tube line-output stage with "feedback through transform

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It's not a new idea. McIntosh did it with their famous "unity-coupled" circuit. Look around for literature about their amps and how they worked.

Here's a couple to start:

http://www.pat2pdf.org/patents/pat2477074.pdf

http://www.berners.ch/McIntosh/Downloads/MC275CE_own.pdf
 
This is feedback using a tertiary winding. Mc Intosh's "Unity Coupling" is a totally different topology using a trifilar-wound output xfmr to float the tube's relationship to the xfmr by placing an identical winding in the cathode circuit and using cross-coupling of the screens. This requires a Very fancy high voltage, bootstrapped driver circuit!

It solves the problem of the unused side of a winding in class ab being de-energized during the opposite side's conduction and creating a back emf from the de-energization. Or DC asymmetry in the xfmr. It keeps the opposite side "tensioned" during its partner's duty cycle.
 
> line stateges instead of power amps

There is no difference, except scale. A Mac makes a fine VERY BIG line-driver. Line drivers are often used to drive headphones, and I have driven speakers with +36dBm line-amps.

> how would one caluclate the transformer winding ratio

The total plate+cathode turns to the load turns should reflect a good load for the tube.

The ratio of cathode turns to total turns affects your negative feedback, but also your grid drive voltage.

For low-Mu triodes like 2A3 and 6EM7, your grid drive voltage is already uncomfortably high for a driver working on the same power supply as the output tube. The plate peak output voltage for a triode amp is typically 2/3 to 3/4 of supply voltage, say 0.7. Gain is typically 0.6*Mu. Good power triodes have Mu near 5. (Higher Mu means less power.) So the peak grid drive voltage is (B+)*0.7*5*0.6= (B+)*0.2. The peak undistorted output of a voltage-amp tends to be 20% of B+. We see that we can barely smack a M=5 power triode to full output, we can't be wasting drive voltage in cathode feedback.

Power pentodes break the relation between gain and power. The Mu of interest is the Screen grid's Mu. For audio power pentodes this ranges from 8 to 18. So it is reasonable to add some cathode feedback. However even if Mu were infinite, if cathode winding were 20% of the total, we could not drive to full power. In real situations we end up with maybe 6dB of local feedback, and considerable theory suggests this sounds worse than zero feedback or high feedback.

However a pentode does not like having its cathode bobbed up and down unless the screen voltage bobs the same way. Most simple ways of doing this are problematic.

Mac puts half the total load in the cathode. This gives lots of feedback, but needs heroic driver and screen systems. Actually there is an elegant trick for the screen, if it can live at the same DC voltage as the plate. Mac's first driver was transformer coupled, which gave enough drive, but (since the local feedback is not enough) complicates the overall feedback loop. The transformerless driver has a bit more bandwidth but is bootstrapped like a spider in her web.

No, build your line amps like a conventional loudspeaker amp. Feedback from the output back to the first volt-amp tube. Since you want a floating output for the load, you want a tertiaty winding for the feedback. For loudspeakers we normally run a 20:1 resistor divider between the speaker impedance winding and the input stage; as long as we are making a dedicated winding, we may as well use only as many turns as we need. For example: want gain of 40dB or 1:100 from first grid to 600 ohm winding, say the 600 ohm winding works out to 1,000 turns to give desired bass response, the feedback winding can be 10 turns. Since this delivers nearly no power, it can be a thin-gauge winding. If you find a tranny with the right plate to load ratio, on which you can see the winding, and the bobbin is not stuffed to the brim, you can probably sneak on 10 turns of #32 wire for your feedback winding.
 
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