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Simple Tube Gain Calculations:By Steve Bench
The simple (the way my mind works) way of determining stage gain for either triodes or pentodes (or anything else, for that matter) is an extension of the notes you've listed. Namely, regardless of whether the cathode is bypassed or not, stage gain is simply the effective plate circuit resistance (including plate resistance, load resistance etc) divided by cathode circuit resistance (where transconductance is translated to cathode res).
1. 12AX7 circuit, biased at 2 mA, with a plate load resistor of 100k, next stage grid resistor 470k. Cathode biased using 1k bypassed with a big capacitor + 470 ohms additional unbypassed resstor. (At this point the tube rp is about 62k and the gm is about 1600 umhos).
Effective Plate resistance = 62k, paralleled by 100k, paralleled by 470k, or about 35.4k.
Effective Cathode resistance = 620 (1/gm) in series with 470 ohms, or about 1090 ohms. (bypassed part of cathode is treated as 0 ohms).
Stage gain = 35400/1090 = 32.4.
gm = 3000 umhos, plate resistance 260k.
Plate load resistor = 220k.
Next stage grid resistor = 470k.
Cathode resistor is not bypassed at all, and is 1.6k.
Solution: Effective plate resistance = 260k paralleled by 220k, paralleled by 470k, or about 95k.
Effective Cathode resistance = 333 (1/gm) + 1600 or about 1933 ohms.
Stage gain = 49.1.
Note that if the cathode resistor were bypassed, the stage gain would be about 285 (95000/333).
3. Differential amplifier (phase splitter etc) made from a 12AX7.
The plate load resistors are 150k each, and the next stage grid resistors are 220k. The cathodes are tied together and fed from a 2 mA current source.
What is the gain from either input to either output?
Solution: Effective Plate Resistance = 71k (plate resistance for these bias conditions) paralleled by 150k, paralleled by 220k or about 39k.
Effective cathode resistance. The current source is assumed to have a very high resistance, so may be neglected. Each cathode is "looking at" the other cathode. The gm under these bias conditions is about 1400 umhos (710 ohms). Therefore the cathode resistance is 710 + 710 or about 1420 ohms.
Stage gain = 39000/1420 = 27.5. Note that the differential gain (one input to both outputs) is double this (55), since equal and opposite output is provided by each plate.
This method also works for sand state devices as well.
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