotor of motor as will be
discussed in the following section. This analysis is
done by the algorithm shown in Fig.16.
Start
Calculation of static magnetic
field and initial mechanical angle
of the rotor position
Coupling of magnetic field equations
and electric circuit equations Fig. 18. The model used to optimize the IPM
synchronous motor
Calculation of electromagnetic field
The vector of design variables is indicated by:
Calculation of flux linkages, three phase X =[ρ ρ ρ r r r θ θ θ]T
Currents and instantaneous torque 1 2 3 1 2 3 1 2 3 (9)
t=t+∆t Solving the two motion equations, i.e., Χ=[X X X ...X ]T (10)
1 2 3 n .
rotation of rotor equations(speed and
torque angle equations) Where parameters ρ,r,θare shown in Fig.18.
i i i
The design variables are subject to constraint with
The rotation of rotor: Moving mesh,
modification of element coordinate upper and lower limits, that is:
systems on each permanent magnet
X ≤ X ≤ X
i i i
(11)
Write related new constraint equations according i =1,2,3,...,n,n =9

to new node positions corresponding to mid
airgap length
g (Χ)≤ g i =1,2,...,m
i i 1
No t > tmax h ≤h(Χ) i =1,2,...,m (12)
i i 2
Yes
w ≤ w(Χ)≤ w i =1,2,...,m
i i i 3
Stop

A first order optimization method is used and this
Fig. 16. Transient Finite Element Method
constrained problem is translated into an

unconstrained one using penalty functions. Each
At last, convergence of the procedure is checked
iteration is composed of sub-iterations that include
by the velocity, magnetic vector potential and back-
search direction and gradient (i.e., derivatives)
EMF errors. Fig. 17 (at the end of chapter) shows the
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computations. The unconstrained objective function
is formulated as follows:
⎛ f ⎞ ⎛ n ⎞
Q(Χ,q)=⎜ ⎟+⎜∑Ρ (X )⎟+
⎜⎝ f0 ⎟⎠ ⎝i=1 X i ⎠ (13)
⎛ m1 m2 m3 ⎞
q*⎜∑Ρ (g )+∑Ρ (h )+∑Ρ (w )⎟
g i h i w i
⎝ ⎠
i=1 i=1 i=1
And for each optimization iteration(j), a search
direction vector,d , is calculated. The next iteration

(j+1) is obtained from the following equation
Fig. 19. Diagram for applying desired currents of the
[18,19]: IPM synchronous motor [12]
Χ(j+1) = Χ(j) +s d(j) (14)
j
S is the line search step size and d(j) is given
j
by:
d(j) =−∇Q(Χ(j),q )+r d(j−1) (15)
k j−1
Where,
[∇Q(Χ(j),q )−∇Q(Χ(j−1),q )]T∇Q(Χ(j),q )
r = k k k . (16)
j−1 ∇Q(Χ(j−1),q )2
k
In this paper, the objective function is defined as:
T −T
Ψ = max min (17)
T
avg

Fig. 20. Cross section of the IPM synchronous motor

based on different phase mmf axes

4.4 Motor-Drive Control Strategy
The d-q components of the reference currents that
were calculated according to the method described in
section 4-1, are used for estimation of phase currents.
So the reference currents, i*(t), i*(t) and i*(t)
A B C
will be applied to the FEM model, according to Figs.
19,20. For example, i*(t)can be estimated as:
A
i*(t) = 2(−i* cos(θ)+i*sin(θ))
A 3 d q (18) Fig. 21. Equipotential lines: flux distribution (full

load condition) at time t=0.556 S, after creating
Fig.21 shows equi-potential lines for the magnetic optimal circular holes
vector potential solutions at a time obtained by time
stepped FEM. Fig. 22 presents different wave forms It can be seen that there is a valuable
of motor torque, corresponding to different rotor improvement both in performance index, torque
shapes, for the operational point A. Fig. 23 shows the pulsations and saliency ratio. This new shape of the
spectrum of curves shown in Fig. 22. In this Fig., rotor with optimized holes, has the advantage of
curve 1 is the motor torque and the rotor has no increasing the maximum speed for the field
holes, curve 2 presents the output torque of motor weakening region from almost 1.83 p.u. to 1.96 p.u.
when three equal-area circles are created on the rotor above the base speed. Optimal shape design of rotor
and curve 3 shows the rotor torque when three has large effect on reduction of torque pulsations of
optimized circles are drilled into the rotor. IPM synchronous motor.

