International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)
THD Analysis for Different Levels of Cascade Multilevel
Inverters for Industrial Applications
Dr. Jagdish Kumar
Department of Electrical Engineering, PEC University of Technology, Chandigarh
Abstract — Cascade Multilevel Inverters are very popular
and have many applications in electric utility and for
industrial drives. When these inverters are used for industrial
drive directly, the THD contents in output voltage of inverters
is very significant index as the performance of drive depends
very much on the quality of voltage applied to drive. In this
article, the THD contents of 7, 11 and 15 level cascade
multilevel inverters have been analysed. The THD depends on
the switching angles for different units of multilevel inverters,
therefore, the switching angles are calculated first by using
N-R method where certain number of harmonic components
has been eliminated. Using calculated switching angles, THD
analysis is carried out analytically as well as using MATLAB
simulation (both results are in close agreement). As per
IEEE-519 standard, the output voltage produced should
satisfy the limit on THD. It has been found that the fifteen
level CMLI satisfies this limit while seven and eleven level
CMLI violates this limit.
High magnitude sinusoidal voltage with extremely low
distortion at fundamental frequency can be produced at
output with the help of multilevel inverters by connecting
sufficient number of dc levels at input side. There are
mainly three types of multilevel inverters; these are a)
diode- clamped, b) flying capacitor and c) cascade
multilevel inverter (CMLI). Among these three, CMLI has
a modular structure and requires least number of
components as compared to other two topologies, and as a
result, it is widely used for many applications in electrical
engineering [1], [5].
To produce multilevel output ac voltage using different
levels of dc inputs, the semiconductor devices must be
switched on and off in such a way that the fundamental
voltage is obtained as desired along with the elimination of
certain number of higher order harmonics in order to have
least harmonic distortion in the ac output voltage. For
switching the semiconductor devices, proper selection of
switching angles is must. The switching angles at
fundamental frequency, in general, are obtained from the
solution of non linear transcendental equations
characterizing harmonics contents in the output ac voltage;
these equations are known as selective harmonic
elimination (SHE) equations. As the SHE equations are
non linear transcendental in nature, their solutions may
have simple, multiple and even no roots for a particular
value of modulation index (m), moreover, it is difficult to
solve these equations. A big challenge is how to get all
possible solution sets where they exist using simple and
less computationally complex method. Once these solution
sets are obtained, the switching angles producing minimum
total harmonic distortion (THD) in the output ac voltage are
selected for switching of the power electronics devices.
In [3]-[4], iterative numerical techniques have been
implemented to solve the SHE equations producing only
one solution set, and even for this, a proper initial guess
and starting value of m for which the solutions exist are
required, in general, it is difficult to guess the initial
solution and the value of m for which solution exist.
Keywords — Cascade multilevel inverter, harmonic
elimination, modulation index, switching angles, total
harmonic distortion.
I. INTRODUCTION
Since past decade, multilevel inverters have drawn
increasing attention because of their promising applications
in power systems and industrial drives. They can be
efficiently used in the distributed energy systems in which,
output ac voltage is obtained by connecting dc sources such
as batteries, fuel cells, solar cells, rectified wind turbines
etc at input side of the inverters. The ac output voltage
obtained from the inverters can be fed to a load directly or
interconnect to the ac grid without voltage balancing
problems. In addition, the multilevel inverters are used as
voltage source inverters (VSIs) in the static synchronous
compensator (STATCOM), a reactive power compensating
device used for voltage regulation in power systems [1]-[7].
The multilevel inverters offer several advantages as
compared to the hard-switched two-level pulse width
modulation inverters, such as their capabilities to operate
at high voltage with lower dv/dt per switching, high
efficiency, low electromagnetic interference etc [2]-[4].
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Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)
In [8]-[9], theory of resultants of polynomials and the
theory of symmetric polynomials has been suggested to
solve the polynomial equations (these polynomial
equations are obtained from the transcendental equations).
A difficulty with these approaches is, for several Hbridges connected in series, the order of the polynomials
become very high, thereby making the computations of the
solutions of these polynomials very complex. Moreover,
these techniques have been applied up to 11-level
multilevel inverters only due to the computational
complexity associated with these techniques. Optimization
technique based on Genetic Algorithm (GA) is proposed
for computing switching angles only for 7-level inverter in
[6], and even for the implementation of this method, proper
selection of certain parameters such as initial population
size, mutation rate etc are required, thereby
implementation of this method becomes difficult for
higher level multilevel inverters.
