Abstract − In this paper the static (flux linkage and torque) and steady state characteristics (speed, current and torque) of the switched reluctance motor (SRM) is investigated for four magnetic structures with various combinations of stator and rotor poles: 8/6 (basic configuration), 16/12 ('double poles' configuration), 16/14 and 16/18 ('toothed poles' configurations). The static characteristics are calculated for different rotor positions and excitation currents using numerical finite element method (FEM) for magnetostatic field analysis. Dynamic characteristics are calculated using specially developed software routines in package Matlab/Simulink. The purpose of this paper is to show the effects of the magnetic circuit structures on static and steady state characteristics of the SRM. Validation of the computational methods was carried out by comparing the calculated static and steady state characteristics with the ones measured on 8/6 SRM prototype.

Index Terms – Switched reluctance motor (SRM), Characteristics, Modeling, Finite Element Method (FEM).

I. INTRODUCTION

The switched reluctance motor is an electric motor in which torque is produced by the tendency of its rotor to move to a position where the inductance of the excited phase is max-imized. The SRM consists of a salient pole stator with concen-trated excitation windings and a salient pole rotor with no windings or permanent magnets.

The switched reluctance drive (SRD) consists of a SRM, ap-propriate power converter, controller and rotor position-sensing device. Since the first industrial applications of SRD in the 1970s, a variety of SRMs configurations, power conver-ters topology and control methods have been investigated [1]-[6]. Fig. 1. shows the basic structure of the SRD.

Fig. 1. Basic structure of the switched reluctance drive

The advantages of an SRM have been presented in [4], [7]:

low manufacturing cost owing to simple construction; stator windings with short end-turn, thus reducing machine’s length and copper losses; rotor is robust and well-suited for high speed operation; long constant power range; high reliability due to absence of excitation on the rotor; fault tolerance opera-tion due to low cross-coupling effects between phases; cooling

is simple as the major sources of heat are on the stator; low maintenance; resilience to harsh operating condition.

A major disadvantage of SRMs is large torque ripple, which produces acoustic noise and vibrations. However, there are other disadvantages: EMI noise generation; numerous wires between motor and converter; special converter topology; highly nonlinear behavior.

The operation of the SRM is inherently linked with torque pulsations due to double saliency and switched form of supply. An increased interest for SRM in variable speed applications imposes the question about appropriateness of this motor in servo drive applications. In these applications the torque ripple of the SRM has to be limited to minimal possible level. The shape of the torque and level of torque ripple are dependent on the magnetic circuit structure, saturation level and phase cur-rent waveform.

There are two approaches that can be used for torque ripple minimization of the SRD: 1) design approach dealing with the magnetic circuit geome-

try [8]-[11]; 2) control approach dealing with the shape of the phase cur-

rent waveform [12], [13]. To reduce the torque ripple of the SRM, the optimization of

control method and design of motor geometry must be per-formed simultaneously, bearing in mind that actions leading to minimization of torque ripple can be at the expense of some output characteristics of the SRM, e.g. efficiency.

SRMs are typically designed such that the rotor and stator poles are symmetrical about their centerlines and equally spaced around the circumference (regular SRM). Various combinations of rotor and stator poles, with one or more phas-es and machine’s multiplicities, are proposed over the years. Also, research includes irregular spacing and shapes of stator and/or rotor poles [6].

Four regular motor configurations with four phases and two or four stator poles per phase are considered in this paper. Ba-sic 8/6 configuration was designed to reduce fringing flux and to have a more linear inductance profile, so the torque ripple is reduced. Other three configurations were designed during the research of influence of geometrical parameters on the charac-teristics of the SRM, especially on the torque ripple at low speed [14]-[16].

