Reading some old and recent threads which mention chaos in feedback systems and the limitations of FFT distortion analysis:
http://www.diyaudio.com/forums/showthread.php?postid=418718#post418718
http://www.diyaudio.com/forums/showthread.php?postid=711636#post711636
Got me thinking a bit on this. Intermodulations of complex signals can lead to an apparent increase in the noise floor on FFTs. But this correlated or coherent "noise" may be quite audible in contrast to just ordinary noise.
Varying device capacitances with signal level may cause phase intermodulation distortions. (which normally would appear as spectral lines in an FFT, similar to amplitude distortions)
Usually we think of a varying capacitance as just affecting signal phase and impedance. But looking at the equation for group velocity on a transmission line shows group velocity as proportional to 1/SQRT(L*C) (L = inductance, C = capacitance) So group propagation velocity thru the amplifier may be varying slightly.
High bandwidth feedback amplifiers allow very high harmonics to circulate. Varying delay time thru the amplifier may cause high harmonics to recirculate in non periodic times, ie chaotic. The inverse of time delay also acts as another source for intermodulation distortion. Intermodulation between these higher harmonics will bring some artifacts back down into the audible range.
Chaotic recirculation times will likely cause spectral lines to widen. Since when has anyone bothered to look at the WIDTH of FFT spectral lines in distortion plots? This may be a very audible distortion to the ear, since the spectral lines are generally well above the noise floor. (ie, similar looking FFTs to sine wave testing may sound quite different with sounds)
We need some test procedure to isolate what is left in the signal after the primary harmonic distortions and the signal are removed. Listening to the remaining grunge would give some indication as to its harmfulness.
The obvious fixups would be operating devices in regions where capacitance is either minimized or stable.
And how about some Static Induction Transistors with linear transfer functions by the way too. (and slightly OT, since these somehow manage linearity and are SS versions of tubes, what needs to be fixed with tube design to make them linear instead of 3/2 power? - - well, must be the varying electron velocity in tubes versus saturated velocity in SS)
Another idea might be some way to limit the bandwidth of Neg. feedback signals. Who needs out of audio band NFB anyway? (OK, I'm waiting for mikeks usual lengthy reply on this one: "no"
)
Don
http://www.diyaudio.com/forums/showthread.php?postid=418718#post418718
http://www.diyaudio.com/forums/showthread.php?postid=711636#post711636
Got me thinking a bit on this. Intermodulations of complex signals can lead to an apparent increase in the noise floor on FFTs. But this correlated or coherent "noise" may be quite audible in contrast to just ordinary noise.
Varying device capacitances with signal level may cause phase intermodulation distortions. (which normally would appear as spectral lines in an FFT, similar to amplitude distortions)
Usually we think of a varying capacitance as just affecting signal phase and impedance. But looking at the equation for group velocity on a transmission line shows group velocity as proportional to 1/SQRT(L*C) (L = inductance, C = capacitance) So group propagation velocity thru the amplifier may be varying slightly.
High bandwidth feedback amplifiers allow very high harmonics to circulate. Varying delay time thru the amplifier may cause high harmonics to recirculate in non periodic times, ie chaotic. The inverse of time delay also acts as another source for intermodulation distortion. Intermodulation between these higher harmonics will bring some artifacts back down into the audible range.
Chaotic recirculation times will likely cause spectral lines to widen. Since when has anyone bothered to look at the WIDTH of FFT spectral lines in distortion plots? This may be a very audible distortion to the ear, since the spectral lines are generally well above the noise floor. (ie, similar looking FFTs to sine wave testing may sound quite different with sounds)
We need some test procedure to isolate what is left in the signal after the primary harmonic distortions and the signal are removed. Listening to the remaining grunge would give some indication as to its harmfulness.
The obvious fixups would be operating devices in regions where capacitance is either minimized or stable.
