sharing-loses-stored-energy - Electronic Circuits XV & XVI: Transformers and Switched-Capacitor Conversion Step by Step

A sharing-loses-stored-energy scan holds the two capacitances fixed and varies the source voltage across three rows, checking that charge closes while the stored-energy gap delta-U = U_i - U_f grows row by row. The arithmetic is exact only for the stated capacitances, voltages, stored energies, energy losses, and rows in the ideal capacitor and ideal-switch model. Real circuits add ESR, switch on-resistance, parasitics, dielectric loss, switching-timing errors, probe/calibration limits, finite precision, and measurement uncertainty.

Charge conservation does not mean stored capacitor energy is conserved. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Sharing keeps charge but opens an energy ledger

The same source snapshots that conserve charge also compute initial and final stored energy.

U_i=243\ \text{J}\quad U_f=162\ \text{J}

Larger initial voltage makes a larger dissipated difference

Every row conserves charge first, then compares the initial and final stored energy.

\begin{array}{c|c|c|c}V_a&U_i&U_f&\Delta U\\3\ \text{V}&27\ \text{J}&18\ \text{J}&9\ \text{J}\\6\ \text{V}&108\ \text{J}&72\ \text{J}&36\ \text{J}\\9\ \text{V}&243\ \text{J}&162\ \text{J}&81\ \text{J}\\\end{array}

Charge can close while stored energy falls

The loss row is explicit; this lesson does not claim lossless charge sharing.

243\ \text{J}\rightarrow162\ \text{J}\quad243\ \text{J}\mathbin{-}162\ \text{J}=81\ \text{J}