Modified Sine-Wave
Inverter Enhanced
By James H. Hahn, Associate Professor Emeritus, University of
Missouri-Rolla Engineering Education Center, St. Louis
Altering the waveform produced by a modified
sine-wave inverter reduces distortion products,
while still permitting use of efficient switching
techniques.
W
ith the increasing popularity of alternate
power sources, such as solar and wind,
the need for static inverters to convert dc
energy stored in batteries to conventional
ac form has increased substantially. Most
use the same basic concept: a dc source of relatively low
voltage and reasonably good stability is converted by a highfrequency oscillator and stepup transformer to a dc voltage
with magnitude corresponding to the peak of the desired
ac voltage. A power stage at the output then generates an ac
voltage from the higher-voltage dc. Conceptually, the operation is illustrated in Fig. 1.
market today. One category is the “pure sine-wave” inverter,
which produces sine waves with total harmonic distortion
(THD) in the range of 3% (-30 dB). These are typically used
when there is a need for clean, near-sine-wave outputs for
medical, instrument and other critical applications.
Some, for example, are used in boats and RVs as the
main source of electricity, and some feed energy back into
the utility power grid. Waveforms approaching sine waves,
with minimal distortion, are required in any case. These
inverters are available in sizes up to several thousand watts
and typical costs are in the range of $0.50 per watt (visit
www.invertersrus.com/digitalpuresineinverters.html for
an example). Early techniques for designing these true sinewave inverters incorporated significant linear technology,
reducing their efficiency and contributing to their higher
cost.
More recent designs used pulse-width modulation
(PWM) to produce a pulsed waveform that can be filtered
relatively easily to achieve a good approximation to a sine
wave (for example, see U.S. patent numbers 4,742,441;
4,600,984; 6,980,450; and 4,466,052). The significant advantage of the PWM approach is that switching techniques
are used in the power stages, resulting in relatively high
efficiency.
However, PWM, with the pulse width made to vary according to the amplitude of a sine wave, requires significant
control circuitry and high-speed switching. This is because
the frequency of the PWM signal has to be much higher than
that of the sine wave to be synthesized if the PWM signal is
to be filtered effectively. So the PWM approach introduces
significant complexities and switching losses.
The second category consists of relatively inexpensive
units, producing modified sine-wave outputs, which could
logically be called “modified square waves” instead. They are
basically square waves with some dead spots between positive
and negative half-cycles. Switching techniques rather than
Current State of the Art
There are basically two kinds of dc-ac inverters on the
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Fig. 1. Most static-power inverters used in solar- and wind-power
applications convert dc to ac using the architecture shown here.
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Fig. 2. Modified sine-wave inverters actually generate square waves
with dead spots between the half-cycles, allowing switching techniques rather than linear circuits to be used in the power stage.
Power Electronics Technology August 2006
20
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Voltage amplitude
linear circuits are used in the power stage, because switching techniques are more efficient and thus less expensive.
These inverters require no high-frequency switching, as
the switching takes place at line frequency. Their costs are
generally in the range of $0.10 per watt (for an example, see
www.invertersrus.com/inverters.html).
The typical modified sine-wave inverter has a waveform
as shown in Fig. 2. It is evident that if the waveform is to be
considered a sine wave or a modified sine wave, it is a sine
wave with significant distortion.
Time
Fig. 3. The square wave provides a benchmark against which we may
compare the THD of the modified sine wave and the waveform generated by the proposed inverter architecture.
Analysis of Current Technology
Voltage amplitude
It is well known that any periodic waveform such as that
mentioned previously can be represented by a Fourier series,
an infinite sequence of sines and cosines, at the fundamental
frequency of the waveform and its harmonics. These harmonics can cause trouble in several areas—particularly in
motors and sensitive applications—and the data sheets for
the inverters frequently caution the user that certain devices
may not work with the inverter. Furthermore, even though
the root-mean-square (RMS) value of the waveform may
be a nominal 115 V or 120 V, the peak will be different than
that of a true sine wave, and that factor can cause trouble in
applications that depend on the peak value.
The actual percent distortion is not usually quoted in
the specifications for inverters other than the pure sinewave versions, so it is instructive to compute the distortion
products to get a feel for the relative distortion involved
with the different approaches. For purposes of comparison,
let us look first at a conventional square wave (Fig. 3). The
coefficients of the Fourier series are computed with a pair of
integrals that produce the coefficients of the sine and cosine
terms in the series.
For a signal f(x) with a zero dc component, the integrals
are:
�
2�
2�
for the conventional square wave becomes:
π
2
 4
bn =   ∫ f (x ) sin(n
nxx )d
dx =
 π 0
4
for odd values of n only.
nπ
The series is then (4/π)sin x + (4/3π)sin(3x)+
(4/5π)sin(5x) + . . .
The standard measure of distortion is THD defined as:
2 1/ 2
 Σ(distortion products) 
fundamental
n>0
 nΣ=∞(b )2 
 n=3 n 
=
b1
1/ 2
.
Numerical evaluation of the coefficients for the square
wave indicates that if the square wave is to be considered
a sine wave with distortion, the THD is in the range of
45% (-7 dB). The third harmonic, the hardest to filter out,
is one-third the magnitude of the fundamental (-10 dB).
