Here is what i think is the maths of negative feedback amplifiers.
http://s7.postimage.org/jkcsqkwpn/untitled12121.jpg
As you can see by the 5th iteration we reach close to the new gain as per the formula, which can be actually derived in the similar way. These iterations happen speed of electrons, close to speed of light and the final system equilibrium is reached.
Negative feedback : Operational Amplifiers
Please let me know if I am wrong
2ndly here is my alternative method of negative feedback.
using this method the gain of the amplifier will not reduce but i am not sure if it will work
instead of providing a -ve feedback = feedback factor * output
If i provide -ve feedback as = input - amplification factor * output
http://s7.postimage.org/jkcsqkwpn/untitled12121.jpg
it seems the system oscillates between distortion and distortion free state.
What will happen.
Any ideas... great minds
-Amit_112db
-4yrs EE bachelor graduated and not so bad with maths
http://s7.postimage.org/jkcsqkwpn/untitled12121.jpg
As you can see by the 5th iteration we reach close to the new gain as per the formula, which can be actually derived in the similar way. These iterations happen speed of electrons, close to speed of light and the final system equilibrium is reached.
Negative feedback : Operational Amplifiers
Please let me know if I am wrong
2ndly here is my alternative method of negative feedback.
using this method the gain of the amplifier will not reduce but i am not sure if it will work
instead of providing a -ve feedback = feedback factor * output
If i provide -ve feedback as = input - amplification factor * output
http://s7.postimage.org/jkcsqkwpn/untitled12121.jpg
it seems the system oscillates between distortion and distortion free state.
What will happen.
Any ideas... great minds
-Amit_112db
-4yrs EE bachelor graduated and not so bad with maths
Last edited:
Hi,
Feedback is not an interative process, it simply happens in realtime.
(Properly if you've met the stability criteria, and frequency dependent,
which varies according to the type of compensation scheme employed.)
Long time since I did my EEE degree but I really can't see
how you've come to approaching it in such a manner.
rgds, sreten.
Feedback is not an interative process, it simply happens in realtime.
(Properly if you've met the stability criteria, and frequency dependent,
which varies according to the type of compensation scheme employed.)
Long time since I did my EEE degree but I really can't see
how you've come to approaching it in such a manner.
rgds, sreten.
My new feedback theory may not work as I said. its just an idea that popped up
But my theory of how the classical negative feedback works is not flawed. It gives absolutely the correct answer with the correct algebra. The output values of gains match with the gain formula from wikipedia.
sreten- my approach to feedback is not interactive at all.. just iterative
peranders- you may be absolutely correct. There may be only one kind of negative feedback. Its the way i am analyzing/viewing it suing iterations. I would love to know how its flawed.
Pure and simple math algebra of how a system is trying to achieve a stable state.
Howver you all may be right. I will definitely read op amps/amps n other stuff and think of it again.
Last edited:
Thats really funny... but i always sick up for friends that want to learn and don't know how.
@Amit
The best reference on feedback theory is probably still the fundamental work by H.W. Bode. It is available on-line.
There is so much half-baked nonsense written about feedback that you will have to be careful if you want to be taken seriously. Your OP has both links the same, so it is not clear what the difference is supposed to be. The thumbnails are too small to be much help.
Best wishes
David
The best reference on feedback theory is probably still the fundamental work by H.W. Bode. It is available on-line.
There is so much half-baked nonsense written about feedback that you will have to be careful if you want to be taken seriously. Your OP has both links the same, so it is not clear what the difference is supposed to be. The thumbnails are too small to be much help.
Best wishes
David
Just want to toss this into the ring:
First, feedback theory is well worked out, and isn't really hard to understand, once you read completely through a couple of different sources. In the very easiest (ironically) presentation, the effect of feedback on a theoretically perfect operational amplifier makes it almost completely clear how it flies.
A "near perfect" operational amplifier has near-infinite gain, no stability issues, and infintessimally fast response time. It is obvious that if any signal at all is put on the "+" pin with the "-" pin connected to ground - at near-infinite gain, the output will swing instantly to the + or - rail. Not a very useful device (as an amplifier. A great comparator!)
