This paper presents an implementation of selective harmonic elimination (SHE) modulation for a singlephase 13level transistorclamped Hbridge (TCHB) based cascaded multilevel inverter. To determine the optimum switching angle of the SHE equations, the NewtonRaphson method is used in solving the transcendental equation describing the fundamental and harmonic components. The proposed SHE scheme used the relationship between the angles and a sinusoidal reference waveform based on voltageangle equal criteria. The proposed SHE scheme is evaluated through simulation and experimental results. The digital modulator basedSHE scheme using a fieldprogrammable gate array (FPGA) is described and has been implemented on an Altera DE2 board. The proposed SHE is efficient in eliminating the 3
^{rd}
, 5
^{th}
, 7
^{th}
, 9
^{th}
and 11
^{th}
order harmonics, which validates the analytical results. From the results, it can be seen that the adopted 13level inverter produces a higher quality with a better harmonic profile and sinusoidal shape of the stepped output waveform.
I. INTRODUCTION
Multilevel inverter technology has drawn tremendous interest among researchers from industry and academia in recent years due to its superior performance. In contrast to conventional twolevel inverters, they are more efficient and better suited for applications requiring high power and high voltage levels. Among the various types of multilevel inverter topologies, the cascaded Hbridge (CHB) has attracted special attention due to its modular structure, which provides high reliability and better fault tolerance. Increasing the number of levels is also easier with minimal modifications in the hardware and control algorithm. Therefore, the CHB multilevel inverter has become popular in renewable energy (solar/wind power inverters), reactive power compensation (STATCOMs) and motordrive applications up to MegaWatt (MW) power levels
[1]

[6]
. For these applications, the converter’s output voltage must fulfill the requirement for maximum voltage and current THD as specified in IEEE Std.5191992
[7]
.
It is essential to produce an effective power converter from the perspective of cost, efficiency and output quality. These factors have leads for developing a new family of multilevel inverters known as transistorclamped converters (TCC)
[5]
,
[8]
. By adding additional devices (such as power switches, power diodes and capacitors) to an existing Hbridge topology, it is possible to increase the number of output voltage levels and produce a better sinusoidal output waveform. The TCC has a reduced number of dc power supplies and switches when compared to the conventional CHB topology designed for the same number of voltage levels.
The modulation strategy has a profound impact on the performance of multilevel inverters, since it determines the switching losses as well as the voltage and current harmonics. Generally, multilevel modulation strategies can be classified, according to the switching frequency, into two categories: highfrequency switching and lowfrequency (fundamental frequency) switching methods. The most popular modulation schemes discussed in the literature for multilevel inverters are multireference/multicarrierbased PWM
[4]
,
[5]
,
[9]
,
[10]
, multilevel spacevector PWM (SVM)
[11]
,
[12]
and multilevel selective harmonic elimination (SHE)
[7]
,
[13]
. The multilevel carrierbased PWM and SVM techniques are considered highswitching frequency schemes, whereas SHE falls within the lowswitching frequency group. Each solution has its unique advantages and drawbacks. As a result, the choice of the modulation technique usually depends on the inverter topology and its application.
Among the many modulation strategies, SHE is commonly adopted in high power applications where the switching frequency has to be low enough to minimize the switching losses. Since the effectiveness of the SHE method depends heavily on the switching angle, various algorithms have been developed for determining the optimum switching angles
[14]
,
[15]
. Usually, this is done offline, using optimization techniques such as the NewtonRaphson (NR) method
[14]

[17]
. More complex techniques such as the genetic algorithm (GA)
[15]
,
[18]

[20]
and the particle swarm optimization (PSO)
[15]
,
[21]
,
[22]
have also been demonstrated. Although these new techniques are fast in determining the optimized angle, the solutions only minimize the harmonics rather than eliminate them. Moreover, as a prerequisite, the engineers need to understand advance control and mathematics algorithms
[6]
.
