Bob Cordell Interview: Error Correction

[traderbam]

PB2 wrote:

What Bob did in his original article was to show a linear system, with the distortion injected as an additive signal.

This isn't a correct mathmatical way to model the distortion of element A. I showed this earlier in this thread. Linear addition and cancellation does not work in a non-linear system. Rodolfo and I are considering the sensitivity of the CL gain to the transfer function of A.

I believe that the key word/difference between traditional feedback and EC, is null.

The so-called "nulling" is just maximizing the PFB gain (the forward path gain). There is no null in the sense of an "inverse" error signal being added to the original error signal. The same effect could be achieved by varying the gain of a normal gain element...except the output error would continue to reduce until the system became unstable.

We null out the signal, in order to extract the error in isolation of the signal (Ideal case - with an excellent approximation in real hardware) which can then be fed back without this path setting the gain of the system.

No, I realise that is what you would like to think the system is doing. But it isn't. It is impossible to achieve this in a feedback system. Changing the "null" changes the forward gain. You can only "null" in the way you are describing in a feed-forward system.

I've been around this loop enough times now that I feel I have offered an exhaustive body of evidence. At least I am exhausted! I have shown equivalence in the mathmatics. I have shown measurements from simulations. I have given common-sense reasoning based on the obvious inability to correct a non-linear device by means of a linear sum and difference system. Bob Cordell has provided JAES evidence from Vanderkooy and Lipshitz.

I apologize if any of you are still seeking a pursuasive argument but I am all out of proofs.

 

[Nelson Pass]

Did you ever worry about "gm doubling", as described by Self?

In your view, what was the main contribution to performance made by your sliding bias scheme? For example, was it more in the nature of smoothing the gm transition curve with respect to static crossover distortion, or was it some dynamic high-speed benefit that arose from not fully turning off the output transistors, or was it something else altogether?

The thoughts I had with respect to "gm doubling" are limited to
the diagram in the 1978 "A40" article in Audio Amateur, where
I have a graphic representing the open loop output impedance
of a complementary follower in Class B, AB, and A. (Interestingly
the bias network for that amplifier showed up as an example
discussed by Slone).

In any case, the network in the 800A, 400A and 4000 smoothed
the static curve a little better than the raw bias, but its biggest
impact was to create a maelstrom of imitators, none of whom
wanted to put more than about 100 mA of bias on the output
stage.

 

[jan.didden]

[snip]There is no null in the sense of an "inverse" error signal being added to the original error signal. The same effect could be achieved by varying the gain of a normal gain element...except the output error would continue to reduce until the system became unstable. [snip]

Brian,

I guess there are still some of us that don't agree to your 'its just a clever pos/neg feedback combo' argument.
Observe: in the equation of a classical neg feedback loop, the cl gain depends on the forward gain. We casually say: if we make that gain large enough, the feedback will determine the gain. But that's not true! Even in the case that the forward gain is infinite, STILL there is a ol gain dependency in the cl equation (the '1' factor in the denominator).
So, even in theory, even with forward gain infinite, impossible to zero out the nonlinearity. That applies also to your transformation of H.ec where you introduce the pos feedback 'infinite gain' block: still even with an infinite gain pos fb block, still no zero distortion.

This is in start contrast with H.ec topology where in theory it IS possible to null the error, without resorting to infinite forward gain, real or synthesised with pos feedback.

The inescapable conclusion is that the two ARE fundamentally different.

Now, you can transfor one into the other in the sense that at black-box level they look, from the outside, the same. Just like a voltage source and a series resistance black box, from the outside, cannot be distinguished from a current source with a parallel resistor. But the way that that particular black-box look is made is fundamentally different.

[snip]I have given common-sense reasoning based on the obvious inability to correct a non-linear device by means of a linear sum and difference system. [snip][/B]

The ec signal is subjected to the same non-linearity it tries to correct. This results from the fact that the ec system 'bootstraps it's own input' counteracted by the ec signal going itself through the non-linearity. So it is not just a linear sum&difference system to correct a non-linear system. The non-linear system itself is part of the process, and that is why it actually produces a null.

