sensitivity analysis of several geometrical parameters on linear switched reluctance motor...
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Proceedings of the 2008 International Conference on Electrical Machines Paper ID 1105
978-1-4244-1736-0/08/$25.00 2008 IEEE 1
Sensitivity Analysis of Several GeometricalParameters on
Linear Switched Reluctance Motor Performance
J. G. Amoros 1, P. Andrada 2, L. Massagues 1, P. Iiguez 11DEEEA Dept., University Rovira i Virgili
Av. Pasos Catalans, 26, Tarragona, Spain 2 EPSEVG, DEE, GAECE, UPC Technical University of Catalonia
E-mail [email protected]
Abstract- This paper studies the sensitivity of severalgeometrical parameters on the performance of a linear switchedreluctance motor (LSRM). The analysis is made in twodimensions using the Finite Element Method. The study shows thestrong influence of the width of the stator pole (bp) and the widthof the moving pole or translator pole (bs) on inductance and forceprofiles. The results of this study could be a useful tool foroptimizing the geometry of a LSRM.
I. INTRODUCTIONVarious papers regarding the sensitivity of several
geometrical parameters on rotating switched reluctance motors
(SRM) have been published [1] [2]. The aim of this paper is to
analyze in detail the sensitivity of a 4-phase linear switched
reluctance motor (LSRM). The sensitivity study compares the
inductance and force profile for different stator pole widths
(bp) and translator pole widths (bs) (see Fig. 2). A method
based on the lumped parameter magnetic circuit model allows
us to obtain analytical expressions that connect geometrical
parameters with the inductance and the force developed by the
LSRM [3]. However, these expressions are not simple, and if
saturation has to be taken into account, an iterative process is
required. Therefore, finite element method (FEM) is the
preferred method used in the study. In order to save computing
time, the whole LSRM is broken down into the minimum
repetition pattern that guarantees the same results as full
LSRM. To do this, suitable boundary conditions must be
established.
The study uses a two-dimensional finite element solver [4].
Is well known that 2-D solvers are not particularly appropriate
for accurately studying 3-D devices because the end effects arenot taken account, but they can be used effectively to optimize
lamination geometry.
II. TWO DIMENSIONAL MODELThe whole LSRM was presented by Amoros J.G. et al.
(2007) [5], and is formed by three identical sections, each one
of which has 8 primary poles (Np), 6 secondary poles (Ns) and
is double sided. Fig. 1 shows one section of the LSRM and the
piece being studied. The number of phases (m=4) and the
stroke (PS=4mm) are design parameters, and let us obtain the
primary pole pitch (Tp) and secondary pole pitch (Ts). The
design parameters lp , ls andgare fixed.
Fig. 2 shows a piece of one section of the LSRM that can be
considered the minimum repetition, in which the geometrical
parameters bp , cp , lp , bs , cs , ls ,gare shown. The windings
are placed in the stator, and are located in the inter-polar area
(cpxlp). The translator does not have any current density.
Assigning the boundary conditions is fundamental to solving
the field problem (see Fig. 3). The first boundary condition is
the homogeneousDirichletcondition that generally equals the
magnetic vector potential, A, at zero. This condition is
equivalent to an external material with null magnetic
permeability; and therefore any flux line can cross this
boundary.
Fig. 1. One section of the whole LSRM
Fig. 2. Main dimensions for the minimum study pattern
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Fig. 3. Boundary conditions for the minimum study pattern. Poles completely
unaligned (x=0)
The Neumann condition imposes a value to the normalderivative ofA on the boundary. When this value is zero, it isequivalent to an external material with infinite magneticpermeability.
Under these conditions (Fig. 3), the results for the piece ofthe LSRM are 6% less than the full LSRM. The cross section(Fig. 3) is meshed with a uniform mesh size of 0.25mm that
gives 62,476 elements and 31,765 nodes. The distance S (seeFig. 2) between aligned and unaligned positions is given by:
( ) / 2s sS b c= + (1)
The variablex showed in Fig. 2 equalsx=0 when the poles
are fully unaligned, and x=S, when the poles are completely
aligned. Between these two positions we take equidistant
points that are separated by x ( x=S/32=0.25mm). . These
equidistant positions are computed, giving 33 computations in
total for each combination ofbp and bs.
