The wavelet PWM (WPWM) technique has been applied in twolevel inverters successfully, but directly applying the WPWM technique to threelevel inverters is impossible. This paper proposes a WPWM technique suitable for a singlephase threelevel inverter. The work analyzes the control strategy with the WPWM and obtains the design of its parameters. Compared with the SPWM technique for a singlephase threelevel inverter under the same conditions, the WPWM can obtain high magnitudes of the output fundamental frequency component, low total harmonic distortion, and simpler digital implementation. The feasibility experiment is given to verify of the proposed WPWM technique.
I. INTRODUCTION
In recent years, Saleh has proposed and developed the wavelet modulation techniques on different twolevel converters
[1]

[5]
. The wavelet modulation technique is based on establishing a nondyadic type multi resolution analysis (MRA), which is required to support a nonuniform recurrent sampling–reconstruction process. The merits of this approach includes simpler realization by digital algorithm, higher magnitudes of the fundamental output voltage, and lower harmonic contents better than other types of modulation techniques. In
[1]
and
[2]
, the manner of implementation of a wavelet modulation technique for singlephase voltage source inverters was proposed. The manner of implementation of a wavelet modulation technique for threephase voltagesource sixpulse inverters was proposed in
[3]
and
[4]
. In
[5]
, the manner of implementation of a wavelet modulation technique for AC–DC converters was proposed. Hence, the present research on the WPWM technique focuses on the twolevel inverter.
Compared with the twolevel inverter, the threelevel inverter is a new type of highvoltage large capacity power converter with advantages of having improved voltage waveform on the AC side, smaller filter size, lower electromagnetic interference, and lower acoustic noise
[6]
. Therefore, threelevel inverter options are attracting greater attention in the fields of the grid interconnection, new energy, fuel electromagnetic
[7]
,
[8]
. Because of the wide application of threelevel inverters, the study of its control strategy has been increasingly highlighted
[9]

[15]
. One of commonly used control strategies is sinusoidal pulse width modulation (SPWM). The SPWM technique for twolevel inverters only needs a modulating signal and a carrier signal, but the conventional SPWM technique for threelevel inverters needs a modulating signal, two carrier signals, and a square signal. Thus, directly applying the WPWM technique to threelevel inverters is impossible because the WPWM technique to twolevel inverters can only generate two unipolarcontrolled signals or two bipolarcontrolled signals.
Thus, this paper presents the development and performance testing of the WPWM technique for singlephase threelevel inverters. The singlephase threelevel inverter with SPWM technique is reviewed in Section II. The singlephase threelevel inverter with WPWM technique is proposed in Section III. The analysis of the WPWM technique for the singlephase threelevel inverter is provided in Section IV. The experimental results are obtained in Section V. Conclusions are given in Section VI.
II. SINGLEPHASE THREELEVEL INVERTER WITH SPWM TECHNIQUE
Fig. 1
shows the circuit schematic of an asymmetric singlephase threelevel inverter
[17]
,
[18]
. The circuit is composed of a twolevel bridge and a threelevel bridge.
C
_{1}
and
C
_{2}
are the DC side filter capacitor. U
_{d}
is the DC voltage source. When U
_{d}
=E, Uao has three levels, i.e., +E/2, 0, and –E/2, and U
_{bo}
has two levels: +E/2, and –E/2, which, in total, gives output voltage U
_{ab}
five levels. The operation states of singlephase threelevel inverter are listed in
TABLE I
.
Scheme of the singlephase threelevel inverter.
OPERATION STATES OF THE SINGLEPHASE THREELEVEL INVERTER
OPERATION STATES OF THE SINGLEPHASE THREELEVEL INVERTER
A conventional SPWM scheme is shown in
Fig. 2
[18]
, which has a reference rectified sine wave (
V
_{ref}
) and two carrier signals (
v
_{tri1}
and
v
_{tri2}
). The comparison result of
V
_{ref}
and
v
_{tri1}
is the control signal of
A
_{1}
; the comparison result of
V
_{ref}
and
v
_{tri2}
is the control signal of
B
_{1}
; the comparison result of
V
_{ref}
and zero is the control signal of
C
_{1}
. Then, the control signals for switches
S
_{1}
–
S
_{6}
can be derived by
A
_{1}
,
B
_{1}
, and C
_{1}
, as shown in
Fig. 3
, where
S
_{1}
=
A
_{1}
and
;
SPWM operation principle of the threelevel inverter.