But simply connect the output back to the - input ... and it becomes clear that the output will exactly track the input: if it becomes even a tiny bit more positive, that will be fed to the negative input, and it'll reverse back to matching. Likewise, if it is a tiny bit shy of exactly the same, then the + pin is greater than the - input, and the output will again rapidly swing more positive.
Lesson "2" is that the "really large gain" aspect - as well as slew time - cannot be infinite, or an undefined condition will exist leading to no solution. The math works out with a few more complicated terms, but "really large gain" serves to make the smaller compensation terms so small as to be generally negligible.
Lesson "3" is that combinations of voltage-dividers can be used in the feedback loop to give the overall device a numerically consistent, rational gain figure from the ratios of the external resistance elements themselves. Again, the op-amp's gain hardly figures in before the 4th or 5th decimal place of accuracy.
Lesson "4" is that time-domain reactive components (inductors, and especially capacitors) serve to make the feedback far less simple mathematically. Things rapidly get complex. Unusual circuit topologies can exploit the near-perfect integration abilities of capacitors to simulate inductors, to effect filter elements and more. Point though is, you realized after this that "damn, it can get complicated".
Lesson "5" introduces notions of bandwidth limitations, stability, input differential bias, and so on. By this time, one can break away from the op-amp learnin', and start to apply the theory to conventional multistage discrete component amplifier design.
I recommend this path-to-understanding to one and all. The lessons are rendered so much clearer by the strikingly unique and outstanding mathematical simplification of the op-amp model. Real-world is different of course, but the foundation ... is 75% of the battle.
GoatGuy
First, feedback theory is well worked out, and isn't really hard to understand, once you read completely through a couple of different sources. In the very easiest (ironically) presentation, the effect of feedback on a theoretically perfect operational amplifier makes it almost completely clear how it flies.
A "near perfect" operational amplifier has near-infinite gain, no stability issues, and infintessimally fast response time. It is obvious that if any signal at all is put on the "+" pin with the "-" pin connected to ground - at near-infinite gain, the output will swing instantly to the + or - rail. Not a very useful device (as an amplifier. A great comparator!)
But simply connect the output back to the - input ... and it becomes clear that the output will exactly track the input: if it becomes even a tiny bit more positive, that will be fed to the negative input, and it'll reverse back to matching. Likewise, if it is a tiny bit shy of exactly the same, then the + pin is greater than the - input, and the output will again rapidly swing more positive.
Lesson "2" is that the "really large gain" aspect - as well as slew time - cannot be infinite, or an undefined condition will exist leading to no solution. The math works out with a few more complicated terms, but "really large gain" serves to make the smaller compensation terms so small as to be generally negligible.
Lesson "3" is that combinations of voltage-dividers can be used in the feedback loop to give the overall device a numerically consistent, rational gain figure from the ratios of the external resistance elements themselves. Again, the op-amp's gain hardly figures in before the 4th or 5th decimal place of accuracy.
Lesson "4" is that time-domain reactive components (inductors, and especially capacitors) serve to make the feedback far less simple mathematically. Things rapidly get complex. Unusual circuit topologies can exploit the near-perfect integration abilities of capacitors to simulate inductors, to effect filter elements and more. Point though is, you realized after this that "damn, it can get complicated".
Lesson "5" introduces notions of bandwidth limitations, stability, input differential bias, and so on. By this time, one can break away from the op-amp learnin', and start to apply the theory to conventional multistage discrete component amplifier design.
I recommend this path-to-understanding to one and all. The lessons are rendered so much clearer by the strikingly unique and outstanding mathematical simplification of the op-amp model. Real-world is different of course, but the foundation ... is 75% of the battle.
GoatGuy
An article i read said its not an iterative process.
Although iterations lead to same result. I am yet not sure how iterations are wrong.
Then the article goes on to say that
"In our ideal device, the change is instant, in a real device it is possible to measure the time it takes for the correction(i call iterations) to be made. "
So the iterations are instant. As i said at the speed of electrons.
Wikipedia didn't help much and I will try to find something with more maths
Quoting from
Distortion and Feedback (This is the URL)
This is not an iterative process, which is to say that the amplifier does not keep feeding the input signal (meaning a significant part of the input waveform) into the inverting input to be re-amplified, re-distorted and re-compared. This is where some of those who criticise negative feedback have made their first error.
