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Old 5th January 2017, 06:09 AM   #11
PRR is offline PRR  United States
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And FWIW the AC30 cited has a 220K plate resistor. Half as deadly (not) as a 100K Fender.
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Click the image to open in full size.

I'm sure the designer of the AC30 never looked at this that way. He'd probably take it funny if you showed it to him. If you bought the beer, he might mentor you on understanding tubes "intuitively" before you reach for the algebra. Most questions can be answered by thinking "if this goes up, that goes down, and about so much".

> reason why the AC components of i1 and i2 would have similar magnitude or shape to each other

Input is applied to V3a grid? NO, input is always 2 wires! Input is applied from V3a grid to V3b grid's cap to ground. To a first approximation, V3a and V3b split the input signal across their grid-cathode ends. V3a(gk) = V3b(gk). And their plate circuits are equal. So the output signals are equal.

Wait. V3a(gk) = V3b(gk) is incomplete because R21 and friends leak dynamic current which does flow in V3a but not in V3b. How much? A caveman approximation is to find the cathode impedance, 1/Gm, of V3b. We should know that 12AX7 at Fender-like conditions has Gm of 1uMho and thus Rk about 1K Ohms. So there is a 47K:1K split. V3b gets 98% of V3a dynamic current. There is a few-percent difference in output. As our EL84s are liable to be up to 30% apart, and we are not humping for point-oh-oh THD, this is not critical (and not worth a picture full of math). Although there is a "fix" in many Fenders using a similar stage.

> is there any point at all to matching the output tubes?

I think many new-age g-amp designers polish poop. And I never see them "match" for FULL output, where it *might* affect the sound. They obsess about idle matching, but the amp makes no sound at idle. We do want some equality from the teeniest sounds to the MAX, but it does not have to be a "match". Especially in a self-bias amp which can hardly get out of class A.
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Old 5th January 2017, 04:03 PM   #12
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Quote:
Originally Posted by Malcolm Irving View Post
I think Vbias should be R22 ( i1 + i2 )

Also, I think your first two equations need to allow for AC variation in Vk
which affects Vgk ( = Vg Vk ) for both triodes. Vg for the first triode caries the input signal and Vg for the second triode is AC grounded.
Hmm, I see: the grid voltage should be Vin(AC) + R21(i1+i2)(DC) and the cathode voltage should be (R21 + R22)(i1+i2). So the difference is more complicated than I thought. Sorry for being so sloppy.

I will call the differential inputs v+ and v- from now on, even though v- is at AC ground in this particular circuit.

Click the image to open in full size.

So the unreasonably large dependence on mu2 and what seemed like a very small AC magnitude for i2 are both simply because I was missing important terms.

Click the image to open in full size.

Okay, so... I think I can see that nothing is seriously broken in the most simplified case. This 12AX7 spec sheet claims nominal transconductance of 1.7mA/V (which I believe I can plug in for k') and nominal amplification factor of 92 (which I believe I can plug in for mu). So I get .6mA/V for k'', or differential gain of around 60, and, umm, "other" gain of 2.

Does this seem correct?

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Originally Posted by PRR View Post
Input is applied to V3a grid? NO, input is always 2 wires! Input is applied from V3a grid to V3b grid's cap to ground. To a first approximation, V3a and V3b split the input signal across their grid-cathode ends. V3a(gk) = V3b(gk). And their plate circuits are equal. So the output signals are equal.

Wait. V3a(gk) = V3b(gk) is incomplete because R21 and friends leak dynamic current which does flow in V3a but not in V3b. How much? A caveman approximation is to find the cathode impedance, 1/Gm, of V3b. We should know that 12AX7 at Fender-like conditions has Gm of 1uMho and thus Rk about 1K Ohms. So there is a 47K:1K split. V3b gets 98% of V3a dynamic current. There is a few-percent difference in output. As our EL84s are liable to be up to 30% apart, and we are not humping for point-oh-oh THD, this is not critical (and not worth a picture full of math). Although there is a "fix" in many Fenders using a similar stage.
I don't think I really appreciated at all that R21+R22 is big and Rk is small until I understood your post (and without thinking about that, seeing why i1 ~= -i2 is much more convoluted...).

Next, I'm trying to understand how nonlinearities affect the shape of the outputs. It seems that any asymmetrical nonlinearities (in the sense of positive peaks being shaped differently than negative peaks) in the response would be greatly diminished by the symmetry of the circuit. If positive peaks were supposed to be pointy and negative peaks were supposed to be rounded, then that raises questions about what would happen if an inverted version of the signal were applied to the inverting input instead. So I don't think this can happen, other than to the <5% "i1+i2" term?

And then there is the potential for nonlinearity that is symmetric about 0. Is there a useful known way to model how it may affect the signal in the 3/2-law range?
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Old 5th January 2017, 05:53 PM   #13
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Quote:
Originally Posted by chaerin View Post
... the grid voltage should be Vin(AC) + R21(i1+i2)(DC) and the cathode voltage should be (R21 + R22)(i1+i2). ...
Sorry if this is just a quibble, or if I am misunderstanding something, but I'm not sure why you have R21(i1+i2)(DC) in the above. I think that voltage term has both a DC and an AC component, i.e. (i1 + i2) goes up slightly when Vg1 goes up.
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Old 6th January 2017, 03:19 AM   #14
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> any asymmetrical nonlinearities ... greatly diminished by the symmetry of the circuit

Yes. If the tail resistor is infinite, distortion is nearly cancelled.

