Transcript
Introduction to Interharmonics and RMS Variation
Hello, everyone, and welcome to today’s white paper webinar. Today, we’re talking about interharmonics and how interharmonic frequencies can cause a continuously varying RMS voltage. For the fact that they’re not synchronous with 60 hertz, you get an effect where the RMS voltage is different from cycle to cycle.
This just doesn’t happen with harmonic distortions. It’s not inherent in waveform distortion that you get this RMS variation. It’s something that arises when you have non-synchronous frequencies present. Landon here with me has written an interesting white paper on a mathematical treatment of this issue and how it relates to flicker. So I’ll hand this over to Landon.
RMS Calculation with a Sum of Sine Waves
Hello. What I’ve done is I’ve used the standard root mean squared calculation, and we assume that a signal can be broken up into a sum of a bunch of different sine waves. So we can do as many sine waves as are present in the signal. I apply this sum of sine waves into this RMS formula, where we square it, integrate over time, and then divide by the period, and then take the square root.
So Equation 1 is just plugging this sum into this expression. Equation 2 is simplifying this a little bit. So I’m squaring everything, writing it as the RMS squared, rearranging the sums and the integrals, and factoring out all of the amplitudes of the sine waves, and then we’re just left with this giant expression.
The main part of this expression is this. We have an integral from X minus one up to X of some random sine wave times another random sine wave. And so this is mainly what I’m analyzing, is this integral, this inside integral right here, and that’s written out in Equation 3. So Equation 3 is just integral over a one-cycle time period of a sine wave times that of a sine wave.
Equation 5 and the F Function
After plugging this into a computer algebra system and going through a lot of algebra steps, we end up with Equation 5 right here, where this F is defined to be one over M times the sine of M over two times the cosine of M times X minus one plus B.
So we have the difference of the two frequencies. There’s M sub I and M sub J right here. So this term here is M sub I minus M sub J. This term is B sub I minus B sub J. This term is M sub I plus M sub J. And this term here is B sub I plus B sub J. So we’re taking the difference of the two parameters here, and we’re taking the sum of the two parameters here.
Graphing the Results
And then this expression here has a lot of interesting results. If I go to my graph here, I have one sine wave defined on this purple trace, and I have a different sine wave defined on this blue trace. And then this red graph is this expression right here, this Equation 5, where I have the F function of the difference minus the F function of the sum.
So what you see is you have one sine wave with a frequency of one, and you have another sine wave with a frequency of 1.2. This term here has a frequency of the difference of the two. So if we look here, that’s only a frequency of 0.2. So this is 1.2 minus one, and so we get this main period that you see right here, and then the much faster period, which is the sum of the two. So that’s M equals one plus M equals 1.2. So you have one sine wave which is 0.2, and then you have another sine wave which is 2.2 on this graph here.
I can adjust any of these parameters. It’ll tell me the evaluation of this integral. This is at some point X, you’re integrating over the last cycle from X of this product of the sine waves, and then the result is the red curve here.
Special Cases of the F Function
A couple of interesting things to note. If I make M1 go to zero, then we’re just left with a straight line. So the first thing to notice is that when the frequency of the sine wave goes to zero, this is just a straight line. In normal RMS calculations, when your sampling period of the RMS is the same or an integer multiple of the 60 hertz line, then it doesn’t matter where in the cycle you start sampling, consistent throughout the entire sample.
Another thing to see is that if this is not zero, so let’s make this one, then it’s just a sine wave expression. So this F function is just another sine wave. It’s the same frequency as the original, but it’s different amplitude and it’s different phase shift. If I change my original phase shift, it’s the same phase offset as the original.
Now, one interesting thing that can happen when our sampling is the same frequency as the 60 hertz line. So for example, let’s do a pi and a pi here. Notice how we get another DC offset because this term here with the difference contributes to the DC offset, and this term here with the sum goes away. So this one has a frequency of zero. This one has a frequency of two times the sine. And so this term in the F expression makes this vanish.
So when your RMS period is an integer multiple of the 60 hertz line, this term vanishes and you’re just getting a DC offset. But if I would make it 3.5, then we get something very, very different from that.
Detailed Analysis of Sums and Differences
All of my results about what is happening with this expression here, this is detailed in all of these paragraphs. So this paragraph talks about how you’re always getting a difference in frequency, and you’re getting a sum frequency. So if you start with a 25 hertz and a 40 hertz, you’re gonna end up with a 15 hertz signal, which is the difference of the two, and you’re gonna end up with a 65 hertz signal, which is the sum of the two.
When the two input waves have the same frequency, you have a DC offset here, and then you have a frequency which is double. So if I make each of these three, notice that when both frequencies are three, then we get a DC offset, and then we get just a pure sine wave of double the frequency. So this is oscillating twice as fast as the main frequencies. We’re just getting pure DC offset, and then we’re also getting a pure sine wave.
