Multi-cell array

What does ganging cells in series or parallel do to the capacitance and the resonance?

C_{series} = \frac{c}{n} \qquad C_{parallel} = n \cdot c \qquad f = \frac{1}{2\pi\sqrt{L \cdot C}}

Method

1. For cells in series, divide the single-cell capacitance by the number of cells. Capacitors in series combine like resistors in parallel — the total is always smaller than the smallest.
2. For cells in parallel, multiply instead. Parallel capacitors add, because the plate area effectively adds.
3. Resonate the result against the choke in the usual way: one over 2π times the square root of L times C.
4. Compare against the single-cell frequency. Because frequency goes as one over the square root of capacitance, n cells in series ring √n times higher and n in parallel ring √n times lower — nine in series is three times the frequency of one.
5. Divide the applied voltage by the number of cells for the nominal per-cell share. In series this is what each cell nominally sees; in parallel every cell sees the whole voltage.

Assumptions

• Every cell is identical. They are not. Series cells divide the voltage in inverse proportion to their capacitance, so the cell with the widest gap — or the most gas on its plates — takes the largest share and breaks down first. A stack fails at its weakest cell, not at its average.
• The cells do not interact. Adjacent tubes in a common bath share electrolyte, and a conducting path through the water puts an unintended resistance across parts of the array.
• Leakage is ignored. Every cell has a resistance across it, and in series those add while the capacitances divide — so a long series string is a worse capacitor and a better resistor than one cell.
• The wiring contributes nothing. At the frequencies and voltages involved, the inductance and capacitance of the interconnects between nine tubes is not always negligible.
• In parallel, every cell sees the full applied voltage — so the breakdown limit of the array is the breakdown limit of one cell, not n times it.