Torque Ripple Reduction Method for SRM based on Mathematical Modelconsidering Voltage Limitation
Takahiro Kumagai, Daisuke Sato, 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, itoh@vos.nagaokaut.ac.jp
URL: http://itohserver01.nagaokaut.ac.jp/itohlab/index.html

Keywords«Switched reluctance motor», «Torque ripple reduction», «Instantaneous current control»

Abstract
This paper proposes a torque ripple reduction method for Switched Reluctance Motor (SRM). Large
torque ripple occurs in SRM during switching phase interval. In order to reduce the torque ripple, theideal current waveform, which reduces completely the torque ripple, is applied as a command value
controlling the instantaneous torque. In the proposed method, the ideal current waveform is derived byconsidering voltage limitation based on mathematical model. The algorithm for determining the torquecommand is introduced to derive current command which is possible to generate by the power supply.
In addition, the maximum torque which is possible to output under condition of the zero torque ripple isclarified. A three-phase 18S/12P type SRM is used in simulation and experiment in order to validate theproposed method. As a result, the torque with no torque ripple is achieved by employing the proposed
algorithm in the simulation. Furthermore, the fundamental components of the torque ripple is reduced
by 91.5% on average by applying the proposed method in the experiment.

I. Introduction
Switched reluctance motor (SRM) has several advantages; rare earth element free, low cost, suitability for high speed, rugged construction, and maintenance free. Therefore, SRM is expected to apply into
hybrid electric vehicle (HEV) and electric vehicle (EV). However, a large torque ripple occurs comparedwith other motors, e.g. an induction motor or a permanent magnetic synchronous motor, due to the useof only the reluctance torque generated by the magnetic saliency. This results in large noise and vibration.
In order to solve these problems, many control methods to reduce torque ripple, noise, and vibration
have been studied actively with the development of the power electronics technology in recent years[1]-[4].
One-phase excitation mode is used as a general control method [5]. In this method, the motor voltage
is applied in order that a current flows into only one set of the winding in accordance with the commutation sequence. However, a large torque ripple occurs because it is impossible to generate sufficiently
the torque during the switching phase interval by this method. Therefore, the control method of using
two-phase excitation mode, the motor voltage in which is applied in order that a current flows into two
sets of the winding during the switching phase interval in order to reduce torque ripple, is proposed [1][4]. In the two-phase excitation mode, the ideal current waveform which reduces the torque ripple isderived to apply as a command value controlling the instantaneous torque. In this case, it is necessary
to model the magnetic characteristic of SRM which is nonlinear.
Several modeling methods have been proposed so far in order to deal with this nonlinear characteristic. One of the modeling method is to use a look-up table based on the measurement result or the
analysis result using finite element method (FEM) [1]-[2]. Nevertheless, this method is time-consuming
because it is necessary to build the look-up table of the magnetic characteristic for every single motor
in advance. On the other hand, another method is to obtain the magnetic characteristic by deriving amagnetic circuit from the shape of SRM [3]. In this method, the effect of the torque ripple reduction islimited due to the inaccuracy of the modeling. Besides, neural network algorithm has been applied as
this solution [4]. However, it seems that this method is complex and requires a long computation time[6]. In summary, the common disadvantage of those above method is that it is time-consuming in orderto optimize the current waveform, because the optimization needs to be repeated many times. In addition,
look-up table has to build every times if the motor is changed.
This paper proposes a modeling method by mathematical expression based on only the measured
parameters of the magnetic characteristic of SRM in order to avoid the problems of time-consuming andlack of accuracy. The derivation method of the ideal current waveform without repeating tests is proposed by utilizing the mathematical expression.
This paper is organized as follows; first, the derivation method of the ideal current waveform basedon the SRM’s mathematical model is explained. Next, the algorithm for determining the torque command to derive current command which is possible to generate by the power supply. Finally, the maximum torque which is possible to output under condition of the zero torque ripple is clarified.

