Reduction Method of Current RMS Value,
DC Current Ripple, and Radial Force Ripple for
SRM based on Mathematical Model of
Magnetization Characteristic
Takahiro Kumagai
Dept. of Science of Technology Innovation
Nagaoka University of Technology
Nagaoka, Japan
kumagai_t1125@stn.nagaokaut.ac.jp

Keisuke Kusaka
Dept. of Electrical, Electronics, andInformation Engineering
Nagaoka University of Technology
Nagaoka, Japan
kusaka@vos.nagaokaut.ac.jp

Jun-ichi Itoh
Dept. of Science of Technology Innovation
Nagaoka University of Technology
Nagaoka, Japan
itoh@vos.nagaokaut.ac.jp

Abstract— This paper proposes a motor current RMS value,
DC current ripple, and radial force ripple reduction method
under zero torque ripple for switched reluctance motor (SRM).
In the proposed method, the current waveform to achieve
constant torque is derived based on a mathematical model ofmagnetization characteristic. In order to optimize multiple
objective functions considering a trade-off relationship, theweight function is introduced. In particular, quantitativeevaluation functions are derived from the mathematical model
for avoidance of time-consuming numerical process. In the
proposed method, it is possible to derive the desired current
waveform for the motor current RMS value, the DC current
ripple, and the radial force ripple by setting appropriate weights.
The reduction of the motor current RMS value, DC current
ripple, and radial force ripple by 18%, 45%, and 37%,
respectively, are confirmed.
Keywords—Switched reluctance motor, Instantaneous current
control, Multi-objective optimizationI.

INTRODUCTION

Switched Reluctance Motor (SRM) has potential for theapplication of hybrid electric vehicle (HEV) and electricvehicle (EV) thanks to its rare earth-element-free and low
manufacturing cost, and suitability for high speed. In SRM,
continuous rotation is achieved by excitation of each phasestator winding at an appropriate timing. However, a large
torque ripple occurs during the switching phase interval due
to insufficient torque. Therefore, the excitation of two-phasestator windings during the switching phase interval is applied
to reduce the torque ripple [1]-[10]. In those methods, several
current waveforms which generate the constant torque are
obtained. The current waveform which generates the constanttorque is referred as the constant-torque current waveform in
Ref.[9]. Nevertheless, this method increases the current rootmean-square (RMS) value because the torque is generated
during the interval of small torque per current value. In
addition, DC current ripple occurs in the input of the inverter
because the magnetic energy oscillation occurs between loadand power supply even if the output torque is constant.
Furthermore, the radial force ripple occurs in SRM due to the

978-1-7281-3153-5/19/$31.00 ©2019 IEEE

excitation during the alignment state where the radial force percurrent value is large. In order to solve the above problems,
the constant-torque current waveform should be optimized inconsideration of all above criteria.
In general, the desired constant-torque current waveformis selected from several waveforms derived by the variation of
the parameters value, determining the shape of the current
waveform [3], [4]. However, the optimization is timeconsuming and needs to be conducted whenever the motor ischanged. In Ref.[5]-[8] , the weight factors for the voltage
limitation and the current reduction are derived. However, theDC current ripple and the radial force ripple are not consideredin the optimization of the constant-torque current waveform.
In addition, the computation requires time-consuming
numerical process due to the large amounts of data of flux
linkage and torque. Meanwhile, the optimization method forthe motor current RMS value reduction under zero torqueripple has also been proposed [9]. However, the DC current
ripple and the radial force ripple increase when the motor
current RMS reduction method is applied. In other words, thedesign method considering all above criteria has not been
considered.
This paper proposes a multi-objective optimization basedon a mathematical model for the constant-torque current
waveform considering the motor current RMS value, the DC
current ripple, and the radial force ripple. In order to conduct
the multi-objective optimization for the current waveform, theweight functions are introduced for each criteria. The
originality of this paper is that the desired current waveformconsidering the motor current RMS value, the DC current
ripple, and the radial force is achieved with setting appropriate
weights. In particular, the relationship between the parameters
value determining the shape of the current waveform and allabove criteria is clarified. In addition, the optimization isconducted without any time-consuming numerical process byderiving quantitative evaluation functions of the motor current
RMS value, the DC current ripple, and the radial force basedon the mathematical model of the magnetization characteristic.

