Dave, please review the Thevenin theorem. Wikipedia has a good discussion at wikipedia.org/wiki/Thévenin's_theorem . The first screen shot would be sufficient.
When you do, you'll see that it states something very simple. For any two nodes of any linear circuit, the impedance can be calculated by dividing the open circuit voltage between the nodes by the short circuit current that flows between them when you short them. (Of course, with a Cathodyne, you'd typically use a large cap for an AC signal instead of a short so as to not disturb the triode bias.)
Thevenin says this works for any pair of nodes in any linear circuit. No exceptions, no special pleading for circuits like Cathodynes. Thevenin doesn't care a fig about "boundary conditions."
There are three pairs of Thevenin equivalent resistances you might want to measure in a Cathodyne: those between P&K, P&gnd, and K&gnd. When you measure them, you'll find typical values of a little less than 2/gm for Zpg (useful when considering the balanced load operating condition of the Cathodyne), Zp-gnd, a little less than the plate resistor's resistance, and Zk-gnd, a bit more than 1/gm. These last two are useful for understanding what happens during unbalanced operation, such as when output tube grid current flows.
Consider that Thevenin has been around over 100 years and is well proven, utilized and accepted. Where in the literature other than what SY and associates have written do you find anything about "boundary conditions" as they apply to impedance measurements? And even if you did, the application of it could not in any way contradict Thevenin. And I have already shown how to apply Thevenin to obtain results that contradict those that are obtained by using "boundary conditions."
If you can read the Wikipedia summary of the Thevenin theorem and come up with an interpretation other than the one I gave, I'd be interested in hearing it.
When you do, you'll see that it states something very simple. For any two nodes of any linear circuit, the impedance can be calculated by dividing the open circuit voltage between the nodes by the short circuit current that flows between them when you short them. (Of course, with a Cathodyne, you'd typically use a large cap for an AC signal instead of a short so as to not disturb the triode bias.)
Thevenin says this works for any pair of nodes in any linear circuit. No exceptions, no special pleading for circuits like Cathodynes. Thevenin doesn't care a fig about "boundary conditions."
There are three pairs of Thevenin equivalent resistances you might want to measure in a Cathodyne: those between P&K, P&gnd, and K&gnd. When you measure them, you'll find typical values of a little less than 2/gm for Zpg (useful when considering the balanced load operating condition of the Cathodyne), Zp-gnd, a little less than the plate resistor's resistance, and Zk-gnd, a bit more than 1/gm. These last two are useful for understanding what happens during unbalanced operation, such as when output tube grid current flows.
Consider that Thevenin has been around over 100 years and is well proven, utilized and accepted. Where in the literature other than what SY and associates have written do you find anything about "boundary conditions" as they apply to impedance measurements? And even if you did, the application of it could not in any way contradict Thevenin. And I have already shown how to apply Thevenin to obtain results that contradict those that are obtained by using "boundary conditions."
If you can read the Wikipedia summary of the Thevenin theorem and come up with an interpretation other than the one I gave, I'd be interested in hearing it.
There is a differential output impedance and there is a common mode output impedance.
The "outputs" are also inputs, asymmetrically coupled to the grid.
We're sitting here arguing like Talmudic scholars about how many angels can dance on the head of a pin. I've heard that the reason academic arguments are so bitter is because there's so little worth arguing about.
Thanks,
Chris
The "outputs" are also inputs, asymmetrically coupled to the grid.
We're sitting here arguing like Talmudic scholars about how many angels can dance on the head of a pin. I've heard that the reason academic arguments are so bitter is because there's so little worth arguing about.
Thanks,
Chris
(1) OK...
(2) unbalanced has never been an option
(3) see # 2. what happens under imbalance has never been a concern.
(4) I'd love to have a parallel discussion about the load presented by a 5K output to each tube in a class A PP pair. It will have a very similar trend about currents to ground. (BTW, each tube will see 1/2 the A-A impedance of the transformer, yet i suspect if you attempt to apply Thevenin, you will get 1/4 A-A.
dave
(2) unbalanced has never been an option
(3) see # 2. what happens under imbalance has never been a concern.
(4) I'd love to have a parallel discussion about the load presented by a 5K output to each tube in a class A PP pair. It will have a very similar trend about currents to ground. (BTW, each tube will see 1/2 the A-A impedance of the transformer, yet i suspect if you attempt to apply Thevenin, you will get 1/4 A-A.
dave
Chris, yes, there are common mode and differential impedances. But when grid current flows, the impedance it represents is neither of these. How could it be? It flows into only one output, and not the other!
