Influence of Geometrical Parameters
of Switched Reluctance Motor on Electromagnetic Torque
Krzysztof Bieńkowski, Jan Szczypior, Bogdan Bucki, Adam Biernat, Adam Rogalski
Institute of Electrical Machines, Warsaw University of Technology, Poland
Nowowiejska 20 A, 00-661 Warsaw, Poland, phone: (+4822) 660-73-35,
email: K.Bienkowski@ime.pw.edu.pl
Abstract - Researches on torque characteristic of switched
reluctance motor using Finite Elements Method are presented
in this paper. Self-prepared additional software makes
possible to find optimal set of constructional parameters of
the motor. Characteristics of torque versus rotor position are
presented for different values of some constructional
parameters. Influence of internal diameter of stator,
stator’s and rotor’s pole breadth, rotor’s pole height
and air gap on electromagnetic torque and torque
ripple coefficient are presented.
An Effective design of SRM requires determination of
the set of geometrical parameters, which enable producing
maximum electromagnetic torque. For that reason only
development of CAD procedures allows to utilize all
anticipated advantages of these motors.
The aim of presented researches is to formulate
remarks concerning designing process and methods of
determining these parameters leading to optimal SRM
construction.
I. INTRODUCTION
II. FEM MODEL OF SWITCHED RELUCTANCE MOTOR
Switched Reluctance Motor is an electric machine,
which produces torque in consequence of variability of
reluctance for magnetic flux excited by currents flowing in
the stator windings. Reluctance changes during rotation of
the rotor for the reason of properly formed stator and rotor
cores [1].
To the effective operation of SR Motor commutator
should to precisely determine position of the rotor. On this
signal it must calculate the time of connection and
disconnection of each phase band to source voltage. These
moments are not constant but depend of the rotational
speed and load of the machine. Only development of
power electronics, CAD systems and microprocessor
control methods caused fully using of advantages of those
motors [2].
Concentrated coils are placed around the stator’s poles.
Coils placed on reciprocal poles are series connected and
creating phase band. Each band of the stator winding is
connected to supply by transistor switches. Current in each
band should flow at such positions of rotor, to assure
continuous rotation with minimum torque ripples. It can be
archived through such reconnecting phase’s bands which
causing current flowing during increasing inductance of
phase band. If this condition is not fulfilled some breaking
torque will appear.
Electromagnetic torque depends from changes of
reluctance caused by the rotor rotation. Reluctance as
function of the rotor position depends on flux excitation –
current linkage of the winding, (because of non-linear
characteristic of magnetisation of core) and on
constructional parameters determining magnetic circuit. At
given current, number of turns, B versus H curve and
selected variant of stator construction, the reluctance
depends on rotor pole breadth, it heights and thickness of
air gap.
Available commercial FEM processors deliver
possibility for modelling electromechanical converter with
any construction [3], but pre-processing of the model is
arduous and time-consuming. Result of single solution is
distribution of magnetic potential in the whole area of the
model.
To solve problem of determining torque characteristic
special software based on PC-OPERA package was
created. In result of the software operation the set of torque
for different positions of the rotor ware obtained. The
structure chart of this software is presented on Fig.1.
Edition of data file
Preprocesing of the first task
Solution of the task
Preprocesing
of the next task
Torque calculations,
saving of the value
Checking of the
finish conditions
Not filled
Filled
Finish of the calculations,
preparation of the results,
export to Matlab environent
Figure 1. Structure diagram of the software.
The software was in detail described in [6].
The problem of determination of static torque
characteristic versus angular position of the rotor amounts
to calculation of the set of values of electromagnetic torque
for given set of angular position of the rotor. This
characteristic of SRM is periodic due to periodicity of the
magnetic circuit structure, so the set of angular position of
the rotor could be reduced to one pole pith. If the stator and
rotor poles are symmetrical the range of calculations could
be reduced even to the half of pole pith.
Single value of torque, at fixed position of rotor, can be
calculated by integrating of Maxwell stress tensor in the air
gap region. Simplifying assumption of constant magnetic
flux distribution along the length of the core leads to
solution of the two-dimensional magnetostatic problem.
Therefore the expression on electromagnetic torque has
form:
2π
T =r l∫
2
0
Where: r
l
Bn,Bt
Bn Bt
µ0
dα
(1)
– radius of air gap circle,
– length of the core,
– components of flux density.
