A Switched Reluctance Motors Analyse using Permeance ... · Switched Reluctance Motors (SRM) are fast becoming a popular alternative to Induction Motors (IM) in the variable speed

International Journal of Applied Engineering Research ISSN 0973-4562 Volume 1 Number 2 (2006) pp. 137-152 (c) Research India Publications http://www.ripublication.com/ijaer.htm

A Switched Reluctance Motors Analyse using Permeance Network Method

B. Bekkouche1, A. Chaouch2 and Y. Mezari3

1Department of Electronic, University of Mostaganem, 27000, Algeria, (E-mail: ikchaouchdz@ yahoo.fr).

2Department of Electronic, University of Mostaganem, 27000, Algeria, (E-mail: [email protected]).

3Sciences and Technologie University «Mohamed BOUDIAF» BP 1505 EL M'naouer Oran, Algeria.

Abstract

This paper presents a permeance network model for use in the dynamic simulation of Switched Reluctance Motors. The modeling is based on the concept to divide the magnetic circuit of the machine in each zone which are described by her own reluctance, therefore we get an equivalent electric circuit and according to laws of ciruits analysis as law of meshing, so we calculate differents permeances in each part of the machine. The advantage in this method, we calculate the airgap permeance in different rotor position while using triangular method. it enables to see the influence in configuration and width of the airgap on currents and torque forms. The modeling procedures are described and simulation results are presented.

Index Terms— Network, Permeance, Switched reluctance motors, Triangular.

Introduction Switched Reluctance Motors (SRM) are fast becoming a popular alternative to Induction Motors (IM) in the variable speed drive market. Though the first SRM was built in 1838, it did not find widespread use until the late 1970’s . This was due to the difficulty in controlling the machine. Since the 1960’s, with the advent of power electronics and highpower semiconductor switches, control of the SRM has become much easier and there has been a renewed interest in SRM drives [1,2].

138 B. Bekkouche et al

The advantages of the SRM are summarized below. ��Simple and robust structure. ��There are no windings on the rotor of the machine and hence it is inherently less

expensive. ��High torque-to-inertia ratio. It has high starting torque without the problem of in-

rush currents and its power density is comparable to that of an induction machine. ��Maximum operating speed and maximum rotor temperatures it can withstand are

higher when compared to other machines of similar ratings. ��Each phase winding of the SRM is independent of the other phase windings and

this makes the machine highly reliable. ��Motor torque is independent of the phase current polarity. Thus, the converter for

the SRM drive requires only one switch per phase. ��Has inherent four-quadrant operation. ��Has a wide constant torque/power region in the torque-speed characteristics. ��High efficiency throughout the entire torque-speed range. ��The disadvantages of SRM are summarized below. ��Pulsed nature of torque production, which leads to torque ripple and acoustic

noise. ��To have high torque-to-volume ratio, the airgap between the stator and rotor

should be very small. This leads to less manufacturing tolerances and increased acoustic noise.

Overall, SRM has many advantages and thus can be successfully used in variable speed drives. The major disadvantage of having a high torque ripple can be overcome by using suitable control methodologies. Of late, SRM has become a better alternative in a host of applications including electric vehicles, traction, industrial drives, position servo, robotics, textiles, aerospace, office automation, machine tools and appliances.

Several studies concerning dynamic simulation have been achieved using the C language, Fortran progrmming, and also using differential equation-based as FDM (Finite difference method) and FEM (Finite element method) [3,4,5]. In this paper, the SRM is studied using a permeance network méthod (PNM) with C language programming. This method permits to take into consideration non linear and movement of systems then facilate dynamics studies. Model of the Switched Reluctance Motor The Switched Reluctance Motor A Switched Reluctance Motor (SRM) is an electric motor in which torque is produced by the tendency of its moveable part to move to a position of least reluctance, which corresponds to the position of maximum inductance. It is a doubly salient, singly excited motor. That is, the SRM has salient poles on both the rotor and the stator, but only the stator poles carry windings.

The rotor tries to get to a position of minimum reluctance by aligning itself with the stator magnetic field. In the presence of a rotating magnetic field, the rotor tries to rotate along with the rotating magnetic field to always be in a position of minimum reluctance. Thus, exciting the stator phase windings of the motor in a particular

A Switched Reluctance Motors Analyse 139

sequence and consequently, controlling the rotating magnetic field, we can control the movement of the rotor.

Figure 1 shows a typical 6/4 SRM. It is a three-phase machine and has 6 poles on the stator and 4 poles on the rotor. The number of poles on the stator and on the rotor is usually not equal. This is to avoid the eventuality of the rotor being in a state of producing no initial torque, which occurs when all the rotor poles are locked in with the stator poles. Here, the diametrically opposite stator pole windings are connected in series and they form one phase. Thus, the six stator poles constitute three phases. When the rotor poles are aligned with the stator poles of a particular phase, the phase is said to be in an aligned position (fig1. 1-A, 1’-A’). Similarly, if the inter-polar axis of the rotor is aligned with the stator poles of a particular phase, the phase is said to be in an unaligned position (fig 2.2-B, 2’-B’) .

