= Cockcroft-Walton multiplier model
{c}
{title2=$V_{\mathrm{ideal}}=2NV_p$}
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,
$$
V_k=2kV_p,\qquad k=0,1,\ldots,N,\qquad V_{\mathrm{ideal}}=2NV_p.
$$
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 $0\leq\lambda<1$. 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 $j$, then
$$
d_{j+1}=\lambda(d_j+a),\qquad
d_+=\frac{\lambda a}{1-\lambda},\qquad
d_-=\frac{a}{1-\lambda}.
$$
Here $d_+$ and $d_-$ are the equilibrium deficits just after and just before replenishment. Linear discharge between events gives the average output and peak-to-peak variation
$$
\overline V_{\mathrm{out}}=2NV_p-N\frac{I}{fC}\frac{1+\lambda}{2(1-\lambda)},\qquad
V_{\mathrm{pp}}=N\frac{I}{fC}.
$$
For 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 $I\to0$ recovers the ideal ladder; increasing $C$ or $f$ reduces droop in this model. As $\lambda\to1$, 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 $2NV_p$. 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>.
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