The output of the amplifier simply keeps changing in the appropriate direction until the error amp detects that the voltages are again identical, at which point the output of the error amp ideally just stops where it is, and so does the rest of the chain. In reality, there will be a small amount of instantaneous correction as the two voltages approach equality, but this must happen much faster than the input signal can change with normal programme material.
The fact that the correction is usually done well before the input voltage has even changed significantly clearly means that no part of the feedback signal is fed through the amplifier over and over again - that just doesn't happen. In our ideal device, the change is instant, in a real device it is possible to measure the time it takes for the correction to be made. For an audio amplifier, the correction must be completed faster than the highest frequency of interest can change - how much faster is open to some conjecture, and that will be looked at later in this article.
Although iterations lead to same result. I am yet not sure how iterations are wrong.
Then the article goes on to say that
"In our ideal device, the change is instant, in a real device it is possible to measure the time it takes for the correction(i call iterations) to be made. "
So the iterations are instant. As i said at the speed of electrons.
Wikipedia didn't help much and I will try to find something with more maths
Quoting from
Distortion and Feedback (This is the URL)
This is not an iterative process, which is to say that the amplifier does not keep feeding the input signal (meaning a significant part of the input waveform) into the inverting input to be re-amplified, re-distorted and re-compared. This is where some of those who criticise negative feedback have made their first error.
The output of the amplifier simply keeps changing in the appropriate direction until the error amp detects that the voltages are again identical, at which point the output of the error amp ideally just stops where it is, and so does the rest of the chain. In reality, there will be a small amount of instantaneous correction as the two voltages approach equality, but this must happen much faster than the input signal can change with normal programme material.
The fact that the correction is usually done well before the input voltage has even changed significantly clearly means that no part of the feedback signal is fed through the amplifier over and over again - that just doesn't happen. In our ideal device, the change is instant, in a real device it is possible to measure the time it takes for the correction to be made. For an audio amplifier, the correction must be completed faster than the highest frequency of interest can change - how much faster is open to some conjecture, and that will be looked at later in this article.
will lookup H.W. Bode,Thanks david,
There is a url link below each tumbnail.
the 2nd thumbnail in orignal post is duplicate.
here is the thumbnail/pic for the alternative feedback design which i was curious about.
http://s7.postimage.org/9nml3ywt7/untitled2222222.jpg
Amit:
There is nothing wrong with your approach, except that your math approximation method is just a naive first try. Conveniently, you have terms and "errors" that converge to ever lower values. (this is because you used 1/20th the 'feedback'). Had you chose a larger number, it wouldn't have converged.
Numerically (since i do this stuff for a living!) successive approximation using differential equations are used to get arbitrary-precision solutions to such problems (when one doesn't want to work out the fancy algebra or calculus). It is a very similar idea, except that one uses "time intervals" with the system reacting - like a real one - with specified slew rates, response times, frequency and time domain reactiveness. It isn't even that complex - once the basic maths are worked out. Indeed - whole books (and graduate school programs) are dedicated to it.
You can keep fiddling with simple spreadsheets (and learn what makes stable feedback in these gross equations), or, you can get ahold of the SPICE family of programs .. that allow you to do many, many useful things "nearly for free", with near-magical results. You can draw the circuits using the packages, and connect the nodes, then ask to run time-domain simulations that show actual predicted response waveforms. You can take those and do Fourier transforms, to get frequency and phase domain plots. And you can change circuit values, add components, take 'em away, with a few clicks of the mouse. And run the circuit sims again.
At this point, at university, there are NO "hands on" courses that don't require sim work first.
GoatGuy
There is nothing wrong with your approach, except that your math approximation method is just a naive first try. Conveniently, you have terms and "errors" that converge to ever lower values. (this is because you used 1/20th the 'feedback'). Had you chose a larger number, it wouldn't have converged.