When the two input waves have the same frequency, M, and the same phase shift, then this DC offset that was cosine of the difference just becomes one, and then you just get one half for the DC offset. That’s when the two phase shifts are aligned.
And then this paragraph talks about how because of this two factor, pure sampling is a multiple of 30 hertz, then it also goes away. So a frequency of two pi for each one makes this flat lined. But if I make it sine of pi, which is half, then it also just goes to a flat line as well.
Interharmonic Signals and Equation 6
This paragraph says that interharmonic signals, which are basically when you have a signal present on the line which is not an integer multiple of the 60 hertz, when that is present, then you don’t get this nice cancellation out that all of your variation goes away.
So we’re basically left with this equation six, where I’m substituting in this result into this expression into equation two. So I’m solving this integral in equation two, and then the answer is equation five, and then I’m substituting that into equation six right here to get the result.
Worked Example
And then I’ve done a worked example. My red signal is a 120 RMS signal. So this is your standard 120 volt signal. I have a 5% signal at the third harmonic, and I have a 3% signal at the fifth harmonic. That’s in the red graph right here.
And then for blue graph, I have the same as the red, but I have an additional 2% signal at 175 hertz. So this is just a little bit off from this 180 hertz signal. So with this example, I’ve gone through this calculation. This is the H of T that we’re talking about. And then we’re going to substitute this to equation six into this equation right here, and we’re going to compute all of the sums and all the differences and all the product of the waveforms.
Plugging this in directly, we get this whole expression right here. So you get basically from this parentheses all the way down to here. And then after doing as many simplifications as we can do, all of the places where we go 60 hertz part. So for example, this term right here is going to go away because it is a direct integer multiple of 60 hertz.
But what you’re gonna be left with is all of this simplifies down to all of this, where you get all of these terms. All of those come from just the DC part of each individual sine wave. And then you’re also gonna get a bunch of the sums and differences:
- A 115 hertz component
- A 235 hertz component
- A five hertz component
- A 355 hertz component
- A 125 hertz component
- A 475 hertz component
- A 350 hertz component
RMS Variation and the Five Hertz Signal
So I’ve went ahead and plotted all of those together. That’s in this black graph here. It’s basically just oscillating very quickly between two different sine waves. This is a low frequency five hertz sine wave right here, and this is a low frequency sine wave right here that are offset by just a little bit.
This expression describes how the RMS of this signal is changing at any point in time over the last cycle of measurement. So we’re computing the RMS over a single cycle of the 60 hertz line. You can see there’s a lot of high frequency variation, but the problem that we care about is the low frequency five hertz signal, because if you notice, it’s going from down here, which is roughly -0.4%, up to up here, which is roughly 0.41%.
So this is oscillating on a five hertz basis between 1.0041 times 120, and then going down to 0.9996 of 120. So it’s very quickly oscillating between this point here and this point here on a five hertz basis.
Flicker and the IEEE Standard 141
One of the problems that you see with that is that the flicker standard — this graph is from the IEEE Standard 141 that describes the borderline of visibility for flicker. So you have dips per second, dips per minute, and dips per hour. This is the tolerable limit for what is noticeable, and then this line here is the limit for what is irritable and very, very annoying to look at.
With this example, our red dot — we’re oscillating about 0.45%, basically we’re oscillating half a percent in RMS over a five hertz basis. So our five hertz line is right here, and we’re right about half, and that’s well inside of the borderline of visibility curve for the IEEE standard. So with this particular example, it would be very, very obvious. It would be very annoying to look at with lights.
Just an example of how interharmonic variation with just a little 2% signal at 175 hertz — we’re injecting just a little bit of extra signal into the power line, and we get a massive variation in the RMS, which would lead to a massive problem with flicker.
Key Takeaways on Interharmonics and Flicker
Thanks, Landon. The key here is that if you have these non-synchronous frequencies, interharmonics, you’re gonna get variations in RMS voltage that are very quick here. Like Landon said, we’re at about five hertz. Anywhere between the two and 10 hertz range is where people are absolutely the most sensitive to flicker.
So as you can see in this graph, if you’re in this region, the allowable limit before you see the threshold of irritation is really small. We’re way under 1% is the limit. It’s more like half percent. That’s not a lot of RMS voltage variation allowed, and it’s easy to get that kind of variation when you have even just small levels of these interharmonics because of the way they create very rapid variations from cycle to cycle.
Unlike harmonic components, which even though harmonics are waveform distortion, they’re the same RMS value from cycle to cycle because they’re synchronous. And so harmonics may produce other problems on the power line, but they’re not going to introduce flicker like these interharmonics do.
Contact Information
If you have any questions about the paper, feel free to give us a call anytime at 1-800-296-4120 or send an email to support@powermonitors.com. Thanks for tuning in, everyone. Have a great day.