II. DERIVING IDEAL CURRENT BASED ON MATHMATICAL
MODEL
A. Generated torque in SRM
This subsection describes the relationship between the torque, the current, and the rotor position
because the generated torque in SRM is determined by their instantaneous values. Note that SRM generates only reluctance torque. In order to formulate this reluctance torque, the magnetic co-energy is
considered. The magnetic co-energy Wc equals to the surface areas surrounded by the magnetization
curve and the current axis of the magnetic characteristic, and expressed as in (1),i

WC (i,  m )   (i ' ,  m )di'(1)

0

The reluctance torque per one phase is expressed by the variation of magnetic co-energy Wc according to the change of the rotor position, and expressed as in (2),T (i, m ) iWC (i, m )(i ' , m )di' m
 m 0(2)

The derivation of the flux linkage  in (2) is important to obtain the instantaneous torque.
Table 1 shows the motor parameters of the investigated SRM. A three-phase 18S/12P type SRM ischosen to the example to investigate.
Fig. 1 shows a magnetic characteristic in the SRM. The magnetic characteristic consists of a linear
region limited by a small value of the current and a saturation region with a non-linear characteristic in
the range of large current. The flux linkage  is a function of the position (electrical angle) e and thecurrent i. Therefore, the flux linkage  is expressed by two trigonometric functions in relation to position, where the first function is defined as a linear function expressing the magnetic characteristic of thelinear region, and the second function is defined as an exponential function representing the magnetic
characteristic of the saturation region.
In the linear region, the relationship between the flux linkageand the current i is expressed as in(3) [7],(i, m )  Lu i  f (e )( La i  Lu i)(3)
where La is the initial aligned stator inductance and Lu is the unaligned stator inductance. Then, theperiodic function f(e) is approximated by the fundamental harmonic and the second-to-tenth-order harmonics of the cosine function as shown in (4),10f ( e ) 

1  cos(e)   hn((1) n1  cos(ne))n22(1  h3  h5  h7  h9 )(4)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 thecomplete unalignment state. The relationship between the rotation electrical angle e and the rotation
mechanical angle m is expressed as in (5), e  N r m  2

Nr
x
Ns

(5)

where Nr is the number of the rotor’s poles, Ns is the number of the stator’s poles, and x=0…(Ns/2-1) isthe coefficient representing stator’s phase. Substituting (3) into (2), the generated torque in the linear
region is expressed as in (6),T (i, m ) f ( e ) ( La  Lu )i 2
x

2

(6)

In the saturation region, the relationship between the flux linkage  and the current i is expressed asin (7),(i, m )  Lu i  f ( e )  sat 1  (1  Ki)e i  Lsati  Lu i(7)where Lsat is the saturated aligned stator inductance, sat is the aligned saturation flux linkage, and 
is the 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 the inductance in the low current range and (3), K is expressed as in (8),K  

La  Lsat
 sat

(8)

Substituting (2) into (3) and (7), the generated torque in the saturation 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 valueof the generated torque at the boundary T0 is expressed as in (10),To 

La  Lsat 2
K    KIo Io 
Io   sat  Io 
e 
2
2



(10)

Fig. 2 depicts the T-i-m characteristic obtained by calculating the formula T(i,m) at the values of thecurrent from 60A to 540A and the rotor position. It is observed from Fig. 2 that a required current value
for any torque at any rotor position can be calculated. On the other words, the proper current value toreduce the torque ripple can also be calculated from Fig. 2.
TABLE I. MOTOR PARAMETERS OF SRM.

Rated mechanical power Pm
Rated speed wn
Rated torque Tn
Number of poles
Winding resistance R
Number of coil turns
0.15

2.3 kW
2000 r/min
11.0 N
Rotor 12, Stator 18
0.011W
24 turns

Interlinkage flux [Wb]Saturation Aligned position 15 deg (mech.)current

0.12
0.09
0.06

Unaligned position
0 deg (mech.)

0.03

every 1deg

0
0Fig. 1. Magnetization characteristic.

