Thd

[dr.power]

Hi guys,

When I was reading through several papers regarding to THD to learn more about it I noticed something strange. To be honest it seems there is AT lEAST 2 definitions for THD which are not the same!
One defination is at the begining of page #297 of this:
https://books.google.com/books?id=y...resnum=10&ved=0CF4Q6AEwCQ#v=onepage&q&f=false

The other defination is here:
https://docs.google.com/viewer?a=v&...GA_ho-&sig=AHIEtbQyYpfnDa85EGyc4f3eKdQUpGYOfg

If we suppose that the Voltage of both the fundamental frequncy and the secondharmonic for instance happens to be 4v, then for the formula of the first link the THD would be:
THD= √16/√32 =0.707 =70.7%

But regarding to the formula of the second link it would be:
THD= √16/4 =1=100%

I myself think that the formula of the second link is correct as it gives 100% of THD for when both fundamental and the second harmonic of a waveform which contains just both of them with the same amplitude would be 100% not 70.7%.

So any idea plz?

The other question is that why we must use "RMS" values of a Voltage when want to calculte the THD of a signal? You know that V1p/V2p of a signal is equal to its V1rms/V2rms. so?

Thanks

 

[misterT]

The other question is that why we must use "RMS" values of a Voltage when want to calculte the THD of a signal?

You don't have to use RMS values. You can use any method you think that gives the best information you need.

 

[crutschow]

I agree that the equation in the second reference is less logical, and not a common definition of THD.

 

[dr.power]

I agree that the equation in the second reference is less logical, and not a common definition of THD.

Altough I think that the first one is correct but regarding to the example which I gave. think that the equation in the second reference is More logical. Plz take another look at my example.
Are you thinking that by having a signal equal in magnitude to the second harmonic we would get a THD of 70.7%?
I think we should get a THD f 100%. Am I wrong?

 

[squishy36]

There are numerous definitions of distortion. As always, such things are used to quantify a behavior in a single number -- and you often use those numbers to help you make decisions. Choose the definition that best helps with your decision-making process -- and only you can define the criteria that are important to you.

 

[MrAl]

Hi drpower,


As you probably already know, we need to compare the power in of the harmonics to get a true idea what is going on, and so RMS voltage values allow us to compare the power in two signals without having to actually calculate the power. This means that for a sine wave we can use the RMS values OR we can use the Peak values to do the comparison because in the division we cancel the conversion factor. If the wave is not sinusoidal however then it makes more sense to specify this as the RMS value so that it still relates to the power and not simply the voltage. I've never encountered a non sinusoidal case however.

Apparently for audio work they prefer the formula:
(DEFINITION 1) sqrt(sum of square harmonics 2 to infinity)/fundamental

over:
(DEFINITION 2) sqrt(sum of square harmonics 2 to infinity)/sqrt(sum of square harmonics 2 to infinity plus fundamental squared)

Note that DEF 1 satisfies both yours and my intuitive thoughts about a 2nd harmonic being equal to the fundamental should be deemed "100 percent distortion".
It makes sense that if you have a signal that is 1 and distortion that is 1 that the distortion is equal to the signal so it's 100 percent of the signal.

It's also interesting however that if we look at the addition in time of the fundamental and the second harmonic only, we only get an RMS value of 1, not 1.4142, so it appears that the addition of the single 2nd harmonic did not contribute to the total value equally but only as 0.7071 times as much, which is the result from the second definition above.
So it could be that the overall effect is that the second harmonic does not appear to be as great an influence as the fundamental even though it is equal.
For a very rough view of how this works, you can look at the peaks of the addition of sin(w*t) and sin(2*w*t) and see that the peak of that resulting wave does NOT reach as high as 2, but quite less than that, and also the resulting general width of the main body of the resulting wave is not as wide as the fundamental. These two observations taken together suggest that the power of the signal is not twice as much power as the fundamental, but only some fraction of twice as much. Thus, it would not make as much sense to say that the distortion is 100 percent.
To put it another way, with 70.71 percent of the fundamental added to 70.71 percent of the second harmonic, the result is not 1.4142 but is only 1.0 . If we then compare the distortion (71) to the total signal now (100) we find only 71 percent distortion.

