Torque Ripple Reduction Method with Minimized Current RMS Value for
SRM Based on Mathematical Model of Magnetization Characteristic
Takahiro Kumagai, Keisuke Kusaka, Jun-ichi Itoh
Nagaoka University of Technology
1603-1 Kamitomioka-cho, Nagaoka city Niigata, Japan
Tel., Fax: +81 / (258) – 47.9533.
E-Mail: kumagai_t1125@stn.nagaokaut.ac.jp, kusaka@vos.nagaokaut.ac.jp,
itoh@vos.nagaokaut.ac.jp
URL: http://itohserver01.nagaokaut.ac.jp/itohlab/index.html

Keywords«Switched reluctance drive», «Electrical drive», «Ohmic Losses»

Abstract
This paper proposes a torque ripple reduction method with minimum current RMS value forswitched reluctance motors (SRM). In the proposed method, the current waveform to achieve constanttorque is derived based on a mathematical model of magnetization characteristics. In particular, thecurrent waveform is optimized in term of the reduction of the current RMS value and the satisfactionof the input voltage limitation. The algorithm for determining the torque command is introduced toderive current command which is possible to generate by the input voltage and lead to the design ofthe maximum torque per current RMS value without any complex and time-consuming computation
process. A three-phase 18S/12P type SRM is used in simulation and experiment in order to validate
the proposed method. As a result, the reduction of torque ripple by 65.8% and the reduction of thecurrent RMS value by 19.2% are achieved by proposed method.

I. Introduction
In the last decade, researches on switched reluctance motor (SRM) have been accelerated due to anincreasing demand of electric vehicle. SRM consists of only an iron core and concentrated windings,
which can achieve rare-earth-element-free and low manufacturing cost. In addition, the rotor is builtwith a solid salient pole core structure, which is suitable for high speed rotation with low inertia. In
SRM, one-phase excitation mode is applied as a general control method [1][2]. In this method, thecontinuous rotation of the motor is achieved by the excitation of each phase stator winding at an
appropriate timing. However, a large torque ripple still occurs in the torque since the torque decreases
during the switching phase interval. In order to overcome this problem in SRM, many control methods
have been studied actively with the development of the power electronics technology in recent years[3]-[10].
The control method is proposed in order to reduce the torque ripple by the excitation of two phasestator windings during the switching phase interval [3]-[10]. In this method, several current
waveforms which obtains an instantaneous torque without any ripple, i.e. the constant torque, isobtained from the magnetization characteristic. In this paper, the current waveform which obtains theconstant torque is referred to as the constant-torque current waveform.
However, in the conventional method, the current root-mean-square (RMS) value significantly
increases due to the high current peak occurring in the motor current during the switching phaseinterval [10]. This is because the conventional method chooses the constant-torque current waveformwhich is not optimized in order to both eliminate the torque ripple and reduce the current RMS value.
In particular, the switching phase interval starts when the generable torque from one phase has alreadybecome very small. As a result, the current from the other phase has to be large in order to generatesufficient torque and maintain the constant torque. In order to reduce the current peak, the starting
point is designed so that the generable torque from one phase is still large. Nevertheless, in this case,the required slope of the current from the other phase might not be sufficient due to the limitation of
the input voltage. If the constant-torque current waveform which is impossible to generate by the input
voltage is applied, torque ripple still occurs in the torque.
In SRM, it is not clear how to properly obtain the constant-torque current waveform which satisfies
both the voltage limitation and the current reduction of the inverter due to the nonlinear magnetization
characteristic of SRM. In order to achieve the above objectives, the constant-torque current waveformis optimized by considering both current reduction and voltage limitation of the inverter. In general,
first, several constant-torque current waveforms are derived by the variation of the parameters value
determining the shape of the waveform; next, the optimized current waveform is selected [7]-[9].
However, the common disadvantage of these conventional methods is that the derivation of theconstant-torque is time-consuming and has to be conducted whenever the motor is changed.
This paper proposes a torque ripple reduction method considering the current RMS value and the
input voltage limitation. The constant-torque current waveforms are derived by the mathematicalmodel of the magnetization characteristic, which reduces the computation time [10]. In addition, theconstant-torque current waveform is optimized in term of the reduction of the current RMS value andthe satisfaction of the input voltage limitation. This leads to the design of the maximum torque percurrent RMS value without any complex and time-consuming computation process.
This paper is organized as follows; first the derivation method of the constant-torque current
waveforms based on the mathematical model of the magnetization characteristic is explained. Next,
the algorithm for the determination of the ideal constant-torque current waveform which is possible toachieve simultaneously the torque ripple reduction and the current RMS value reduction. Finally, thesimulation and experimental results are shown in order to confirm the validity of the proposed method.

