TELKOMNIKA, Vol.10, No.4, August 2012, pp. 670~679
e-ISSN: 2087-278X
accredited by DGHE (DIKTI), Decree No: 51/Dikti/Kep/2010
670
Non-Sinusoidal PWM Method for Cascaded Multilevel
Inverter
Shahrin Md Ayob*, Zainal Salam, Abdul Halim Mohamed Yatim
Power Electronics and Drives Research Group Faculty of Electrical Enginering
Universiti Teknologi Malaysia 81310 UTM Skudai, Johor, Malaysia
e-mail: shahrin@fke.utm.my*
Abstrak
Makalah ini menyajikan teknik pensaklaran yang baru untuk inverter berundak dan bertingkat
jamak (CMI). Metode baru ini menggunakan sinyal non-sinusoidal sebagai sinyal modulasinya.
Keuntungan utama penggunaan sinyal semacam ini adalah kemampuan untuk memberikan nilai
komponen dasar yang lebih tinggi dan pada saat yang sama mempunyai beberapa orde harmonik yang
rendah pada tegangan keluarannya sebelum rangkaian penapis; hal ini secara alamiah dikarenakan
adanya sinyal modulasi non-sinusoidal. Di dalam tulisan ini, sinyal trapezoidal digunakan sebagai sinyal
modulasi untuk CMI. Profil harmonik untuk kemiringan sudut yang berbeda telah dianalisis. Dari analisis
o
tersebut, didapati bahwa sudut 36 mampu menghilangkan harmonik ketiga dan kelima namun masih
mengandung nilai komponen dasar yang tinggi. Untuk menguji rancangan yang diusulkan maka purwa
rupa CMI fasa tunggal telah dibuat dan ALTERA FPGA digunakan untuk mengimplementaskan algoritma
pensaklaran yang diusulkan. Hasil pengujian menunjukkan bahwa analisis yang dilakukan adalah benar.
Kata kunci: FPGA, harmonik orde rendah, inverter berundak dan bertingkat jamak, modulasi lebar pulsa
Abstract
This paper presents a new switching method for cascaded multilevel inverter (CMI). The new
method uses multiple non-sinusoidal signals as the modulating signal. The main advantage of using nonsinusoidal modulating signal is that it provides higher fundamental component but at the cost of having a
numbers of low-order harmonics in its unfiltered output voltage; naturally contributed by the non-sinusoidal
modulating signal. In this paper, the Trapezoidal waveform is employed as the non-sinusoidal modulating
signal for the CMI. Its harmonics profile of different slope angles is analysed. From the analysis, it was
o
found that the trapezoidal with the slope angle of 36 eliminates the third and fifth harmonics yet yields a
high fundamental magnitude. To verify, a single-phase CMI prototype is constructed and an ALTERA
FPGA is deployed to implement the proposed method’s algorithm. From the results, it was shown that
analysis is validated
Keywords: cascaded Multilevel Inverter, FPGA, low-order harmonics, pulse width modulation
Copyright © 2012 Universitas Ahmad Dahlan. All rights reserved.
1. Introduction
Recently, multilevel inverter topologies have become very popular especially in
renewable source energy. A large body of literatures over the past decades has testified its
viability especially for high power application. It is an effective and practical solution to reduce
switching losses and to improve harmonic profile.
In general, issues in multilevel inverter technology can be divided into two issues. The
first issue is the topology and second is on the imposed switching strategy that produces the
PWM gating signals. For multilevel inverter, there are three main topologies that are usually
cited, namely Flying-Capacitor multilevel inverter (FCMI), Diode-Clamp Multilevel Inverter
(DCMI) and Cascaded Multilevel Inverter (CMI).
These topologies have their own merits and limitations. The decision on selecting the
appropriate topologies largely depends on the particular application and the nature of the dc
supply that feed to the inverter. The CMI is known to be the simplest topology and less
component counts. Moreover, additional full-bridge inverters can be added up to meet an
increased power demand due to its modular structure. However, the topology requires that the
voltage source to be separated. The structure of CMI is illustrated in Figure 1.
