CPI2007-148-BenAmor

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Optimisation of a linear stepping motor

force by geometrical parameters control

Abdessattar Ben Amor * - Sondes Hajri ** - Moncef Gasmi *

Unité de Recherche en Automatique et Informatique Industrielle, (URAII).

*Institut National des Sciences Appliquées et de Technologie, (INSAT). **Ecole Nationale d’Ingénieurs de Tunis, (ENIT).

[email protected], [email protected] , [email protected]

ABSTRACT . In this paper, a method for improving a stepping motor prototype design is presented. Firstly, a detailed electric and magnetic study is considered to reduce Joule losses.Several numerical simulations results, based on the electromagnetic constraint analysis, had showed the influence of the dental ratio on the saturation and clutter reduction.Secondly, the comparison of developed forces for various dental structure dimensionsallowed the optimisation of the structure parameters. The best choice of the parameters had resulted in an optimal developed force.

RÉSUMÉ . Dans ce papier, une méthode d’amélioration de la conception d’un prototypeexistant de moteur pas à pas est présentée. Au début, la tendance à réduire les pertes Jouledans ce moteur a nécessité une étude conjointe sur les deux plans électrique et magnétique. L’analyse des contraintes électrique et magnétique rencontrées ont permis d’illustrer

l’influence du rapport dentaire sur la réduction de la saturation et de l’encombrement dumoteur. Des résultats de simulation, basés sur une comparaison des efforts développés encorrespondance avec des variantes dimensionnelles de la structure dentaire, ont permis deretenir les dimensions optimales conduisant à un effort optimal développé.

KEYWORDS: losses, dental ratio, induction threshold, saturation, force.

MOTS-CLÉS : pertes, rapport dentaire, seuil d’induction, saturation, force.

Nom de revue. Volume X /2007. p 1-15

mailto:[email protected]






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1. Introduction

The heating, in an electric machine, is very important factor that causes the

destruction of its constituents and limit eventual magnets functioning [CAV 01],

[HOA 95]. Joule losses, created in coils, are essentially the origin of electric

machine heating. Hence, several researchers oriented theirs efforts to the reduction

of Joule losses in numerous types of actuators to avoid theirs effects [HOA 95].

Yet, linear electromagnetic actuators, frequently, suffer from a low force volume

ratio. In general, to increase this ratio the number of pole is increased or the polar

step is reduced [CAV 01].

So, a new approach presented in this paper permits to attenuate the heating and

leads to the augmentations of the force volume ratio in a linear tubular stepping

motor.

This approach consists in modification of horizontal geometric parameters

dimensions with considering constant the dental step. So the augmentation of notch

width will be accompanied by the reduction of tooth width.

Also, this dimensional modification will reduce the motor clutter what is agree with

the new research axe directed to actuators miniaturisation in order to be integrated in

small systems.

In this paper, we will start by studying Joule losses in the concerned stepping motor;

however we will analyse electric and magnetic constraint opposing the reduction of

losses. Next we will present several curves in an effort to illustrate the effects of the

indicated modification on the motor clutter and the critic induction levels.

Finally, we will examine the force variations according to the notch width in order

to reach the optimum.

2. Joule losses expressions

Joule losses are the most major ones in an electric machine, they are created in

coils.

These losses, expressed by [1] are functions of the coil middle resistance R, the

current density J and the copper section S c [GRE 00], [JUF 79],[CHE 06].

[1] S J R p 2c

2 J ⋅⋅=

The coil middle resistance can be expressed by [BEN AM], [GRE 00], [JUF 79]:

N S

l R 2

c

moy.c⋅=

ρ [2]

Where:

ρc: is the copper resistivity,

l moy: is the spires middle length,

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Optimisation of a linear stepping motor force by geometrical parameters control 3

N : is the spires number.

The copper section is equal to the product of winding coefficient k b by the winding

area S b.

A module of the tubular stepping motor, which we are interested in its amelioration,

is presented in figure1. So, we can write the expressions of the winding surface S b

and the spire middle length lmoy relative to this motor with geometric parameters

illustrated in figure1:

( ) r b 34 r S b −×= [3]

( )34 r r l moy +×=π [4]

Where:

b: is the notch width,

r 3: is the coil interior radius,r 4: is the coil exterior radius.