And how about some Static Induction Transistors with linear transfer functions by the way too. (and slightly OT, since these somehow manage linearity and are SS versions of tubes, what needs to be fixed with tube design to make them linear instead of 3/2 power? - - well, must be the varying electron velocity in tubes versus saturated velocity in SS)
Another idea might be some way to limit the bandwidth of Neg. feedback signals. Who needs out of audio band NFB anyway? (OK, I'm waiting for mikeks usual lengthy reply on this one: "no"
)Don
A possible test procedure
One way to test for non-harmonic distortions might be to use a delay line oscillator to generate a picket fence harmonic signal spectrum across the audio spectrum. (Maybe an adjustable windowing wide bandpass filter too.) Then a similar delay line filter would generate a picket fence filter to remove all of these from the amplifier output. Anything left would be non-harmonic chaotic distortion which could be listened to or put onto a FFT plot.
Don
One way to test for non-harmonic distortions might be to use a delay line oscillator to generate a picket fence harmonic signal spectrum across the audio spectrum. (Maybe an adjustable windowing wide bandpass filter too.) Then a similar delay line filter would generate a picket fence filter to remove all of these from the amplifier output. Anything left would be non-harmonic chaotic distortion which could be listened to or put onto a FFT plot.
Don
smoking-amp said:
Another idea might be some way to limit the bandwidth of Neg. feedback signals. Who needs out of audio band NFB anyway? (OK, I'm waiting for mikeks usual lengthy reply on this one: "no"
Don
No.
The reason: The distortion products have much higher frequency than the original frequency, especially crossover distortion artefacts.
For example, a 5th order harmonic from a 10khz will be 50khz. If you limit feedback to 20khz, you will only be able to attenuate the 2nd harmonic from a 10khz signal.
These harmonics above 20khz will not be audible directly, but only for a single sinewave. IM products will fold down these distortions again into audible range, when amp is fed with complex signals like music.
Intermodulation is not a product from NFB, it always happens when a complex signal gets distorted.
Mike
"NFB can also create IMs additionaly..."
My point exactly! The crossover HF distortions are generated in the last stage of the amplifier (no non-linear stages left to affect them), why let them recirculate to intermodulate back down to the audio band. Once an ultrasonic distortion gets generated, just kill it with a filter.
Maybe could put the filter after the input diffl. stage, since the amplifier would be effectively operating open loop above the lowpass filter frequency if placed in the feedback only.
Don
My point exactly! The crossover HF distortions are generated in the last stage of the amplifier (no non-linear stages left to affect them), why let them recirculate to intermodulate back down to the audio band. Once an ultrasonic distortion gets generated, just kill it with a filter.
Maybe could put the filter after the input diffl. stage, since the amplifier would be effectively operating open loop above the lowpass filter frequency if placed in the feedback only.
Don
as for OP post, this is interesting. Are you concidering the effects of changing capacitance. it does require energy to change capacitance, so i'm not sure the linear equations can be used.
IMD can take higher frequencies and convert them back down, as it is essentailly caused by mixing. this is a known phenomenon in inamps where high RF signals become LF noise on the inamp.
IMD can take higher frequencies and convert them back down, as it is essentailly caused by mixing. this is a known phenomenon in inamps where high RF signals become LF noise on the inamp.
Checking the data sheets for most any bipolar or mosfet device shows varying capacitance across junctions or terminals versus applied voltages. An unavoidable effect. In the amplifier case I would expect this to mainly affect phase of high frequencies, particularly ultrasonic ones, as large amplitude low frequencies swing voltages around. If delay time is also affected via the dielectric affect on signal velocity, then further changes in phase/freq. of the HF gremlins will occur. Making more annoying possibilities for IMD to generate audible products.
Maybe one could put back to back varicaps across offending junctions and control them with opposite phase audio to compensate capacitance. Probably difficult due to non-linear cap. curves.
Don
Maybe one could put back to back varicaps across offending junctions and control them with opposite phase audio to compensate capacitance. Probably difficult due to non-linear cap. curves.
Don
smoking-amp said:
That was exactly my point, this thinking works only for single sinewaves. For complex signals this "ultrasonic distortion" creates IM that folds down to audible distortions. This effect has NOTHING to do with feedback. You can't kill them with a filter.
You need to avoid/reduce ANY distortions, they become all audible with complex signals because of IM. Even in the Mhz range.