TORQUE RIPPLE REDUCTION METHOD

(i, m )  u (i)  f (e )( a (i)   u (i))(1)

where f(e) is the periodic function which represents theflux linkage characteristic between aligned state andunaligned state. The torque T is generated by the change ofmagnetic co-energy Wc depending on the saliency between thestator and rotor which is caused by the change of the rotor
angle m.
T (i, m ) f (e )( a (i)  u (i))di ' m 0i(2)Note that Wc equals to the surface areas surrounded by themagnetization curve and the current axis in the magnetization
characteristic.
Fig. 2 depicts the T-i-m characteristic obtained by
calculating the formula T(i,m) at the values of the current
from 20A to 250A and the rotor position. It is observed from
Fig. 2 that the required current value for any torque at any
rotor position can be calculated.
B. Torque sharing function (TSF)
Fig. 3 shows the example of the torque sharing function(TSF) fTx(m). In this paper, the TSF fTx(m) which represents
the sharing rate of each phase torque according to the rotor
position, is used to derive the constant-torque current
waveforms. In order to make the sum of each phase’s torqueresult in the constant torque, fTx becomes 1 during one-phaseconduction period, whereas the sum of two-phase’s fTx equals
to 1 during two-phase conduction period (the period lap in the
figure). It is observed that various functions can be selected
for the torque sharing function. In this paper, the quadratic
function is utilized to avoid a steep rise. Therefore, the TSF isexpressed as in (3),fTx ( m ) 2( m   0 ) 2
 0   m   0  0.5lap
lap 21  f ( )   0.5    
Tx 1
m
0
lap
m
f0

1
 f 0   m   fc

1  fTx 1 ( m )  fc   m   fc  0.5 lap 2(   ) 2mc
 fc  0.5lap   m   c2
  lap0
otherwise

(3)

where lap represents the two-phase conduction period,and f0 indicates the initial angle of the one-phase conduction
period. In the SRM with number of phases m, number ofstator pole Ns, and number of rotor poles Nr structure, since
the conduction period of the adjacent phase shifts the angleof 2/mNr rad in the mechanical angle, the turn-on angle 0,

MOTOR PARAMETERS OF SRM.
SymbolValue

Rated mechanical power

Pm

5.5kW

Maximum speed

n12000r/min

Maximum torque

Tn

9.3Nm

Input voltage

Vdc

48V

Number of poles

Ns/Nr

12/18

Winding resistanceR

0.011

Number of turns

N

12turns

0.12Interlinkage flux [Wb]Fig. 1 shows the magnetization characteristic of theevaluated SRM. The flux linkage changes periodicallybetween the aligned flux linkage a and the unaligned flux
linkage u, and expressed as in (1) [9],TABLE I.

Aligned position 180 deg (Elec.)

0.09

Line:Eq.(1)Bot:FEM

0.06

every 30deg

0.03
Unaligned position
0 deg (Elec.)

0
0

50

100
150
Current[A]

200

250

Fig. 1. Magnetization characteristic.

Torque [Nm]A. Torque generation
Table I shows the motor parameters of the evaluated SRM.
A three-phase 18S/12P type SRM is chosen as an example toevaluate.

16
14
12
10
8
6
4
2
0

T-e characteristic is expressedwith currenti resolutio of 50A/div
250A

Line:Eq.(2)Bot:FEM

060
120Electrical angle [deg.]

50A

180

Fig. 2. T-i-m characteristic.

Torque sharing function

II.

1.2

U phase1

V phaseW phase

Total

20

30

0.8

0.6
0.4
0.2

0
0

5

10

lap
0 f0

15

mid

lap
fc c25

Position [deg.]

Fig. 3. Torque sharing function.

the ending angle fc of the one-phase conduction period, andthe turn-off angle c are expressed as dependent variables oflap and f0. The torque of each phase is controlled accuratelyby the multiplication of (3) with the torque command T*,
eliminating completely the torque ripple.
In order to optimize the constant-torque current waveform,
the only two dependent variables of lap and f0 are required toC. Motor Current RMS value minimization
The derivation method of the constant-torque current
waveform with minimum motor current RMS value has been
proposed in [9]. In this method, the margin M is defined as
the difference between the maximum current slope, which ispossible to generate by the input voltage, and the constanttorque current slope. The constant-torque current waveform
is applicable when the margin M is positive. In this method,
the torque per current value during the conduction period is
maximized while shortening the conduction period under thecondition that M≧0 is satisfied. However, the DC current
ripple increases because the current slope becomes steep. In
addition, the radial force ripple also increases because thetorque is obtained during the alignment state where the radial
force per current value is large.
III.