This is where Zp-gnd and Zk-gnd come it.
Please read Thevein's theorem: wikipedia.org/wiki/Thévenin's_theorem and explain to me how it is in accord with your statement that these impedances don't exist. It clearly states that you can measure the impedance of any two nodes in any linear circuit.
This is where Zp-gnd and Zk-gnd come it.
Please read Thevein's theorem: wikipedia.org/wiki/Thévenin's_theorem and explain to me how it is in accord with your statement that these impedances don't exist. It clearly states that you can measure the impedance of any two nodes in any linear circuit.
From the earth to the moon, close.
The nature of the balance is IN the output! You can't have a singular reference to a multiple input impedance.
You just devide by 2 or as many input nodes as you have.
They wouldn't be unbalanced if they are truly balanced, and so they exist with or without permission.
Saying the same thing 5 times in different ways does not create 5 different supporting points.
Funny definition. Again, divide by 2...etc. Create a reflected output with a 2 node junction and the difference voltages will reveal themselves across the inputs.
Model it with MathLab and run a Fourier Transform. You'll see it really can be done.
20.. Merry Christmas
didn't I say this in post 598???
we all agree what happens during unbalanced conditions. The whole point is what happens during balanced conditions. That is what we want to know. I'm fine with the concept of the differential output impedance. What is of interest to me is rather than considering the effects of imbalance, figuring out a way to assure it doesn't happen so you can take advantage of the differential output impedance.
I wonder how the cathodyne would do on A2 biased grids? what about cathode drive? Maybe increase the loads to swamp out possible variations. If the diff Z-out is really 160 ohms then 1600 ohms as a load is within audiophile consideration.
by your contention and applying Thevenin it isn't even worth looking at these ideas however I find it interesting being able to consider it. Other people look at the Thevenin results and insist build out resistors are needed.
Again I have never disputed the validity of your analysis, I have just insisted it doesn't apply under the conditions we are trying to look at.
dave
My application is I want to use a cathodyne splitter to drive a pair of differential 6SN7's which in turn will drive the grids of a pair of PP 2A3's.
Since this is a bass amp, I want to place a high pass filter inbetween the cathodyne and the grids of the 6SN7.
A) where is the grid current and why must I concern myself with it?
B) what source impedance do I use to calculate the filter sections?
C) are the filter sections identical?
dave
dave
Same answer you'd get if you wanted to drive EL84 or EL34 or KT88 or whatever in class A or AB1 (i.e., what the vast majority of amp designers do). A) it's negligible; B) the source impedances given by my model, which give correct answers for any pair of loads, and C) of course they are.
The hyperventilation and word games are just an added bonus.
Dave, I'm sorry I misinterpreted you.
I think I'm with you now, but you'll let me know if not.
I can see two ways to go here. One is a solution that occured to me that even SY recommended, although we disagree on the reasons that it helps. That is to use larger than usual (matched of course) grid stop resistors. That would reduce the large currents that the low impedance cathode dumps into the grid coupling capacitor, messing up the output tube biasing until that current slowly bleeds off. Unfortunately, global feedback will just try to drive the cathode harder. Still, there should be some benefit from this approach.
Another approach that has occured to me that I haven't quite thought through, but may have occured to others, is to apply very low mu cathode followers between the Cathodyne and the output tubes. The CF's would be direct-coupled to the output tube grids. The CF's would be biased so that they could drive significant grid current into the output tubes before drawing grid current themselves.
Anytway, that's the extent of my thoughts on the matter.
I think I'm with you now, but you'll let me know if not.
I can see two ways to go here. One is a solution that occured to me that even SY recommended, although we disagree on the reasons that it helps. That is to use larger than usual (matched of course) grid stop resistors. That would reduce the large currents that the low impedance cathode dumps into the grid coupling capacitor, messing up the output tube biasing until that current slowly bleeds off. Unfortunately, global feedback will just try to drive the cathode harder. Still, there should be some benefit from this approach.
Another approach that has occured to me that I haven't quite thought through, but may have occured to others, is to apply very low mu cathode followers between the Cathodyne and the output tubes. The CF's would be direct-coupled to the output tube grids. The CF's would be biased so that they could drive significant grid current into the output tubes before drawing grid current themselves.