II. SRMS CONFIGURATIONS

The SRMs configurations that have been created during the research of influence of geometrical parameters on the charac-teristics of the SRM, are [14]: • basic SRM 8/6 configuration with 8 stator and 6 rotor

poles (stator pole arc/rotor pole arc: 20°/25°);

The Effects of Magnetic Circuit Geometry on Characteristics of Switched Reluctance Motors

Šemsudin Mašić, Senior Member, IEEE, Senad Smaka, Member, IEEE, Iris Salihbegović, and Mirsad Ćosović

POWER SUPPLY

CURRENT AND

POSITIONSENSING

SWITCHED

RELUCTANCE MOTOR

C O N T R O L L E R

POWER ELECTRONIC CONVERTER

2011 IEEE International Electric Machines & Drives Conference (IEMDC)

978-1-4577-0061-3/11/$26.00 ©2011 IEEE 1427

• SRM 16/12 double poles configuration which is the basic 8/6 geometry with a multiplicity of 2 (stator pole arc/rotor pole arc: 10°/12,5°);

• SRM 16/14 toothed configuration with 8 stator poles, each of them having 2 teeth (tooth arc/total pole arc: 9°/34,7°) and 14 rotor poles (pole arc: 7°);

• SRM 16/18 toothed configuration with 8 stator poles, each of them having 2 teeth (tooth arc/total pole arc: 9°/29°) and 18 rotor poles (pole arc: 7°).

The coils on two opposite stator poles are connected in se-ries, thus forming short-pitch phase winding in the 8/6, 16/14 and 16/18 configurations. In the 16/12 configuration, the coils on four poles spaced in 90° (mechanical) are connected in series to form phase winding.

Fig. 2. shows magnetic circuit structures and main geometry characteristics of the considered SRM configurations. All four machines have equal main dimensions: outside diameter of stator core, rotor diameter and core length. The numbers of turns per phase winding are determined so that maximum use of available interpolar space.

Other geometry parameters are given in the Table I.

Fig. 2. SRM configurations

TABLE I GEOMETRY CHARACTERISTICS

Geometry parameter SRM configuration8/6 16/12 16/14 16/18

Rotor pole pitch [°] 60,0 30,0 25,7 20,0 Stator yoke width c [mm] 17 8,5 13 12Stator outside radius r0 [mm] 120 120 120 120Rotor outside radius rr [mm] 69,8 69,8 69,8 69,8Rotor yoke radius rry [mm] 48,6 48,6 48,6 48,6Bore radius rg [mm] 70,3 70,3 70,3 70,3Shaft radius rsh [mm] 25 25 25 25Core length lstk [mm] 200 200 200 200Turns per phase winding N 440 440 380 400Phase resistance R [Ω] 4,72 3,35 4,36 4,10

III. CHARACTERISTICS CALCULATION A. Static Characteristics

The calculation of the flux linkage of one phase and static torque characteristics as a function of rotor position θ (me-chanical angle) and excitation current i is the first step in per-formance prediction of the SRM. These characteristics can be obtained by two approaches. During design exploration, an analytical approach is usually applied. Analytical models for static characteristics calculation can be derived either from the machine geometry and magnetic theory, from the previously obtained data using numerical methods or directly from the SRM physical information (flux tube, magnetic equivalent circuits models). The second approach, commonly used during the final validation stage of the design process, is to use 2D FEM or 3D FEM for magnetostatic field analysis to derive the phase flux linkage and static torque characteristics in the form of a lookup tables ψ(θ,i) and T(θ,i). B. Dynamic Characteristics

The mathematical model that enables prediction of the dy-namic and steady state characteristics of the SRMs is de-scribed by the following set of equations:

d θ

dψ

± = ⋅ + k k kk

( ,i )V R i

t (1)

d θdω = −∑

phn

k k k Lk=1

J T ( ,i ) Tt

(2)

dθd

= ωt

(3)

θ 1 ε= θ + − ⋅k (k ) . (4)

The voltage V has sign + if switch is "on" and sign − if phase switch is "off". Angular displacement between adjacent phases or step angle is ε = 2π/(Nr⋅nph), where Nr is number of rotor poles. J is inertia torque, TL is load torque and ω is me-chanical angular speed.

Mathematical model enables computation of instantaneous values of phase and total currents and instantaneous value of phase and total torque and speed. The instantaneous total elec-tromagnetic torque is obtained by summing instantaneous phase torques for all nph phases at instantaneous angular posi-tion.

