Nick's little project

[NickK]

Not built but currently in the design stage - a valve headphone amp.

The phones I'm targeting are being able to support a 32ohm impedence headphone rather than the normal 600ohm that tube users use. The unofficial industry standard seems to be settling on 120ohms - the phones I have are 55ohm AKG420mk2s.

Low impedances use lower voltages (smaller voltage swings peak-to-peak) and higher current with solid state amplification it's easier to run more current than attempt to support higher voltage. This is the reason there is a trend to low impedance with iPod etc.. This also means the headphone coils are heavier to cope with the current (a larger cross section needed to prevent heating up).

High impedance use higher voltage (larger voltage swings peak-to-peak) and small current. These originally appeared as valves (tubes) can support big voltages but can't really supply current. Hence the headphones with high impedance are made up of thinner coils as less current is used.

First thing todo is to understand the headphones - 55ohm stated, in fact this impedance depends on the frequency, and the phones actually measure between 55 and 200 with a 84ohm average. However you need to support the lowest impedance. These are for each headphone channel, these values are all 104dB/Vrms sensitivity targeting 110dB max:

32ohm = 2Vrms drawing 62.5mA so 125mW of power.
55ohm = 2Vrms, 36mA so 73mW
84ohm = 2Vrms, 24mA so 48mW
200ohm = 2Vrms, 10mA so 20mW

Note the rms value - 2Vrms is 5V peak to peak (ie +2.5V to -2.5V). Also the fun that is milli-amps (or 1mA=0.001A). The valves I'm using - the pre-amp only outputs 10mA max and the output tube delivers 110mA max in an amplifier situation. This is why you'll see a lot of valve amps with a MOSFET output to deliver current.

The next bit is that valves' input needs to be in a certain range - to be amplified to the right level. That amplification factor means you need a couple of stages to boost the voltages. The input of a valve is all about voltage and not current.

So you end up with a chain that needs to be designed. However the characteristics of a valve aren't entirely linear, and have a number of variables - plate voltage (the voltage applied across the valve), grid voltage (this is a negative voltage to the cathode that controls the valve) and the cathode (which provides the electrons)

I won't get into the workings of all the different types of valves but for the ones I'm thinking of using - triodes - these work by applying high voltage across the cathode and plate, in the region of 100-300V, then supplying a signal (+/- voltage wave) to the gid. In reality there are a couple of addition points:
* current bias - for constant current flow (CCS) is an improvement over standard simple valve setups to reduce distortion.
* tube bias - this is the negative voltage that carries the signal to control the valve. The more negative the more the valve is shut off - with 0V being no control and normally a bad thing.

So working back from the headphones - I need to look at valve data sheets and understand the the characteristics to get a minimum distortion, ensure we get low harmonics that are even rather than odd plus sit in a range of being the most linear. The subject for the next post.

 

[BF-Bert]

I may be wrong but I was always told tube amps are better with high impedance headphones.

 

[NickK]

I may be wrong but I was always told tube amps are better with high impedance headphones.

The reason is simply down to the higher voltage used with high impedance being easier for valves.

This is what makes this project interesting - it'll be an output transformerless (OTL) amp. so the valves rather than a step down output transformer will need to provide the current. The design decision then becomes what output valves support that current, or, design a set of parallel output valves to provide more current. The down side with parallel tubes is that the grid input capacitance then increases which then causes it's own problems. It's the reason that hybrid valve amps are more popular - the valve makes the sound and then the power/output section is a solid state MOSFET.

Whilst designing I'm using LTSpice to model the circuit - this allows me to put in a 1KHz tone and see how the system behaves, including voltages and current.

 

[BF-Bert]

The reason is simply down to the higher voltage used with high impedance being easier for valves.

This is what makes this project interesting - it'll be an output transformerless (OTL) amp. so the valves rather than a step down output transformer will need to provide the current. The design decision then becomes what output valves support that current, or, design a set of parallel output valves to provide more current. The down side with parallel tubes is that the grid input capacitance then increases which then causes it's own problems. It's the reason that hybrid valve amps are more popular - the valve makes the sound and then the power/output section is a solid state MOSFET.