II. TORQUE GENARATION MECHANISM IN SRM
In SRM, the torque T is generated by the change of magnetic co-energy Wc depending on thesaliency between the stator and rotor which is caused by the change of the rotor position m, [1][11]iWC (i,  m )
(1)
T (i,  )  (i ',  )di ' m 0

 m

m

m

where  is the flux linkage, i is the current, and i’ is the integral variable defined on the interval [,i].
Note that Wc equals to the surface areas surrounded by the magnetization curve and the current axis in
the magnetization characteristic. In order to obtain the instantaneous torque, the flux linkage  in (1)is mathematically derived based on the method which has already been proposed in [10].
Table I shows the motor parameters of the evaluated SRM. A three-phase 18S/12P type SRM ischosen as an example to investigate.
Fig. 1 depicts the magnetization characteristic of the evaluated SRM. The flux linkage  is afunction of the position (electrical angle) e and the current i. The flux linkage changes periodicallybetween the aligned flux linkage a and the unaligned flux linkage u, and expressed as in (2), [10](2)(i,m )  u (i)  f (e )(a (i)  u (i))
The periodic function f(e) is approximated by the fundamental harmonic and the second-to-tenthorder harmonics of the cosine function as shown in (3),10f (e ) 1  cos( e)   hn((1) n1  cos(n e))n22(1  h3  h5  h7  h9 )(3)where hn is the content rate of the n-order harmonic component. Next, the rotation electrical angle e isconsidered in the range of -to 0, where e=0 indicates the alignment state, and e =- indicates thecomplete unalignment state. Substituting (2) into (1), the torque is expressed as in (4).
if ( e )(4)
T (i,  ) ( (i )   (i ))di '
m

 m



a

u

0

In the linear region, the aligned flux linkage a(i) and the unaligned flux linkage u(i) are
expressed as in (5),u (i)  Lu i a (i)  Lai(5)

where La is the initial aligned stator inductance and Lu is the unaligned stator inductance. Substituting(5) into (4), the torque in the linear region is expressed as in (6).
f (e ) ( La  Lu )i 2
(6)
T (i, m ) 2In the saturation region, the aligned flux linkage a(i) and the unaligned flux linkage u(i) are
expressed as in (7),u (i )  Lu i i a (i)   sat 1  (1  Ki)e   Lsat i(7)where Lsat is the saturated aligned stator inductance, sat is the aligned saturation flux linkage, and  isthe coefficient which is determined to minimize the sum of the squares of the residuals between themagnetic characteristic and (7) in the alignment state. In order to maintain the continuity of theinductance in the low current range and (3), K is expressed as in (8),L  Lsat(8)
K   a sat

Substituting (7) into (4), the torque in the saturated region is expressed as in (9).
T (i, m ) f (e )  K    K i  i  Lsat  LU 2e i  To  sat  i 2
 m 

2




(9)

Let the current value of the boundary of the linear region and the saturation region be I0 , the value ofthe torque at the boundary T0 is expressed as in (10),La  Lsat 2
K    K Io  Io 

(10)
To 
Io  
Io 
e
2

sat







2

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 300A and the rotor position. It is observed from Fig. 2 that a required current
value for any torque at any rotor position is calculated.
TABLE I. MOTOR PARAMETERS OF SRM.
Rated mechanical power Pm
Rated speed wn
Rated torque Tn
Input voltage
Number of poles
Winding resistance R
Number of coil turns

Flux linkage [Wb]0.15

5.5 kW
12000 r/min
9.3 Nm
48V
Rotor 12, Stator 18
.11W
12 turns

Aligned position 15deg (mech.)

0.12
0.09

0.06

Unaligned position
0deg (mech.)

0.03

every 1deg
0
0

200

Current[A]

400

600

Fig. 1. Magnetization characteristic.
20

Torque T[Nm]

300A

15
10
5
every 20A
0
0

Fig. 2. T-i-mcharacteristic.

