Sensitivity Analysis of Several Geometrical Parameters on Linear Switched Reluctance Motor Performance

8/4/2019 Sensitivity Analysis of Several Geometrical Parameters on Linear Switched Reluctance Motor Performance

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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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Proceedings of the 2008 International Conference on Electrical Machines

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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

[1] Arumugam R. Lindsay J. F., Sensitivity of pole arc/pole pitch ratio on switched reluctance motor performance, IEEE Industry ApplicationsSociety Annual Meeting. 1988, vol.1, pp.50 - 54

[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.

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Sensitivity Analysis of Several Geometrical Parameters on Linear Switched Reluctance Motor Performance