Valve
Circuit
Bias resistor + tail resistor (AC)
How it works
The long-tailed pair (LTP) splits one signal into two antiphase signals for a push-pull output stage. Two triodes share a common "tail" resistance. Grid 1 receives the signal; grid 2 is held at AC ground by a capacitor. As V1 conducts more, the shared cathode rises, which turns V2 off, so V2's plate moves in phase with the input while V1's moves in antiphase.
Modelling each triode as µ·Vgk in series with rp and solving the node equations gives (per volt of input):
Because the tail is not an ideal current source, a little signal "leaks" into it and the non-inverting side is always slightly weaker. The traditional fix, seen in Fender's 82 k / 100 k pair, is to make Ra2 a little larger. A longer tail (bigger Rt) or higher-µ valve improves natural balance. The red dashed trace shows the residual imbalance signal.
The calculator finds the exact Ra2 for perfect AC balance by bisection.