Logic control scheme for switches S1–S6.
III. WPWM TECHNIQUE FOR SINGLEPHASE THREELEVEL INVERTER
 A. Principle of the WPWM Technique
The WPWM technique is based on sampling–reconstructing a referencemodulating signal in a nonuniform recurrent manner using sets sampling and synthesis basis functions
[1
,
2]
. These sampling basis functions are generated as dilated and translated versions of the scalebased linearlycombined scaling function
φ
_{(j,k)}
(
t
) . Furthermore, synthesis basis functions are generated as dilated and translated versions of the scalebased linearly combined synthesis scaling function
The scalebased linearlycombined scaling function is defined at scale j as
and
φ
_{(j,k)}
(
t
)=
φ
(2
^{j}
t

k
), where
j
=0, 1, 2, 3, … and
ϕ_{H}
(
t
) is the Harr scaling function that is given by
Moreover, synthesis scaling function
associated with
φ
(
t
) can be defined as
Using these two dual scaling functions, a continuoustime signal
x_{c}
(
t
) can be expanded as
where
j
,
k
∈ Z, where Z is the set of integer numbers. Such form of signal processing suggests that a continuoustime signal <
x_{c}
(
t
),
φ
(2
^{j}
t

k
)> can be recovered from its samples using sets of synthesis functions
The work on
[10]
proved that the switching pulses for the inverter can be generated by using dilated and shifted versions of synthesis scaling function
When each cycle of
x_{c}
(
t
) is divided by a finite number of sample groups
D
, the length of the time interval of the sample group [
t
_{d1}
,
t
_{d2}
] changes as scale j changes, where
In addition, based on the the procedure on how to implement the WPWM technique given in
[1]
, the flowchart for WPWM can be shown as
Fig. 4
[16]
, where Tm is the period of the reference sine wave.
Flowchart of the WPWM technique implementation.
 B. WPWM Strategy for the SinglePhase ThreeLevel Inverter
According to the above flowchart of WPWM, once
T
_{m}
,
j
0 and
D
are given, the time points (
t
_{d1}
and
t
_{d2}
) of each sample group can be calculated, and the driving pulses can be generated by the time points in each sample group, which can be integrated into two unipolarcontrolled signals (
W
_{1}
and
W
_{2}
). However, the signals (
W
_{1}
and
W
_{2}
) cannot be used to control the switches of singlephase threelevel inverter directly. Thus, based on
Figs. 2
and
3
, the WPWM operation principle for the threelevel inverter can be shown as
Fig. 5
.
W
_{1}
and
W
_{2}
are generated according to the flowchart of the WPWM technique shown in
Fig. 4
. Pulse
P
_{1}
has a halfcycle symmetry property, its frequency is the double of reference sine wave, and its pulse width can be varied to adjust the distribution of the output voltage levels, which will be discussed in detail in the following. Pulse
C
_{1}
is a square signal, and its frequency is the same as the reference sine wave. Then, the control signals for switches
S
_{1}
–
S
_{6}
can be derived by the specific logic relationship among
W
_{1}
,
W
_{2}
,
P
_{1}
, and
C
_{1}
, as shown in
Fig. 6
, where
and
WPWM operation principle of threelevel inverter.
Logic control scheme for the switches S_{1}~S_{6} with WPWM technique.
Moreover,
Fig. 6
shows that the WPWM control strategy for the singlephase threelevel inverter can be implemented simply by a digital algorithm.
IV. ANALYSIS OF THE WPWM TECHNIQUE FOR THE SINGLEPHASE THREELEVEL INVERTER
To verify the control strategy of the WPWM technique for the singlephase threelevel inverter, a MATLAB/SIMULINK model is built and simulation is made by selecting
D
=30,
f
_{m}
=50 Hz (
f
_{m}
is the frequency of the reference sine wave),
j
0=0, the pulse width of
P
_{1}
is 50%, the simulation results of signals
W
_{1}
,
W
_{2}
,
W
_{3}
,
P
_{1}
,
C
_{1}
, and
S
_{1}
–
S
_{6}
can be obtained, as shown in
Fig. 7
. When input voltage U
_{d}
=50 V, output voltage U
_{ab}
is shown in
Fig. 8
.
