What is harmonic resonance and how does it work?

A very important part of harmonics is what is harmonic resonance, specifically parallel resonance. 

If we look at harmonic resonance, the most famous example of resonance in terms of mechanical resonance is the Tacoma Narrows Bridge. If you've ever seen videos of that bridge, it was built as a suspension bridge and it oscillated because the wind started blowing and moved the bridge. And because it didn't have enough damping, it oscillated and eventually ripped itself apart. That mechanical resonance example is similar to what we've seen with electrical resonance, with harmonics on power systems. In the case of that bridge, the wind induced the vibrations. And, after a little bit of time, it ripped itself apart. In our case, harmonics are a lot like the wind in terms of exciting the resonance point. Then what happens is the harmonics in the power system create resonance between two components. In this case, a capacitor and a transformer. And when that occurs, it creates a very significant amount of current – its current amplifier situation. 

So if we take this situation, we look at the equation – KVA short circuit divided by the capacitor size on the system. We put in our numbers for this situation. We have 1500 KVA divided by 0.05. So, 5% impedance – that’s 30,000 divided by 600 is basically the square root of 50. So, the resident point for harmonics is around the seven point first harmonic. Now, what that means is basically if I have any seventh harmonic on the system, I might put out 10 amps, but it might amplify it to a hundred amps or a thousand amps. So that problem becomes a real issue if that current's interchanging between the capacitor and the utility power source. Basically, through the transformer. As we look at that - what's important is there's an initiation. Usually when something occurs, like we turn on a drive, we turn on some load that creates harmonics and then the resonant effect occurs. 

What do we have to worry about if we have a capacitor on the system and we have a harmonic source  just like the bridge? It would've been fine if the wind didn't blow, but since we had both the bridge started moving and if it didn't have enough damping, it ripped itself apart. In our case, we create resonance because of the power system components, the transformer, the capacitor, and resonance. If the right harmonic frequency comes by, it's going to excite that resonant point. You have to check for series and parallel resonance. In this case specifically, we're looking at parallel resonance, which is usually the most damaging series resonance we’ll talk about in passive harmonic filter design. And that's actually how you design a filter, but parallel resonance. We look at the power system impedance, including the transformer and upstream impedance and that electrical impedance in parallel with the capacitor to put on the power system creates that resonance. 

You can see in this graphic where there's some change in resonant point and the larger the capacitor size, the more that frequency goes down. And of course, that's a problem because when we talk about harmonics, the lower order harmonics are usually the higher magnitude harmonics. So with a bigger capacitor, we excite resonance at a lower frequency. Now, if you had a switch capacitor bank, and you had 50 K bar steps, with every step, you'd have to be concerned with calculating that resident point. We did it once for our system. I'm going to basically turn on a power factor, correction, capacitor with a variable frequency drive and show you what I mean. Let me switch over here and we'll take a look. I'll turn on a variable frequency drive. 

I'll put the drive on at first, without any harmonics or without any capacitors on the system. And, we'll see what the drive current looks like. So this is going to be a standard six pulse variable frequency drive. And now I'm going to turn on a capacitor that's about 75 KVAR. As you look at this, you can actually count which frequency is exacerbated or excited higher. You can see here, start here 1, 2, 3, 4, 5, and then it repeats. If I look at my harmonic spectrum, my current, my fifth harmonic right now on that system is about 35 or 36 amps. And if I turn off the capacitor bank and I look at the resident point after that, or if I look at the fifth harmonic after that, it's going to be about 12 or 13 amps. 

You can see it triples the current, just with that size. Now, what I'm going to do is turn off a few of the capacitors, go down to maybe 40 capacitors or 40 KR. And you can see then that the fifth starts to go down in terms of magnitude. Looks like it's going up, but it's because it's a percent THD, but the actual current goes down to about 27 amps and the seventh harmonic current goes up. If I go down to 20 K V R again, you can see the seventh is coming up and the fifth is going down. So, as we change and the 11th is actually coming up. This is a very common situation. Again, going back to this situation, the smaller, the capacitor bank, the higher the frequency that you're going resonate. And a final thing I'll draw that out on a piece of paper for you here. 

If I have my power system and I have a variable frequency drive here, VFD, or any other harmonic load, I create a current that normally wants to flow back to the utility source. However, if I put a capacitor here, that capacitor in parallel with that transformer looks like this. We have a harmonic current source here. We have an inductor, which is the transformer. We have a capacitor here. And, that parallel combination is what we're talking about. Let's use the numbers J 10 and minus J 10 for the capacitor and the inductor in parallel. And if I have, let's say 10 amps of harmonic current here, and I have this number and this number in parallel X parallel, the parallel combination is the product over the sum. So, it's J 10 times minus J 10, over J 10 plus minus J 10. 

You can see quickly that the bottom part of this equation ends up being essentially zero. So, some number divided by zero starts to approach infinity. The problem is if I look at that on a curve, in terms of my frequency versus impedance graphic, it's going to have what we call a parallel resonant point or very high impedance at some frequency. If that frequency happens to correspond to some frequency that this drive creates, we have a problem – which frequencies are we going to create? Fifth, seventh, 11th, 13th. And, in our previous example, let's say that this is the seventh harmonic. So we're going to amplify the seventh harmonic because we have a high impedance. Let's say it's a thousand OS. If it's a thousand OS and we have 10 amps of current going into this parallel combination, that's 1000 OS 10 amps times a thousand OS. 

The voltage across here is going to be 10,000 volts. Well, that's not going happen for very long because something's going to blow up. So, what happens is it excites resonance and these two start to oscillate together. You get current that flows that might be 10,000 volts divided by how many homes for this branch. Basically we have 10. So, we have a thousand amps flowing through each part of this system circulating back and forth. And what that really means is – I have current going back and forth here, a thousand amps, even though I'm only putting out 10 amps here. So you can see pretty quickly that hopefully if you're lucky, you blow a fuse in that capacitor, otherwise if you damage or blow up the capacitor, you could have a real problem. So harmonic resonance – specifically parallel resonance – is a significant problem. 

I'm going to show you one last picture here. Again, harmonic resonance – specifically parallel resonance –  is a significant problem. And if you put on harmonic sources – meaning variable frequency drives or anything else that produces harmonics – and you have capacitors on the system. You have to check for resonance, or you blow up capacitors. Now, if you see the inside of a capacitor, that's not a good thing. So we want to avoid harmonic resonance and we won't blow fuses and capacitors on a power system or even worse, damage your transformers.