I0

200

Current[A]

400

600

B. Torque sharing function
In this subsection, the control method of each phase’s generated torque, which can remove the torqueripple from the motor shaft torque is explained. In this paper, the torque sharing function fTx(m) which
represents the sharing rate of each phase’s generated torque according to the rotor position, is used toderive each phase’s ideal current waveform generating the flat instantaneous torque. Generally, the application of the torque sharing function fTx(m) is proposed in the conventional derivation method [8].
Fig. 3 shows an example of the torque sharing function. The sum of each phase’s generated torqueis the constant torque, i.e. a flat torque. Note that the subscript x indicates the phase in the torque sharing
function fTx(m). In order to generate the command torque for obtaining the flat generated torque, (11)and (12) need to be satisfied,W

Ttotal  T *   f Tx ( m )  T

(11)

x U

W

 f ( )  1
x U

Tx

(12)

m

The torque sharing function can have many shapes as long as (12) is satisfied. In Fig.3, lap represents
the two-phase conduction period and f0 indicates the initial angle of one-phase conduction period. Since
the conduction period of the adjacent phase shifts the angle of /Ns rad in the mechanical angle in the
SRM with the Ns/Nr pole structure, the turn-on angle 0, the ending angle fc of the one-phase conduction
period, and the turn-off angle c are expressed by (13) as dependent variables of lap and f0,

 0   f 0   lap , fc   0 , c   f 0 
Ns
Ns

(13)

Therefore, the torque sharing function fTx(m) can be determined as the independent variable of lap andf0. The torque sharing function becomes unity during the one-phase conduction period from f0 to fc,
which implies that the torque is generated by only one phase. On the other hand, various functions canbe selected during the two-phase conduction period from 0 to f0. In this paper, the cosine function is
utilized in order to avoid a steep rise [8]. Therefore, the torque sharing function used in this paper isexpressed as in (14), 1 m  f 0
)  1 cos( lap 2 
1
f Tx ( m )   m   fc

 1 
cos(
)


1


 2
 lap




0

0  m   f 0
 f 0   m   fc

(14)

 fc   m   cotherwise

40
30
20
10
0
-10
-20
-30
-40

Torque contour function

Torque T[Nm]

If the generated torque of each phase is controlled exactly by multiplying (14) to the torque command
T*, the torque ripple can be completely removed from the motor shaft torque.
C. Derivation of ideal current waveform
540A

60A
every 60A
0

10

Fig. 2. T-i-mcharacteristic.

20
Position [deg]

30

1.2

U phaseV phaseW phase

Total

20

30

1
0.8
0.6
0.4
0.2
0
0

5

lap 10
0 f0Fig. 3. Torque sharing function.

15

lap

fc

c

25
Angle[deg]In this subsection, the derivation method of the ideal current waveform from the T-i- characteristic
and the torque sharing function is explained. The applied sharing command torque T*(m) is calculated
from command torque T* and ftx and express as ftxT*. The derivation method of command current torealize T*(m) for any rotor position is explained as following.
In order to obtain the flat torque, the generated torque should be controlled exactly to the sharing
command torque. Consequently, the following (15) should be satisfied.

T (i * , m )  T * fTx ( m )(15)Then, the instantaneous command current value is calculated from (15). If the current completely follows the command current, the generated torque becomes theoretically a constant value. Next, in thelinear region, substituting (6) into (15), the command value i*(m) of the ideal current waveform isexpressed as in (16),2T *i ( m ) 
La  Lu*fTx ( m )f ( e )
 m

(16)

On the other hand, in the saturation region, the command value i*(m) of the ideal current waveform canbe calculated from (15) based on (9) by Newton’s method.

III. VOLTAGE LIMITATION FOR IDEAL CURRENT CONTROL
A. Maximum controllable current gradient
The current command which is possible to generate by the power supply is limited. Therefore, thederivation method of the ideal current waveform responding to the voltage limitation is explained.
Fig. 4 shows the circuit diagram of the multiple phase half bride inverter which is generally used as
the driving circuit for SRM. In general, the hysteresis comparator is used for the current control. In thiscontrol method, the switch S1 is turned off in order that the current circulates through D1 when thecurrent exceeds an upper threshold. Then, S1 and S2 are turned on in order that the current is supplied
from the power supply when the current falls below a lower threshold. The motor current is regulated
to the command value by repeating the ON/OFF operation. However, it should be noted that the current
gradient is limited by the inductance of SRM and DC voltage of the inverter. On the other words, it is
impossible for the current to follow the current command, the gradient of which surpasses the limit.
Therefore, when deriving the ideal current waveform, it is necessary that the ideal current waveform’s
gradient should not surpass the maximum current gradient.
First, the maximum gradient which can be followed by the current is derived from the differential
equation of SRM at the element’s state of ON or OFF. The differential equation of SRM is expressed asin (17),d (i, m ) Ri  Eddt(17)d (i, m )  i   m  i 




