Theoretical and Performance Analysis of PWM
Control-Based Variable Switching Frequency for
Torque Ripple Reduction in SPMSM Drive Systems
Mohamed G. Hussien
Dept. of Electrical Power and Machines
Engineering, Faculty of Engineering,
Tanta University, Tanta, Egypt
mohamed.hussien3@f-eng.tanta.edu.eg
P. Sanjeevikumar
Dept. of Energy Technology,
Aalborg University,
Esbjerg, Denmark
san@et.aau.dk
Jens Bo Holm-Nielsen
Center for Bioenergy and Green Engineering,
Dept. of Energy Technology,
Aalborg University, Esbjerg, Denmark
jhn@et.aau.dk
Abstract— In this paper, the three-phase current ripples and
also the torque ripple of a surface-mounted permanent magnet
synchronous motor (SPMSM) were thoroughly analyzed and
minimized based on variable switching frequency PWM
(VSFPWM) method. The basic SPMSM model is developed and
simulated using MATLAB/SIMULINK environment. For
control the torque ripple, VSFPWM method was applied with
its control diagram, which consists of a torque ripple calculation
module and switching period updating module. Effectiveness of
the presented control method are verified, some of the obtained
simulation results are discussed, and the comparison of
switching frequency caused by different PWM method is
analyzed.
Keywords—Surface-mounted PM synchronous motor
(SPMSM), torque ripple prediction, variable switching frequency
PWM (VSFPWM).
Zbigniew Leonowicz
Faculty of Electrical Engineering,
Wroclaw University of Technology,
Wroclaw, Poland.
leonowicz@ieee.org
Lucian Mihet-Popa
Smart Energy Science and Technology.
Østfold University College,
Fredrikstad, Norway
lucian.mihet@hiof.no
the fundamental output frequency of the inverter. It can be
approximated in any sector by the weighted average
combination of two adjacent vectors and a null state vector 0
or 7. For better performance of the motor drives, the torque or
current ripples should be minimized in most of the industrial
applications [18]-[19].
This paper aims to apply the variable switching frequency
PWM (VSFPWM) method for torque ripple control. The
simulation process is presented as follows: the motor is
initially driven by an inverter with 2.5 kHz constant switching
frequency PWM (CSFPWM) under 25.8 N.m load to achieve
rated speed. When the process of dynamic response has been
finished, then the control program is turned to a variable
switching frequency PWM algorithm. The control target for
VSFPWM is the identical torque ripple peak value at stable
operation conditions.
I. INTRODUCTION
Three-phase inverters provide to the stator the currents and
voltages of variable magnitude and frequency needed for
motor drives. Depending on the type of d.c. source supplying
the inverters, they can be classified as voltage source inverters
(VSI) [1]-[14], shown in Fig. 1, or current source inverters
(CSI) [15]-[16]. A rectifier whose output is fed into an LC
filter constitutes the d.c. source of the inverter and is called d.c
link. It is also possible to have a battery instead of the rectifier,
but the LC filter is used to smooth the high-frequency
components that could damage the battery.
To achieve control of the magnitude and frequency of the
output voltage pulse-width modulation (PWM) can be used.
A PWM generator produces modulated pulses to control the
switches of the inverter in such a way that the impressed stator
voltage will be proportional to a given input reference signal.
The most basic design is known as sine triangle-modulation
because a triangle carrier is compared with a sinusoidal
reference to create the modulated PWM output [10], [17]. The
maximum phase voltage that can be produced by this type of
modulation approach is one half of the d.c. link amplitude and
it is possible to achieve higher phase voltages by using other
types of modulation such as third harmonic and space vector
modulation (SVPWM). The central point of this technique
resides in the fact that a balanced three-phase system can be
represented as a vector rotating with the angular velocity of
Fig. 1. Inverter-fed SPMSM system.
II. PREDICTION OF CURRENT AND TORQUE RIPPLES
A. Current Ripple Prediction
The detailed principle of current ripple prediction can be
acquired in [18]-[19].
For three-phase converters, the average formula of the
output voltage can be written as
978-1-7281-7455-6/20/$31.00 ©2020 IEEE
Authorized licensed use limited to: University of Missouri Libraries. Downloaded on October 31,2022 at 11:31:04 UTC from IEEE Xplore. Restrictions apply.
′
=
′
=
⋅
1
(1)
=
′
With the aid of the principles of SVPWM, in Fig. 2,
including the seven zones of switching cases, the changes in
the ripple of current in each one cycle will also consider seven
states as illustrated in Fig. 3.