In the present work, 7-level, 11-level and 15-level
CMLIs are employed to generate ac output voltage
producing different magnitudes of THD at different values
of modulation indices for comparison purpose. The THD
produced by 7-level inverter is more than IEEE-519
standard [10] for all values of m, THD produced by
11-level inverter is satisfying this standard for some values
of m (not for all values) while THD produces by 15-level
inverter is well within the limits imposed by IEEE-519
standards for complete working range of m. The switching
angles have been computed by the implementation of
Newton Raphson (N-R) numerical technique in a particular
way producing complete solutions for working range of
modulation index without much computational complexity.
II. CASCADE MULTILEVEL INVERTER
The cascade multilevel inverter consists of a number of
H-bridge inverter units with separate dc source for each
unit and is connected in cascade or series as shown in Fig.
1. Each H-bridge can produce three different voltage
levels: +Vdc, 0 and –Vdc by connecting the dc source to ac
output side by different combinations of the four switches
S1, S2, S3, and S4. The ac output of each H-bridge is
connected in series such that the synthesized output voltage
waveform is the sum of all of the individual H-bridge
outputs. By connecting sufficient number of H-bridges in
cascade and using proper modulation scheme, a nearly
sinusoidal output voltage waveform can be synthesized.
Fig. 1. Single-phase cascade multilevel inverter topology.
The number of levels in the output phase voltage and
line voltage are 2s+1 and 4s+1 respectively, where s is the
number of H-bridges used per phase. For example, three Hbridges, five H-bridges and seven H-bridges per phase are
required for 7-level, 11-level and 15-level multilevel
inverter respectively. Fig. 2 shows a typical waveform
produced by 7-level CMLI.
The magnitude of the ac output phase voltage is the sum
of the voltages produced by H-bridges.
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In the Fig. 2, α1, α2 and α3 are the switching angles for
three H-bridges in each phase, and β1, β2 and β3 are
corresponding supplementary angles for α1, α2 and α3. The
magnitude and THD content of output voltage depends
very much on these switching angles, therefore, these
angles need to be selected properly.
The modulation index, m, is defined as the ratio of the
fundamental output voltage (V1) to the maximum
obtainable
fundamental
voltage.
The
maximum
fundamental voltage is obtained when all the switching
angles are zero i.e. V1max = 4sVdc/π, therefore,
m = πV1/4sVdc [1].
For 7, 11 and 15-level cascade multilevel inverters,
s = 3, s = 5 and s = 7 respectively. Number of degrees of
freedom available is equal to s; one degree of freedom is
used to choose the value of V1 and the remaining degrees of
freedom are used to eliminate the lower order harmonic
components. For example, in case of 7-level CMLI, only
two harmonic components (in general, 5th and 7th) can be
eliminated, similarly for 11-level CMLI, four harmonic
components (i.e. 5th, 7th, 11th and 13th) and in 15-level
CMLI, six harmonic components (i.e. 5th, 7th, 11th, 13th, 17th
and 19th) can be eliminated.
From equation (1), the expressions for fundamental
voltage in terms of m, and lower order harmonic
components, when they are eliminated, can be written as
for 7-level CMLI:
3Vdc
1  2  3
3
2
1 
2 t
-3Vdc
Fig. 2. Output phase voltage waveform for 7-level CMLI.
III. SELECTIVE HARMONIC ELIMINATION EQUATIONS
In general, the Fourier series expansion of the staircase
output voltage waveform as shown in Fig. 2 is given by [4].
cos(1 )  cos( 2 )  cos( 3 )  3m

4Vdc
v an ( wt )  
(cos(k 1 )  cos(k 2 )  ...
k 1, 3, 5,... k
 cos(k s )) sin(kt )
cos(51 )  cos(5 2 )  cos(5 3 )  0
cos(71 )  cos(7 2 )  cos(7 3 )  0
(1)
For 11-level CMLI, corresponding equations are as
follows:
Where s is the number of H-bridges connected in
cascade per phase and k is order of harmonic components.