 

[carbonzit]

Note that DEF 1 satisfies both yours and my intuitive thoughts about a 2nd harmonic being equal to the fundamental should be deemed "100 percent distortion".
It makes sense that if you have a signal that is 1 and distortion that is 1 that the distortion is equal to the signal so it's 100 percent of the signal.

Hmm; I was always taught that 1 + 1 = 2.

But seriously, shouldn't that example be termed 50% distortion? Because half the signal (the fundamental) is not distorted, and the other half is.

Now, if the 2nd harmonic somehow completely swamped the fundamental, then you'd have 100% distortion.

There are numerous definitions of distortion. As always, such things are used to quantify a behavior in a single number -- and you often use those numbers to help you make decisions. Choose the definition that best helps with your decision-making process -- and only you can define the criteria that are important to you.

Isn't this the very definition of apples-vs.-oranges comparisons? (Or cherry-picking, to use another metaphor.) What possible value could there be to THD as a figure of merit if you're allowed to choose the method by which you compute it?

 

[The Electrician]

Hi guys,

When I was reading through several papers regarding to THD to learn more about it I noticed something strange. To be honest it seems there is AT lEAST 2 definitions for THD which are not the same!
One defination is at the begining of page #297 of this:
https://books.google.com/books?id=y...resnum=10&ved=0CF4Q6AEwCQ#v=onepage&q&f=false

Both definitions are given in this reference. For some reason, they used %D of the left side of the first and the other has %THD on the left side.

You'll notice that the numerator of both expressions is the same; the first has the fundamental amplitude in the denominator and the other has the amplitude of the total signal in the denominator.

It's not an issue of which is correct--it's simply a matter of convention.

The electric power industry uses the one with the fundamental in the denominator. The audio industry uses the other.

Some instruments, such as, for example, the Fluke Power Quality analyzer, can measure both kinds of distortion:
https://www.fluke.com/fluke/usen/Power-Quality-Tools/Single-Phase/Fluke-43B.htm?PID=56080

The other question is that why we must use "RMS" values of a Voltage when want to calculte the THD of a signal? You know that V1p/V2p of a signal is equal to its V1rms/V2rms. so?

Thanks

What happens if there is more than one harmonic? Then you can't just add the peak values; you must use the square root of the sum of the squares method, which works only with RMS values.

 

[MrAl]

What happens if there is more than one harmonic? Then you can't just add the peak values; you must use the square root of the sum of the squares method, which works only with RMS values.

Hi Electrician,

That's not really true is it? We have the basic form:
sqrt(a+b+c)/sqrt(a+b+c+d)

and if we multiply every variable by a constant K, we get of course:
sqrt(Ka+Kb+Kc)/sqrt(Ka+Kb+Kc+Kd)

(where i didnt use multiplication signs for simplicity)

That can be factored into:
sqrt(K*(a+b+c))/sqrt(K*(a+b+c+d))

and simplifying a little we get:
(sqrt(K)*sqrt(a+b+c))/(sqrt(K)*sqrt(a+b+c+d))

or:

sqrt(K)/sqrt(K) * sqrt(a+b+c)/sqrt(a+b+c+d)

which of course simplifies back to the original expression:
1 * sqrt(a+b+c)/sqrt(a+b+c+d) = sqrt(a+b+c)/sqrt(a+b+c+d)


Carbon:
When we add the two waves in time and then find the RMS value, we dont get the sum of the two peaks or the sum of the two RMS values.
That's where the second definition comes from, because the TOTAL compared to the individual harmonic is not 1 when they are both the same.
I'll post some graphs.
In the diagram, you can see that both sine waves have peaks equal to 1, but the sum does not have a peak of 2.
You can also note that the area above and below zero is less than twice the area of the fundamental.

 

[The Electrician]

Hi Electrician,

That's not really true is it?

