equal-series-caps-split-rail - Electronic Circuits XV & XVI: Transformers and Switched-Capacitor Conversion Step by Step

An equal-series-caps-split-rail scan requires two equal capacitances and varies the source rail across three rows, checking that the midpoint V_mid = V_s / 2 splits the rail only when C_a = C_b row by row. The arithmetic is exact only for the stated source voltages, midpoint voltages, capacitances, and rows in the ideal capacitor, switch, and phase-topology model. Real switched-capacitor converters add switch resistance, ESR, parasitic capacitance, charge injection, timing skew, load current, probe/calibration limits, finite precision, and measurement uncertainty.

A symmetric split is accepted only from equal capacitor sources. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A split row first proves the two capacitances match

The midpoint is accepted only from two equal-capacitance sources across the same rail.

C_a=C_b=2\ \text{F}\quad V_s=8\ \text{V}

Equal split keeps the midpoint at one half of the rail

Each row rebuilds the split from two equal capacitors and a different source rail.

\begin{array}{c|c|c}V_s&V_{\text{mid}}&\text{sources}\\8\ \text{V}&4\ \text{V}&\text{equal}\\10\ \text{V}&5\ \text{V}&\text{equal}\\12\ \text{V}&6\ \text{V}&\text{equal}\\\end{array}

Equal source capacitors split the rail

The split row cites two equal capacitance sources before accepting the midpoint.

8\ \text{V}/2=4\ \text{V}