For a given desired fundamental peak voltage V1, it is
required to determine the switching angles such that 0 ≤ α1
< α2 … < αs ≤ π/2 and some predominant lower order
harmonics of phase voltage are zero. Among s number of
switching angles, generally one switching angle is used for
fundamental voltage selection and the remaining (s-1)
switching angles are used to eliminate certain
predominating lower order harmonics. In three-phase
power system, triplen harmonic components are absent in
line-to-line voltage, as a result, only non-triplen odd
harmonic components are present in line-to-line voltages
[4-5].
From equation (1), the expression for the fundamental
voltage in terms of switching angles is given by
4Vdc

(3)
cos(1 )  cos( 2 )  ...  cos( 5 )  5m
cos(51 )  cos(5 2 )  ...  cos(5 5 )  0
cos(71 )  cos(7 2 )  ...  cos(7 5 )  0
cos(111 )  cos(11 2 )  ...  cos(11 5 )  0
cos(131 )  cos(13 2 )  ...  cos(13 5 )  0
(4)
Similarly for 15-level CMLI equations are given below:
cos(1 )  cos( 2 )  ...  cos( 7 )  7m
cos(51 )  cos(5 2 )  ...  cos(5 7 )  0
cos(71 )  cos(7 2 )  ...  cos(7 7 )  0
cos(111 )  cos(11 2 )  ...  cos(11 7 )  0
(cos(1 )  cos( 2 )  ...  cos( s ))  V1 (2)
cos(131 )  cos(13 2 )  ...  cos(13 7 )  0
cos(171 )  cos(17 2 )  ...  cos(17 7 )  0
Moreover, the relation between the fundamental voltage
and the maximum obtainable voltage is given by
modulation index.
cos(191 )  cos(19 2 )  ...  cos(19 7 )  0 (5)
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The equations (3-5) are transcendental equations, known
as selective harmonic elimination (SHE) equations, where
unknown parameters are switching angles. The first
equation of the set of equations given by (3-5) determines
the magnitude of fundamental voltage for a given value of
m, and the remaining equations eliminate selective
harmonic components [11]. The equations (3-5) are to be
solved by employing N-R method in such a way that all
possible solutions for a given value of m are obtained
without much computational effort. The algorithm for the
solution of equations (3-5) is given in [11].
The THD plots for solution are plotted as a function of m
in Fig. 4. It can be seen from the Fig. 4, that THD is more
than 5% for all values of modulation indices, hence not
satisfying IEEE-519 standard. This type of output voltage
cannot be used directly for power system and industrial
drive applications due to high THD.
n  49
THD 
V
n x
V1
2
n
 100
(6)
TABLE I
Some Values of Switching Angles (in radians) for 7-level CMLI
IV. COMPUTATIONAL RESULTS
α1
0.8120
0.6881
0.2064
0.2729
0.1394
m
0.2710
0.5000
0.6000
0.8400
0.9200
By using N-R method, all possible solution sets for a 7,
11 and 15-level CMLIs are computed and a complete
analysis is presented.
A. 7-Level CMLI
For 7-level CMLI, two switching angles have been
calculated as shown in Fig. 3.It can be seen from Fig. 3 that
for some values of m solutions do not exist, for some other
values of m, multiple solutions exist and simple solution
exists for most of the values of m. Some typical values of
switching angles (in radians) are given in Table I.
α2
1.4755
0.9818
0.7280
0.3273
0.2672
α3
1.5410
1.3980
1.4960
0.9146
0.6348
20
15
100
2
3
80
Switching Angles (Degrees)
THD (%)
1
90
70
10
60
50
40
5
30
20
0.5
0.6
0.7
Modulation Index (m)
0.8
0.9
1
Fig. 4. THD verses m for combined solution of 7-level CMLI.
10
0
0.4
0.4
0.5
0.6
0.7
Modulation Index(m)
0.8
0.9
B. 11-level CMLI
In similar way as discussed in case of seven-level CMLI,
switching angles are calculated by using equation (4). In
this case five switching angles are obtained eliminating 5 th,
7th, 11th and 13th (x= 17 in equation 6) order harmonic
components. Fig. 5 shows
different solution sets and
Fig. 6 shows corresponding THD for the combined solution
producing least THD. It can be seen from Fig. 6 that THD
level is higher at lower and upper ends of modulation
indices.