Signals of W_{1}, W_{2}, W_{3}, P_{1}, C_{1}, and S_{1}–S_{6} at D=30, f_{m}=50 Hz, j0=0.
Output voltage U_{ab}.
According to the control strategy of the WPWM technique for the singlephase threelevel inverter, the width and position of the pulses (
W
_{1}
and
W
_{2}
) generated by the WPWM technique are determined when
D
and
j
0 are given, and
C
_{1}
is a determined square wave when the frequency of the reference sine wave is given. Therefore, the only way of changing the control signals for switches
S
_{1}
–
S
_{6}
is by adjusting the pulse width of
P
_{1}
, the distribution of the output voltage levels is affected.
To analyze the effects of pulse
P
_{1}
on the distribution of the output voltage levels, this study selects
D
=30,
f
_{m}
=50 Hz,
j
0=0, and input voltage U
_{d}
=50 V as a sample object to be simulated based on the MATLAB/SIMULINK model of a singlephase threelevel inverter, as shown in
Fig. 2
. The pulse width of
P
_{1}
is chosen in the range of 10%–90%. The simulation results of the total harmonic distortion (THD) and the amplitude of fundamental voltage
V
_{1}
for output voltage U
_{ab}
are shown in
Figs. 9
and
10
, respectively.
Fig. 9
shows that the THD is smallest when the pulse width of
P
_{1}
is about 62%.
Fig. 10
shows that
V
_{1}
increases as the pulse width of
P
_{1}
increases, and
V
_{1}
can be larger than the input voltage when the pulse width is larger than 50%.
THD of U_{ab} vs. the pulse width of P_{1}.
V_{1}of U_{ab} vs. the pulse width of P_{1}.
V. EXPERIMENTAL RESULTS
To verify the analysis of the WPWM technique for the singlephase threelevel inverter, the algorithm of the WPWM technique is implemented by using DSP (TMS320LF2812), and the input voltage of the singlephase threelevel inverter is 0V
V_{dc}
=50V , MOSFET IRFPE40 is selected as switch, TLP250 is used as driver, and pure resistance
R
= 50 Ω is used as the load. A photograph of the experimental setup is shown in
Fig. 11
. Note that the value of the THD is tested by Fluke Norma 5000 Power Analyzer.
Photograph of the experimental setup.
First, the experiments have been performed by choosing
f
_{m}
=50 Hz,
j
0=0,
D
=30, and the pulse width of
P
_{1}
=40%, 50%, 60%, 62%, 63%, 70%, 80%, and 90%. The experimental results are shown in
Fig. 12
.
Fig. 12
(a) shows that the THD is smallest when the pulse width of
P
_{1}
is about 62%, and
Fig. 12
(b) shows that
V
_{1}
increases as the pulse width of
P
_{1}
increases, which are consistent with the simulation results shown in
Figs. 9
and
10
.
Experimental results of the THD and V_{1}vs. the pulse width of P_{1} at j0=0, f_{m}=50 Hz, D=30.
Second, the experiments have been performed by choosing
f
_{m}
=50 Hz,
j
0=0, the pulse width of
P
_{1}
=62%, and
D
=20, 30, 40. The experimental results are shown in
Figs. 13
(a)13(c).
Fig. 13
shows that using the WPWM technique to control the singlephase threelevel inverter is effective.
Output voltage U_{ab} and its spectrum of the singlephase threelevel inverter by WPWM.
Finally, to validate the performance of the singlephase threelevel inverter with the WPWM technique, this paper compares the WPWM with conventional SPWM, as shown in
Fig. 2
. The algorithm of the conventional SPWM is implemented by using DSP (TMS320LF2812), and the experiments have been performed by choosing
f
_{m}
=50 Hz,
m
=1.0 (
m
denotes the modulation index), the switching frequency
f
_{s}
=1 kHz (
D
=20),
f
_{s}
=1.5 kHz (
D
=30), and
f
_{s}
=2 kHz (
D
=40). The experimental results are shown in
Fig. 14
(a), (b), (c) respectively.
Output voltage U_{ab} and its spectrum of the singlephase threelevel inverter by the SPWM.