More recently, online methods have been proposed where the realtime calculations of the switching angles are made possible using a high speed processor. In
[23]
, a mathematical calculation based on a trigonometric function was proposed to obtain the switching angles. In
[24]
, a simple and improved realtime algorithm for calculating the switching angles has been introduced and then proven by mathematical derivation. In
[6]
, the authors proposed a fourequation method based on harmonic injection and equal area criteria, regardless of the number of inverter output levels. In
[3]
, an artificial neural network (ANN) was used for obtaining the optimal switching angles for multilevel inverters in solar applications. However, due to the simplifications used, these online methods cannot fully eliminate the harmonics. Hence, they generally yield a significantly higher THD and are inferior to the offline methods.
SHE analyses for 5level to 13level CHB inverters with an unequal dc sources considering the minimum THD with or without elimination of the lowest order harmonics are discussed in
[25]
. However, the real implementation results are discontinued for 11level and 13level cases. For the 13level case, only simulation studies have been reported so far
[26]
. In this paper, real implementation of the SHE method on a 13level inverter is demonstrated.
Normally, digital implementation of SHE modulation involves two steps. First, the switching angles are calculated offline through solving a set of nonlinear and transcendental equations. Then, the switching angles are stored in a lookup table (LUT) for realtime application
[27]
.
In order to implement realtime control systems in power electronics applications, the system designers have many choices. Microcontrollers, microprocessors and DSPs are softwarebased devices, which come with efficient software compilers and programs usually written in C or assembly language. Although these technologies are matured and usually have dedicated PWM generation blocks, they have limited sampling rates and limited speed due to their natural sequence based operation (the programs are executed line by line, not simultaneously). This limitation can be resolved with a fieldprogrammable gate array (FPGA) as an alternative to programmable logic device (PLD) and application specific integrated circuit (ASIC) technologies. FPGAs are digital hardwarebased devices and they have become an increasingly popular technology in digital prototyping for multilevel inverters
[15]
,
[28]
due to their speed and flexibility.
In this paper, SHE modulation is suggested for a 13level transistorclamped Hbridge (TCHB) inverter based cascaded multilevel inverter topology. The NewtonRaphson method is used to calculate switching angles with the capability to eliminate the lowest order harmonics while maintaining the fundamental component. In order to generate an optimum stepped output waveform, a simple SHE modulation is defined according the voltageangle equal criteria. Real implementation of SHE modulation for a TCHB inverter using an FPGA is presented. The analytical results are validated through both simulation and experimental results.
II. THE POWER CIRCUIT
Fig. 1
shows the studied inverter configuration based on a 5level transistorclamped Hbridge (TCHB) multilevel inverter. The 5level TCHB is modified by adding one bidirectional switch to the Hbridge module. A very attractive feature of the bidirectional switch is that it allows for a bidirectional current flow and enables five output voltage levels of 0, ±½
V_{DC}
and ±
V_{DC}
. Even though such a topology has been discussed in
[9]
, the SHE modulation method is used instead of the multicarrier modulation method.
The 13level TCHB based cascaded multilevel inverter topology.
The inverter is supplied by three independent dc sources, three Hbridge modules and three bidirectional switches to produce a 13level output. Multiple dc sources at the input of the inverter may be obtained from constant dc supplies, batteries, super capacitors, photovoltaics or fuel cells. Generally, the number of output levels for the inverter is given by 4
i
+ 1, where
i
is the number of TCHB cells.
Through combinations of the on state of the switches (
S
_{i1}

S
_{i5}
), the cell output voltage
V_{i}
can be expressed as:
The modes of operation, the switches to be turned on and the corresponding output voltage levels are summarized in
Table I
. The 13level TCHB inverter’s operation involves 14 switching states, and all the operating states are illustrated in
Fig. 2
.
SWITCHING STATES AND VOLTAGE LEVELS OF THE 13LEVEL TCHB INVERTER
SWITCHING STATES AND VOLTAGE LEVELS OF THE 13LEVEL TCHB INVERTER
Operating states of the 13level TCHB inverter. (a) State 1. (b) State 2. (c) State 3. (d) State 4. (e) State 5. (f) State 6. (g) State 7. (h) State 8. (i) State 9. (j) State 10. (k) State 11. (l) State 12. (m) State 13. (n) State 14.