Anytway, that's the extent of my thoughts on the matter.
Two more considerations if there's to be loop feedback: the time constants of the RC coupling networks should be staggered, with the smallest time constant at the point where grid current occurs (because this also determines "bleed-down" time).
Output transformer inductance varies with signal level, increasing for most of the way up, so its time constant must be kept away from the dominant RC network's.
Thanks,
Chris
HI All, and congratulations to Stuart on the publication of his interview in AudioXpress magazine. Stuart, you must be very pleased!
I’m posting today after this long cooling off period because I think it’s both important and possible for us to come to an agreement on Cathodyne (Cdyne) output impedances.
It’s important because determining output impedance is fundamental, and if we can’t agree on the fundamentals, we can’t agree on anything. Also, as a practical matter, an understanding here might help explain why a bench test shows that when grid current is drawn from a P(late), Cdyne output voltages are so different from when it is drawn from the K(athode).
It is possible if we agree to respond only rationally and logically to all arguments with which we disagree. Simply ignoring an argument is inconsistent with intellectual integrity, denying us an opportunity to reason together. And answering with an ad hominem attack just introduces irrelevancies and obfuscation. (Even if it were true that my arithmetic skills were inferior to those of the average four year old, that would not invalidate my claims - although it would probably make them easier to refute!)
So for easy reference, I’ll enclose my arguments in curly brackets { } and name them within square ones [].
You may recall from our prior postings that Stuart named a price for breaking his silence on matters Cdyne: disclosing loads which confound the Cdyne equal output impedance model (Equal_Z). Presumably, the premise here is that the absence of such loads confers some sort of validity on that model. [Problematic Premise:] { It seems clear that if the absence of loads which confound Equal_Z validates Equal_Z, then the absence of loads which confound Diff_Z (different Cdyne output impedances) must validate Diff_Z. Since I am unable to identify any loads which alone confound either model, by the Problematic Premise, I must confirm both models. But that’s nonsense – the models conflict with one another! The Premise has led to a contradiction. By basic logic, therefore, it is false and must be abandoned. So the absence of confounding loads is of no consequence to either model.} I think Stuart would be justified in breaking his vow of silence in light of this. Stuart?
But even if that doesn’t happen, may I remind that we were promised a response to what was revealed by the current mirror circuit? [Mirror, mirror:] { If we drive the input of a unity gain current mirror from a voltage source through an impedance Z and terminate its output with a separate but identical Z, then both Z’s will pass the same current and develop the same voltage. Since the input impedance of a mirror is far less than that of its output, this is all accomplished despite the fact that the Z’s are driven from sources of very different impedances. This falsifies the assertion of the second argument supporting Equal_Z in the Linear Audio article “Split the difference…”, namely, that within a circuit, identical loads which develop equal voltages demand to be driven by sources of identical impedance.}
With this second argument down, we can focus on the first and only remaining argument for Equal_Z in the article. But let’s save that for a subsequent posting.
Looking forward to hearing from those still willing to engage!
I’m posting today after this long cooling off period because I think it’s both important and possible for us to come to an agreement on Cathodyne (Cdyne) output impedances.
It’s important because determining output impedance is fundamental, and if we can’t agree on the fundamentals, we can’t agree on anything. Also, as a practical matter, an understanding here might help explain why a bench test shows that when grid current is drawn from a P(late), Cdyne output voltages are so different from when it is drawn from the K(athode).
It is possible if we agree to respond only rationally and logically to all arguments with which we disagree. Simply ignoring an argument is inconsistent with intellectual integrity, denying us an opportunity to reason together. And answering with an ad hominem attack just introduces irrelevancies and obfuscation. (Even if it were true that my arithmetic skills were inferior to those of the average four year old, that would not invalidate my claims - although it would probably make them easier to refute!)
So for easy reference, I’ll enclose my arguments in curly brackets { } and name them within square ones [].
You may recall from our prior postings that Stuart named a price for breaking his silence on matters Cdyne: disclosing loads which confound the Cdyne equal output impedance model (Equal_Z). Presumably, the premise here is that the absence of such loads confers some sort of validity on that model. [Problematic Premise:] { It seems clear that if the absence of loads which confound Equal_Z validates Equal_Z, then the absence of loads which confound Diff_Z (different Cdyne output impedances) must validate Diff_Z. Since I am unable to identify any loads which alone confound either model, by the Problematic Premise, I must confirm both models. But that’s nonsense – the models conflict with one another! The Premise has led to a contradiction. By basic logic, therefore, it is false and must be abandoned. So the absence of confounding loads is of no consequence to either model.} I think Stuart would be justified in breaking his vow of silence in light of this. Stuart?