Received June 26, 2012; Revised July 23, 2012; Accepted July 29, 2012
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S11
E1
671
S21
+
–
V1
S31
Vphase
(Vout)
S41
Module 1
S12
E2
S22
+
_
V2
S32
S42
Module 2
S1M
EM
S2M
+
_
VM
S3M
0
S4M
Module M
Figure 1. Structure of cascaded multilevel
The second issue is on the switching strategy of the inverter. In the past decades, a
number of switching schemes have been proposed to improve the inverter performance
especially its harmonic profile [1], [2]. The simplest strategy is the square wave though it
exhibits the poorest harmonic profile. Then this is followed by quasi-square wave in which the
harmonic profile can be improved by selecting the appropriate firing angle. By far, however the
most popular switching is Pulse Width Modulation (PWM) techniques.
Basically PWM technique is based on the triangulation method; a physical comparison
between a sinusoidal signal with triangular carrier signal. The intersections between the signals
produce the PWM pulses. This technique helps the improvement of harmonic profile by locating
all undesired harmonic to a higher frequency range. The analog version of this technique has
been introduced by Schonung and Stemmler [3]. It however, suffers from serious problem of
drift and ageing of the components. The digital version i.e. regular-sampling has been
introduced by Bowes in 1975 [4]. It has become the impetus for the proliferation of several
important digital PWM techniques until the present day. Several currently popular PWM
schemes are Optimized PWM, Selective Harmonic Elimination PWM (SHEPWM) and more
recently the Space Vector PWM (SVPWM) [5-7].
Originally, the abovementioned schemes are applied to the two-level inverter.
However, by some modification the schemes can be extend to be applied in multilevel inverter.
For a carrier-based multilevel sinusoidal PWM in particular, a method has been proposed by
using a different arrangement of triangular carrier. They are Phase Opposition Disposition
(POD), Phase Disposition (PD) and alternatively in Phase Opposition (APOD) [8]. These
techniques employ multiple carriers with a single reference signal i.e. sinusoidal. Each method
has its own unique spectra and specific application. However, these methods suffered from a
low fundamental magnitude.
In this paper, a new switching scheme is proposed for cascaded multilevel inverter
(CMI). The scheme uses multiple modulating signals with single carrier signal which is contrary
to the abovementioned switching schemes. Moreover, this paper will analyze the utilization of
non-sinusoidal modulating signal i.e. trapezoidal waveform. The waveform is choosen due to its
simple signal generation. Moreover, its harmonics profile is changeable by varying its slope
Non-Sinusoidal PWM Method for Cascaded Multilevel Inverter (Shahrin Md Ayob)
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angle (α). The study will analyze on the viability of the new switching scheme in synthesizing
sinusoidal output voltage. A CMI prototype will be constructed to verify the viability. For the
switching algorithm, a cost-effective ALTERA FPGA is employed. It is expected that by using
Trapezoidal as the modulating signals, a higher fundamental magnitude can be achieved.
2. Trapezodial PWM Switching Scheme
The heart of any multilevel inverter is undoubtedly the switching scheme used to
generate the switching edges of the PWM waveform pattern. In the past few decades, a large
number of PWM switching strategies have been proposed, starting from analogue-based
system through digital based and more recently ROM-based and microprocessor-implemented
controls schemes [9], [10]. This section is dedicated to describe the Trapezoidal PWM
modulation scheme applied to the Cascaded Multilevel Inverter (CMI). The main impetus of the
proposed modulation scheme is its simplicity and suitability to be implemented in a full digital
circuit such as FPGA.
2.1. The Waveform
Trapezoid waveform as shown in Figure 2 can be digitally generated using FPGA. The
waveform itself simply comprises of two linear sections; a slope line and a horizontal line
section. The shape and hence, the harmonics depend on the location of slope angle (α), which
0
0
0
can be varied within 0 to 90 . It forms a square shape if α located at 0 and a triangular shape
0
if α located at 90 [11].
