Torque Ripple Minimization of Switched Reluctance Motor Using
Modified Torque Sharing Function
Milad Dowlatshahi*, Seyed Morteza Saghaeian Nejad**, Jin-Who Ahn***
*Electrical and Computer Engineering Department, Isfahan University of Technology(IUT), Isfahan, Iran,
Dolatshahi@ec.iut.ac.ir
** Electrical and Computer Engineering Department, Isfahan University of Technology(IUT), Isfahan, Iran,
saghaian@cc.iut.ac.ir
*** Department of Mechatronics Engineering, Kyungsung University, Busan, Korea, jwahn@ks.ac.kr
Abstract: Torque pulsation mechanism and highly nonlinear
magnetic characterization of switched reluctance motors(SRM)
lead to unfavourable torque ripple and limit the variety of
applications in industry. In this paper, a modification method
proposed for torque control of SRM based on conventional
torque sharing functions(TSF) to improve maximum speed of
torque ripple-free operation considering converter voltage
limitation. Due to increasing phase inductance in outgoing
phase during the commutation region, reference current
tracking can be deteriorated especially when the speed
increased. Moreover, phase torque production in incoming
phase may not be reached to the reference value near the turnon angle in which the incremental inductance would be
dramatically decreased. Torque error for outgoing phase can
cause increasing the resultant motor torque while it would be
negative for incoming phase and yields reducing the motor
torque. In the proposed modification method, phase torque
tracking error for each phase under the commutation added to
the other phase so that the resultant torque remained in
constant level. This yields to enhance motor efficiency
compared than the conventional receding turn-on and turn-off
angles. An experimental four phase 1.6 KW, 8/6 SRM set of
magnetic characterization data used to simulate and validate
the effectiveness of the proposed method.
Keywords: Torque control, switched reluctance motor,
torque sharing function, high speed operation.
1.
Introduction
Switched reluctance motor (SRM) is a double salient
motor with no winding or permanent magnet on rotor. It
has rugged and fault tolerant structure with the most
robust and reliable constructions [1]. Due to the simple
structure, low manufacturing and repairing costs as well
as ability to operate in wide range of temperature, SRM
has been taking into accounts more recently. One of the
most drawbacks in switched reluctance applications is
high torque-ripple compared than other ac type machines
[2]. Torque pulsation mechanism and high saturated
nature of magnetic characterization can cause undesirable
torque ripple and deteriorate drive performance especially
in high precise industrial applications. In order to
suppress torque ripple many investigations have been
done recently and many torque control techniques
introduced [3]-[9]. Even though design approaches were
taking into consideration in [3],[4] to produce constant
torque and minimizing torque ripple, control techniques
have been more considered to control the motor torque in
conventional switched reluctance motors. One of the
most convenient approaches is to coordinate incoming
and outgoing phase torque so that the resultant torque
remains constant during the commutation between
phases. Torque Sharing Function (TSF) is a well known
torque control alternative in which reference torques for
individual phases defined assuming the total torque stays
at constant level. Direct instantaneous torque control
(DITC) has also the simple torque control concept.
However, implementation of switching rules is
complicated. Some of the conventional TSFs such as
linear, cosine, cubic and exponential types reported in
literature [5]. One of the most drawbacks of TSF method
in industrial applications is the limitation of maximum
speed with acceptable torque ripples. Effective and peak
phase currents, power loss and voltage requirements
during the commutation would be some possible
secondary objective functions to make an optimal TSF.
[6]introduced θ ci , the rotor position where two adjacent
phases produce the same torque at the same phase current
and demonstrated that balanced commutation can
guarantees the higher efficiency and reduced power loss
during the commutation area. However, minimum
required commutation interval depends on rotor speed. In
other words, increasing the operating speed would cause
to extending the minimum required commutation area.
vujicic[7], proposed a family of TSFs to extend the
torque ripple-free speed range. However, the method was
based on motor parameters and efficiency improvement
can’t be guaranteed. Negative torque compensation using
a modified TSF also presented by [8], in which reference
phase torque in incoming phase regulated to compensate
the negative torque rises from the tail of outgoing phase
current. However, receding turn-off angle can eliminate
negative torque especially in mid-speed range of
operation. In this paper, a simple and convenient method
is introduced to modify the conventional TSF in order to
minimize torque ripple and enhance torque-speed
characteristics. Phase torque tracking error in incoming
phase conduction, compensated by outgoing phase torque
as well as excess of torque in outgoing phase caused by
converter voltage saturation and limitation decreased the
reference torque of incoming phase. Efficiency
improvement, feasibility and effectiveness of the
proposed method will be demonstrated by simulation
results.