III. SENSITIVITY ANALYSISWith the aim of getting dimensionless variables, the poles
widths are normalized for stator pole width, obtaining p and s
defined as:/p p P b T = (2)
/s s P b T = (3)
The interval of variation for p and s is limited by the
Lawrenson criterion [6] for feasible configurations. These
physical constraints define a triangle given by:
p s (4)
2 /p sN (5)
/p s S P T T + (6)
In order to obtain high resolution in the scanning area, the
triangle is framed in a dotted rectangle (Fig 4).
Each combination explored ofp and s is represented by a
dot in Fig 4. The normalized poles widths (p ,s) are increased
in steps of 1/48, giving p a range from 1/3 to 2/3.
Fig 4. Triangle for feasible configurations and exploration area
The range for s is from 1/3 to 1. The total number of
computed problems is 17x33x33=18,573 which means a strong
computational effort. For each computed problem the current
density (J) is a constant value.
The first sensitivity analysis investigates the influence ofs
on the force (Fx) and inductance (L), for a given configuration
with a ratio ofp=0.5
(see Fig 4). The computed problems of
static force for each s are presented in Fig. 5, where, for
clarity, only five of the thirty three profiles are showed.
From these results we must obtain a parameter that evaluates
the goodness of each static force characteristic. There are
several parameters that can do this, e.g. peak force, rise or
down slope, average force etc. We take the average force to
evaluate the influence of the geometrical parameter s. The
average force is calculated for each profile from the integration
of static force profiles. The average force for a fixed value of
p=0.5 is plotted in Fig. 5.
The average force reaches maximum fors[0.417,0.5], as
is shown in Fig. 5. Therefore, we can conclude that forp=0.5
there is an interval fors that optimizes the average force.For the inductance study, Fig. 5 shows the influence ofs on
the profiles of inductance versus position for a given p=0.5.
In order to evaluate the inductance profiles, the inductance
ratio (La/Lu) between alignment and unalignment is taken as
parameter independent of position. The inductance ratio is
shown in Fig. 5. The optimal values of s that maximize the
inductance ratio (La/Lu) are given by s[0.333,0.417].
The intervals that optimize average force and inductance
ratio overlap when s=0.417. Therefore, in this case the
geometry of LSRM can be optimized for both parameters, for
the average force and for the inductance ratio, but this cannot
always be achieved.
Thus we can conclude that the optimal pole widths areobtained for p=0.5 and s=0.417. Although this point is
outside the physical constraints defined in (4), (5) and (6), this
does not imply that the configuration is not possible. The
symmetrical triangle generated about (4), and softly shaded in
Fig 4, represents the configurations with widerbp and narrower
inter-polar area, meaning larger cooper losses and therefore a
non practical design.
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The second sensitivity analysis investigates the influence of
p on the force (Fx) and inductance (L) of a fixed secondary
normalized width (s=0.5). Fig. 6 shows several static force
profiles as well as the average force. In this case, the static
force is dramatically reduced when p increases, because of the
wide primary pole reduces space for copper. Therefore, for a
fixed current density, the reduction in the current and force is
in direct proportion to the increase in bp. Small values of bp
produce a wide dead zone that reduces the slope and therefore
the average force. Summarizing, fors=0.5 the optimal range
of values forp are given by p[0.4,0.52].
As before, the inductance profiles and inductance ratio are
shown in Fig. 6, where no optimal is reached.
Fig. 5. Sensitivity of force and inductance profiles, fixing p=0.5
Fig. 6. Sensitivity of force and inductance profiles, fixing s=0.5
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The optimum poles shape has to satisfy various opposing
requirements. On the one hand, wide primary poles increase
the aligned inductance and inductance ratio, which is good for
motor performance. On the other hand, high efficiency designs
needs maximum copper area, thus narrow primary poles have
to be chosen.
In order to get a full description of the average force and
inductance ratio, Fig. 7 and Fig. 8 show the complete analysis
made for each dot painted in Fig 4.The contour lines are represented on the p-s plane of
average force (Fig. 7) as well as the area closed by the physical
constraints (4), (5) and (6) (see Fig 4). As it can be seen, the
maximum average force lies near the line given by (4). For the
sensitivity of inductance ratio curves (see Fig. 8) no maximum
is achieved near the triangular area and therefore it can not be
optimized for this parameter.