But even if that doesn’t happen, may I remind that we were promised a response to what was revealed by the current mirror circuit? [Mirror, mirror:] { If we drive the input of a unity gain current mirror from a voltage source through an impedance Z and terminate its output with a separate but identical Z, then both Z’s will pass the same current and develop the same voltage. Since the input impedance of a mirror is far less than that of its output, this is all accomplished despite the fact that the Z’s are driven from sources of very different impedances. This falsifies the assertion of the second argument supporting Equal_Z in the Linear Audio article “Split the difference…”, namely, that within a circuit, identical loads which develop equal voltages demand to be driven by sources of identical impedance.}
With this second argument down, we can focus on the first and only remaining argument for Equal_Z in the article. But let’s save that for a subsequent posting.
Looking forward to hearing from those still willing to engage!
With reference to Post 616 and no objections heard to [Mirror, Mirror] (which contests the second of two arguments for equal Cathodyne output impedance in the Linear Audio article “Split the Difference…”), let’s address the remaining argument in the article.
This is based on an unusual circuit for testing short circuit currents to determine impedances. It shorts three nodes together simultaneously, P(late), K(athode) and G(round), instead of the usual two. The result is three shorted current paths, P-K, P-G and K-G, rather than the usual one due to only two shorted nodes (e.g., a single output circuit’s output and ground.) I’ll refer to this circuit and its associated analysis and conclusion as TripleShort.
[This game is rigged!] {Before a single calculation or measurement of TripleShort is made, it can be seen that it is constitutionally incapable of concluding anything but Equal_Z. This is because, in the absence of grid current, triode P and K currents must always be exact opposites when measured simultaneously as they are here- even when they are short circuit currents. Since the Cdyne P and K unshorted voltages are also exact opposites, these (impedance) ratios of voltage to current for the P-G and K-G paths must be identical. And so we see that TripleShort assumes the result it presumes to test. Although this doesn’t necessarily mean that it reaches the wrong conclusion, it does mean that it has little value as an unbiased arbiter of the truth.}
[It’s against the Law!] {What does make TripleShort wrong, however, is that it violates Thevenin’s Theorem. To see why, let’s review what Thevenin explicitly promises. This theorem applies to all linear circuits consisting of no more than resistors and controlled and independent sources. It states that the voltage and current characteristics of any two nodes in such a circuit are indistinguishable from those of a specific voltage source in series with a specific resistor. The voltage of the source is the voltage between the two nodes. The value of the resistor can be obtained by shorting the two nodes together, noting the current that flows between them, and dividing the source voltage by that current. That’s the entire theorem. There are no addenda, and there are no exclusions from these rules. In particular, there is no special pleading for circuits with multiple outputs having specific interrelationships. Does anyone believe that if Mr. T intended there to be such, he would have failed to list the exceptions and describe how to deal with them? Clearly, he neither listed nor described. So it is reasonable to conclude that he intended no such things and that we can apply his theorem as stated here without reservation.
If we apply it between the P and K, we arrive at the uncontroversial result of a bit less than 2/gm. But if we apply it to the P and G, the result is between 50 and 100% of the plate resistor, a value in stark contrast to Equal_Z’s approximately 1/gm, which we see must therefore be rejected as violating Thevenin.}
Let me anticipate objections to this line of reasoning. They might go as follows. [Two Criticalities:] { Cdyne output voltages are exact opposites. If only the plate is grounded, this property is violated: the circuit is no longer a Cdyne, and so cannot be used to determine Cdyne characteristics.
Let us consider the consequences of this admittedly seductive reasoning. First, we might note that the output voltage of an amplifier is by definition greater than that of its input. And so, if we short its output to ground to determine its short circuit current, it is no longer an amplifier. Therefore, in accordance with the seductive, it cannot be employed to determine the characteristics of one. Here we see a hint that the seductive argument may have a problem.