w
dt
i t  m t
i t  m(18)

where  is the flux linkage, Ed is the power supply voltage, and R is the winding resistance. The leftside first term of (17) is rewritten into the partial differential equation as in (18) because the flux linkage
 is the function of the current i and the rotor position m,

The maximum gradient is then derived from (17) and (18). When the current is supplied fromthe power supply at the element’s state of ON, the change rate of the current, i.e. the maximum command current gradient, is expressed as in (19),

i

t

Ed 


w  Ri
 m

i

(19)

When the current flows to the power supply at the element’s state of OFF, the change rate of the current
is then expressed as in (20),

i

t

 Ed 

w  Ri
 m

i

(20)

B. Margin of current gradientIn this subsection, in order to regulate the current to the command current, the appropriation methodof the variables f0 and lap, which determine the two-phase period, is explained.
The margin M is defined as the difference between the maximum current change rate and the ideal
current change rate. When the margin M is positive, the current to achieve the flat torque can be generated. Otherwise, the flat torque cannot be achieved. If the influence of the speed electromotive force isignored for simplicity, the angle for minimizing M is either the initial angle of the one-phase conduction
period or the turn-off angle. The margin M is calculated from the characteristic of the linear region inboth cases because the current is around zero at those angles. Deriving the equation (16) with respect totime, the gradient of the ideal current waveform in the linear region is expressed as in (21),
1
3




 f ( e )  2  2 f ( e ) 
i 1T* 1 fTx  f ( e )  2
  fTx 



2 
t w 2( La  Lu )  fTx  m   m  m 
m





(21)

When the current rises, substituting (3) into (19) and (21), the margin Mf is expressed as in (22), df ( f 0   lap ) 
Ed
w
T*


Mf 

Lu  f ( f 0   lap )( La  Lu )  lap 2( La  Lu ) d m1 / 2(22)

On the other hand, when the current falls, substituting (3) into (20) and (21), the margin Ml is given by
(23),

S1
Ed

・・・・

S2n-1

D2

D2n

・・・・

D1
S2

D2n-1
S2n

・・・・

Fig. 4. Multiple phase inverter.
Start
T*

f0 =0f0 >/2Ns

Yes
No

Calculation
lap by Eq.(23)(Ml=0)

f0 =f0 +d

Calculation
Mf by Eq.(22)Nod:increment
Ns:Number of stator poles

Fig. 5. Generation flow for ideal current waveform.

Mf≧0
Calculationi*() by Eq.(15)
End

Yes

 df ( N r   f 0 ) 
Ed
w
T*


Ml 

Lu  f ( N r   f 0 )( La  Lu )  lap 2( La  Lu ) d m1 / 2(23)

Fig. 5 shows the algorithm to maximize the torque under the condition of the zero torque ripple. In orderto derive the ideal current waveform, lap is determined with Mf≦0 by adjusting f0 when Ml=0.