′
=
′
′
′
′
′
′
′
(4)
(5)
′
(6)
With the aid of the specified circuits in Fig. 4 for the
associated vectors cases, the corresponding slope equation of
ripples can be summarized as in Table.1. With the specified
relations in Table 1, the peak level of the slope can be given
as in (7) with the values x and y to be plus or minus signs.
=
=
.
.
(7)
.
B. Torque Ripple Prediction
It is necessary to clarify the relationship between current
ripple and PWM torque ripple in each switching cycle [18].
Given the fact that the phase current is consist of the
fundamental component and the ripple component, which is
needed to be captured with the target of PWM torque ripple.
By transferring the current ripple from abc coordinate to d-q
coordinate, d-q real-time current ripple is derived.
Fig. 2. Seven zones in one switching cycle in SVPWM.
_
_
=
(8)
_
_
_
where,
2
3
2
3
2
=
3
2
3
2
3
where
,
and
are a real-time
_
_
_
and
current ripple in abc coordinate,
_
_
are the d-q components. The angle, indicates the d-axis
position.
Furthermore, the PWM torque ripple can be expressed in
(9) as a function of q-axis current ripple.
=
⋅
_
(9)
Fig. 3. Ripple current variation in one switching cycle in SVPWM.
As shown in Fig. 4, there are eight vectors can be regarded
with the equivalent circuit and obtained aided with (1) and
given as in (3).
=
=
(2)
′
′
′
(3)
In the second zone, and the current slope is derived as (4).
In the third zone, the current slope is derived as (5). In the
fourth zone, the slope ia is derived in (6).
) is predicted, and the
Finally, PWM torque ripple (
locus of PWM torque ripple is achieved simultaneously.
Real-time prediction can be made in each switching cycle,
and variable switching frequency PWM (VSFPWM) can be
done through this prediction.
To confirm the effectiveness of the presented method, for
torque ripple prediction, some of the obtained results are
introduced using constant switching frequency PWM (fsw =
2.5 kHz) under the loading condition of rated torque while the
SPMSM is accelerated at rated speed. Fig. 5 shows the speed
response with a reference speed of 1000 rpm. Also, Fig. 6
shows the response of electromagnetic torque with its ripple
contents. Moreover, Fig. 7 shows the three-phase stator
currents.
Authorized licensed use limited to: University of Missouri Libraries. Downloaded on October 31,2022 at 11:31:04 UTC from IEEE Xplore. Restrictions apply.
Fig. 4. Switch combination of eight different voltage vectors and their Thevenin equivalent circuits.
TABLE. 1. RIPPLE CURRENT SLOPE FOR THE THREE-PHASES A, B, C WITH DIFFERENT VOLTAGE VECTORS.
Vector
Phase_A
′
000
100
′
′
′
′
111
010
011
101
′
′
1
110
001
Phase_B
′
′
′
′
′
′
′
′
′
′
′
1
′
′
′
′
′
′
′
′
′
′
′
1
′
′
′
1
′
′
1
′
′
1
1
′
′
′
′
′
′
1
′
′
′
′
′
1
′
′
1
Phase_C
′
′
′
′
′
′
′
′
′
′
′
′
′
′
′
′
′
1
′
′
′
1
Authorized licensed use limited to: University of Missouri Libraries. Downloaded on October 31,2022 at 11:31:04 UTC from IEEE Xplore. Restrictions apply.
Fig. 5. SPMSM speed response (under constant switching frequency PWM
[fsw = 2.5 kHz]).
Fig. 8. VSFPWM control diagram.
Fig. 6. Response of electromagnetic torque (load torque = 25.8 N.m)
(under constant switching frequency PWM [fsw = 2.5 kHz]).
Fig. 9. Switching period calculation module.
60
40
20
The updated switching cycle is calculated in (10) where
the PWM torque ripple is proportional to the switching cycle.
The updated switching period will control the maximum
PWM torque ripple to be equal with the PWM torque ripple
requirement in each switching cycle.
50
0
0
-50
0.6
-20
0.62
0.64
-40
-60
=
0
0.1
0.2
0.3
0.4
0.5
0.6
⋅
(10)
0.7
Time (s)
Fig. 7. Three-phase stator currents (under constant switching frequency
PWM [fsw = 2.5 kHz]).
III. VARIABLE SWITCHING FREQUENCY PWM (VSFPWM)
METHOD FOR TORQUE RIPPLE CONTROL
The control target for VSFPWM is the identical torque
ripple peak value at stable operation conditions. In each
switching period, the torque ripple of the system model can be
calculated under the d-q axis frame through the phase current
ripple prediction method aided with (9).
VSFPWM for torque ripple control of SPMSM focus on
the PWM method and still based on vector control method.