In order to justify the use of a TCHB inverter, performance comparisons are made against a conventional CHB inverter with the same 13level voltage outputs. In general, the output voltage quality for both inverters is the same. The most obvious advantage of the adopted inverter is that it has a lower component count when compared to a CHB inverter with the same number of output levels. This results in lower total power losses produced by the TCHB inverter when compared to the CHB inverter
[5]
. Since the CHB inverter requires two series connected cells to produce 5 voltage levels, a total of 24 switches are needed to generate a 13level voltage. On the other hand, the TCHB inverter requires only 15 switches to produce the same number of voltage levels. Moreover, the provision of isolated sources is the main disadvantage of the CHB topology. In this case, a total of six isolated dc sources is needed for the CHB, whereas the TCHB inverter requires three isolated dc sources. Owing to a bidirectional switch connected at the midpoint of the dc link to the TCHB inverter output, both of the topologies require six capacitors. However, a larger capacitance is required in the TCHB inverter to prevent a capacitor voltage imbalance
[5]
,
[9]
. A few practical approaches for balancing capacitor voltages are by replacing the capacitors with isolated dc sources, applying a backtoback intertie system
[29]
or the use of auxiliary dcdc converter circuits
[30]
. It is expected that the overall cost of the TCHB inverter will be lower than that of the conventional CHB inverter due to the reduced switch count and the lower number of isolated dc sources required.
III. BACKGROUND AND SOLUTION FOR SHE MODULATION
 A. Harmonic Elimination Technique
In general, the inverter output voltage
V_{inv}
waveform (see
Fig. 3
) can be expressed in the Fourier series as:
The inverter output voltage V_{inv} waveform.
It is worth noting that the actual waveform consists of sine terms only. The even harmonics are also absent due to the quarterwave symmetry of the output voltage along the
x
axis. Thus, both
a_{0}
and
a_{n}
are zero, and (2) can be simplified into:
Note that, when three TCHB cells are connected in cascade they provide a total output voltage
V_{inv}
as given by:
For each 5level TCHB module in the adopted inverter, the output waveform has a minimum step of ½
V_{DC}
. Thus, the Fourier series expansion of the inverter output
V_{inv}
waveform as seen in
Fig. 3
is
[4]
:
From (5), the fundamental output voltage
V
_{1}
can be expressed as:
and as indicated in
Fig. 3
, the switching angles
θ
_{1}
–
θ
_{6}
must satisfy the following condition:
Assuming that the expression of the modulation index
M
is given by:
where
s
is the number of positive steps in a quarter waveform.
The main objective of this paper is to produce an output waveform with the ability to eliminate the 3
^{rd}
, 5
^{th}
, 7
^{th}
, 9
^{th}
and 11
^{th}
order harmonics. In order to generate a staircase waveform, the switching angles can be obtained by solving the following equations:
A different approach is used to solve the harmonic problem of the adopted inverter. There are three steps involved in this method. A brief description of the implementation steps are given as follows:

1) The switching angle calculations are performed through the NewtonRaphson (NR) method.

2) The voltageangle equal criteria of the sinusoidal reference waveform is used to find the threshold voltage corresponding to the calculated angles.

3) A sinusoidal voltage reference is compared with the threshold voltages and combination logics are used to generate the gating signals.
 B. NewtonRaphson Method
The NewtonRaphson (NR) method is a commonly used iterative method for solving equations which are difficult to solve analytically. Here, the NR method is used in Matlab to solve the set of transcendental equations in (8). The solutions, which are the set of switching angles for the 13level inverter, are illustrated in
Fig. 4
. These switching angle are then examined for their corresponding total harmonic distortion (THD) given by:
The switching angles solutions for 13level inverter.
where
V_{n}
is the rms value of the
n
th harmonic component and
V
_{1}
is the rms value of the fundamental component.
IV. SHE MODULATION AND ITS IMPLEMENTATION
The 13level TCHB inverter topology is built with three isolated dc supplies. It is worth noting that there are a total of six equal steps (½
V_{DC}
,
V_{DC}
, 1½
V_{DC}
, 2
V_{DC}
, 2½
V_{DC}
and 3
V_{DC}
) in a quarter of the output waveform as depicted in
Fig. 3
. Taking into account the instances where output voltage is zero, this gives a total of thirteen steps in the output voltage waveform, hence the name 13level TCHB.
