Unidimensional Modulation Technique for
Cascaded Multilevel Converters
J. I. Leon, Member, IEEE, S. Vazquez, Member, IEEE, S. Kouro, Member, IEEE, L. G. Franquelo, Fellow
Member, IEEE, J. M. Carrasco, Member, IEEE and J. Rodriguez, Member, IEEE
Abstract—This paper presents a simple and low computational
cost modulation technique for multilevel cascaded H-bridge
converters. The technique is based on geometrical considerations
considering an unidimensional control region to determine the
switching sequence and the corresponding switching times. In
addition, a simple strategy to control the dc voltage ratio between
the H-bridges of the multilevel cascaded converter is presented.
Examples for the two-cell topology are shown but the proposed
technique can be applied to develop modulation methods for a
higher number of H-bridges. Experimental results are presented
to validate the proposed technique.
I. INTRODUCTION
T
HE development of new simple and efficient modulation
techniques for multilevel converters is a focus of research
in the last 25 years. Pulse width modulation (PWM) and
space vector modulation (SVM) are the most common ways
to obtain the modulated voltage of a multilevel converter
generating the reference voltage averaged over a switching
period [1]–[3]. The large number of output voltage levels is
an important constraint leading to complex hardware systems
using multiple triangle carriers (in PWM techniques) or a
high computational burden (in SVM techniques). Among the
existing multilevel converter topologies, the cascaded H-bridge
converter (CHB) is one of the most interesting ones due to
its high modularity, fault tolerant capability, reduced number
of power devices and high efficiency [4]–[6]. In Fig. 1, a
single-phase two-cell CHB is shown. The output phase voltage
V
ab
is obtained as the addition of the output voltage of
each H-bridge (also called cell) of the converter. In general,
the dc voltage ratio of the two-cell CHB converter can be
defined as k:1 (being k a real positive number) meaning that
the voltage of the upper cell V
C1
is k times higher than
the voltage of the lower cell V
C2
. In this paper, a simple
and intuitive modulation method for CHB is presented. This
modulation technique is based on geometrical considerations
leading to the determination of the switching sequence and the
switching times by very simple calculations. In addition, the
Manuscript received August 22, 2008. Accepted for publication Febru-
ary 14, 2009. Copyright
c
° 2009 IEEE. Personal use of this material is
permitted. However, permission to use this material for any other pur-
poses must be obtained from the IEEE by sending a request to pubs-
permissions@ieee.org. J. I. Leon, S. Vazquez, L. G. Franquelo and J. M.
Carrasco are with the Electronic Engineering Department, University of
Seville (Spain), (e-mail: jileon@gte.esi.us.es). S. Kouro is with Department of
Electrical and Computer Engineering, Ryerson University, Toronto (Canada)
(e-mail: samir.kouro@ieee.org). J. Rodriguez is with the Electronic Engineer-
ing Department, Universidad T
´
ecnica Federico Santa Mar
´
ıa (Chile), (e-mail:
jrp@usm.cl).
b
S
3
C
1
C
2
a
V
C1
V
C2
V
ab
S
3
S
4
S
4
S
1
S
1
S
2
S
2
upper cell
lower cell
I
ab
Fig. 1. Two-cell cascaded H-bridge converter.
ab
V
3
3
a)
b)
ab
V
Fig. 2. 1D control region of the two-cell CHB (where V
C2
= E volts) with
dc voltage ratio equal to a) 1:1 b) 2:1. The redundant switching states are
chosen reducing the switching losses.
proposed method considers directly the redundant switching
states in order to reduce the switching losses. This paper is
an updated version of [7] including new experimental results
and introducing improvements in the implementation of the
proposed modulation technique.
II. 1D CONTROL REGION
As was introduced in [8], [9], a possible way to represent the
switching states of multilevel converters is to plot the possible
output voltages of the converter using a one dimensional (1D)
control region. For example, the control region of the two-cell
CHB with dc voltage ratio 1:1 and 2:1 are shown in Fig. 2.
Each cell of the converter can obtain three different output
voltages, −V
Ci
, 0 and V
Ci
, defined as H-bridge states 0, 1
and 2 respectively. In this figure, a state XY corresponds to
the upper H-bridge having state X and the lower H-bridge
having state Y.
III. GEOMETRICAL MODULATION TECHNIQUE
The proposed modulation strategy generates the reference
voltage (V
∗
ab
) as a linear combination of the two nearest switch-
ing states of the control region. Therefore, this calculation is
reduced to a geometrical search of V
∗
ab
in the control region.
