THD question

[shooter]

I’m still crawling here, so don’t hit me with grown-up stuff…yet.

Trying to understand THD, using only a single frequency, symmetrical, no clipping.

Got this quote from pages of wiki data;

“Triodes (and MOSFETs) produce a monotonically decaying harmonic distortion spectrum.[clarification needed] Even-order harmonics and odd-order harmonics are both natural number multiples of the input frequency”

If I amplify a single sine wave, one cycle, then watch the spectrum either side, I see decaying harmonics of the fundamental?

[HotBluePlates]

Trying to understand THD, using only a single frequency, ... no clipping.

Got this quote from pages of wiki data;

“Triodes (and MOSFETs) produce a ... decaying harmonic distortion spectrum.[clarification needed] ...

"Decaying Spectrum" = Higher harmonics are lower amplitude.

... Even-order harmonics and odd-order harmonics are both natural number multiples of the input frequency” ...

1st Harmonic: 100Hz Signal (Test tone; "Fundamental")
2nd Harmonic: 200Hz ("Octave")
3rd Harmonic: 300Hz ("Octave + 5th")
4th Harmonic: 400Hz ("Octave + 5th + 4th", "2 Octaves")
5th Harmonic: 500Hz ("2 Octaves + Maj 3rd")
6th Harmonic: 600Hz ("2 Octaves + Maj 3rd + Min 3rd", "2 Octaves + 5th")
7th Harmonic: 700Hz ("2 Octaves + Maj 3rd + Min 3rd + Sub-Minor 3rd", "Out-of-tune Flat-7")
8th Harmonic: 800 Hz ("3 Octaves")

All the "even numbers" above are the "even harmonics".  All the "odd numbers" above are "odd harmonics".  Harmonic distortion goes only in one direction: higher than the fundamental/test tone.

Musical instruments don't play a single, pure tone (flutes can come pretty close, as do some others).  The harmonics inherent in instruments played notes are often called "overtones" (they are "over" or higher than the named-note being played), and contribute to an instrument's sound quality ("timbre").

If I amplify a single sine wave, ... then watch the spectrum either side, I see decaying harmonics of the fundamental?

If you see a spectrum plot with blips on both sides of the test tone, you're seeing intermodulation distortion.  Harmonic distortion would only be higher, and only whole-number multiples of the test frequency.  Intermodulation is when 2 or more tones are amplified by a non-linear system, and generate sum and difference frequencies.

So blips below your test tone are difference frequencies, and point to I.M. distortion.  If you have any I.M. distortion, you have more than one tone being applied to the system.  "More than one tone" wouldn't be due to I.M. of the pure test tone, because the sum/difference frequencies always result in other frequencies already present  due to harmonic distortion (try adding/subtracting any of the frequencies in the list above).

[HotBluePlates]

... Even-order harmonics and odd-order harmonics ...

If I amplify a single sine wave, ... then watch the spectrum either side, ...

A spectrum analyzer is the easiest/best tool for "seeing" distortion.

But in case you're looking at the wave on a scope, this link should help you identify what even & odd harmonic distortion looks like.

In a nutshell, even harmonic often shows up as a flattening of one side of the wave & and narrowing of the other side.  When you're looking at loadlines and see the tube will limit peak excursion on one side & not the other, you might guess that will result in even harmonic distortion.  The is also "asymmetrical" distortion.  Triodes tend toward even harmonic distortion because the grid curve crowd together at low plate current, and this is often the first limit reached before severe overdrive.

Odd harmonic distortion tends to flatten the waveform on both sides; it is "symmetrical".  When you run into tube characteristics which will limit the peak excursion on both sides of the wave, you can guess you'll have odd harmonic distortion.  Pentodes have grid lines which bunch together below the knee of the grid curves, as well as crowding of grid curves at low plate current.  So there are two places at either end of the plate voltage swing which can contribute to distortion, and a greater possibility of odd harmonic distortion.

[shooter]

 

Harmonic distortion would only be higher

I knew that one

Intermodulation is when 2 or more tones are amplified by a non-linear system, and generate sum and difference frequencies.