Whilst designing I'm using LTSpice to model the circuit - this allows me to put in a 1KHz tone and see how the system behaves, including voltages and current.

The trouble with valves is the high output impedance which ideally for your headphones would be in the 10 - 20 ohm range, maybe even lower.

Great project though, will be keeping an eye on it.

 

[NickK]

The trouble with valves is the high output impedance which ideally for your headphones would be in the 10 - 20 ohm range, maybe even lower.

Great project though, will be keeping an eye on it.

That too - the valves I'm eyeing up are low plate resistance, which mean low output impedance which restricts the choices somewhat - which why looking at an 6AS7. The 6AS7 is a bit of a beast, but has a low amplification factor, so needs a decent pre-amp. The current design is a parallelised phase inverter/differential amp driving the output push pull - per channel. So two 6SN7GTB giving headroom and one 6AS7G per channel.

The 6SN7 will take a line in signal, biased to -8V input, a 270-300V plate voltage and running it at 5-8mA to maximise the voltage swing. Max current from a single 6AS7 is 150mA, but 110mA max in push-pull audio amp with a suggested idle of 80mA. I've also switched from a cathode CCS to an Anode/Plate CCD - this provides a reduction in distortion. The advantage of using LTSpice - to play around with setups before committing.

Once I have a basic design - I will still need to build in some 'tunability' given tubes can vary as much as 30% from the datasheet.

On the plus side at least we're not talking 450V on one side of capacitor to the headphones - only 1/2 that

Edit- I forgot to say, I wanted this design also to have a little unique aspect a guitar connection at the back. So essentially I can either play clean or crank it up. Not sure how the headphones will like a cranked up & soft clip distortion but.. that's in my head along with this.

But I hear - Nick, you can't make different valve amp..



That is stunning and I'm thinking of something similar - perhaps a walnut butcher blocked repurposed. A CNC milled plate - this could do two things (a) provide a convection air through the gap for the valves but also - I could fix the CCS current control chips to the underside, creating a heat sink on the top.
I would still use a metal case inside to provide the best shielding but I love the power transformer in it's shield peaking through..

 

[NickK]

Been doing *a lot* of research on this. How to reduce the impedance, the impedance matching between stages, etc.

I've started to settle on the theoretical design currently it's looking like this as multiple stages - I just need to set this out in LTSpice to see how it works:

6SN7 input stage (providing line amplification) with a solid state CCS
6SN7 phase splitter - this has changed from a long tailed pair to a straight phase splitter to maintain the balance easier
6SN7 driver - voltage amplification for the 6AS7 if needed - allowing a ramping up of the signal

Output section could be:
6AS7 cathode following push-pull, the push pull configuration provides the best current drive and lower impedance
EL34 cathode follower/mu-follower to the 6AS7. This acts as a CCS and further reduces the impedance without increasing the miller capacitance to the previous driver stage (unlike running parallel valves)

The 6AS7 is running 80mA idle with 110mA max with 130mA limit. The EL34 also is able to run at 120mA but has a higher amplification factor but the follower configuration needs high voltage rails - which is precisely what the 6SN7 needs too. The configuration with Push-Pull also lowers the noise and distortion.

Lastly a global feedback also means the output attempt to match the impedance, causing negative feedback to drive the gain.

Valves tubes typically have two 'valves' (sections) per tube. So in Push-pull you use the upper and lower in the same tube. So one 6AS7, one EL34 per channel, the 6SN7s I could use either shared tubes for the stages (ie one input+splitter and one as driver) .

There's also some 'safety' that's needed during startup as tubes take a few seconds to reach their correct operation (this causes voltages to run amok - especially with cathode biasing). Also it's good practice to get the heaters up to temp before applying high voltages, I can do this as part of a soft start. 6SN7 takes about 10 seconds for this.
The output will have a big output cap - it needs to cope with the high voltage and sized to pass low frequencies (the cap will act as a filter)

 

[Kei]

Looks like it'll be a good project. I will be watching with interest.