The switching sequence is formed by two switching states
XY called upper
1
-lower
1
and upper
2
-lower
2
with switching
times t
1
and t
2
respectively. The switching times are also
determined using very simple mathematical expressions. If
the two nearest switching states of the control region have
redundancies, those that reduce the number of commutations
are selected. For example, in the case of CHBs with dc
voltage ratios different of 1:1, the redundant switching states
are chosen reducing the switching of the high dc voltage
cell. The transitions between the different redundancies are
illustrated with arrows in Fig. 2. In addition, the order of the
switching states in the switching sequence is chosen according
to the premises presented in [10], [11] in order to improve the
harmonic performance of the output waveforms. The proposed
modulation strategy is based on the following steps:
1) Normalization of the reference phase voltage V
∗
ab
using
the dc voltage of the lower voltage cell (E volts) using
expression
a =
V
∗
ab
E
. (1)
2) Determination of a
i
factor as floor(a) where operator
floor(x) rounds the elements of x to the nearest integer
towards minus infinity.
3) Determination of the switching times per unit of the two
switching states applying
t
1
= a − a
i
t
2
= 1 − t
1
. (2)
4) Geometrical search of the reference phase voltage V
∗
ab
in the 1D control region using factor a
i
.
As an example, the flow diagram of the proposed geometri-
cal modulation technique for the two-cell CHB with dc voltage
ratio 1:1 and 2:1 are shown in Fig. 3. The flow diagram of the
dc voltage ratio 3:1 case was shown in the previous version of
this paper [7]. As can be observed, the resulting flow diagrams
are simple and consequently the computational cost is very
low.
The proposed technique can be extended to CHB with
more than two power cells. The analysis to extend this
modulation technique for converter with more power cells rises
proportionally in complexity, since the number of cases to be
studied increases. However, this study can be done offline
and it can be noticed that the online calculations needed
to execute the corresponding modulation technique do not
increase significantly.
IV. DC VOLTAGE RATIO CONTROL
An interesting application of the CHB is the grid connection
without use of the input transformer that provides the isolated
dc sources or also as active filter. However, the dc voltages
control is a challenge in these applications [12]–[15]. A simple
strategy to control the dc voltage ratio of the two-cell CHB
is introduced with the proposed geometrical modulation to
be applied to existing rectifier or grid connection control
strategies. The controller used in this paper to manage the sum
of the dc voltages of the two-cell CHB was introduced in [16].
However, the total dc voltage is shared and controlled between
the cells of the CHB applying the proposed modulation with
simple additional dc voltage ratio considerations.
The output of the controller is the reference phase voltage
V
∗
ab
which is the input of the proposed geometrical modulation
technique. The technique to control the dc voltage ratio
between the cells is based on the elimination of the switching
states which tend to unbalance the dc voltage ratio [17]. In
Table I, the switching states to be eliminated of the 1D control
region are summarized depending on the actual dc voltage
ratio and the sign of the phase current I
ab
. These switching
states are eliminated from the 1D control region shown in Fig.
2, leading to new flow diagrams to develop the geometrical
modulation technique. For instance, the dc voltage ratio 2:1
has been studied in this paper. The flow diagrams to carry out
the modulation achieving the control of the dc voltage ratio
2:1 are introduced in Fig. 4. The 1:1 case and the 3:1 case
were presented in the previous version of this paper [7].
V. EXPERIMENTAL RESULTS
Firstly, the proposed geometrical modulation technique for
the two-cell CHB working as an inverter is applied generating
a pure 50 Hz sinusoidal reference with modulation index equal
to 0.9. The switching frequency is 2 kHz and the CHB is
connected to a RL load (R=20 Ω, L=15 mH). The obtained
results are shown in Fig. 5 for dc voltage ratios equal to
1:1 (V
C1
=V
C2
=90 V), 2:1 (V
C1
=200 V, V
C2
=100 V) and 3:1
(V
C1
=270 V, V
C2
=90 V). Note that for dc voltage ratios 2:1
and 3:1, the commutations and consequently the switching
losses of the high power cell are reduced. It can be noticed
that in the 1:1 case (Fig. 5 a) both cells equally share (in
average) the power demanded by the load due to the nature of
the proposed modulation and control strategy. In other cases
such as 2:1 and 3:1, as the same current flows through all the
cells and each cell contributes with a voltage proportional to
its dc-link voltage, the H-bridges share the power in a ratio
similar to the dc voltage ratio of the cells of the CHB (Fig. 5 b
and Fig. 5 c). This phenomenon could be seen as a drawback
in terms of switch usage and loss of modularity. However, the
benefits are a strong reduction in the switching losses and an
improvement of the power quality with same number of power
semiconductors.