1
Fig. 3. Switching angles verses modulation index for 7-level CMLI.
For each of the multiple solution sets as computed
above, total harmonic distortion (THD) in percentage is
computed according to equation (6), for 7-level CMLI
x=11. The set of switching angles among multiple solutions
which produce least THD is selected for switching of
semiconductor devices, and these are termed as combined
solutions.
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90
80
70
Switching Angles (Degrees)
It can be observed from Fig. 8 that THD level in output
voltage is strictly satisfying IEEE-519 standard for normal
working range of modulation indices. Some of calculated
switching angles have been shown in Table III for
reference. By bringing THD level below threshold value is
very significant as THD produces various losses and
undesirable effects in electric power systems and
equipments [12-14].
It can be observed from Figs. 3, 5 and 7 that the range of
modulation index for which solution for switching angles
exist decreases with the increase in number of levels but
the occurrence of multiple solutions increase with the
increase in number of level i.e. for 7-level inverter only two
sets of solution exist, for 11-level four sets of solution exist
and for 15-level nine sets of solution are obtained thereby
making computation more complex with number of levels.
Fig. 9 shows the comparison of THD produced by 7, 11
and 15 level CMLIs. It can be observed from this figure
that the THD reduces with the increase in number of levels.
1
2
3
4
5
60
50
40
30
20
10
0
0.4
0.5
0.6
0.7
Modulation Index (m)
0.8
0.9
Fig. 5. Switching angles verses modulation index for 11-level CMLI.
11
10
TABLE II
Some Values of Switching Angles (in radians) for 11-level CMLI
9
THD(%)
8
7
6
5
m
α1
α2
α3
α4
α5
0.4410
0.6247
0.8391
1.0615
1.3305
1.5706
0.5490
0.0718
0.6486
0.7353
1.3836
1.5480
0.6000
0.4650
0.7667
0.8994
1.0891
1.2654
0.6580
0.1563
0.3292
0.6048
1.0123
1.5675
0.7320
0.0782
0.2101
0.4618
0.7124
1.5378
0.8460
0.1604
0.2027
0.4227
0.6225
1.0017
4
3
0.4
0.5
0.6
0.7
Modulation Index (m)
0.8
1
2
3
4
5
6
7
90
0.9
80
Fig. 6. THD verses m for combined solution of 11-level CMLI.
Switching Angles (Degrees)
C. 15-Level CMLI
Lastly, seven switching angles were calculated for
15-level CMLI for its 7 H-bridges using N-R method. The
calculated switching angles (including multiple solutions)
are shown in Fig. 7. In this case, more number of harmonic
components (six) as compared to 7 and 11 level CMLIs
THD contents in output voltage reduces appreciably. The
harmonic components eliminated are 5th, 7th, 11th, 13th, 17th
and 19th. The output voltage THD content is calculated
using equation (6) with x=23 and the corresponding plot is
shown in Fig. 8. It is to be noted here that among different
solution sets of switching angles computed as above, the
THD is computed by using those switching angles which
are producing minimum THD (this applies for the values of
m where multiple solution exist).
70
60
50
40
30
20
10
0
0.45
0.5
0.55
0.6
0.65
0.7
0.75
Modulation Index (m)
0.8
0.85
0.9
Fig. 7. Switching angles verses modulation index for 15-level CMLI.
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6
V. SIMULATION RESULTS
In order to validate the analytical results, three-phase, 7,
11 and 15-level CMLIs have been simulated on
MATLAB/SIMULINK platform using SimPower Systems
BlockSets [15]. For each of the H-bridges in the CMLI,
12V dc source is used. The switching device used is 400V,
10A MOSFET.
Simulated line voltage and THD analysis is presented in
Fig. 10. The analytical and simulated values of THD are
11.17% and 11.51% respectively at m = 0.8400. These two
values of THDs are in close agreement hence validating
analytical result.
5
THD (%)
4
3
2
1
0
0.45
0.5
0.55
0.6
0.65
0.7
Modulation Index (m)
0.75
0.8
0.85
Fig. 8. THD verses m for combined solution of 15-level CMLI.
