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In Reply to: RE: more on output impedance posted by Ralph on January 15, 2015 at 10:08:07
"I like to use the black box approach similar to how one measures the impedance of a speaker. In this case, the amp has an unknown impedance but we can easily find out what it is by simply connecting it to a variable load. We then find out at what impedance it makes its maximum power and then continue to reduce the load until that value is cut in half. At that point, half of the power is being dissipated in the output circuit itself (causing it to heat up) and the other half in the load. At this point the load must be equal to that of the output circuit."The thing that worries me about your definition of output impedance is that it doesn't give a sensible answer if we test it on the classic example of an idealised device that is modelled as a perfect voltage source (i.e. zero impedance and voltage V) in series with a resistance R. This device unambiguously has an output impedance R.
If we now drive this into an external impedance Z, the power dissipated in the external load is P= V^2 Z/(R+Z)^2, and this is, of course, maximised at Z = R. Let's call this Z1, so we have Z1 = R. We also have Pmax = V^2/(4R). If we now follow your rule, and ask for the smaller of the two values of Z for which P = Pmax/2, we get Z = (3 - 2* sqrt2) R. Calling this value Z2, we have approximately Z2 = 0.17 R.
Thus in this case of an idealised output device, for which all conventional ways of calculating the output impedance will give the answer Zout = R, your way of defining the output impedance will give the answer Zout = 0.17 R, which is about one sixth of the correct answer.
And, by the way, the value of Z for which the power dissipated in the device equals the power dissipated in the external load is also given by Z = R. When Z = 0.17 R, which is your definition of the output impedance when applied to this idealised case, the power dissipated in the external load is about one sixth of the power dissipated in the device itself.
I suppose your argument must be that the idealisation of the output device as a perfect zero-impedance voltage source in series with a resistor R is not an accurate model of a real amplifier output stage. But it does seem to be troubling that your definition gives a result that is so much at odds with the actual answer, if applied to the usual idealisation.
Chris
Edits: 01/17/15Follow Ups:
Thus in this case of an idealised output device, for which all conventional ways of calculating the output impedance will give the answer Zout = R, your way of defining the output impedance will give the answer Zout = 0.17 R, which is about one sixth of the correct answer.
In practice this is not borne out. I stated earlier what the differences are. However I should point out that its also impractical to assume that we are going for a perfect voltage source. In fact we are not, in our case we are going more for a power source. All amplifiers were power sources in the old days- SETs still are, as are many tube amps without loop feedback.
Obviously a perfect voltage source will have infinate power at zero ohms. It does not exist.
Its a different way of looking at it. What I recommend is that you play with the equipment and take measurements as they will reveal more than the math seen so far. Or you could trust that the numbers I suggest are right- I've done these measurements many times. Did you read the article at the link? There is a reason why someone would want to design an amplifier that has a higher output impedance like this- oddly, its because you can get less coloration- quite the opposite of the 'accepted wisdom'. This is because of how the human brain processes sound, which ultimately is the final arbiter.
"Did you read the article at the link? There is a reason why someone would want to design an amplifier that has a higher output impedance like this- oddly, its because you can get less coloration- quite the opposite of the 'accepted wisdom'. This is because of how the human brain processes sound, which ultimately is the final arbiter."Yes, I read the article you linked. It seems to me there are several distinct questions here. Some of them, such as the voltage paradigm vs the power paradigm, can be discussed even within the conventional model of an amplifier output stage as a perfect voltage source in series with an internal impedance R.
What is certainly true is that if the output impedance R is tiny, such as with a typical solid state amplifier, then for a given signal level the power delivered into a typical loudspeaker load impedance Z falls approximately inversely with Z. On the other hand, if the output impedance R is comparable with the typical loudspeaker impedance Z, then one is "sitting on the top of the curve" in the plot of power dissipated into the load as a function of Z. Thus, to a good approximation, moderate variations in Z around the value Z=R will then not cause the power into the load to change by too much. This is illustrated in the first plot here, of power into the load Z, as a function of Z, for the case of output impedance R=8 Ohms. (x-axis is Z, y-axis is power into load Z.)
By contrast, in the case of a very low output impedance amplifier, the power into Z=16 Ohms would be one quarter of the power into Z=4 Ohms, as can be seen in this second plot (normalised to the same power at Z=8 ohms):
In other words, you don't need to invoke any non-standard definition of "output impedance" in order to make your point about the power paradigm vs the voltage paradigm. It is already visible, just with the usual definition, and the usual model of an output device. If the output impedance is tiny, then for a load impedance Z in the range between 4 and 16 ohms, the power lies in a large range, with Pmin being 25% of Pmax. On the other hand, if the output impedance is 8 ohms then for the same range of load impedances the power lies in a relatively small range, with Pmin almost 89% of Pmax.
Another rather different question is whether the usual model of an amplifier output stage as a perfect voltage source in series with an internal resistance R is a good one or not. I tried various experiments in the past with my OTL amplifiers, and it seemed to me that the model worked reasonably well. Now, in all of my amplifiers there is quite a lot of negative feedback. I am interested now to experiment a bit more, in particular with how well the idealised model works if I turn off the negative feedback. I suppose my expectation would be that the model would still be reasonable, but now the value of R would be a lot larger. But I will keep an open mind until I have done the experiments.
Concerning the non-achievability of the "perfect" zero impedance voltage source, yes, of course I agree with you it cannot literally be realised in practice. But that doesn't mean it isn't a useful concept in circuit analysis. As long as its "actual" output impedance is very small compared to that of the internal resistance R that one adds in the model of the output stage, then the imperfection of the "perfect" source is insignificant. I still feel that if a definition of "output impedance" gives a radically wrong answer when tested on the conventional model of an output device, then there is reason to question the validity of that definition. But then, as I said above, I don't think you even need your non-standard definition of output impedance in order to make your point about the power vs voltage paradigms.
Anyway, I shall try to carry out some further experiments soon.
Chris
Edits: 01/21/15 01/21/15 01/21/15 01/21/15 01/21/15
The non-standard output impedance method is actually based on the formula for calculating the output impedance of a cathode follower. But as you point out, the graphs really make the point.
The problem is that quite often the application of loop negative feedback contributes to higher ordered harmonic content (re.: Norman Crowhurst). Since the ear uses such harmonics to calculate sound pressure it is very sensitive to these harmonics; so much so that even trace amounts that are hard to measure are readily audible, which is why transistor amps tend to sound bright. The brightness is not measurable as a frequency response variation, in just the same way that the 'warmth' of a tube amp is not.
Quite literally the application of feedback causes a violation of one of the most fundamental rules of human hearing perception. The Power Paradigm is a means of getting around that problem. The idea is that while frequency response is likely to not be as accurate, colorations due to distortion (which are interpreted by the ear as tonality) will be reduced. Since no speaker is really flat with any source, the actual real-world result is that you can get what sounds like very uncolored sound.
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