Figure 2: Inductance profile of one phase of an SRM

Figure 1: A typical 6/4 SRM with 6 rotor poles and 4 stator poles

A

A’

B’ B


In a three-phase SRM, the direction of rotation of the rotor is opposite to the direction of the switching sequence of stator poles. Principle of Operation When current flows in a phase, the resulting torque tends to move the rotor in a direction that leads to an increase in the inductance. Provided that there is no residual magnetization of steel, the direction of current flow is immaterial and the torque always tries to move the rotor to the position of highest inductance. Positive torque is produced when the phase is switched on while the rotor is moving from the unaligned position to the aligned position (fig.2).

The torque (T) produced in the SRM is given by the formula :

θddLT i

2

21= (1)

Thus positive torque is produced when the phase is switched on during the rising inductance. Consequently, if the phase is switched on during the period of falling inductance, negative torque will be produced.

An SRM is normally operated by getting a feedback of the rotor position and firing the phases depending on this feedback. The phase is switched on when the rotor is in the unaligned position and switched off before it reaches the aligned position. This is done so that the current (and flux) decays to zero before the inductance begins to fall, resulting in negative torque. This in turn maximizes the effective torque. The angle between turn-on and turn-off is called the dwell angle. The turn-on, turn-off and dwell angles are used to control the torque output of the motor. Having a negative voltage across the winding would help in getting the flux back to zero quickly. This way, the dwell angle can be increased and the torque ripple can be minimized.

Mathematical Model The instantaneous voltage across the terminals of a phase of an SR motor winding is related to the flux linked in the winding by Faraday's law as

(2)

where V is the terminal voltage, I is the phase current, R is the phase winding resistance, and ψ �� is the flux linked by the winding. Because of the double salience construction of the SR motor and the magnetic saturation effects, the flux linked in an SRM phase varies as a function of rotor position θ and the phase current. Equation (1) can be expanded as

(3)


where I∂

∂ψis defined as L (θ,I) , the instantaneous inductance, and term

dt

dθθψ

∂∂

is the

instantaneous back e.m.f. The SRM can be described by a convex function that only depends on rotor

position θ and currents in the n phases I = (I1, I2 , …, In)t . This function is the co-

energy W (I, θ). In a similar manner, the function energy W (ψ,θ), whose variables are the fluxes of n phases ψ = (ψ1, ψ2,…, ψn)

t and the rotor position, also permits to describe the SRM. Whatever are the vectors ψ �and I, the functions of co-energy and energy, verify the following inequality:

( ) ( ) IWIWtψθψθ ≥+ ,, (4)

Thanks to its double saliency, the SRM can have a variation of the magnetic energy and therefore torque production. The partial derivative of the energy function in relation to the rotor position gives the machine torque T:

( ) ( ).,,...,,,...,11

θθψθ ψψψψ

nnT

∂∂= (5)

Applying this relation to the 6/4 SRM, one has

( ) ( ).,,,,,,321321

θθψθ ψψψψψψ ∂

∂=T (6)

When one energizes one phase, the torque appears so that the rotor evolves in the direction where the inductance increases. Therefore, the torque will be in direction of the nearest aligned position. The Permeance Network Method (PNM) This method is based on the concept of the subdivision for magnetic circuit (M.C), of an electromechanical system into a sufficient number of elements, called tubes of flux, each tube present the main field lines of, and this is descibed by a permeance which depends on material dimension and its saturation state. The P.N. M has been applied, with success on different electric machine types. There results are in agreement with experimental when using a simple network taking into account the movement and the saturation.

The modeling asynchronous machine, both in transient and permanent regimes with saturation has been developed by B.Bekkouche [6]. The P.N.M also permits the 3D modelisation, which was already the object of the study applied to the transformer and alternators of cars [7]. Basic Concept The equivalent magnetic circuit The stator and the rotor of an SRM both have salient poles and the magnetic field can be subdivided in two components:


� The transverse component transits the air gap and forms the mutual permeance between the stator and the rotor.

� The tangent component of the field which describes the permeance of slot shown in Fig. 3.