Got the sum/dif part, didn't know that was IM, thanks

instruments don't play a single, pure tone

This go round I'm shooting for replication of the input, amplified.

Ok, so  "decaying spectrum" is inevitable, a normal process of a decaying sin wave?

In my old life we called it a FID (frequency in decay, IIRC)

The pic below is from a previous build, 1.8K sine wave, is my notation...close then from your explanation?

[HotBluePlates]

... Ok, so  "decaying spectrum" is inevitable, a normal process of a decaying sin wave? ...

No, because you're no longer in the time/amplitude domain.  You're in the frequency domain, and the spectrum plot could be what you see when you montior the output of a tube stage with a continuous, steady-amplitude sine wave applied as input.

"Decaying spectrum" is because most of the time upper harmonics are generated at lower % than the 2nd-harmonic distortion.  So successively-higher harmonics are lower % (lower amplitude) than earlier harmonics.

The passage you quoted is technically-accurate, but could have been written very much clearer.

... The pic below is from a previous build, 1.8K sine wave, is my notation...close then from your explanation? ...

No.  The 1.8kHz signal is "1" or "1st harmonic".  An easier way to remember it is "1*Applied Frequency".  So "2nd Harmonic" is "2*Applied Frequency" and also the first harmonic distortion component above the applied frequency.

The blips below your applied frequency are not d.c. offset.  Remember, d.c. is effectively a frequency of zero, so it won't be displayed in the frequency domain plot of a spectrum analyzer.  More likely, you are seeing difference frequencies due to I.M. of the test tone with a second "applied" tone, like hum.  The amount of difference in frequency between you applied tone and the blips will tell you for sure.

And you'll see some amount of noise floor depicted on a spectrum plot.  You'll have to get used to knowing what "looks normal" with your amp & system.

[shooter]

what "looks normal" with your amp & system.

That is where I'm at

You're in the frequency domain

Got that,
what I still don't have clear is "normal"
harmonic "image" from a fundamental tone, which I think is coming into focus, thank you.

The 1.8kHz signal is "1" or "1st harmonic"

now I can count

the spectrum plot could be what you see when you monitor the output of a tube stage with a continuous, steady-amplitude sine wave applied as input.

That's what I'm doing, I can do a single pulse, but I wanna get what's normal sine wave.

So I guess I'm still trying to understand, the harmonics are created "normally" in a Ideal amp?  Or a result of real limitations of analog amplification, In other words would they still be there in a perfect "world"?  I'm getting hung up on the Distortion part of THD

d.c. is effectively a frequency of zero

In my old life, "that blip" was the DC coming on, 0 to +DC, was a newbee mistake if you tried to "calibrate" to that blip.  It was simply called the DC offset, remember, I was a Service Engineer, not a real one

[HotBluePlates]

... So I guess I'm still trying to understand, the harmonics are created "normally" in a Ideal amp?  Or a result of real limitations of analog amplification, In other words would they still be there in a perfect "world"?  I'm getting hung up on the Distortion part of THD ...

A perfect 1kHz sine wave is composed only of the single frequency 1kHz.  If 50mV of that perfect 1kHz was applied to a "perfect world ideal triode" stage with a gain of 50, the output would be 2.5v of only 1kHz and no other frequency.

Look at a real triode's grid and transfer curves.  Both have bends in the low-current area of the curves.  Additionally, the grid curves are not equally-spaced for all grid voltages.  They come pretty close over portions of grid voltage from 0v to -2v, but are more closely-spaced with more grid bias.

If you lay a load line over those curves and pick an idle point (perhaps -2v), then you could assess the distortion.  You could assume a signal input of 2v peak (or 4v peak-to-peak), and you'd see that th amount of plate voltage change along the loadline is less when going from the -2v gridline to the -4v gridline than it is when moving from the -2v gridline to the 0v gridline.  This tells you the plate output voltage waveform is flattened when plate voltage increases, but not when it decreases, which points to asymmetrical even harmonic distortion.

The output wave is distorted because the triode does not have a linear change of plate voltage for a corresponding change of grid voltage all the time.  Since the output voltage is smaller on one side than the other, despite the input voltage being the same size on both side, the shape of the output voltage is different than the input and is therefore distorted.