Mike
"creates IM that folds down to audible distortions"
So you are saying that a single transistor stage could generate some HF harmonics which could immediately intermodulate to cause audible IMD. I was hoping that 1st pass products would have to re-enter the input to interact again to generate 2nd order IMD products. (I'm not saying that higher order products couldn't be generated by a bad transfer curve straight away, I'm just looking at say signal1 -> 2nd H1, and signal2 -> 2nd H2, then interacting down to 2nd H1- 2nd H2)
If so, then we need to kick some semi manufacturers B--ts! Where are some linear silicon SITs to work with!
Don
So you are saying that a single transistor stage could generate some HF harmonics which could immediately intermodulate to cause audible IMD. I was hoping that 1st pass products would have to re-enter the input to interact again to generate 2nd order IMD products. (I'm not saying that higher order products couldn't be generated by a bad transfer curve straight away, I'm just looking at say signal1 -> 2nd H1, and signal2 -> 2nd H2, then interacting down to 2nd H1- 2nd H2)
If so, then we need to kick some semi manufacturers B--ts! Where are some linear silicon SITs to work with!
Don
I should clarify the last model scenario, since a quartic transfer function could obviously generate 2nd H1 -2nd H2 straight away.
(sig1 + sig2)**4
Lets say we have a square law transfer function like Mosfets. Then (sig1 + sig2)**2 can generate 2nd H1 and 2nd H2. Would it not require a second pass thru the device then to get (2nd H1 + 2nd H2)**2 --> 2nd H1 - 2nd H2.
Since varying junction capacitances would mainly affect HF signals, one could hope that such 2nd harmonics and above would be out of band, and hence filterable. Of course this still would not prevent Sig1-Sig2 intermod.
Likewise, varying propagation delay effects that widen FFT lines would also become serious only if multiple passes thru the amplifier are permitted.
Don
(sig1 + sig2)**4
Lets say we have a square law transfer function like Mosfets. Then (sig1 + sig2)**2 can generate 2nd H1 and 2nd H2. Would it not require a second pass thru the device then to get (2nd H1 + 2nd H2)**2 --> 2nd H1 - 2nd H2.
Since varying junction capacitances would mainly affect HF signals, one could hope that such 2nd harmonics and above would be out of band, and hence filterable. Of course this still would not prevent Sig1-Sig2 intermod.
Likewise, varying propagation delay effects that widen FFT lines would also become serious only if multiple passes thru the amplifier are permitted.
Don
Yes, quite simple, a mixture of let's say 10+11khz sinewaves distorted by "2nd harmonic" only, will not only create a 20khz and 22khz harmonic, it also creates 1khz "harmonic" (and some more). That's IM. This has nothing to do with feedback. This interaction acts immediately.
Not each frequency gets distorted on its own, its the whole sum. This creates completely different spectras.
As Leach said, any distorting amp creates IM, it's just a question of how.
Yes, a quadratic transfer function (for example) on non pure sinus signals immediately creates IM.
Try it out in a simulator for example, feed a single stage without feedback with a sum of 2 sinus signals and observe the products.
(Simply put 2 AC-sources in series)
Now, the good news: if you compensate the distortion that originally created the IM-harmonics, these harmonics also disappear.
Mike
Not each frequency gets distorted on its own, its the whole sum. This creates completely different spectras.
As Leach said, any distorting amp creates IM, it's just a question of how.
Yes, a quadratic transfer function (for example) on non pure sinus signals immediately creates IM.
Try it out in a simulator for example, feed a single stage without feedback with a sum of 2 sinus signals and observe the products.
(Simply put 2 AC-sources in series)
Now, the good news: if you compensate the distortion that originally created the IM-harmonics, these harmonics also disappear.
Mike
I agree that 10 +11 KHz sine signals will give you 20KHz, 22 KHz and 1 KHz thru a square law transfer. But one should not get 2 KHz unless the 20 and 22 KHz passes around a feedback loop again, right? (Well, yes, the 1 KHz IMD will get doubled to two KHz next time around. But at least by preventing the HF dist. from recirculating we will cut the 1KHz amount in half. And the NFB is generally quite good at cleaning up LF dist. anyway )
It still seems to me that putting a 20KHz low pass after the input diffl. amp should clean up a lot of junk, particularly multi-pass non harmonic related gunk. The NFB is still allowed to fix up the 1 KHz IMD or anything in-band, right? Or am I missing something?