Numerically (since i do this stuff for a living!) successive approximation using differential equations are used to get arbitrary-precision solutions to such problems (when one doesn't want to work out the fancy algebra or calculus). It is a very similar idea, except that one uses "time intervals" with the system reacting - like a real one - with specified slew rates, response times, frequency and time domain reactiveness. It isn't even that complex - once the basic maths are worked out. Indeed - whole books (and graduate school programs) are dedicated to it.
You can keep fiddling with simple spreadsheets (and learn what makes stable feedback in these gross equations), or, you can get ahold of the SPICE family of programs .. that allow you to do many, many useful things "nearly for free", with near-magical results. You can draw the circuits using the packages, and connect the nodes, then ask to run time-domain simulations that show actual predicted response waveforms. You can take those and do Fourier transforms, to get frequency and phase domain plots. And you can change circuit values, add components, take 'em away, with a few clicks of the mouse. And run the circuit sims again.
At this point, at university, there are NO "hands on" courses that don't require sim work first.
GoatGuy
Thoughts on distortion, damping and great sound...
Um... actually, there is a very, very interesting thing I learned about this recently that makes me both partially agree with the statement, and on a different level strongly disagree.
Here it is: damping factor
[what?, he asks...]
For the longest time, I, being an electrical engineer, and in particular having a specialty in analog 2nd and 3rd order control circuits, thought to myself, Jeez... the job of an amplifier is solely to reproduce a faithful, but amplified version of the input voltage waveform, with sufficient power that the speakers can damned well do whatever they like in representing that as "sound". Period.
That kind of thinking then leads directly to, "whaddya mean, feedback compensation is bad? No such thing!"
Technically, this is true but not complete. The guiding principle behind so many uber-high-end amplifier setups is fine linearity, sweet near-limit clipping dynamics, and wide bandwidth throughout. Yet, it is truly the case that one amplifier with demonstrable ability to produce output of say "10 watts" at 0.001% distortion [typically a solid state monster] and another amp, again producing identically 10 watts at say 0.1% 'tube' distortion will sound quite a bit different. WAY more than the 0.001% versus 0.1% distortion, too. Indeed, one can take the attenuated output of the tube amp, run it into the input of the superduper amp, adjust volume knob to get exactly 10 watts out (now with the imbedded 0.1% tube distortion), and use an A:B switch on a set of speakers ... and you will almost immediately hear a noticeable difference.
Why?
Well, its actually both simpler and more complex at the same time!
The finely crafted super-low THD (total harmonic distortion) amplifier will also very likely have many layers of negative feedback both within stages and globally (end-to-end, or really end-back-to-beginning). This global (and especially - last-stage!) feedback serves to markedly decrease technical distortion, but at the same time, it markedly raises "damping factor" of the amplifier. By comparison, the ultra-fidelity tube amplifier almost never has end-to-beginning global feedback, so its damping factor is WAY less, not by spec, but real-life attached to speakers.
And what is damping factor?
There are some good articles on it. But in a really easy nutshell: damping factor is the "force" with which the amplifier insists its output voltage should be, from moment to moment, a faithful copy of the input voltage! Speakers, as it turns out, are REACTIVE devices, with massy speaker cones wiggling in and out of the magnetic field, pushing and pulling air, being rebuffed by the air, and doing all sorts of things to "make music". However, they are powered by strong magnetic coils, in even stronger magnetic fields, (called "motors", go figure), and as such, generate their own electricity too as the cones relax and fling about along with the music program.
This "back-EMF" from the speakers is transmitted down the speaker cables, back to the amplifier, where it adds to the amplifier's own view of "output". The tighter the damping factor, the more that amplifier tries to tug the output voltage from Output+EMF back to just Output. The interesting thing is this: that tug-of-war between the speaker-cones and the amplifier in turn causes the speaker cones to be more tightly bound to their position as authorized by the amplifier.
NOW here's the jump: This is not what the microphones are doing! that originally picked up the sound. Or the pickups on guitars, or outputs of synthesizers, or the mix of the mixing board. Voltage ... corresponds to change-in-pressure, not position-of-speaker-cone.
