Step charging accumulation

Does the cell voltage climb over a gated pulse train, or fall back between pulses?

The formula

T = R \cdot C r = e^{- p / T} U = v \cdot \frac{1 - r^{n}}{1 - r}

LaTeX

T = R \cdot C \qquad r = e^{-p/T} \qquad U = v \cdot \frac{1 - r^{n}}{1 - r}

Work it out

v Voltage per pulse V

What one pulse would add to an uncharged cell.

n Pulses in the gate

How many pulses arrive before the gate closes.

p Pulse period µs

Time between pulses within the burst — one over the drive frequency.

C Cell capacitance nF

The cell as a capacitor.

R Leakage resistance kΩ

Resistance across the cell — the path the accumulated charge drains through. This is the water, and it is the whole question.

Multiply the leakage resistance by the cell capacitance for the discharge time constant. In SI: ohms times farads gives seconds.
Divide the pulse period by that time constant and take e to the minus that. The result is the fraction of the accumulated voltage still there when the next pulse arrives — the retention.
The staircase is then a geometric series: each pulse adds its own volts and keeps the retained fraction of everything before it. Summing n terms gives v × (1 − rⁿ) ÷ (1 − r).
Compare that against n × v, which is what perfect retention would give. The ratio is how much of the intended accumulation actually survives.
If the retention is close to 1, the series is nearly linear and the staircase climbs as intended. If it is well below 1, the sum converges on v ÷ (1 − r) and adding more pulses achieves nothing.

The cell is a capacitor with a resistor across it. It is not: water’s effective resistance depends on frequency and on the field across it, so the leakage during a high-voltage pulse is not the leakage measured with a meter.
Each pulse deposits the same increment regardless of the voltage already present. In a real circuit the increment falls as the cell voltage approaches the drive voltage, so this overstates the top of the staircase.
The blocking path is perfect during the gate-off period. A real diode or choke leaks, and that leakage adds to the water’s.
No breakdown. If the accumulated voltage takes the field past what the gap will hold, the cell discharges through an arc and the staircase resets — which this arithmetic will not show.
This is a derived model, not a formula from Meyer’s filings. The claim taken from the patents is that gating allows accumulation; the geometric series is the ordinary consequence of an RC discharge between pulses.