M ULTIPLE OBJECTICVE OPTIMIZATIONIn this section, the derivation method of the constanttorque current waveform considering the motor current RMS
value, the DC current ripple, and the radial force ripple
reduction is explained. As explained above, in the derivation
method of the constant-torque current waveform withminimized motor current RMS, the DC current ripple and theradial force ripple increase inversely with the motor current
RMS value reduction. First, the relationship between allabove criteria and two variables of lap and f0 is clarified.
Next, in order to optimize multiple objective functions
considering a trade-off relationship, the weight function is
introduced. For avoidance of time-consuming numerical
process, quantitative evaluation functions of the motor
current RMS value, the DC current ripple, and the radial force
ripple are derived from the mathematical model. Finally, thederivation method of the desired constant-torque current
waveform considering the motor current RMS value, the DC
current ripple, and the radial force ripple is explained.
A. Motor current RMS value
In this subsection, the evaluation function of the motor
current RMS value is derived. The motor current RMS value
affects the motor efficiency. Generally, the motor current
RMS value is evaluated by numerically computing the root
mean square of the constant-torque current waveform in the
TSF methods [5]-[8]. In this paper, the quantitativeevaluation function is mathematically derived based on thecharacteristics of the linear region for the sake of simplicity.
Fig. 4 shows the evaluated results and simulation resultsof motor current RMS value. The constant-torque current
waveform in linear region is expressed as in (4)[9].
i* ( m ) 

2T *La  LufTx ( m )f ( e )
 m

(4)

Motor current RMS[p.u.]

be designed in consideration of the motor current RMS value,
the DC current ripple, and the radial force. In the constanttorque current waveform, lap affects the smoothness of thecurrent during switching phase interval, meanwhile f0 affects
the excitation timing. First, the relationship between all above
criteria and two variables of lap and f0 is clarified. Next, theoptimization process of lap and f0 is explained based on thederived quantitative evaluation functions of all above criteria.

1/max=0.75p.u.

0.95

/max=0.5p.u.

0.9

/max=0.25 p.u.

0.85
0.8

0.75
0

1
2
3
4
Two-phase conduction period lap[deg]5Fig. 4. Evaluated results and simulation results of Motor current RMS
value.

where La is the initial aligned stator inductance and Lu is
the unaligned stator inductance. It is mathematically
impossible to formulate accurately the motor current RMS
value of the constant-torque current waveform; therefore, themotor current RMS value is approximately derived asfollowing. First, it is assumed that lap is sufficiently shorter
than one cycle. The change of the inductance during lap isconstant compared to the change of TSF. In addition, since
the influence of the air gap is dominant in the inductance in
one-phase period, the change of the inductance can beassumed as constant. In the approximation, the average value
during each period of the inductance change is adopted, andthe motor current RMS value is expressed as in (5).
i* RMS 

S1 

S2 
S3 

2T *
La  LuS1  S2  S3
2 N rlap 2 2
f ( f 0 )  f ( 0 )2 mN r  lapf ( fc )  f ( f 0 )lap 2 2
f (c )  f ( fc )

(5)
(6)
(7)
(8)

where S1, S2, and S3 are the integral of the square of thecurrent in the two-phase conduction period (rise), the onephase conduction period, and the two-phase conduction
period (fall), respectively. The motor current RMS value isapproximated only by TSF parameters and motor parameters.
In order to reduce the motor current RMS value, lap should
be short and the torque per current value during the
conduction period is maximized[9].
B. DC current ripple RMS
In this subsection, the evaluation function of the DC
current ripple is derived. The DC current ripple affects thelifespan of the battery. In this paper, the low frequency DC
current ripple generated by the cycle of the phase current,
which is related to the rotation speed of SRM, is focused. The
high frequency DC current ripple generated by the switchingof the converter can be reduced by applying the free-wheelingmode of the conversion [10]. In Ref. [11], the instantaneous
DC current ripple is suppressed with specified current profile.
However, the DC current ripple RMS cannot evaluated for not
specified current profile such as the constant-torque current
waveform. In this paper, the quantitative evaluation function
is mathematically derived based on the characteristics in the
TSF method in order to simplify the optimization process.