Fig. 9 shows better the contour lines of average force than
those roughly displayed on the p-s plane of Fig. 7. It also
shows conditions (4)-(5)-(6). The optimum values forsandp
are clearly showed in Fig. 9 in light color.
Numerically the optimal values can be defined through a
bounded square within the ellipse, p[0.42, 0.52] and
s[0.42, 0.52], although any inner point of the contour line
(691.5N) can be considered an optimal configuration.
Fig. 7. Average force vs. s and p . Fx,avg=f(sp),
Fig. 8. Inductance ratio vs. s and p . La/Lu=f(sp),
Fig. 9. Contour lines of average force.Lawrenson criterion.
The previous studies have looked for a constant high level of
saturation (J=15 A/mm2).
The latest study investigates the influence of the current
density on the average force and the inductance ratio, for four
levels of current density. (J=5A/mm2 , J=10A/mm2,
J=15A/mm2, J=20A/mm
2).
As can be seen, the optimal region goes up, increasing the
value p proportionally to the current density increase. (see Fig.
10 and Fig. 11). The optimal region is partially located in (4),
(5) and (6), for the average force in all cases. This means that
an optimal configuration can be achieved for the average force.
This does not occur for the inductance ratio. Only for low
current density (J=5A/mm2) can an optimal region be achieved
for the inductance ratio.
Fig. 10. Contour lines of average force for several current density values
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Fig. 11. Contour lines of inductance ratio for several current density values
IV. CONCLUSIONSThis paper reports a detailed analysis of sensitivity carried
out on a 4-phase LSRM.
The analysis was done on a piece of an LSRM and therefore
computational time is saved in proportion to the reduction of
the area. This simplification does not affect appreciably the
results that we would have obtained had we considered the
whole LSRM.
The study shows the strong influence of the width of the
stator pole (bp) and the width of translator pole (bs) on
inductance and force profiles as well as the average force and
inductance ratio. The current density is also considered for thesensitivity analysis on the average force and inductance ratio.
The best parameter to estimate the optimum geometry is the
average force since optimum values are always reached. The
optimum lies near the line s=0.5 for all current densities. For
the case J=5A/mm2 the primarys wide pole is narrow and is
situated underp0.4167. For high current density values (10-
20A/mm2) a general rule for the optimum is ps=0.5,
although there are many points shown in the bright areas in
Fig. 10. The following table summarizes the optimal intervals
forp and s that have been obtained from Fig. 10.
TABLE ISUMMARIZED OPTIMUM INTERVALS FORAVERAGE FORCE
p s
J=5A/mm2 [0.333 , 0.417] [0.417 , 0.583]
J=10A/mm2 [0.375 , 0.500] [0.417 , 0.542]
J=15A/mm2 [0.417 , 0.542] [0.375 , 0.542]
J=20A/mm2 [0.458 , 0.542] [0.375 , 0.583]
From the results presented it can be seen that this sensitivity
analysis can give guidelines to improve the design procedures
of the LSRM.
ACKNOWLEDGEMENT
This study has been done with the support of Spanish
Ministry of Science and Innovation under the projects numberENE2005-06934 and DPI2006-09880.
REFERENCES
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[2] Murthy S. S., Singh B., Sharma V. K., Finite element analysis toachieve optimum geometry of switched reluctance motor, TENCON '98.
IEEE Region 10th International Conference on Global Connectivity inEnergy, Computer, Communication and Control , vol.2, No., pp.414-418.1998
[3] R. Krishnan, Switched Reluctance Motor Drives. CRC Press 2001,pp.138-167.
[4] D.C. Meeker,Finite Element Magnetics Method. Version 4.0.1(03Dec2006 Build). http://femm.foster-miller.net
[5] Amoros J. G., Andrada P., Massagus L., Iiguez P., Motor lineal dereluctancia conmutada de doble cara para aplicaciones de elevadadensidad de fuerza, (In Spanish) Book of Abstracts XCLEEE 2007 X
Portuguese Spanish Congress in Electrical Engineering, pp. 3.43-3.46,5-7 July 2007, Madeira Island, Portugal.
[6] Lawrenson P. J, Stephenson J. M., Blenkinsop P. T., Corda J., Fulton N. N., Variable-speed switched reluctance motors, IEE Proceedings-Belectric power applications. vol.127 (No.4) July 1980, pp. 253-265.