Don
It still seems to me that putting a 20KHz low pass after the input diffl. amp should clean up a lot of junk, particularly multi-pass non harmonic related gunk. The NFB is still allowed to fix up the 1 KHz IMD or anything in-band, right? Or am I missing something?
Don
Yes, you are missing one detail. As the speaker also distorts, any hf-junk entering the speaker will cause again IM, making the HF-junk audible. Because of that an amp must not put out any of these hf-harmonics. You could filter them out, but you would need some 80db/octave filter for that.
Also, you missed this "good news" thing, correcting the original distortion that created the IM-products, will also remove these.
For this beeing possible the feedback must be able to operate at the freq of the harmonics beeing generated. Another good reason for flat and high openloop bandwidth.
Mike
Also, you missed this "good news" thing, correcting the original distortion that created the IM-products, will also remove these.
For this beeing possible the feedback must be able to operate at the freq of the harmonics beeing generated. Another good reason for flat and high openloop bandwidth.
Mike
Ahhh! The speaker. Good point. Very difficult to make such a filter and at high power.
Maybe this is an application for a type of feedforward error correction. Low signal level active filter to sample HF distortions and invert them thru a feedforward error corrector amp. in series with the main output. No doubt some problems will arise with phasing issues though.
Hmmm, actually, we don't need no fancy filter. The correction signal is already available. Just use the full bandwidth error signal from the diffl. stage (before LP filter) , high pass filter it with a simple RC to remove the audio band, use that to drive the feedforward. This looks do-able: IC amplifier chip error feedforward, $2.
Some half- a-- alternatives:
Its only the tweeter that will receive significant HF dist. energy if there is a crossover network (more octaves available to suppress ultrasonics on lower freq. drivers). Maybe can use some ultrasonic capable driver. Drive pets out of the house. Mosquitos too.
If only the tweeter has these HF distortion energies at significant level, it may be incapable of radiating much low freq. IMD. So maybe then one could get away with a simpler ultrasonic stop filter.
Don
Maybe this is an application for a type of feedforward error correction. Low signal level active filter to sample HF distortions and invert them thru a feedforward error corrector amp. in series with the main output. No doubt some problems will arise with phasing issues though.
Hmmm, actually, we don't need no fancy filter. The correction signal is already available. Just use the full bandwidth error signal from the diffl. stage (before LP filter) , high pass filter it with a simple RC to remove the audio band, use that to drive the feedforward. This looks do-able: IC amplifier chip error feedforward, $2.
Some half- a-- alternatives:
Its only the tweeter that will receive significant HF dist. energy if there is a crossover network (more octaves available to suppress ultrasonics on lower freq. drivers). Maybe can use some ultrasonic capable driver. Drive pets out of the house. Mosquitos too.
If only the tweeter has these HF distortion energies at significant level, it may be incapable of radiating much low freq. IMD. So maybe then one could get away with a simpler ultrasonic stop filter.
Don
Problems
Hmmm, another problem exists with this LP filtered feedback scheme. The ultrasonic distortion from early stages will have the full open loop gain of the amplifier available to boost them since no (effective) "anti ultrasonic distortion" will be present at the input (at least after the LP filter).
The early gain stages in the amp. would have to be very clean for this idea to work this way. This can be fixed, I think, by moving the LowPass filter further downstream. Say, just before the output stage where most HF crossover dist. is generated anyway. (could be just an RC LP filter) Then the ultrasonic distortion will typically only get unity gain from the output stage.
In-band audio distortion still gets corrected by conventional NFB.
This will require a separate input/output error subtraction for the IC feedforward ultrasonic error corrector then, at the LP filter I/O points (ie, we want to get the part discarded by the LP filter, maybe just use another RC HP filter?). Most of these IC chip amps look like a diffl. input Op. Amp anyway. The ultrasonic error correction (High Pass filtered to remove the audio band) can then be inserted into the output line with a small ferrite xfmr. in series.
The ultrasonic error correction is for the benefit of the speaker.
This seems workable now. Anyone see any other problems?
Don
Hmmm, another problem exists with this LP filtered feedback scheme. The ultrasonic distortion from early stages will have the full open loop gain of the amplifier available to boost them since no (effective) "anti ultrasonic distortion" will be present at the input (at least after the LP filter).