α
00
900
1
0
0
Figure 2. Trapezoidal signal over 90 of fundamental cycle
0
0
The harmonic contents of the trapezoid signal, having an angle α within 0 and 90 can
be found by using standard Fourier analysis. By assuming quarter-wave symmetry, only the
odd sine terms are present. Thus, the sine-term Fourier coefficients are given by:
An =
4 π /2
∫ F (θ ).sin(nθ )d θ
(1)
π θ =0
where F(θ) represents trapezoidal signal. Equation (1) can be expand to be
α

4  −θ kosnθ  1
An =

 +
απ 
n
0 n
+
4
π
α

0

∫ kosnθ d θ 
(2)
π /2
∫ (1) sin nθ d θ
α
Equation (2) can be simplified down to the general expression:
 4   sin(n α ) 
An =  2  

 n π  α

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(a)
673
(b)
Figure 3. (a) Harmonics magnitude (1, 3, 9, and 15) with different slope angles. (b) Harmonics
magnitude (5, 7, 11, and 13) with different slope angles
The harmonics magnitude versus trapezoidal slope angle, α, are shown in Figs. 3(a)
0
and 3(b). From the figures, it can be observed that triangular shape (α = 90 ) has less
harmonics than trapezoidal waveform. However the latter offers higher fundamental component
as compared to the triangular. Thus, the trapezoidal is more desirable.
2.2. The Principle of the Switching Scheme
Figure 4 illustrates the proposed switching scheme. The proposed scheme is based on
the classical unipolar PWM switching. The main idea behind this method is to compare several
trapezoidal modulation signals m(t) that have equal amplitude (Am) and frequency (fs) with a
single triangular carrier signal c(t). The carrier signal is a train waveform with a frequency fc and
amplitude Ac. The intersection between modulation signals and carrier defines the switching
instant of the PWM pulses. The number of modulation signals required is equal to the number
of modules in CMI (M).
Figure 4. Trapezoidal modulation signals and single carrier signal
The relationship between the number of output level (N) and M is described in (4),
M=(N-1)/2
(4)
Modulation index (mi) for N-level inverter with M modules is defined by equation (5):
mi =
Am
A
= m
(N −1)
MAc
Ac
2
(5)
Therefore, if Ac is defined at a fixed 1 p.u, then mi will range between 0 and 1 and Am
will range between 0 and M. The definition for mf is similar to the conventional two-level output
inverter and can be rewritten as
Non-Sinusoidal PWM Method for Cascaded Multilevel Inverter (Shahrin Md Ayob)
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mf =
fc
fm
(6)
In equation (6), fc is defined as the carrier’s frequency and fm is the trapezoidal
modulation signal’s frequency.
Figure 5 illustrates the principal of the proposed scheme of five-level CMI with mi = 0.8.
The modulation ratio, mf is set at 40. For five-level CMI, the number of modulating signals is
equal to the number of modules. Thus, by referring to equation (4), two trapezoidal signals are
required and are arranged as shown in Figure 5.
Figure 5. Principal of the proposed modulation scheme for mi = 0.8 and mf = 20
As depicted in Figure 5, the intersections between trapezoidal signal and carrier signal
define the PWM switching edges. The intersections between m1(t) with carrier produce the
lower level output voltage and the intersections between m2(t) produce the upper level of
voltage output.
3. Analysis of the Proposed Scheme
This section presents the harmonic analysis of the output voltage of the proposed
scheme with different Trapezoid’s slope angle. The Matlab/Simulink software is used to conduct
the simulation of the proposed scheme in five-level CMI model. The simulation parameters are
tabulated in Table 1.
Table 1. Parameters for Simulation
Parameter
DC source
Fundamental frequency
Carrier frequency
Modulation Index
Modulation Frequency
Symbol
Value
E
fs
fc
mi
mf
100 V
50 Hz
2k Hz
0.8
40
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However, due to pages constraint, only trapezoidal with four different lope angles will be
o
o
o
analyzed. The selection of the angle is arbitrary; the selected slope angles are 30 , 36 , 60 and
o
72 . Figs. 6, 7, 8 and 9 show the output waveform and its harmonics spectrum for each slope
angle, respectively.
Figure 6. Output voltage and harmonics
o
spectrum for α = 30
Figure 7. Output voltage and harmonics
o
spectrum for α = 36
Figure 8. Output voltage and harmonics
o
spectrum for α = 60
Figure 9. Output voltage and harmonics
o
spectrum for α = 72
Table 2 tabulates the fundamental magnitude and low-order harmonics for different
o
slope angles. The table clearly shows that trapezoidal signal with slope angle of 30 offers the
o
highest fundamental magnitude as compared to the other slope values. Moreover, the 30
trapezoidal is easier to be digitally generated with minimal digital circuit [11]. However, it should
be noted that it also has the highest low-order harmonics content.