 A. Proposed SHE Modulation
Fig. 5
illustrates the proposed SHE modulation for the 13level TCHB inverter which is derived from one sinusoidal reference signal and several triggered voltage levels (
V_{H}
and
V_{L}
). Voltages
V_{H}
and
V_{L}
are classified as the triggered voltage levels; which correspond to the switching angles and switching states of the related cell.
The reference signal and triggered voltages for the respective angles.
According to
Fig. 5
, the reference voltage
V_{ref}
waveform is given by:
where
V_{m}
is an arbitrary (peak) value. For the developed SHE technique, only a halfwave diagram is illustrated and the modulus operation of a sinusoidal waveform has been included due to the fact that the steps in the other regions are sequential.
There are six voltage levels (
V
_{L,Cell1}
,
V
_{H,Cell1}
,
V
_{L,Cell2}
,
V
_{H,Cell2}
,
V
_{L,Cell3}
, and
V
_{H,Cell3}
) during the positive halfcycle of the modulating signal, where two voltage levels are specified for each cell. By using the voltageangle equal criteria, a simple calculation for the triggered voltage levels is derived from the calculated switching angles (
θ
_{1}
–
θ
_{6}
), which can be computed as:
Equation (12) is a generalized set of trigger voltages corresponding to the switching angles in order to perform the SHE modulation for the adopted inverter. The sequence is chosen such that cell 1 turns on and off at angles
θ
_{1}
&
θ
_{4}
, cell 2 at
θ
_{2}
&
θ
_{5}
, and cell 3 at
θ
_{3}
&
θ
_{6}
, respectively. This is to ensure that each cell has approximately the same turn on time, and to avoid overburdening of any particular cell. This concept can be applied to any angle, number of voltage levels and type of multilevel inverter topology.
 B. Hardware Implementation Using FPGA
Fig. 6
shows a block diagram of the SHE scheme implemented in an FPGA. The controller functional model was designed by means of Verilog HDL code and schematic design entries in Quartus II software. There are four submodules in the modulator architecture. One module is designated for the sine lookup table (LUT) for the reference
V_{ref}
signal and three logic modules are derived for the switches in each 5level inverter cell.
Illustration of FPGAbased SHE modulator.
In order to generate a sine reference waveform, only data for a quarter sine wave is needed. The sinusoidal reference generator of a 50 Hz signal is stored in the LUT with a 1 MHz sampling frequency. This requires only 5,000 sample data to be stored in the LUT, reducing the utilization of the FPGA devices without compromising the performance of the controller. The sinewave LUT (see
Fig. 6
) displayed data value of 650 used with Verilog code as the chosen peak reference throughout the simulations and hardware implementation. The 180˚ reference signal is generated based on the first 10,000 counter in logic low, and the next 10, 000 sequences in logic high. This will be continuously repeated.
The logic modules consist of the assigned voltage levels as representatives of the switching angles, comparators and combinational logics. The switching patterns of each cell in the 13level inverter were then formed by logic combinations from the reference and the calculated threshold voltage comparisons. Switches
S
_{i1}
,
S
_{i3}
, and
S
_{i5}
(
i
is the cell number) operate by comparing the reference signal with the triggered voltage levels and through the combinational logic gates, whereas
S
_{i2}
and
S
_{i4}
operate complementarily in a half cycle of the reference signal (i.e 50 Hz). Therefore, signals
S
_{12}
,
S_{22}
and
S_{32}
have the same pattern, as well as the signals
S_{14}
,
S
_{24}
and
S
_{34}
.
The general logic expression (simple combinations of AND, OR and NOT logics) of the gating signals are given as follows:
where
C
_{i1}
and
C
_{i2}
are the outputs of the comparators and the
p
180
ref
signal is the 180° reference based on the 50 Hz pulse signal which is also identified as the
S
_{i4}
signal. This control scheme can be extended to any numbers of 5level TCHB modules.