The early gain stages in the amp. would have to be very clean for this idea to work this way. This can be fixed, I think, by moving the LowPass filter further downstream. Say, just before the output stage where most HF crossover dist. is generated anyway. (could be just an RC LP filter) Then the ultrasonic distortion will typically only get unity gain from the output stage.
In-band audio distortion still gets corrected by conventional NFB.
This will require a separate input/output error subtraction for the IC feedforward ultrasonic error corrector then, at the LP filter I/O points (ie, we want to get the part discarded by the LP filter, maybe just use another RC HP filter?). Most of these IC chip amps look like a diffl. input Op. Amp anyway. The ultrasonic error correction (High Pass filtered to remove the audio band) can then be inserted into the output line with a small ferrite xfmr. in series.
The ultrasonic error correction is for the benefit of the speaker.
This seems workable now. Anyone see any other problems?
Don
Re-entrant feedback distortion
What you are really discussing here is so-called re-entrant distortion in feedback amplifiers. It is absolutely true that it exists, but one needs to do the math and do measurements to see how big an issue it is.
Assume you have an amplifier with purely second order distortion of 1% in the open loop forward path, it has 26 dB gain, and it is producing 20V rms at its output. Assume its input is 1 V rms at 1 kHz. The output will have a 2 kHz spectral component in it equal to 1% of 20V, or 0.2V rms.
Now let's increase the forward path gain by 20 dB and apply 20 dB of negative feedback, so we once again have an amplifier with 26 dB of gain. The second harmonic distortion is now reduced by a factor of 10 to 0.1%. The 2 kHz spectral component at the output is now 20 mV rms. This is fed back to the input via the 20:1 attenuation of the feeback path to arrive at the input as 1 mV rms. This 1 mV rms mixes with the 1 kHz 1V rms input at the second-order nonlinearity in the forward path.
Such IM products from a second order nonlinearity are proprtional in amplitude to the product of the two signals's amplitudes, times the degree of the nonlinearity. The 1 kHz input signal will mix with the 2 kHz feedback component to form a new spectral line at 3 kHz. This formation of a new spectral line by the negative feedback re-entrant distortion process is what people get all excited about.
However, the amplitude of the new product created is going to be on the order of 1 mV/1V * 1%, or on the order of 0.001%. Very small, indeed. This is the part that many people miss. YES, the NFB process created a new, higher-order distortion product, but when you do the math (and confirm it by measurement), it is very small.
In reality, just about all nonlinearities in an amplifier can be modeled by a Taylor series that has non-zero coefficients out to pretty high order. In other words, just about any real amplifier is producing 2nd, 3rd, 4th, 5th, etc order distortion products in decreasing values with increasing order. You will usually find that the higher order distortion components that were there in the first place are bigger than their cousins that were created by the re-entrant phenomena under most conditions.
Interestingly, the 3 kHz product in the example above will be that much smaller when a LARGER amount of NFB is applied, since the recirculating 2 kHz first-round-created distortion component will be much smaller.
Lastly, you mentioned chaos and the generation of non-harminically-related distortion spectra. This definitely happens in clipping, but happens very little below clipping in any reasonably designed amplifier.
We recently put on a series of workshops at the Rocky Mountain Audio Fest, and they are summarized at my web site at www.cordellaudio.com.
In the amplifier measurement workshop, we demonstrated measurement on a Super Gain Clone amplifier, and did spectral analysis of its output during clipping. This amplifier was equipped with a defeatable soft clip circuit at its front end, outside the negative feedback loop. The clipping distortion spectra clearly showed the chaos-like distortion spectra on both THD and twin tone 19 kHz + 20 kHz spectral analysis when the soft clip circuit was not engaged. This grunge between the harmonic spectral lines disappeared when the soft clip circuit was turned on.
Bob Cordell
What you are really discussing here is so-called re-entrant distortion in feedback amplifiers. It is absolutely true that it exists, but one needs to do the math and do measurements to see how big an issue it is.
Assume you have an amplifier with purely second order distortion of 1% in the open loop forward path, it has 26 dB gain, and it is producing 20V rms at its output. Assume its input is 1 V rms at 1 kHz. The output will have a 2 kHz spectral component in it equal to 1% of 20V, or 0.2V rms.