2.
Fig.2 represents the conventional profiles of four typical
TSFs which are linear, cosine, cubic and exponential
forms.
Conventional Torque Sharing Functions
Energy conversion from magnetic to mechanical type has
strong relationship with rotor position and phase current
in SRM. The produced torque of each phase due to
flowing of ik at rotor position of θ can be calculated
Reference Torque
from (1):
Te ,k =
∂ ∫ λk (ik , θ )dik
∂θ
Stroke Angle)
Cubic TSF
k = 1,2,....., m
(1)
where m is the number of phases. Generally, the
relationship between flux and phase current is not linear.
Therefore obtaining a distinctive formula to calculate
exact phase torque as a function of phase current and
rotor position will not be possible. However, in many
references, (2) is used to find phase torque which comes
from linear relationship of flux and current.
1 ∂L(θ , ik ) 2
Te ,k = .
.ik
2
∂θ
(2)
Simple concept along with feasibility to implement in
almost every kinds of SRM caused TSF to be considered
dramatically in both research and industrial applications.
Generally, the conduction period of each phase is divided
into two regions. One is single phase conduction and the
other is two phase or commutation between adjacent
phases. In TSF method the reference torque of incoming
phase considered as a specific function so that total
produced torque by outgoing and entering phase
remained constant. Fig.1 shows the conventional block
diagram for TSF control method for a 3-phase SRM. The
input reference torque divided into individual phase
torques through the TSF functions. Phase reference
currents obtained from “torque to current” look up tables
at each rotor position and PWM or hysteresis control
block applied to get the desired currents. The “torque to
Flux” look up tables can also be used to control phase
flux in inner control loop [9].
Tref
on
on+ ov
off
Rotor Position
Reference Torque
Exponential
TSF
Stroke Angle)
Tref
on
on+ ov
off
Rotor Position
Fig. 2: Conventional TSF profiles: Cosine, linear, cubic and exponential
Reference torques are defined for incoming and outgoing
phases during the commutation interval as follows:
TPh* = Tref . f Linear
f Linear
⎧ (θ − θ on )
⎪
⎪ θ ov
⎪
= ⎨1
⎪
⎪ (θ off − θ )
⎪⎩ θ ov
θ on ≤ θ ≤ θ on + θ ov
θ on + θ ov ≤ θ ≤ θ off −θ ov
θ off − θ ov ≤ θ ≤ θ off
TPh* = Tref . f Co sin e
f Co sin e
⎧ 2 ⎛ π ⎛ θ − θ on ⎞ ⎞
⎟⎟ ⎟
θ on ≤ θ ≤ θ on + θ ov
⎪Sin ⎜⎜ ⎜⎜
⎟
⎪
⎝ 2 ⎝ θ ov ⎠ ⎠
⎪
θ on + θ ov ≤ θ ≤ θ off −θ ov
= ⎨1
⎪
⎪Sin 2 ⎛⎜ (θ off − θ ) ⎞⎟
θ off − θ ov ≤ θ ≤ θ off
⎜ θ
⎟
⎪
ov
⎝
⎠
⎩
*
= Tref . fCubic
TPh
⎧3
2
2
3
θon ≤ θ ≤ θon + θov
⎪ 2 (θ − θon ) − 3 (θ − θon )
θ
θ
ov
ov
⎪⎪
θon + θov ≤ θ ≤ θoff −θov
fCubic = ⎨1
⎪
3
⎪1 − (θ − θoff + θov )2 − 2 (θ − θoff + θov )3 θoff − θov ≤ θ ≤ θoff
⎪⎩ θov2
θov3
Fig. 1: Basic torque control block diagram using TSF
(3)
(4)
(5)
3.