In my AES paper I took the approach to add a feedback loop 1/A to null the error. Then you realise that 1/A is actually the input to the (non-linear) A block: the non-linear block processes it's own (inverted) error and in the end it comes out linear. Neat, huh?

Jan Didden

 

[traderbam]

This is in start contrast with H.ec topology where in theory it IS possible to null the error, without resorting to infinite forward gain, real or synthesised with pos feedback.

Jan, what theory? I've shown the proof that Hec depends on infinite gain. You prove me wrong! Show me your mathmatical proof. You are making arguments with words and not equations. The statement "The non-linear system itself is part of the process, and that is why it actually produces a null" has no mathmatical substance, so I cannot analyze it.

I'd like to drop this debate and continue to analyze Bob's output stage and see if there are opportunities to improve it. There really ought to be after 22 years.

 

[traderbam]

Attached is Andy_c's LTSpice model of Bob's output circuit. I have inserted a voltage signal generator V6 in the main feedback loop.

The voltage gain in any circuit loop can be measured by inserting a 0Vdc voltage source in a wire connection. A small ac signal is applied (small enough to avoid undesirable large signal effects at any point in the circuit). The ratio of voltage at the + end of the source to that at the - end is the loop gain x -1.

For V6 I chose 1uV ac. You can see from the graph that the loop gain is about 41dB up to a knee at 30kHz. The unity-gain frequency is 3MHz with a phase margin of 88 degrees.
Andy has used 724 ohm resistors for the "S2" summers. Bob's paper shows 680 ohms. With 680 ohm resistors the loop gain drops to about 22dB. I would guess the real circuit to be optimum somewhere between these two...the stability margin of the real circuit will be smaller because the simulation does not include inductance and psu impedance and so on.

This is reassuringly consistent with post #743 where Bob measured a 30dB improvement in the real circuits THD performance with the feedback correction applied.

I am going to adjust the S2 resistors to give a loop gain of 30dB to match the real circuit result and then measure other performance parameters of the stage.

Bob, can you confirm the transistor types you used in the real circuit and the resistor values?

 

[PB2]

The so-called "nulling" is just maximizing the PFB gain (the forward path gain). There is no null in the sense of an "inverse" error signal being added to the original error signal. The same effect could be achieved by varying the gain of a normal gain element...except the output error would continue to reduce until the system became unstable. No, I realise that is what you would like to think the system is doing. But it isn't. It is impossible to achieve this in a feedback system. Changing the "null" changes the forward gain. You can only "null" in the way you are describing in a feed-forward system.

I've been around this loop enough times now that I feel I have offered an exhaustive body of evidence. At least I am exhausted! I have shown equivalence in the mathmatics. I have shown measurements from simulations. I have given common-sense reasoning based on the obvious inability to correct a non-linear device by means of a linear sum and difference system. Bob Cordell has provided JAES evidence from Vanderkooy and Lipshitz.

I apologize if any of you are still seeking a pursuasive argument but I am all out of proofs.

I've not read all that you've posted traderbam, and I certainly understand that Bob's linear model, with error injection is an approximation to real hardware, however these are the diagrams that have been under discussion here, at least from what I've seen. I think it's best if we simply agree to disagree.

I do want to point out that Bob's and Andy's examples when nulled do not provide a positive feedback signal since that also nulls. This is easy to show through the equations, and indeed the distortion also nulls, when the elements outside the non-linear one under consideration are linear.

Pete B.

 

[mikeks]

Having demonstrated the general criteria for error cancellation in respect of an output stage with an arbitrary nominal gain K, an even more general solution may be obtained by simple manipulation.

From this it is clear that this is a simple negative feedback system, subject to certain conditions.

Discuss.

 

[ingrast]

.........

Now I see it again, well, look Michael, Fig. 1 and Fig 2 are not equivalent, will leave for your to find out.