Now let's increase the forward path gain by 20 dB and apply 20 dB of negative feedback, so we once again have an amplifier with 26 dB of gain. The second harmonic distortion is now reduced by a factor of 10 to 0.1%. The 2 kHz spectral component at the output is now 20 mV rms. This is fed back to the input via the 20:1 attenuation of the feeback path to arrive at the input as 1 mV rms. This 1 mV rms mixes with the 1 kHz 1V rms input at the second-order nonlinearity in the forward path.
Such IM products from a second order nonlinearity are proprtional in amplitude to the product of the two signals's amplitudes, times the degree of the nonlinearity. The 1 kHz input signal will mix with the 2 kHz feedback component to form a new spectral line at 3 kHz. This formation of a new spectral line by the negative feedback re-entrant distortion process is what people get all excited about.
However, the amplitude of the new product created is going to be on the order of 1 mV/1V * 1%, or on the order of 0.001%. Very small, indeed. This is the part that many people miss. YES, the NFB process created a new, higher-order distortion product, but when you do the math (and confirm it by measurement), it is very small.
In reality, just about all nonlinearities in an amplifier can be modeled by a Taylor series that has non-zero coefficients out to pretty high order. In other words, just about any real amplifier is producing 2nd, 3rd, 4th, 5th, etc order distortion products in decreasing values with increasing order. You will usually find that the higher order distortion components that were there in the first place are bigger than their cousins that were created by the re-entrant phenomena under most conditions.
Interestingly, the 3 kHz product in the example above will be that much smaller when a LARGER amount of NFB is applied, since the recirculating 2 kHz first-round-created distortion component will be much smaller.
Lastly, you mentioned chaos and the generation of non-harminically-related distortion spectra. This definitely happens in clipping, but happens very little below clipping in any reasonably designed amplifier.
We recently put on a series of workshops at the Rocky Mountain Audio Fest, and they are summarized at my web site at www.cordellaudio.com.
In the amplifier measurement workshop, we demonstrated measurement on a Super Gain Clone amplifier, and did spectral analysis of its output during clipping. This amplifier was equipped with a defeatable soft clip circuit at its front end, outside the negative feedback loop. The clipping distortion spectra clearly showed the chaos-like distortion spectra on both THD and twin tone 19 kHz + 20 kHz spectral analysis when the soft clip circuit was not engaged. This grunge between the harmonic spectral lines disappeared when the soft clip circuit was turned on.
Bob Cordell
smoking-amp said:Reading some old and recent threads which mention chaos in feedback systems and the limitations of FFT distortion analysis:
http://www.diyaudio.com/forums/showthread.php?postid=418718#post418718
http://www.diyaudio.com/forums/showthread.php?postid=711636#post711636
Got me thinking a bit on this. Intermodulations of complex signals can lead to an apparent increase in the noise floor on FFTs. But this correlated or coherent "noise" may be quite audible in contrast to just ordinary noise.
Varying device capacitances with signal level may cause phase intermodulation distortions. (which normally would appear as spectral lines in an FFT, similar to amplitude distortions)
Usually we think of a varying capacitance as just affecting signal phase and impedance. But looking at the equation for group velocity on a transmission line shows group velocity as proportional to 1/SQRT(L*C) (L = inductance, C = capacitance) So group propagation velocity thru the amplifier may be varying slightly.
High bandwidth feedback amplifiers allow very high harmonics to circulate. Varying delay time thru the amplifier may cause high harmonics to recirculate in non periodic times, ie chaotic. The inverse of time delay also acts as another source for intermodulation distortion. Intermodulation between these higher harmonics will bring some artifacts back down into the audible range.
Chaotic recirculation times will likely cause spectral lines to widen. Since when has anyone bothered to look at the WIDTH of FFT spectral lines in distortion plots? This may be a very audible distortion to the ear, since the spectral lines are generally well above the noise floor. (ie, similar looking FFTs to sine wave testing may sound quite different with sounds)
We need some test procedure to isolate what is left in the signal after the primary harmonic distortions and the signal are removed. Listening to the remaining grunge would give some indication as to its harmfulness.
The obvious fixups would be operating devices in regions where capacitance is either minimized or stable.