Secondly, the application of the proposed method for a
CHB controlled rectifier connected to the grid through an
TABLE I
SWITCHING STATES TO BE AVOIDED TO CONTROL THE DC VOLTAGE RATIO
k:1
Voltage Forbidden states Forbidden states
unbalance with I
ab
> 0 with I
ab
< 0
V
C1
> kV
C2
20,21,10 02,12,01
V
C1
< kV
C2
02,12,01 20,21,10
a
i
< 0
ab
*
/
a
i
=floor(a)
t
1
=a-a
i
t
2
=1-t
1
upper
1
=0
=0
=a
i
YES
NO
a=V
E
upper
2
lower
1
lower
2
+2
=a
i
+3
upper
1
=
upper
2
lower
1
lower
2
2
=
2
=a
i
+1
=a
i
a
i
> 0
YES
NO
a
i
> -2
YES
NO
upper
1
upper
2
=
1
=
1
upper
1
=2
=2
=a
i
upper
2
lower
1
lower
2
-1
=a
i
=a
i
lower
1
lower
2
+1
=a
i
+2
upper
1
upper
2
=
0
=
0
=a
i
lower
1
lower
2
+3
=a
i
+4
a)
b)
ab
*
/
a
i
=floor(a)
t
1
=a-a
i
t
2
=1-t
1
a=V
E
Fig. 3. Flow diagram of the proposed geometrical modulation technique (where V
C2
= E volts) for the two-cell CHB with dc voltage ratio equal to a) 1:1
b) 2:1
a
i
< -1
a
i
=floor(a)
YES
NO
a
i
= -1
YES
NO
a
i
= 0
NO
YES
upper
1
=
upper
2
lower
1
lower
2
2
=
1
=
1
=
0
upper
1
=0
=0
=a
i
upper
2
lower
1
lower
2
+3
=a
i
+4
t
1
=a-a
i
t
2
=1-t
1
t
1
=a+1
t
2
=1-t
1
t
1
=a
t
2
=1-t
1
upper
1
=
upper
2
lower
1
lower
2
1
=
2
=
1
=
1
t
2
=1-t
1
upper
1
=
upper
2
lower
1
lower
2
2
=
2
=
2
=
1
t
1
=
a-1
2
a
i
> 0
a
i
=floor(a)
YES
NO
a
i
= 0
YES
NO
a
i
= -1
NO
YES
upper
1
=
upper
2
lower
1
lower
2
1
=
0
=
2
=
1
upper
1
=2
=2
=a
i
upper
2
lower
1
lower
2
-1
=a
i
t
1
=a-a
i
t
2
=1-t
1
t
1
=a
t
2
=1-t
1
t
1
=a+1
t
2
=1-t
1
upper
1
=
upper
2
lower
1
lower
2
0
=
1
=
1
=
1
t
2
=1-t
1
upper
1
=
upper
2
lower
1
lower
2
0
=
0
=
1
=
0
t
1
=
a+3
2
a)
b)
ab
*
/
a=V
E
ab
*
/
a=V
E
Fig. 4. Flow diagram of the geometrical modulation technique (where V
C2
= E volts) for a dc voltage ratio control equal to 2:1 when the eliminated
switching states are a) 20, 21 and 10 b) 02, 12 and 01.
inductance (L=3 mH) is tested. The switching frequency is 2.5
kHz. Experimental results are presented achieving dc voltage
ratios 1:1, 2:1 and 3:1 in Fig. 6. In the experiments, a load
step from no load to connecting unbalanced resistive loads
(R
1
=20 Ω and R
2
=10 Ω for upper and lower cells respectively)
to the dc voltages of the CHB is shown. As conclusion, a
high quality dynamic behavior of the dc voltage ratio control
strategy is achieved. It is clear that the proposed geometrical
modulation with the dc voltage ratio strategy achieves an
accurate dc voltage control of each cell, despite that the
external rectifier controller is only in charge of the total dc
voltage control. If the capacitor voltages are not perfectly
controlled a possible solution to minimize the related distortion
in the output waveforms is to apply a feed-forward modulation
(a)
(b)
(c)
Fig. 5. Experimental results for inverter operation with dc voltage ratio: a)
1:1, b) 2:1 and c) 3:1. In all the figures from bottom to top: Channel 1: Output
voltage of upper cell; Channel 2: Output voltage of lower cell; Channel 3:
Phase voltage V
ab
; Channel 4: Phase current I
ab
technique [17].