Fig. 8 shows the VSFPWM control diagram. In the controller,
duty cycles (da, db, dc) are calculated in every interrupt cycle,
and torque ripple prediction module reads duty cycles after
receiving a sampling signal. PWM torque ripple can be
predicted in each switching cycle. Then, based on the torque
ripple peak value requirement, switching period is updated.
A sampling signal will be generated when a complete carrier
waveform has been sent to the comparator, and the VSFPWM
control will enter the next update. Fig. 9 shows the switching
period calculation module, the concrete key of VSFPWM.
is the PWM torque ripple peak value in the
where
whole fundamental period, with the nominal switching period
. The time is the updated switching period, and satisfies
that PWM torque ripple peak value of every switching period
will be equal with
.
To confirming the capability of the presented VSFPWM
method, for torque ripple control, some of the obtained results
are introduced in comparison with the constant switching
frequency PWM method. The system is started typically using
constant switching frequency PWM (fsw = 2.5 kHz) under the
loading condition of rated torque while the SPMSM is
accelerated at rated speed. Then, at t = 0.7 s, the switch over
process is executed to activate the presented VSFPWM
method for torque ripple control.
Fig. 10 shows the response of electromagnetic torque with
its ripple contents (during the switch over process between
constant switching frequency PWM and VSFPWM).
Besides, Fig. 11 shows the corresponding three-phase stator
currents during the control transition (the switch over process
is obtained at t = 0.7 s). Furthermore, the variation of the
switching frequency during the switch over process is shown
in Fig. 12. The switching frequency is started with [fsw = 2.5
kHz] for constant switching frequency PWM method and
then changed according to (10) for VSFPWM method as
shown in Fig. 12.
Authorized licensed use limited to: University of Missouri Libraries. Downloaded on October 31,2022 at 11:31:04 UTC from IEEE Xplore. Restrictions apply.
Electromagnetic Torque (N.m)
REFERENCES
[1]
[2]
[3]
[4]
Fig. 10. The response of electromagnetic torque (Switch over process
between constant switching frequency PWM and VSFPWM).
[5]
[6]
[7]
Fig. 11. Three-phase stator currents (Switch over process).
[8]
[9]
[10]
[11]
Fig. 12. Switching frequency variation.
The obtained simulation results confirm the effectiveness
of the presented VSFPWM control method for torque ripple
minimization, as illustrated in Fig. 10. Henceforth, ensures the
capability of VSFPWM method for torque ripple control
purpose.
[12]
[13]
IV. CONCLUSION
In this paper, the torque ripple of a surface-mounted
permanent magnet synchronous motor (SPMSM) has been
thoroughly analyzed and minimized based on variable
switching frequency PWM (VSFPWM) method. The main
control diagram of VSFPWM has been described, which
consists of a torque ripple calculation module and switching
period updating module. Effectiveness of the presented
VSFPWM control method for torque ripple control, verified
by obtained simulation results and been discussed, and the
comparison of switching frequency caused by the constant
switching frequency PWM and VSFPWM method has been
thoroughly analyzed. The obtained results have verified the
effectiveness of the presented method for current and torque
ripple prediction. Moreover, the applied VSFPWM method
has been successfully implemented for torque ripple reduction
based on the predicted waveforms. This has ensured the
capability of VSFPWM control method for torque ripple
control target.
[14]
[15]
[16]
[17]
[18]
[19]
Kiran M.Pandav, S.B.Mahajan, P.Sanjeevikumar, Sunita M.Badave,
Ruchi Pachagade , “2.4 kW Three-Phase Inverter for Aircraft
Application-Hardware Implementation”, Lecture Notes in Electrical
Engineering, Springer Journal Publications, Mar. 2017.
M. K. Kazimierczuk, Pulse-width Modulated dc-dc Power Converters.
John Wiley & Sons, 2008.
V. Vorp´erian, “Simplified analysis of PWM converters using the
model of the PWM switch: Parts I and II,” IEEE Trans. Aerospace and
Electronic Systems, vol. 26, no. 3, pp. 490–505, May 1990.
V. Vorperian, R. Tymerski, and F. C. Y. Lee, “Equivalent circuit
models for resonant and PWM switches,” IEEE Trans. Power
Electron., vol. 4, no. 2, pp. 205–214, Apr. 1989.
M. G. Hussien and A. E. Hassan, “Mathematical Analysis of the Small
Signal Model for Voltage-Source Inverter in SPMSM Drive Systems,”
2019 21st International Middle East Power Systems Conference
(MEPCON), Cairo, Egypt, Dec. 2019, pp. 540-549.
