Modified Sine-Wave Inverter Enhanced

Aug 1, 2006 12:00 PM
By James H. Hahn, Associate Professor Emeritus, University of Missouri-Rolla Engineering Education Center, St. Louis

Analysis of Current Technology

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 sine-wave 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:

where the a_n and b_n 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 for the conventional square wave becomes:

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:

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 2α 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 π/2, we have:

Evaluation of this expression for various values of a indicates that the minimum harmonic distortion occurs at α = 0.352π, 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 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.