Something to think about regarding negative feedback and push pull designs. Push-pull naturally cancels even order harmonics, meaning you are more likely to end up with 3rd & 5th order harmonics being dominant. That said, if it's at a low enough level, it'll be no worse than any good solid state design. If you are aiming for a "valve" sound, push pull coupled with negative feedback will not easily give you that. Negative feedback can be a serious challenge to implement as it can require some effort to get things stable in order for it not to oscillate. Without an output transformer, it shouldn't be quite so hard as the phase shifts are less significant and it's easier to model in LTSpice.

A long tail pair with identical tight tolerance plate resistors and a CCS in the tail forces AC balance which makes life much easier. I've used this on my own valve amp project this year. (after doing it the old fashioned way first time around)

A good manufacturer of toroidal transformers is a polish firm called Toroidy, They make a very good potted transformer with a classy looking cover.

 

[NickK]

Looks like it'll be a good project. I will be watching with interest.

Something to think about regarding negative feedback and push pull designs. Push-pull naturally cancels even order harmonics, meaning you are more likely to end up with 3rd & 5th order harmonics being dominant. That said, if it's at a low enough level, it'll be no worse than any good solid state design. If you are aiming for a "valve" sound, push pull coupled with negative feedback will not easily give you that. Negative feedback can be a serious challenge to implement as it can require some effort to get things stable in order for it not to oscillate. Without an output transformer, it shouldn't be quite so hard as the phase shifts are less significant and it's easier to model in LTSpice.

A long tail pair with identical tight tolerance plate resistors and a CCS in the tail forces AC balance which makes life much easier. I've used this on my own valve amp project this year. (after doing it the old fashioned way first time around)

A good manufacturer of toroidal transformers is a polish firm called Toroidy, They make a very good potted transformer with a classy looking cover.

Awesome - I'll have to have a full read later after work.

The noise cancellation from the PS was also something that's attractive with push pull, although the front end will be class A, it's simply the back end that will be AB1 to give flexibility given the low impedance. With a 50ohm impedance it'll be a 16V 330mA max peak swing (big wattage!). With paralleled output tubes that should be possible for 220mA which will be loud enough. Just need to work through the data sheets.
I've done away with the cathode follower, running both 6AS7 and EL34 output tubes in parallel. My thinking later on is the EL34 gives some opportunity to tine with the other grids.

I do know that a two 6AS7 in parallel into 50ohms does work. I just need the previous stages to ramp up the grid enough. Just needs some decent capacity for peaks in the power supply. I am tempted to run the heaters from a switched mode power supply (with filtering), then use a linear regulated for the main power demands.

Reading up - the LTP works, although it needs balancing, but if the anode resistors aren't balanced you start getting distortion. I found that one side of the signal weaker and then you're into playing around balancing. The design is setup in LTSpice with named network tags for input and output, that way it's easy to replace a section or focus on a section to get it working correctly. That was the reason for switching to a simpler splitter.

The nice thing about the DIY element (and this will be point-to-point wired) is the simplicity to simply change the setup.

 

[NickK]

Playing with the spice model, due to the fact that this is more current rather than voltage based, I'm not sure I need an input and a driver stage. I was getting 110V peak-to-peak out of one stage so if need be that can be added back in - although currently there's no feedback which will reduce the gain.

For example - a 7V peak to peak into the inverter comes out and through the power stage, currently I have 2.5V at 80mA just with that linein into the inverter. Late now to continue playing but I think that should be easy enough to adjust to be 24V and more current.

 

[NickK]

Getting there. The signal is an output voltage and current at a 32ohm load with a 1KHz input. The FFT with the driver seems less clear without - so I have some work to go.



Frequency response is a bit brutal so I think rolling off above 25K will help.