5

Position [deg]

10

20A

15

A. Torque sharing function
In this subsection, the control method of each phase torque, which removes the torque ripple fromthe motor shaft torque is explained. In this paper, the torque sharing function fTx(m) [3] which
represents the sharing rate of each phase torque according to the rotor position, is introduced to derive
the constant-torque current waveforms. Generally, the application of the torque sharing function
fTx(m) is proposed in the conventional derivation method [6]-[10].
Fig. 3 shows the example of the torque sharing function fTx(m). The sum of each phase torque isthe constant torque. Note that the subscript x indicates the phase in the torque sharing function fTx(m).
Each individual phase generates the torque during the one-phase conduction period, whereas twophases generate the torque during the two-phase conduction period, i.e. the switching phase interval.
In order to make the sum of each phase torque result in the torque with no ripple, i.e. a constant torque,fTx becomes 1 during one-phase conduction period, whereas the sum of two-phase fTx equals to 1
during two-phase conduction period (the period lap in the figure). It is observed that various functions
is selected for the torque sharing function. In this paper, the cosine function is utilized in order toavoid a steep rise [4]. Therefore, the torque sharing function is expressed as in (11),1 m   f 0)  1  0   m   f 0
 cos(
2

 


lap


1
 f 0   m   fc
fTx ( m )   m   fc
1)  1  fc   m   c
 2 cos( 


lap

 

0
otherwise

(11)

where lap represents the length of the two-phase conduction period, and f0 indicates the initial angleof the next one-phase conduction period, i.e. the end point of the two-phase conduction period. In the
SRM with number of phases m, number of stator pole Ns, and number of rotor poles Nr structure, since
the conduction period of the adjacent phase shifts the angle of 2/mNr rad in the mechanical angle, theturn-on angle 0, the ending angle fc of the one-phase conduction period, and the turn-off angle c are
expressed by (12) as dependent variables of lap and f0,
2
2
(12)
     ,   ,   
0

f0

lap

fc

0

mN r

c

f0

mN r

Torque sharing function

Therefore, the torque sharing function fTx(m) can be determined as the independent variable of lap andf0. The torque of each phase is controlled accurately by the multiplication of (11) with the torquecommand T*, eliminating completely the torque ripple. Consequently, the following (13) should besatisfied.
(13)
T (i* ,m )  T * fTx (m )The instantaneous command current value is calculated from (13). If the current follows exactly thecommand current, the torque becomes theoretically constant.
1.2

U phase1

V phaseW phase

Total

20

30

0.8
0.6
0.4
0.2

0
0

5

0

10
lap
f0

15

lap
fc c25

Angle[deg]Fig. 3. Torque sharing function which represents the sharing rate of each phase torque according to
the rotor position.

III. DERIVATION OF IDEAL CURRENT WAVEFORM
A. Consideration of Input Voltage Limitation
The torque ripple still occurs in the torque if the constant-torque current waveform which is
impossible to generate by the input voltage is employed. In this subsection, in order to derive theconstant-torque current waveform considering the input voltage limitation, the derivation method of
the variables f0 and lap, which determine the two-phase conduction period, is explained [10].
The margin M is defined as the difference between the maximum current slope, which is possible togenerate by the input voltage, and the constant-torque current slope. The constant-torque current
waveform is applicable when the margin M is positive. Otherwise, the torque is not achieved with theselected constant-torque waveform. The angle where the margin M becomes the minimum is either the
initial angle of the one-phase conduction period or the turn-off angle. Therefore, first, M is derived atthese two angles. Next, it is determined that the variables f0 and lap which satisfies M≧0 at these twoangles.
The margin Mf is expressed as in (14) when the current rises,
Mf 

Ed
w

Lu  f ( f 0  lap )( La  Lu ) lap

 df ( f 0  lap ) T*2( La  Lu ) 
d m


1/2

(14)

On the other hand, the margin Ml is given by (15) when the current falls,
Ml 

Ed
w

2
Lu  f (  f 0 )( La  Lu ) lap
mN r

2


 df ( mN   f 0 ) Tr2( La  Lu ) 
d m






1/2

(15)