TPh* = Tref . f Expon
⎧
2
⎛
⎞
⎪1 − Exp⎜ − (θ − θ on ) ⎟
θ on ≤ θ ≤ θ on + θ ov
(6)
⎜
⎪
θ ov ⎟⎠
⎝
⎪⎪
θ on + θ ov ≤ θ ≤ θ off −θ ov
f Expon = ⎨1
⎪
⎛ (θ off − θ ov − θ )2 ⎞
⎪
⎜
⎟
θ off − θ ov ≤ θ ≤ θ off
⎪Exp⎜ −
⎟
θ
ov
⎪⎩
⎝
⎠
In order to increase the efficiency of TSF, it would be
desired to operate in maximum torque per Ampere
(MTPA) condition. Many investigations have been done
to find the optimum overlap period and turn-on angle
such that minimum copper loss occurred during the
commutation period. [6] Introduced θ ci the angle where
two adjacent phases produce the same torque at same
current. It is shown that if turn-on and turn-off angle are
determined such that the intersection point of reference
torques take place in θ ci the optimum performance and
minimum torque ripple factor would be achieved. Torque
ripple factor obtained from the following equation.
T −T
T .R% = Max Min *100
Tave
(7)
Neglecting the stator resistance with some simple
manipulations it can be shown that the minimum overlap
period required for commutation between two adjacent
phases at torque T and speed of ω obtained from (8):
⎧
θ ov ,min = max ⎨
T
⎩ ak (θ )
.
Lk (θ )ω ⎫
⎬
Vdc ⎭
(8)
In the above equation:
ak (θ ) =
1 ∂L (θ , i )
2 ∂θ
(9)
One of the most drawbacks of TSF methods in torque
control problem as well as maximizing torque per ampere
ratio is that the maximum speed, in which the torque
ripple-free performance can be obtained. This value will
be much lower than base speed especially at nominal
torque value. It is obvious that some constraints on
raising and falling time may cause deteriorations in
torque control performance. In other words, if it is desired
to achieve the rated torque at the rated speed, the turn-on
and turn-off angle must be changed in advance and it
would cause to decrease torque per ampere ratio
dramatically. Moreover, receding turn-on angle may
cause to increase the peak current at the starting
conduction of phase in order to produce the desired
torque. This may pull down the efficiency and imposes
higher copper loss in commutation intervals. Setting back
the turn-off angle can prevents tail current to enter to the
negative torque region. In this paper, simple
modifications are proposed to reduce torque ripples and
increase maximum torque ripple-free speed based on
conventional TSFs. This novel method can be applied on
any torque sharing function methods to enhance the
efficiency with voltage limitation considerations.
Modified Torque Sharing Functions
One of the most important constraints for phase torque
tracking in TSF method is the limitation of converter
voltage. During the excitation of incoming phase, phase
current rises up from zero to the required value to
produce reference torque. This will be achieved
instantaneously at start up only where the speed is zero
and the magnitude of induced voltage can be negligible.
(8) Indicates that in order to obtain desirable torque ripple
factor at constant torque, applied converter voltage
should be increased when the speed comes up. In other
words, increasing the operating speed imposes higher
converter voltage at constant torque and commutation
interval. Due to limitation in dc-link voltage torque
control performance will be deteriorated using TSF
method when the speed increased. Neglecting stator
resistance, phase voltages are calculated from the
following equation:
vk ≈
∂λk
.ω
∂θ
(10)
According to equation (10) maximum speed with torque
ripple-free operation can be determined from (11).
⎛
⎞
⎜
⎟
V
V
−
+
DC
DC
(11)
⎜
⎟
,
ωmax = min
⎜
⎛ ∂λ ⎞ ⎟
⎛ ∂λ ⎞
⎜ Max⎜ ⎟ Min⎜ ⎟ ⎟
⎝ ∂θ ⎠ ⎠
⎝ ∂θ ⎠
⎝
From the above equation if the turn-on angle reduced
toward the unaligned position the phase current peak
value will be increased to produce the same rate of
change for stator flux. This will yield to rise up copper
losses and reduce drive efficiency. In addition, high
inductance near the aligned rotor position may cause to
produce phase torque greater than the desired reference
value. In the proposed method, deficiency of produced
phase torque in incoming phase will be compensated by
excess of phase torque generated from the error of
outgoing phase current. Fig 3 shows the concept of
modified procedure for a simple linear TSF.