As Malcolm says, "(i1 + i2) goes up slightly when Vg1 goes up", so linearity is somewhat better if input is balanced.

> nonlinearity that is symmetric about 0

To a first order, the rise of Gm on one side is cancelled by the drop of Gm on the other side. Typically Gm goes up slower than it goes down, so there is some 3d order compression, quite small.

Hasn't the long-tail been beaten to death? Extensive analyses have been published. Carry your salt- some are flawed. Here is a bibliography from 1947:
http://www.americanradiohistory.com/...=%22schmitt%22

Broskie, Merlin and Kuehnel have writ-up the long-tail. It was a standard page in 1950s EE textbooks because most op-amps used it. Parallel studies have been done on the BJT long-tail, the foundation of all chip opamps. There's no need to start from scratch.
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Old 6th January 2017, 05:51 AM   #15
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Originally Posted by PRR View Post
...studies have been done on the BJT long-tail, the foundation of all chip opamps.
The BJT analysis is considerably simpler, actually; the differential in / differential out voltage transfer function is simply the tanh() function, the hyperbolic tangent of the input signal voltage. (Assuming perfectly matched devices, and a constant tail current.)

Because of the cancellation between the exponential transfer functions of the two individual BJTs, this transfer function is linear to first order for small voltage excursions - much more so than one individual BJT.

And, as PRR says, the first distortion term is cubic, so in principle, there is no 2nd harmonic distortion. In fact, all even harmonics disappear from the output. Only odd harmonics are present. Much like the hypothetical perfectly balanced push-pull output stage.

Do a Taylor Series expansion on the tanh() function, and this result drops right out.

-Gnobuddy
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Old 6th January 2017, 12:22 PM   #16
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Originally Posted by Malcolm Irving View Post
Sorry if this is just a quibble, or if I am misunderstanding something, but I'm not sure why you have R21(i1+i2)(DC) in the above. I think that voltage term has both a DC and an AC component, i.e. (i1 + i2) goes up slightly when Vg1 goes up.
My thinking was that R19/R20 are too big to pass AC signals to the grid because C11a/C12 act as a short circuit. So any dependence on (i1+i2)(AC) in the grid-cathode voltage would be due to variation in the cathode-ground voltage, making the grid-cathode voltage (Vin(AC) + R21(i1+i2)(DC)) - ((R21 + R22)(i1+i2)) in total.

Quibbles are super welcome. I didn't study EE in school, so if I write something that is wrong on a technicality, it's likely it's because I actually don't understand it.

Quote:
Originally Posted by PRR View Post
> any asymmetrical nonlinearities ... greatly diminished by the symmetry of the circuit

Yes. If the tail resistor is infinite, distortion is nearly cancelled.

As Malcolm says, "(i1 + i2) goes up slightly when Vg1 goes up", so linearity is somewhat better if input is balanced.

> nonlinearity that is symmetric about 0

To a first order, the rise of Gm on one side is cancelled by the drop of Gm on the other side. Typically Gm goes up slower than it goes down, so there is some 3d order compression, quite small.

Hasn't the long-tail been beaten to death? Extensive analyses have been published. Carry your salt- some are flawed. Here is a bibliography from 1947:
http://www.americanradiohistory.com/...=%22schmitt%22

Broskie, Merlin and Kuehnel have writ-up the long-tail. It was a standard page in 1950s EE textbooks because most op-amps used it. Parallel studies have been done on the BJT long-tail, the foundation of all chip opamps. There's no need to start from scratch.
I checked out Broskie's web site, and it seems quite interesting. I am already looking into getting the Merlin and Kuehnel books.

As for tube-era literature, is there a good source of it online anywhere? I imagine there are a lot of expired copyrights, but I don't find much when I search for it.
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Old 6th January 2017, 01:09 PM   #17
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This is the library:
Technical books online
You'll love them, there is a lot of math in there. I think in pictures so I don't, much. Pages and pages on LTP, Pierce oscillators, astable multivibrators and such, devices you can buy now four for a dollar. Or find them on the curb left out for the trash pickup.
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Last edited by indianajo; 6th January 2017 at 01:11 PM.
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Old 6th January 2017, 08:59 PM   #18
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Originally Posted by chaerin View Post
My thinking was that R19/R20 are too big to pass AC signals to the grid because C11a/C12 act as a short circuit. ... .
OK. I see what you mean now. Thanks.
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Old 6th January 2017, 11:58 PM   #19
PRR is offline PRR  United States
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Here is a 1939 audio paper reviewing the common splitter/driver schemes, and describing the long-tail as "new". 1.8MB PDF

Here is a scan of Cathode Phase Inversion, O H Schmitt, Journal of Scientific Instruments, Volume 15, Number 3, March 1938. 250KB PDF The scan is not great. You can buy the article from the current copyright owner, but 20 GB Pounds! ($33)

Otto did a bunch of stuff.
https://en.wikipedia.org/wiki/Otto_Schmitt
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Old 7th January 2017, 12:27 AM   #20
PRR is offline PRR  United States
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> Merlin and Kuehnel books.

I encourage supporting authors; but both have extensive "free tease" websites. Have you found them?

If you really love math, ask Kuehnel for the first edition of 5F6a. He solved the whole thing with a matrix method I had never seen, known but impractical in the 1950s when a "computer" was a grad student.
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