The transfer curves show this as well (bending at the low current part of the curve, which corresponds to the positive-half of the plate voltage output).  For them to represent a distortionless triode, they would have to be straight lines.

d.c. is effectively a frequency of zero

In my old life, "that blip" was the DC coming on, 0 to +DC, was a newbee mistake if you tried to "calibrate" to that blip.  It was simply called the DC offset ...

Upon double-checking, at least some spectrum analyzers will show a d.c. offset blip.  When they do, it's at the far left-edge of the display; there are no additional blips below it (because there's no "negative frequency").

[PRR]

> analyzers will show a d.c. offset blip.  When they do, it's at the far left-edge of the display

His display is very strange to me.

It *appears* to *center* the strongest component.

As you say, it is conventional and convenient to set DC on the left border. (Cuz nothing can be "lower than DC".) This display does not do that (in the shots we have seen).

Assuming this, the five blips in the latest shot are DC, 1.8KHz Fund, 3.6KHz 2nd, 5.4KHz 3rd, and 7.2KHz 4th. (Not DC, Fund, 1st, 2nd, 3rd as crayoned.) There IS a discrepancy here: Musicians count Fund then the octave-up is 1st OVER-tone, Physicists count the Fund as 1 and the octave-up as 2nd harmonic. The analyzer probably should be used with "physical" lingo as it is a dumb mechanical thing. The overtone-series studied in music-theory makes sense in the musical world.

As confirmation: under complete analysis, there IS a DC shift due to even-order partials, and for simple distorters its amplitude is similar to the 2nd harmonic. This is what the shot shows. I forget if the DC=2nd, or there is a "2" in there; or how you do the sum if the 4th is significant. And the 1-dot difference in the shot may just be round-off in the display.

For gitar-amp studies, _I_ would try to scale DC far left, get 100-2KHz in the left half of the screen, distortion products in the right. Some sine testing is useful. Vary the level and notice how the relative levels change from small to loud. But (IF the analyzer is fast enough!) dynamic testing with pluck and decay may tell even more about how the amp "plays".

[PRR]

I set up a very simple transistor amplifier, knowing it would give very high 2nd harmonic distortion.

The DC component is 1.6 and the 2nd is 1.7. The 4th is negligible.

FOURIER COMPONENTS OF TRANSIENT RESPONSE V(out)
DC COMPONENT =  -1.6
 HARM   FREQ    FOURIER   
    NO     (HZ)    COMPONENT   
     1     1KHz    7.7
     2     2KHz    1.7 (22% of Fund)
     3     3KHz    0.22 (3% of Fund)
     4     4KHz    0.006
     5     5KHz    0.003

TOTAL HARMONIC DISTORTION = 23% THD

The "DC Component" is readily seen. I start 1 cycle of no signal, you see the DC bias is about 17V. With tone, the wave is very lop-sided. The average (DC) of the signal is lower than the average when under no-signal condition.

The same effect is seen when load-plotting audio power tubes, except they call it "rectification" (perhaps a poor choice of word, derived from signal-detectors for AM radio). It is why a SE stage may idle at 39mA then shift to 42mA at full power.

[shooter]

the output would be 2.5v of only 1kHz and no other frequency.

Thank you, now I have a starting place to crawl from

It *appears* to *center* the strongest component.

I centered the fundamental freq with the time base knob, It defaults to the DC component at far left.  I'm just so used to having the fundamental centered on the screen.

latest shot are DC, 1.8KHz Fund, 3.6KHz 2nd, 5.4KHz 3rd, and 7.2KHz 4th

That's how I knew it in my old life, non musical, just freq.

TOTAL HARMONIC DISTORTION = 23% THD

I see where you got the 23% from, Fourier math was left to an array processor in my old life.
but that leads to my next homework, what/how do I "compare" the harmonics to the fundamental, db's, volts, db converted to volts....

thanks kind Sirs for your knowledge.
In my old world, there were no harmonics to worry about since the fund. was 64Mhz, and the BW was 250Khz, audio wants to make it complicated!