And how about some Static Induction Transistors with linear transfer functions by the way too. (and slightly OT, since these somehow manage linearity and are SS versions of tubes, what needs to be fixed with tube design to make them linear instead of 3/2 power? - - well, must be the varying electron velocity in tubes versus saturated velocity in SS)
Another idea might be some way to limit the bandwidth of Neg. feedback signals. Who needs out of audio band NFB anyway? (OK, I'm waiting for mikeks usual lengthy reply on this one: "no"
Don
Hi Don
That's wonderful, that you raised this issue.
It's always made me think of amplifiers as circuits which are both dynamic and non-linear, therefore described by nonlinear differential equations.
Feedback causes the input signal to be an attractor of output signal in chaos jargon.
Note, that it takes only feedback, nonlinearity and LTI network to produce a Chua oscillator.
But, yes, varying capacitance and dynamic phase shifts are key to nonlinear dynamics in feedback circuits in my view.
Chua oscillator:
Attachments
IMD math seems to confuse people, you do see difference frequencies, below the frequency of the input even without feedback
the -30dB 1KHz line in this sim is due to 2nd order nonlinearity alone, not feedback
http://www.diyaudio.com/forums/showthread.php?postid=424592#post424592
you can of course look at feedback with an "ideal" sq law device and find new, odd order distortion products:
http://www.diyaudio.com/forums/showthread.php?s=&postid=920583&highlight=#post920583
you could change the input in this sim to see what feedback does for IMD with the two-tone test signal
avoiding chaos in audio amplifiers is as simple as avoiding oscillation -no oscillation no chaos, the output always decays to a steady state
the -30dB 1KHz line in this sim is due to 2nd order nonlinearity alone, not feedback
http://www.diyaudio.com/forums/showthread.php?postid=424592#post424592
you can of course look at feedback with an "ideal" sq law device and find new, odd order distortion products:
http://www.diyaudio.com/forums/showthread.php?s=&postid=920583&highlight=#post920583
you could change the input in this sim to see what feedback does for IMD with the two-tone test signal
avoiding chaos in audio amplifiers is as simple as avoiding oscillation -no oscillation no chaos, the output always decays to a steady state
re: Bob Cordell
"one needs to do the math and do measurements to see how big an issue it is"
Yes, I'm probably chasing neutrino-like ghosts here. But at least it seems there is a relatively easy way to minimize this recirculating junk. Although I worry there may be a big price to pay as far as sensitivity to power supply noise when not correcting ultrasonic distortions too. At this point my emphasis is on determing whether this has audible effect or not to "golden ears" (not mine though). So a defensively constructed test amplifier could be useful for testing.
re: darkfenriz and jcx
I probably am bending the technical definitions for chaos theory here a bit, since I would expect this effect to be minute in most amplifiers. No actual chaotic oscillations, just insidious line broadening of FFT sine wave testing spectrums. But the question remains as to whether anyone can hear these effects. The usefulnes of the picket fence harmonic deletion test mentioned earlier, or a test amplifier with defensive measures taken, could be useful in deciding this.
I also wonder if there could be some other mechanisms to generate non-harmonic distortions. For example, acoustic feedback to components, as occurs in most tube amplifiers. Of course RF intrusions, as recently hashed over in the Cordell interview thread.
Don
"one needs to do the math and do measurements to see how big an issue it is"
Yes, I'm probably chasing neutrino-like ghosts here. But at least it seems there is a relatively easy way to minimize this recirculating junk. Although I worry there may be a big price to pay as far as sensitivity to power supply noise when not correcting ultrasonic distortions too. At this point my emphasis is on determing whether this has audible effect or not to "golden ears" (not mine though). So a defensively constructed test amplifier could be useful for testing.
re: darkfenriz and jcx
I probably am bending the technical definitions for chaos theory here a bit, since I would expect this effect to be minute in most amplifiers. No actual chaotic oscillations, just insidious line broadening of FFT sine wave testing spectrums. But the question remains as to whether anyone can hear these effects. The usefulnes of the picket fence harmonic deletion test mentioned earlier, or a test amplifier with defensive measures taken, could be useful in deciding this.
I also wonder if there could be some other mechanisms to generate non-harmonic distortions. For example, acoustic feedback to components, as occurs in most tube amplifiers. Of course RF intrusions, as recently hashed over in the Cordell interview thread.
Don
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