Then, Joule losses p J relative to the concerned motor can be written according to the

geometric and electric parameters of this motor by:

( ) ( )34

22

34 r r bk J N r r p bc J −⋅⋅⋅⋅⋅+⋅⋅= π ρ [5]

Our improvement tendency is to reduce Joule losses by modification of geometric

parameters maintaining constant the electric ones.

To minimize these losses, we have to consider two constraints: electric and

magnetic.

Figure1. Geometric characteristics of the existing motor

The electric constraint obliged us to maintain constant the magneto motive force

( N.J ). Or, there is a relation between spires number N and copper section S c:( )

f

b

f

c

S

r r bk

S

S N 34 −⋅⋅

== [6]

Where:

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S f : is the copper wire section.

That leads from [6] to maintain constant S c so that the product (b. (r 4 – r 3)) must

be constant. So, the reduction of Joule losses requires minimizing the difference

(r 4 – r 3) and the sum (r 4 + r 3) and maintaining constant the area (b. (r 4 –r 3)).

The magnetic constraint imposes the level of inductions top values in different

regions of iron volume.

So, several numerical simulations using finite elements software “MAXWELL”

served to obtain the optimal dimensions to reduce Joule losses.

3. Electric constraint

As it was noted in the previous part, the electric constraint imposes that the area

(b. (r 4 – r 3)) must be constant. But, the reduction of Joule losses necessitates

minimizing the difference (r 4 – r 3). So we have to increase the value of notch width b.

The equation [7] permits us to get values of the radius r 4 corresponding to

several values of notch width b considering that the interior radius of stator r 3 has a

constant value equal to 14.1mm and the winding area S b is equal to 157mm2.

34 r b

S r b += [7]

The exterior radius r 5 of the motor is the sum of the radius r 4 and the breechthickness ec:

cer r += 45 [8]

Figure2 shows the variation of the exterior radius r 5 according to the variation of

notch width b.

37

38

39

40

41

42

43

44

45

46

5,1 5,2 5,2 5,3 5,3 5,4 5,4 5,5 5,6 5,6 5,7 5,8 5,8 5,9 6,0 6,0 6,1 6,2 6,2 6,3 6,4

b (mm)

r5 (mm)

Figure2. Variations of exterior radius r 5

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function of notch width “b”.We note that the exterior radius of the motor can be more reduced when we

increase the notch width of stator and movable part. So, the increase of notch width

allows the clutter reduction. In addition, the clutter reduction and the tooth width

attenuation entail the reduction of iron volume.

4. Magnetic constraint

For the choice of magnetic circuit measurements we have to fix a level of

induction achieving a compromise between the technical constraints and the

economic aspects.

In fact, we have to work in a level of induction inferior to the saturation limit

situated in the extreme part of the induction curve linear domain and we have to

reduce the iron volume.

Then, maintaining the same iron material of the existing motor prototype (soft

steel) we tend, in this part, to choose the optimal horizontal dimensions that can

satisfy technical constraints and economic constraints.

Several numerical simulations give us induction values in different points of

magnetic circuits having different horizontal dimensions.

Figure3 is an example of simulation result, it shows the values of induction in a

magnetic circuit where the notch width b is equal to the tooth width a(a=b=5.08mm).

Figure3. Induction values in a module magnetic circuit

having the same width of notch and tooth.

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4.1. Characteristics of the used material

The material used for the construction of the motor is the soft steel. Its

characteristic curve B=f(H) is given by figure4.

Figure4. Characteristic curve B=f(H) of soft steel.

The saturation induction value of soft steel is equal to 1.6T, so the induction values

obtained by simulations must do not exceed this value.

4.2. Induction variation in the movable part long path

To study the variation of induction in the movable part long path according to

notch width variations we identify a point in the middle of this path for every

dimension and we simulate the induction value in this point, Figure5.

Figure5. Identification of induction simulation point

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in the movable part long path.

Figure6 presents the induction variations in the identified point according to

notch width variations. We notice in this figure that the values of induction decrease

when values of notch width increase. So, it’s obvious that increasing notch width

encourages our tendency to reduce induction values.

B(T)

0

0,2

0,4

0,6

0,8

1

1,2

1,4

1,6

5,1 5,2 5,2 5,3 5,3 5,4 5,4 5,5 5,6 5,6 5,7 5,8 5,8 5,9 6,0 6,0 6,1 6,2 6,2 6,3 6,4

b (mm)

Figure6. Variation of induction in the movable part long path

according to notch width variations.