Read that back again - it is the most important "ah-hah!" I've had in over 40 years of doing this stuff. When the damping factor increases, in essence, the amplifier is trying to control the POSITION of the speaker cone more and more precisely in an absolute sense. All the reactive components of the speaker, the wires, the cross-over in the cabinet, etc.. are all being clamped tight by the amplifier. In a tube-rig, with much lower damping factor, the amplifier is "directing the speaker" where to go, but not trying so damned hard to ensure that it "gets there" and "stays there" and "doesn't wiggle".
The net result speaks for itself: tube amps will sound much sweeter for a given simple, straight-forward, high-definition speaker system over the exact same power (and distortion, per above!) fed to them from a high-damping factor (typically solid state) rig. Period.
Yet, there are some segments of the market which are so impressed with either "more" or "less" numbers, on data sheets, that the factual bug (too high damping factor) becomes a virtual feature. I hear people all the time asking about it.
As an experiment, I took one of my friend's really high end solid-state monoblock amplifiers (which was broken, which I was fixing anyway...), and modified it with a complicated little switch that moved the 'global' feedback away from the final stage, and tapped into the pre-final stage instead. With a little 4-pole switch, I also adjusted the change-in-gain so that there would be absolutely no change in output power. I predicted that freed of tight (high damping) output, the amplifier would immediately become more musical through the same speakers.
This proved to be true - not to the degree of a full tube amp, but still ... the theory was borne out.
Finally - the big thing to remember about tube amplifiers is that they almost inevitably have a massive output transformer in the output stage, to convert from relatively high voltage swings of the output tubes (800+ volts! but low amps) to something speakers like better (30-40 volts, plus higher amps). Because "massive transformers" are also "massive inductors" and store a LOT of power as magnetic field, they are able to source quite a bit more out-of-phase current to the speakers than any direct-coupled solid state amplifier can ever achieve. EVER. This then is the second important effect that affects the perception of "sweet sound" from tube-block amplifiers. I'd even go so far as to claim that the ONLY two important things in a tube amplifier are really high voltage output tubes, and a lack of significant end-to-input feedback to "control" distortion.
I'm trying to set aside time to test this theory: that a really magnificent amplifier can be built with very high voltage tubes in the output, in class AB mode, with nothing but solid state pre-amplification in the front end. Careful selection of soft-clipping topologies and devices (like N-JFETS) ought to do the trick. Very little (perhaps no) global feedback, certainly no output-to-input feedback, and modest amplification at each stage internally. Symmetric amplification of the input and its complimentary (voltage inverted) form, all along the path.
I'm betting it sounds great.
GoatGuy
I must be the only one who might suggest that no negative feedback is part of a perfect amplifier (in theory)?
Um... actually, there is a very, very interesting thing I learned about this recently that makes me both partially agree with the statement, and on a different level strongly disagree.
Here it is: damping factor
[what?, he asks...]
For the longest time, I, being an electrical engineer, and in particular having a specialty in analog 2nd and 3rd order control circuits, thought to myself, Jeez... the job of an amplifier is solely to reproduce a faithful, but amplified version of the input voltage waveform, with sufficient power that the speakers can damned well do whatever they like in representing that as "sound". Period.
That kind of thinking then leads directly to, "whaddya mean, feedback compensation is bad? No such thing!"
Technically, this is true but not complete. The guiding principle behind so many uber-high-end amplifier setups is fine linearity, sweet near-limit clipping dynamics, and wide bandwidth throughout. Yet, it is truly the case that one amplifier with demonstrable ability to produce output of say "10 watts" at 0.001% distortion [typically a solid state monster] and another amp, again producing identically 10 watts at say 0.1% 'tube' distortion will sound quite a bit different. WAY more than the 0.001% versus 0.1% distortion, too. Indeed, one can take the attenuated output of the tube amp, run it into the input of the superduper amp, adjust volume knob to get exactly 10 watts out (now with the imbedded 0.1% tube distortion), and use an A:B switch on a set of speakers ... and you will almost immediately hear a noticeable difference.
Why?
Well, its actually both simpler and more complex at the same time!
The finely crafted super-low THD (total harmonic distortion) amplifier will also very likely have many layers of negative feedback both within stages and globally (end-to-end, or really end-back-to-beginning). This global (and especially - last-stage!) feedback serves to markedly decrease technical distortion, but at the same time, it markedly raises "damping factor" of the amplifier. By comparison, the ultra-fidelity tube amplifier almost never has end-to-beginning global feedback, so its damping factor is WAY less, not by spec, but real-life attached to speakers.
