Spectrum of a full wave rectifier loading a power line

Harmonic Currents

Published on April 15, 2020

Copyright © Dan P. Bullard

I almost cringe writing such words, harmonic currents. In the power industry, this is a common obfuscation (To make so confused or opaque as to be difficult to perceive or understand) similar to telling a little kid that "The rain is God's tears." But now you have to explain why God is crying, and boy, that brings up a ton of issues, doesn't it?

Dr. Surya Santoso, professor at the University of Texas at Austin wrote a book on the topic of power quality named Electrical Power Systems Quality. I have been referred to this book over and over again in various online articles and posters on Quora to explain how harmonics are created from harmonic currents (shudder again!). Very often, the articles talk about florescent lights, motors and rectifiers causing harmonic currents, which in turn causes harmonic voltages. I realized that I could easily simulate the effect of an analog full wave rectifier from the perspective of a load. What I did was to knock the transfer function down to 0.9 all the way from -1.0 to -0.12, then I left it at 1.0 all the way to +1.2 where I went back to 0.9 for the rest of the transfer function all the way to +1.0. This simulates having a power supply with a 17:1 transformer that puts out a maximum of 10V peak, which is then applied to a standard full wave bridge rectifier which would remove a total of 1.2 volts (0.6V on each leg) reducing the voltage to about 8.8 volts peak. By programming the transfer function down to 0.9 for most of the transfer function I am assuming that the current draw of the circuitry attached to the power supply pulls the power voltage down by 10% while the diodes are conducting, and are not pulling anything when the diodes are not conducting, which should make sense.

So, how does that impact the voltage on the AC Input side? Well, the current draw pulls the voltage down on the source side and that causes the source voltage to sag, and the waveform should look like this:

Now you can barely see it, but if you look at the zero crossings (both of them) you can see a slight "bump" in the wave. Let me show you a closup:

So, the power supply is causing the power line to be distorted just like the good professor posits. So what is the effect? Here is the spectrum.

Only Odd harmonics that start low, then rise up out of the noise floor, hump over, then go back down to zero, then hump back over, and over and over. We have seen this before, have we not?

Yes, we have seen this before. This is the spectrum of a Class B amplifier causing a crossover distortion right at the zero crossing. We also know that it's symmetrical because otherwise it would have Even harmonics. Here is a look at the time domain version of the wave.

The distortion is quite small so I had to circle it in purple, but it's a similar kind of distortion. As I point out in my books, it's not the kind of distortion that causes specific types of harmonic signatures, it's the angle at which the sine wave impacts the distortion. It doesn't matter whether the signal comes from a bridge rectifier, a Class B amplifier, tube or transistor, fish or ham sandwich. What matters is at what phase the sine wave is at when it impacts the distortion (Bullard Laws of Harmonics #2). It doesn't matter whether it takes a bite out of the wave at that point, or boosts it up (as in the rectifier example above). The spectrum would look the same either way. Remember Bullard Laws of Harmonics #4 (which some people think doesn't need to be stated). The spectrum for these waves:

Look like this:

The biggest bump positive and the biggest bump negative both share the same spectrum because what you are not seeing in this spectrum is the phase. The phase of the harmonic determines which way the signal goes, whether it bumps up or bumps down. But almost no spectrum analyzer will show you that, which is why I invented the Bullard plot.

So, whether the zero crossing is chopped out, like in the Class B crossover distortion example, or is bumped out because the diodes in a rectifier turn off and reflect a high impedance back across the transformer windings back to the incoming power line, the effect is the same. And that interesting pattern (let me show you again) -

is one reason I find harmonics so interesting. Look at that harmonic signature! If there were any Even harmonics, they would start high near the fundamental, but because this distortion happens near the zero crossing, the Odd harmonics start low, then rise up out of the noise floor and then peak later. That is one of the biggest reasons why I named my first book Distortion: The Cause Of Harmonics And The Lie Of THD. THD is a lie, because most often nobody goes above the 9th harmonic. Follow me (by twos). The 3rd harmonic is -73dB, the 5th is -69dB, the 7th harmonic is -66dB and the 9th is -64dB. After that the harmonics continue to rise up to -63dB, -62dB, -62dB -61.7dB and then it starts humping downward again. But why are we ignoring all those values? They would significantly increase the THD if they were added in. Oh, well, there's the rub! That would make the THD look worse than it would if we just keep it down to only the first 8 harmonics. So counting above the 9th harmonic would make the device (amplifier, mux, etc) look bad, and we can't have that!

Thinking about this, I decided to try something else. What if one of the diodes in my diode bridge had a voltage drop of 0.8V instead of 0.6V? That would mean that in one direction, the cut-off voltage would be 1.4 volts instead of 1.2 volts, making the distortion asymmetric. Here is a look at the time domain version of this wave.

Notice that I also used a trick to smooth the transition to be more lifelike. That won't make much difference to the lower harmonics. And if you look closely you can see that the top starts to flatten out at about 1.6V but the bottom of the distortion happens around 1.4V. The difference is 0.2 volts, but it seems a little odd because I used a smoothing trick which modified the voltages you might have expected. It doesn't matter much though, the main point is that it's asymmetrical now, so we get this spectrum:

Now you can see the Even harmonics come up the way we have gotten used to them, starting high and arching over to a low, then humping over again and again. So in this simulation, Bullard Laws of Harmonics #3 still applies, asymmetry releases the Even harmonics. What I have proven there is that Bullard Laws of Harmonics apply whether the distortion happens in an amplifier or as a voltage reflection from a power using appliance back to the power company. Maybe somebody should let Professor Santoso know about this.

So, do current harmonics cause voltage harmonics? Well, you could say that, but it still doesn't answer the question. Why is God crying? What did you do to him? How many virgins do we need to sacrifice to appease him? See where this leads? Better to just answer the bloody question honestly than to make stuff up.