m W f i W f T
t m t i(9)where Idc is the input DC current, idcRip. is the input DC
current ripple, Vdc is the input voltage, Wf is the magnetic
energy, and  is the rotation speed. Note that Wf equals to thesurface areas surrounded by the magnetization curve and theflux axis in the magnetization characteristic (cf., Fig. 1). It is
assumed that the effect of the magnetic saturation is small;hence, Wc is same as Wf. As a result, the first term in the left
side of (9) is considered as same as the third term which isconstant value. Therefore, the input power ripple idcRip.Vdc is
calculated by the second term in the left side of (9). Since theperiodic integral of the fluctuation of Wf which represents thefirst and second terms in the left side of (9) is zero,DC current ripple RMS[p.u.]

Pin  ( I dc  idcRip. )Vdc 

0.8

/max=0.75p.u.
/max=0.5p.u.

0.6

/max=0.25 p.u.

0.4
0.2

0
0

1
2
3
4
Two-phase conduction period lap[deg]5Fig. 5. Evaluated results and simulation results of DC current RMS value.
250
200
Current [A]Fig. 5 shows the evaluated results and the simulation
results of the DC current ripple RMS value. The input power
Pin is expressed by (9).

150
100
50

idcRip.Vdc d  T 

2
mN r

Since the current change is almost zero except for lap, theintegral range of the left term is lap. It is assumed that idcRip.
is a linear function (triangle). According to (10), the surfaceof the triangle is T 2 /mNrVdc and the width is lap.
Therefore, the DC current ripple RMS is expressed as in (11).
idcRip. RMS 

T

2
3

Vdc

2 mN rC. Radial force ripple
In this subsection, the evaluation function of the radial
force ripple is derived. The radial force ripple affects theacoustic noise and vibration. The radial force ripple isreduced by decreasing the sum of the radial forces generated
from all phases of SRM. Generally, the radial force ripple isevaluated by simulating the instantaneous radial force withradial force model and calculated its peak-to-peak [12].
However, large amount of calculation is needed. In thispaper, the quantitative evaluation function is mathematicallyderived based on the characteristics in the TSF method inorder to simplify the optimization process.
Fig. 6 shows the constant-torque current waveform andthe radial forces of three phases. As shown in Fig. 6, in theconstant-torque current waveform, the excitation is continued
to the alignment state when the radial force per current value
is large, and then the current falls. Therefore, the radial force
is maximized at the ending angle of the one-phase conduction
period fc=f0-lap+2/mNr when the current starts to fall.
Meanwhile, the radial force is minimized at the turn-off angle
c (the initial angle of the one-phase conduction period of
adjacent phase) when the current is zero. Therefore, the radial
force ripple is expressed as in (12),FrSUMRip.  Fr (i ( fc ), fc )  Fr (i ( f 0 ), f 0 )
*

(12)

300
250

200
150

FrSUMRip.

100
50
0

(11)

lap

The DC current ripple RMS values is approximated byonly TSF parameters and experimental condition. In order toreduce the DC current ripple, lap should be long which iscontradictory to the motor current RMS reduction.

*

0
350

(10)
Radial force sum [N]

0

lap
0
10
0
f0 5
Rotor position [deg]Unaligned
position

fc

lap

c 15

AlignedpositionFig. 6. Current waveform and radial force (f0=4deg, lap=2deg).
1.6
Radial force ripple [-]



l ap

1.5

/max=0.25 p.u.

1.4
1.3
1.2

/max=0.5p.u.

1.1
1

/max=0.75p.u.

0.9
0

1
2
3
4
Two-phase conduction period lap [deg.]