Rodolfo

 

[mikeks]

Totally equivalent.

See last arrangement on page nine (fig. 6), here.

Else try this.

 

[andy_c]

Attached is Andy_c's LTSpice model of Bob's output circuit.

As jcx mentioned before, my sim contains an error. It's missing a ".options plotwinsize=0" directive. This directive disables the data compression that degrades the accuracy of the distortion sim.

For AC analysis, the voltage amplitude is not relevant except for scaling of displayed data. For AC analysis, SPICE replaces all nonlinear devices by their linear equivalent circuits, making 1V a nice convenient value to use for injected voltages.

Also, I basically pulled the device types out of a hat for this sim .

For determining the depth of the null, the distortion sim should work well, but it needs the ".options plotwinsize=0". Of course, when you change collector resistor values, you'll need to rebias for 150 mA output stage current.

In Hawksford's equations for the component values of his error correction circuit, he neglected the effect of the error correction transistor r_e. I think that's why I changed the collector resistor value back when I originally set the sim up. One possible thing to try might be replacing the error correction transistors by CFPs. This will eliminate the r_e contribution, so a smaller collector resistor would be required for a null.

 

[moamps]

Totally equivalent.

It's 20 past 1a.m. here so I might be wrong but...

S1out=Vo-Vo/K1
=Vo(1-1/K1)

S2(-in)=K2(1-1/K1)*Vo


Regards,
Milan

 

[mikeks]

To continue..

To continue from here, and equating terms here and here, it is clear that the following is true at balance:

 

[mikeks]

or:

 

[moamps]

Thank you mikeks for editing the error in your previous posts.

Regards,
Milan

 

[mikeks]

Which error?

 

[moamps]

Need a screenshot?
Carry on, I'm out of your hair now.

Regards,
Milan

 

[mikeks]

Carry on.....

Hellllll No!

I am off to hit the sack; it's 00:49Hrs here!

 

[traderbam]

Of course, when you change collector resistor values, you'll need to rebias for 150 mA output stage current.

Andy, you can always insert 150mA CCSs in series with each FET source, each one bypassed with a huge cap, like 1F, to avoid having to keep iterating the bias resistor.

Hopefully Bob will provide the authentic part values. Meanwhile your model is fine for getting the right "ball park" estimate of the shape of the loop gain vs f and for exploring how the circuit works. If you do a distortion measurement with and without the feedback connected you ought to find the distortion ratio is pretty close to the feedback ratio.

One possible thing to try might be replacing the error correction transistors by CFPs.

Interesting idea. We can also take a look at why Bob added that extra NFB path from the gates. I think it is just a convenience for biasing the FETs but is not strictly a la Hawksford either.

Brian

 

[andy_c]

Andy, you can always insert 150mA CCSs in series with each FET source, each one bypassed with a huge cap, like 1F, to avoid having to keep iterating the bias resistor.

Hehe. Where there's a will, there's a way!

We can also take a look at why Bob added that extra NFB path from the gates. I think it is just a convenience for biasing the FETs but is not strictly a la Hawksford either.

Not sure which NFB path you're referring to. If it's the bias resistor between the error correction transistor bases, I believe the AC voltages on both sides of this resistor are equal, bootstrapping it out of the AC circuit. I could be wrong though.

 

[PB2]

I just noticed that traderbam quoted me in a previous post and his text came up in my quote. I did not say most of what's in the quote. Would you or a moderator please fix this.

PB2 wrote:The so-called "nulling" is just maximizing the PFB gain (the forward path gain). There is no null in the sense of an "inverse" error signal being added to the original error signal. The same effect could be achieved by varying the gain of a normal gain element...except the output error would continue to reduce until the system became unstable. No, I realise that is what you would like to think the system is doing. But it isn't. It is impossible to achieve this in a feedback system. Changing the "null" changes the forward gain. You can only "null" in the way you are describing in a feed-forward system.

Moderator's edit: Fixed the quote by deleting "PB2 wrote" from the box. Regards, Milan