When you need a few kilovolts from a modest AC source — driving a CRT, a Geiger tube, a photomultiplier, or an electrostatic precipitator — winding a giant step-up transformer is often the wrong answer. The Cockcroft-Walton (CW) multiplier gives you the same output using nothing but small diodes and capacitors. Cockcroft and Walton famously used it in 1932 to split the lithium atom with only an 800 V transformer feeding a stack that reached 800 kV.

The topology is a ladder. Each stage consists of two diodes and two capacitors. On the negative half-cycle, the "pump" (coupling) capacitor charges through one diode to the peak input voltage V_p. On the positive half-cycle, that stored charge is transferred through the second diode into a "smoothing" capacitor stacked on top of the previous stage's output. Each additional stage adds another 2·V_p to the DC output. An N-stage multiplier fed by a sine wave of peak V_p produces an ideal no-load output of V_out = 2N·V_p.

The catch is loaded behavior. Every cycle, a load current I_L drawn at frequency f steals charge from every pump capacitor. The voltage droop under load is approximately:

• Ripple: ΔV_ripple ≈ (I_L·N)/(f·C)
• Sag from ideal: ΔV_sag ≈ (I_L/(f·C)) · (2N³/3 + N²/2 − N/6)

Notice that sag grows as N-cubed. This is why CW multipliers work brilliantly for high voltage at low current (µA to low mA), but fall on their face if you try to draw serious current. Doubling the stage count quadruples-plus your voltage drop under load.

Concrete example: Suppose you want ~4 kV to bias a photomultiplier from a 500 V_p transformer at 60 Hz. You need N = 4 stages (4·2·500 = 4000 V ideal). At a 10 µA load with 22 nF capacitors: ripple ≈ (10 µA · 4)/(60 · 22 nF) ≈ 30 V. Sag ≈ (10 µA/(60·22 nF)) · (2·64/3 + 8 − 4/6) ≈ 380 V. Real output: ~3.6 kV. Bump the frequency to 20 kHz (drive it from a switching inverter) and both ripple and sag drop by more than 300×.

Rule of thumb: Cockcroft-Walton is great for stages up to N ≈ 6–8; beyond that, either raise the drive frequency or increase the capacitance dramatically. Voltage-rate the caps and diodes for at least 2·V_p each, with margin — the pump caps swing through 2·V_p every cycle. Ultra-fast diodes are mandatory above ~10 kHz to avoid reverse-recovery losses cooking the stack.