 Cockcroft-Walton multiplier model

Define a stage as one storage increment of $2V_p$, with $V_p$ the alternating input's peak electric potential difference relative to its reference, not its peak-to-peak value. Assume ideal rectifiers, no load, no leakage or stray capacitance, a stiff alternating source and enough cycles to reach equilibrium. For $N$ such stages,The factor two follows from adding the input's next opposite excursion to an already stored peak. The subsequent series increments add. This ideal argument concerns equilibrium voltage; it supplies neither the charge-up time nor the useful load electric current.

To isolate the effect of drawing charge, consider the following original reservoir model. Each storage stage has equal capacitance $C$; a constant output electric current $I$ removes $\Delta q=I/f$ between replenishment events of frequency $f$. Every series reservoir loses this charge, hence its voltage falls by $a=I/(fC)$ per interval. Suppose replenishment removes a fixed fraction $1-\lambda$ of each reservoir's accumulated deficit, with $0\le \lambda <1$. This is an explicit phenomenological pump rule, not a solution of a particular multiplier's rectifier network. If $d_j$ is the deficit immediately after event $j$, thenHere $d_{+}$ and $d_{-}$ are the equilibrium deficits just after and just before replenishment. Linear discharge between events gives the average output and peak-to-peak variationFor chosen dimensionless settings $N=3$, $a/V_p=0.020$ and $\lambda =1/2$, the output varies between $5.88V_p$ and $5.94V_p$, averaging $5.91V_p$, instead of the unloaded $6V_p$. Starting with zero deficit, the recurrence approaches equilibrium geometrically with factor $\lambda$. Taking $I\to 0$ recovers the ideal ladder; increasing $C$ or $f$ reduces droop in this model. As $\lambda \to 1$, effective replenishment fails and the small-deficit assumption fails too.

Real multiplier stages have unequal charge-transfer burdens; their mean deficits need not be equal and depend on topology, input impedance and parasitic capacitance. Thus this reservoir model explains droop and voltage variation without claiming their exact dependence on stage count for a real Cockcroft-Walton generator. In particular, adding more stages cannot be judged solely by the unloaded $2N{V}_p$. For a beam sampling a slowly varying output, the electrostatic acceleration energy gain follows that instantaneous potential drop, so voltage variation also produces a spread of beam kinetic energies.