5

Fig. 7. Evaluated results and simulation results of radial force ripple.

where Fr(im,) of (12) is one phase radial force at the valueof the current i and the rotor position . In order to derivate
the radial force ripple, Fr(i,) is formulated. The radial force
is expressed as in (13) from Maxwell's stress.
Fr (i, m ) g (i, m ) 2
B(i, m ) 2
Salig ( m ) 
2 0
20 Salig ( m )(13)

where B is the magnetic flux density, g is the flux of air
gap, 0 is the vacuum permeability, and Salig is the surface ofaligned teeth. Salig is calculated from the tooth width of therotor and stator, core thickness, and rotor position. g is
calculated from the magnetization characteristic.
Fig. 7 shows the evaluated results and simulation resultsof the radial force ripple. In order to reduce the radial force
ripple, both the decrease in the maximum point of fc=f0-

lap+2/mNr and the increase in the minimum point of f0 are
necessary. To achieve both requirements, lap should be long.

Start
T*

D. Derivation of ideal constant-torque current waveform
The motor current RMS value becomes high when the DC
current ripple RMS value and radial force ripple are small, i.e.
the two-phase conduction period lap is long. Therefore, it isclear that there is a trade-off relationship between thesecriteria. In order to simultaneously optimize multiple
objective functions considering the trade-off relationship, theweight function is introduced. The weight function g isdefined as (14), (15). Note that each evaluation function isnormalized by using the maximum and minimum values.

where w1+w2+w3=1, and 0≦wn≦1 are satisfied. In thispaper, the constant-torque current waveform is optimized
considering the motor current RMS value, the DC current
ripple, and the radial force ripple by using (14) and setting
appropriate weight factors w1, w2, w3.
Fig. 8 depicts the flowchart for the derivation of the ideal
constant-torque current waveform. It is assumed that thecurrent RMS value has been already minimized [9]. Under
the consideration of the minimized current RMS value
method, the DC current ripple and the radial force ripple are
maximized since lap is the shortest. In addition, the margin
Mf when the current rises is positive, but the margin Ml when
the current falls is derived as 0. As a result, the turn-off angle
c cannot be increased any further. Therefore, while fixing
the turn-off angle c obtained by the minimized current RMS
value method, lap is varied to minimize the weight function.
IV.

M ULTIPLE OBJECTICVE OPTIMIZATION

A. Simulation results
The effectiveness of the ideal constant-torque current
waveform is verified by simulation. The motor parameters
and the magnetization characteristic of the evaluated SRM
are as shown in Table I and Fig.1. The simulation isconducted at / max=0.25, T*=4.91Nm. Note that the
maximal torque-ripple-free speed (TRFS) max is the
maximum speed which can still achieve the constant torque[6].
Fig. 9 shows the constant-torque command currents, thecurrents, the torque, the DC current, and the radial force
generated from all the phases of SRM, in two cases (a) which
focuses more on the current RMS value reduction and (b)which focuses more on the radial force ripple reduction. In
both cases shown in Fig. 9, an instantaneous torque without
any ripple, i.e. the constant torque, is achieved. Comparing Fig.
9(a) and (b), the motor current RMS value is reduced in caseof (a) which focuses more on the motor current RMS value
reduction, while the radial force ripple is reduced in case of(b) which focuses more on the radial force ripple reduction.
Note that the DC current ripple is also reduced even if w3 islarge. The reason is because the longer the lap is, the smaller
the DC current ripple RMS value becomes.

Mf≧0

Yes
No

No
Yes

Calculation
g by Eq.(14)No

gmax >glap_gmax =lap, gmax=g
d:increment
Ns:Number of stator poles
m:Number of phases

Yes

Calculation
i*()
End

Fig. 8. Generation flow for ideal current waveform.
200

Torque[Nm] Current[A](15)

Calculation
Mflap =lap +d

200

RMS=
86.8A

100
0
6 0

10

20

30

0
6 0

4

2

2

0
30 0
20
10
0
-10 0
-20
400

200

10

10

RMS=
100.4A100

4DC
current[A](14)f  min( f )where Z ( f ) max( f )  min( f )lap >(m-2)/mNr

Radialforce[N]g  w1Z (i*RMS )  w2 Z (iDCRip. RMS )  w3Z ( FrRip. )f0 =f0_Wmax, lap =lap_Wmax

0
30 0
20
10
0
20
30 -10 0
-20
400
p-p=
295N
200

20
30
RMS=
7.4A

10

20

30

10

20
30
RMS=
6.2A

10

20

30

p-p=
202N

0

0
0

10
20
Angle [deg.]

30

0

10
20
Angle [deg.]

