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Bootstrapped Cathode Coupled Amplifier

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The gain should still be [Plate load]/[unbypassed cathode resistance]. Assuming a high enough mu anyway. Basically, it's the same gain equation as a regular grounded cathode. The input impedance is just much higher.

~Tom

Is that the quick and easy equation? I've always used:

A = (u * Plate load) / (Plate Load + anode resistance + Cathode resistance*(u + 1))

In either case, I get a similar answer for my values:

Tube: 12AX7
Plate Load: 100K
Cathode Resistance: 1.5K + 10K
A ~= 8
 
A circuit diagram would help ensure that we are all talking about the same circuit. The OP in post 1 seems mention two quite different circuits: cathode coupled (first sentence) or cathode degeneration (second sentence)?

Boostrapping the grid resistor raises input impedance but has almost no effect on gain.

Assuming we are talking about cathode degeneration, post 3 gives the correct formula for gain. An alternative view of that is to regard valve mu and Ranode/Rcathode as two separate gain calculations, then treat them as though they were resistors in parallel: so gain = mu*(Ra/Rk) / (mu + (Ra/Rk) ). However, Rk should strictly be the external cathode resistor plus 1/gm (i.e. the 'internal' cathode resistor).
 
A circuit diagram would help ensure that we are all talking about the same circuit. The OP in post 1 seems mention two quite different circuits: cathode coupled (first sentence) or cathode degeneration (second sentence)?

Cathode degeneration is a new term for me and perhaps why I could never find the proper circuit when searching bootstrapped cathode coupled amp.

Attached is the schematic from Fun With Tubes.
 

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That is a grounded cathode amp, with cathode degeneration and grid resistor partial bootstrapping. With equal cathode and anode resistors it becomes a cathodyne phase splitter.

With smaller (or zero) anode resistor and output taken from the cathode it becomes a cathode follower.

The cathode coupled amp has two valves with cathodes connected together, rather like the long tail pair. I sometimes call it a short tail pair.
 
Putting a resistor in series with the cathode bypass cap is an alternative way of providing cathode degeneration.

To calculate the frequency response you need to use circuit theory to arrive at an expression for the net (frequency dependent) impedance at the cathode, and then use that in the expression for gain. What you get is an LF shelf cut. The corner frequencies are proportional to 1/C, but the exact relationship with the resistor values depends on gm too so the algebra can be a bit messy. To some extent it depends on the anode resistor too, although this is often overlooked.
 
Is that the quick and easy equation? I've always used:

A = (u * Plate load) / (Plate Load + anode resistance + Cathode resistance*(u + 1))

Yeah. It's the quick and dirty equation. In practice, the gain will be slightly lower than Rload/Rcathode as the gain of the tube isn't anywhere near infinite.

I tend to take an intuitive approach to circuit design. I make a couple of quick back-of-the-envelope calculations to get started and throw it into a simulator for the finer detail. I follow that up with the final optimization in the lab. For most work, the quick & easy equations get you close enough.

~Tom
 
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