When I said "...which works only with RMS values. ", I meant it in the most general sense. When combining quantities RMS wise (square root of the sum of the squares), and if said quantities are not all sinusoidal (or possessed of some other shared waveshape) such that the ratio of peak to RMS is the same for each of the individual quantities, then one must combine RMS wise using the RMS values of the quantities, not their peak values.

If one had a peak value for the sum of several harmonics and then wanted to add another harmonic to the mix, adding (RMS wise) the peak value of the next single harmonic to the peak value of the previously existing sum of several harmonics won't necessarily give the peak value of the total. This is because, of course, the ratio of peak to RMS (crest factor) for a sum of several harmonics is not necessarily the same as the crest factor for a single sinusoid.

In the particular case where the individual components to be added together are all sinusoidal and are to be combined all at once then each component may be multiplied by any constant (such as peak to RMS) and then combined RMS wise. But this only works if all the components have the same crest factor.

Using the RMS wise addition of RMS components always works to give an RMS sum no matter what the waveshape of the components. It's better to get in the habit of using the RMS value of individual components when combining RMS wise.

Hmm; I was always taught that 1 + 1 = 2.

1 + 1 does equal 2 when you use the right point of view.

When RMS quantities are used, the underlying consideration is power. That is the rationale for using RMS quantities in the first place. The RMS value (voltage perhaps) of a complex waveform is that voltage which would dissipate the same power in a resistor as a DC voltage of that same value.

So, when you have a waveform consisting of a fundamental of 10 volts and a second harmonic of 10 volts, the RMS value of the combination is 14.14 volts. Notice that the fundamental and second harmonic, individually would each dissipate 1 watt in a 100 ohm resistor. The combination of fundamental and second harmonic has an RMS value of 14.14 volts, which would dissipate 2 watts in a 100 ohm resistor; this is double the power dissipated by the fundamental alone.

The distortion using the second (the audio version) distortion definition for this waveform would be 70.7%; this is a ratio of RMS voltages. This says that the power dissipated in a resistor by the harmonics (in this case, only a second harmonic of the same value as the fundamental) would be 1/2 the power dissipated by the total waveform (fundamental and second harmonic). In other words (using the second definition), the distortion power is proportional to the square of the distortion percentage (taken as a fraction) because the power a voltage E dissipates in a resistor R is given by (E^2)/R.

 

[MrAl]

Hi again Electrician,

Makes sense

I think what happens is most of the time we are dealing with sine waves so shortcuts come up. I'm pretty sure drpower is dealing with pure sine waves here, but i wont try to guess too hard and rather ask him if that's what he meant.

So drpower are you using only sine waves? You're working with some form of audio right?

I forgot to mention that the diagram i posted a couple posts back shows the addition of fundamental and second harmonic, but i meant to say that the 'peaks' dont add up to 2. The peak of the result only reaches up to approximately 1.76 and that is not strange because sine waves dont add up perfectly like apples or oranges when the phase is different.

 

[dr.power]

Hi again Electrician,

Makes sense

I think what happens is most of the time we are dealing with sine waves so shortcuts come up. I'm pretty sure drpower is dealing with pure sine waves here, but i wont try to guess too hard and rather ask him if that's what he meant.

So drpower are you using only sine waves? You're working with some form of audio right?

I forgot to mention that the diagram i posted a couple posts back shows the addition of fundamental and second harmonic, but i meant to say that the 'peaks' dont add up to 2. The peak of the result only reaches up to approximately 1.76 and that is not strange because sine waves dont add up perfectly like apples or oranges when the phase is different.

Hi Al,

Sorry I have to spend more time to read this thread again so that write several things. But Now I am not sure what do you mean by asking if I am using only the sine waves?!
Anyway to see the THD behavior of a circuit I just use pure sine waves, But for the goal which I have (you know what it is) I surely will use audio and even maybe music

 

[MrAl]

Hi drpower,

Yes, so that means you are using sine waves so you can use the peak voltage shortcut, but beware as Electrician pointed out this may not work for every waveshape you encounter in the future. Sine waves yes, other shapes depend on other things too. Some may work, some may not. RMS always works. That's all really.