The values of torque ripple and efficiency are calculated as:

( ) ( )

(%) 100−

= ⋅i ir

avg

max T min TT

T (5)

⋅ω

η =⋅

avg avg

avg avg

TV I

, (6)

where are: max(Ti) and min(Ti) – maximum and minimum values of instantaneous torque, Tavg – average electromagnetic torque, ωavg – average speed, Iavg – average total current, Vavg – average voltage.

c

s

βs βr

r0 rr rry rsh

rg 2

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IV. CALCULATED STATIC CHARACTERISTICS

Figs. 3-6 show flux lines at unaligned position, flux linkage and static torque characteristics obtained with 2D FEM for all considered SRMs. Rotor positions are defined by electrical degrees (θel = Nr⋅θ ). Angle θel = 0° represents unaligned posi-tion and angle θel = 180° depicts aligned position.

At unaligned position, the phase inductance has minimum value due to large airgap. Magnetic saturation is unlikely to occur at this position and hence the flux linkage shows linear behavior until a pair of stator and rotor poles exhibits overlap.

In the partial and fully overlap regions, magnetic field den-sity tends to saturate tip and body of the poles at high current

levels making flux linkage a nonlinear function of position and current.

The area enclosed between flux linkage characteristic for unaligned and aligned position is greatest for the SRM 8/6. As a consequence of smaller area between these two characteris-tics other machines have reduced static torques. Minimum torque per ampere has SRM 16/18. Analysis of static torque profiles can provide indication of the torque ripple in dynamic states. If mean value of the static torque is equal to 35,8 Nm for all considered SRMs, the lowest static torque ripple has SRM 16/18 (38,9 %). SRM 8/6 has static torque ripple equal to 53,4 %, SRM 16/12 has 61,9 % and static torque ripple is highest for SRM 16/14 (70,2 %).

a. Flux lines at the unaligned position b. Flux linkage characteristics c. Static torque profiles

Fig. 3. Configuration 8/6 a. Flux lines at the unaligned position b. Flux linkage characteristics c. Static torque profiles

Fig. 4. Configuration 16/12

a. Flux lines at the unaligned position b. Flux linkage characteristics c. Static torque profiles

Fig. 5. Configuration 16/14

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a. Flux lines at the unaligned position b. Flux linkage characteristics c. Static torque profiles

Fig. 6. Configuration 16/18

V. DYNAMIC SIMULATIONS

Dynamic simulation is carried out based on the model men-tioned in section III. The equations (1)-(4) are solved simulta-neously using software Matlab/Simulink. A. Operating Conditions

Operating conditions for all SRMs configurations are listed in the Table II and they are briefly described here:

• equal load torque on the shaft; • all machines are subjected to single-pulse operation

mode with the supply voltage variation between posi-tive and negative DC bus voltage (hard switching);

• the turn-on angle θon is in the unaligned (minimum in-ductance) position and the turn-off angles θoff, ex-pressed in mechanical degrees, are determined in order to minimize torque ripple.

TABLE II

OPERATING CONDITIONS

Operating parameter SRM configuration 8/6 16/12 16/14 16/18

Supply Voltage [V] 145 135 220 240Load torque [Nm] 35,8 35,8 35,8 35,8Step angle ε [°] 15 7,5 6,43 5 Switch-on angle θon [°] 0 0 0 0 Switch-off angle θoff [°] 19,66 11,19 11,66 8,10

B. Simulation Results

Figs. 7-9 show the results of dynamic simulations. Instanta-neous total electromagnetic torque and speed waveforms in steady state are showed.

Computed values of average total torque, speed, current, torque ripple and efficiency in steady state for all considered SRMs are shown in the Table III.

TABLE III RESULTS COMPARISON

Results SRM configuration 8/6 16/12 16/14 16/18

Average torque [Nm] 35,9 35,8 35,8 35,9Average speed [rpm] 149,8 150,4 150,2 150,0Torque ripple [%] 22,5 32,6 40,6 16,6Efficiency [%] 71,7 62,8 65,7 55,0

Fig. 7. Total torque waveforms for SRM 8/6 and SRM 16/12

Fig. 8. Total torque waveforms for SRM 16/14 and SRM 16/18

Fig. 9. Speed waveforms for all considered SRMs

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Machines with two teeth per stator pole have lower effi-ciency. The shape of the stator poles restricted the winding area available. Due to the lower number of turns per phase winding, these SRMs requires higher currents to produce the same magneto motive force as SRM 8/6, which in turn can increase the copper losses. Also, an increase in the number of teeth per stator pole requires a commensurate increase in the resolution of the position sensing and an increase in switching frequency resulted in the higher core and switching losses.