The FFT of 30 seconds and with the driver in it's noisy, even with feedback - not sure why it's also decided to only give me below 1KHz.. seems something is messed up. Also I need to work out the operating points on the tubes a bit better and it may be my handling of AB1 with that tube operating point.



Vs without the driver stage and feedback:



Edit I've added in the input stage now having bypassed it previously - it boosts the final signal as was originally intended, so output is ~ ±16V (32Vp2p) and 600mA.. which will fry the tubes, so some play needed..

Edit2: I've been mapping out the voltages and current for each stage after disabling the feedback and it looks like there's a couple of issues:

a) the phase splitter is outputting a 50.76V p2p in phase and 49.04V p2p out of phase which is a little unbalanced however the signal is clean and no clipping. The current is running between 4.69-7.2mA so nicely in the zone for the 6SN7. The idea of this phase is that it should be zero amplification anyway (55.98V p2p input -> ~50V p2p is ok).

b) the driver section is outputting 92.2V p2p and 88.69V p2p but the voltage shows a clipping and when you look at the current, you can see why - the current drops from 5.5mA to 0mA in the swing, causing the voltage clip.
This looks like a bias issue - mismatching the operating point of the tubes, where it should be capable of running 8mA to 2.2mA for example.

c) the output section then simply amplifies this clipping plus this section still needs a look it's still pushing 200mA per tube, the max should be 110mA. As it is at the moment the output is 4.4W into 32ohm. With the current at realistic levels, this should then drop to 2.2W or there about.. still that's shedloads for headphones - then using a smaller input signal that should then be into the normal realms. Fingers crossed.

Spice is crawling now 3ms/sec processing speed.. on a beefy Mac mini using wine (so threading sucks!)

 

[NickK]

I did a little adjustment, focusing on the biasing.

The input stage I've bypassed, with an 8V peak-to-peak, without gain needed for any feedback just yet, this causes problems. Once the feedback is added and the gain drops this may be needed.

The phase splitter now acts almost at unity - no amplification, but phase split out with a -4V grid bias for the 8V peak-to-peak input and output.

I've also done the same to the driver stage, which feed the output stage. This now takes a 8V peak-to-peak and boosts it to 100V peak-to-peak.

The output stage is still in need of some work - I've focused on the pre-stages.

This is the current performance into a 32ohm headphone load.

Decent frequency performance, I can boost the bass response simply by making the capacitor larger on the phase-splitter (it's acting as a filter with the resistors).


The traces seem decent a tube frying (and headphone igniting) 600mA at 20V. So with a little tweaking that backend will have a vice like grip on the headphones.



Now to the FFT - a 1KHz tone. Getting better. I'm hoping a little that some work on the output stage will help this a little more.

 

[NickK]

So looking more into the impedance piece, this is a very good summary: aria.html

Output impedance = Resistance of the tube plate / (1 + tube gain)
Or another way is = 1 / transconductance

So to get a low impedance, selection of tubes that can provide the current required but have a low output impedance, for example
So a single 6AS7 = 1 / 7000 = 142 ohm output impedance
A single 5998 = 1 / 14000 = 70ohm for example

Then to add to that you can use the topology to reduce further with multiple tubes.

An improved flutterman impedance is plate resistance / 2(1+tube gain), then with doubled tubes the impedance drops. With four 6C33C-B tubes that would be 6.8ohms output (according to the link) in open gain (no feedback). The 6C33C-B IIRC is straight out of a MIG fighter jet.

Lastly on noise: https://www.aikenamps.com/index.php/resistor-types-does-it-matter

 

[NickK]

I been annoyed of getting a DC offset - the system seems decent but I've also been wanting to experiment with Brokesies "Brazilian OTL" this is 6C33C based 8ohm amp that uses mosfets to push beyond the wattage (this gives out something like 64W). Or another way of looking at it - line level input (0.447V peak consumer levels) gives a blistering 30V peak-to-peak if you push more then it'll start clipping the front end but really kicks out current. The nice thing is the solid state only kicks in when needed.