30(a) w1=0.7, w2=0.15, w3=0.15 (b) w1=0.15, w2=0.15, w3=0.7
Fig. 9. Simulation results of the currents, the torque, the DC current, andthe radial force at (T*=4.91Nm, /max=0.25).

B. Effectiveness of weight factor w
Fig. 10(a)-(c) show the relationship between the weightw and the motor current RMS value, the DC current ripple
RMS value, and the radial force ripple. The horizontal axis is
the weight of the corresponding value. The rotation speed is/ max=0.25. In Fig. 10(a)-(c), the motor current RMS value,
the DC current ripple RMS value, and the radial force ripple
are reduced by 18%, 45%, 37 % at most when the respective
weight factors w1, w2, w3 increase. Note that the DC current
ripple and the radial force ripple are small even if w3 is large
as shown in Fig. 10(b) and w2 is large in Fig. 10(c). The
reason is because the longer the lap is, the smaller the radial
force ripple and the DC current ripple RMS value become. In
Fig. 10(b), the DC current ripple tends to increase in theregion of large weight factor, because the evaluation functionof the DC current ripple is derived assuming that lap issufficiently short. Therefore, when lap increases, theapproximation error increases, resulting in that the tendencies
do not match.

Max. i*RMS value:
1.22p.u.

0.8

0.4

0
0

0.2

1.15
1.10
1.05

1.02

1.04

1.06

0.2

0.4
0.6
0.8
Weighting factor w1 [-]10

Weighting factor w3 [-]0.4
0.6 0.
6 60

0.9
6

0
0

0.2

iw

iv

iu

Time(4ms/div)

0.90

Torque (2Nm/div)

0.80
0.70

0.6 0.
0 66

0.4
0.6
0.8
Weighting factor w2 [-]Trip.=75.3%(a) Single pulse-wise current method (Ipeak=144.7A, IRMS=83.5A)

1p.u.=9.4A
Min. iDCRip.RMS value:
0.55p.u.
(Reduced by 45%)
0.65

0.6

0.2

Phase Current (50A/div)
iw
iu
iw
iu
iv

1.00

Max. iDCRip.RMS value:
1p.u.

0.8

0

1.00
[p.u.](a) Motor current RMS value
1

Torque (2Nm/div)

1.20

1p.u.=81.0A
Min. i*RMS value:
1p.u.
(Reduced by 18%)

0.6
1.22

Weighting factor w2 [-]

1

0.60

1

[p.u.]

0Phase Current (50A/div)
iw
iu
iw
iu
iv

Trip.=15.9%
iw

iv

iu

(b) DC current ripple RMS value

Weighting factor w2 [-]1

1.00
Max. FrRip. value:
1p.u.

0.8

1p.u.=323.1N
0.90

Min. FrRip. value:
0.63p.u.
(Reduced by 37%)

0.6
0.4
0.2

0.9

0
0

5

0.2

0.7 0
0 .6

0

Time(4ms/div)(b) w1=0.7, w2=0.15, w3=0.15 (Ipeak=160.1A, IRMS=85.8A)

0.80

Torque (2Nm/div)

0.70
5

0.4
0.6
0.8
Weighting factor w3 [-]1

Phase Current (50A/div)[p.u.]

0iw(c) Radial force ripple

C. Experimental results
The effectiveness of the ideal constant-torque current
waveform is verified experimentally. In the experiment, themaximum hysteresis error designed to be 10A regulated by
the switching of the switching devices. In order to evaluate
properly the torque ripple due to the commutation, it isnecessary to avoid the torsional resonance. In the test bench,
the re sonant frequency is 171Hz. Therefore, the experimentis conducted at 345 r/min, 4.91Nm in which the first orderfrequency is 69 Hz and second order frequency is 138 Hz.
Fig. 11 and Fig.12 show the current waveforms, the torquewaveforms, and the harmonic components of the torque with(a) single pulse-wise current method, the torque ripple
reduction methods based on the ideal constant-torque current
in two case s (b) w1=0.7, w2=0.15, w3=0.15 and (c) w1=0.15,
w2=0.15, w3=0.7. The torque ripples of the constant-torquecurrent waveform as shown in Fig. 11 (b) and (c) are reduced
by 78.9% and 74.1 % compared with the single pulse-wisecurrent in Fig. 11 (a), respectively. In addition, thefundamental component of the torque ripple is reduced by

iv

iw

iu

0

iv

iw

iu

Time(4ms/div)(c) w1=0.15, w2=0.15, w3=0.7 (Ipeak=209.5A, IRMS=96.9A)
Fig. 11. The current waveforms, the torque waveforms at T*=4.91Nm,
Rotation speed: 345rpm.