And what is damping factor?
There are some good articles on it. But in a really easy nutshell: damping factor is the "force" with which the amplifier insists its output voltage should be, from moment to moment, a faithful copy of the input voltage! Speakers, as it turns out, are REACTIVE devices, with massy speaker cones wiggling in and out of the magnetic field, pushing and pulling air, being rebuffed by the air, and doing all sorts of things to "make music". However, they are powered by strong magnetic coils, in even stronger magnetic fields, (called "motors", go figure), and as such, generate their own electricity too as the cones relax and fling about along with the music program.
This "back-EMF" from the speakers is transmitted down the speaker cables, back to the amplifier, where it adds to the amplifier's own view of "output". The tighter the damping factor, the more that amplifier tries to tug the output voltage from Output+EMF back to just Output. The interesting thing is this: that tug-of-war between the speaker-cones and the amplifier in turn causes the speaker cones to be more tightly bound to their position as authorized by the amplifier.
NOW here's the jump: This is not what the microphones are doing! that originally picked up the sound. Or the pickups on guitars, or outputs of synthesizers, or the mix of the mixing board. Voltage ... corresponds to change-in-pressure, not position-of-speaker-cone.
Read that back again - it is the most important "ah-hah!" I've had in over 40 years of doing this stuff. When the damping factor increases, in essence, the amplifier is trying to control the POSITION of the speaker cone more and more precisely in an absolute sense. All the reactive components of the speaker, the wires, the cross-over in the cabinet, etc.. are all being clamped tight by the amplifier. In a tube-rig, with much lower damping factor, the amplifier is "directing the speaker" where to go, but not trying so damned hard to ensure that it "gets there" and "stays there" and "doesn't wiggle".
The net result speaks for itself: tube amps will sound much sweeter for a given simple, straight-forward, high-definition speaker system over the exact same power (and distortion, per above!) fed to them from a high-damping factor (typically solid state) rig. Period.
Yet, there are some segments of the market which are so impressed with either "more" or "less" numbers, on data sheets, that the factual bug (too high damping factor) becomes a virtual feature. I hear people all the time asking about it.
As an experiment, I took one of my friend's really high end solid-state monoblock amplifiers (which was broken, which I was fixing anyway...), and modified it with a complicated little switch that moved the 'global' feedback away from the final stage, and tapped into the pre-final stage instead. With a little 4-pole switch, I also adjusted the change-in-gain so that there would be absolutely no change in output power. I predicted that freed of tight (high damping) output, the amplifier would immediately become more musical through the same speakers.
This proved to be true - not to the degree of a full tube amp, but still ... the theory was borne out.
Finally - the big thing to remember about tube amplifiers is that they almost inevitably have a massive output transformer in the output stage, to convert from relatively high voltage swings of the output tubes (800+ volts! but low amps) to something speakers like better (30-40 volts, plus higher amps). Because "massive transformers" are also "massive inductors" and store a LOT of power as magnetic field, they are able to source quite a bit more out-of-phase current to the speakers than any direct-coupled solid state amplifier can ever achieve. EVER. This then is the second important effect that affects the perception of "sweet sound" from tube-block amplifiers. I'd even go so far as to claim that the ONLY two important things in a tube amplifier are really high voltage output tubes, and a lack of significant end-to-input feedback to "control" distortion.
I'm trying to set aside time to test this theory: that a really magnificent amplifier can be built with very high voltage tubes in the output, in class AB mode, with nothing but solid state pre-amplification in the front end. Careful selection of soft-clipping topologies and devices (like N-JFETS) ought to do the trick. Very little (perhaps no) global feedback, certainly no output-to-input feedback, and modest amplification at each stage internally. Symmetric amplification of the input and its complimentary (voltage inverted) form, all along the path.
I'm betting it sounds great.
GoatGuy
You should first read about classical feedback control system theory, not involving op amps.