*

where Ed is input voltage. In order to derive the constant-torque current waveform that satisfies the
input voltage limitation, lap is determined under the condition that Mf≦0 by the adjustment of f0 at
Ml=0.
B. Consideration of Current Limitation
In this subsection, the design of the variables f0 and lap, resulting in the constant-torque current
waveform with minimum current RMS value, is explained.
Fig. 4 depicts the flowchart for the derivation of the ideal constant-torque current waveform. In
order to achieve the constant torque with minimum current RMS value during the two-phaseconduction period, it is necessary to generate the torque during the interval where the generable torqueis still high under the condition that M≧0 is satisfied. First, the ideal two-phase conduction period,
which is achieves the maximum torque per current value, is calculated under the condition that the
input voltage is limitless, i.e. neglecting the limitation of the input voltage. Next, the actual two-phaseconduction period satisfying M≧0, i.e. the consideration of the input voltage limitation, is designed as
Start
T*

f0 =0, mid_max =0
Yes

f0 >(m-2)/mNr

No
Calculation
lap which satisfies Ml=0

f0 =f0 +d

Calculation
Mf
No

Mf≧0

Yes

mid =(2f0-lap+2/mNr)/2
No
d:increment
Ns:Number of stator poles
m:Number of phasesmid_Smax >mid

Yes

Calculation
i*()
End

Fig. 4. Generation flow for ideal current waveform which is possible to generate by the input
voltage and lead to the design of the maximum torque per current RMS value.

close as possible to the calculated ideal two-phase conduction period. In particular, the calculation of
the ideal two-phase conduction period is explained as follows.
Fig. 5 shows the relationship between the variables f0 and the magnetic co-energy W neglecting the
input voltage limitation. The magnetic co-energy W is the integral of the average torque over one cycle(2). Therefore, the maximum torque per current value is achieved when the magnetic co-energy W
becomes the largest. Meanwhile, the magnetic co-energy W in one cycle of current path equals to thesurface area enclosed by the current path as shown in Fig. 2. Hence, the magnetic co-energy W during
the conduction period [f0,f0+2/mNr] is expressed in (10),I(16)
W  ( f ( f 0  2 mN r )  f ( f 0 )) ( a (i)  u (i))di0Note that the term of the integral in (10) indicates the magnetic co-energy Wo between the alignment
state and the unalignment state. This magnetic co-energy Wo is constant and depends only on motor
parameters. Therefore, the magnetic energy W becomes the largest by the maximization off(f0+/mNr)-f(f0). As shown in Fig. 5, the maximum magnetic energy W is achieved at f0_Wmax of
3.98 deg.. Then, the middle angle of the conduction periodmid_Wmax which can maximize W isexpressed as in (11).
(17)
mid _W max   f 0_ max   mNr
On the other hand, the middle angle of conduction period mid as shown in Fig. 3 is expressed as in(12).
2    2  mN r
(18)
 mid  f 0 lap
2

Under the condition of the limitless input voltage, the selection of f0_Wmax of 3.98 deg. results inmid_Wmax of 8.98 deg.. Therefore, when considering the input voltage limitation, the actual f0 should beselected such that the actual mid becomes as close as possible to mid_Wmax.

Ⅳ. SIMULATION AND EXPERIMENTAL RESULTS

Magetic energy W/WO[-]

A. Validation of proposed algorithm
A three-phase 18S/12P type SRM is used in order to validate the proposed method. The hysteresis
comparator is employed as the current control. In the simulation, the maximum hysteresis error isdesigned to be small enough to evaluate the torque ripple caused only by the follow or non-follow of
the current to the ideal current waveform.
Fig. 6 shows the relationship between f0 and Mf when varying the value of f0 during the former
half of the conduction period, i.e. the period when the current rises. Note that the input voltage
limitation is considered. At each value of f0, first, lap is derived by (9) when Ml=0; then the margin
Mf is calculated by substituting the values of f0 and lap into (8). As shown in Fig. 6, the region whereMf≧0 represents the values of f0, the applicable region of the constant-torque current waveform. On
the other hand, the region where Mf<0 represents the values of f0 where there is no applicable
constant-torque current waveform.
Fig. 7 shows the relationship between f0 and mid when varying the value of f0. If f0 of 4.65 deg. isselected, midbecomes the closet value to mid_Wmax. Hence, this selection of f0 leads to the constanttorque current waveform with the minimum current RMS current value applicable with the given input
voltage.
0.9

●

0.85
Selected point

0.8

0.75
0.7

f0_Wmax=3.98 deg.