In the cross section of the motor model, there are
following regions: stator and rotor cores, stator windings
(coils), air regions.
The motor following parameters:
Number of stator/rotor pole pairs ps/pr = 3/2
External diameter of stator core dse = 80 mm
Internal diameter stator
d = 32 mm
Effective length core
lFe = 100 mm
Breadth of stator pole
bps = 8 mm
Breadth of rotor pole
bpr, = 9 mm
Height of rotor pole
hpr, = 5 mm
Air gap
δ = 0. 3 mm
Researches of influence of geometrical parameters on
electromagnetic torque were carried out with changing
values of breadth and height of rotor pole and the air gap.
During changing of one parameter the others were
changing too, assuming the constant copper loses in stator
windings, but external diameter of stator core was still
constant.
Family of torque characteristics at different values of
breadth of rotor pole is presented on Fig. 3. Torque values
between computed points (marked with suitable marks)
were obtained by spline interpolation method.
1,6
Beside of produced torque, the torque ripples
coefficient is very important parameter to assess the
construction. This coefficient is defined as:
t rpl =
Where: Tmax
Tmin
Tav
Tmax − Tmin
2Tav
bpr =9 mm
1,4
1,2
(2)
1
– the maximum value of torque,
– the minimum value of torque,
– the average value of torque.
bpr = 8 mm
0,8
0,6
0,4
bpr = 7 mm
III. RESULTS
The construction of the first researched motor is
schematically represented on Fig. 2.
Stator core
Shaft
Rotor
core
Air
regions
Stator
winding
Figure 2. Cross section of researched motor.
Torque T [Nm]
II.1 The static torque characteristics and torque
ripple coefficient
0,2
0
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Rotor position angle α [deg]
Figure 3. Family of torque characteristics for different pole
breadth of rotor.
Change of breadth of rotor pole, in examined range,
practically does not influence on the maximum value of
torque. Increasing of breadth of rotor pole causes only
changes of maximum torque rotor position. Worth
mentioning is fact of existence of optimal value of breadth
of rotor pole for which the average torque has maximum
value.
The average torque and torque ripples coefficient is
calculated for 30° operational range of the phase band
(between -35° and -5°), with the assumptions of constant
phase current, what is fulfilled at motor operation with low
speed (under the base speed).
The average torque and torque ripple coefficient for
different values of breadth of rotor pole is presented on
Fig. 4.
The smallest value is defined by mechanical conditions
(bearing clearances, tolerance of brackets, shaft bending
and stretching deformations of stator core). The torque
ripples can be slightly reduced by growing air gap (Fig. 7).
1,4
2
Tav
1,6
1
0,8
1,4
δ = 0,3 [mm]
0,6
1,2
1
trpl
0,4
0,8
δ = 0,4 [mm]
0,2
0,6
0
0,4
6
7
8
9
0,2
10
0
bpr [mm]
-45
-40
-35
The influence of height of rotor pole on the maximum
torque is shoved on Fig. 5. There are presented the
maximum torque value because height of rotor pole
increasing does not influence rotor position angle with
maximum of torque.
-30
-25
-20
-15
-10
-5
0
Rotor position angle α [°]
Figure 4. Average torque and torque ripple coefficient for
different values of breadth of rotor.
Figure. 6. Family of torque characteristics for different air
gap.
1,5
Tav
1,2
1,6
1,4
0,9
T [Nm], trpl [-]
1,2
1
T [Nm]
1,8
T [Nm]
Tav [Nm], trpl [-]
1,2
δ = 0,2 [mm]
0,8
0,6
0,6
trpl
0,3
0
0,4
0,1
0,2
0,2
0,3
0,4
0,5
bpr [mm]
0
0
1
2
3
4
5
6
hpr [mm]
Figure 5. Maximum torque for different values of height of rotor
pole.
Height of pole increasing from 0 to 5δ caused torque
increasing from 0 to 1.3 Nm but further torque increasing
is very slow. Enlarging the height of rotor pole above
10÷15δ is practically not justified.
The family of torque characteristics at different values
of air gap is presented on Fig. 6.In SRM the air gap should
be as small as possible.
Figure. 7. Average torque nad torque ripple coefficient for
different values of air gap.