As can be observed from Fig. 7, the elimination the un-
balancing switching states leads to a slight distortion in the
phase voltage V
ab
when the desired dc voltage ratio is not
1:1. This is the trade-off to achieve asymmetric dc voltage
ratios. In order to analyze this phenomenon, the total harmonic
distortion (THD) of V
ab
is studied depending on the value
of the modulation index for dc voltage ratios 1:1, 2:1 and
3:1. The switching frequency is 2 kHz and the total active
power provided by the two-cell CHB is 6 kW. The converter
is connected to resistive loads in such a way that the power
ratio between the cells coincides with the dc voltage ratio.
The obtained results are shown in Fig. 8 where the THD
is calculated considering harmonic order up to 49
th
. In this
figure, the modulation index is defined as the ratio between the
(a)
(b)
(c)
Fig. 6. Experimental results of the proposed geometrical modulation
technique with control of the dc voltage ratio: a) 1:1 (
V
∗
C1
=
V
∗
C2
= 40
V), b) 2:1 (V
∗
C1
= 2V
∗
C2
= 50 V) and c) 3:1 (V
∗
C1
= 3V
∗
C2
= 60 V). Grid
voltage V
s
= 50 V
rms
. In all the figures from bottom to top: Channels 1-2:
Voltage of the cells V
C1
and V
C2
, Channel 3: Grid voltage V
s
, Channel 4:
Phase current I
ab
amplitude of the fundamental component of the phase voltage
and maximum possible output dc voltage of the converter.
From this result, it can be seen that the dc voltage ratio 2:1 is
the best one in order to obtain a better performance in terms
of THD. Ratio 1:1 has no additional distortion since the elim-
inated switching states have redundant allowed states keeping
the five output voltage levels. In ratio 2:1, there is a better
THD compared to 1:1 despite of the eliminated switching
states (some of them have no redundancy) because 2:1 has
seven output levels, compensating the distortion introduced
by the dc voltage ratio algorithm. On the other hand, ratio 3:1
presents higher THD since there are no redundancies available
for all switching states leading to a higher distortion when
they are eliminated. The possible nine output levels that can
−100
0
100
−100
0
100
0 0.01 0.02 0.03 0.04 0.05
−100
0
100
Time (s)
V
ab
(V)
a)
b)
c)
Fig. 7. Phase voltage V
ab
using the proposed geometrical modulation
technique with control of the dc voltage ratio. Dc voltage ratio a) 1:1 b)
2:1 c) 3:1
0.6 0.7 0.8 0.9 1
15
20
25
30
35
40
45
Modulation index
Vab THD(%)
ratio 1:1
ratio 2:1
ratio 3:1
Fig. 8. Total harmonic distortion versus modulation index of the obtained
phase voltage V
ab
depending on the desired dc voltage ratio.
be generated using ratio 3:1 do not compensate this distortion.
VI. CONCLUSIONS
A simple and low computational cost modulation method
for multilevel cascaded converters has been presented. The
modulation method determines the switching sequence and the
switching times based on geometrical considerations using an
unidimensional control region. The reference phase voltage
V
∗
ab
is generated using a linear combination of the two nearest
switching states in the control region reducing the switching
losses. Several examples using a two-cell CHB have been
introduced depending on the dc voltage ratio of the converter.
Experimental results are shown in order to validate the pro-
posed strategies. The same method can be applied to CHB
with a higher number of cells extending the control region and
developing similar flow diagrams to determine the switching
sequence and the switching times.
In addition, a simple strategy to control the dc voltage ratio
of each cell of the two-cell CHB has been introduced. This
technique is based on the elimination of the inappropriate
switching states from the 1D control region. Once these
switching states are eliminated, new flow diagrams are used
to carry out the geometry-based modulation using reduced
versions of the 1D control regions.
The phase voltage V
ab
quality achieved by the proposed
modulation strategy is similar to other well-known PWM tech-
niques such us level-shifted PWM or hybrid PWM techniques.