B. Chokkalingam, M. S. Bhaskar, S. Padmanaban, V. K.
Ramachandaramurthy, and A. Iqbal, “Investigations of multi-carrier
pulse width modulation schemes for diode free neutral point clamped
multilevel inverters,” J. Power Electron., vol. 19, no. 3, pp. 702–713,
May 2019.
C.Bharatiraja, S.Raghu, J.L Munda, P.Sanjeevikumar, “Analysis,
Design and Investigation on a New Single-Phase Switched Quasi ZSource Inverter for Photovoltaic Application”, Intl. Journal of Power
Electronics and Drive Systems (IJPEDS), Institute of Advanced
Engineering and Science (IAES) Publication, Indonesia, vol. 8, no. 2,
pp. 853-860, Jun. 2017.
Pandav K. M., S.B.Mahajan, P.Sanjeevikumar, Frede Blaabjerg,
“Three Phase Z-Source Multilevel Inverter System for Renewable
Energy Application”, Intl. Conf. on Renewable Energy and Resources,
OMICS International Publishing, USA, Jul. 24-25, 2017.
P.Sanjeevikumar, M.S.Bhaskar, Kiran M.Pandav, Pierluigi Siano,
Valentin Oleschuk, “Hexuple-Inverter Configuration for Multilevel
Nine-Phase Symmetrical Open- Winding Converter”, Conf. Proc.,
IEEE First Intl. Conf. Power Electron., Intelligent Control and Energy
System, IEEE-ICPEICES’16, pp. 1837–1844, India, 4–6 Jul. 2016.
P.Sanjeevikumar, Pierluigi Siano, Ahmet H.Ertas, S.Rajasekar, Kiran
M. Pandav, “Single-Phase Seven-Level Stack Multicell Converter
Using Level Shifting SPWM Technique”, Conf. Proc. of 16 IEEE Intl.
Conf. on Environment and Electrical Engg., Italy, 7-10 Jun. 2016.
W. Xu, M. G. Hussien, Y. Liu, and S. M. Allam, “Sensorless control of
ship shaft stand-alone BDFIGs based on reactive-power MRAS
observer,” IEEE J. Emerg. Sel. Topics Power Electron., 2020, in press.
W. Xu, M. G. Hussien, Y. Liu, M. R. Islam, and S. M. Allam,
“Sensorless voltage control schemes for Brushless Doubly-Fed
Induction Generators in stand-alone and grid-connected applications,”
IEEE Trans. Energy Convers., 2020, in press.
R.Chinthamalla, P.Sanjeevikumar, R.Karampuria, Sachin Jain, Ahmet
H.Ertas, Viliam Fedak, “A Solar PV Water Pumping Solution Using a
Three-Level Cascaded Inverter Connected Induction Motor Drive”,
Engineering Science and Technology: An International Journal
(JESTECH), vol. 19, no. 4, pp. 1731–1741, Dec. 2016.
R.Gunabalan, P.Sanjeevikumar, Frede Blaabjerg, Olorunfemi
Ojo, V.Subbiah, “Analysis and Implementation of Parallel Connected
Two Induction Motor Single Inverter Drive by Direct Vector Control
for Industrial Application”, IEEE Trans. Power Electron., vol. 30, no.
12, pp. 6472–6475, Dec. 2015.
M. Hombu, S. Ueda, and A. Ueda, “A current source GTO inverter with
sinusoidal Output voltage and current,” IEEE Trans. Ind. Appl., vol.
21, no. 2, pp. 1192-1198, Sep./Oct 1985.
M. Hombu, S. Ueda, and A. Ueda, “A current source GTO inverter with
sinusoidal input and outputs,” IEEE Trans. Ind. Appl., vol. 23, no. 2,
pp. 247-255, Mar./Apr 1987.
Michael A. Boost and Phoivos D. Ziogas, “State of the Art Carrier
PWM Techniques: A Critical Evaluation,” IEEE Trans. Ind. Appl.,
vol.24, no. 2, pp. 271-280, Mar./Apr. 1988.
Dong Jiang and Fei Wang, "Current Ripple Prediction for Three-phase
PWM Converters," IEEE Trans. Ind. Appl., vol. 50, no. 1, Jan.-Feb.
2014, pp.531-538.
Dong Jiang and Fei Wang, "Variable Switching Frequency PWM for
Three-phase Converters Based on Current Ripple Prediction," IEEE
Trans. Power Electron., vol. 28, no. 11, Nov. 2013, pp. 4951-4961.
Authorized licensed use limited to: University of Missouri Libraries. Downloaded on October 31,2022 at 11:31:04 UTC from IEEE Xplore. Restrictions apply.