Workbench / Gain Stages

Cathode Follower

Gain, output and bootstrapped input impedance.

Valve

Preset

Circuit

Cathode load Rk

External load RL

e.g. tone stack or next stage

Grid return

Grid leak Rg

Bias resistor Rb

Bootstrapped cathode follower

Voltage gain

0.932

-0.61 dB

Output Z

424 Ω

≈ 1/gm = 453 Ω

Input Z

10.4 MΩ

10.4× Rg

Transconductance

2.21 mA/V

Voltage gainlimit µ/(µ+1) = 0.944

Engineering Notes

How it works

The cathode follower takes its output from the cathode instead of the plate. All of the output voltage is fed back in series with the input (100% negative feedback), so the gain is slightly below unity, but output impedance becomes very low. It is the classic way to drive a tone stack (Marshall), a long cable or an effects loop.

A = µ·Rk' / (rp + (µ+1)·Rk') where Rk' = Rk ∥ RL

Zout = rp/(µ+1) ∥ Rk ≈ 1/gm

Even with an infinite load, gain can never exceed µ/(µ+1), so high-µ valves get closer to unity.

Bootstrapping

With a large cathode resistor, the cathode sits at a high DC voltage. Returning the grid leak to a tap between a small bias resistor Rb and Rk lets the valve self-bias properly, and as a bonus the bottom of Rg moves with the signal. Very little signal current flows through Rg, so it appears much larger:

Zin = Rg / (1 − A·Rk/(Rk + Rb))

This is how a humble 1 MΩ resistor can present a 5–10 MΩ input impedance.

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Cathode Bias & Bypass

Long-Tailed Pair