In order to implement the modulator on an FPGA, a cost effective Altera DE2 development board has been chosen to generate the digital gating signal. A Cyclone II 2C35 FPGA device has been included with aid of various hardware resources and components incorporated onboard, including 33 216 logic elements (LE), 483 840 RAM bits, 35 embedded multipliers, and 475 user I/O pins. The FPGA utilizations for the proposed system are summarized in
Table II
.
THE FPGA USAGE (ALTERA CYCLONE II  EP2C35F672C6)
THE FPGA USAGE (ALTERA CYCLONE II  EP2C35F672C6)
V. SIMULATION AND EXPERIMENTAL RESULTS
 A. Simulation Results
For verification of the FPGAbased design, simulations are done on ModelSim software. Two sets of switching angles are used: one is a set of angles which yields the minimum THD. The other case uses a set of angles which yields the maximum THD. The used switching angles are presented in
Table III
.
Fig. 7
shows the simulated gating signals of the minimum THD for the TCHB inverter switches in the ModelSim simulation environment. The highlighted line in
Fig. 7
is the location of the peak sine wave at a duration of 0.005 second, which has been determined by (11) and the respective gating signals.
THE SWITCHING ANGLES SET
Waveform of switching signals.
The THCB inverter is then simulated using Matlab/Simulink in order to verify the SHE modulation. The total dc voltage input is set to 360 V, so that the dclink for each 5level TCHB inverter module is 120 V. Two types of loads are tested, i.e. a purely resistive load and an RL load. The former uses only a 130 Ω resistor at the inverter output, while the later adds an 81 mH inductor in series with the resistor. These RL values are chosen to be the same as those in the experimental setup to facilitate the analysis and performance comparison between the simulation and experimental results.
Fig. 8
shows the simulated results of the inverter output voltage
V_{inv}
when the adopted inverter operates with the switching angles of the minimum THD.
Fig. 9
a shows the inverter output voltage THD of 6.77% for the minimum THD case. Meanwhile,
Fig. 9
b shows the inverter voltage THD of 8.13% for the maximum THD case. Both THD results show that the specific 3
^{rd}
, 5
^{th}
, 7
^{th}
, 9
^{th}
and 11
^{th}
harmonic orders have been eliminated.
The inverter output voltage V_{inv} using solution set with minimum THD.
Harmonic spectrum. (a) Solution set with minimum THD. (b) Solution set with maximum THD.
 B. Experiment Results
A prototype inverter is built to verify the effectiveness of the proposed SHE scheme. The experimental setup has been constructed to be similar to the simulation setup and it operates using the same switching angles shown in
Table III
. Insulated gate bipolar transistors (IGBTs) with ultrafast soft recovery diodes (IRG4PC50UD) and power diodes (30CPF12PBF) are used in the prototype inverter. The inverter was supplied by three GW Instek (GPC6030D) isolated dc sources
V_{DC}
= 120 V for each of the cells. Six 3300 μF electrolyte capacitors are used as dclink capacitors (
C
_{1}

C
_{6}
). The large capacitance is used to reduce the effects of voltage unbalance
[5]
, so that the voltage across each capacitor is maintained at approximately ½
V_{DC}
throughout the experiment. The prototype is tested under two load conditions as in the simulation: the resistive load case,
R
= 130 Ω, and the RL load case,
R
= 130 Ω and
L
= 81 mH.
Figs. 10

12
illustrate the waveform results of the adopted inverter in the test experiments using the solution angles set of the minimum THD.
Figs. 10
a
10
c show the output voltage for module 1, module 2 and module 3 (
V
_{L1}
,
V
_{L2}
,
V
_{L3}
) together with the resultant output voltage
V_{inv}
of the 13level inverter.
Fig. 11
shows a zoomin of the waveform of the 13level inverter output.
Figs. 12
a and
12
b show the inverter output voltage
V_{inv}
and the load current
I_{o}
waveforms when the inverter is tested with a highly resistive load and an RL load, respectively.
Experiment results with respect to 13level output. (a) Output voltage V_{L1}. (b) Output voltage V_{L2}. (c) Output voltage V_{L3}.