So what if we modified the design from 12AT7/12AX7 with 6C33C running Class A into a 6SN7 / 6AS7 running Class AB1 with mosfets so (a) using less gain tubes and (b) using less current as we're running headphones... I will keep you updated..

Also what if the heaters were running a switched mode power supply with some additional filtering, that way it would be more efficient..

 

[NickK]

I've been playing with the Brazilian OTL circuit;
a) the original Brazilian OTL design had some errata in the values of the resistors and capacitors
b) I swapped out the 6C33Cs straight for 6AS7s (I've played with the bias and it improves things but breaks other bits - these figures are simply a swap)
c) a 5 minute run but not counting the muted first 30 seconds of startup is a good idea.
d) the amount of feedback that this amp uses to keep the gain in check is interesting (the front end will produce a square wave and clip if it's not attached).

Annoyingly the FFT window seems to only show me the first 100Hz, not sure what is going on there.. but still.

N-Period=1
Fourier components of V(output)
DC component:2.77989

Harmonic    Frequency     Fourier     Normalized     Phase      Normalized
 Number       [Hz]       Component     Component    [degree]    Phase [deg]
    1        1.000e+3     1.349e+1     1.000e+0       -0.42°        0.00°
    2        2.000e+3     5.819e-3     4.313e-4       78.52°       78.94°
    3        3.000e+3     2.890e-1     2.143e-2     -168.56°     -168.14°
    4        4.000e+3     1.206e-2     8.936e-4       62.39°       62.81°
    5        5.000e+3     5.380e-2     3.988e-3       -0.42°        0.00°
    6        6.000e+3     6.753e-3     5.006e-4      104.76°      105.19°
    7        7.000e+3     1.473e-2     1.092e-3      -34.57°      -34.15°
    8        8.000e+3     3.097e-3     2.296e-4     -147.64°     -147.22°
    9        9.000e+3     8.660e-3     6.419e-4     -174.08°     -173.66°
Total Harmonic Distortion: 2.185938%(2.207875%)

So the THD is a pretty bad 2.2% at 20.8W full volume and load still ignoring the tube current and speaker level output - 32ohm (each PP side putting out 520mA 20V peak or 10W). Once the updates are done, the bias/tube swaps etc, the output should drop and the current approach something more sensible the THD should drop too.

Edit- sure enough, the anode move-140V to -114 upsets the bias, now trying a slightly different bias for the AS7 so the current is in a better area. Also I can segregate the ground returns.

 

[Psycho Sonny]

I have a box full of valves I probably spent £200 on but never used a decade ago.

Probably worth even more now. I really should eBay them.

 

[NickK]

I have a box full of valves I probably spent £200 on but never used a decade ago.
Probably worth even more now. I really should eBay them.

As long as the barium 'getter' on the inside hasn't turned white, they don't degrade when not used. If they've been hammered then over time the cathode, grids and anodes will degrade. If you have 300B, etc then you could be onto a nice little earner.

 

[NickK]

This guy shows how to create plot diagrams etc for LTSpice:
https://www.youtube.com/watch?v=VV3e_mNQ-dQ

Here's my 6AS7 and 6SN7 plot for the LTSpice model using a random Rload value:


This won't be completely accurate to the valve but it will do for a initial LTSpice model of the amp.

If we look at this diagram a different way - symmetrical non-linear distortions between the input (the blue gate voltage) and the output (the axis) give rise to even harmonics. When the output distortions are non-symmetrical you get odd harmonics - the weird curve change near the bottom of the graph.

The red line is the maximum power dissipation from the tube anode plate. The further above that line the faster you reduce the tube lifetime.

The lower the output impedance for the 6AS7 (say 32ohm headphones), the more vertical the green load line becomes - the range of current goes up but the voltage swing reduces.
The higher output impedance for the 6AS7 (say 600ohm) - especially with Class A SET where a anode resistor is used to drive the voltage swing, it relates directly to the Rload green line here, where the higher Rload results in a flatter load line - thus less current swing and more voltage (Vplate) swing.