The compared results between WPWM and SPWM from
Figs. 13
and
14
are listed in
TABLE II
. From
TABLE II
, it can be concluded that (1) WPWM can get higher magnitudes of the fundamental component than SPWM; (2) WPWM technique can get smaller THD and more disperser spectrum than SPWM.
COMPARISON OF RESULTS BETWEEN WPWM AND SPWM FROMFIGS. 13AND14
COMPARISON OF RESULTS BETWEEN WPWM AND SPWM FROM FIGS. 13 AND 14
VI. CONCLUSION
This study has developed the WPWM technique for singlephase threelevel inverters. The design of the parameter for the WPWM is obtained by analyzing the magnitudes of the fundamental frequency component and harmonic distortion of its output voltage. The simulation and experimental results have shown that the proposed WPWM for the threelevel inverter can obtain higher magnitudes of the output fundamental frequency component, lower THD, and simpler digital implementation than the SPWM, which will promote the application of WPWM technique in power electronics converters.
Acknowledgements
This research was supported in part by the Key Program of the National Natural Science Foundation of China under Grant 51437005, the National Natural Science Foundation of China under Grant 51277079, and the Guangzhou Science and Technology Project under Grant 201510010238.
BIO
ChunFang Zheng was born in China in 1978. She received her B.S. degree in Electrical Engineering from Nanchang University, Jiangxi, China, in 2000, the M.S. and Ph.D. degree in the Power Electronics and Power Drives from South China University of Technology, Guangzhou, China, in 2003,and 2006, respectively. From 2006 to 2010, she was an Engineer Supervisor in Emerson Network Power ASTEC Power Supply (Shenzhen) Co., Ltd., China, where she worked on the development of telecom power supply. Since 2013, she has been a Teacher in the School of Shipbuilding Engineering, Guangzhou Maritime Institute, Guangzhou, China. Her main research interests include design and control of power converters and inverters.
Bo Zhang was born in Shanghai, China, in 1962. He received his B.S. degree in Electrical Engineering from Zhejiang University, Hangzhou, China, in 1982, his M.S. degree in Power Electronics from Southwest Jiaotong University, Chengdu, China, in 1988, and his Ph.D. degree in Power Electronics from Nanjing University of Aeronautics and Astronautics, Nanjing, China, in 1994. He is currently a Professor and the Vice Dean of the School of Electric Power, South China University of Technology, Guangzhou, China. He is the author or coauthor of more than 400 papers and the owner of 24 patents. His current research interests include nonlinear analysis and control of power electronics and AC drives.
DongYuan Qiu was born in China in 1972. She received her B.S. and M.S. degrees from South China University of Technology, Guangzhou, China, in 1994 and 1997, respectively, and her Ph.D. degree from City University of Hong Kong, Kowloon, Hong Kong, in 2002. Since 2010, she has been a Professor in the School of Electric Power, South China University of Technology. Her main research interests include design and control of power converters, fault diagnosis, and sneak circuit analysis of power electronic systems.
XiaoHui Zhang was born in Fujian, China. She received her B.S. degrees in Electrical Engineering and Automation from Xiamen University, Xiamen, China, in 2014. She is currently working toward her M.S. degree in Power Electronics and Power Drives at South China University of Technology, Guangzhou, China. Her research interests include wavelet modulation.
LeMing Xiao was born in China in 1962. He received his B.S. degree in Shipbuilding Electronics from DaLian Maritime University, Liaoning, China, in 1983. He is currently a Professor and the Vice Dean of the Department of Shipbuilding Engineering, Guangzhou Maritime Institute, Guangzhou, China. His main research interests include marine electric energy conservation and its intelligent control.
Saleh S. A.
,
Moloney C. R.
,
Rahma M. A.
“Developing a nondyadic MRAS for switching DC–AC inverters,”
in Proc. IEEE 12th DSP Conf, Jackson Lake Lodge
WY
Sep. 2006
544 
549
Saleh S. A.
,
Moloney C. R.
,
Rahman M. A.
2009
“Development and testing of wavelet modulation for singlephase inverters,”
IEEE Trans. Ind. Electron.
56
(7)
2588 
2599
DOI : 10.1109/TIE.2009.2019776
Saleh S. A.