Figure 3: Defining of magnetic field in teeth and slots

Figure 4: Defining path of magnetic field

For the electric machine, the set of the inductive contours are closed, therefore creating a dependence of each contour with the another. It permits also to establish the equivalent magnetic circuit (E.M.C) of the machine [8]. Each slots of the equivalent magnetic circuit (E.M.C) are located on the surfaces delimiting zones of the M.C. A mesh of the stator or the rotor of the equivalent circuit correspond to the contour of a tooth will be composed of zones belonging to this mesh, as well as of a voltage source represent a slot magnétomotive force (MMF). The mesh of the air gap materializes the different paths browsed by the field of the air gap. Fig. 4 shows the differents path of the field which can be represented by reluctance whose value depends on the geometry of the air gap and the position stator/rotor.


Figure 5: Defining permeance network of magnetic circuit

Fig.5 shows the final configuration of the E.M.C can be gotten with a fine subdivision of portions of the M.C in under zone as well as a choice of path of the field in the air gap.

This approach permits a better meshing of the domain study, so we write a different equation based on electrical circuit theory like mesh law. In order to reduce the time of calculation, we must limit the number of mesh, what requires the regrouping of the under zones and therefore the obtaining for each of her only reluctance, as well as the limitation of the interaction of a tooth with these simplifications one gets a relatively simple equivalent circuit.

Calculation of Parameters of the Equivalent Circuit Our E. M.C is subdivided into three parts: magnetics-iron, slots (statorique and rotorique) and air gap. In this paper we will limit the calculation on the reluctance of slot rotorique and air gap. Stator slot and parts magnetics-iron, are presented in references [9, 10]. a) Magnetics-Iron Permeance The reluctance of the magnetic parts for motor are calculated from the following formula

(7)

where l , h, L, kf and � are average length of flux, the width of the magnetic part, motor length, foisonnement coefficient and permeability respectively.

µLhlR

k f

=


b) Slots Permeance Let's consider the geometric shape of the rotor represented by Fig- 6 and it is for example the case of a switched reluctance motor 6/4.

The elementary permeance of interpolaire zone is given by:

(8)

where µ0 is air permeability.

After integration, the expression (8) becomes:

( ) ( )( )

−−

= ππ

πµ

/2/22 0

ArcdrraArcdrRLnLkP rrfu

sr (9)

Figure 6: Rotor polar opening permeance Figure 7: Statorique slot where Rr is rotor radius and ra is tree radius.

With similar calcul we get statorique slot permeance.

+=

bh

bhkP

e

e

c

cfs

L30

µ (10)

where hc , he , bc and be are shown in fig 7. c) Airgap Permeance The geometric shape of the air gap with double tooth in an electric machine is very complex. In order to makes difficult the calculation of airgap permeance particularly when the rotor is in movement and therefore the variation of the airgap geometry. In order to facilitate the calculation during the movement, we subdivised our calcul in two steps:

bc

hc

he

be


We transform the real airgap of the machine with lenght e in an equivalent airgap of CARTER (fig-8a) with lenght ec, this is given by:

ec=kcs kcr e (11)

where kcs, kcr are coefficients of carter statorique and rotorique respectively (appendix I).

Figure 5: The Transformation process of the air gap.

2-Calculation of the triangular slot permeance (fig-8-b) Pt is expressed by

Pt = Pc – Pd (12)

where Permeance Pc and Pd are Carter and tooth, respectively.

The air gap in his new configuration will give some various geometric shapes on all his length. These shapes depend on the number of slot situated on every side of airgap, of their relative position and their opening width. In order to facilitate calculation of airgap permeance, we subdivise his length in several way says angular recoverements as shown in Fig. 9.

Figure 9: Calcul of airgap permeance.


For a machine 6/4 airgaps include ten tooth step on all sides so ten recoverements delimited by axes of slot. Every recoverement include one or a several zones whose permeance is expressed by

( )xCC

dxCLRkdp

xx

mf

+=/

0µ (13)

where Rm is means radius, C and Cx are constants (Appendix II). The total perméance is given by

( )

( )

( )

( )∫ +=∫++

=j

j xx

mfj

jt

xCCdx

CLRkdpP

α

α

α

αµ

10

1 / (12)

where (j) and (j+1) are limits of integration counted from the origin of the studied domain as shown in Fig. 6. This equation is used to calculate the permeance of each zone for differents values of the constant C and Cx with the exception of the rectangular zone, his permeance is given by

( ) ( )( )103 +−= jjeLRkP mf ααµ (13)

The completed circuit model for example SRM 8/6 is proposed here is shown in Fig.10.

Figure 10: The magnetic circuit model of SRM 8/6

The application of the mesh method permits us to get the flux of each branch.


Results and Discussion Our simulation are summarized in the organigramme presented in appendix III. Studied of the reluctance, the current and the torque along the air gap The curve in Fig. 11 shows the variation of the réluctance versus positions, in the first positions, the reluctance of main airgap is maximal and it decreases to the following positions, to beyond the length of this path, the reluctance keeps a minimal value then constant.