It is very often POSSIBLE to iterate to see the correct behavior of dynamic systems, since it is usually possible to simulate a differential equation(s) numerically, which involves iteration. Or you can convert the differential equation to a difference equation, where the iterative nature is more clear. However, an opamp amplifier with no capacitor or inductor is modeled by a simple algebraic equation and no iteration is necessary. You would probably still be able to use an iterative technique to arrive at the correct gain but I am actually not sure about that and don't feel like figuring it out, at the moment.
But anyway, your math appears to be wrong, in your original post. For example, X - (1/20)10x should equal 0.5x, not 5x.
For correct, practical opamp-based amplifier gain equations (and many useful circuit examples), download AN-31 and AN-20 from
http://www.ti.com/lit/an/snla140a/snla140a.pdf
and
http://www.ti.com/lit/an/snoa621b/snoa621b.pdf
It is very often POSSIBLE to iterate to see the correct behavior of dynamic systems, since it is usually possible to simulate a differential equation(s) numerically, which involves iteration. Or you can convert the differential equation to a difference equation, where the iterative nature is more clear. However, an opamp amplifier with no capacitor or inductor is modeled by a simple algebraic equation and no iteration is necessary. You would probably still be able to use an iterative technique to arrive at the correct gain but I am actually not sure about that and don't feel like figuring it out, at the moment.
But anyway, your math appears to be wrong, in your original post. For example, X - (1/20)10x should equal 0.5x, not 5x.
For correct, practical opamp-based amplifier gain equations (and many useful circuit examples), download AN-31 and AN-20 from
http://www.ti.com/lit/an/snla140a/snla140a.pdf
and
http://www.ti.com/lit/an/snoa621b/snoa621b.pdf
Just because the experimental result didn't falsify the theory doesn't automatically mean the theory was correct - some other explanation could fit. Did you check with just a series resistor to increase the damping factor?
I predict that with just the series resistor and the feedback from the original point (the output stage) that you won't get as much additional musicality as with moving the feedback point. I have another hypothesis which I believe fits your results
Just because the experimental result didn't falsify the theory ... I have another hypothesis which I believe fits your results
Have at it, bro! What theory might this be? I took the time to type clearly mine... love to read yours.
I wasn't sure that you'd be interested so I didn't share it at first, now I know you're curious, here goes...
What's my current hypothesis for sound quality (musicality) in amps is noise modulation. How this works is that when music is playing (as opposed to testing with sines) the low level intermodulation performance is what matters, not high level THD. I reckon by moving the feedback point you improved the low level IMD even though you probably worsened the high level measured THD.
The reason you improved low-level IMD is because of output stage crossover distortion - there's a band close to the zero point where many output stages have an effective 'dead band' or alternatively they have a window where there's 'gm doubling'. The degree to which its one or the other depends on the thermal characteristics of the bias tracking/compensation network. Either way the linearity suffers at low levels, and when feedback is closed around this I reckon the IMD distortion products in total increase, but only when the stimulus is low level tones.
Another possibility here is that in moving the feedback point you changed the low-level RF being fed to the input stage, which is normally an LTP. The speaker cable being an antenna picks up all kinds of RF hash, the LTP input stage is very susceptible to this. Moving the feedback take-off point to an internal node means less RF gets into the sensitive LTP stage. Lower RF translates to lower low-level IMD.
So although I originally mentioned just one hypothesis, there are actually two here, and both might be operating to differing degrees.
What's my current hypothesis for sound quality (musicality) in amps is noise modulation. How this works is that when music is playing (as opposed to testing with sines) the low level intermodulation performance is what matters, not high level THD. I reckon by moving the feedback point you improved the low level IMD even though you probably worsened the high level measured THD.
The reason you improved low-level IMD is because of output stage crossover distortion - there's a band close to the zero point where many output stages have an effective 'dead band' or alternatively they have a window where there's 'gm doubling'. The degree to which its one or the other depends on the thermal characteristics of the bias tracking/compensation network. Either way the linearity suffers at low levels, and when feedback is closed around this I reckon the IMD distortion products in total increase, but only when the stimulus is low level tones.