Non-Sinusoidal PWM Method for Cascaded Multilevel Inverter (Shahrin Md Ayob)
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o
The third harmonic is eliminated for the 60 slope angle. In addition, the fundamental
component is 2.97 % higher than when Sinusoidal PWM is used (2miVdc = 160 V). The loworder harmonics content is still large i.e. fifth harmonic component is high in amplitude and
could not be eliminated.
Table 2. Harmonics magnitude according to the slope
Harmonics
magnitude (V)
Fundamental
3
5
7
9
30
195.17
49.47
12.42
1.05
3.34
36
188.72
35.65
0.00
0.87
4.31
Slope Angle (α)
60
164.74
0.00
8.95
1.74
4.48
o
72
153.56
8.72
0.00
1.65
1.94
o
By choosing the slope angle to be 36 and 72 , the fifth harmonic is eliminated.
However, the third harmonic component is high for both slope angles. To counter that problem,
it is beneficial to take advantage of Triplen-effect in three-phase system. In Triplen-effect, the
harmonics number with the multiple of three can be automatically eliminated if the carrier
frequency is odd and in multiple of three. For example, by setting mf = 39 (odd and multiple of
rd
th
th
three), the harmonic with multiple number of three i.e. 3 , 9 , 15 , ... will be eliminated.
o
o
Based on the analysis, it was shown that the trapezoidal with 36 and 72 slope angles
seem suitable for the modulating waveform. Both angles provide better low-order harmonics
profile. However, the former slope offers higher fundamental than the latter. Thus, based on
o
that judgment, the trapezoidal with the slope angle of 36 is selected for the implementation.
4. Hardware Implementation
The proposed switching scheme is implemented using UP1 board from Altera Inc. The
UP1 board contains two FPGA processors namely EPM7128S and EPF10K70 devices.
However, due to limited numbers of logic gates and problems in configuring EPM7128S, only
EPF10K70 device is used. With up to 70,000 of typical gates with 25MHz system performance
at 2.5 V logic supply, EPF 10K70 device is ideal for intermediate to advance digital design.
Altera Inc. allows one to program the FPGA board either by using graphical approach or
Verilog Hardware Description Language (VHDL) via Max-Plus II software. The former is simpler
where the users only need to click and drag the model that available in the software library. The
latter is more towards customizing or re-creates a new model or algorithm. For this work, the
hierarchal approach, which is the combination of both approaches, is used. Figure 10 shows
the gating signals for the 5-level CMI, generated using Max-Plus II software.
Figure 10. Gate signals waveform from MAX-Plus II
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5. Results and Discussions
Figures 11 and 12 show the simulation and the experiment results of the output voltage,
respectively. The figures showed that the experimental result is in good agreement with the
simulation result. Both figures yields five-level output voltage. The harmonics spectrum is
discussed in the following section.
Figure 11. Simulation result of 5-level CMI
Figure 12. Experimental result for 5-level CMI
6. Harmonics Spectrum Analysis
This section discussed on the harmonic spectrum of a Cascaded Multilevel Inverter
0
(CMI) for mi = 0.8, mf = 40 and α = 36 . The simulation result of the output voltage harmonic
spectrum is shown in Figure 13. The experimental result of harmonics spectrum for the similar
case is shown in Figure 14. It can be clearly seen from both figures that the experimental result
is in close agreement with the simulation.
The proposed scheme produces odd harmonics. Furthermore, harmonics at the carrier
and the multiples of the carrier frequency do not exist at all. This criterion is similar with the
Phase Opposition Disposition (POD) technique as described in literature.
Figure 13. Harmonic spectrum of output
voltage from simulation
Figure 14. Harmonic spectrum of output
voltage from prototype
Non-Sinusoidal PWM Method for Cascaded Multilevel Inverter (Shahrin Md Ayob)
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As depicted in Figures 13 and 14, the spectrum has a wide range of side-band
0
harmonics. For the case of mi = 0.8, mf = 40 and α = 36 , the side-band harmonics for the first
significant high-order harmonic group (mf) is mf ±15. Table 3 tabulates the harmonics
magnitude for the first significant high-order harmonic group (mf) with respect to their trapezoid
slope angle (α). The table is created using the calculated value by MATLAB.