Figure 11: Characteristic of the main reluctance

Consequently, the current as shown in Fig. 12 establish quickly during the first positions where the e.m.f of auto - induction is small then reached constant value until the instant of his extinction.

Figure 12: Caracteristic of phase currents


The torque of the Fig. 13. increases quickly in the zone which has a strong variation of reluctance (zone I) in order to reach his maximum, then he decreases until the position of extinction (zone II) from which it decreases enough quickly. The zone III corresponds at the instant when a tooth rotorique is inside a tooth statorique.

Figure 13: Torque caracteristic at rotor touth The influence of the airgap configuration The influence of the airgap configuration will make from the variation of teeth widths statoriques and rotoriques.

The widening of tooth rotorique involve the shortening of the window between extremities of teeth, from where a reduction of the path browsed by the flux and values of reluctances decrease shown in Fig. 14-a. Therefore the values of currents and torques decrease as shown in Fig. 14-b and in Fig. 14 c.

a b


Figure 14: SRM caracteristics with different wide of teeth

Figure 15: Reluctance and curent as function of position with different width of airgap

Conclusion While this new machine exhibits a potential performance advantage over existing technology, it cannot be seriously considered until its performance is well characterized. The first step to develop an accurate nonlinear magnetic model that can quickly evaluate candidate designs. This paper provides such a model.

The P.N.M enables us to caracterize SRM with computational time was reduces. All results and method presented herein can be applied to any SRM; it permits to know the flux shape of the sensitive parts of the machine while taking account of the real geometry, the movement and the saturation. The application of this method presented in reference [3] gave some satisfactory results.

c d

a b


The simplicity of the permeance network for the SRM facilitates the use of this method. According to the simulation, we note that the essential parameters of dimensionnement and optimization are the geometry of the airgap and instants of phase commutation. The other parameters as the length, the diameter of the machine, the applied voltage…etc, are well on influencing but affects the order of size of the torque mainly and therefore the out power [4]. We conclude that for every tooth step statorique corresponds a tooth step rotorique that gives the best torque in the shape and quality. Our perspectives are to establish a complete algorithm for SRM redimensionnement in order to especially for torque ripple minimisation. References [1] M. Rodrigues, P. J. Costa Branco and W. Suemitsu, “Fuzzy logic control for a

switched reluctance motor,” Proc.ISIE’97, vol. 2, pp.527-531, 1997. [2] L. Henriques, L. Rolim, W. Suemitsu, “Torque Ripple Minimisation in

switched reluctance motor Drive by Neuro-Fuzzy Compensation,” IEEE Trans. Magn., vol. 36, no. 5, Sept.2000.

[3] R. Arumugan, D. A. Lowther, R. Krishnan, and J. F. Lindsay, “ Magnetic field analysis of a switched reluctance motor using a two dimensional finite element method,” IEEE Trans. Magn., vol. May-21, pp.1883-1885, Sep.1985.

[4] J. F. Lindsay , R. Arumugan, and R. Krishnan, “Finite-element analysis characterization of a switched reluctance motorwith multitooth per stator pole,” IEE Proc., pt.B, vol. 133, no. 6, pp. 347-353, Nov. 1986.

[5] M. Omekanda and M. Renglet, “Calculation of the electromagnetic parameters of a switched reluctance motor using an improved FEM-BIEM-application to different models for the torque calculation,” IEEE Trans. Industry Applicat., vol. 33, no. 4, pp. 914-918, July/Aug. 1997.

[6] Bekkouche,“Calculation of Magnetic Field with Permeance network Method in Induction Motors”, Magister Thesis, 1995.

[7] R. Inderka, M. Menne, and R. W. De Doncker, “Generator Operation of a Switched Reluctance Machine Drive for Electric Vehicles,” in Proc. 1999 European Power Electronics and Applications Conf., Lausanne.

[8] J. M. Kokernak and D.A.Torrey,“Magnetic Circuit Model for the Mutually Coupled Switched Reluctance Machine,” IEEE Trans. Magn., vol. 36, no. 2, Mar.2000.


Appendix I

s

ss

scs

bebT

Tk

+−=

*52

r

rr

rcr

bebT

Tk

+−=

*52

where Ts and Tr are step of stator and rotor respectively. bs and br are openned width slot of stator and rotor respectively.

Appendix II

( ) ( ) xhxhxhxh

es

s

ess θθ =⇒= 11

( ) xCCxe x111 +=

( ) xhexees

s

θ+=1

THEN C eC =1

es

sx

hC θ=1


Appendix - III

Begin

Initial position

Current and flux prediction

Correction of current and

flux

Convergence

Calcul of the Torque

Us position

End

Following position

No

No

Yes

Yes

A Switched Reluctance Motors Analyse using Permeance ... · Switched Reluctance Motors (SRM) are fast becoming a popular alternative to Induction Motors (IM) in the variable speed

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