Loop gain T, β, R1/R2 divider, AC bypass cap, ripple atten in dB.
Module 04 locked Vout against mains drift, but Zout is still 1/gm and ripple atten is still in the tens of dB. Adding an error amplifier closes the loop: every change in Voutgets amplified, inverted, and pushed back into the pass tube's grid. The math is the same as op-amp feedback — only the tubes are bigger.
Ripple attenuation follows the same factor. A loop gain T = 50 cuts Zout by 34 dB and ripple by 34 dB. With Aol = 100 and a β = 0.5 divider (Vref = Vout/2), T = 50 is your starting point.
Σ DerivationSeries + error amp (classic) — closed-loop math
The R1/R2 divider samples V_out and feeds it to the error amp's grid. The error amp's cathode sits on V_ref. Any difference (V_out·β − V_ref) is amplified by A_ol, inverted at the pass tube's grid, and counter-acts the change.
Feedback fraction (DC, no AC bypass):
Open-loop gain of a triode error amp in CF/cathode-input mode:
Loop gain:
Closed-loop output impedance:
Ripple attenuation (dB):
Line regulation (DC):
ConceptThe AC bypass cap trick
Plant a 100 nF cap across R1 (upper divider) and you decouple the AC and DC operating points: at audio frequencies the cap is a short, β jumps from 0.5 to 1, and Tac = Aol regardless of the DC ratio. Ripple atten gains another 20 dB or so for free.
ConceptGrounded-cathode variant — more loop gain
Swap the common-cathode triode (12AX7) error amp for a pentode (EF86). The cathode sits on Vref (clamped by the VR tube — hence the historical name "grounded cathode") and the grid takes Vfb. Open-loop gain rises to Aol ≈ 80–150 — the pentode is no longer capped by µ like the triode (≈ µ ≈ 60–100) — at the cost of one extra pole to compensate.
WarningHigher loop gain → less phase margin
Every dB of loop gain you add pushes the crossover up and erodes phase margin. A T of 100 might give beautiful Zout, but if you have a 47 µF Cload on the output node, you'll meet it as ringing or oscillation. See chapter 8 for compensation.