TABLE III
Some Values of Switching Angles (in radians) for 15-level CMLI
m
α1
α2
α3
α4
α5
α6
α7
0.42
0.601
0.757
0.920
1.093
1.292
1.540
1.554
0.47
0.590
0.733
0.873
1.028
1.190
1.386
1.565
0.60
0.249
0.585
0.681
0.914
1.027
1.164
1.474
0.70
0.108
0.374
0.558
0.747
0.866
1.073
1.300
0.75
0.119
0.319
0.427
0.621
0.867
0.994
1.185
0.80
0.126
0.228
0.364
0.484
0.683
0.952
1.095
Fig. 10. (a) Simulated line voltage and (b) THD analysis for 7-level
CMLI at m = 0.8400.
Similarly, the simulated phase and line voltage
waveforms for 11-level CMLI for the modulation index
equals to 0.6200 are shown in Figs. 11 and 12 respectively.
Again there is a close agreement between analytical and
simulated values of THDs and these are 5.93% and 5.97%
respectively. It is to be noted that in Fig. 11 (b), THD
shown is 32.90%, the reason for this is that the THD shown
is for phase voltage which includes triplen harmonic
components while analytical value is for line voltage which
excludes triplen harmonic components.
20
7-level
11-level
15-level
18
16
THD (%)
14
12
10
8
6
4
2
0.4
0.5
0.6
0.7
Modulation Index (m)
0.8
0.9
1
Fig. 11. (a) Simulated phase voltage and (b) THD analysis for 11-level
CMLI at m = 0.6200.
Fig. 9. Comparison of THD for different levels of CMLI.
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The experimentally obtained wave form for phase
voltage and its THD spectrum is shown in Fig. 14 for m =
0.6200. It can be seen from Figs. 11 and 14 that waveforms
as well as harmonic spectrum of simulated and
experimental results are approximately identical hence
validating analytical and simulated results. Further it is to
be noticed that in Figs. 11 and 14 triplen harmonics are
present as these waveforms are for phase voltage. The
analytical, simulated and experimentally obtained values of
THD upto 13th order and 49th order have been shown in
Table IV for comparison purpose. It can be seen from the
Table IV that all results are in close agreement.
Fig. 12. (a) Simulated line voltage and (b) THD analysis for 11-level
CMLI at m = 0.6200.
The simulated result for line voltage and its THD
spectrum is shown in Fig. 13 for a 15-level CMLI at
modulation index equals to 0.7400. The calculated and
simulated values of THD are 3.97% and 4.02%
respectively. These values of THD are in close agreement
hence validating results. It is to be noted that the
modulation index for different level of CMLIs have been
chosen randomly.
Fig. 14. (a) Experimentally obtained phase voltage and (b) harmonic
spectrum for 11-level CMLI at m = 0.6200.
TABLE IV
Analytical, Simulated and Experimental values of THDs for
m = 0.6200
THDs
Analytical
Fig. 13. (a) Simulated line voltage and (b) THD analysis for 15-level
CMLI at m = 0.7400.
Simulated
Experimental
Up to 13 order
0.00
0.15
0.20
Up to 49th order
5.93
5.97
5.80
th
VII. CONCLUSION
VI. EXPERIMENTAL RESULTS
The switching angles for cascade multilevel inverters of
7, 11 and 15-level have been computed for analysis of total
harmonic distortions produced in output voltage and
complexity in computation of these angles. It has been
found that complexity in computation of switching angles
increases with increase in number of levels as more sets of
solution are produced but the operating range of
modulation index goes down. On other part, the THD in
output voltage decrease and output voltage increase with
increase in number of levels. Analytical results are
validated with simulation results for all level of CMLIs for
11-level CMLI the results also validated with experiment.
A prototype single-phase 11-level CMLI has been
built using 400V, 10A MOSFET as the switching device.
Five separate dc supply of 12V each is obtained using step
down transformers with rectifier unit. Pentium 80486
processor based PC with clock frequency 2MHz with timer
I/O card is used for firing pulse generation. Firing pulses to
the switching devices are given through a delay circuit
which provides 5µsec delay.
In order to validate the analytical and simulated
results, an 11-level single-phase output voltage at m =
0.6200 was synthesized at fundamental frequency (f =
50Hz) producing output voltage.
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