Inverter output V_{inv} waveform for solution set with minimum THD.
Experiment results for solution set with minimum THD. (a) The waveforms for R load. (b) The waveforms for RL load.
The inverter output voltage
V_{inv}
waveform when conducted with the solution angles set of the maximum THD is shown in
Fig. 13
. The output voltage THD for the minimum and maximum THD cases are found to be 6.6% and of 8.0% respectively, and they are shown in
Fig. 14
. These THD values demonstrate that the low order harmonics have been almost completely eliminated. They are generally in good agreement with the simulation results as depicted in
Fig. 15
, with slight deviations (not more than 0.3% THD).
Inverter output V_{inv} waveform for solution set with maximum THD.
The harmonic spectrum. (a) For minimum THD. (b) For maximum THD.
Comparison of inverter output voltage THD between simulation and experiment.
VI. CONCLUSIONS
This paper has presented a modular method for implementing SHE modulation in an FPGA for a singlephase 13level transistorclamped Hbridge (TCHB) based cascaded multilevel inverter. When more TCHB cells are cascaded, double voltage levels are produced when compared to the conventional CHB topology. As a result, the output quality can be improved with a lower number of switches. Using NewtonRaphson’s method, the optimum switching angles for eliminating the low order harmonics are obtained. The proposed SHE method produced a simple and precise implementation in practical applications. The analytical results were verified through simulation and experimental results, where the 3
^{rd}
, 5
^{th}
, 7
^{th}
, 9
^{th}
and 11
^{th}
order harmonics have been successfully eliminated. The proposed SHE modulation can be extended to others multilevel topologies at any level of output.
Acknowledgements
This work has been funded by University of Malaya (UM) through UM postgraduate research grants (PS112 2009C and PV037/2011B). The authors thank the Ministry of Higher Education Malaysia (MOHE) and Universiti Teknikal Malaysia Melaka for sponsoring the research work.
BIO
Wahidah Abd. Halim was born in Kuala Lumpur, Malaysia, in 1977. She received her B. Eng. degree (with Honors) in Electrical Engineering from the Universiti Teknologi Malaysia, Johor, Malaysia, in 2001, and her M.Sc. degree in Electrical Power Engineering from the Universiti Putra Malaysia, Serdang, Selangor, Malaysia, in 2005. She is presently working toward her Ph.D. degree at the University of Malaya, Kuala Lumpur, Malaysia. She is a Lecturer in the Faculty of Electrical Engineering, Universiti Teknikal Malaysia Melaka (UTeM), Melaka, Malaysia.
Nasrudin Abd. Rahim was born in Johor, Malaysia, in 1960. He received his B.S. (with Honors) and M.S. degrees from the University of Strathclyde, Glasgow, UK, in 1985 and 1988, respectively, and his Ph.D. degree from HeriotWatt University, Edinburgh, UK, in 1995. He is presently working as a Professor at the University of Malaya, Kuala Lumpur, Malaysia, where he is also the Director of the UM Power Energy Dedicated Advanced Centre (UMPEDAC). At present he is an Adjunct Professor at King Abdulaziz University, Jeddah, Saudi Arabia. Professor Rahim is a Senior Member of the IEEE, a Fellow of the Institution of Engineering and Technology, UK, and a Fellow of the Academy of Sciences Malaysia. He is also a Chartered Engineer.
Maaspaliza Azri was born in Malacca, Malaysia, in 1977. She received her B.Eng. degree (with Honors) in Electrical Engineering from Universiti Teknologi Mara, Shah Alam, Selangor, Malaysia, in 2001, and her M.Sc. degree in Electrical Power Engineering from the Universiti Putra Malaysia, Serdang, Selangor, Malaysia, in 2004. She is presently working toward her Ph.D. degree at the UM Power Energy Dedicated Advanced Centre (UMPEDAC), University of Malaya, Kuala Lumpur, Malaysia. She is a Lecturer in the Department of Power Electronics and Drives, Faculty of Electrical Engineering, Universiti Teknikal Malaysia Melaka (UTeM), Melaka, alaysia.
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