0.65
0.6
0

1

2
3
4
Torque sharing function f0[deg]Fig. 5. Calculated magnetic co-energy W/Wo = f(f0+/mNr)-f(f0)5

Fig. 8 shows the relationship between rotation speed and the middle angle of the conduction period

mid. The graph is divided by a dashed line at mid_Wmax of 8.98 deg.. The constant-torque current
waveforms under this dashed line are applicable at low rotation speed, which means that mid should bechosen as close as possible to this dashed line in order to minimize the current RMS value. However,at high rotation speed, the applicable mid becomes lower because the current slope during the twophase conduction period is limited by the input voltage as shown in point (a). In the conventional
method, the same mid is chosen over entire rotation speed as point (b) shown in Fig. 8, whereas in the
proposed method, mid is chosen to be as close as possible to mid_Wmax at every single rotation speed as
points (c) in Fig. 8. Therefore, the current RMS value can be reduced at low rotation speed with theproposed method.
Fig. 9 shows the current paths in the magnetization characteristic for each designed points (a), (b),and, (c) of Fig. 8. The ideal current path with the minimum current RMS value is shown as a dashedline. The current path with the designed points (a) and (b) of Fig. 8 do not follow the ideal current path,leading to the increase in the current peak. On the other hand, the designed point (c) of Fig. 8 follows
the ideal current path, resulting in the minimum current peak.
Fig. 10 shows the waveforms of the current, and the torque at three designed points (a), (b) and (c)of Fig. 8. As shown in Fig.10, in all cases, an instantaneous torque without any ripple, i.e. the constanttorque, is achieved. Comparing Fig. 10(b) and (c), the proposed method suppresses the high current
Uncontrollable

10.0

Controllable

Middle angle mid[deg]Margin Mf[A/deg]

1500
1000

500
0
0
-500

1

2

3

4

5

mid_Wmax=8.98 deg.

9.0

●

8.0

Selected point

7.0
6.0

f0=4.65 deg.

5.0
0

1

2

3

4

5

Torque sharing function f0[deg]Torque sharing function f0[deg]Fig. 6. Margin of the current slop in former half Fig. 7. Calculated value of the middle angle whenof conduction period.
varying the value of torque sharing function
Middle angle mid[deg]9.5
Proposed method: Applied

9.0

(c)

●

mid_Wfmax=8.98deg.

8.5

Evaluated points

8.0
7.5

(b)
●

7.0
0

0.1

0.2

(a)

Proposed method: Not applied
0.3 0.4 0.5 0.6 0.7
Rotation speed w/wmax [-]

0.8

●

0.9

1

Fig. 8. Relationship between rotation speed and middle angle mid, the rotation speed 1 p.u. isdefined as the maximum rotation speed under no torque ripple at rated torque, i.e. Mf and Ml are 0,
and further optimization is impossible.(Tn=4.91Nm, Vn=48V, ωmax=98.3[rad/s])
Flux linkage [Wb]0.12

Ideal conduction period[3.98deg, 13.98deg]

0.1
0.08

(c)

0.06
0.04

0.02

(a), (b)

0
0

50

100

150
200
Current[A]Fig. 9. Current paths in magnetization characteristic

250

300

peak occurring during the switching interval at low rotation speed, leading to the reduction of thecurrent RMS current value.
Fig. 11 shows the relationship between the rotation speed and the current RMS value with theconventional and proposed method. As shown in Fig. 11, comparing the conventional method, thecurrent RMS value reduction with the proposed method becomes more effective at low rotation speed.
In particular, the current RMS value is reduced by 21.2% at most. Thus, the effectiveness of the
proposed method is confirmed by these results.
Fig. 12 shows the relationship between the rotation speed and the current peak value with theconventional and proposed method. As shown in Fig.12, the current peak value is reduced, and the
maximum reduction is 43.2%. Note that there is a trade-off between the RMS value and peak value ofcurrent. Therefore, it is not concluded that these peak values are the minimum values. However, thepeak value is reduced in all range. Thus, the effectiveness of the proposed method is confirmed bythese results.
4.92

Torque4.91

300

IRMS=103.2A

200

4.90
4.89

iW

iV

iU

4.88

100
0

Torque[Nm]Current[A]

400

4.87

0

10

Position[deg.]

30

20

0

(a) Rotation speed: 1 p.u.
4.91

300

IRMS=103.2A

200

4.90
4.89

iW

iV

iU

4.88

100
0

4.87

0

(b) Rotation speed: 0.25 p.u.

10

Position[deg.]

30

20

4.92

Torque4.91
IRMS=81.3A

200

4.90
iW

iV

iU

100

4.89
4.88

Torque[Nm]

300

0

0

Conventional method

400

Current[A]4.92

TorqueTorque[Nm]Current[A]

400

4.87

0

10

Position[deg.]