Fig. 3: Concept of proposed modification method for linear TSF
In the above figure, “incoming effect” refers to the lack
of torque production in incoming phase due to low value
of incremental inductance and generated torque near the
unaligned rotor position. Similarly, “outgoing effect”
refers to the excess of produced torque by outgoing phase
due to high inductance value near the aligned rotor
position which caused to negative phase current error in
inner control loop represented by fig 1. Fig 4 displays the
concept of incoming and outgoing effects from the rate of
changing flux point of view.
Rate of changing in
flux respect to rotor
position
Starting Effect (P)
on
on+ ov
off
Rotor Position
Ending Effect (D)
Fig. 4: Typical rate of flux changing respect to rotor position
considering conventional TSFs
The above figure illustrates that starting and ending
effect would not take place at the same rotor position.
The magnitude of P and D depends on turn-on and turnoff angles as well as the desired torque for each phases
and the corresponding rotor position. Considering this
principle, limitation due to starting effect can be
compensated by outgoing phase. In the proposed method,
incoming torque phase will be compensated by the
outgoing phase and additional outgoing torque phase will
decrease the reference torque of incoming phase.
Compared with receding turn-on and turn-off angle
the proposed method is more efficient due to using feed
forward control procedure and reduce phase peak current.
Fig 5 represents the control block diagram for modified
torque sharing function. The current control loop may
consist of hysteresis or simple pi controller. When the
speed starts to increase, the torque error in incoming
phase would be positive, while torque error for outgoing
phase will be negative.
Fig. 5: Control block diagram for modified TSF
Generally, receding turn-on and turn-off angles
introduced in literature to overcome the negative torque
production due to continuing of the outgoing phase
current. However, this method my cause to decrease the
efficiency and increase phase peak current. One of the
most advantages of the proposed modification method is
to enhance the torque control performance and efficiency
considering constant conduction angles.
4.
Simulation Results
In order to verify the proposed scheme, a four phase
8/6, 1.6 KW SRM experimental data is considered to
simulate the introduced approach. Each phase has 1.005
ohm resistance and the nominal rated voltage is 280 V.
This motor is an industrial type of SRM with 8(N.m)
rated torque and has impressive nonlinearity in magnetic
characteristic. The motor is considered to operate at
nominal load torque and simulations were undertaken
using MATLAB software based on the concept
represented by fig 5. Turn-on and off angles are selected
so that rated speed without negative phase torque
obtained. In this case, phase conduction will be started at
32 degree and ended at 52degree. Fig 6 shows the phase
currents, phase torques and the resultant torque using
linear TSF without the proposed modification at 1500
rpm.
Fig. 6: Phase Current, Phase Torques and resultant torques using linear
TSF without using the modification scheme at rated load torque and
speed of 1500 rpm
It’s clear from the above figure that torque control
performance deteriorated due to starting and ending
effects explained before. Fig 7 shows phase torques and
motor torque obtained from conventional linear, cosine,
cubic and exponential TSFs. Simulation results
demonstrated that torque control performance and torque
ripple factor would not be in acceptable level due to
ignoring voltage limitation in control process.
Considering converter voltage saturation at the starting of
phase conduction can reduced the dip magnitude in
resultant motor torque. Moreover, overshoots caused by
torque control error in the ending of the phase
commutation will be suppressed. Fig 8 represents the
reference phase torques, actual and resultant torques for
four conventional already mentioned TSFs considering
the proposed modifications scheme.
a
a
b
b
c
c
d
d
Fig. 7: Reference, actual phase and resultant torques considering
rated load torque and the speed of 1500 rpm using conventional TSFs,
a)linear, b)cosine, c) cubic and d)exponential
Torque ripple factor defined by (7) is considered to
show the effectiveness of torque control approaches. This
parameter obtained from the motor torque curves which
are 66.6, 64.9, 65.2 and 87.7 for linear, cosine, cubic and
exponential TSFs, respectively. However, torque ripple
factor calculated for four TSFs using the proposed
modification method are around 6.25%. It can be shown
in fig 8 that torque error for incoming phase added to
reference torque of outgoing phase, while the
corresponding torque error of outgoing phase affects on
the reference torque of incoming phase during the
commutation period. It is almost shown that reference
phase torques converge to similar TSF using this
modification method when the speed increased.