,
Rahman M. A.
2010
“Development and testing of a new controlled wavelet modulated inverter for IPM motor drives,”
IEEE Trans. Ind. Appl.
46
(4)
1630 
1643
DOI : 10.1109/TIA.2010.2049814
Saleh S.A.
,
Moloney C. R.
,
Rahman M. A.
2010
“Analysis and development of wavelet modulation for three phase voltage source inverters,”
IEEE Trans. Ind. Electron.
58
(8)
3330 
3348
DOI : 10.1109/TIE.2010.2081957
Saleh S. A.
,
Rahman M. A.
2011
“Development and experimental validation of resolutionlevel controlled wavelet modulated inverters for three phase induction motor drives,”
IEEE Trans. Ind. Appl.
47
(4)
1958 
1970
DOI : 10.1109/TIA.2011.2156375
Saleh S. A.
2013
“The implementation and performance evaluation of 3φVS wavelet modulated ACDC converters,”
IEEE Trans. Power Electron.
28
(3)
1096 
1106
DOI : 10.1109/TPEL.2012.2205945
Lai J.S.
,
Peng F. Z.
1996
“Multilevel converters  a new breed of power converters,”
IEEE Trans. Ind. Appl.
32
(3)
509 
517
DOI : 10.1109/28.502161
Chang G. W.
,
Lin H.W.
,
Chen S.K.
2004
“Modeling characteristics of harmonic currents generated by highspeed railway traction drive converters,”
IEEE Trans. Power Deliv.
19
(2)
766 
773
DOI : 10.1109/TPWRD.2003.822950
Chao K.H.
,
Chen P.Y.
,
Cheng C.H.
“A threelevel converter with output voltage control for highspeed railway tractions,”
The 33rd Annual Conf. IEEE Ind. Electron. Society (IECON)
Taipei, Taiwan
Nov. 2007
1793 
1798
Tallam R. M.
,
Naik R.
,
Nondahl T. A
2005
“A carrierbased PWM scheme for neutralpoint voltage balancing in threelevel inverters,”
IEEE Trans. Ind. Appl.
41
(6)
1734 
1743
DOI : 10.1109/TIA.2005.858283
Song W.
,
Feng X.
,
Xiong C.
“A neutral point voltage regulation method with SVPWM control for singlephase threelevel NPC converters,”
in Proc. IEEE VPPC Conference
Sep. 2008
1 
4
Salehi R.
,
Farokhnia N.
,
Abedi M.
,
Fathi S. H.
2011
“Elimination of low order harmonics in multilevel inverter using genetic algorithm,”
Journal of Power Electronics
11
(2)
132 
139
DOI : 10.6113/JPE.2011.11.2.132
Li Z.
,
Wang P.
,
Zhu H.
,
Chu Z.
,
Li Y.
2012
“An improved pulse width modulation method for choppercellbased multilevel inverters,”
IEEE Trans. Power Electron.
27
(8)
3472 
3481
DOI : 10.1109/TPEL.2012.2187800
Marzoughi A.
,
Imaneini H.
,
Moeini A.
2013
“An optimal selective harmonic mitigation technique for high power converters,”
International Journal of Electrical Power and Energy Systems
49
34 
39
DOI : 10.1016/j.ijepes.2012.12.007
Zhang M.
,
Huang L.
,
Yao W.
,
Lu Z.
2014
“Circulating harmonic current elimination of a CPSPWMbased modualr mulitlevel converter with a plugin repetitivecontroller,”
IEEE Trans. Power Electron.
29
(4)
2083 
2097
DOI : 10.1109/TPEL.2013.2269140
Zheng C. F.
,
Xu X. M.
,
Zhang B.
,
Qiu D. Y.
“Inverter’s characteristic analysis under different parameters of Wavelet PWM Technique,”
EPE’14 ECCE
Europe
Aug. 2014
1 
7
Wang X.H.
,
Ruan X. B.
2005
“SPWM control singlephase threelevel inverter,”
inProc. of the CSEE
25
(1)
73 
76
Wu F. J.
,
Sun B.
,
Peng H. R.
2011
“Singlephase threelevel SPWM scheme suitable for implementation with DSP,”
Electronics Letters
47
(17)
994 
996
DOI : 10.1049/el.2011.1969