20

30

0

(c) Rotation speed: 0.25 p.u.
Proposed method
Fig. 10 Current waveform and Output torque.
1.10

Conventional method

Current peak value [p.u.]

Current RMS value [p.u.]

1.05
1.00
0.95

Reducedby 21.2%

0.90
0.85
0.80

Proposed method

IRMS:1p.u.=103.2A

0.75

Conventional method

1.00
0.90

Reducedby 43.2%

0.80
0.70

Proposed method: Applied
IPeak:1p.u.=263.1A

0.60

0.50
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Rotation speed w/wmax [-]1

Fig. 11 Relationship between rotation speed andcurrent peak value0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Rotation speed w/wmax [-]1

Fig. 12 Relationship between rotation speed andcurrent peak valueB. Experimental Results of ideal current waveform
The effectiveness of the proposed method is confirmed by experiment. In order to realize highspeed current control and reduce the torque ripple caused by the hysteresis error, current-hysteresis
controller is constructed by analog circuit. In the experiment, the maximum hysteresis error designed
to be 10A regulated by the switching of the switching devices. In addition, in order to evaluate thetorque ripple excluding the component caused by the hysteresis error, the low-pass filter (LPF) with acutoff frequency of 1.6kHz is applied to the torque signal.
Fig. 13 shows the current waveforms, the torque waveforms, and the harmonic components of thetorque with one-phase excitation mode, the conventional and proposed ripple torque reduction method
based on two-phase excitation mode. Large torque ripple occurs in SRM during switching phaseinterval. Therefore, the fundamental frequency of the torque ripple is 3NrN/60=207Hz in theexperimental condition The torque ripples of the two-phase excitation mode as shown Fig. 13 (b) and(c) are reduced by 67.0% and 65.8 % compared with the one-phase excitation mode in Fig. 13 (a),respectively. In addition, the fundamental component of the torque ripple is reduced by 88.6% and93.2% respectively. Furthermore, the current RMS value of the proposed torque reduction method in
Fig. 13 (c) is reduced by 19.2% compared with that of the conventional torque reduction method in
Fig. 13 (b). Furthermore, the current peak value is reduced by 32.1%. Therefore, the ideal current

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

0

iv

iw

iu

Amplitude of torque ripple [%]Torque (2Nm/div)

20
The fundamental frequency is 207Hz

15
10

5
0
1st

Time(4ms/div)

2nd

3rd

Frequency [order]

4th

5th

Torque (2Nm/div)
Phase Current (50A/div)
0

iw

iu

iv

iw

iu

0

iv

iw

iu

Amplitude of torque ripple [%](a) One-phase excitation mode (IRMS=80.8A, Ipeak=140A, Torque ripple=77.9%)
5
The fundamental frequency is 207Hz

4

3
2

1
0
1st

Time(4ms/div)

2nd

3rd

Frequency [order]

4th

5th

Torque (2Nm/div)00Phase Current (50A/div)
iw

iu

iv

iw

iu

iv

iw

iu

Time(4ms/div)

Amplitude of torque ripple [%](b) Conventional method (Command: IRMS=94.8A, Ipeak=215.4A, Torque ripple=25.7%)
5The fundamental frequency is 207Hz

4

3
2
1
0
1st

2nd

3rd

Frequency [order]

4th

5th

(c) Proposed method (Command: IRMS=79.5A, Ipeak=146.1A, Torque ripple=26.6%)
Fig. 13 The current waveforms, the torque waveforms, and the harmonic components of the torqueat T*=4.91Nm, Rotation speed: 345rpm.

waveform to achieve the constant torque with the minimized current RMS value is confirmed. Notethat the torque ripple still occurs even if the constant-torque current waveform is applied. This isbecause the torque ripple is caused by model inaccuracies of magnetization characteristic.

V. Conclusion
This paper proposed the torque ripple reduction method considering the input voltage limitationand the minimization of the current RMS. In the proposed method, the design algorithm was employed
to determine the ideal constant-torque current waveform which was possible to achieve simultaneously
constant torque and the current RMS value reduction. It was confirmed from the simulation andexperiment results that the reduction of torque ripple by 65.8% and the reduction of the current RMS
value by 19.2% are achieved by proposed method. In other words, the copper loss is reduced by
34.7％ by proposed method.

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