[PRR]

> the fund. was 64Mhz, and the BW was 250Khz

"Narrow band". Width <0.01 of center. Yes, then centered center is the only useful way.

Audio is wide-band. 100-10KHz is 100:1 of range. We generally round the low-edge to "zero", set that left, let the high end go to the right side.

I would start by just eyeballing "height". If stuff you did not put in comes out near as big as the desired signal, that's huge THD. If small, small THD. Definition of large/small depends on context.

Did you try the experiment of varying level? For most amplifiers, the distortion falls faster than the fundamental.

[shooter]

Did you try the experiment of varying level

The screen shot and others I have was just to get "back" to what I knew n did.  I hope to run a full test on my KT88 B.B. build this weekend.  Still tweaking and moding.

Not to "study" IM yet, but once I have HD, that allows for IM? since I now have more than 1 freq to beat against?

start by just eyeballing "height".

That's where I figured I'd start, just express for myself as a % of the fundamental.
also what conditions make it less or more.  HBP gave me a good start with where the signal operates along the load line

[HotBluePlates]

Not to "study" IM yet, but once I have HD, that allows for IM? since I now have more than 1 freq to beat against? ...

Pretend you have only a single amplifier stage.  If you apply a single, pure tone to its input, the only thing you might get out is harmonic distortion.

If you apply 2 tones to the single amplifier stage, now you have the opportunity for I.M.  You shouldn't get meaningful I.M. just from the harmonic distortion of a single tone, because the possible I.M. products just amount to strengthening tones already developed by harmonic distortion.

... "More than one tone" wouldn't be due to I.M. of the pure test tone, because the sum/difference frequencies always result in other frequencies already present  due to harmonic distortion (try adding/subtracting any of the frequencies in the list above).

I hoped you'd have tried "sum & difference" already of any of the harmonic distortion components of the 100Hz tone.

How about 200Hz (2nd harmonic) & 300Hz (3rd harmonic)?  200+300 = 500Hz (same as 5th harmonic), and 300-200 = 100Hz (same as fundamental/1st harmonic).

How about 700Hz (7th harmonic) & 400Hz (4th harmonic)?  700+400 = 1.1kHz (same as 11th harmonic), and 700-400 = 300Hz (same as 3rd harmonic).

You hopefully see you'll need a 2nd tone which is not simply harmonic distortion of the 1st tone to get I.M. products which sound materially different than harmonic distortion.

[shooter]

"sum & difference"

II was just reading about test standards and one used for IMD is 1024hz n 60hz.
I'm a ways away, but getting closer.  Need to "shield" my bread-board
Then progress probably;
find cleanest "max" signal both sine-wave shape and harmonic content, playing with gain n NFB to "adjust"

Run a audio spectrum test for flatness
figure out SNR
plug in speakers and listen for awhile
get lazy, set aside, procrastinate 

[jjasilli]

Riffing on some thoughts.

 [IMD] test standards and one used for IMD is 1024hz n 60hz.
Yes IMD is worse, or is more apparent to the human ear, when the two fundamental frequencies are far apart - especially if one is well into the bass region (below 300Hz) and the other is well into the mid region.  Another example is LFO.  E.g., an amp may have a 5Hz oscillation.  Though 5Hz itself is inaudible, it can cause an audible frequency to warble; i.e., unwanted tremolo.

Need to "shield" my bread-board. . .figure out SNR
Yes.  As used in the "sound industry" the term THD tends to include everything not in the input signal, including hiss, noise, hum, etc. -- not only harmonic distortion per se.

plug in speakers and listen for awhile
Remember that speaker cones add their own IMD, etc.; not to mention room effect, etc. -- which things are not the fault of or in the province of the amp.

[shooter]

Thanks JJ, I will probably test into DL, tuned to give me best power at the tubes V n I
I'm not trying out for sound engineer, just passing time til things start to look green outside

[shooter]

I got around to establishing a base-line for the 6SN7 out.
Here's 2 shots, same sig, nothing changed except Bandwidth of image.

I started with max clean sine, but didn't adjust gain to see if the HD changed
maybe tonight.  Then i'll run multi freq across spectrum n see what I see.