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In this part, the shape design optimization is
carried out by drilling internal circular holes of
optimal radius in the flux path at rotor surface. The
torque curves of the optimized motor show lower
pulsating torque and higher average torque. Another
advantage is that the field weakening region has been
extended for optimized motor. Although the shape
optimization is done at nominal operation point, the
performance evaluation of optimized motor at other
operation conditions shows improvement too. The
new shapes are easily applicable in the factory by
drilling the holes of different radius at predetermined

positions.
Fig. 22. Comparison of electromagnetic torque

calculated by FEM for different rotor structures in

point A, curve 1: no holes on the rotor, curve 2:
5 Interior Permanent-Magnet
same holes on the rotor and curve 3: optimized holes
on the rotor Synchronous-Induction Motor
In this part, a synchronous-induction motor has
been considered and a rotor cage for self starting of
the IPM synchronous motor is included in the rotor.
Fig. 24 shows this new topology. Time stepping
FEM is used to simulate this new machine. Fig. 25
shows the equipotential lines for this motor. Figs. 26
and 27 show the results obtained for the motor at
startup from standing using a three-phase 50Hz
voltage source under 15 N.m. load torque obtained
by time stepped finite element method and d-q model
respectively. In these waveforms it is evident that the
results of two dimensional filed modeling, i.e., time
stepping FEM has contained the rotor and stator slots
and rotor saliency affected by permanent magnet
shapes. Using a cage on the rotor of a permanent
magnet motor has this advantage that the motor can
be started directly as an induction motor and it needs
not to have a frequency control process for starting
[23-24].

Fig. 23. Spectrum of three curves shown in Fig.22,
curve 1: no holes on the rotor, curve 2: same holes on

the rotor and curve 3: optimized holes on the rotor Fig. 24. Interior permanent-magnet synchronous-
induction motor
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At last, a new topology, i.e., Interior Permanent-
Magnet Synchronous-Induction Motor has also been
completely studied. Among these different rotor
topologies for these PM machines, it can be seen that
the last one is the best from the point of view of
efficiency and pulsating torque.

Acknowledgment
The authors would like to really appreciate decent
considerations of all people who engaged on different

parts of this work. Unless they had accompanied and
Fig. 25. Equipotential lines for magnetic vector
had helped us, the work could not have been finished
potential: full-load conditions, interior permanent-
and finalized. At last, we would say that the work
magnet synchronous induction motor
presents the results of almost 8 years of

interpretations and main conclusions of different
research and educational projects in the field of FEM
analysis of PM machines.

References
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distribution in permanent magnet machines",
IEEE Transactions on Industry Applications,
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[2] N. Boules, "Two-dimensional field analysis of
cylindrical machines with permanent magnet
excitation", IEEE Transactions on Industry

Fig. 26. Electromagnetic torque obtained by time Applications, Vol.IA-20, pp. 1267-1277,
stepping finite element method September-October 1984.
[3] Z. Q. Zhu, D. Howe, "Instantaneous magnetic
field distribution in brushless permanent
magnet DC motors, part III: Effect of stator
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[4] J. J. Cathey, Electric machines, analysis and
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January 2000.
Fig. 27. Electromagnetic torque obtained by
[6] US Steel Manual, Non-Oriented Electrical
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Steel Sheets, Pittsburgh, Pa. 15230.
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[7] ANSYS 5.6 Theory and APDL Reference

Manual, 2003.

[8] W. Cai, D. Fulton and K. Reichert, "Design of
6 Conclusion
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[9] K. F. Rasmussen, et al., "Analysis and
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10