Amplitude of torque ripple [%]Fig. 10. Relationship with weighting factors w. Note that 1 p.u. is thevalue when applying minimized current RMS value method. w1: Motorcurrent RMS value, w2: DC current ripple RMS value, w3: Radial force
ripple.

iu

Trip.=19.5%

16
14
12
10
8
6
4
2
0

(a) Single pulse-wise current(b) w1=0.7, w2,w3=0.15(c) w1,w2=0.15, w3=0.7

1st frequency is 69Hz
Reducedby 91.7%

1st

2nd

Reducedby 90.4%

3rd

4th

5th

6th

Frequency [order]
Fig. 12. The harmonic components of the torque ripple at T*=4.91Nm

thth-9.1dB 24 25-19.1dB10095(a) Single pulse-wise current(b) w1=0.7, w2,w3=0.15(c) w1,w2=0.15, w3=0.7-15.2dB-23.3dB

Vibration [dB]

90

85
80
75

70
65

1st frequency is 69Hz

60
1st

5th

10th

15th

20th

25th

30th

35th

40th

Frequency [order]
Fig. 13. The current waveforms, the torque waveforms at T*=4.91Nm, Rotation speed: 345rpm.

Increasedby 133%

Amplitude of DC current ripple [A]9Increasedby 104%(a) Single pulse-wise current(b) w1=0.7, w2,w3=0.15(c) w1,w2=0.15, w3=0.7

87Increasedby 9.8%

6
5

Reducedby 8.0%

4
32Reducedby 79%

1st frequency is 69Hz

Reducedby 3.5%

1
0
1st

5th

10th

15th

20th

25th

30th

35th

40th

Frequency [order]
Fig. 14. The harmonic components of the DC current ripple at T*=4.91Nm, Rotation speed: 345rpm.

91.7% and 90.4% respectively as shown in Fig. 12. In
addition, comparing Fig. 9(b) and (c), the current RMS value
is reduced by 11.8% in case of (b) which focuses more on thecurrent RMS value reduction. Furthermore, the current peakvalue is reduced by 23.6%. Note that the torque ripple still
occurs even if the constant-torque current waveform isapplied. This is because the torque ripple is caused by themodel inaccuracies. However, the harmonic component of
the main torque ripple has already significantly reduced byaround 90%.
Fig. 13 and Fig.14 show the harmonic components of thevibration and the DC current ripple generated from all thephases of SRM with (a) single pulse-wise current method, thetorque ripple reduction methods based on the ideal constanttorque current in two cases (b) w1=0.7, w2=0.15, w3=0.15 and(c) w1=0.15, w2=0.15, w3=0.7. The 25th harmonic components

which is the main vibration is reduced by 23.3 dB and 8.1 dB
in case of (c) which focuses more on the radial force ripple
reduction compared with (a) the single pulse-wise current andthe case of (b) which focuses more on the current RMS value
reduction. On the other hand, as shown in Fig. (14), the DC
current ripple RMS value is not reduced in the experiment.
Especially, the low frequency components of the DC current
ripple are increased, although the high frequency componentof the DC current ripple are reduced. As shown in Fig. 9 (b),the time width of the DC current ripple is increased, although
the peak of the DC current ripple is decreased. As a result,
low frequency components of DC current ripple increases
instead of reducing the high frequency components.
V.

CONCLUSION

This paper proposed the current RMS value, the DC
current ripple, and the radial force ripple reduction method

under zero torque ripple for SRM. In order to optimize
multiple objective functions considering the trade-off
relationship, the weight function was introduced. In particular,
quantitative evaluation functions of the motor current RMS
value, the DC current ripple, and the radial force ripple arederived from the mathematical model for avoidance of timeconsuming numerical process. The reduction of the current
RMS value, the DC current ripple, and the radial force ripple
by 18%, 45%, and 37%, respectively, were achieved
respectively by the proposed method.
In the future work, the accuracy of the objective functions
will be improved. In addition, the experimental results will beobtained in order to validate the effective of the DC current
ripple reduction.
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