But let’s return to Cdynes. Clearly, it is critical that Cdyne output voltages are exact opposites. But that’s hardly the only critical Cdyne characteristic. A Cdyne’s outputs must also be proportional to its input. Otherwise, we don’t even have an audio circuit, let alone a Cdyne. Now, I see no reason to privilege either of these requirements over the other. Either both must be preserved during testing, or neither need be. If both must be, we note that TripleShort violates output proportionality by grounding outputs (zero is not proportional to an input signal.) It is therefore not a Cdyne, and so cannot be used to determine Cdyne characteristics. But if neither requirement need be honored, then we needn’t employ TripleShort to ensure exactly opposite output voltages during test. In fact, we are required to reject it, as we have proven that it violates Thevenin. Either way, TripleShort cannot be used to justify Equal_Z. And as a result, we see that there is no justification left for Equal_Z in the Linear Audio article.
I’d be happy to hear from anyone with thoughts on the foregoing.
This is based on an unusual circuit for testing short circuit currents to determine impedances. It shorts three nodes together simultaneously, P(late), K(athode) and G(round), instead of the usual two. The result is three shorted current paths, P-K, P-G and K-G, rather than the usual one due to only two shorted nodes (e.g., a single output circuit’s output and ground.) I’ll refer to this circuit and its associated analysis and conclusion as TripleShort.
[This game is rigged!] {Before a single calculation or measurement of TripleShort is made, it can be seen that it is constitutionally incapable of concluding anything but Equal_Z. This is because, in the absence of grid current, triode P and K currents must always be exact opposites when measured simultaneously as they are here- even when they are short circuit currents. Since the Cdyne P and K unshorted voltages are also exact opposites, these (impedance) ratios of voltage to current for the P-G and K-G paths must be identical. And so we see that TripleShort assumes the result it presumes to test. Although this doesn’t necessarily mean that it reaches the wrong conclusion, it does mean that it has little value as an unbiased arbiter of the truth.}
[It’s against the Law!] {What does make TripleShort wrong, however, is that it violates Thevenin’s Theorem. To see why, let’s review what Thevenin explicitly promises. This theorem applies to all linear circuits consisting of no more than resistors and controlled and independent sources. It states that the voltage and current characteristics of any two nodes in such a circuit are indistinguishable from those of a specific voltage source in series with a specific resistor. The voltage of the source is the voltage between the two nodes. The value of the resistor can be obtained by shorting the two nodes together, noting the current that flows between them, and dividing the source voltage by that current. That’s the entire theorem. There are no addenda, and there are no exclusions from these rules. In particular, there is no special pleading for circuits with multiple outputs having specific interrelationships. Does anyone believe that if Mr. T intended there to be such, he would have failed to list the exceptions and describe how to deal with them? Clearly, he neither listed nor described. So it is reasonable to conclude that he intended no such things and that we can apply his theorem as stated here without reservation.
If we apply it between the P and K, we arrive at the uncontroversial result of a bit less than 2/gm. But if we apply it to the P and G, the result is between 50 and 100% of the plate resistor, a value in stark contrast to Equal_Z’s approximately 1/gm, which we see must therefore be rejected as violating Thevenin.}
Let me anticipate objections to this line of reasoning. They might go as follows. [Two Criticalities:] { Cdyne output voltages are exact opposites. If only the plate is grounded, this property is violated: the circuit is no longer a Cdyne, and so cannot be used to determine Cdyne characteristics.
Let us consider the consequences of this admittedly seductive reasoning. First, we might note that the output voltage of an amplifier is by definition greater than that of its input. And so, if we short its output to ground to determine its short circuit current, it is no longer an amplifier. Therefore, in accordance with the seductive, it cannot be employed to determine the characteristics of one. Here we see a hint that the seductive argument may have a problem.
But let’s return to Cdynes. Clearly, it is critical that Cdyne output voltages are exact opposites. But that’s hardly the only critical Cdyne characteristic. A Cdyne’s outputs must also be proportional to its input. Otherwise, we don’t even have an audio circuit, let alone a Cdyne. Now, I see no reason to privilege either of these requirements over the other. Either both must be preserved during testing, or neither need be. If both must be, we note that TripleShort violates output proportionality by grounding outputs (zero is not proportional to an input signal.) It is therefore not a Cdyne, and so cannot be used to determine Cdyne characteristics. But if neither requirement need be honored, then we needn’t employ TripleShort to ensure exactly opposite output voltages during test. In fact, we are required to reject it, as we have proven that it violates Thevenin. Either way, TripleShort cannot be used to justify Equal_Z. And as a result, we see that there is no justification left for Equal_Z in the Linear Audio article.
I’d be happy to hear from anyone with thoughts on the foregoing.
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