4.3. Variation of induction in the stator long path

The simulation of induction in the middle point of the stator long path, as shown

in figure7, permits us to study the variation of induction in this point for different

values of notch width, Figure8.

Figure7. Identification of induction simulation point in the stator long path.

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0,56

0,58

0,6

0,62

0,64

0,66

0,68

0,7

5,1 5,2 5,2 5,3 5,3 5,4 5,4 5,5 5,6 5,6 5,7 5,8 5,8 5,9 6,0 6,0 6,1 6,2 6,2 6,3 6,4b (mm)

B (T)

Figure8. Variations of induction in the stator long path

according to values of notch width b.

We note from figure8 that the values of induction in the stator long path increase

with the notch width, but this increase of induction is not significant because all

simulated values of induction in the indicated point are smaller than 0.7T.

4.4. Variation of induction in the stator breech

We need to get the induction value in a point situated in the breech middle as

indicated in figure9 to study the induction variation in this point according to notch

width variations, Figure 10.

Figure9. Identification of induction simulation point

in the stator breech.

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0,34

0,35

0,36

0,37

0,38

0,39

0,4

0,41

0,42

0,43

5,1 5,2 5,2 5,3 5,3 5,4 5,4 5,5 5,6 5,6 5,7 5,8 5,8 5,9 6,0 6,0 6,1 6,2 6,2 6,3 6,4

b (mm)

B (T)

Figure10. Variations of induction in the stator breech

according to notch width values.

We note from figure10 that the values of induction in the stator breech decrease

when values of notch width increase, which agrees with our tendency to reduce

induction values.

4.5. Variation of induction in the airgap

Figure11 shows the induction level in the middle point of the airgap.

We had to simulate the induction in this point for different values of notch width to

study the induction variation in the airgap according to this geometric parameter,

Figure12.

Figure11. Identification of induction

simulation point in the airgap.

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Nom de la revue. 10

1,55

1,56

1,57

1,58

1,59

1,6

1,61

5,1 5,2 5,2 5,3 5,3 5,4 5,4 5,5 5,6 5,6 5,7 5,8 5,8 5,9 6,0 6,0 6,1 6,2 6,2 6,3 6,4

b (mm)

Be (T)

Figure12. Variations of induction value in the airgap

according to notch width values.

We notice that induction values in the airgap increase with notch width values

but these values are usually inferior to the limit induction value that is equal to 1.6T

[JUF 79], where notch width value is inferior to 6.4mm.

In these previous parts of work we have increased the values of notch width

respecting the electric and magnetic constraints.

As a result of simulations we note that the increase of notch width is favourablefor the reduction of iron volume and it is agreed with our tendency to reduce the

induction values. This permits us to have other actions to more ameliorate this

motor.

Then we consider that it will be very profitable to increase the notch width “b” and

decrease the tooth width “a” maintaining the equality a+b=10,16mm. But how we

can limit these values? The response to this question will be done in the next part

where we try to study the force variation function of the notch width values and to

find the value of notch width corresponding to the optimal value of force.

5. Effect of notch width variation on the force

If we consider a system characterized by k electric circuits marked by the j

indication (j=1 to k), to every circuit j we associate a current i j, a tension u j and a

flux ψ j, the expression of the electromagnetic force created by this system for a

displacement Δx of the mobile part is given by [MUL 04], [MAY 00], [GRE 00],

[JUF 79]:

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.. x2

1 F p j

k

1 j

k

1 p

jpθ θ

Δ

ΔΡ ∑∑

= =

= [9]

Where:

θ : is the magnetomotive force,

P: is the permeance.

5.1. Force created by the reluctant stepping motor

In the conception of the tubular linear stepping motor we intend to separate its

modules by a perfect magnetic insulation that permits us to annul the mutual permeances in the expression of the electromagnetic force that becomes [MUL 04],

[MAY 00], [GRE 00], [JUF 79]:

2 j

k

1 j

j

x

P

2

1 F θ

Δ

Δ⋅= ∑

=

[10]

The flux circulating in the motor is divided in a flux crossing iron parts, a flux

crossing the airgap and a small number of leakage flux lines, Figure13.