However, using the proposed modulation technique with dc
voltage ratio 1:1, an equal usage of the power semiconductors
under all possible values of the modulation index is obtained.
In addition, for other dc voltage ratios, the commutations of
the higher voltage H-bridge have been reduced leading to a
reduction of the commutation losses of the system making
the proposed technique very attractive improving the overall
efficiency of the converter. On the other hand, when the CHB
is operating as a rectifier, the proposed modulation strategy has
been modified in order to control the dc voltage ratio using
the redundant switching state concept what cannot directly
considered by other PWM techniques.
Experimental results are presented to show the operation of
the proposed modulation technique with the dc voltage ratio
control for a two-cell CHB. The results show that the proposed
modulation technique with the dc voltage control achieve high
quality results with very low computational cost.
ACKNOWLEDGMENT
The authors gratefully acknowledge financial support pro-
vided by the Spanish Ministry of Science and Technology
under project TEC2006-03863, by the Chilean National Fund
of Scientific and Technological Development (FONDECYT),
under grant no. 1080582 and by the Universidad T
´
ecnica
Federico Santa Mar
´
ıa.
REFERENCES
[1] D. G. Holmes and T. A. Lipo, “Pulse Width Modulation for Power
Converters - Principles and Practice,” IEEE Press, 2003.
[2] R. Naderi and A. Rahmati, “Phase-Shifted Carrier PWM Technique
for General Cascaded Inverters,” IEEE Trans. Power Electron., vol. 23,
no. 3, pp. 1257–1269, May 2008.
[3] Y. Wenxi, H. Haibing and L. Zhengyu, “Comparisons of Space-Vector
Modulation and Carrier-Based Modulation of Multilevel Inverter,” IEEE
Trans. Power Electron., vol. 23, no. 1, pp. 45–51, Jan. 2008.
[4] J. Rodriguez, S. Bernet, Bin Wu, J. O. Pontt and S. Kouro, “Multilevel
Voltage-Source-Converter Topologies for Industrial Medium-Voltage
Drives,” IEEE Trans. Ind. Electron., vol. 54, no. 6, pp. 2930–2945, Dec.
2007.
[5] L. G. Franquelo, J. Rodriguez, J. I. Leon, S. Kouro, R. Portillo and M.
M. Prats, “The age of multilevel converters arrives,” IEEE Trans. Ind.
Electron. Magazine, vol. 2, no. 2, pp. 28–39, June 2008.
[6] D. Krug, S. Bernet, S. S. Fazel, K. Jalili and M. Malinowski, “Com-
parison of 2.3-kV Medium-Voltage Multilevel Converters for Industrial
Medium-Voltage Drives,” IEEE Trans. Ind. Electron., vol. 54, no. 6, pp.
2979–2992, Dec. 2007.
[7] J. I. Leon, S. Vazquez, A. J. Watson, P. W. Wheeler, L. G. Franquelo
and J. M. Carrasco, “A simple and low cost modulation technique
for single-phase multilevel cascade converters based on geometrical
considerations,” IEEE International Conference on Industrial Technology
2008 (ICIT’08), pp. 1–6, 21-24 April 2008, Chengdu (China).
[8] J. I. Leon, R. Portillo, L. G. Franquelo, S. Vazquez, J. M. Carrasco and E.
Dominguez, “New space vector modulation technique for single-phase
multilevel converters,” IEEE International Symposium on Industrial
Electronics (ISIE’07), pp. 617–622, 4-7 June 2007, Vigo (Spain).
[9] J. I. Leon, R. Portillo, S. Vazquez, J. J. Padilla, L. G. Franquelo
and J. M. Carrasco, “Simple Unified Approach to Develop a Time
Domain Modulation Strategy for Single-Phase Multilevel Converters,”
IEEE Trans. Ind. Electron., vol. 55, no. 9, pp. 3239–3248, Sept. 2008.
[10] S. Kouro, J. Rebolledo and J. Rodriguez, “Reduced Switching-
Frequency-Modulation Algorithm for High-Power Multilevel Inverters,”
IEEE Trans. Ind. Electron., vol. 54, no. 5, pp. 2894–2901, Oct. 2007.
[11] B. P. McGrath, D. G. Holmes and T. Lipo, “Optimized space vector
switching sequences for multilevel inverters,” IEEE Trans. Power Elec-
tron., vol. 18, no. 6, pp. 1293–1301, Nov. 2003.