Another possibility here is that in moving the feedback point you changed the low-level RF being fed to the input stage, which is normally an LTP. The speaker cable being an antenna picks up all kinds of RF hash, the LTP input stage is very susceptible to this. Moving the feedback take-off point to an internal node means less RF gets into the sensitive LTP stage. Lower RF translates to lower low-level IMD.
So although I originally mentioned just one hypothesis, there are actually two here, and both might be operating to differing degrees.
And here comes my theory:
Without global negative feedback the level of harmonic components of the distortion is proportional (not necessarily linearly) to the signal level. Also lower order harmonics rise more than higher order harmonics. With negative feedback this is not rue. Push-pull stages with crossover distortion might be exception.
Without global negative feedback the level of harmonic components of the distortion is proportional (not necessarily linearly) to the signal level. Also lower order harmonics rise more than higher order harmonics. With negative feedback this is not rue. Push-pull stages with crossover distortion might be exception.
Sweet...
Door #2 is more believable, actually ... because antenna wires (I mean speaker cables LOL) do transmit back a fair amount of common-mode RF juju back toward the amp.
[Goat sighs, sits, thinks... finds an objection...]
But wait. The RF chatter is almost entirely common-mode, as the speaker wires are colinear, and attached further to a low-impedance load called "the crossover" (as well as the drivers). Hmmm... I suppose in a conventional split power supply (not "H" class, but just regular class AB, direct coupled, transistor/MOSFET type) amp, common mode or not, the end-to-front feedback tap would be picking up only one side ... and not subtracting out the common mode.
OK. It'd still propagate. I guess that's why so many global feedback loops have capacitive and resistive HF/RF clamping. No need exacerbating RF chatter amplification.
Let's see... Door #1 - noise (IM) modulation. Yah, yah, yah... I suppose its possible, but why then when (some years ago) I did the output-of-tube-amp-thru-attenuator-then-feeding-a-wowie-grade-semiconductor amp that was raved about as being utterly neutral and in no way showing acoustic signature ... why was the speaker tone so damned differnnt? The IM distortion (from the tube amp) was clearly being fed INTO the other semiconductor channel ... and its absolute linearity was gospel-truth just amplifying whatever came in to something larger on the output.
It was because of that experience that I actually over the years have gone on to more or less dismiss the "IM distortion view". While no experiment (per the above LF chatter) of this kind necessarily falsifies (or even validates entirely) a theory ... there still could be other reasons. But the ones that seem most likely (Occam's Razor in action) are, [1] damping and [2] damping. Possibly [3] damping. But one would have a hard time saying which.
Off to bed, this goat is.
GoatGuy
Door #2 is more believable, actually ... because antenna wires (I mean speaker cables LOL) do transmit back a fair amount of common-mode RF juju back toward the amp.
[Goat sighs, sits, thinks... finds an objection...]
But wait. The RF chatter is almost entirely common-mode, as the speaker wires are colinear, and attached further to a low-impedance load called "the crossover" (as well as the drivers). Hmmm... I suppose in a conventional split power supply (not "H" class, but just regular class AB, direct coupled, transistor/MOSFET type) amp, common mode or not, the end-to-front feedback tap would be picking up only one side ... and not subtracting out the common mode.
OK. It'd still propagate. I guess that's why so many global feedback loops have capacitive and resistive HF/RF clamping. No need exacerbating RF chatter amplification.
Let's see... Door #1 - noise (IM) modulation. Yah, yah, yah... I suppose its possible, but why then when (some years ago) I did the output-of-tube-amp-thru-attenuator-then-feeding-a-wowie-grade-semiconductor amp that was raved about as being utterly neutral and in no way showing acoustic signature ... why was the speaker tone so damned differnnt? The IM distortion (from the tube amp) was clearly being fed INTO the other semiconductor channel ... and its absolute linearity was gospel-truth just amplifying whatever came in to something larger on the output.
It was because of that experience that I actually over the years have gone on to more or less dismiss the "IM distortion view". While no experiment (per the above LF chatter) of this kind necessarily falsifies (or even validates entirely) a theory ... there still could be other reasons. But the ones that seem most likely (Occam's Razor in action) are, [1] damping and [2] damping. Possibly [3] damping. But one would have a hard time saying which.
Off to bed, this goat is.
GoatGuy
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