Fig. 8: Reference, actual phase and resultant torques considering
rated load torque and the speed of 1500 rpm using proposed TSFs,
a)linear, b) cosine, c) cubic and d) exponential
5.
Conclusions
In this paper, a novel modification method introduced
which has the ability to be applied on any torque sharing
functions. The proposed scheme considered voltage
saturation and limitation of SRM converter. Due to low
capability of torque production at the starting of incoming
phase and converter voltage constraints during the
commutation area, incoming phase torque control failed
to operate properly especially in mid-high speed region.
This yields dip ripples on the resultant motor torque and
deteriorates torque average and ripple factor dramatically.
Similarly, torque error in outgoing phase torque
controller may cause some tips on produced motor
torque. In the proposed modification method, torque error
in incoming phase added to the reference torque of
outgoing phase as a positive value to regulate total torque
during the starting of phase conduction. While, phase
torque error of outgoing phase added to the reference
torque of incoming phase as a negative value.
Simulations were done and results demonstrated the
effectiveness of proposed modification scheme. It
confirmed the superiority of novel TSF compared than
conventional types in torque ripple minimization and
improvements in torque-speed characteristics.
Acknowledgements
This work was supported by the National Research
Foundation of Korea (NRF) grant funded by Korea government
(Grant No. 2010-0014172).
References
[1] S.K. Sahoo, S. Dasgupta, S.K. Panda, J.X. Xu, “A Lyapunov
Function Based Robust Direct torque Controller for Switched
Reluctance Motor Drive System”, IEEE Trans. On Power
Electronics, vol. 27, pp. 555-564, Feb. 2012.
[2] R. Krishnan, Switched Reluctance Motor Drives: Modeling,
Simulation, Analysis, Design and Applications. Boca Rotan, FL:
CRC Press, 2001.
[3] Y. K. Choi, H.S. Yoon, C. S. Koh, “Pole-shaped optimization of a
switched reluctance motor for torque ripple reduction”, IEEE
Trans. Magn., vol. 43, no. 4, pp. 1797-1800, Apr.2007.
[4] S. I. Nebeta, I. E. Chabu, L. Lebensztajn, D. A. P. Correa, W. M.
Dasilva, K. Hameyer, “Mitigation of the torque ripple of a
switched reluctance motor througha ultiobjective optimization”,
IEEE Trans. Magn., vol. 44, no. 6, pp. 1018-1021, Jun. 2008.
[5] I. Husain, “Minimization of toruqe ripples in SRM drives”, IEEE
Trans. Ind. Electron., vol. 49, no.1, pp. 28-39, Feb. 2002.
[6] R. S. Wallace, D.G. Taylor, “A balanced commutatorfor switched
reluctance motors to reduce torque ripple”, IEEE Trans. Power
Electron., vol. 7, no. 4, pp. 617-626, Oct. 1992.
[7] V. P. Vujicic, “Minimization of torque ripple and copper losses in
switched reluctance drive”, IEEE Trans. Pow. Electron., vol. 27,
no. 1, Jan. 2012.
[8] S.Y., Ahn, J.W., Ahn, D.H., Lee, “A Novel Torque Controller
Design for High Speed SRM Using Negative Torque
Compensator”, 8th International Conference on Power ElectronicsECCE Asia, PP.937-944, May 30-June 3, 2011.
[9] D. H. Lee, J. Liang, Z. G. Lee, J. W. Ahn, “A simple nonlinear
logical torque sharing function for low toruqe ripple SR drives”,
IEEE Trans. Ind. Electron., vol. 56, no. 8, pp. 3021-3028, Aug.
2009.