Next video worth looking at is Dennis' video on class of amplification and how it relates to this diagram - it also shows how the input waveform is mapped from the grid to the output voltage and current:

So you can see why a class A amp outputs more even harmonics vs class B and AB1/AB2 why these result in odd harmonics as the wave uses the load line into bottom of the non-symmetrical non-linear output.

Now the push-pull architecture of the amp uses two sets of waves out of phase by 180degree, so symmetrical differences are cancelled out, however non-symmetrical differences will not be cancelled out, hence we get odd harmonics remaining.

There's a technique where we replace the slope of the load line with a flat horizontal line (the tree line in my diagrams) - this is call constant current source (CCS) it basically means we keep a baseline constant current at the side - making the load line flat. This helps in situations where the peak-to-peak input voltage results in a output that starts entering the crappy non-symmetrical lines under about 5mA (for the 6SN7). This helps reduce bad distortions.

So the next step is to design a chain of these diagrams, from the input line in to the headphone output. We match the voltage swings from the output into the next stage inputs. The input line in becomes the blue lines (grid) of the first valve, the output on the horizontal voltage becomes the output that feeds into the grid of the next valve and so on. Then we split and we use the modified AB1 approach with the biasing to then drive the output.

The output stage will be the interesting one - this really moves away from 'voltage' amplification to current driving for the low impedance headphones however to support this we need to use designs that reduce impedance - such as parallelising the tubes (ie resistance in parallel reduces the total resistance presented) and other techniques.

 

[NickK]

So let's have a look at the waveform and the AB1 amplification that John Broskie does with his Brazilian OTL does:



So you can see the Class A if the amplitude sits below a certain level. When the wave amplitude hits the wall of 0mA at the bottom, the bottom tetrode (the second of the two acting in push-pull) takes over. Finally if the current pulls more, then there is the NPN/PNP network that takes over for some class C action simply following the valve in class AB but providing more current as a "follower". They also serve to act as a resistance/non-resistance (playing around with the transconductance on the crossover) if I read that right but that I will get into later.

So basically the system acts like a push pull class A, then into Class AB1 then into Class AB1+Class C depending on the load and the music playing. Lastly - it may look like there's lots of gain (x38 for each 6SN7 stage, 0.98 for the 6AS7) but there will be quite a lot disappearing in the feedback loop which we want to drop the output impedance further. This could be 20dB of feedback so ultimately reduces the overall resulting gain of the amp. If I want a volume control, I will put it between the input stage and the splitter which should help the signal-to-noise ratio at the expense of higher voltage (but not current) resistors.

Brazilian OTL: https://tubecad.com/2018/03/blog0416.htm and some errata/update here: blog0417.htm

If you look at the gain of the tubes chained together it looks nuts but the gain is (a) used to bring mW to tens of watts and is used to drive the 6C33C tube to give big current.. a headphone amp doesn't need quite that.
The Brazilian as a gain chain like this: 12AT7 (x60) -> 12AX7 (x100) -> ECC99 (x22) -> 6C33C (x2.7) = x60x200x22x2.7 = 356,400x, where 2V becomes 32V and about 2A or 64W into 8ohm speakers

The Nick version of that is looking something like: 6SN7 (x38) -> 6SN7 (x38) -> 6SN7 (x38) -> 6AS7 (x1.7 but x0.98 in reality) = 93,282x, where 2V becomes about 2-16V and about 70mA.. in total 200mW into 55ohm headphones. If needed I could slot in a 6SL7 for a 100x but it can cause saturation and clipping very easily so it depends the amount of feedback.. The majority of the problem we're solving is not voltage/power but current and low impedance.