Table 3. Harmonics magnitude for the first significant high-order harmonic group (mf)
Slope angle, α
Fundamental (normalized)
mf ± 1
mf ± 3
mf ± 5
mf ± 7
mf ± 9
mf ± 11
mf ± 13
mf ± 15
mf ± 17
300
0.968
0.1874
0.0463
0.0147
0.001
0.0130
0.0230
0.0292
0.0262
0.0148
360
0.952
0.1818
0.0383
0.0120
0.0146
0.0262
0.0324
0.0270
0.0135
600
0.840
0.1661
0.0237
0.0352
0.0514
0.0222
0.0157
0.0134
720
0.768
0.1554
0.0235
0.0593
0.0416
0.0156
0.0154
0.0108
From the table, it can be shown that the number of sidebands for the first high-order
harmonic group is not solely depends on the modulation index (mi), but also relies on the slope
angle. The number of sidebands with constant mi decreased from mf ± 17 down to mf ± 13
0
when the slope angle is 72 . This may be due to the fact that the low-order harmonics for that
particular slope angle is small (referring to Figs. 3(a) and 3(b)).
It can also be seen that the fundamental component amplitude is not linear with the
modulation index (mi); it is dependent to the slope angle. Higher fundamental component is
achieved for smaller slope angle. In the case of mi = 0.8, the highest fundamental components
0
magnitude is given by slope angle of 30 , which is 0.968 p.u.
In SPWM method, the amplitude of the side-band harmonic for constant mi is
determined by the modulation frequency (mf) alone. However, in this proposed scheme is was
shown that the amplitude of the harmonics is also on the slope angle (α).
7. Total Harmonics Distortion
Total Harmonic Distortion (THD) is one of the common performance index to describe
the quality of the output waveform. It can be compute using
2
V 
∑ nh 

n =2 
THD =
V1
∞
(6)
In equation (6), n denotes the harmonic order and 1 is the fundamental quantity. For
inverter application, THD represents how close the ac output waveform with a pure sinusoidal
waveform. A high quality output should have low THD.
The proposed method used trapezoidal signal as a modulation signal, which naturally
contains low-order harmonics. For harmonics improvement, some of the harmonics should be
eliminated by choosing an appropriate trapezoid slope angle. Since triplens are allowed in
0
0
three-phase system, slope angle of 36 and 72 is selected, so as to eliminate the fifth
harmonics. This allows reduction in the output waveform THD.
0
As discussed earlier, trapezoid slope angle of 72 offers the smallest number of sideband harmonic and amplitude. However, the angle does not provides a high fundamental
0
component as compared to slope angle of 36 . Thus, the latter is more preferable. The large
0
number and amplitude of the sidebands harmonic produce by trapezoid slope angle of 36 can
be reduced by switching the inverter to the higher frequency range. Table 4 tabulates the
predicted and measured results of the significant low-order harmonics. The comparison shows
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that the error is quite small. This validates the proposed scheme designed in FPGA with the
simulation.
Table 4. Predicted and measured values of the significant low-order harmonics
0
for mi = 0.8, mf = 40, α =36
Harmonic
order
3
5
7
9
11
13
Frequency (Hz)
Harmonic Magnitude (p.u)
Predicted
150
250
350
450
550
650
0.204
0.00
0.035
0.010
0.014
0.017
Measured
0.210
0.00
0.033
0.010
0.011
0.018
Note that the fifth harmonic is eliminated. The third harmonic yields the highest
fundamental component. However, the third harmonic and its multiple can be eliminated in
three-phase if mf is selected to be odd and triplen. With the elimination of the third and fifth
harmonics, the low-order harmonic by considering only the first thirteen low-order harmonics
can be compute as 0.53%. The harmonics profile can be further improved to 0.182 % by filtering
the seventh harmonic.
8. Conclusion
In this paper, a new modulation scheme using trapezoidal waveform as the modulating
signal is proposed. The scheme is applied to cascaded multilevel inverter (CMI). Its viability is
verified by constructing a hardware prototype of CMI. The scheme has been implemented in
Altera UP1 FPGA board. Based on the results, it was shown that by appropriate selection of
slope angle, better harmonics profile with high fundamental magnitude of the output voltage can
be obtained.
Acknowledgements
This work was supported by Universiti Teknologi Malaysia
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Non-Sinusoidal PWM Method for Cascaded Multilevel Inverter (Shahrin Md Ayob)