Turning now to the modified sine wave, let us define the
width of the positive and negative portions as 2 as depicted
in Fig. 4. Again noting that the waveform has both half-wave
symmetry and quarter-wave symmetry, and carrying out the
integration from 0 to  /2, we have:
2π
n > 0,
where the an and bn terms are the coefficients of the
cosine and sine terms, respectively, in the series. The Fourier
series is then:
f(x) = a1cos x + a2cos 2x + a3cos 3x + . . . + b1sin x + b2sin
2x + b3sin 3x + . . .
The complete background on Fourier series, as well as
treatment of special cases, is covered in several textbooks
on networks or engineering mathematics, and will not be
repeated here. We will just note that because both the square
wave and the modified sine wave have both half-wave
symmetry and quarter-wave symmetry, integration is
required only over one-quarter of the waveform, and
further that only the sine terms and odd harmonics are
required. Thus, the integral used to compute the coefficients
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Fig. 4. Defining the width of the positive and negative portions of the
modified sine wave as 2 enables us to calculate its Fourier coefficients
and then determine its minimum values of THD.
and
 1
bn =   ∫ f (x ) sin(n
nxx )dx
dx
 π 0
0
Time
2π
 1
a n =   ∫ f (x ) cos(n
nxx )dx
dx
 π 0
2�
 4
bn =   sin(nα) for odd values of n only.
 nπ 
Evaluation of this expression for various values of 
indicates that the minimum harmonic distortion occurs at
 = 0.352, where the THD is 23.8% (-12 dB), about half
that of the square wave. The third harmonic is about 6.5%
(-24 dB) of the fundamental, also a significant improvement
21
Power Electronics Technology August 2006
SINE WAVES
2�
Voltage amplitude
B
Voltage A
Switch #1
Voltage B
Switch #2
GND
Switch #3
Voltage -B
Switch #4
Voltage -A
Switch #5
2�
VOUT
A
0
��2
GND
�
-A
-B
2�
2�
Fig. 6. The proposed enhancement to the modified sine wave inverter is
implemented by connecting the output lead to a specific voltage level
at the correct time with electronic switches such as power MOSFETs.
Time
Fig. 5. Adding another level to the modified sine-wave results in a Fourier
series with four variables that may be varied to minimize distortion,
though in practice “B” will be set equal to 2 A.
but any filtering applied to reduce the third through ninth
harmonics will be even more effective on those above the
ninth. Therefore, the higher-order harmonics are ignored
in this analysis.]
over the square wave. However, these figures indicate that
the modified sine wave is far from being a true sine wave,
and suggest that improvement is in order.
Implementation
Proposed Improvement
As demonstrated here, the modified-sine-wave inverter
can be modified further to produce a much closer approximation to a sine wave, at a relatively small increase in
manufacturing costs, simply by incorporating another level
into the waveform. The design still uses switching technology in the power stage, assuring high efficiency. A patent
application has been submitted for the approach described
in this article.
The switching stage could be implemented with a combination of bridge and half-bridge components commonly
used in power switching applications. To produce the proposed multiple-level waveform, several implementations
are possible. In general, they all involve connecting the
output lead to a specific voltage level with switches such as
power MOSFETs capable of handling substantial current.
Consider the block diagram shown in Fig. 6 where the
voltages A and B correspond to the voltage levels defined
previously.
Appropriate digital logic and timing circuits will be used
to activate each switch at the correct time to achieve the
 and  pulse widths. A table can be developed to indicate which switches must be closed for each section of the
output waveform. Note that Switch #3 in Fig. 6 will need
to be a bidirectional switch, since it must switch the output
lead VOUT to ground regardless of any voltage present in the
load. All other switches can be unidirectional.
Unlike conventional PWM-inverter designs, which switch
at high frequencies, the proposed inverter design switches
at just three times the line frequency. As a consequence, the
proposed inverter design will reduce switching losses from
that of the PWM-controlled inverter and will save power
regardless of the output power level.
For further details on implementation, contact the author
at jhahn@umr.edu.
PETech
Consider now a further modification—the addition of
another level. The waveform is shown in Fig. 5. Again using
the fact that the waveform has both half-wave and quarterwave symmetry, we carry out the integration over the period
0 to /2, with the result that:
 4
bn =   {(A sin (nβ) + (B-A)sin (nα)}
 nπ 
for odd values of n only.
This result has four variables, of which all could theoretically be varied to achieve minimum distortion. However, one
particularly efficient approach is to choose a very simple
set of values for A and B—namely B = 2A—and then
optimize the values of  and  for minimum distortion.
This approach requires only two positive and two negative
power-supply voltages, all of which can be generated from a
single transformer in the high-frequency oscillator. (Other
values of A versus B may be useful, but were not investigated
because the simple relationship of B = 2A had very good
results as discussed later.)
With this restriction, evaluation of the Fourier coefficients indicates that the minimum distortion that can
be achieved is about 6.5% (-24 dB), and occurs at  =
0.42 and  = 0.248. The third harmonic is only about
0.17% (-55 dB) of the fundamental, suggesting that minimal low-pass filtering would greatly reduce the fifth and
higher-order harmonics and produce a relatively clean sine
wave. The third harmonic can be eliminated entirely, with
 = 0.42 and  = 0.246698, at the expense of slightly
higher THD.
[Note: The Fourier analysis was carried out through the
ninth harmonic for all three types of waveforms considered
in this article. Harmonics above the ninth are not negligible,
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