As static torque profiles analysis indicates, configuration SRM 16/18 has lower torque ripple in steady states.

VI. EXPERIMENTAL VERIFICATION

An experiment was designed such that the flux linkage and

static torque curves of the tested SRM could be measured at discrete rotor positions from the unaligned position to the aligned position and for several excitation currents. Also, ex-perimental setup enables controlled loading of the machine and storing of phase currents and torque waveforms in steady state. All experiments are conducted on the basic SRM 8/6 configuration with rated data 750 W, 200 rpm. The photography of the SRM with testing and measuring equipment is shown on Fig. 10.

Fig. 10. Experimental setup

The SRM was loaded with magnetic powder brake MO-BAC, 6 kW, 0-400 Nm. The torque and speed are obtained from a shaft mounted brushless strain gauge transducer HBM-TSN200, 200 Nm, 6000 rpm. The phase currents were meas-ured by compensated Hall-effect current transducers and stored on a digital oscilloscope. A. Static Characteristics

The flux linkage characteristics and static torque characte-ristic of the SRM 8/6 was obtained over an angular displace-ment in range from 0° to 30° (mechanical) in 3° increments and for different DC excitation currents. The rotor was blocked at different positions with a large moment from mag-netic powder brake for a given current. The measurement of magnetization curves is realized using the direct method, i.e. the inductance of the machine’s bifilar winding is measured using ballistic method and flux linkage is calculated on the

basis of the measured excitation current and inductance. Static torque is obtained from a shaft mounted strain-gauge torque transducer.

Fig. 11. shows comparison between calculated and meas-ured flux linkage characteristics for six different rotor posi-tions expressed in electrical degrees. Fig. 12. shows compari-son between calculated and measured static torques for four excitation currents.

Fig. 11. Calculated and measured flux linkage characteristics ψ(θ,i)

Fig. 12. Calculated and measured static torque characteristics T(θ,i)

It should be noted very good match between calculated and measured static characteristics. B. Steady State Characteristics

The steady state characteristics of phase current and total torque are measured using the equipment shown in Fig. 10. Comparison between measured and calculated steady state characteristics of phase current and total torque are shown in Figs. 13-16. These characteristics are obtained for V = 105 V, TL = 36 Nm, J = 0,57 kgm2, n = 100 rpm (steady state average speed), θon = 0°, θoff = 30°.

Again, the results of the calculations show good match with the measurement. Developed model of the SRM can be used

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for calculation of steady state and dynamic characteristics of the machines with different magnetic circuit structures.

Fig. 13. Measured steady state phase current; current scale: 1,25 V/div = 2 A/div, time scale: 20 ms/div

Fig. 14. Calculated steady state phase current

Fig. 15. Measured total steady state torque; torque scale: 1,00 V/div = 20 Nm/div, time scale: 20 ms/div

Fig. 16. Calculated total steady state torque

VII. CONCLUSIONS

The influence of the magnetic circuit structure, including special toothed topology of stator poles, on the characteristics of the four phases SRM is investigated in this paper. The static characteristics are calculated by using 2D FEM. The steady state characteristics are calculated by using described ma-chine’s model.

Multiple-tooth-per-stator pole structures has relatively low inductance ratio due to increased unaligned inductance and therefore limited static torque.

SRM with two teeth per stator pole also has limited space for winding placement, which in turn may lead to a higher phase currents for the same magneto motive force production as in single-tooth-per pole designs. As a consequence mul-tiple-tooth machines can have higher copper losses. Increased number of poles also leads to higher core losses.

Specific toothed topology of the stator poles offer the possi-bility to improve the SRM performances regarding torque rip-ple.

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Time (second)

Time (second)

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