Modified Sine Wave Inverter: Harmonic Elimination

In inverter designs, we see the concept of a modified (or quasi) sine wave inverter - "better" than a square wave but still very easy to implement by adding a deadtime. By adjusting the duty cycle of the square wave, the output RMS value can be adjusted - and this is often used in inverters for feedback regulation.

One concept that is tied to how a modified sine wave output is better than a square wave has to do with harmonic elimination.

To start off with, let's remind ourselves that a square wave is composed of an infinite number of sine waves at odd multiples of the wave's frequency. Here is a great reference for this: Fourier Series--Square Wave -- from Wolfram MathWorld

Now, if we consider the modified sine wave, what happens to the harmonic contents?

We can solve for the nth harmonic (where n=1, 3, 5, 7, ...) which gives a Fourier coefficient of:

Why is this interesting? Can we fix the duty cycle so that we eliminate certain harmonics? Yes! We can make the cos term go to zero by picking nδ = π/2 = 90°. We could then pick δ = 30° and eliminate the 3rd harmonic! Or we could pick δ = 18° and eliminate the 5th harmonic. If we had to pick just one, it would then make sense to eliminate the 3rd harmonic which has a higher Fourier coefficient than the 5th! This will contribute to a better THD (total harmonic distortion).

We can of course time shift this waveform and retain the same benefits - this would then correspond to a standard (edge-aligned) PWM. If we are using a full-bridge to generate this waveform, this would correspond to a duty cycle (for each pair of diagonal switches) of (180°-30°-30°)/360° = 33.33%. The off time would, of course, reduce the output RMS voltage compared to a square wave, so you would have to keep that in mind!

What about multiple harmonic eliminations? We can come up with a scheme where we add multiple notches and eliminate multiple harmonics. And, in fact, this is what SPWM achieves by forming these notches in a "sinusoidally optimal" fashion and eliminating several lower order harmonics. I have several articles where I have discussed SPWM (such as Generation and Implementation of Sine Wave Table) albeit not in the context of harmonic elimination.

Can we do this in one-stage inverters like with an iron core? No. Unfortunately, in these one-stage inverters, the output voltage regulation is obtained by altering the duty cycle and so you can't have a fixed δ and optimize for a particular harmonic elimination.

Can we do this with two-stage inverters such as ones where the input voltage is first stepped up with a ferrite transformer at high frequencies followed by an H-bridge to generate the output? Yes - if we use feedback to control the output of the first stage, we can use this constant δ on the output of the second stage.