IV. Simulation and experimental Results
A. Validation of proposed algorithm
The validation of the proposed method is confirmed by simulation. A three-phase 18S/12P type SRM
is used. The hysteresis comparator is employed as the current control. 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 comparison between the ideal command current’s gradient and the maximum gradient. The steps to derive the ideal command current’s gradient is that, first let Ml=0, then varying f0 to
calculated lap which satisfies Mf≦0. As shown in Fig. 6, the ideal command current’s gradient which
can be followed in the latter half of the conduction period, i.e. the period when the current falls, can becalculated by (23).
Fig. 7 shows the relationship between f0 and Mf when varying the value of f0 during the former halfof the conduction period, i.e. the period when the current rises. Note that at each value of f0, first lap isderived by (23) when Ml=0, then the margin Mf is calculated by substituting the values of f0 and lap
into (22). As shown in Fig. 7, the region where Mf≧0 represents the values of f0 where the current canbe regulated to the ideal current in order to generate the flat torque. On the other hand, the region whereMf<0 represents the values of f0 where the current cannot be regulated to the ideal current.
Fig. 8 shows the ideal command currents, the currents, and generated torque in two cases as shownin Fig. 7, where the current can or cannot be regulated to the ideal current. As shown in Fig. 8(a), a large
torque ripple occurs when the current cannot follow the command current, because the ideal commandcurrent’s gradient is higher than the maximum gradient, i.e. the negative margin Mf. On the other hand,
as shown in Fig. 8(b), the torque without torque ripple is achieved when the current can follow thecommand current because the ideal command current’s gradient is lower than the maximum gradient,i.e. the positive margin Mf. The validation of the proposed method is confirmed by these results.
50

10

11

12

13

14

Margin Mf[A/deg]

0
15

-50
-100
-150

iU

iV

0.99

iW

0.985

50
0

0.98
0

10

Pisition[deg.]

20(a) Uncontrollable condition (f0=3.00[deg]).
Fig. 8. Current waveform and Output torque.

30

Evaluated points
10
●

●

-10 3

3.5

4

4.5

0

1.005

Torque

1.00

150

0.995

100

iU

iV

0.99

iW

0.985

50
0

0.98
0

10

Pisition[deg.]2030(b) Controllable condition (f0=3.51[deg])0Torque[Nm]

0.995

100

30

200

1.00
Torque[Nm]Current[A]

1.005

150

uncontrollable

Torque contour function's parameter θf0[deg]Fig. 7. Margin in former half of conduction period.

Angle[deg]Fig. 6. Current gradient in latter half of conduction period.

Torquecontrollable

-50

-200

200

uncontrollable-30

command currrent
maximum controlable current

Current[A]Current gradient [A/deg]

50

B. Effectiveness evaluation
The effectiveness of the proposed method is confirmed by experiment. In order to realize high-speed
current control and reduce the torque ripple caused by the hysteresis error, current-hysteresis controller
is constructed by analog circuit and the current command value i* of the ideal current waveform is
calculated by digital signal processor (DSP) and field-programmable gate array (FPGA).
Fig. 9 shows the currents when applying one-phase excitation mode used as a general mode andproposed method. In this evaluation, command torque T* is 1.0Nm and rotation speed N is 600rpm.
Fig. 10 shows the harmonic components of the generated torque under the same conditions as Fig. 9.
Large torque ripple occurs in SRM during switching phase interval. Therefore, the fundamental frequency of the torque ripple is 3NrN/60=360Hz in the experimental condition. As shown in the Fig. 10,
the fundamental components of torque ripple is reduced from 0.127Nm to 0.011Nm by applying the
proposed method. In addition, the second components is reduced from 0.137Nm to 0.042Nm.
Fig. 11 shows the fundamental components and the second components of the generated torque atfrom T*=0.2Nm to T*=1.2Nm. As shown in Fig. 11, the fundamental components of the torque ripple isreduced by 91.5% on average by applying the proposed method. On the other hand, the effectiveness on
the second components of the torque ripple is lower than that on the fundamental components. This isbecause the torque ripple caused by model inaccuracies occurs especially in both cases that the current
falls and rises, i.e. the frequency of the torque ripple caused by model inaccuracies is twice as high asthat of the torque ripple caused during switching phase interval.