The important point here is the bias of the grids for each tube as shown above. There's a couple of ways todo this
a) cathode where the bias tracks the wave form by being linked to the cathode - it tracks the tube itself, down side is that it can sound compressed and the tube may unbalance itself in relation to the other tubes in the push pull making a kink in the crossover and thus adding distortion.
b) fixed bias using resistors from the rails - the Brazilian does this (it's part of the driver section with the 7.5 and 11.5K resistors from the rails) which then directly output into the grid of the output tube. There's a diode on the driver section and one on the input for two reasons - stopping the grid from going positive above the cathode on startup/operation and keeps the bias in alignment with the cathode during that time. Once the rails and DC servo are up and running they set the bias for the driver section. The rest seems fixed bias (input and LTP phase splitter).

So with this the next steps are:
a) Determine the 6AS7 load line and resulting bias/voltages to give the headphone current and voltage (with Rload/Rcathode) that follow the wave form for Class A/AB1.
b) Determine the driver stage 6SN7 load lines and bias voltages
c) Determine the phase splitter 6SN7 load lines bias voltages

 

[NickK]

Been doing some proper maths calculations.

The AKGs are 55ohm but using 32ohm and 200mW, using 10x output to reduce distortions means 2W. Which equates to 8V peak at 250mA into 32ohms.

For the impedance, 32ohms is low and if there’s 4 devices attached to the output for the push pull (1/R=1/R+...) so the current is 1/4 on each, keeping the same 8V means each sees 4x the impedance.
The output impedance then I believe is reduced too (8ohm).

With a push pull - two triodes on each side when on each side of the wave are conducting, but in AB1 the bias overlap causes all 4 to conduct until the voltage moves outside results in a difference between 2 and 4 devices causing a problem. The addition of the NPN and PNP transistors then kick in beyond the bias overlap so 4 devices are always conducting and so the impedance stays the same throughout the waveform.

So the amp as a system has to go from 445mV line in to 8V which Gain = u * Rload / (Rload + dV/dI) = Vout/Vin.
Working through this results in a need for u=17.9 + 2 for the 6AS7 which only needs 4.004V with each device running 63mA. So the amp running open loop is ~25dB.

 

[NickK]

So.. my thinking of the calculations..

Impedance is basically the voltage out with a voltage divider.


Then by using parallel'd devices the current is 1/4 from each, and reduces the impedance by 1/4 for the same current:

So if I have 4 devices conducting in parallel the impedance is 1/4Z.

In addition negative feedback works to reduce the relative source impedance:

This simply increases current due to the difference from the expected output voltage. As V=IR this pumps the voltage again..

So back to the calculations. This is ignoring frequency but is good enough for general figures (impedance varies with frequency depending on the headphone). As impedance increases the voltage required goes up but the current goes down for the same power, however if we reduce impedance we need more current.

A 32ohm headphone, with 200mW needing a capacity of 10x for example to reduce distortion, gives 2W. So P=I^2R, rearranging = sqrt(2/32) = .250A or 250mA. V=IR, 0.250*32 = 8V peak. Or 250mA at 8V into 32ohm.

Now if we're 1/4 impedance now by paralleling the output devices, then we're into 8ohms. sqrt(2/8) = 500mA at 4V into 8ohms. However if we use negative feedback and use the rule of 8, 32/8 = 4ohm source impedance into our 32ohm headphones, then we should be able to calculate the negative feedback required to drive up the current to reduce the apparent impedance. Into 4 ohms, we'd need sqrt(2/4) = 707mA at 2.82V. (remember our triode valves max out at 120mA each in class A!).

Gain = Vout / Vin. So we can take the different of the voltage out and the output we have here - in other words how much do we need to get from 2.82 back to the 8V (I assume it's the full amount but it could be 4V). So the gain needed to pump up the error in voltage would be 8/2.82 = 2.837x to correct the difference in voltage. This negative feedback difference will then cause more current to flow to support the 4ohm impedance.

So if we take the gain required to amplify 447mV to 8V output first this x17.897. Then we need additional gain to cover the voltage needed into feedback into the negative feedback we can calculate it (https://www.electronics-tutorials.ws/systems/negative-feedback.html) .. but I have dinner and the Mrs is complaining I'm sat here.

In short I think I need about 20x possibly 50x but I need to finish the calculation.