Figure13. Flux lines circulating in one studied motor module

Then, the total permeance is the sum of different iron parts permeances P iron, airgap

permeances P airgap and leakage permeances P leakage. The force expression can be

written :

jironleakageleakage

airgapairgapairgapironiron

k

j x F ⎟

⎟ ⎠

⎞⎜⎜⎝

⎛ Ρ+Ρ+Ρ

Δ

Δ= ∑ ∑∑∑

=

222

1

...2

1θ θ θ

[11]

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Nom de la revue. 12

By tendency to the simplification of this expression we impose these hypotheses:

- magnetomotive force θ iron corresponding to every element of iron circuit is

constant during the displacement.

- The permeance of leakage remains constant during the displacement.

Hence, the variation of coenergy is situated mainly in the airgap. So the useful

equation for the calculation of the electromagnetic force will be given by:

jairgap

2airgapairgap

k

1 j

. x2

1 F ⎟

⎟ ⎠

⎞⎜⎜⎝

⎛ = ∑∑

=

θ Ρ Δ

Δ[12]

In addition, if we want to aliment the conceived motor phases successively, for

every impulse only one phase is alimented, the expression of the force will be

simplified:

x.

2

1 F airgap

airgap

2airgap

Δ

ΔΡ θ ∑= [13]

The application of this formula to the studied system gives this expression:

( ) ( )[ 22

2

5

2220δ δ

δ ]

π μ r r ehr ba K J F cd b ⋅−−−⋅+⋅⋅⋅

⋅= [14]

Where:

r δ: is the airgap middle radius,

δ: is the airgap,

a: is the tooth width,

hd: is the tooth height.

5.2. Force variations function of notch width values

It’s clear in equation [14] that the force depends on geometric parameters: notch

width “b”, teeth width “a”, and the exterior radius r 5. Or these three parameters are

linked by relation [7] and the indicated equality a+b=10,16mm.

So we can concentrate the study of the force variations as a function of all these

parameters in the study of force variations only for notch width values. Figure14

presents the result of force simulations.

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0

2

4

6

8

10

12

14

16

18

20

5,1 5,2 5,2 5,3 5,3 5,4 5,4 5,5 5,6 5,6 5,7 5,8 5,8 5,9 6,0 6,0 6,1 6,2 6,2 6,3 6,4

b (mm)

F (N)

Figure14. Force variations according to notch width values “b”.

It’s clear in figure14 that the optimal value of force equal to 18.48N is relative to a

notch width value 6.2mm, to which correlates a tooth width value equal to 3.96mm

and an exterior radius value equal to 40.4mm. So we will consider these values in

the next stages of the motor design.

6. Conclusion

In this paper, a theoretical study of Joule losses in a linear stepping motor has

been illustrated. The problem of heating caused by Joule losses is resolved due to a

modification of horizontal dimensions.

The numerical simulations using finite elements software have served to highlight

the advantages of these modifications, what is shown in the reduction of clutter and

induction level. Hence, the concerned motor can be integrated in more small

systems. The attenuation of induction value let an available margin for other

dimensions improvements.

Furthermore, the comparison of the force evolution as a function of actuator

horizontal dimensions has been conducted in an effort toward the optimisation of

these dimensions.

In the next work, we will realize other kinds of dimensions improvements in order to more ameliorate the characteristics of the concerned stepping motor.

7. REFERENCES

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Nom de la revue. 14

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[BEN AH 03] Ben Ahmed H., Multon B., Gabsi M., Cavarec P.E., “Machines électriques-

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[GRE 00] Grellet G., Clerc G., “ Actionneurs Electriques”, Editions Eyrolles, Paris, 2000.

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[KOS 79] Kostenko M., “Machines électriques”, Edition Mir Moscou,1979.

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[MAN 99] Manual V., “Contribution aux microsystèmes magnétiques : Micromoteur asynchrone à palier magnétique”, doctorate thesis, Polytechnic National Institute of Grenoble,

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[MAY 00] Maye P., “Moteurs électriques pour la robotique”, Edition Dunod, Paris, 2000.

[MUL 04] Multon B., “Pertes mécaniques”, Master STS/IST, University Paris - Sud/ENS

Cachan, 2004.

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[PIN 04] Pinard R., “Commande électronique des moteurs électriques”, Edition Dunod, Paris

2004.

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