Phase Current (20A/div)

Phase Current (20A/div)
iw

iu

iv

iw

0

iu

iw

iv

Time(2ms/div)

Amplitude of torque ripple [Nm](a) Conventional method.
Fig. 9. Current waveform at T*=1.0Nm, N=600rpm
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1
2

iu

iv

0

iw

iu

iv

Time(2ms/div)(b) Proposed method

Conventional method
Proposed method

4Second components of
torque ripple [Nm]

5
6
7
8
9 10
Frequency [order]
Fig. 10. Harmonic components of the generated torque at T*=1.0Nm, N=600rpm.
0.16
0.18
Conventional method
Conventional method
0.16
0.14
ProposedmethodProposed method
0.14
0.12
0.12
0.1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
0
0.2
0.4
0.6
0.8
1
1.2
0.2
0.4
0.6
0.8
1
1.2
Command torque [Nm]Command torque [Nm](a) Fundamental components of torque ripple.
(b) Second components of torque ripple
Fig. 11. Characteristics of command torque and harmonic components of the generated torque.
By the employment of the proposed method, the fundamental components of torque ripple is reduced by 91.5% and the Second components of torque ripple is reduced by 29.5% on average.

Fundamental components of
torque ripple [Nm]3

C. Clarification of maximum torque command
The algorithm shown in Fig.6 is the method for deriving the torque sharing function’s parameterconsidering the power supply voltage, when the flat torque can be achieved. In contrast, it is possible toderive the limit of the command torque, which can be completely followed to obtain the flat torque under
any condition of the power supply voltage, by comparing the value of Ml when varying the value of T*.
Fig. 12 shows the relationship between f0 and Mf when varying the value of T*. As shown in Fig.
12, max(Mf)=0 is reached with T*=1.08 Nm. Therefore, the current cannot follow the command current
if T* is more than 1.08 Nm, which implies that the flat torque cannot be achieved at the torque above
1.08 Nm. On the other hand, the conditions when the flat torque can be obtained, can be derived by the
proposed algorithm if T* is less than 1.08 Nm.
D. Maximum command torque with different poles
In order to discuss the optimal shape design for the proposed method, the maximum torque achieving
zero ripple is compared with several motors having different number of poles. In particular, 6S/4P type,
12S/8P type, and 18S/12P type SRMs are evaluated.
Fig. 13 shows SRMs with different number of poles. Note that the motors are designed to have thesame parameters such as; structure, output power, driven supply voltage, and space factor. Furthermore,
the ratio of pole pitch and pole arc is set to be similar at the pole shape design step. The smaller the
initial aligned stator inductance is, the more the number of poles becomes. It is observed that when themagnetic path length is shorter, and the cross-sectional area is bigger, the magnetic resistance becomeslarger.
10

mergin Mf[A/deg]

0
-10

3

3.5

4

T*=1.06Nm

4.5

T*=1.08Nm-20

T*=1.10Nm

-30
-40
-50

Torque contour function's parameter θf0[deg]Fig. 12. Evaluation of maximum controllable torque.

6S/4P
Fig. 13. Motors having different number of poles.
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

12S/8P

18S/12P

Maximum Torque [p.u.]

18S/12P
12S/8P
6S/4P

Torque:1p.u.=47.7[Nm]Linked flux:1p.u.=1.35[Wb]18S/12P:La=10.3[mH]
12S/8P:La=14.6[mH]
6S/4P:La=27.3[mH]

0

1
2
3
Linked flux=Ed/w・wn/Vn [p.u.]

4

5

Fig. 14. Comparison of maximum controllable torque with motors having different number of poles.

Fig. 14 shows the relationship between the flux linkage generated by the power supply voltage andthe maximum torque Tmax with the zero torque ripple in the motors having different number of poles.
Tmax is directly proportional to the square of flux linkage. Therefore, Tmax is directly proportional to thesquare of the power supply voltage and inversely proportional to the square of the rotation speed. Besides, the more the number of poles are, the larger Tmax becomes. This is because the initial aligned statorinductance is smaller. Therefore, the motor design with many multipolar is advantageous if the torquewith the small ripple is required.

V. Conclusion
This paper proposed the derivation method of the ideal current waveform based on SRM’s mathematical model. In the proposed method, a design algorithm was employed to determine the proper torquecommand to derive current command which is possible to generate. Finally, the maximum torque which
was possible to generate under condition of the zero torque ripple was clarified. As a result, the maximum torque was directly proportional to the square of the power supply voltage and inversely proportional to the square of the rotation speed.

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