The second researched model has the same structure as
presented on Fig. 2. and following geometrical parameters:
Number of stator/rotor pole pairs ps/pr = 3/2
External diameter of stator core dse = 300 mm
Internal diameter of stator
d = 180 mm
Effective length of core
lFe = 150 mm
Breadth of stator pole
bps = 49 mm
Breadth of rotor pole
bpr, = 56 mm
Height of rotor pole
hpr, = 20 mm
Air gap
δ = 1 mm
The family of torque characteristics at different values
of internal stator diameter d is presented on Fig.8.
0,6
0,5
250
0,4
trpl [-]
300
150
T [Nm]
200
0,3
0,2
0,1
100
d
0
150
190
50
160
170
0
-50
-40
-30
-20
Rotor position angle α [°]
-10
0
180
190
d [mm]
180
200
170
Figure. 10. Torque ripple coefficient for different values of
internal diameter of stator.
160
The family of torque characteristics at different values
of breadth of rotor pole is presented on Fig. 11. Breadth of
stator pole, in examined range, considerably influence on
the maximum value of torque and shape of the
characteristic. The torque ripples can be slightly reduced
by growing breadth of stator pole. At bps = 49 mm the
average torque has maximum value.
Figure. 8. Family of torque characteristics for different internal
stator diameter.
The change of internal stator diameter d, in examined
range, influence on the maximum value of torque and
shape of the characteristic. At d = 180 mm the average
torque has maximum value, assuming the constant copper
loses in stator windings. Torque ripple coefficient can be
slightly reduced by growing internal diameter of stator.
The average torque and torque ripple coefficient for
different values of breadth of rotor pole are presented on
Fig. 9. and 10.
300
Bps = 42 mm
Bps = 49 mm
250
200
200
T [Nm]
150
Tav [Nm]
Bps = 57 mm
190
100
180
50
170
0
-50
160
150
150
-40
-30
-20
-10
0
Rotor position angle α [deg]
Figure.11. Family of torque characteristics for diferent breadth of
stator pole.
160
170
180
190
200
d [mm]
Figure. 9. Average torque for different values of internal diameter
of stator.
The average torque and torque ripple coefficient for
different values of breadth of stator pole is presented on
Fig. 12 and 13.
Set of motor’s parameters which allow to produce high
torque is not suitable in respect of torque ripple. Criterion
of maximum efficiency is contradicted to criterion of
minimum torque ripples.
The influence of rotor’s pole height and air gap on
electromagnetic torque are presented for switched
reluctance motor with dse = 80 mm. Enlarging the height of
rotor pole above 10÷15δ is practically not justified. To
archive high torque the air gap should be as small as
possible for mechanical reasons.
The influence of internal diameter of stator, and
breadth of stator pole on electromagnetic torque are
presented for switched reluctance motor with dse = 300
mm. Breadth of stator pole significantly influence on the
shape of torque versus angular position of rotor.
200
190
Tav [Nm]
180
170
160
150
40
45
50
55
60
bps [mm]
0,6
Acknowledgement
This research project is supported by The State
Committee for Scientific Research of Poland within years
2003-2006.
0,5
REFERENCES
Figure. 12. Average torque for different breadth of stator pole.
trpl [-]
0,4
0,3
0,2
0,1
0
40
45
50
55
60
bps [mm]
Figure. 13. Torque ripple coefficient for different breadth of
stator pole.
IV. CONCLUSIONS
Worked out software makes possible to choose set of
constructional parameters of the SRM which produce
maximum of electromagnetic torque or minimum of torque
ripple coefficient at the constant copper losses.
Breadth of rotor pole significantly influence on the
average torque and torque ripple therefore, the shape of
rotor core should focussing designer’s attention.
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Control, 1993; Magna Physics Publishing
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Press LLC, London, 2001
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Bruce A. Kalan Design Optimization of a Switched
Reluctance Motor by Electromagnetic and Thermal FiniteElement Analysis, 3334 IEEE Transations on Magnetics,
VOL. 39, NO. 5, SEPTEMBER 2003
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SRM for Electric Vehicle Propulsion”, ICEM proceedings
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on electromagnetic torque”, Berichte und Informationen
HTW Dresden 1/2002 ISSN1433-4135
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