MIT's take on the harmonic content of a square wave

The ignorance of experts II

Published on July 16, 2019

Copyright © Dan P. Bullard

As I said in a previous article, I relish finding idiotic content on harmonics everywhere, but when I find it at a renown center of higher learning, like MIT, I feel like I've hit paydirt, as they used to say in my home state of California. I used to search the web for real information on harmonics, thinking that certainly, a Navy vet and college dropout can't be the only one who figured out how harmonics work, but every time I use DuckDuckGo to search for harmonics (I gave up on Google, tired of being tracked) I am once again treated to inane and moronic diatribes that read more like voodoo incantations.

I recently created a cool new Excel tool that lets you create waves in time domain by specifying which harmonics you want, and I really like it, especially because, unlike my other tools that use inverse FFTs, this one requires no additions to your Excel (meaning that Excel knockoffs can run it) and you don't have to suffer the long delays inherent in running an iFFT in Excel. With this tool, one of the first things I did was to replicate the waves in MIT's (Morons In Training) embarrassingly bad courseware on Signals and Systems, a ramp wave and a square wave. Above you can see MIT's take on a square wave. You might have noticed that this diagram is completely flawed, as the square wave is out of phase with the fundamental and the harmonics. And not just that, as you will soon see. Here is my take on the square wave using my new tool.

You can see that my square wave is in phase with the fundamental (the blue sine wave centered in the plot) and you can also see how the Odd (blue) harmonics conspire to square up the otherwise smooth sine wave. For example, you can see how the 3rd harmonic peaks negative in the center of the first half cycle to push the sine wave down in the center, and on the right side, you can see how the 3rd harmonic is peaking positive to mash the negative peak of the sine wave back up to make it square. You can also see in the center of the plot how all of the harmonics are moving downward, to push the negative edge of our square wave down, faster and faster. The more harmonics, the faster it goes down. That my friends is known as bandwidth. The more bandwidth, the more harmonics and the faster the edges. Now, let's compare it to MIT's diagram.

Again, notice how the square wave is completely out of phase with the fundamental, and if you try to reconcile the wave with the harmonics like we did before, there is no way you can. Notice also that this diagram shows Even harmonics too, along with the Odd harmonics. If you have read any of my articles, you will know that 50% duty cycle square waves contain only Odd harmonics, shown in blue in my plot above, but no Even harmonics in red. Only when the duty cycle gets away from 50% do you start seeing Even harmonics. So why is the MIT diagram so screwed up? Because the course developer cheated, he stole the harmonics diagram from the ramp wave!

Look closely at the harmonics in both diagrams above. They are exactly the same! But the lower one, the ramp matches my plot pretty well.

Notice here that the color coding really helps. You can see the fundamental in blue near the center of the plot, then at the top, is the 2nd harmonic in red, then the 3rd harmonic in blue, etc. Notice that it matches the MIT ramp diagram pretty well! All the harmonics start at a zero degree phase offset, and then naturally fall out of phase over time, to finally end back in phase at the far right again. The conspiracy of all the harmonics lining up to create the one and only fast edge in a ramp wave is what creates a ramp wave. Both Odd and Even harmonics in proportions of 1/harmonic number, as opposed to a square wave which has Odd harmonics only in the same proportions. So why do the MIT ramp wave and the MIT square wave diagrams look so similar? Because the course developer cheated by taking the ramp wave diagram and cropping out the ramp wave, then pasting in the square wave, but in the wrong phase, making it easy for me to find this pair of really stupid mistakes.

Now, people pay good money to attend MIT, and yet MIT can't even be bothered to create simple tools like mine in Excel, MatLab, or even bloody C to create useful and accurate training material. I spent years doing training, and I can tell you that I would be extremely embarrassed if I had screwed up this badly. And yet, I'm not a professor, I don't even have a college degree. And after looking at MIT's coursewave, I am very happy that I don't have a college degree, especially from the morons at MIT!